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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="Congestion game Bounded capabilities Randomness" lang="en" name="keywords"/> <base href="/html/2308.08047v3/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS1" title="In 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Our results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS2" title="In 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Our techniques</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS2.SSS0.Px1" title="In 1.2 Our techniques ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Representing graphs as flow networks</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS2.SSS0.Px2" title="In 1.2 Our techniques ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Approximating the PoU with a simplified PoU</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS2.SSS0.Px3" title="In 1.2 Our techniques ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Reduction of strategy space</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS3" title="In 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.3 </span>Organization</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S2" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Related Work</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S2.SS0.SSS0.Px1" title="In 2 Related Work ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Team-maxmin equilibria</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S2.SS0.SSS0.Px2" title="In 2 Related Work ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Computing TMEs</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S2.SS0.SSS0.Px3" title="In 2 Related Work ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">TMECor and PoU</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S2.SS0.SSS0.Px4" title="In 2 Related Work ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Uniform mixed equilibria</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S3" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Preliminaries</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S3.SS1" title="In 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Notations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S3.SS2" title="In 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>The model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S3.SS3" title="In 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>General lemmas</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>SUM Cost</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.SS1" title="In 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>General <math alttext="s-t" class="ltx_Math" display="inline"><semantics><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow><annotation-xml encoding="MathML-Content"><apply><minus></minus><ci>𝑠</ci><ci>𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex">s-t</annotation><annotation encoding="application/x-llamapun">italic_s - italic_t</annotation></semantics></math> graphs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.SS2" title="In 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Disjoint paths</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>MAX Cost</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.SS1" title="In 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>General <math alttext="s-t" class="ltx_Math" display="inline"><semantics><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow><annotation-xml encoding="MathML-Content"><apply><minus></minus><ci>𝑠</ci><ci>𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex">s-t</annotation><annotation encoding="application/x-llamapun">italic_s - italic_t</annotation></semantics></math> graphs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.SS2" title="In 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Disjoint paths</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A1" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.A </span>Proofs of General Lemmas</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A2" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.B </span>Connections of Previous Results to Our Model</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A2.SS0.SSS0.Px1" title="In Appendix 0.B Connections of Previous Results to Our Model ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Converting payoffs to costs</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A3" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.C </span>Omitted Proofs in Section <span class="ltx_text ltx_ref_tag">4</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4" title="In Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.D </span>Omitted Details in Section <span class="ltx_text ltx_ref_tag">5</span></span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.SS1" title="In Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.D.1 </span>Illustrating example</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.SS2" title="In Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.D.2 </span>More definitions</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.SS3" title="In Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.D.3 </span>Expressions for MAX cost</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.SS4" title="In Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">0.D.4 </span>Omitted proofs</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.SS4.SSS0.Px1" title="In 0.D.4 Omitted proofs ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Case <math alttext="i<m-1" class="ltx_Math" display="inline"><semantics><mrow><mi>i</mi><mo><</mo><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><lt></lt><ci>𝑖</ci><apply><minus></minus><ci>𝑚</ci><cn type="integer">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">i<m-1</annotation><annotation encoding="application/x-llamapun">italic_i < italic_m - 1</annotation></semantics></math>:</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.SS4.SSS0.Px2" title="In 0.D.4 Omitted proofs ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_title">Case <math alttext="i=m-1" class="ltx_Math" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><eq></eq><ci>𝑖</ci><apply><minus></minus><ci>𝑚</ci><cn type="integer">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">i=m-1</annotation><annotation encoding="application/x-llamapun">italic_i = italic_m - 1</annotation></semantics></math>:</span></a></li> </ol> </li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\addbibresource</span> <p class="ltx_p" id="p1.2">bib.bib <span class="ltx_note ltx_role_institutetext" id="p1.2.1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_note_type">institutetext: </span>MIT, Cambridge MA 02139, USA <br class="ltx_break"/><span class="ltx_note ltx_role_email" id="p1.2.1.1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_note_type">email: </span>{iorzech,rinard}@csail.mit.edu</span></span></span></span></span></span></p> </div> <h1 class="ltx_title ltx_title_document">Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Edan Orzech </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Martin Rinard </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id4.4">We consider team zero-sum network congestion games with <math alttext="n" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mi id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><ci id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_n</annotation></semantics></math> agents playing against <math alttext="k" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mi id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><ci id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_k</annotation></semantics></math> interceptors over a graph <math alttext="G" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mi id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><ci id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">italic_G</annotation></semantics></math>. The agents aim to minimize their collective cost of sending traffic over paths in <math alttext="G" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mi id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><ci id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">italic_G</annotation></semantics></math>, which is an aggregation of edge costs, while the interceptors aim to maximize the collective cost by increasing some of these edge costs. To evade the interceptors, the agents will usually use randomized strategies. We consider two cases, the correlated case when agents have access to a shared source of randomness, and the uncorrelated case, when each agent has access to only its own source of randomness. We study the additional cost that uncorrelated agents have to bear, specifically by comparing the costs incurred by agents in cost-minimal Nash equilibria when agents can and cannot share randomness.</p> <p class="ltx_p" id="id8.8">We consider two natural cost functions on the agents, which measure the invested energy and time, respectively. We prove that for both of these cost functions, the ratio of uncorrelated cost to correlated cost at equilibrium is <math alttext="O(\min(m_{c}(G),n))" class="ltx_Math" display="inline" id="id5.5.m1.4"><semantics id="id5.5.m1.4a"><mrow id="id5.5.m1.4.4" xref="id5.5.m1.4.4.cmml"><mi id="id5.5.m1.4.4.3" xref="id5.5.m1.4.4.3.cmml">O</mi><mo id="id5.5.m1.4.4.2" xref="id5.5.m1.4.4.2.cmml">⁢</mo><mrow id="id5.5.m1.4.4.1.1" xref="id5.5.m1.4.4.cmml"><mo id="id5.5.m1.4.4.1.1.2" stretchy="false" xref="id5.5.m1.4.4.cmml">(</mo><mrow id="id5.5.m1.4.4.1.1.1.1" xref="id5.5.m1.4.4.1.1.1.2.cmml"><mi id="id5.5.m1.2.2" xref="id5.5.m1.2.2.cmml">min</mi><mo id="id5.5.m1.4.4.1.1.1.1a" xref="id5.5.m1.4.4.1.1.1.2.cmml">⁡</mo><mrow id="id5.5.m1.4.4.1.1.1.1.1" xref="id5.5.m1.4.4.1.1.1.2.cmml"><mo id="id5.5.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="id5.5.m1.4.4.1.1.1.2.cmml">(</mo><mrow id="id5.5.m1.4.4.1.1.1.1.1.1" xref="id5.5.m1.4.4.1.1.1.1.1.1.cmml"><msub id="id5.5.m1.4.4.1.1.1.1.1.1.2" xref="id5.5.m1.4.4.1.1.1.1.1.1.2.cmml"><mi id="id5.5.m1.4.4.1.1.1.1.1.1.2.2" xref="id5.5.m1.4.4.1.1.1.1.1.1.2.2.cmml">m</mi><mi id="id5.5.m1.4.4.1.1.1.1.1.1.2.3" xref="id5.5.m1.4.4.1.1.1.1.1.1.2.3.cmml">c</mi></msub><mo id="id5.5.m1.4.4.1.1.1.1.1.1.1" xref="id5.5.m1.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="id5.5.m1.4.4.1.1.1.1.1.1.3.2" xref="id5.5.m1.4.4.1.1.1.1.1.1.cmml"><mo id="id5.5.m1.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="id5.5.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="id5.5.m1.1.1" xref="id5.5.m1.1.1.cmml">G</mi><mo id="id5.5.m1.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="id5.5.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="id5.5.m1.4.4.1.1.1.1.1.3" xref="id5.5.m1.4.4.1.1.1.2.cmml">,</mo><mi id="id5.5.m1.3.3" xref="id5.5.m1.3.3.cmml">n</mi><mo id="id5.5.m1.4.4.1.1.1.1.1.4" stretchy="false" xref="id5.5.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="id5.5.m1.4.4.1.1.3" stretchy="false" xref="id5.5.m1.4.4.cmml">)</mo></mrow></mrow><annotation-xml 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xref="id5.5.m1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m1.4c">O(\min(m_{c}(G),n))</annotation><annotation encoding="application/x-llamapun" id="id5.5.m1.4d">italic_O ( roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G ) , italic_n ) )</annotation></semantics></math>, where <math alttext="m_{c}(G)" class="ltx_Math" display="inline" id="id6.6.m2.1"><semantics id="id6.6.m2.1a"><mrow id="id6.6.m2.1.2" xref="id6.6.m2.1.2.cmml"><msub id="id6.6.m2.1.2.2" xref="id6.6.m2.1.2.2.cmml"><mi id="id6.6.m2.1.2.2.2" xref="id6.6.m2.1.2.2.2.cmml">m</mi><mi id="id6.6.m2.1.2.2.3" xref="id6.6.m2.1.2.2.3.cmml">c</mi></msub><mo id="id6.6.m2.1.2.1" xref="id6.6.m2.1.2.1.cmml">⁢</mo><mrow id="id6.6.m2.1.2.3.2" xref="id6.6.m2.1.2.cmml"><mo id="id6.6.m2.1.2.3.2.1" stretchy="false" xref="id6.6.m2.1.2.cmml">(</mo><mi id="id6.6.m2.1.1" xref="id6.6.m2.1.1.cmml">G</mi><mo id="id6.6.m2.1.2.3.2.2" stretchy="false" xref="id6.6.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id6.6.m2.1b"><apply id="id6.6.m2.1.2.cmml" xref="id6.6.m2.1.2"><times id="id6.6.m2.1.2.1.cmml" xref="id6.6.m2.1.2.1"></times><apply id="id6.6.m2.1.2.2.cmml" xref="id6.6.m2.1.2.2"><csymbol cd="ambiguous" id="id6.6.m2.1.2.2.1.cmml" xref="id6.6.m2.1.2.2">subscript</csymbol><ci id="id6.6.m2.1.2.2.2.cmml" xref="id6.6.m2.1.2.2.2">𝑚</ci><ci id="id6.6.m2.1.2.2.3.cmml" xref="id6.6.m2.1.2.2.3">𝑐</ci></apply><ci id="id6.6.m2.1.1.cmml" xref="id6.6.m2.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m2.1c">m_{c}(G)</annotation><annotation encoding="application/x-llamapun" id="id6.6.m2.1d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G )</annotation></semantics></math> is the mincut size of <math alttext="G" class="ltx_Math" display="inline" id="id7.7.m3.1"><semantics id="id7.7.m3.1a"><mi id="id7.7.m3.1.1" xref="id7.7.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="id7.7.m3.1b"><ci id="id7.7.m3.1.1.cmml" xref="id7.7.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="id7.7.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="id7.7.m3.1d">italic_G</annotation></semantics></math>. This bound is much smaller than the most general case, where a tight, exponential bound of <math alttext="\Theta((m_{c}(G))^{n-1})" class="ltx_Math" display="inline" id="id8.8.m4.2"><semantics id="id8.8.m4.2a"><mrow id="id8.8.m4.2.2" xref="id8.8.m4.2.2.cmml"><mi id="id8.8.m4.2.2.3" mathvariant="normal" xref="id8.8.m4.2.2.3.cmml">Θ</mi><mo id="id8.8.m4.2.2.2" xref="id8.8.m4.2.2.2.cmml">⁢</mo><mrow id="id8.8.m4.2.2.1.1" xref="id8.8.m4.2.2.1.1.1.cmml"><mo id="id8.8.m4.2.2.1.1.2" stretchy="false" xref="id8.8.m4.2.2.1.1.1.cmml">(</mo><msup id="id8.8.m4.2.2.1.1.1" xref="id8.8.m4.2.2.1.1.1.cmml"><mrow id="id8.8.m4.2.2.1.1.1.1.1" xref="id8.8.m4.2.2.1.1.1.1.1.1.cmml"><mo id="id8.8.m4.2.2.1.1.1.1.1.2" stretchy="false" xref="id8.8.m4.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="id8.8.m4.2.2.1.1.1.1.1.1" xref="id8.8.m4.2.2.1.1.1.1.1.1.cmml"><msub id="id8.8.m4.2.2.1.1.1.1.1.1.2" xref="id8.8.m4.2.2.1.1.1.1.1.1.2.cmml"><mi id="id8.8.m4.2.2.1.1.1.1.1.1.2.2" xref="id8.8.m4.2.2.1.1.1.1.1.1.2.2.cmml">m</mi><mi id="id8.8.m4.2.2.1.1.1.1.1.1.2.3" xref="id8.8.m4.2.2.1.1.1.1.1.1.2.3.cmml">c</mi></msub><mo id="id8.8.m4.2.2.1.1.1.1.1.1.1" xref="id8.8.m4.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="id8.8.m4.2.2.1.1.1.1.1.1.3.2" xref="id8.8.m4.2.2.1.1.1.1.1.1.cmml"><mo id="id8.8.m4.2.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="id8.8.m4.2.2.1.1.1.1.1.1.cmml">(</mo><mi id="id8.8.m4.1.1" xref="id8.8.m4.1.1.cmml">G</mi><mo id="id8.8.m4.2.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="id8.8.m4.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="id8.8.m4.2.2.1.1.1.1.1.3" stretchy="false" xref="id8.8.m4.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="id8.8.m4.2.2.1.1.1.3" xref="id8.8.m4.2.2.1.1.1.3.cmml"><mi id="id8.8.m4.2.2.1.1.1.3.2" xref="id8.8.m4.2.2.1.1.1.3.2.cmml">n</mi><mo id="id8.8.m4.2.2.1.1.1.3.1" xref="id8.8.m4.2.2.1.1.1.3.1.cmml">−</mo><mn id="id8.8.m4.2.2.1.1.1.3.3" xref="id8.8.m4.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="id8.8.m4.2.2.1.1.3" stretchy="false" xref="id8.8.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id8.8.m4.2b"><apply id="id8.8.m4.2.2.cmml" xref="id8.8.m4.2.2"><times id="id8.8.m4.2.2.2.cmml" xref="id8.8.m4.2.2.2"></times><ci id="id8.8.m4.2.2.3.cmml" xref="id8.8.m4.2.2.3">Θ</ci><apply id="id8.8.m4.2.2.1.1.1.cmml" xref="id8.8.m4.2.2.1.1"><csymbol cd="ambiguous" id="id8.8.m4.2.2.1.1.1.2.cmml" xref="id8.8.m4.2.2.1.1">superscript</csymbol><apply id="id8.8.m4.2.2.1.1.1.1.1.1.cmml" xref="id8.8.m4.2.2.1.1.1.1.1"><times id="id8.8.m4.2.2.1.1.1.1.1.1.1.cmml" xref="id8.8.m4.2.2.1.1.1.1.1.1.1"></times><apply id="id8.8.m4.2.2.1.1.1.1.1.1.2.cmml" xref="id8.8.m4.2.2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="id8.8.m4.2.2.1.1.1.1.1.1.2.1.cmml" xref="id8.8.m4.2.2.1.1.1.1.1.1.2">subscript</csymbol><ci id="id8.8.m4.2.2.1.1.1.1.1.1.2.2.cmml" xref="id8.8.m4.2.2.1.1.1.1.1.1.2.2">𝑚</ci><ci id="id8.8.m4.2.2.1.1.1.1.1.1.2.3.cmml" xref="id8.8.m4.2.2.1.1.1.1.1.1.2.3">𝑐</ci></apply><ci id="id8.8.m4.1.1.cmml" xref="id8.8.m4.1.1">𝐺</ci></apply><apply id="id8.8.m4.2.2.1.1.1.3.cmml" xref="id8.8.m4.2.2.1.1.1.3"><minus id="id8.8.m4.2.2.1.1.1.3.1.cmml" xref="id8.8.m4.2.2.1.1.1.3.1"></minus><ci id="id8.8.m4.2.2.1.1.1.3.2.cmml" xref="id8.8.m4.2.2.1.1.1.3.2">𝑛</ci><cn id="id8.8.m4.2.2.1.1.1.3.3.cmml" type="integer" xref="id8.8.m4.2.2.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.8.m4.2c">\Theta((m_{c}(G))^{n-1})</annotation><annotation encoding="application/x-llamapun" id="id8.8.m4.2d">roman_Θ ( ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G ) ) start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> on the ratio is known. We also introduce a set of simple agent strategies which are approximately optimal agent strategies. We then establish conditions for when these strategies are optimal agent strategies for each cost function, showing an inherent difference between the two cost functions we study.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Keywords: </h6>Congestion game Bounded capabilities Randomness </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Network congestion games are central to the field of game theory <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">koutsoupias1999worst</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">rosenthal1973class</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">roughgarden2002bad</span>]</cite>. Recently, there has been work on team network congestion games, where a team of agents cooperate in a network congestion game against an adversary <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">harks2022multi</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">bilo2023uniform</span>]</cite>. Team network congestion games are motivated by the problem of fare evasion <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correa2017fare</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">harks2022multi</span>]</cite>. There, agents attempt to reach their destination while minimizing the amount of fare they pay, while some central authority attempts to counter this phenomenon with a policy that minimizes fare evasion on the network. These games are also motivated by security applications (see <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">sinha2018stackelberg</span>]</cite> for a survey). There, the agents attempt to attack a set of targets (for example entering a train station via a set of possible entrances), while a team of interceptors attempts to defend those targets with limited resources (for example by conducting inspections).</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">We consider a team zero-sum network congestion game in which a team of agents sends traffic on a network of roads between two locations. Traversing each road segment incurs a cost, with the team attempting to minimize an aggregation of the segment costs. We consider two versions: the SUM version (Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a>) and the MAX version (Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a>). In the SUM version, the agents minimize the total traversal cost, called the SUM cost. In the MAX version, the agents minimize the total traversal cost, called the MAX cost. Intuitively, the SUM cost measures the amount of total expended resources (such as transportation energy or tolls), while the MAX cost measures the amount of time required to transport the agents between the two locations. On the other side, a team of interceptors acts against the agents by imposing costs on selected road segments, causing the agents to expend additional resources to traverse these segments.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.12">We model the road network with a directed acyclic graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_G</annotation></semantics></math>, possibly with multiple edges between a pair of vertices, with a source <math alttext="s" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mi id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_s</annotation></semantics></math> and a sink <math alttext="t" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><ci id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_t</annotation></semantics></math>. The edges model roads, the vertices <math alttext="s,t" class="ltx_Math" display="inline" id="S1.p3.4.m4.2"><semantics id="S1.p3.4.m4.2a"><mrow id="S1.p3.4.m4.2.3.2" xref="S1.p3.4.m4.2.3.1.cmml"><mi id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml">s</mi><mo id="S1.p3.4.m4.2.3.2.1" xref="S1.p3.4.m4.2.3.1.cmml">,</mo><mi id="S1.p3.4.m4.2.2" xref="S1.p3.4.m4.2.2.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.2b"><list id="S1.p3.4.m4.2.3.1.cmml" xref="S1.p3.4.m4.2.3.2"><ci id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1">𝑠</ci><ci id="S1.p3.4.m4.2.2.cmml" xref="S1.p3.4.m4.2.2">𝑡</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.2c">s,t</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.2d">italic_s , italic_t</annotation></semantics></math> model the origin and destination, while the remaining vertices are intersections. There is a team of <math alttext="n" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mi id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><ci id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">italic_n</annotation></semantics></math> agents and a team of <math alttext="k" class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><mi id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><ci id="S1.p3.6.m6.1.1.cmml" xref="S1.p3.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">italic_k</annotation></semantics></math> interceptors playing on <math alttext="G" class="ltx_Math" display="inline" id="S1.p3.7.m7.1"><semantics id="S1.p3.7.m7.1a"><mi id="S1.p3.7.m7.1.1" xref="S1.p3.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p3.7.m7.1b"><ci id="S1.p3.7.m7.1.1.cmml" xref="S1.p3.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p3.7.m7.1d">italic_G</annotation></semantics></math>. Each agent selects an <math alttext="s-t" class="ltx_Math" display="inline" id="S1.p3.8.m8.1"><semantics id="S1.p3.8.m8.1a"><mrow id="S1.p3.8.m8.1.1" xref="S1.p3.8.m8.1.1.cmml"><mi id="S1.p3.8.m8.1.1.2" xref="S1.p3.8.m8.1.1.2.cmml">s</mi><mo id="S1.p3.8.m8.1.1.1" xref="S1.p3.8.m8.1.1.1.cmml">−</mo><mi id="S1.p3.8.m8.1.1.3" xref="S1.p3.8.m8.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.8.m8.1b"><apply id="S1.p3.8.m8.1.1.cmml" xref="S1.p3.8.m8.1.1"><minus id="S1.p3.8.m8.1.1.1.cmml" xref="S1.p3.8.m8.1.1.1"></minus><ci id="S1.p3.8.m8.1.1.2.cmml" xref="S1.p3.8.m8.1.1.2">𝑠</ci><ci id="S1.p3.8.m8.1.1.3.cmml" xref="S1.p3.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.8.m8.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S1.p3.8.m8.1d">italic_s - italic_t</annotation></semantics></math> path in <math alttext="G" class="ltx_Math" display="inline" id="S1.p3.9.m9.1"><semantics id="S1.p3.9.m9.1a"><mi id="S1.p3.9.m9.1.1" xref="S1.p3.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p3.9.m9.1b"><ci id="S1.p3.9.m9.1.1.cmml" xref="S1.p3.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p3.9.m9.1d">italic_G</annotation></semantics></math>; each interceptor selects an edge of the graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p3.10.m10.1"><semantics id="S1.p3.10.m10.1a"><mi id="S1.p3.10.m10.1.1" xref="S1.p3.10.m10.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p3.10.m10.1b"><ci id="S1.p3.10.m10.1.1.cmml" xref="S1.p3.10.m10.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.10.m10.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p3.10.m10.1d">italic_G</annotation></semantics></math>. Traversing an edge imposes a one time unit cost of 1, with an additional cost of <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p3.11.m11.1"><semantics id="S1.p3.11.m11.1a"><mi id="S1.p3.11.m11.1.1" xref="S1.p3.11.m11.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p3.11.m11.1b"><ci id="S1.p3.11.m11.1.1.cmml" xref="S1.p3.11.m11.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.11.m11.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p3.11.m11.1d">italic_α</annotation></semantics></math> on each edge that an interceptor selects. Selected edge costs are additive; i.e., if two interceptors select the same edge, the additional cost is <math alttext="2\alpha" class="ltx_Math" display="inline" id="S1.p3.12.m12.1"><semantics id="S1.p3.12.m12.1a"><mrow id="S1.p3.12.m12.1.1" xref="S1.p3.12.m12.1.1.cmml"><mn id="S1.p3.12.m12.1.1.2" xref="S1.p3.12.m12.1.1.2.cmml">2</mn><mo id="S1.p3.12.m12.1.1.1" xref="S1.p3.12.m12.1.1.1.cmml">⁢</mo><mi id="S1.p3.12.m12.1.1.3" xref="S1.p3.12.m12.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.12.m12.1b"><apply id="S1.p3.12.m12.1.1.cmml" xref="S1.p3.12.m12.1.1"><times id="S1.p3.12.m12.1.1.1.cmml" xref="S1.p3.12.m12.1.1.1"></times><cn id="S1.p3.12.m12.1.1.2.cmml" type="integer" xref="S1.p3.12.m12.1.1.2">2</cn><ci id="S1.p3.12.m12.1.1.3.cmml" xref="S1.p3.12.m12.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.12.m12.1c">2\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p3.12.m12.1d">2 italic_α</annotation></semantics></math>. By this we model settings where the interceptors can increase the cost at road segments, either by collecting additional tolls or by increasing congestion.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">As usual, the agents and interceptors can use randomization to mix their strategies. We consider two simple communication assumptions between the agents. In the first case, agents are unable to communicate any random bits to each other (i.e., they are uncorrelated). In the second case, they can (i.e., they can be correlated). These two cases model scenarios where the interceptors may or may not be able to eavesdrop on the agents’ communication. These cases also model scenarios where the agents’ ability to communicate during the actual play is either disabled or not. We allow the interceptors to communicate freely and securely during the game. Therefore, they can be seen as one interceptor. However, for the sake of presentation, we choose to treat them as separate interceptors who make single and coordinated actions, rather than once interceptor who makes multiple actions.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.2">To understand the impact of different (team) randomization capabilities, we measure the <span class="ltx_text ltx_font_italic" id="S1.p5.2.1">price of uncorrelation</span> (PoU) of the game, an inefficiency index that was first introduced by <span class="ltx_ERROR undefined" id="S1.p5.2.2">\citet</span>basilico2017team. There, the authors define PoU to be the ratio of the cost at a team-maxmin equilibrium (TME) of the game, to the cost at a correlated TME (TMECor) of the game. A TME (<cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">von1997team</span>]</cite>) is a maxmin equilibrium between a team of uncorrelated agents and a single adversary, while in a TMECor the team agents can be correlated (see Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S2" title="2 Related Work ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">2</span></a> for more details). The PoU of the model characterizes the inapproximability (in terms of cost) of correlated agent strategies using uncorrelated strategies: If the PoU is <math alttext="r" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mi id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><ci id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_r</annotation></semantics></math>, then the best approximation factor of any algorithm, regardless of time or space complexity, is exactly <math alttext="\frac{1}{r}" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mfrac id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mn id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml">1</mn><mi id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml">r</mi></mfrac><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1"><divide id="S1.p5.2.m2.1.1.1.cmml" xref="S1.p5.2.m2.1.1"></divide><cn id="S1.p5.2.m2.1.1.2.cmml" type="integer" xref="S1.p5.2.m2.1.1.2">1</cn><ci id="S1.p5.2.m2.1.1.3.cmml" xref="S1.p5.2.m2.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">\frac{1}{r}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">divide start_ARG 1 end_ARG start_ARG italic_r end_ARG</annotation></semantics></math> (this follows from the definition).</p> </div> <div class="ltx_para" id="S1.p6"> <span class="ltx_ERROR undefined" id="S1.p6.4">\citet</span> <p class="ltx_p" id="S1.p6.3">basilico2017team show that in general, normal-form games, when the <math alttext="n" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mi id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><ci id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">italic_n</annotation></semantics></math> team agents have <math alttext="m" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mi id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><ci id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">italic_m</annotation></semantics></math> available strategies each, they can incur an exponential PoU of <math alttext="m^{n-1}" class="ltx_Math" display="inline" id="S1.p6.3.m3.1"><semantics id="S1.p6.3.m3.1a"><msup id="S1.p6.3.m3.1.1" xref="S1.p6.3.m3.1.1.cmml"><mi id="S1.p6.3.m3.1.1.2" xref="S1.p6.3.m3.1.1.2.cmml">m</mi><mrow id="S1.p6.3.m3.1.1.3" xref="S1.p6.3.m3.1.1.3.cmml"><mi id="S1.p6.3.m3.1.1.3.2" xref="S1.p6.3.m3.1.1.3.2.cmml">n</mi><mo id="S1.p6.3.m3.1.1.3.1" xref="S1.p6.3.m3.1.1.3.1.cmml">−</mo><mn id="S1.p6.3.m3.1.1.3.3" xref="S1.p6.3.m3.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p6.3.m3.1b"><apply id="S1.p6.3.m3.1.1.cmml" xref="S1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p6.3.m3.1.1.1.cmml" xref="S1.p6.3.m3.1.1">superscript</csymbol><ci id="S1.p6.3.m3.1.1.2.cmml" xref="S1.p6.3.m3.1.1.2">𝑚</ci><apply id="S1.p6.3.m3.1.1.3.cmml" xref="S1.p6.3.m3.1.1.3"><minus id="S1.p6.3.m3.1.1.3.1.cmml" xref="S1.p6.3.m3.1.1.3.1"></minus><ci id="S1.p6.3.m3.1.1.3.2.cmml" xref="S1.p6.3.m3.1.1.3.2">𝑛</ci><cn id="S1.p6.3.m3.1.1.3.3.cmml" type="integer" xref="S1.p6.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.3.m3.1c">m^{n-1}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.3.m3.1d">italic_m start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. As we discuss next, in our model the PoU is linear. This means that in the settings that we model, correlated randomness and communication in general between the agents is less valuable than in the games considered by <span class="ltx_ERROR undefined" id="S1.p6.3.1">\citet</span>basilico2017team. Intuitively, the reason is as follows: We model scenarios where interceptors can increase the cost at road segments by only a finite amount (but not completely deny agents the ability to use the road segment). On the other hand, <span class="ltx_ERROR undefined" id="S1.p6.3.2">\citet</span>basilico2017team model scenarios where interceptors can completely prevent agents from traversing the graph (if an agent attempts to traverse a road segment that was essentially removed by an interceptor). <br class="ltx_break"/> <br class="ltx_break"/><span class="ltx_text ltx_font_bold" id="S1.p6.3.3">Illustrating example </span> We illustrate the PoU in our game in the following example.</p> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample1"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 1</span></h6> <div class="ltx_para" id="Thmexample1.p1"> <p class="ltx_p" id="Thmexample1.p1.3">Consider the graph in Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.F1" title="Figure 1 ‣ Example 1 ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1</span></a>, with <math alttext="m" class="ltx_Math" display="inline" id="Thmexample1.p1.1.m1.1"><semantics id="Thmexample1.p1.1.m1.1a"><mi id="Thmexample1.p1.1.m1.1.1" xref="Thmexample1.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="Thmexample1.p1.1.m1.1b"><ci id="Thmexample1.p1.1.m1.1.1.cmml" xref="Thmexample1.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p1.1.m1.1d">italic_m</annotation></semantics></math> disjoint paths between <math alttext="s" class="ltx_Math" display="inline" id="Thmexample1.p1.2.m2.1"><semantics id="Thmexample1.p1.2.m2.1a"><mi id="Thmexample1.p1.2.m2.1.1" xref="Thmexample1.p1.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="Thmexample1.p1.2.m2.1b"><ci id="Thmexample1.p1.2.m2.1.1.cmml" xref="Thmexample1.p1.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p1.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p1.2.m2.1d">italic_s</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="Thmexample1.p1.3.m3.1"><semantics id="Thmexample1.p1.3.m3.1a"><mi id="Thmexample1.p1.3.m3.1.1" xref="Thmexample1.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="Thmexample1.p1.3.m3.1b"><ci id="Thmexample1.p1.3.m3.1.1.cmml" xref="Thmexample1.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p1.3.m3.1d">italic_t</annotation></semantics></math> of the same length.</p> </div> <figure class="ltx_figure" id="S1.F1"><svg class="ltx_picture ltx_centering" height="118.26" id="S1.F1.pic1" overflow="visible" version="1.1" width="350.11"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,118.26) matrix(1 0 0 -1 0 0) translate(17.57,0) translate(0,43.89)"><path d="M 17.3 0 C 17.3 9.55 9.55 17.3 0 17.3 C -9.55 17.3 -17.3 9.55 -17.3 0 C -17.3 -9.55 -9.55 -17.3 0 -17.3 C 9.55 -17.3 17.3 -9.55 17.3 0 Z M 0 0" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 -3.24 -2.98)"><foreignobject height="5.96" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.49"><math alttext="s" class="ltx_Math" display="inline" id="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">s</annotation><annotation encoding="application/x-llamapun" id="S1.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_s</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 152.29 -3.46)"><foreignobject height="12.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.38"><math alttext="\vdots" class="ltx_Math" display="inline" id="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1" mathvariant="normal" xref="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">⋮</mi><annotation-xml encoding="MathML-Content" id="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1">⋮</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1c">\vdots</annotation><annotation encoding="application/x-llamapun" id="S1.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1d">⋮</annotation></semantics></math></foreignobject></g><path d="M 332.26 0 C 332.26 9.55 324.51 17.3 314.96 17.3 C 305.41 17.3 297.66 9.55 297.66 0 C 297.66 -9.55 305.41 -17.3 314.96 -17.3 C 324.51 -17.3 332.26 -9.55 332.26 0 Z M 314.96 0" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 312.46 -4.26)"><foreignobject height="8.51" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="5"><math alttext="t" class="ltx_Math" display="inline" id="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1d">italic_t</annotation></semantics></math></foreignobject></g><path d="M 14.39 10.08 C 105.81 74.09 209.15 74.09 297.85 11.98" style="fill:none"></path><g transform="matrix(0.81915 -0.57358 0.57358 0.81915 297.85 11.98)"><path d="M 3.32 0 C 1.94 0.28 -0.55 0.83 -2.21 2.08 C -0.83 0.55 -0.83 -0.55 -2.21 -2.08 C -0.55 -0.83 1.94 -0.28 3.32 0 Z" style="stroke:none"></path></g><path d="M 16.97 4.55 C 122.83 32.91 192.13 32.91 294.78 5.41" style="fill:none"></path><g transform="matrix(0.96593 -0.25882 0.25882 0.96593 294.78 5.41)"><path d="M 3.32 0 C 1.94 0.28 -0.55 0.83 -2.21 2.08 C -0.83 0.55 -0.83 -0.55 -2.21 -2.08 C -0.55 -0.83 1.94 -0.28 3.32 0 Z" style="stroke:none"></path></g><path d="M 16.51 -6.01 C 119.83 -43.61 195.13 -43.61 295.33 -7.15" style="fill:none"></path><g transform="matrix(0.9397 0.34203 -0.34203 0.9397 295.33 -7.15)"><path d="M 3.32 0 C 1.94 0.28 -0.55 0.83 -2.21 2.08 C -0.83 0.55 -0.83 -0.55 -2.21 -2.08 C -0.55 -0.83 1.94 -0.28 3.32 0 Z" style="stroke:none"></path></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Illustrating example</figcaption> </figure> <div class="ltx_para" id="Thmexample1.p2"> <p class="ltx_p" id="Thmexample1.p2.7">Suppose there are <math alttext="n" class="ltx_Math" display="inline" id="Thmexample1.p2.1.m1.1"><semantics id="Thmexample1.p2.1.m1.1a"><mi id="Thmexample1.p2.1.m1.1.1" xref="Thmexample1.p2.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmexample1.p2.1.m1.1b"><ci id="Thmexample1.p2.1.m1.1.1.cmml" xref="Thmexample1.p2.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p2.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p2.1.m1.1d">italic_n</annotation></semantics></math> agents and one interceptor, playing with respect to the SUM cost function. Fix <math alttext="\alpha>0" class="ltx_Math" display="inline" id="Thmexample1.p2.2.m2.1"><semantics id="Thmexample1.p2.2.m2.1a"><mrow id="Thmexample1.p2.2.m2.1.1" xref="Thmexample1.p2.2.m2.1.1.cmml"><mi id="Thmexample1.p2.2.m2.1.1.2" xref="Thmexample1.p2.2.m2.1.1.2.cmml">α</mi><mo id="Thmexample1.p2.2.m2.1.1.1" xref="Thmexample1.p2.2.m2.1.1.1.cmml">></mo><mn id="Thmexample1.p2.2.m2.1.1.3" xref="Thmexample1.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmexample1.p2.2.m2.1b"><apply id="Thmexample1.p2.2.m2.1.1.cmml" xref="Thmexample1.p2.2.m2.1.1"><gt id="Thmexample1.p2.2.m2.1.1.1.cmml" xref="Thmexample1.p2.2.m2.1.1.1"></gt><ci id="Thmexample1.p2.2.m2.1.1.2.cmml" xref="Thmexample1.p2.2.m2.1.1.2">𝛼</ci><cn id="Thmexample1.p2.2.m2.1.1.3.cmml" type="integer" xref="Thmexample1.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p2.2.m2.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p2.2.m2.1d">italic_α > 0</annotation></semantics></math>. When agents are correlated, the resulting equilibrium is one where agents pick together a path that is sampled from the <math alttext="n" class="ltx_Math" display="inline" id="Thmexample1.p2.3.m3.1"><semantics id="Thmexample1.p2.3.m3.1a"><mi id="Thmexample1.p2.3.m3.1.1" xref="Thmexample1.p2.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmexample1.p2.3.m3.1b"><ci id="Thmexample1.p2.3.m3.1.1.cmml" xref="Thmexample1.p2.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p2.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p2.3.m3.1d">italic_n</annotation></semantics></math> paths of the graph uniformly at random. The interceptor also picks a path uniformly at random. Then, the expected cost to the agents is <math alttext="1+\frac{\alpha}{m}" class="ltx_Math" display="inline" id="Thmexample1.p2.4.m4.1"><semantics id="Thmexample1.p2.4.m4.1a"><mrow id="Thmexample1.p2.4.m4.1.1" xref="Thmexample1.p2.4.m4.1.1.cmml"><mn id="Thmexample1.p2.4.m4.1.1.2" xref="Thmexample1.p2.4.m4.1.1.2.cmml">1</mn><mo id="Thmexample1.p2.4.m4.1.1.1" xref="Thmexample1.p2.4.m4.1.1.1.cmml">+</mo><mfrac id="Thmexample1.p2.4.m4.1.1.3" xref="Thmexample1.p2.4.m4.1.1.3.cmml"><mi id="Thmexample1.p2.4.m4.1.1.3.2" xref="Thmexample1.p2.4.m4.1.1.3.2.cmml">α</mi><mi id="Thmexample1.p2.4.m4.1.1.3.3" xref="Thmexample1.p2.4.m4.1.1.3.3.cmml">m</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmexample1.p2.4.m4.1b"><apply id="Thmexample1.p2.4.m4.1.1.cmml" xref="Thmexample1.p2.4.m4.1.1"><plus id="Thmexample1.p2.4.m4.1.1.1.cmml" xref="Thmexample1.p2.4.m4.1.1.1"></plus><cn id="Thmexample1.p2.4.m4.1.1.2.cmml" type="integer" xref="Thmexample1.p2.4.m4.1.1.2">1</cn><apply id="Thmexample1.p2.4.m4.1.1.3.cmml" xref="Thmexample1.p2.4.m4.1.1.3"><divide id="Thmexample1.p2.4.m4.1.1.3.1.cmml" xref="Thmexample1.p2.4.m4.1.1.3"></divide><ci id="Thmexample1.p2.4.m4.1.1.3.2.cmml" xref="Thmexample1.p2.4.m4.1.1.3.2">𝛼</ci><ci id="Thmexample1.p2.4.m4.1.1.3.3.cmml" xref="Thmexample1.p2.4.m4.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p2.4.m4.1c">1+\frac{\alpha}{m}</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p2.4.m4.1d">1 + divide start_ARG italic_α end_ARG start_ARG italic_m end_ARG</annotation></semantics></math>. Now suppose that the agents are uncorrelated. Then one can show that the optimal strategy for the agents is having each agent independently pick a path uniformly at random, and the interceptor’s strategy is unchanged. Then, the expected cost to the agents is <math alttext="m\left(1-\left(1-\frac{1}{m}\right)^{n}\right)\left(1+\frac{\alpha}{m}\right)" class="ltx_Math" display="inline" id="Thmexample1.p2.5.m5.2"><semantics id="Thmexample1.p2.5.m5.2a"><mrow id="Thmexample1.p2.5.m5.2.2" xref="Thmexample1.p2.5.m5.2.2.cmml"><mi id="Thmexample1.p2.5.m5.2.2.4" xref="Thmexample1.p2.5.m5.2.2.4.cmml">m</mi><mo id="Thmexample1.p2.5.m5.2.2.3" xref="Thmexample1.p2.5.m5.2.2.3.cmml">⁢</mo><mrow id="Thmexample1.p2.5.m5.1.1.1.1" xref="Thmexample1.p2.5.m5.1.1.1.1.1.cmml"><mo id="Thmexample1.p2.5.m5.1.1.1.1.2" xref="Thmexample1.p2.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample1.p2.5.m5.1.1.1.1.1" xref="Thmexample1.p2.5.m5.1.1.1.1.1.cmml"><mn id="Thmexample1.p2.5.m5.1.1.1.1.1.3" xref="Thmexample1.p2.5.m5.1.1.1.1.1.3.cmml">1</mn><mo id="Thmexample1.p2.5.m5.1.1.1.1.1.2" xref="Thmexample1.p2.5.m5.1.1.1.1.1.2.cmml">−</mo><msup id="Thmexample1.p2.5.m5.1.1.1.1.1.1" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.cmml"><mrow id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.2" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml"><mn id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.2" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.1" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.cmml"><mn id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.2" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.3" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.3.cmml">m</mi></mfrac></mrow><mo id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.3" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmexample1.p2.5.m5.1.1.1.1.1.1.3" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Thmexample1.p2.5.m5.1.1.1.1.3" xref="Thmexample1.p2.5.m5.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample1.p2.5.m5.2.2.3a" xref="Thmexample1.p2.5.m5.2.2.3.cmml">⁢</mo><mrow id="Thmexample1.p2.5.m5.2.2.2.1" xref="Thmexample1.p2.5.m5.2.2.2.1.1.cmml"><mo id="Thmexample1.p2.5.m5.2.2.2.1.2" xref="Thmexample1.p2.5.m5.2.2.2.1.1.cmml">(</mo><mrow id="Thmexample1.p2.5.m5.2.2.2.1.1" xref="Thmexample1.p2.5.m5.2.2.2.1.1.cmml"><mn id="Thmexample1.p2.5.m5.2.2.2.1.1.2" xref="Thmexample1.p2.5.m5.2.2.2.1.1.2.cmml">1</mn><mo id="Thmexample1.p2.5.m5.2.2.2.1.1.1" xref="Thmexample1.p2.5.m5.2.2.2.1.1.1.cmml">+</mo><mfrac id="Thmexample1.p2.5.m5.2.2.2.1.1.3" xref="Thmexample1.p2.5.m5.2.2.2.1.1.3.cmml"><mi id="Thmexample1.p2.5.m5.2.2.2.1.1.3.2" xref="Thmexample1.p2.5.m5.2.2.2.1.1.3.2.cmml">α</mi><mi id="Thmexample1.p2.5.m5.2.2.2.1.1.3.3" xref="Thmexample1.p2.5.m5.2.2.2.1.1.3.3.cmml">m</mi></mfrac></mrow><mo id="Thmexample1.p2.5.m5.2.2.2.1.3" xref="Thmexample1.p2.5.m5.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample1.p2.5.m5.2b"><apply id="Thmexample1.p2.5.m5.2.2.cmml" xref="Thmexample1.p2.5.m5.2.2"><times id="Thmexample1.p2.5.m5.2.2.3.cmml" xref="Thmexample1.p2.5.m5.2.2.3"></times><ci id="Thmexample1.p2.5.m5.2.2.4.cmml" xref="Thmexample1.p2.5.m5.2.2.4">𝑚</ci><apply id="Thmexample1.p2.5.m5.1.1.1.1.1.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1"><minus id="Thmexample1.p2.5.m5.1.1.1.1.1.2.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.2"></minus><cn id="Thmexample1.p2.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample1.p2.5.m5.1.1.1.1.1.3">1</cn><apply id="Thmexample1.p2.5.m5.1.1.1.1.1.1.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample1.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1">superscript</csymbol><apply id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1"><minus id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.1"></minus><cn id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.2">1</cn><apply id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3"><divide id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3"></divide><cn id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.1.1.1.3.3">𝑚</ci></apply></apply><ci id="Thmexample1.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="Thmexample1.p2.5.m5.1.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmexample1.p2.5.m5.2.2.2.1.1.cmml" xref="Thmexample1.p2.5.m5.2.2.2.1"><plus id="Thmexample1.p2.5.m5.2.2.2.1.1.1.cmml" xref="Thmexample1.p2.5.m5.2.2.2.1.1.1"></plus><cn id="Thmexample1.p2.5.m5.2.2.2.1.1.2.cmml" type="integer" xref="Thmexample1.p2.5.m5.2.2.2.1.1.2">1</cn><apply id="Thmexample1.p2.5.m5.2.2.2.1.1.3.cmml" xref="Thmexample1.p2.5.m5.2.2.2.1.1.3"><divide id="Thmexample1.p2.5.m5.2.2.2.1.1.3.1.cmml" xref="Thmexample1.p2.5.m5.2.2.2.1.1.3"></divide><ci id="Thmexample1.p2.5.m5.2.2.2.1.1.3.2.cmml" xref="Thmexample1.p2.5.m5.2.2.2.1.1.3.2">𝛼</ci><ci id="Thmexample1.p2.5.m5.2.2.2.1.1.3.3.cmml" xref="Thmexample1.p2.5.m5.2.2.2.1.1.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p2.5.m5.2c">m\left(1-\left(1-\frac{1}{m}\right)^{n}\right)\left(1+\frac{\alpha}{m}\right)</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p2.5.m5.2d">italic_m ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ( 1 + divide start_ARG italic_α end_ARG start_ARG italic_m end_ARG )</annotation></semantics></math>. So the PoU is <math alttext="m\left(1-\left(1-\frac{1}{m}\right)^{n}\right)=\Theta(\min(m,n))" class="ltx_Math" display="inline" id="Thmexample1.p2.6.m6.5"><semantics id="Thmexample1.p2.6.m6.5a"><mrow id="Thmexample1.p2.6.m6.5.5" xref="Thmexample1.p2.6.m6.5.5.cmml"><mrow id="Thmexample1.p2.6.m6.4.4.1" xref="Thmexample1.p2.6.m6.4.4.1.cmml"><mi id="Thmexample1.p2.6.m6.4.4.1.3" xref="Thmexample1.p2.6.m6.4.4.1.3.cmml">m</mi><mo id="Thmexample1.p2.6.m6.4.4.1.2" xref="Thmexample1.p2.6.m6.4.4.1.2.cmml">⁢</mo><mrow id="Thmexample1.p2.6.m6.4.4.1.1.1" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.cmml"><mo id="Thmexample1.p2.6.m6.4.4.1.1.1.2" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.cmml">(</mo><mrow id="Thmexample1.p2.6.m6.4.4.1.1.1.1" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.cmml"><mn id="Thmexample1.p2.6.m6.4.4.1.1.1.1.3" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.3.cmml">1</mn><mo id="Thmexample1.p2.6.m6.4.4.1.1.1.1.2" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.2.cmml">−</mo><msup id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.cmml"><mrow id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.cmml"><mo id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.2" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.cmml"><mn id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.2" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.1" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.cmml"><mn id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.2" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.3" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.3.cmml">m</mi></mfrac></mrow><mo id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.3" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.3" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Thmexample1.p2.6.m6.4.4.1.1.1.3" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmexample1.p2.6.m6.5.5.3" xref="Thmexample1.p2.6.m6.5.5.3.cmml">=</mo><mrow id="Thmexample1.p2.6.m6.5.5.2" xref="Thmexample1.p2.6.m6.5.5.2.cmml"><mi id="Thmexample1.p2.6.m6.5.5.2.3" mathvariant="normal" xref="Thmexample1.p2.6.m6.5.5.2.3.cmml">Θ</mi><mo id="Thmexample1.p2.6.m6.5.5.2.2" xref="Thmexample1.p2.6.m6.5.5.2.2.cmml">⁢</mo><mrow id="Thmexample1.p2.6.m6.5.5.2.1.1" xref="Thmexample1.p2.6.m6.5.5.2.cmml"><mo id="Thmexample1.p2.6.m6.5.5.2.1.1.2" stretchy="false" xref="Thmexample1.p2.6.m6.5.5.2.cmml">(</mo><mrow id="Thmexample1.p2.6.m6.5.5.2.1.1.1.2" xref="Thmexample1.p2.6.m6.5.5.2.1.1.1.1.cmml"><mi id="Thmexample1.p2.6.m6.1.1" xref="Thmexample1.p2.6.m6.1.1.cmml">min</mi><mo id="Thmexample1.p2.6.m6.5.5.2.1.1.1.2a" xref="Thmexample1.p2.6.m6.5.5.2.1.1.1.1.cmml">⁡</mo><mrow id="Thmexample1.p2.6.m6.5.5.2.1.1.1.2.1" xref="Thmexample1.p2.6.m6.5.5.2.1.1.1.1.cmml"><mo id="Thmexample1.p2.6.m6.5.5.2.1.1.1.2.1.1" stretchy="false" xref="Thmexample1.p2.6.m6.5.5.2.1.1.1.1.cmml">(</mo><mi id="Thmexample1.p2.6.m6.2.2" xref="Thmexample1.p2.6.m6.2.2.cmml">m</mi><mo id="Thmexample1.p2.6.m6.5.5.2.1.1.1.2.1.2" xref="Thmexample1.p2.6.m6.5.5.2.1.1.1.1.cmml">,</mo><mi id="Thmexample1.p2.6.m6.3.3" xref="Thmexample1.p2.6.m6.3.3.cmml">n</mi><mo id="Thmexample1.p2.6.m6.5.5.2.1.1.1.2.1.3" stretchy="false" xref="Thmexample1.p2.6.m6.5.5.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmexample1.p2.6.m6.5.5.2.1.1.3" stretchy="false" xref="Thmexample1.p2.6.m6.5.5.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample1.p2.6.m6.5b"><apply id="Thmexample1.p2.6.m6.5.5.cmml" xref="Thmexample1.p2.6.m6.5.5"><eq id="Thmexample1.p2.6.m6.5.5.3.cmml" xref="Thmexample1.p2.6.m6.5.5.3"></eq><apply id="Thmexample1.p2.6.m6.4.4.1.cmml" xref="Thmexample1.p2.6.m6.4.4.1"><times id="Thmexample1.p2.6.m6.4.4.1.2.cmml" xref="Thmexample1.p2.6.m6.4.4.1.2"></times><ci id="Thmexample1.p2.6.m6.4.4.1.3.cmml" xref="Thmexample1.p2.6.m6.4.4.1.3">𝑚</ci><apply id="Thmexample1.p2.6.m6.4.4.1.1.1.1.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1"><minus id="Thmexample1.p2.6.m6.4.4.1.1.1.1.2.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.2"></minus><cn id="Thmexample1.p2.6.m6.4.4.1.1.1.1.3.cmml" type="integer" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.3">1</cn><apply id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.2.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1">superscript</csymbol><apply id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1"><minus id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.1"></minus><cn id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.2">1</cn><apply id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3"><divide id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3"></divide><cn id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.1.1.1.3.3">𝑚</ci></apply></apply><ci id="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.3.cmml" xref="Thmexample1.p2.6.m6.4.4.1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmexample1.p2.6.m6.5.5.2.cmml" xref="Thmexample1.p2.6.m6.5.5.2"><times id="Thmexample1.p2.6.m6.5.5.2.2.cmml" xref="Thmexample1.p2.6.m6.5.5.2.2"></times><ci id="Thmexample1.p2.6.m6.5.5.2.3.cmml" xref="Thmexample1.p2.6.m6.5.5.2.3">Θ</ci><apply id="Thmexample1.p2.6.m6.5.5.2.1.1.1.1.cmml" xref="Thmexample1.p2.6.m6.5.5.2.1.1.1.2"><min id="Thmexample1.p2.6.m6.1.1.cmml" xref="Thmexample1.p2.6.m6.1.1"></min><ci id="Thmexample1.p2.6.m6.2.2.cmml" xref="Thmexample1.p2.6.m6.2.2">𝑚</ci><ci id="Thmexample1.p2.6.m6.3.3.cmml" xref="Thmexample1.p2.6.m6.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p2.6.m6.5c">m\left(1-\left(1-\frac{1}{m}\right)^{n}\right)=\Theta(\min(m,n))</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p2.6.m6.5d">italic_m ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) = roman_Θ ( roman_min ( italic_m , italic_n ) )</annotation></semantics></math>. We note that in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>]</cite>, a game is considered which is played on the graph in Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.F1" title="Figure 1 ‣ Example 1 ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1</span></a>, but there the PoU is <math alttext="m^{n-1}" class="ltx_Math" display="inline" id="Thmexample1.p2.7.m7.1"><semantics id="Thmexample1.p2.7.m7.1a"><msup id="Thmexample1.p2.7.m7.1.1" xref="Thmexample1.p2.7.m7.1.1.cmml"><mi id="Thmexample1.p2.7.m7.1.1.2" xref="Thmexample1.p2.7.m7.1.1.2.cmml">m</mi><mrow id="Thmexample1.p2.7.m7.1.1.3" xref="Thmexample1.p2.7.m7.1.1.3.cmml"><mi id="Thmexample1.p2.7.m7.1.1.3.2" xref="Thmexample1.p2.7.m7.1.1.3.2.cmml">n</mi><mo id="Thmexample1.p2.7.m7.1.1.3.1" xref="Thmexample1.p2.7.m7.1.1.3.1.cmml">−</mo><mn id="Thmexample1.p2.7.m7.1.1.3.3" xref="Thmexample1.p2.7.m7.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="Thmexample1.p2.7.m7.1b"><apply id="Thmexample1.p2.7.m7.1.1.cmml" xref="Thmexample1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="Thmexample1.p2.7.m7.1.1.1.cmml" xref="Thmexample1.p2.7.m7.1.1">superscript</csymbol><ci id="Thmexample1.p2.7.m7.1.1.2.cmml" xref="Thmexample1.p2.7.m7.1.1.2">𝑚</ci><apply id="Thmexample1.p2.7.m7.1.1.3.cmml" xref="Thmexample1.p2.7.m7.1.1.3"><minus id="Thmexample1.p2.7.m7.1.1.3.1.cmml" xref="Thmexample1.p2.7.m7.1.1.3.1"></minus><ci id="Thmexample1.p2.7.m7.1.1.3.2.cmml" xref="Thmexample1.p2.7.m7.1.1.3.2">𝑛</ci><cn id="Thmexample1.p2.7.m7.1.1.3.3.cmml" type="integer" xref="Thmexample1.p2.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p2.7.m7.1c">m^{n-1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p2.7.m7.1d">italic_m start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Our results</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.1">For both SUM and MAX costs, we establish bounds on the PoU (Sections <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a>), summarized as follows:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Theorem 1.1</span> (Informal version of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem3" title="Theorem 4.3 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.3</span></a> and Corollary <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmcorollary2" title="Corollary 2 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">2</span></a>)</h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem1.p1.5.5">The PoU, as a function of <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.1.m1.1"><semantics id="S1.Thmtheorem1.p1.1.1.m1.1a"><mi id="S1.Thmtheorem1.p1.1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.1.m1.1b"><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.1.m1.1d">italic_α</annotation></semantics></math>, denoted <math alttext="r(\alpha)" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.2.m2.1"><semantics id="S1.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem1.p1.2.2.m2.1.2" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.cmml"><mi id="S1.Thmtheorem1.p1.2.2.m2.1.2.2" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.2.cmml">r</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.1.2.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.1.2.3.2" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.cmml"><mo id="S1.Thmtheorem1.p1.2.2.m2.1.2.3.2.1" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.cmml">(</mo><mi id="S1.Thmtheorem1.p1.2.2.m2.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml">α</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.1.2.3.2.2" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.2.m2.1b"><apply id="S1.Thmtheorem1.p1.2.2.m2.1.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.2"><times id="S1.Thmtheorem1.p1.2.2.m2.1.2.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.1"></times><ci id="S1.Thmtheorem1.p1.2.2.m2.1.2.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.2.2">𝑟</ci><ci id="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.2.m2.1c">r(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.2.m2.1d">italic_r ( italic_α )</annotation></semantics></math>, satisfies <math alttext="r(\alpha)=\Theta(\min(m_{c}(G),n))" 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)</annotation></semantics></math>, where <math alttext="m_{c}(G)" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.4.4.m4.1"><semantics id="S1.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S1.Thmtheorem1.p1.4.4.m4.1.2" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml"><msub id="S1.Thmtheorem1.p1.4.4.m4.1.2.2" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2.cmml"><mi id="S1.Thmtheorem1.p1.4.4.m4.1.2.2.2" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2.2.cmml">m</mi><mi id="S1.Thmtheorem1.p1.4.4.m4.1.2.2.3" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2.3.cmml">c</mi></msub><mo id="S1.Thmtheorem1.p1.4.4.m4.1.2.1" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem1.p1.4.4.m4.1.2.3.2" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mo id="S1.Thmtheorem1.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml">(</mo><mi id="S1.Thmtheorem1.p1.4.4.m4.1.1" xref="S1.Thmtheorem1.p1.4.4.m4.1.1.cmml">G</mi><mo id="S1.Thmtheorem1.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.4.4.m4.1b"><apply id="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2"><times id="S1.Thmtheorem1.p1.4.4.m4.1.2.1.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.1"></times><apply id="S1.Thmtheorem1.p1.4.4.m4.1.2.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.4.4.m4.1.2.2.1.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2">subscript</csymbol><ci id="S1.Thmtheorem1.p1.4.4.m4.1.2.2.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2.2">𝑚</ci><ci id="S1.Thmtheorem1.p1.4.4.m4.1.2.2.3.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2.3">𝑐</ci></apply><ci id="S1.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.4.4.m4.1c">m_{c}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.4.4.m4.1d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G )</annotation></semantics></math> is the size of a minimum cut of <math alttext="G" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.5.5.m5.1"><semantics id="S1.Thmtheorem1.p1.5.5.m5.1a"><mi id="S1.Thmtheorem1.p1.5.5.m5.1.1" xref="S1.Thmtheorem1.p1.5.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.5.5.m5.1b"><ci id="S1.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem1.p1.5.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.5.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.5.5.m5.1d">italic_G</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.1">This implies that in our games there is a polynomial-time <math alttext="\Theta\left(\frac{1}{\min(m_{c}(G),n)}\right)" class="ltx_Math" display="inline" id="S1.SS1.p2.1.m1.4"><semantics id="S1.SS1.p2.1.m1.4a"><mrow id="S1.SS1.p2.1.m1.4.5" xref="S1.SS1.p2.1.m1.4.5.cmml"><mi id="S1.SS1.p2.1.m1.4.5.2" mathvariant="normal" xref="S1.SS1.p2.1.m1.4.5.2.cmml">Θ</mi><mo id="S1.SS1.p2.1.m1.4.5.1" xref="S1.SS1.p2.1.m1.4.5.1.cmml">⁢</mo><mrow id="S1.SS1.p2.1.m1.4.5.3.2" xref="S1.SS1.p2.1.m1.4.4.cmml"><mo id="S1.SS1.p2.1.m1.4.5.3.2.1" xref="S1.SS1.p2.1.m1.4.4.cmml">(</mo><mfrac id="S1.SS1.p2.1.m1.4.4" xref="S1.SS1.p2.1.m1.4.4.cmml"><mn id="S1.SS1.p2.1.m1.4.4.6" xref="S1.SS1.p2.1.m1.4.4.6.cmml">1</mn><mrow id="S1.SS1.p2.1.m1.4.4.4.4" xref="S1.SS1.p2.1.m1.4.4.4.5.cmml"><mi id="S1.SS1.p2.1.m1.2.2.2.2" xref="S1.SS1.p2.1.m1.2.2.2.2.cmml">min</mi><mo id="S1.SS1.p2.1.m1.4.4.4.4a" xref="S1.SS1.p2.1.m1.4.4.4.5.cmml">⁡</mo><mrow id="S1.SS1.p2.1.m1.4.4.4.4.1" xref="S1.SS1.p2.1.m1.4.4.4.5.cmml"><mo id="S1.SS1.p2.1.m1.4.4.4.4.1.2" stretchy="false" xref="S1.SS1.p2.1.m1.4.4.4.5.cmml">(</mo><mrow id="S1.SS1.p2.1.m1.4.4.4.4.1.1" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.cmml"><msub id="S1.SS1.p2.1.m1.4.4.4.4.1.1.2" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.cmml"><mi id="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.2" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.2.cmml">m</mi><mi id="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.3" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.3.cmml">c</mi></msub><mo id="S1.SS1.p2.1.m1.4.4.4.4.1.1.1" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.p2.1.m1.4.4.4.4.1.1.3.2" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.cmml"><mo id="S1.SS1.p2.1.m1.4.4.4.4.1.1.3.2.1" stretchy="false" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.cmml">(</mo><mi id="S1.SS1.p2.1.m1.1.1.1.1" xref="S1.SS1.p2.1.m1.1.1.1.1.cmml">G</mi><mo id="S1.SS1.p2.1.m1.4.4.4.4.1.1.3.2.2" stretchy="false" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS1.p2.1.m1.4.4.4.4.1.3" xref="S1.SS1.p2.1.m1.4.4.4.5.cmml">,</mo><mi id="S1.SS1.p2.1.m1.3.3.3.3" xref="S1.SS1.p2.1.m1.3.3.3.3.cmml">n</mi><mo id="S1.SS1.p2.1.m1.4.4.4.4.1.4" stretchy="false" xref="S1.SS1.p2.1.m1.4.4.4.5.cmml">)</mo></mrow></mrow></mfrac><mo id="S1.SS1.p2.1.m1.4.5.3.2.2" xref="S1.SS1.p2.1.m1.4.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.1.m1.4b"><apply id="S1.SS1.p2.1.m1.4.5.cmml" xref="S1.SS1.p2.1.m1.4.5"><times id="S1.SS1.p2.1.m1.4.5.1.cmml" xref="S1.SS1.p2.1.m1.4.5.1"></times><ci id="S1.SS1.p2.1.m1.4.5.2.cmml" xref="S1.SS1.p2.1.m1.4.5.2">Θ</ci><apply id="S1.SS1.p2.1.m1.4.4.cmml" xref="S1.SS1.p2.1.m1.4.5.3.2"><divide id="S1.SS1.p2.1.m1.4.4.5.cmml" xref="S1.SS1.p2.1.m1.4.5.3.2"></divide><cn id="S1.SS1.p2.1.m1.4.4.6.cmml" type="integer" xref="S1.SS1.p2.1.m1.4.4.6">1</cn><apply id="S1.SS1.p2.1.m1.4.4.4.5.cmml" xref="S1.SS1.p2.1.m1.4.4.4.4"><min id="S1.SS1.p2.1.m1.2.2.2.2.cmml" xref="S1.SS1.p2.1.m1.2.2.2.2"></min><apply id="S1.SS1.p2.1.m1.4.4.4.4.1.1.cmml" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1"><times id="S1.SS1.p2.1.m1.4.4.4.4.1.1.1.cmml" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.1"></times><apply id="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.cmml" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.1.cmml" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.2">subscript</csymbol><ci id="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.2.cmml" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.2">𝑚</ci><ci id="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.3.cmml" xref="S1.SS1.p2.1.m1.4.4.4.4.1.1.2.3">𝑐</ci></apply><ci id="S1.SS1.p2.1.m1.1.1.1.1.cmml" xref="S1.SS1.p2.1.m1.1.1.1.1">𝐺</ci></apply><ci id="S1.SS1.p2.1.m1.3.3.3.3.cmml" xref="S1.SS1.p2.1.m1.3.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.1.m1.4c">\Theta\left(\frac{1}{\min(m_{c}(G),n)}\right)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.1.m1.4d">roman_Θ ( divide start_ARG 1 end_ARG start_ARG roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G ) , italic_n ) end_ARG )</annotation></semantics></math>-approximation algorithm for playing a TMECor agent strategy using uncorrelated strategies. This is much smaller compared to the exponential (in agents) factor proved in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>]</cite> in their abstract class of games.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.2">Next, we focus on analyzing more carefully the TME strategies of our game with the two cost functions. In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.SS2" title="4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.2</span></a>, we show that there is a set of simple strategies that admits optimal agent strategies over graphs with disjoint <math alttext="s-t" class="ltx_Math" display="inline" id="S1.SS1.p3.1.m1.1"><semantics id="S1.SS1.p3.1.m1.1a"><mrow id="S1.SS1.p3.1.m1.1.1" xref="S1.SS1.p3.1.m1.1.1.cmml"><mi id="S1.SS1.p3.1.m1.1.1.2" xref="S1.SS1.p3.1.m1.1.1.2.cmml">s</mi><mo id="S1.SS1.p3.1.m1.1.1.1" xref="S1.SS1.p3.1.m1.1.1.1.cmml">−</mo><mi id="S1.SS1.p3.1.m1.1.1.3" xref="S1.SS1.p3.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.1.m1.1b"><apply id="S1.SS1.p3.1.m1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1"><minus id="S1.SS1.p3.1.m1.1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1.1"></minus><ci id="S1.SS1.p3.1.m1.1.1.2.cmml" xref="S1.SS1.p3.1.m1.1.1.2">𝑠</ci><ci id="S1.SS1.p3.1.m1.1.1.3.cmml" xref="S1.SS1.p3.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.1.m1.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.1.m1.1d">italic_s - italic_t</annotation></semantics></math> paths with respect to the SUM cost function. In those strategies, agents roughly mix strategies uniformly over disjoint paths. Call these strategies <span class="ltx_text ltx_font_italic" id="S1.SS1.p3.2.1">uniform strategies</span>, and denote them with <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S1.SS1.p3.2.m2.1"><semantics id="S1.SS1.p3.2.m2.1a"><msub id="S1.SS1.p3.2.m2.1.1" xref="S1.SS1.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS1.p3.2.m2.1.1.2" xref="S1.SS1.p3.2.m2.1.1.2.cmml">𝒰</mi><mi id="S1.SS1.p3.2.m2.1.1.3" xref="S1.SS1.p3.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.2.m2.1b"><apply id="S1.SS1.p3.2.m2.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.2.m2.1.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1">subscript</csymbol><ci id="S1.SS1.p3.2.m2.1.1.2.cmml" xref="S1.SS1.p3.2.m2.1.1.2">𝒰</ci><ci id="S1.SS1.p3.2.m2.1.1.3.cmml" xref="S1.SS1.p3.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.2.m2.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.2.m2.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> (see Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S3" title="3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">3</span></a> for the formal definition).</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Theorem 1.2</span> (Informal version of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem6" title="Theorem 4.6 ‣ 4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.6</span></a>)</h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.5"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.5.5">For every graph <math alttext="G" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.1.m1.1"><semantics id="S1.Thmtheorem2.p1.1.1.m1.1a"><mi id="S1.Thmtheorem2.p1.1.1.m1.1.1" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.1.1.m1.1b"><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.1.1.m1.1d">italic_G</annotation></semantics></math> where the <math alttext="s-t" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.2.2.m2.1"><semantics id="S1.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem2.p1.2.2.m2.1.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S1.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">s</mi><mo id="S1.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">−</mo><mi id="S1.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.2.2.m2.1b"><apply id="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1"><minus id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1"></minus><ci id="S1.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.2">𝑠</ci><ci id="S1.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.2.2.m2.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.2.2.m2.1d">italic_s - italic_t</annotation></semantics></math> paths are disjoint, and for every <math alttext="n\geq 1" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.3.3.m3.1"><semantics id="S1.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">n</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">≥</mo><mn id="S1.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.3.3.m3.1b"><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1"><geq id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1"></geq><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2">𝑛</ci><cn id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.3.3.m3.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.3.3.m3.1d">italic_n ≥ 1</annotation></semantics></math>, <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.4.4.m4.1"><semantics id="S1.Thmtheorem2.p1.4.4.m4.1a"><msub id="S1.Thmtheorem2.p1.4.4.m4.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">𝒰</mi><mi id="S1.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.4.4.m4.1b"><apply id="S1.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1">subscript</csymbol><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.2">𝒰</ci><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.4.4.m4.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.4.4.m4.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> admits an optimal agent strategy for <math alttext="n" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.5.5.m5.1"><semantics id="S1.Thmtheorem2.p1.5.5.m5.1a"><mi id="S1.Thmtheorem2.p1.5.5.m5.1.1" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.5.5.m5.1b"><ci id="S1.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.5.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.5.5.m5.1d">italic_n</annotation></semantics></math> uncorrelated agents.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.1">In contrast, for the MAX cost, and for a larger set of graphs, we prove in Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.SS2" title="5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.2</span></a> that the uniform strategies are almost never optimal.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Theorem 1.3</span> (Informal version of Theorems <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem1" title="Theorem 5.1 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem3" title="Theorem 5.3 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.3</span></a>)</h6> <div class="ltx_para" id="S1.Thmtheorem3.p1"> <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.5"><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.5.1">For a graph with disjoint </span><math alttext="s-t" class="ltx_Math" display="inline" id="S1.I1.i1.p1.1.m1.1"><semantics id="S1.I1.i1.p1.1.m1.1a"><mrow id="S1.I1.i1.p1.1.m1.1.1" xref="S1.I1.i1.p1.1.m1.1.1.cmml"><mi id="S1.I1.i1.p1.1.m1.1.1.2" xref="S1.I1.i1.p1.1.m1.1.1.2.cmml">s</mi><mo id="S1.I1.i1.p1.1.m1.1.1.1" xref="S1.I1.i1.p1.1.m1.1.1.1.cmml">−</mo><mi id="S1.I1.i1.p1.1.m1.1.1.3" xref="S1.I1.i1.p1.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.1.m1.1b"><apply id="S1.I1.i1.p1.1.m1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1"><minus id="S1.I1.i1.p1.1.m1.1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.1"></minus><ci id="S1.I1.i1.p1.1.m1.1.1.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2">𝑠</ci><ci id="S1.I1.i1.p1.1.m1.1.1.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.1.m1.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.1.m1.1d">italic_s - italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.5.2"> paths, and for every </span><math alttext="n>1" class="ltx_Math" display="inline" id="S1.I1.i1.p1.2.m2.1"><semantics id="S1.I1.i1.p1.2.m2.1a"><mrow id="S1.I1.i1.p1.2.m2.1.1" xref="S1.I1.i1.p1.2.m2.1.1.cmml"><mi id="S1.I1.i1.p1.2.m2.1.1.2" xref="S1.I1.i1.p1.2.m2.1.1.2.cmml">n</mi><mo id="S1.I1.i1.p1.2.m2.1.1.1" xref="S1.I1.i1.p1.2.m2.1.1.1.cmml">></mo><mn id="S1.I1.i1.p1.2.m2.1.1.3" xref="S1.I1.i1.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.2.m2.1b"><apply id="S1.I1.i1.p1.2.m2.1.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1"><gt id="S1.I1.i1.p1.2.m2.1.1.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.1"></gt><ci id="S1.I1.i1.p1.2.m2.1.1.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2">𝑛</ci><cn id="S1.I1.i1.p1.2.m2.1.1.3.cmml" type="integer" xref="S1.I1.i1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.2.m2.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.2.m2.1d">italic_n > 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.5.3">, </span><math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S1.I1.i1.p1.3.m3.1"><semantics id="S1.I1.i1.p1.3.m3.1a"><msub id="S1.I1.i1.p1.3.m3.1.1" xref="S1.I1.i1.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i1.p1.3.m3.1.1.2" xref="S1.I1.i1.p1.3.m3.1.1.2.cmml">𝒰</mi><mi id="S1.I1.i1.p1.3.m3.1.1.3" xref="S1.I1.i1.p1.3.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.3.m3.1b"><apply id="S1.I1.i1.p1.3.m3.1.1.cmml" xref="S1.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.I1.i1.p1.3.m3.1.1.1.cmml" xref="S1.I1.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S1.I1.i1.p1.3.m3.1.1.2.cmml" xref="S1.I1.i1.p1.3.m3.1.1.2">𝒰</ci><ci id="S1.I1.i1.p1.3.m3.1.1.3.cmml" xref="S1.I1.i1.p1.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.3.m3.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.3.m3.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.5.4"> never admits an optimal agent strategy for </span><math alttext="n" class="ltx_Math" display="inline" id="S1.I1.i1.p1.4.m4.1"><semantics id="S1.I1.i1.p1.4.m4.1a"><mi id="S1.I1.i1.p1.4.m4.1.1" xref="S1.I1.i1.p1.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.4.m4.1b"><ci id="S1.I1.i1.p1.4.m4.1.1.cmml" xref="S1.I1.i1.p1.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.4.m4.1d">italic_n</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.5.5"> uncorrelated agents if and only if the paths are not all of the same length, and </span><math alttext="\alpha" class="ltx_Math" display="inline" id="S1.I1.i1.p1.5.m5.1"><semantics id="S1.I1.i1.p1.5.m5.1a"><mi id="S1.I1.i1.p1.5.m5.1.1" xref="S1.I1.i1.p1.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.5.m5.1b"><ci id="S1.I1.i1.p1.5.m5.1.1.cmml" xref="S1.I1.i1.p1.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.5.m5.1d">italic_α</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.5.6"> is large enough.</span></p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.7"><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.1">Let </span><math alttext="G" class="ltx_Math" display="inline" id="S1.I1.i2.p1.1.m1.1"><semantics id="S1.I1.i2.p1.1.m1.1a"><mi id="S1.I1.i2.p1.1.m1.1.1" xref="S1.I1.i2.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.1.m1.1b"><ci id="S1.I1.i2.p1.1.m1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.1.m1.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.2"> be a graph where, for </span><math alttext="s-t" class="ltx_Math" display="inline" id="S1.I1.i2.p1.2.m2.1"><semantics id="S1.I1.i2.p1.2.m2.1a"><mrow id="S1.I1.i2.p1.2.m2.1.1" xref="S1.I1.i2.p1.2.m2.1.1.cmml"><mi id="S1.I1.i2.p1.2.m2.1.1.2" xref="S1.I1.i2.p1.2.m2.1.1.2.cmml">s</mi><mo id="S1.I1.i2.p1.2.m2.1.1.1" xref="S1.I1.i2.p1.2.m2.1.1.1.cmml">−</mo><mi id="S1.I1.i2.p1.2.m2.1.1.3" xref="S1.I1.i2.p1.2.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.2.m2.1b"><apply id="S1.I1.i2.p1.2.m2.1.1.cmml" xref="S1.I1.i2.p1.2.m2.1.1"><minus id="S1.I1.i2.p1.2.m2.1.1.1.cmml" xref="S1.I1.i2.p1.2.m2.1.1.1"></minus><ci id="S1.I1.i2.p1.2.m2.1.1.2.cmml" xref="S1.I1.i2.p1.2.m2.1.1.2">𝑠</ci><ci id="S1.I1.i2.p1.2.m2.1.1.3.cmml" xref="S1.I1.i2.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.2.m2.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.2.m2.1d">italic_s - italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.3"> every path there is a disjoint </span><math alttext="s-t" class="ltx_Math" display="inline" id="S1.I1.i2.p1.3.m3.1"><semantics id="S1.I1.i2.p1.3.m3.1a"><mrow id="S1.I1.i2.p1.3.m3.1.1" xref="S1.I1.i2.p1.3.m3.1.1.cmml"><mi id="S1.I1.i2.p1.3.m3.1.1.2" xref="S1.I1.i2.p1.3.m3.1.1.2.cmml">s</mi><mo id="S1.I1.i2.p1.3.m3.1.1.1" xref="S1.I1.i2.p1.3.m3.1.1.1.cmml">−</mo><mi id="S1.I1.i2.p1.3.m3.1.1.3" xref="S1.I1.i2.p1.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.3.m3.1b"><apply id="S1.I1.i2.p1.3.m3.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1"><minus id="S1.I1.i2.p1.3.m3.1.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1"></minus><ci id="S1.I1.i2.p1.3.m3.1.1.2.cmml" xref="S1.I1.i2.p1.3.m3.1.1.2">𝑠</ci><ci id="S1.I1.i2.p1.3.m3.1.1.3.cmml" xref="S1.I1.i2.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.3.m3.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.3.m3.1d">italic_s - italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.4"> path of a different length. Then for every </span><math alttext="n>1" class="ltx_Math" display="inline" id="S1.I1.i2.p1.4.m4.1"><semantics id="S1.I1.i2.p1.4.m4.1a"><mrow id="S1.I1.i2.p1.4.m4.1.1" xref="S1.I1.i2.p1.4.m4.1.1.cmml"><mi id="S1.I1.i2.p1.4.m4.1.1.2" xref="S1.I1.i2.p1.4.m4.1.1.2.cmml">n</mi><mo id="S1.I1.i2.p1.4.m4.1.1.1" xref="S1.I1.i2.p1.4.m4.1.1.1.cmml">></mo><mn id="S1.I1.i2.p1.4.m4.1.1.3" xref="S1.I1.i2.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.4.m4.1b"><apply id="S1.I1.i2.p1.4.m4.1.1.cmml" xref="S1.I1.i2.p1.4.m4.1.1"><gt id="S1.I1.i2.p1.4.m4.1.1.1.cmml" xref="S1.I1.i2.p1.4.m4.1.1.1"></gt><ci id="S1.I1.i2.p1.4.m4.1.1.2.cmml" xref="S1.I1.i2.p1.4.m4.1.1.2">𝑛</ci><cn id="S1.I1.i2.p1.4.m4.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.4.m4.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.4.m4.1d">italic_n > 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.5">, and for a large enough </span><math alttext="\alpha>0" class="ltx_Math" display="inline" id="S1.I1.i2.p1.5.m5.1"><semantics id="S1.I1.i2.p1.5.m5.1a"><mrow id="S1.I1.i2.p1.5.m5.1.1" xref="S1.I1.i2.p1.5.m5.1.1.cmml"><mi id="S1.I1.i2.p1.5.m5.1.1.2" xref="S1.I1.i2.p1.5.m5.1.1.2.cmml">α</mi><mo id="S1.I1.i2.p1.5.m5.1.1.1" xref="S1.I1.i2.p1.5.m5.1.1.1.cmml">></mo><mn id="S1.I1.i2.p1.5.m5.1.1.3" xref="S1.I1.i2.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.5.m5.1b"><apply id="S1.I1.i2.p1.5.m5.1.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1"><gt id="S1.I1.i2.p1.5.m5.1.1.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1.1"></gt><ci id="S1.I1.i2.p1.5.m5.1.1.2.cmml" xref="S1.I1.i2.p1.5.m5.1.1.2">𝛼</ci><cn id="S1.I1.i2.p1.5.m5.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.5.m5.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.5.m5.1d">italic_α > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.6">, the strategy set </span><math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.6.m6.1"><semantics id="S1.I1.i2.p1.6.m6.1a"><msub id="S1.I1.i2.p1.6.m6.1.1" xref="S1.I1.i2.p1.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i2.p1.6.m6.1.1.2" xref="S1.I1.i2.p1.6.m6.1.1.2.cmml">𝒰</mi><mi id="S1.I1.i2.p1.6.m6.1.1.3" xref="S1.I1.i2.p1.6.m6.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.6.m6.1b"><apply id="S1.I1.i2.p1.6.m6.1.1.cmml" xref="S1.I1.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.6.m6.1.1.1.cmml" xref="S1.I1.i2.p1.6.m6.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.6.m6.1.1.2.cmml" xref="S1.I1.i2.p1.6.m6.1.1.2">𝒰</ci><ci id="S1.I1.i2.p1.6.m6.1.1.3.cmml" xref="S1.I1.i2.p1.6.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.6.m6.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.6.m6.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.7"> never admits an optimal agent strategy for </span><math alttext="n" class="ltx_Math" display="inline" id="S1.I1.i2.p1.7.m7.1"><semantics id="S1.I1.i2.p1.7.m7.1a"><mi id="S1.I1.i2.p1.7.m7.1.1" xref="S1.I1.i2.p1.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.7.m7.1b"><ci id="S1.I1.i2.p1.7.m7.1.1.cmml" xref="S1.I1.i2.p1.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.7.m7.1d">italic_n</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.7.8"> uncorrelated agents.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S1.SS1.p5"> <p class="ltx_p" id="S1.SS1.p5.3">Going back to the SUM cost, we define a notion of <span class="ltx_text ltx_font_italic" id="S1.SS1.p5.3.1">uniform PoU</span>, denoted with <math alttext="r_{U}" class="ltx_Math" display="inline" id="S1.SS1.p5.1.m1.1"><semantics id="S1.SS1.p5.1.m1.1a"><msub id="S1.SS1.p5.1.m1.1.1" xref="S1.SS1.p5.1.m1.1.1.cmml"><mi id="S1.SS1.p5.1.m1.1.1.2" xref="S1.SS1.p5.1.m1.1.1.2.cmml">r</mi><mi id="S1.SS1.p5.1.m1.1.1.3" xref="S1.SS1.p5.1.m1.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.1.m1.1b"><apply id="S1.SS1.p5.1.m1.1.1.cmml" xref="S1.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p5.1.m1.1.1.1.cmml" xref="S1.SS1.p5.1.m1.1.1">subscript</csymbol><ci id="S1.SS1.p5.1.m1.1.1.2.cmml" xref="S1.SS1.p5.1.m1.1.1.2">𝑟</ci><ci id="S1.SS1.p5.1.m1.1.1.3.cmml" xref="S1.SS1.p5.1.m1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.1.m1.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.1.m1.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> where the agents can only use strategies in <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S1.SS1.p5.2.m2.1"><semantics id="S1.SS1.p5.2.m2.1a"><msub id="S1.SS1.p5.2.m2.1.1" xref="S1.SS1.p5.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS1.p5.2.m2.1.1.2" xref="S1.SS1.p5.2.m2.1.1.2.cmml">𝒰</mi><mi id="S1.SS1.p5.2.m2.1.1.3" xref="S1.SS1.p5.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.2.m2.1b"><apply id="S1.SS1.p5.2.m2.1.1.cmml" xref="S1.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p5.2.m2.1.1.1.cmml" xref="S1.SS1.p5.2.m2.1.1">subscript</csymbol><ci id="S1.SS1.p5.2.m2.1.1.2.cmml" xref="S1.SS1.p5.2.m2.1.1.2">𝒰</ci><ci id="S1.SS1.p5.2.m2.1.1.3.cmml" xref="S1.SS1.p5.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.2.m2.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.2.m2.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Here, we prove appealing properties of <math alttext="r_{U}" class="ltx_Math" display="inline" id="S1.SS1.p5.3.m3.1"><semantics id="S1.SS1.p5.3.m3.1a"><msub id="S1.SS1.p5.3.m3.1.1" xref="S1.SS1.p5.3.m3.1.1.cmml"><mi id="S1.SS1.p5.3.m3.1.1.2" xref="S1.SS1.p5.3.m3.1.1.2.cmml">r</mi><mi id="S1.SS1.p5.3.m3.1.1.3" xref="S1.SS1.p5.3.m3.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.3.m3.1b"><apply id="S1.SS1.p5.3.m3.1.1.cmml" xref="S1.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS1.p5.3.m3.1.1.1.cmml" xref="S1.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S1.SS1.p5.3.m3.1.1.2.cmml" xref="S1.SS1.p5.3.m3.1.1.2">𝑟</ci><ci id="S1.SS1.p5.3.m3.1.1.3.cmml" xref="S1.SS1.p5.3.m3.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.3.m3.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.3.m3.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem4"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Theorem 1.4</span> (Informal version of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem4" title="Theorem 4.4 ‣ Properties of 𝑟_𝑈 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.4</span></a> and Corollary <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmcorollary1" title="Corollary 1 ‣ Properties of 𝑟_𝑈 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1</span></a>)</h6> <div class="ltx_para" id="S1.Thmtheorem4.p1"> <p class="ltx_p" id="S1.Thmtheorem4.p1.4"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem4.p1.4.4">The function <math alttext="r_{U}(\alpha)" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.1.1.m1.1"><semantics id="S1.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S1.Thmtheorem4.p1.1.1.m1.1.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.cmml"><msub id="S1.Thmtheorem4.p1.1.1.m1.1.2.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.2.cmml"><mi id="S1.Thmtheorem4.p1.1.1.m1.1.2.2.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.2.2.cmml">r</mi><mi id="S1.Thmtheorem4.p1.1.1.m1.1.2.2.3" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.2.3.cmml">U</mi></msub><mo id="S1.Thmtheorem4.p1.1.1.m1.1.2.1" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem4.p1.1.1.m1.1.2.3.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.cmml"><mo id="S1.Thmtheorem4.p1.1.1.m1.1.2.3.2.1" stretchy="false" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.cmml">(</mo><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml">α</mi><mo id="S1.Thmtheorem4.p1.1.1.m1.1.2.3.2.2" stretchy="false" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.1.1.m1.1b"><apply id="S1.Thmtheorem4.p1.1.1.m1.1.2.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.2"><times id="S1.Thmtheorem4.p1.1.1.m1.1.2.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.1"></times><apply id="S1.Thmtheorem4.p1.1.1.m1.1.2.2.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.1.1.m1.1.2.2.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.2">subscript</csymbol><ci id="S1.Thmtheorem4.p1.1.1.m1.1.2.2.2.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.2.2">𝑟</ci><ci id="S1.Thmtheorem4.p1.1.1.m1.1.2.2.3.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.2.2.3">𝑈</ci></apply><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.1.1.m1.1c">r_{U}(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.1.1.m1.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_α )</annotation></semantics></math> is a smooth, increasing function that for a large enough <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.2.2.m2.1"><semantics id="S1.Thmtheorem4.p1.2.2.m2.1a"><mi id="S1.Thmtheorem4.p1.2.2.m2.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.2.2.m2.1b"><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.2.2.m2.1d">italic_α</annotation></semantics></math> satisfies <math alttext="r_{U}(\alpha)=\Theta(\min(m_{c}(G),n))" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.3.3.m3.5"><semantics id="S1.Thmtheorem4.p1.3.3.m3.5a"><mrow id="S1.Thmtheorem4.p1.3.3.m3.5.5" xref="S1.Thmtheorem4.p1.3.3.m3.5.5.cmml"><mrow id="S1.Thmtheorem4.p1.3.3.m3.5.5.3" 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id="S1.Thmtheorem4.p1.3.3.m3.5c">r_{U}(\alpha)=\Theta(\min(m_{c}(G),n))</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.3.3.m3.5d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_α ) = roman_Θ ( roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G ) , italic_n ) )</annotation></semantics></math>. <math alttext="r_{U}(\alpha)" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.4.4.m4.1"><semantics id="S1.Thmtheorem4.p1.4.4.m4.1a"><mrow id="S1.Thmtheorem4.p1.4.4.m4.1.2" xref="S1.Thmtheorem4.p1.4.4.m4.1.2.cmml"><msub id="S1.Thmtheorem4.p1.4.4.m4.1.2.2" xref="S1.Thmtheorem4.p1.4.4.m4.1.2.2.cmml"><mi id="S1.Thmtheorem4.p1.4.4.m4.1.2.2.2" xref="S1.Thmtheorem4.p1.4.4.m4.1.2.2.2.cmml">r</mi><mi id="S1.Thmtheorem4.p1.4.4.m4.1.2.2.3" xref="S1.Thmtheorem4.p1.4.4.m4.1.2.2.3.cmml">U</mi></msub><mo id="S1.Thmtheorem4.p1.4.4.m4.1.2.1" xref="S1.Thmtheorem4.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow 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xref="S1.Thmtheorem4.p1.4.4.m4.1.2.2.3">𝑈</ci></apply><ci id="S1.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.4.4.m4.1c">r_{U}(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.4.4.m4.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_α )</annotation></semantics></math> can be computed in polynomial time in the graph size.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p6"> <p class="ltx_p" id="S1.SS1.p6.1">Finally, in Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A2" title="Appendix 0.B Connections of Previous Results to Our Model ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.B</span></a> we formally connect our model with the model of <span class="ltx_ERROR undefined" id="S1.SS1.p6.1.1">\citet</span>basilico2017team. The first result is a lemma where we show that cost functions can be converted to payoff functions while preserving the PoU:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Lemma 1</span> (Informal version of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma20" title="Lemma 20 ‣ Converting payoffs to costs ‣ Appendix 0.B Connections of Previous Results to Our Model ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">20</span></a>)</h6> <div class="ltx_para" id="Thmlemma1.p1"> <p class="ltx_p" id="Thmlemma1.p1.3"><span class="ltx_text ltx_font_italic" id="Thmlemma1.p1.3.3">For every game instance with a payoff function <math alttext="u" class="ltx_Math" display="inline" id="Thmlemma1.p1.1.1.m1.1"><semantics id="Thmlemma1.p1.1.1.m1.1a"><mi id="Thmlemma1.p1.1.1.m1.1.1" xref="Thmlemma1.p1.1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.1.1.m1.1b"><ci id="Thmlemma1.p1.1.1.m1.1.1.cmml" xref="Thmlemma1.p1.1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.1.1.m1.1d">italic_u</annotation></semantics></math>, where the agents can guarantee positive payoff, there is a cost function <math alttext="C" class="ltx_Math" display="inline" id="Thmlemma1.p1.2.2.m2.1"><semantics id="Thmlemma1.p1.2.2.m2.1a"><mi id="Thmlemma1.p1.2.2.m2.1.1" xref="Thmlemma1.p1.2.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.2.2.m2.1b"><ci id="Thmlemma1.p1.2.2.m2.1.1.cmml" xref="Thmlemma1.p1.2.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.2.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.2.2.m2.1d">italic_C</annotation></semantics></math> such that <math alttext="r(C)=r(u)" class="ltx_Math" display="inline" id="Thmlemma1.p1.3.3.m3.2"><semantics id="Thmlemma1.p1.3.3.m3.2a"><mrow id="Thmlemma1.p1.3.3.m3.2.3" xref="Thmlemma1.p1.3.3.m3.2.3.cmml"><mrow id="Thmlemma1.p1.3.3.m3.2.3.2" xref="Thmlemma1.p1.3.3.m3.2.3.2.cmml"><mi id="Thmlemma1.p1.3.3.m3.2.3.2.2" xref="Thmlemma1.p1.3.3.m3.2.3.2.2.cmml">r</mi><mo id="Thmlemma1.p1.3.3.m3.2.3.2.1" xref="Thmlemma1.p1.3.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="Thmlemma1.p1.3.3.m3.2.3.2.3.2" xref="Thmlemma1.p1.3.3.m3.2.3.2.cmml"><mo id="Thmlemma1.p1.3.3.m3.2.3.2.3.2.1" stretchy="false" xref="Thmlemma1.p1.3.3.m3.2.3.2.cmml">(</mo><mi id="Thmlemma1.p1.3.3.m3.1.1" xref="Thmlemma1.p1.3.3.m3.1.1.cmml">C</mi><mo id="Thmlemma1.p1.3.3.m3.2.3.2.3.2.2" stretchy="false" xref="Thmlemma1.p1.3.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma1.p1.3.3.m3.2.3.1" xref="Thmlemma1.p1.3.3.m3.2.3.1.cmml">=</mo><mrow id="Thmlemma1.p1.3.3.m3.2.3.3" xref="Thmlemma1.p1.3.3.m3.2.3.3.cmml"><mi id="Thmlemma1.p1.3.3.m3.2.3.3.2" xref="Thmlemma1.p1.3.3.m3.2.3.3.2.cmml">r</mi><mo id="Thmlemma1.p1.3.3.m3.2.3.3.1" xref="Thmlemma1.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="Thmlemma1.p1.3.3.m3.2.3.3.3.2" xref="Thmlemma1.p1.3.3.m3.2.3.3.cmml"><mo id="Thmlemma1.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="Thmlemma1.p1.3.3.m3.2.3.3.cmml">(</mo><mi id="Thmlemma1.p1.3.3.m3.2.2" xref="Thmlemma1.p1.3.3.m3.2.2.cmml">u</mi><mo id="Thmlemma1.p1.3.3.m3.2.3.3.3.2.2" stretchy="false" xref="Thmlemma1.p1.3.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.3.3.m3.2b"><apply id="Thmlemma1.p1.3.3.m3.2.3.cmml" xref="Thmlemma1.p1.3.3.m3.2.3"><eq id="Thmlemma1.p1.3.3.m3.2.3.1.cmml" xref="Thmlemma1.p1.3.3.m3.2.3.1"></eq><apply id="Thmlemma1.p1.3.3.m3.2.3.2.cmml" xref="Thmlemma1.p1.3.3.m3.2.3.2"><times id="Thmlemma1.p1.3.3.m3.2.3.2.1.cmml" xref="Thmlemma1.p1.3.3.m3.2.3.2.1"></times><ci id="Thmlemma1.p1.3.3.m3.2.3.2.2.cmml" xref="Thmlemma1.p1.3.3.m3.2.3.2.2">𝑟</ci><ci id="Thmlemma1.p1.3.3.m3.1.1.cmml" xref="Thmlemma1.p1.3.3.m3.1.1">𝐶</ci></apply><apply id="Thmlemma1.p1.3.3.m3.2.3.3.cmml" xref="Thmlemma1.p1.3.3.m3.2.3.3"><times id="Thmlemma1.p1.3.3.m3.2.3.3.1.cmml" xref="Thmlemma1.p1.3.3.m3.2.3.3.1"></times><ci id="Thmlemma1.p1.3.3.m3.2.3.3.2.cmml" xref="Thmlemma1.p1.3.3.m3.2.3.3.2">𝑟</ci><ci id="Thmlemma1.p1.3.3.m3.2.2.cmml" xref="Thmlemma1.p1.3.3.m3.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.3.3.m3.2c">r(C)=r(u)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.3.3.m3.2d">italic_r ( italic_C ) = italic_r ( italic_u )</annotation></semantics></math>, and vice versa.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p7"> <p class="ltx_p" id="S1.SS1.p7.1">This implies that the PoU can be defined with either a cost function or a payoff function, thus establishing that our PoU bounds are indeed comparable to those of <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>]</cite>.</p> </div> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Our techniques</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.1">We now present and discuss the techniques we use in the proofs. Our techniques are novel in the context of the PoU.</p> </div> <section class="ltx_paragraph" id="S1.SS2.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Representing graphs as flow networks</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p1.3">We define the model formally using terminology from flow networks. Sets of disjoint paths are identified with 0-1 flows from <math alttext="s" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p1.1.m1.1a"><mi id="S1.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.1.m1.1b"><ci id="S1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.1.m1.1c">s</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.1.m1.1d">italic_s</annotation></semantics></math> to <math alttext="t" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.2.m2.1"><semantics id="S1.SS2.SSS0.Px1.p1.2.m2.1a"><mi id="S1.SS2.SSS0.Px1.p1.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.2.m2.1b"><ci id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.2.m2.1d">italic_t</annotation></semantics></math>. This view of the graph allows us to gain insights on the behavior of rational agents in our model. In short, for a given <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p1.3.m3.1a"><mi id="S1.SS2.SSS0.Px1.p1.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.3.m3.1b"><ci id="S1.SS2.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.3.m3.1d">italic_α</annotation></semantics></math>, they tend to ignore prohibitively long paths, and spread out over the remaining paths as much as possible. See Sections <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a> for further discussion.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p2"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p2.1">Finally, we use this perspective on the problem to also prove the computational result of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.Thmtheorem4" title="Theorem 1.4 (Informal version of Theorem 4.4 and Corollary 1) ‣ 1.1 Our results ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1.4</span></a>. There, we translate a given game instance to a mincost flow problem instance (see <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">ahuja1988network</span>]</cite> for reference).</p> </div> </section> <section class="ltx_paragraph" id="S1.SS2.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Approximating the PoU with a simplified PoU</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px2.p1.3">To prove the <math alttext="\Theta(\min(m_{c}(G),n))" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p1.1.m1.4"><semantics id="S1.SS2.SSS0.Px2.p1.1.m1.4a"><mrow id="S1.SS2.SSS0.Px2.p1.1.m1.4.4" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.cmml"><mi id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.3" mathvariant="normal" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.3.cmml">Θ</mi><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.2" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.cmml"><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.cmml">(</mo><mrow id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.2.cmml"><mi id="S1.SS2.SSS0.Px2.p1.1.m1.2.2" xref="S1.SS2.SSS0.Px2.p1.1.m1.2.2.cmml">min</mi><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1a" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.2.cmml">⁡</mo><mrow id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.2.cmml"><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.2.cmml">(</mo><mrow id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.cmml"><msub id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.cmml"><mi id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.2" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.2.cmml">m</mi><mi id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.3" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.3.cmml">c</mi></msub><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.3.2" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="S1.SS2.SSS0.Px2.p1.1.m1.1.1" xref="S1.SS2.SSS0.Px2.p1.1.m1.1.1.cmml">G</mi><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.2.cmml">,</mo><mi id="S1.SS2.SSS0.Px2.p1.1.m1.3.3" xref="S1.SS2.SSS0.Px2.p1.1.m1.3.3.cmml">n</mi><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.4" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p1.1.m1.4b"><apply id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4"><times id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.2.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.2"></times><ci id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.3.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.3">Θ</ci><apply id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1"><min id="S1.SS2.SSS0.Px2.p1.1.m1.2.2.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.2.2"></min><apply id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1"><times id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.1"></times><apply id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.1.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2">subscript</csymbol><ci id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.2.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.2">𝑚</ci><ci id="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.3.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.4.4.1.1.1.1.1.1.2.3">𝑐</ci></apply><ci id="S1.SS2.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.1.1">𝐺</ci></apply><ci id="S1.SS2.SSS0.Px2.p1.1.m1.3.3.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p1.1.m1.4c">\Theta(\min(m_{c}(G),n))</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p1.1.m1.4d">roman_Θ ( roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G ) , italic_n ) )</annotation></semantics></math> bound on the PoU, we approximate the PoU with the uniform PoU. The uniform PoU can be seen as a simplified version of the PoU, a version that is simpler both to compute (for example Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.Thmtheorem4" title="Theorem 1.4 (Informal version of Theorem 4.4 and Corollary 1) ‣ 1.1 Our results ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1.4</span></a> for SUM cost) and to understand. The second step of the technique is therefore to bound the uniform PoU. Put together, this two-step technique allows us to obtain asymptotically tight bounds on the PoU, while needing to consider only simple strategies in our calculations. Specifically, this technique enables us to prove (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem2" title="Theorem 4.2 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.2</span></a>) that, for every <math alttext="n\geq 1" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p1.2.m2.1"><semantics id="S1.SS2.SSS0.Px2.p1.2.m2.1a"><mrow id="S1.SS2.SSS0.Px2.p1.2.m2.1.1" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.2" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.2.cmml">n</mi><mo id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.cmml">≥</mo><mn id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.3" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p1.2.m2.1b"><apply id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1"><geq id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1"></geq><ci id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.2">𝑛</ci><cn id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p1.2.m2.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p1.2.m2.1d">italic_n ≥ 1</annotation></semantics></math>, and for sufficiently large <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p1.3.m3.1"><semantics id="S1.SS2.SSS0.Px2.p1.3.m3.1a"><mi id="S1.SS2.SSS0.Px2.p1.3.m3.1.1" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p1.3.m3.1b"><ci id="S1.SS2.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p1.3.m3.1d">italic_α</annotation></semantics></math>:</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\left(1-O\left(\frac{1}{\alpha}\right)\right)r(\alpha)\leq r_{U}(\alpha)\leq% \left(1+O\left(\frac{1}{\alpha}\right)\right)r(\alpha)" class="ltx_Math" display="block" id="S1.Ex1.m1.7"><semantics id="S1.Ex1.m1.7a"><mrow id="S1.Ex1.m1.7.7" xref="S1.Ex1.m1.7.7.cmml"><mrow id="S1.Ex1.m1.6.6.1" xref="S1.Ex1.m1.6.6.1.cmml"><mrow id="S1.Ex1.m1.6.6.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.1.cmml"><mo id="S1.Ex1.m1.6.6.1.1.1.2" xref="S1.Ex1.m1.6.6.1.1.1.1.cmml">(</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.1.cmml"><mn id="S1.Ex1.m1.6.6.1.1.1.1.2" xref="S1.Ex1.m1.6.6.1.1.1.1.2.cmml">1</mn><mo id="S1.Ex1.m1.6.6.1.1.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.1.1.cmml">−</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1.3" xref="S1.Ex1.m1.6.6.1.1.1.1.3.cmml"><mi id="S1.Ex1.m1.6.6.1.1.1.1.3.2" xref="S1.Ex1.m1.6.6.1.1.1.1.3.2.cmml">O</mi><mo id="S1.Ex1.m1.6.6.1.1.1.1.3.1" xref="S1.Ex1.m1.6.6.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1.3.3.2" xref="S1.Ex1.m1.1.1.cmml"><mo id="S1.Ex1.m1.6.6.1.1.1.1.3.3.2.1" xref="S1.Ex1.m1.1.1.cmml">(</mo><mfrac id="S1.Ex1.m1.1.1" xref="S1.Ex1.m1.1.1.cmml"><mn id="S1.Ex1.m1.1.1.2" xref="S1.Ex1.m1.1.1.2.cmml">1</mn><mi id="S1.Ex1.m1.1.1.3" xref="S1.Ex1.m1.1.1.3.cmml">α</mi></mfrac><mo id="S1.Ex1.m1.6.6.1.1.1.1.3.3.2.2" xref="S1.Ex1.m1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S1.Ex1.m1.6.6.1.1.1.3" xref="S1.Ex1.m1.6.6.1.1.1.1.cmml">)</mo></mrow><mo id="S1.Ex1.m1.6.6.1.2" xref="S1.Ex1.m1.6.6.1.2.cmml">⁢</mo><mi id="S1.Ex1.m1.6.6.1.3" xref="S1.Ex1.m1.6.6.1.3.cmml">r</mi><mo id="S1.Ex1.m1.6.6.1.2a" xref="S1.Ex1.m1.6.6.1.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.6.6.1.4.2" xref="S1.Ex1.m1.6.6.1.cmml"><mo id="S1.Ex1.m1.6.6.1.4.2.1" stretchy="false" xref="S1.Ex1.m1.6.6.1.cmml">(</mo><mi id="S1.Ex1.m1.2.2" xref="S1.Ex1.m1.2.2.cmml">α</mi><mo id="S1.Ex1.m1.6.6.1.4.2.2" stretchy="false" xref="S1.Ex1.m1.6.6.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex1.m1.7.7.4" xref="S1.Ex1.m1.7.7.4.cmml">≤</mo><mrow id="S1.Ex1.m1.7.7.5" xref="S1.Ex1.m1.7.7.5.cmml"><msub id="S1.Ex1.m1.7.7.5.2" xref="S1.Ex1.m1.7.7.5.2.cmml"><mi id="S1.Ex1.m1.7.7.5.2.2" xref="S1.Ex1.m1.7.7.5.2.2.cmml">r</mi><mi id="S1.Ex1.m1.7.7.5.2.3" xref="S1.Ex1.m1.7.7.5.2.3.cmml">U</mi></msub><mo id="S1.Ex1.m1.7.7.5.1" xref="S1.Ex1.m1.7.7.5.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.7.7.5.3.2" xref="S1.Ex1.m1.7.7.5.cmml"><mo id="S1.Ex1.m1.7.7.5.3.2.1" stretchy="false" xref="S1.Ex1.m1.7.7.5.cmml">(</mo><mi id="S1.Ex1.m1.3.3" xref="S1.Ex1.m1.3.3.cmml">α</mi><mo id="S1.Ex1.m1.7.7.5.3.2.2" stretchy="false" xref="S1.Ex1.m1.7.7.5.cmml">)</mo></mrow></mrow><mo id="S1.Ex1.m1.7.7.6" xref="S1.Ex1.m1.7.7.6.cmml">≤</mo><mrow id="S1.Ex1.m1.7.7.2" xref="S1.Ex1.m1.7.7.2.cmml"><mrow id="S1.Ex1.m1.7.7.2.1.1" xref="S1.Ex1.m1.7.7.2.1.1.1.cmml"><mo id="S1.Ex1.m1.7.7.2.1.1.2" xref="S1.Ex1.m1.7.7.2.1.1.1.cmml">(</mo><mrow id="S1.Ex1.m1.7.7.2.1.1.1" xref="S1.Ex1.m1.7.7.2.1.1.1.cmml"><mn id="S1.Ex1.m1.7.7.2.1.1.1.2" xref="S1.Ex1.m1.7.7.2.1.1.1.2.cmml">1</mn><mo id="S1.Ex1.m1.7.7.2.1.1.1.1" xref="S1.Ex1.m1.7.7.2.1.1.1.1.cmml">+</mo><mrow id="S1.Ex1.m1.7.7.2.1.1.1.3" xref="S1.Ex1.m1.7.7.2.1.1.1.3.cmml"><mi id="S1.Ex1.m1.7.7.2.1.1.1.3.2" xref="S1.Ex1.m1.7.7.2.1.1.1.3.2.cmml">O</mi><mo id="S1.Ex1.m1.7.7.2.1.1.1.3.1" xref="S1.Ex1.m1.7.7.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.7.7.2.1.1.1.3.3.2" xref="S1.Ex1.m1.4.4.cmml"><mo id="S1.Ex1.m1.7.7.2.1.1.1.3.3.2.1" xref="S1.Ex1.m1.4.4.cmml">(</mo><mfrac id="S1.Ex1.m1.4.4" xref="S1.Ex1.m1.4.4.cmml"><mn id="S1.Ex1.m1.4.4.2" xref="S1.Ex1.m1.4.4.2.cmml">1</mn><mi id="S1.Ex1.m1.4.4.3" xref="S1.Ex1.m1.4.4.3.cmml">α</mi></mfrac><mo id="S1.Ex1.m1.7.7.2.1.1.1.3.3.2.2" xref="S1.Ex1.m1.4.4.cmml">)</mo></mrow></mrow></mrow><mo id="S1.Ex1.m1.7.7.2.1.1.3" xref="S1.Ex1.m1.7.7.2.1.1.1.cmml">)</mo></mrow><mo id="S1.Ex1.m1.7.7.2.2" xref="S1.Ex1.m1.7.7.2.2.cmml">⁢</mo><mi id="S1.Ex1.m1.7.7.2.3" xref="S1.Ex1.m1.7.7.2.3.cmml">r</mi><mo id="S1.Ex1.m1.7.7.2.2a" xref="S1.Ex1.m1.7.7.2.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.7.7.2.4.2" xref="S1.Ex1.m1.7.7.2.cmml"><mo id="S1.Ex1.m1.7.7.2.4.2.1" stretchy="false" xref="S1.Ex1.m1.7.7.2.cmml">(</mo><mi id="S1.Ex1.m1.5.5" xref="S1.Ex1.m1.5.5.cmml">α</mi><mo id="S1.Ex1.m1.7.7.2.4.2.2" stretchy="false" xref="S1.Ex1.m1.7.7.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex1.m1.7b"><apply id="S1.Ex1.m1.7.7.cmml" xref="S1.Ex1.m1.7.7"><and id="S1.Ex1.m1.7.7a.cmml" xref="S1.Ex1.m1.7.7"></and><apply id="S1.Ex1.m1.7.7b.cmml" xref="S1.Ex1.m1.7.7"><leq id="S1.Ex1.m1.7.7.4.cmml" xref="S1.Ex1.m1.7.7.4"></leq><apply id="S1.Ex1.m1.6.6.1.cmml" xref="S1.Ex1.m1.6.6.1"><times id="S1.Ex1.m1.6.6.1.2.cmml" xref="S1.Ex1.m1.6.6.1.2"></times><apply id="S1.Ex1.m1.6.6.1.1.1.1.cmml" xref="S1.Ex1.m1.6.6.1.1.1"><minus id="S1.Ex1.m1.6.6.1.1.1.1.1.cmml" xref="S1.Ex1.m1.6.6.1.1.1.1.1"></minus><cn id="S1.Ex1.m1.6.6.1.1.1.1.2.cmml" type="integer" xref="S1.Ex1.m1.6.6.1.1.1.1.2">1</cn><apply id="S1.Ex1.m1.6.6.1.1.1.1.3.cmml" xref="S1.Ex1.m1.6.6.1.1.1.1.3"><times id="S1.Ex1.m1.6.6.1.1.1.1.3.1.cmml" xref="S1.Ex1.m1.6.6.1.1.1.1.3.1"></times><ci id="S1.Ex1.m1.6.6.1.1.1.1.3.2.cmml" xref="S1.Ex1.m1.6.6.1.1.1.1.3.2">𝑂</ci><apply id="S1.Ex1.m1.1.1.cmml" xref="S1.Ex1.m1.6.6.1.1.1.1.3.3.2"><divide id="S1.Ex1.m1.1.1.1.cmml" xref="S1.Ex1.m1.6.6.1.1.1.1.3.3.2"></divide><cn id="S1.Ex1.m1.1.1.2.cmml" type="integer" xref="S1.Ex1.m1.1.1.2">1</cn><ci id="S1.Ex1.m1.1.1.3.cmml" xref="S1.Ex1.m1.1.1.3">𝛼</ci></apply></apply></apply><ci id="S1.Ex1.m1.6.6.1.3.cmml" xref="S1.Ex1.m1.6.6.1.3">𝑟</ci><ci id="S1.Ex1.m1.2.2.cmml" xref="S1.Ex1.m1.2.2">𝛼</ci></apply><apply id="S1.Ex1.m1.7.7.5.cmml" xref="S1.Ex1.m1.7.7.5"><times id="S1.Ex1.m1.7.7.5.1.cmml" xref="S1.Ex1.m1.7.7.5.1"></times><apply id="S1.Ex1.m1.7.7.5.2.cmml" xref="S1.Ex1.m1.7.7.5.2"><csymbol cd="ambiguous" id="S1.Ex1.m1.7.7.5.2.1.cmml" xref="S1.Ex1.m1.7.7.5.2">subscript</csymbol><ci id="S1.Ex1.m1.7.7.5.2.2.cmml" xref="S1.Ex1.m1.7.7.5.2.2">𝑟</ci><ci id="S1.Ex1.m1.7.7.5.2.3.cmml" xref="S1.Ex1.m1.7.7.5.2.3">𝑈</ci></apply><ci id="S1.Ex1.m1.3.3.cmml" xref="S1.Ex1.m1.3.3">𝛼</ci></apply></apply><apply id="S1.Ex1.m1.7.7c.cmml" xref="S1.Ex1.m1.7.7"><leq id="S1.Ex1.m1.7.7.6.cmml" xref="S1.Ex1.m1.7.7.6"></leq><share href="https://arxiv.org/html/2308.08047v3#S1.Ex1.m1.7.7.5.cmml" id="S1.Ex1.m1.7.7d.cmml" xref="S1.Ex1.m1.7.7"></share><apply id="S1.Ex1.m1.7.7.2.cmml" xref="S1.Ex1.m1.7.7.2"><times id="S1.Ex1.m1.7.7.2.2.cmml" xref="S1.Ex1.m1.7.7.2.2"></times><apply id="S1.Ex1.m1.7.7.2.1.1.1.cmml" xref="S1.Ex1.m1.7.7.2.1.1"><plus id="S1.Ex1.m1.7.7.2.1.1.1.1.cmml" xref="S1.Ex1.m1.7.7.2.1.1.1.1"></plus><cn id="S1.Ex1.m1.7.7.2.1.1.1.2.cmml" type="integer" xref="S1.Ex1.m1.7.7.2.1.1.1.2">1</cn><apply id="S1.Ex1.m1.7.7.2.1.1.1.3.cmml" xref="S1.Ex1.m1.7.7.2.1.1.1.3"><times id="S1.Ex1.m1.7.7.2.1.1.1.3.1.cmml" xref="S1.Ex1.m1.7.7.2.1.1.1.3.1"></times><ci id="S1.Ex1.m1.7.7.2.1.1.1.3.2.cmml" xref="S1.Ex1.m1.7.7.2.1.1.1.3.2">𝑂</ci><apply id="S1.Ex1.m1.4.4.cmml" xref="S1.Ex1.m1.7.7.2.1.1.1.3.3.2"><divide id="S1.Ex1.m1.4.4.1.cmml" xref="S1.Ex1.m1.7.7.2.1.1.1.3.3.2"></divide><cn id="S1.Ex1.m1.4.4.2.cmml" type="integer" xref="S1.Ex1.m1.4.4.2">1</cn><ci id="S1.Ex1.m1.4.4.3.cmml" xref="S1.Ex1.m1.4.4.3">𝛼</ci></apply></apply></apply><ci id="S1.Ex1.m1.7.7.2.3.cmml" xref="S1.Ex1.m1.7.7.2.3">𝑟</ci><ci id="S1.Ex1.m1.5.5.cmml" xref="S1.Ex1.m1.5.5">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex1.m1.7c">\left(1-O\left(\frac{1}{\alpha}\right)\right)r(\alpha)\leq r_{U}(\alpha)\leq% \left(1+O\left(\frac{1}{\alpha}\right)\right)r(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S1.Ex1.m1.7d">( 1 - italic_O ( divide start_ARG 1 end_ARG start_ARG italic_α end_ARG ) ) italic_r ( italic_α ) ≤ italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_α ) ≤ ( 1 + italic_O ( divide start_ARG 1 end_ARG start_ARG italic_α end_ARG ) ) italic_r ( italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px2.p2"> <p class="ltx_p" id="S1.SS2.SSS0.Px2.p2.1">Next, we discuss the techniques used in our equilibrium strategies analyses.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS2.SSS0.Px3"> <h5 class="ltx_title ltx_title_paragraph">Reduction of strategy space</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px3.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px3.p1.1">In a search for optimal strategies, the simple forms of SUM and MAX allow us to simplify significantly the strategy spaces of both the agents and the interceptors. When simplifying the strategy spaces, we always keep at least one optimal strategy in the strategy space.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px3.p2"> <p class="ltx_p" id="S1.SS2.SSS0.Px3.p2.1">In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.SS2" title="4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.2</span></a> we focus on the SUM cost over graphs of disjoint paths. There, the strategy space reduction technique is particularly effective:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Lemma 2</span> (Informal version of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma11" title="Lemma 11 ‣ 4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">11</span></a>)</h6> <div class="ltx_para" id="Thmlemma2.p1"> <p class="ltx_p" id="Thmlemma2.p1.4"><span class="ltx_text ltx_font_italic" id="Thmlemma2.p1.4.4">For every graph <math alttext="G" class="ltx_Math" display="inline" id="Thmlemma2.p1.1.1.m1.1"><semantics id="Thmlemma2.p1.1.1.m1.1a"><mi id="Thmlemma2.p1.1.1.m1.1.1" xref="Thmlemma2.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmlemma2.p1.1.1.m1.1b"><ci id="Thmlemma2.p1.1.1.m1.1.1.cmml" xref="Thmlemma2.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma2.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmlemma2.p1.1.1.m1.1d">italic_G</annotation></semantics></math> of disjoint paths, and for every <math alttext="n\geq 1" class="ltx_Math" display="inline" id="Thmlemma2.p1.2.2.m2.1"><semantics id="Thmlemma2.p1.2.2.m2.1a"><mrow id="Thmlemma2.p1.2.2.m2.1.1" xref="Thmlemma2.p1.2.2.m2.1.1.cmml"><mi id="Thmlemma2.p1.2.2.m2.1.1.2" xref="Thmlemma2.p1.2.2.m2.1.1.2.cmml">n</mi><mo id="Thmlemma2.p1.2.2.m2.1.1.1" xref="Thmlemma2.p1.2.2.m2.1.1.1.cmml">≥</mo><mn id="Thmlemma2.p1.2.2.m2.1.1.3" xref="Thmlemma2.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma2.p1.2.2.m2.1b"><apply id="Thmlemma2.p1.2.2.m2.1.1.cmml" xref="Thmlemma2.p1.2.2.m2.1.1"><geq id="Thmlemma2.p1.2.2.m2.1.1.1.cmml" xref="Thmlemma2.p1.2.2.m2.1.1.1"></geq><ci id="Thmlemma2.p1.2.2.m2.1.1.2.cmml" xref="Thmlemma2.p1.2.2.m2.1.1.2">𝑛</ci><cn id="Thmlemma2.p1.2.2.m2.1.1.3.cmml" type="integer" xref="Thmlemma2.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma2.p1.2.2.m2.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="Thmlemma2.p1.2.2.m2.1d">italic_n ≥ 1</annotation></semantics></math>, the convex hull of <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="Thmlemma2.p1.3.3.m3.1"><semantics id="Thmlemma2.p1.3.3.m3.1a"><msub id="Thmlemma2.p1.3.3.m3.1.1" xref="Thmlemma2.p1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma2.p1.3.3.m3.1.1.2" xref="Thmlemma2.p1.3.3.m3.1.1.2.cmml">𝒰</mi><mi id="Thmlemma2.p1.3.3.m3.1.1.3" xref="Thmlemma2.p1.3.3.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Thmlemma2.p1.3.3.m3.1b"><apply id="Thmlemma2.p1.3.3.m3.1.1.cmml" xref="Thmlemma2.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmlemma2.p1.3.3.m3.1.1.1.cmml" xref="Thmlemma2.p1.3.3.m3.1.1">subscript</csymbol><ci id="Thmlemma2.p1.3.3.m3.1.1.2.cmml" xref="Thmlemma2.p1.3.3.m3.1.1.2">𝒰</ci><ci id="Thmlemma2.p1.3.3.m3.1.1.3.cmml" xref="Thmlemma2.p1.3.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma2.p1.3.3.m3.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma2.p1.3.3.m3.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> admits an optimal agent strategy for <math alttext="n" class="ltx_Math" display="inline" id="Thmlemma2.p1.4.4.m4.1"><semantics id="Thmlemma2.p1.4.4.m4.1a"><mi id="Thmlemma2.p1.4.4.m4.1.1" xref="Thmlemma2.p1.4.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmlemma2.p1.4.4.m4.1b"><ci id="Thmlemma2.p1.4.4.m4.1.1.cmml" xref="Thmlemma2.p1.4.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma2.p1.4.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmlemma2.p1.4.4.m4.1d">italic_n</annotation></semantics></math> uncorrelated agents.</span></p> </div> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px3.p3"> <p class="ltx_p" id="S1.SS2.SSS0.Px3.p3.1">In the case of the MAX cost, this technique’s effectiveness is more limited (since MAX is determined only by the longest path, rather than all paths).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Lemma 3</span> (Informal version of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma13" title="Lemma 13 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">13</span></a>)</h6> <div class="ltx_para" id="Thmlemma3.p1"> <p class="ltx_p" id="Thmlemma3.p1.3"><span class="ltx_text ltx_font_italic" id="Thmlemma3.p1.3.3">For every graph <math alttext="G" class="ltx_Math" display="inline" id="Thmlemma3.p1.1.1.m1.1"><semantics id="Thmlemma3.p1.1.1.m1.1a"><mi id="Thmlemma3.p1.1.1.m1.1.1" xref="Thmlemma3.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmlemma3.p1.1.1.m1.1b"><ci id="Thmlemma3.p1.1.1.m1.1.1.cmml" xref="Thmlemma3.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma3.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmlemma3.p1.1.1.m1.1d">italic_G</annotation></semantics></math>, and for every <math alttext="n\geq 1" class="ltx_Math" display="inline" id="Thmlemma3.p1.2.2.m2.1"><semantics id="Thmlemma3.p1.2.2.m2.1a"><mrow id="Thmlemma3.p1.2.2.m2.1.1" xref="Thmlemma3.p1.2.2.m2.1.1.cmml"><mi id="Thmlemma3.p1.2.2.m2.1.1.2" xref="Thmlemma3.p1.2.2.m2.1.1.2.cmml">n</mi><mo id="Thmlemma3.p1.2.2.m2.1.1.1" xref="Thmlemma3.p1.2.2.m2.1.1.1.cmml">≥</mo><mn id="Thmlemma3.p1.2.2.m2.1.1.3" xref="Thmlemma3.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma3.p1.2.2.m2.1b"><apply id="Thmlemma3.p1.2.2.m2.1.1.cmml" xref="Thmlemma3.p1.2.2.m2.1.1"><geq id="Thmlemma3.p1.2.2.m2.1.1.1.cmml" xref="Thmlemma3.p1.2.2.m2.1.1.1"></geq><ci id="Thmlemma3.p1.2.2.m2.1.1.2.cmml" xref="Thmlemma3.p1.2.2.m2.1.1.2">𝑛</ci><cn id="Thmlemma3.p1.2.2.m2.1.1.3.cmml" type="integer" xref="Thmlemma3.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma3.p1.2.2.m2.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="Thmlemma3.p1.2.2.m2.1d">italic_n ≥ 1</annotation></semantics></math>, there exists an optimal agent strategy for <math alttext="n" class="ltx_Math" display="inline" id="Thmlemma3.p1.3.3.m3.1"><semantics id="Thmlemma3.p1.3.3.m3.1a"><mi id="Thmlemma3.p1.3.3.m3.1.1" xref="Thmlemma3.p1.3.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmlemma3.p1.3.3.m3.1b"><ci id="Thmlemma3.p1.3.3.m3.1.1.cmml" xref="Thmlemma3.p1.3.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma3.p1.3.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmlemma3.p1.3.3.m3.1d">italic_n</annotation></semantics></math> uncorrelated agents where all agents use the same individual mixed strategy.</span></p> </div> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px3.p4"> <p class="ltx_p" id="S1.SS2.SSS0.Px3.p4.1">In other words, the agents’ mixed strategies are distributed iid. This produces optimal strategies which may seem counter-intuitive, where agents might intentionally send traffic on inherently long paths, while ignoring shorter paths that are also available.</p> </div> </section> </section> <section class="ltx_subsection" id="S1.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.3 </span>Organization</h3> <div class="ltx_para" id="S1.SS3.p1"> <p class="ltx_p" id="S1.SS3.p1.1">In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S2" title="2 Related Work ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">2</span></a> we overview related work. In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S3" title="3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">3</span></a> we formally define the problem and state some general lemmas. In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a> we analyze and prove our results for the SUM cost function. In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a> we analyze and prove our results for the MAX cost function. In the Appendices we provide omitted details proofs for some of the theorems and lemmas in this paper, and connect the results of <span class="ltx_ERROR undefined" id="S1.SS3.p1.1.1">\citet</span>basilico2017team to ours as described above.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Related Work</h2> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Team-maxmin equilibria</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p1"> <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px1.p1.1">\citet</span> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p1.2">von1997team study the maxmin value of zero-sum, normal-form games between a team and an adversary, where the team agents aim to maximize their identical payoffs. The team agents are uncorrelated. Von Stengel and Koller show that every team (mixed) strategy that attains the maxmin value of such a game is a part of a NE of the game, and in particular a payoff-maximal NE. Such a team strategy is called a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.2.1">team-maxmin strategy</span>, and such a NE is called a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.2.2">team-maxmin equilibrium</span> (TME). The payoff the team gets from TMEs is unique. <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px1.p1.2.3">\citet</span>von1997team introduce the concept of TME and only consider the case of uncorrelated agents. In contrast, we use the TME notion to model the impact that correlation between the agents has on their performance in our games, and prove results that highlight the extent to which correlation is significant in the games that we study.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p2.1">Some work on determining TMEs and their values has been done in <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.1.1">rendezvous games</span> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">alpern1995rendezvous</span>]</cite>, where two agents seek to meet at a point in space. <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px1.p2.1.2">\citet</span>lim1997rendezvous,lim2000rendezvous,alpern1998symmetric consider versions of the rendezvous game where the agents in addition attempt to avoid being captured by an enemy searcher. <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px1.p2.1.3">\citet</span>lim1997rendezvous,lim2000rendezvous consider correlated team agents playing against an enemy searcher on a clique and a circle (respectively), while <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px1.p2.1.4">\citet</span>alpern1998symmetric consider ‘symmetric’ team agents who cannot correlate their strategies if they want to meet. While these rendezvous games are multi-round games, these games have only two agents and one enemy and are played on restricted types of graphs. In comparison, while our games are single-round games, we allow any number of players on both teams, as well as any <math alttext="s-t" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p2.1.m1.1a"><mrow id="S2.SS0.SSS0.Px1.p2.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.2" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.2.cmml">s</mi><mo id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.1" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.1.cmml">−</mo><mi id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.3" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.1.m1.1b"><apply id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1"><minus id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.1"></minus><ci id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.2">𝑠</ci><ci id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.1.m1.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.1.m1.1d">italic_s - italic_t</annotation></semantics></math> graph. This means that our paper studies the PoU for a much larger set of scenarios.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Computing TMEs</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px2.p1.1">Computational aspects of TMEs have been thoroughly studied, showing both algorithms for finding TMEs (in both normal-form and extensive form games) <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">zhang2020computing</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">kalogiannis2022efficiently</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">celli2018computational</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">anagnostides2023algorithms</span>]</cite> and hardness results <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">anagnostides2023algorithms</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">hansen2008approximability</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">kalogiannis2022towards</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">celli2018computational</span>]</cite>, in particular showing that computing a TME is FNP-hard <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">hansen2008approximability</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>]</cite>. In the computational aspect, we show (Corollary <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmcorollary1" title="Corollary 1 ‣ Properties of 𝑟_𝑈 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1</span></a>) that for SUM cost, the PoU can be approximated in polynomial time. Our focus in this paper is on asymptotic quantification of the PoU rather than an exact quantification. Note that the results mentioned in this paragraph imply that exact quantification is potentially a computationally hard problem, therefore there might not be an explicit and exact expression for the PoU in our games.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px3"> <h5 class="ltx_title ltx_title_paragraph">TMECor and PoU</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px3.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px3.p1.1">When the team agents are correlated, they can be seen as one player, and then the game reduces to a 2-player zero-sum game. Then the TME is often called <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p1.1.1">TMECor</span> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>]</cite>, and can be found in polynomial time via linear programming. As was stated earlier, The PoU is the ratio of the payoff (of the team agents) at a TMECor to the payoff at a TME, and it was studied in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">celli2020coordination</span>]</cite>. <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px3.p1.1.2">\citet</span>celli2020coordination considers the PoU for extensive-form games, and shows bounds on it that depend on the number of leaves in the tree. When the game is in extensive form, one can pick whether the team agents have shared randomness only at the root or at all nodes of the game. This gives rise to refined notions of correlation <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">celli2020coordination</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">zhang2020computing</span>]</cite>. In our games, since they are single-round games, the distinction between these correlation notions does not exist. On the other hand, the extensive-form games mentioned above are abstract, while ours are concrete and motivated by the concrete scenarios mentioned in the introduction.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px3.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px3.p2.10">The PoU was also studied in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">schulman2017duality</span>]</cite> as a difference rather than a ratio. The setting there is even more general, having 2 teams (call them team <math alttext="A" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.1.m1.1"><semantics id="S2.SS0.SSS0.Px3.p2.1.m1.1a"><mi id="S2.SS0.SSS0.Px3.p2.1.m1.1.1" xref="S2.SS0.SSS0.Px3.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.1.m1.1b"><ci id="S2.SS0.SSS0.Px3.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.1.m1.1d">italic_A</annotation></semantics></math> and team <math alttext="B" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.2.m2.1"><semantics id="S2.SS0.SSS0.Px3.p2.2.m2.1a"><mi id="S2.SS0.SSS0.Px3.p2.2.m2.1.1" xref="S2.SS0.SSS0.Px3.p2.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.2.m2.1b"><ci id="S2.SS0.SSS0.Px3.p2.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.2.m2.1d">italic_B</annotation></semantics></math>) playing against each other in a team zero-sum game. The main focus is on the <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.10.1">defensive gap</span> of such games, which is the largest difference between the minmax and the maxmin values of a game of a fixed size, and can be phrased as a sum of additive PoUs. <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px3.p2.10.2">\citet</span>schulman2017duality show that the additive version of PoU for team <math alttext="A" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.3.m3.1"><semantics id="S2.SS0.SSS0.Px3.p2.3.m3.1a"><mi id="S2.SS0.SSS0.Px3.p2.3.m3.1.1" xref="S2.SS0.SSS0.Px3.p2.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.3.m3.1b"><ci id="S2.SS0.SSS0.Px3.p2.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.3.m3.1d">italic_A</annotation></semantics></math> can get as large as <math alttext="(1-\text{Val}T)(1-\hat{m}_{A}^{1-k_{A}})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.4.m4.2"><semantics id="S2.SS0.SSS0.Px3.p2.4.m4.2a"><mrow id="S2.SS0.SSS0.Px3.p2.4.m4.2.2" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.cmml"><mrow id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.cmml"><mn id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.1.cmml">−</mo><mrow id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.3.cmml"><mtext id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.1.1.1.3.2" 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xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3">superscript</csymbol><apply id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3">subscript</csymbol><apply id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.2"><ci id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.2.1">^</ci><ci id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.2.2">𝑚</ci></apply><ci id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.3.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.2.3">𝐴</ci></apply><apply id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3"><minus id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.1"></minus><cn id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.2.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.2">1</cn><apply id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3.2">𝑘</ci><ci id="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3.3.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.2.2.2.1.1.3.3.3.3">𝐴</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.4.m4.2c">(1-\text{Val}T)(1-\hat{m}_{A}^{1-k_{A}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.4.m4.2d">( 1 - Val italic_T ) ( 1 - over^ start_ARG italic_m end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 - italic_k start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT )</annotation></semantics></math>, where <math alttext="\text{Val}T" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.5.m5.1"><semantics id="S2.SS0.SSS0.Px3.p2.5.m5.1a"><mrow id="S2.SS0.SSS0.Px3.p2.5.m5.1.1" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.cmml"><mtext id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.2" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.2a.cmml">Val</mtext><mo id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.1" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.3" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.5.m5.1b"><apply id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1"><times id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.1"></times><ci id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.2a.cmml" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.2"><mtext id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.2">Val</mtext></ci><ci id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.5.m5.1c">\text{Val}T</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.5.m5.1d">Val italic_T</annotation></semantics></math> is the minmax value of the game when the members of each team are correlated, <math alttext="k_{A}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.6.m6.1"><semantics id="S2.SS0.SSS0.Px3.p2.6.m6.1a"><msub id="S2.SS0.SSS0.Px3.p2.6.m6.1.1" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.6.m6.1.1.2" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1.2.cmml">k</mi><mi id="S2.SS0.SSS0.Px3.p2.6.m6.1.1.3" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.6.m6.1b"><apply id="S2.SS0.SSS0.Px3.p2.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1.2">𝑘</ci><ci id="S2.SS0.SSS0.Px3.p2.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.6.m6.1c">k_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.6.m6.1d">italic_k start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> is the size of team <math alttext="A" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.7.m7.1"><semantics id="S2.SS0.SSS0.Px3.p2.7.m7.1a"><mi id="S2.SS0.SSS0.Px3.p2.7.m7.1.1" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.7.m7.1b"><ci id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.7.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.7.m7.1d">italic_A</annotation></semantics></math> and <math alttext="\hat{m}_{A}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.8.m8.1"><semantics id="S2.SS0.SSS0.Px3.p2.8.m8.1a"><msub id="S2.SS0.SSS0.Px3.p2.8.m8.1.1" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.cmml"><mover accent="true" id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.2.cmml">m</mi><mo id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.1" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.1.cmml">^</mo></mover><mi id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.3" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.8.m8.1b"><apply id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1">subscript</csymbol><apply id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2"><ci id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.1">^</ci><ci id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.2.2">𝑚</ci></apply><ci id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.8.m8.1c">\hat{m}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.8.m8.1d">over^ start_ARG italic_m end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> is the geometric mean of the number of strategies of team <math alttext="A" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.9.m9.1"><semantics id="S2.SS0.SSS0.Px3.p2.9.m9.1a"><mi id="S2.SS0.SSS0.Px3.p2.9.m9.1.1" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.9.m9.1b"><ci id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.9.m9.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.9.m9.1d">italic_A</annotation></semantics></math>’s members. This aligns with the <math alttext="m^{n-1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.10.m10.1"><semantics id="S2.SS0.SSS0.Px3.p2.10.m10.1a"><msup id="S2.SS0.SSS0.Px3.p2.10.m10.1.1" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2.cmml">m</mi><mrow id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.2.cmml">n</mi><mo id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.1" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.1.cmml">−</mo><mn id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.10.m10.1b"><apply id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2">𝑚</ci><apply id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3"><minus id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.1"></minus><ci id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.2">𝑛</ci><cn id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.10.m10.1c">m^{n-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.10.m10.1d">italic_m start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> bound of <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px3.p2.10.3">\citet</span>basilico2017team when converting the ratio form into a difference form. As discussed in the introduction, we show a linear PoU for our class of games.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px4"> <h5 class="ltx_title ltx_title_paragraph">Uniform mixed equilibria</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px4.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px4.p1.1">A somewhat similar model was defined in the recent <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">bilo2023uniform</span>]</cite>, who consider congestion games such that team agents can only play uniform strategies, and face an adversary who is able to corrupt edges. As we show, after we change the cost function, uniform distributions will not always be the best strategies for the team agents, hence deviating from <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">bilo2023uniform</span>]</cite>. Our model is different than <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">bilo2023uniform</span>]</cite> in two ways. First, in our model the interceptors can only ‘slow down’ the agents rather than causing link failures. Therefore, the way we model the interceptors models different settings than the settings modeled by <span class="ltx_ERROR undefined" id="S2.SS0.SSS0.Px4.p1.1.1">\citet</span>bilo2023uniform. Second, in our model each agent can play an arbitrary mixed strategy, as opposed to only picking “<math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.1.m1.1"><semantics id="S2.SS0.SSS0.Px4.p1.1.m1.1a"><mi id="S2.SS0.SSS0.Px4.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.1.m1.1b"><ci id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.1.m1.1d">italic_ρ</annotation></semantics></math>-uniform strategies”. In other words, in our model the agents have richer sets of (mixed) strategies, which makes our model more general.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Preliminaries</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Notations</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.27">For a set <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mi id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_X</annotation></semantics></math> and <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">n</mi><mo id="S3.SS1.p1.2.m2.1.1.1" xref="S3.SS1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><in id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1"></in><ci id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2">𝑛</ci><ci id="S3.SS1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, we denote with <math alttext="X^{n}" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><msup id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.p1.3.m3.1.1.2" xref="S3.SS1.p1.3.m3.1.1.2.cmml">X</mi><mi id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">superscript</csymbol><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">𝑋</ci><ci id="S3.SS1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">X^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> the cartesian product of <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mi id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><ci id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">italic_n</annotation></semantics></math> copies of <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mi id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><ci id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">italic_X</annotation></semantics></math> and <math alttext="X^{*}=\bigcup_{n\in\mathbb{N}}X^{n}" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.1"><semantics id="S3.SS1.p1.6.m6.1a"><mrow id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml"><msup id="S3.SS1.p1.6.m6.1.1.2" xref="S3.SS1.p1.6.m6.1.1.2.cmml"><mi id="S3.SS1.p1.6.m6.1.1.2.2" xref="S3.SS1.p1.6.m6.1.1.2.2.cmml">X</mi><mo id="S3.SS1.p1.6.m6.1.1.2.3" xref="S3.SS1.p1.6.m6.1.1.2.3.cmml">∗</mo></msup><mo id="S3.SS1.p1.6.m6.1.1.1" rspace="0.111em" xref="S3.SS1.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S3.SS1.p1.6.m6.1.1.3" xref="S3.SS1.p1.6.m6.1.1.3.cmml"><msub id="S3.SS1.p1.6.m6.1.1.3.1" xref="S3.SS1.p1.6.m6.1.1.3.1.cmml"><mo id="S3.SS1.p1.6.m6.1.1.3.1.2" xref="S3.SS1.p1.6.m6.1.1.3.1.2.cmml">⋃</mo><mrow id="S3.SS1.p1.6.m6.1.1.3.1.3" xref="S3.SS1.p1.6.m6.1.1.3.1.3.cmml"><mi id="S3.SS1.p1.6.m6.1.1.3.1.3.2" xref="S3.SS1.p1.6.m6.1.1.3.1.3.2.cmml">n</mi><mo id="S3.SS1.p1.6.m6.1.1.3.1.3.1" xref="S3.SS1.p1.6.m6.1.1.3.1.3.1.cmml">∈</mo><mi id="S3.SS1.p1.6.m6.1.1.3.1.3.3" xref="S3.SS1.p1.6.m6.1.1.3.1.3.3.cmml">ℕ</mi></mrow></msub><msup id="S3.SS1.p1.6.m6.1.1.3.2" xref="S3.SS1.p1.6.m6.1.1.3.2.cmml"><mi id="S3.SS1.p1.6.m6.1.1.3.2.2" xref="S3.SS1.p1.6.m6.1.1.3.2.2.cmml">X</mi><mi id="S3.SS1.p1.6.m6.1.1.3.2.3" xref="S3.SS1.p1.6.m6.1.1.3.2.3.cmml">n</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.1b"><apply id="S3.SS1.p1.6.m6.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1"><eq id="S3.SS1.p1.6.m6.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1.1"></eq><apply id="S3.SS1.p1.6.m6.1.1.2.cmml" xref="S3.SS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.2.1.cmml" xref="S3.SS1.p1.6.m6.1.1.2">superscript</csymbol><ci id="S3.SS1.p1.6.m6.1.1.2.2.cmml" xref="S3.SS1.p1.6.m6.1.1.2.2">𝑋</ci><times id="S3.SS1.p1.6.m6.1.1.2.3.cmml" xref="S3.SS1.p1.6.m6.1.1.2.3"></times></apply><apply id="S3.SS1.p1.6.m6.1.1.3.cmml" xref="S3.SS1.p1.6.m6.1.1.3"><apply id="S3.SS1.p1.6.m6.1.1.3.1.cmml" xref="S3.SS1.p1.6.m6.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.3.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1.3.1">subscript</csymbol><union id="S3.SS1.p1.6.m6.1.1.3.1.2.cmml" xref="S3.SS1.p1.6.m6.1.1.3.1.2"></union><apply id="S3.SS1.p1.6.m6.1.1.3.1.3.cmml" xref="S3.SS1.p1.6.m6.1.1.3.1.3"><in id="S3.SS1.p1.6.m6.1.1.3.1.3.1.cmml" xref="S3.SS1.p1.6.m6.1.1.3.1.3.1"></in><ci id="S3.SS1.p1.6.m6.1.1.3.1.3.2.cmml" xref="S3.SS1.p1.6.m6.1.1.3.1.3.2">𝑛</ci><ci id="S3.SS1.p1.6.m6.1.1.3.1.3.3.cmml" xref="S3.SS1.p1.6.m6.1.1.3.1.3.3">ℕ</ci></apply></apply><apply id="S3.SS1.p1.6.m6.1.1.3.2.cmml" xref="S3.SS1.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.3.2.1.cmml" xref="S3.SS1.p1.6.m6.1.1.3.2">superscript</csymbol><ci id="S3.SS1.p1.6.m6.1.1.3.2.2.cmml" xref="S3.SS1.p1.6.m6.1.1.3.2.2">𝑋</ci><ci id="S3.SS1.p1.6.m6.1.1.3.2.3.cmml" xref="S3.SS1.p1.6.m6.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">X^{*}=\bigcup_{n\in\mathbb{N}}X^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.1d">italic_X start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ⋃ start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> as usual. For <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m7.1"><semantics id="S3.SS1.p1.7.m7.1a"><mi id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.1b"><ci id="S3.SS1.p1.7.m7.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m7.1d">italic_X</annotation></semantics></math> which is either a set or a tuple <math alttext="(x_{i})_{i\in I}" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m8.1"><semantics id="S3.SS1.p1.8.m8.1a"><msub id="S3.SS1.p1.8.m8.1.1" xref="S3.SS1.p1.8.m8.1.1.cmml"><mrow id="S3.SS1.p1.8.m8.1.1.1.1" xref="S3.SS1.p1.8.m8.1.1.1.1.1.cmml"><mo id="S3.SS1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S3.SS1.p1.8.m8.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.p1.8.m8.1.1.1.1.1" xref="S3.SS1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S3.SS1.p1.8.m8.1.1.1.1.1.2" xref="S3.SS1.p1.8.m8.1.1.1.1.1.2.cmml">x</mi><mi id="S3.SS1.p1.8.m8.1.1.1.1.1.3" xref="S3.SS1.p1.8.m8.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S3.SS1.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.p1.8.m8.1.1.3" xref="S3.SS1.p1.8.m8.1.1.3.cmml"><mi id="S3.SS1.p1.8.m8.1.1.3.2" xref="S3.SS1.p1.8.m8.1.1.3.2.cmml">i</mi><mo id="S3.SS1.p1.8.m8.1.1.3.1" xref="S3.SS1.p1.8.m8.1.1.3.1.cmml">∈</mo><mi id="S3.SS1.p1.8.m8.1.1.3.3" xref="S3.SS1.p1.8.m8.1.1.3.3.cmml">I</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m8.1b"><apply id="S3.SS1.p1.8.m8.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.8.m8.1.1.2.cmml" xref="S3.SS1.p1.8.m8.1.1">subscript</csymbol><apply id="S3.SS1.p1.8.m8.1.1.1.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.8.m8.1.1.1.1.1.2">𝑥</ci><ci id="S3.SS1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S3.SS1.p1.8.m8.1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.SS1.p1.8.m8.1.1.3.cmml" xref="S3.SS1.p1.8.m8.1.1.3"><in id="S3.SS1.p1.8.m8.1.1.3.1.cmml" xref="S3.SS1.p1.8.m8.1.1.3.1"></in><ci id="S3.SS1.p1.8.m8.1.1.3.2.cmml" xref="S3.SS1.p1.8.m8.1.1.3.2">𝑖</ci><ci id="S3.SS1.p1.8.m8.1.1.3.3.cmml" xref="S3.SS1.p1.8.m8.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m8.1c">(x_{i})_{i\in I}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m8.1d">( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="\bigcup X=\bigcup_{x\in X}x" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m9.1"><semantics id="S3.SS1.p1.9.m9.1a"><mrow id="S3.SS1.p1.9.m9.1.1" xref="S3.SS1.p1.9.m9.1.1.cmml"><mrow id="S3.SS1.p1.9.m9.1.1.2" xref="S3.SS1.p1.9.m9.1.1.2.cmml"><mo id="S3.SS1.p1.9.m9.1.1.2.1" xref="S3.SS1.p1.9.m9.1.1.2.1.cmml">⋃</mo><mi id="S3.SS1.p1.9.m9.1.1.2.2" xref="S3.SS1.p1.9.m9.1.1.2.2.cmml">X</mi></mrow><mo id="S3.SS1.p1.9.m9.1.1.1" rspace="0.111em" xref="S3.SS1.p1.9.m9.1.1.1.cmml">=</mo><mrow id="S3.SS1.p1.9.m9.1.1.3" xref="S3.SS1.p1.9.m9.1.1.3.cmml"><msub id="S3.SS1.p1.9.m9.1.1.3.1" xref="S3.SS1.p1.9.m9.1.1.3.1.cmml"><mo id="S3.SS1.p1.9.m9.1.1.3.1.2" xref="S3.SS1.p1.9.m9.1.1.3.1.2.cmml">⋃</mo><mrow id="S3.SS1.p1.9.m9.1.1.3.1.3" xref="S3.SS1.p1.9.m9.1.1.3.1.3.cmml"><mi id="S3.SS1.p1.9.m9.1.1.3.1.3.2" xref="S3.SS1.p1.9.m9.1.1.3.1.3.2.cmml">x</mi><mo id="S3.SS1.p1.9.m9.1.1.3.1.3.1" xref="S3.SS1.p1.9.m9.1.1.3.1.3.1.cmml">∈</mo><mi id="S3.SS1.p1.9.m9.1.1.3.1.3.3" xref="S3.SS1.p1.9.m9.1.1.3.1.3.3.cmml">X</mi></mrow></msub><mi id="S3.SS1.p1.9.m9.1.1.3.2" xref="S3.SS1.p1.9.m9.1.1.3.2.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m9.1b"><apply id="S3.SS1.p1.9.m9.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1"><eq id="S3.SS1.p1.9.m9.1.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1.1"></eq><apply id="S3.SS1.p1.9.m9.1.1.2.cmml" xref="S3.SS1.p1.9.m9.1.1.2"><union id="S3.SS1.p1.9.m9.1.1.2.1.cmml" xref="S3.SS1.p1.9.m9.1.1.2.1"></union><ci id="S3.SS1.p1.9.m9.1.1.2.2.cmml" xref="S3.SS1.p1.9.m9.1.1.2.2">𝑋</ci></apply><apply id="S3.SS1.p1.9.m9.1.1.3.cmml" xref="S3.SS1.p1.9.m9.1.1.3"><apply id="S3.SS1.p1.9.m9.1.1.3.1.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m9.1.1.3.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1">subscript</csymbol><union id="S3.SS1.p1.9.m9.1.1.3.1.2.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1.2"></union><apply id="S3.SS1.p1.9.m9.1.1.3.1.3.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1.3"><in id="S3.SS1.p1.9.m9.1.1.3.1.3.1.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1.3.1"></in><ci id="S3.SS1.p1.9.m9.1.1.3.1.3.2.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1.3.2">𝑥</ci><ci id="S3.SS1.p1.9.m9.1.1.3.1.3.3.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1.3.3">𝑋</ci></apply></apply><ci id="S3.SS1.p1.9.m9.1.1.3.2.cmml" xref="S3.SS1.p1.9.m9.1.1.3.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m9.1c">\bigcup X=\bigcup_{x\in X}x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.9.m9.1d">⋃ italic_X = ⋃ start_POSTSUBSCRIPT italic_x ∈ italic_X end_POSTSUBSCRIPT italic_x</annotation></semantics></math> in the former case, and <math alttext="\bigcup X=\bigcup_{i\in I}x_{i}" class="ltx_Math" display="inline" id="S3.SS1.p1.10.m10.1"><semantics id="S3.SS1.p1.10.m10.1a"><mrow id="S3.SS1.p1.10.m10.1.1" xref="S3.SS1.p1.10.m10.1.1.cmml"><mrow id="S3.SS1.p1.10.m10.1.1.2" xref="S3.SS1.p1.10.m10.1.1.2.cmml"><mo id="S3.SS1.p1.10.m10.1.1.2.1" xref="S3.SS1.p1.10.m10.1.1.2.1.cmml">⋃</mo><mi id="S3.SS1.p1.10.m10.1.1.2.2" xref="S3.SS1.p1.10.m10.1.1.2.2.cmml">X</mi></mrow><mo id="S3.SS1.p1.10.m10.1.1.1" rspace="0.111em" xref="S3.SS1.p1.10.m10.1.1.1.cmml">=</mo><mrow id="S3.SS1.p1.10.m10.1.1.3" xref="S3.SS1.p1.10.m10.1.1.3.cmml"><msub id="S3.SS1.p1.10.m10.1.1.3.1" xref="S3.SS1.p1.10.m10.1.1.3.1.cmml"><mo id="S3.SS1.p1.10.m10.1.1.3.1.2" xref="S3.SS1.p1.10.m10.1.1.3.1.2.cmml">⋃</mo><mrow id="S3.SS1.p1.10.m10.1.1.3.1.3" xref="S3.SS1.p1.10.m10.1.1.3.1.3.cmml"><mi id="S3.SS1.p1.10.m10.1.1.3.1.3.2" xref="S3.SS1.p1.10.m10.1.1.3.1.3.2.cmml">i</mi><mo id="S3.SS1.p1.10.m10.1.1.3.1.3.1" xref="S3.SS1.p1.10.m10.1.1.3.1.3.1.cmml">∈</mo><mi id="S3.SS1.p1.10.m10.1.1.3.1.3.3" xref="S3.SS1.p1.10.m10.1.1.3.1.3.3.cmml">I</mi></mrow></msub><msub id="S3.SS1.p1.10.m10.1.1.3.2" xref="S3.SS1.p1.10.m10.1.1.3.2.cmml"><mi id="S3.SS1.p1.10.m10.1.1.3.2.2" xref="S3.SS1.p1.10.m10.1.1.3.2.2.cmml">x</mi><mi id="S3.SS1.p1.10.m10.1.1.3.2.3" xref="S3.SS1.p1.10.m10.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m10.1b"><apply id="S3.SS1.p1.10.m10.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1"><eq id="S3.SS1.p1.10.m10.1.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1.1"></eq><apply id="S3.SS1.p1.10.m10.1.1.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2"><union id="S3.SS1.p1.10.m10.1.1.2.1.cmml" xref="S3.SS1.p1.10.m10.1.1.2.1"></union><ci id="S3.SS1.p1.10.m10.1.1.2.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2.2">𝑋</ci></apply><apply id="S3.SS1.p1.10.m10.1.1.3.cmml" xref="S3.SS1.p1.10.m10.1.1.3"><apply id="S3.SS1.p1.10.m10.1.1.3.1.cmml" xref="S3.SS1.p1.10.m10.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.1.1.3.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1.3.1">subscript</csymbol><union id="S3.SS1.p1.10.m10.1.1.3.1.2.cmml" xref="S3.SS1.p1.10.m10.1.1.3.1.2"></union><apply id="S3.SS1.p1.10.m10.1.1.3.1.3.cmml" xref="S3.SS1.p1.10.m10.1.1.3.1.3"><in id="S3.SS1.p1.10.m10.1.1.3.1.3.1.cmml" xref="S3.SS1.p1.10.m10.1.1.3.1.3.1"></in><ci id="S3.SS1.p1.10.m10.1.1.3.1.3.2.cmml" xref="S3.SS1.p1.10.m10.1.1.3.1.3.2">𝑖</ci><ci id="S3.SS1.p1.10.m10.1.1.3.1.3.3.cmml" xref="S3.SS1.p1.10.m10.1.1.3.1.3.3">𝐼</ci></apply></apply><apply id="S3.SS1.p1.10.m10.1.1.3.2.cmml" xref="S3.SS1.p1.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.1.1.3.2.1.cmml" xref="S3.SS1.p1.10.m10.1.1.3.2">subscript</csymbol><ci id="S3.SS1.p1.10.m10.1.1.3.2.2.cmml" xref="S3.SS1.p1.10.m10.1.1.3.2.2">𝑥</ci><ci id="S3.SS1.p1.10.m10.1.1.3.2.3.cmml" xref="S3.SS1.p1.10.m10.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m10.1c">\bigcup X=\bigcup_{i\in I}x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m10.1d">⋃ italic_X = ⋃ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in the latter. For a set <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.p1.11.m11.1"><semantics id="S3.SS1.p1.11.m11.1a"><mi id="S3.SS1.p1.11.m11.1.1" xref="S3.SS1.p1.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.11.m11.1b"><ci id="S3.SS1.p1.11.m11.1.1.cmml" xref="S3.SS1.p1.11.m11.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.11.m11.1d">italic_X</annotation></semantics></math>, let <math alttext="\Delta X" class="ltx_Math" display="inline" id="S3.SS1.p1.12.m12.1"><semantics id="S3.SS1.p1.12.m12.1a"><mrow id="S3.SS1.p1.12.m12.1.1" xref="S3.SS1.p1.12.m12.1.1.cmml"><mi id="S3.SS1.p1.12.m12.1.1.2" mathvariant="normal" xref="S3.SS1.p1.12.m12.1.1.2.cmml">Δ</mi><mo id="S3.SS1.p1.12.m12.1.1.1" xref="S3.SS1.p1.12.m12.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p1.12.m12.1.1.3" xref="S3.SS1.p1.12.m12.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.12.m12.1b"><apply id="S3.SS1.p1.12.m12.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1"><times id="S3.SS1.p1.12.m12.1.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1.1"></times><ci id="S3.SS1.p1.12.m12.1.1.2.cmml" xref="S3.SS1.p1.12.m12.1.1.2">Δ</ci><ci id="S3.SS1.p1.12.m12.1.1.3.cmml" xref="S3.SS1.p1.12.m12.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.12.m12.1c">\Delta X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.12.m12.1d">roman_Δ italic_X</annotation></semantics></math> be the set of distributions over <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.p1.13.m13.1"><semantics id="S3.SS1.p1.13.m13.1a"><mi id="S3.SS1.p1.13.m13.1.1" xref="S3.SS1.p1.13.m13.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.13.m13.1b"><ci id="S3.SS1.p1.13.m13.1.1.cmml" xref="S3.SS1.p1.13.m13.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.13.m13.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.13.m13.1d">italic_X</annotation></semantics></math>. For a distribution <math alttext="\sigma\in\Delta X" class="ltx_Math" display="inline" id="S3.SS1.p1.14.m14.1"><semantics id="S3.SS1.p1.14.m14.1a"><mrow id="S3.SS1.p1.14.m14.1.1" xref="S3.SS1.p1.14.m14.1.1.cmml"><mi id="S3.SS1.p1.14.m14.1.1.2" xref="S3.SS1.p1.14.m14.1.1.2.cmml">σ</mi><mo id="S3.SS1.p1.14.m14.1.1.1" xref="S3.SS1.p1.14.m14.1.1.1.cmml">∈</mo><mrow id="S3.SS1.p1.14.m14.1.1.3" xref="S3.SS1.p1.14.m14.1.1.3.cmml"><mi id="S3.SS1.p1.14.m14.1.1.3.2" mathvariant="normal" xref="S3.SS1.p1.14.m14.1.1.3.2.cmml">Δ</mi><mo id="S3.SS1.p1.14.m14.1.1.3.1" xref="S3.SS1.p1.14.m14.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.p1.14.m14.1.1.3.3" xref="S3.SS1.p1.14.m14.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.14.m14.1b"><apply id="S3.SS1.p1.14.m14.1.1.cmml" xref="S3.SS1.p1.14.m14.1.1"><in id="S3.SS1.p1.14.m14.1.1.1.cmml" xref="S3.SS1.p1.14.m14.1.1.1"></in><ci id="S3.SS1.p1.14.m14.1.1.2.cmml" xref="S3.SS1.p1.14.m14.1.1.2">𝜎</ci><apply id="S3.SS1.p1.14.m14.1.1.3.cmml" xref="S3.SS1.p1.14.m14.1.1.3"><times id="S3.SS1.p1.14.m14.1.1.3.1.cmml" xref="S3.SS1.p1.14.m14.1.1.3.1"></times><ci id="S3.SS1.p1.14.m14.1.1.3.2.cmml" xref="S3.SS1.p1.14.m14.1.1.3.2">Δ</ci><ci id="S3.SS1.p1.14.m14.1.1.3.3.cmml" xref="S3.SS1.p1.14.m14.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.14.m14.1c">\sigma\in\Delta X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.14.m14.1d">italic_σ ∈ roman_Δ italic_X</annotation></semantics></math>, let <math alttext="\operatorname{supp}\sigma" class="ltx_Math" display="inline" id="S3.SS1.p1.15.m15.1"><semantics id="S3.SS1.p1.15.m15.1a"><mrow id="S3.SS1.p1.15.m15.1.1" xref="S3.SS1.p1.15.m15.1.1.cmml"><mi id="S3.SS1.p1.15.m15.1.1.1" xref="S3.SS1.p1.15.m15.1.1.1.cmml">supp</mi><mo id="S3.SS1.p1.15.m15.1.1a" lspace="0.167em" xref="S3.SS1.p1.15.m15.1.1.cmml">⁡</mo><mi id="S3.SS1.p1.15.m15.1.1.2" xref="S3.SS1.p1.15.m15.1.1.2.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.15.m15.1b"><apply id="S3.SS1.p1.15.m15.1.1.cmml" xref="S3.SS1.p1.15.m15.1.1"><ci id="S3.SS1.p1.15.m15.1.1.1.cmml" xref="S3.SS1.p1.15.m15.1.1.1">supp</ci><ci id="S3.SS1.p1.15.m15.1.1.2.cmml" xref="S3.SS1.p1.15.m15.1.1.2">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.15.m15.1c">\operatorname{supp}\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.15.m15.1d">roman_supp italic_σ</annotation></semantics></math> be the support of <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS1.p1.16.m16.1"><semantics id="S3.SS1.p1.16.m16.1a"><mi id="S3.SS1.p1.16.m16.1.1" xref="S3.SS1.p1.16.m16.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.16.m16.1b"><ci id="S3.SS1.p1.16.m16.1.1.cmml" xref="S3.SS1.p1.16.m16.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.16.m16.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.16.m16.1d">italic_σ</annotation></semantics></math>. For <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="S3.SS1.p1.17.m17.1"><semantics id="S3.SS1.p1.17.m17.1a"><mrow id="S3.SS1.p1.17.m17.1.1" xref="S3.SS1.p1.17.m17.1.1.cmml"><mi id="S3.SS1.p1.17.m17.1.1.2" xref="S3.SS1.p1.17.m17.1.1.2.cmml">n</mi><mo id="S3.SS1.p1.17.m17.1.1.1" xref="S3.SS1.p1.17.m17.1.1.1.cmml">∈</mo><mi id="S3.SS1.p1.17.m17.1.1.3" xref="S3.SS1.p1.17.m17.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.17.m17.1b"><apply id="S3.SS1.p1.17.m17.1.1.cmml" xref="S3.SS1.p1.17.m17.1.1"><in id="S3.SS1.p1.17.m17.1.1.1.cmml" xref="S3.SS1.p1.17.m17.1.1.1"></in><ci id="S3.SS1.p1.17.m17.1.1.2.cmml" xref="S3.SS1.p1.17.m17.1.1.2">𝑛</ci><ci id="S3.SS1.p1.17.m17.1.1.3.cmml" xref="S3.SS1.p1.17.m17.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.17.m17.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.17.m17.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, let <math alttext="\sigma^{n}" class="ltx_Math" display="inline" id="S3.SS1.p1.18.m18.1"><semantics id="S3.SS1.p1.18.m18.1a"><msup id="S3.SS1.p1.18.m18.1.1" xref="S3.SS1.p1.18.m18.1.1.cmml"><mi id="S3.SS1.p1.18.m18.1.1.2" xref="S3.SS1.p1.18.m18.1.1.2.cmml">σ</mi><mi id="S3.SS1.p1.18.m18.1.1.3" xref="S3.SS1.p1.18.m18.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.18.m18.1b"><apply id="S3.SS1.p1.18.m18.1.1.cmml" xref="S3.SS1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.18.m18.1.1.1.cmml" xref="S3.SS1.p1.18.m18.1.1">superscript</csymbol><ci id="S3.SS1.p1.18.m18.1.1.2.cmml" xref="S3.SS1.p1.18.m18.1.1.2">𝜎</ci><ci id="S3.SS1.p1.18.m18.1.1.3.cmml" xref="S3.SS1.p1.18.m18.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.18.m18.1c">\sigma^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.18.m18.1d">italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> be the product distribution of <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.p1.19.m19.1"><semantics id="S3.SS1.p1.19.m19.1a"><mi id="S3.SS1.p1.19.m19.1.1" xref="S3.SS1.p1.19.m19.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.19.m19.1b"><ci id="S3.SS1.p1.19.m19.1.1.cmml" xref="S3.SS1.p1.19.m19.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.19.m19.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.19.m19.1d">italic_n</annotation></semantics></math> copies of <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS1.p1.20.m20.1"><semantics id="S3.SS1.p1.20.m20.1a"><mi id="S3.SS1.p1.20.m20.1.1" xref="S3.SS1.p1.20.m20.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.20.m20.1b"><ci id="S3.SS1.p1.20.m20.1.1.cmml" xref="S3.SS1.p1.20.m20.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.20.m20.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.20.m20.1d">italic_σ</annotation></semantics></math>. If <math alttext="\sigma\in(\Delta X)^{n}" class="ltx_Math" display="inline" id="S3.SS1.p1.21.m21.1"><semantics id="S3.SS1.p1.21.m21.1a"><mrow id="S3.SS1.p1.21.m21.1.1" xref="S3.SS1.p1.21.m21.1.1.cmml"><mi id="S3.SS1.p1.21.m21.1.1.3" xref="S3.SS1.p1.21.m21.1.1.3.cmml">σ</mi><mo id="S3.SS1.p1.21.m21.1.1.2" xref="S3.SS1.p1.21.m21.1.1.2.cmml">∈</mo><msup id="S3.SS1.p1.21.m21.1.1.1" xref="S3.SS1.p1.21.m21.1.1.1.cmml"><mrow id="S3.SS1.p1.21.m21.1.1.1.1.1" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.cmml"><mo id="S3.SS1.p1.21.m21.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.p1.21.m21.1.1.1.1.1.1" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p1.21.m21.1.1.1.1.1.1.2" mathvariant="normal" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.2.cmml">Δ</mi><mo id="S3.SS1.p1.21.m21.1.1.1.1.1.1.1" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p1.21.m21.1.1.1.1.1.1.3" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.3.cmml">X</mi></mrow><mo id="S3.SS1.p1.21.m21.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.SS1.p1.21.m21.1.1.1.3" xref="S3.SS1.p1.21.m21.1.1.1.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.21.m21.1b"><apply id="S3.SS1.p1.21.m21.1.1.cmml" xref="S3.SS1.p1.21.m21.1.1"><in id="S3.SS1.p1.21.m21.1.1.2.cmml" xref="S3.SS1.p1.21.m21.1.1.2"></in><ci id="S3.SS1.p1.21.m21.1.1.3.cmml" xref="S3.SS1.p1.21.m21.1.1.3">𝜎</ci><apply id="S3.SS1.p1.21.m21.1.1.1.cmml" xref="S3.SS1.p1.21.m21.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.21.m21.1.1.1.2.cmml" xref="S3.SS1.p1.21.m21.1.1.1">superscript</csymbol><apply id="S3.SS1.p1.21.m21.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.21.m21.1.1.1.1.1"><times id="S3.SS1.p1.21.m21.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.1"></times><ci id="S3.SS1.p1.21.m21.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.2">Δ</ci><ci id="S3.SS1.p1.21.m21.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p1.21.m21.1.1.1.1.1.1.3">𝑋</ci></apply><ci id="S3.SS1.p1.21.m21.1.1.1.3.cmml" xref="S3.SS1.p1.21.m21.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.21.m21.1c">\sigma\in(\Delta X)^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.21.m21.1d">italic_σ ∈ ( roman_Δ italic_X ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> we also write <math alttext="\sigma=(\sigma_{1},...,\sigma_{n})" class="ltx_Math" display="inline" id="S3.SS1.p1.22.m22.3"><semantics id="S3.SS1.p1.22.m22.3a"><mrow id="S3.SS1.p1.22.m22.3.3" xref="S3.SS1.p1.22.m22.3.3.cmml"><mi id="S3.SS1.p1.22.m22.3.3.4" xref="S3.SS1.p1.22.m22.3.3.4.cmml">σ</mi><mo id="S3.SS1.p1.22.m22.3.3.3" xref="S3.SS1.p1.22.m22.3.3.3.cmml">=</mo><mrow id="S3.SS1.p1.22.m22.3.3.2.2" xref="S3.SS1.p1.22.m22.3.3.2.3.cmml"><mo id="S3.SS1.p1.22.m22.3.3.2.2.3" stretchy="false" xref="S3.SS1.p1.22.m22.3.3.2.3.cmml">(</mo><msub id="S3.SS1.p1.22.m22.2.2.1.1.1" xref="S3.SS1.p1.22.m22.2.2.1.1.1.cmml"><mi id="S3.SS1.p1.22.m22.2.2.1.1.1.2" xref="S3.SS1.p1.22.m22.2.2.1.1.1.2.cmml">σ</mi><mn id="S3.SS1.p1.22.m22.2.2.1.1.1.3" xref="S3.SS1.p1.22.m22.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.p1.22.m22.3.3.2.2.4" xref="S3.SS1.p1.22.m22.3.3.2.3.cmml">,</mo><mi id="S3.SS1.p1.22.m22.1.1" mathvariant="normal" xref="S3.SS1.p1.22.m22.1.1.cmml">…</mi><mo id="S3.SS1.p1.22.m22.3.3.2.2.5" xref="S3.SS1.p1.22.m22.3.3.2.3.cmml">,</mo><msub id="S3.SS1.p1.22.m22.3.3.2.2.2" xref="S3.SS1.p1.22.m22.3.3.2.2.2.cmml"><mi id="S3.SS1.p1.22.m22.3.3.2.2.2.2" xref="S3.SS1.p1.22.m22.3.3.2.2.2.2.cmml">σ</mi><mi id="S3.SS1.p1.22.m22.3.3.2.2.2.3" xref="S3.SS1.p1.22.m22.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S3.SS1.p1.22.m22.3.3.2.2.6" stretchy="false" xref="S3.SS1.p1.22.m22.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.22.m22.3b"><apply id="S3.SS1.p1.22.m22.3.3.cmml" xref="S3.SS1.p1.22.m22.3.3"><eq id="S3.SS1.p1.22.m22.3.3.3.cmml" xref="S3.SS1.p1.22.m22.3.3.3"></eq><ci id="S3.SS1.p1.22.m22.3.3.4.cmml" xref="S3.SS1.p1.22.m22.3.3.4">𝜎</ci><vector id="S3.SS1.p1.22.m22.3.3.2.3.cmml" xref="S3.SS1.p1.22.m22.3.3.2.2"><apply id="S3.SS1.p1.22.m22.2.2.1.1.1.cmml" xref="S3.SS1.p1.22.m22.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.22.m22.2.2.1.1.1.1.cmml" xref="S3.SS1.p1.22.m22.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.22.m22.2.2.1.1.1.2.cmml" xref="S3.SS1.p1.22.m22.2.2.1.1.1.2">𝜎</ci><cn id="S3.SS1.p1.22.m22.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.22.m22.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS1.p1.22.m22.1.1.cmml" xref="S3.SS1.p1.22.m22.1.1">…</ci><apply id="S3.SS1.p1.22.m22.3.3.2.2.2.cmml" xref="S3.SS1.p1.22.m22.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.22.m22.3.3.2.2.2.1.cmml" xref="S3.SS1.p1.22.m22.3.3.2.2.2">subscript</csymbol><ci id="S3.SS1.p1.22.m22.3.3.2.2.2.2.cmml" xref="S3.SS1.p1.22.m22.3.3.2.2.2.2">𝜎</ci><ci id="S3.SS1.p1.22.m22.3.3.2.2.2.3.cmml" xref="S3.SS1.p1.22.m22.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.22.m22.3c">\sigma=(\sigma_{1},...,\sigma_{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.22.m22.3d">italic_σ = ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>. For an set or event <math alttext="A" class="ltx_Math" display="inline" id="S3.SS1.p1.23.m23.1"><semantics id="S3.SS1.p1.23.m23.1a"><mi id="S3.SS1.p1.23.m23.1.1" xref="S3.SS1.p1.23.m23.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.23.m23.1b"><ci id="S3.SS1.p1.23.m23.1.1.cmml" xref="S3.SS1.p1.23.m23.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.23.m23.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.23.m23.1d">italic_A</annotation></semantics></math> we denote the indicator of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS1.p1.24.m24.1"><semantics id="S3.SS1.p1.24.m24.1a"><mi id="S3.SS1.p1.24.m24.1.1" xref="S3.SS1.p1.24.m24.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.24.m24.1b"><ci id="S3.SS1.p1.24.m24.1.1.cmml" xref="S3.SS1.p1.24.m24.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.24.m24.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.24.m24.1d">italic_A</annotation></semantics></math> by <math alttext="1_{A}" class="ltx_Math" display="inline" id="S3.SS1.p1.25.m25.1"><semantics id="S3.SS1.p1.25.m25.1a"><msub id="S3.SS1.p1.25.m25.1.1" xref="S3.SS1.p1.25.m25.1.1.cmml"><mn id="S3.SS1.p1.25.m25.1.1.2" xref="S3.SS1.p1.25.m25.1.1.2.cmml">1</mn><mi id="S3.SS1.p1.25.m25.1.1.3" xref="S3.SS1.p1.25.m25.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.25.m25.1b"><apply id="S3.SS1.p1.25.m25.1.1.cmml" xref="S3.SS1.p1.25.m25.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.25.m25.1.1.1.cmml" xref="S3.SS1.p1.25.m25.1.1">subscript</csymbol><cn id="S3.SS1.p1.25.m25.1.1.2.cmml" type="integer" xref="S3.SS1.p1.25.m25.1.1.2">1</cn><ci id="S3.SS1.p1.25.m25.1.1.3.cmml" xref="S3.SS1.p1.25.m25.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.25.m25.1c">1_{A}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.25.m25.1d">1 start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="1\left\{x\mid x\in A\right\}" class="ltx_Math" display="inline" id="S3.SS1.p1.26.m26.2"><semantics id="S3.SS1.p1.26.m26.2a"><mrow id="S3.SS1.p1.26.m26.2.2" xref="S3.SS1.p1.26.m26.2.2.cmml"><mn id="S3.SS1.p1.26.m26.2.2.3" xref="S3.SS1.p1.26.m26.2.2.3.cmml">1</mn><mo id="S3.SS1.p1.26.m26.2.2.2" xref="S3.SS1.p1.26.m26.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.p1.26.m26.2.2.1.1" xref="S3.SS1.p1.26.m26.2.2.1.2.cmml"><mo id="S3.SS1.p1.26.m26.2.2.1.1.2" xref="S3.SS1.p1.26.m26.2.2.1.2.1.cmml">{</mo><mi id="S3.SS1.p1.26.m26.1.1" xref="S3.SS1.p1.26.m26.1.1.cmml">x</mi><mo fence="true" id="S3.SS1.p1.26.m26.2.2.1.1.3" lspace="0em" rspace="0em" xref="S3.SS1.p1.26.m26.2.2.1.2.1.cmml">∣</mo><mrow id="S3.SS1.p1.26.m26.2.2.1.1.1" xref="S3.SS1.p1.26.m26.2.2.1.1.1.cmml"><mi id="S3.SS1.p1.26.m26.2.2.1.1.1.2" xref="S3.SS1.p1.26.m26.2.2.1.1.1.2.cmml">x</mi><mo id="S3.SS1.p1.26.m26.2.2.1.1.1.1" xref="S3.SS1.p1.26.m26.2.2.1.1.1.1.cmml">∈</mo><mi id="S3.SS1.p1.26.m26.2.2.1.1.1.3" xref="S3.SS1.p1.26.m26.2.2.1.1.1.3.cmml">A</mi></mrow><mo id="S3.SS1.p1.26.m26.2.2.1.1.4" xref="S3.SS1.p1.26.m26.2.2.1.2.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.26.m26.2b"><apply id="S3.SS1.p1.26.m26.2.2.cmml" xref="S3.SS1.p1.26.m26.2.2"><times id="S3.SS1.p1.26.m26.2.2.2.cmml" xref="S3.SS1.p1.26.m26.2.2.2"></times><cn id="S3.SS1.p1.26.m26.2.2.3.cmml" type="integer" xref="S3.SS1.p1.26.m26.2.2.3">1</cn><apply id="S3.SS1.p1.26.m26.2.2.1.2.cmml" xref="S3.SS1.p1.26.m26.2.2.1.1"><csymbol cd="latexml" id="S3.SS1.p1.26.m26.2.2.1.2.1.cmml" xref="S3.SS1.p1.26.m26.2.2.1.1.2">conditional-set</csymbol><ci id="S3.SS1.p1.26.m26.1.1.cmml" xref="S3.SS1.p1.26.m26.1.1">𝑥</ci><apply id="S3.SS1.p1.26.m26.2.2.1.1.1.cmml" xref="S3.SS1.p1.26.m26.2.2.1.1.1"><in id="S3.SS1.p1.26.m26.2.2.1.1.1.1.cmml" xref="S3.SS1.p1.26.m26.2.2.1.1.1.1"></in><ci id="S3.SS1.p1.26.m26.2.2.1.1.1.2.cmml" xref="S3.SS1.p1.26.m26.2.2.1.1.1.2">𝑥</ci><ci id="S3.SS1.p1.26.m26.2.2.1.1.1.3.cmml" xref="S3.SS1.p1.26.m26.2.2.1.1.1.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.26.m26.2c">1\left\{x\mid x\in A\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.26.m26.2d">1 { italic_x ∣ italic_x ∈ italic_A }</annotation></semantics></math>. For conciseness, by <math alttext="\stackrel{{\scriptstyle\text{def}}}{{=}}" class="ltx_Math" display="inline" id="S3.SS1.p1.27.m27.1"><semantics id="S3.SS1.p1.27.m27.1a"><mover id="S3.SS1.p1.27.m27.1.1" xref="S3.SS1.p1.27.m27.1.1.cmml"><mo id="S3.SS1.p1.27.m27.1.1.2" xref="S3.SS1.p1.27.m27.1.1.2.cmml">=</mo><mtext id="S3.SS1.p1.27.m27.1.1.3" xref="S3.SS1.p1.27.m27.1.1.3a.cmml">def</mtext></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.27.m27.1b"><apply id="S3.SS1.p1.27.m27.1.1.cmml" xref="S3.SS1.p1.27.m27.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.27.m27.1.1.1.cmml" xref="S3.SS1.p1.27.m27.1.1">superscript</csymbol><eq id="S3.SS1.p1.27.m27.1.1.2.cmml" xref="S3.SS1.p1.27.m27.1.1.2"></eq><ci id="S3.SS1.p1.27.m27.1.1.3a.cmml" xref="S3.SS1.p1.27.m27.1.1.3"><mtext id="S3.SS1.p1.27.m27.1.1.3.cmml" mathsize="70%" xref="S3.SS1.p1.27.m27.1.1.3">def</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.27.m27.1c">\stackrel{{\scriptstyle\text{def}}}{{=}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.27.m27.1d">start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP</annotation></semantics></math> we mean “equal by definition”.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.3">Throughout this paper, we will use the continuity of several functions in the joint distributions of uncorrelated agents. Then, we treat these distributions <math alttext="\sigma=(\sigma_{1},...,\sigma_{n})" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.3"><semantics id="S3.SS1.p2.1.m1.3a"><mrow id="S3.SS1.p2.1.m1.3.3" xref="S3.SS1.p2.1.m1.3.3.cmml"><mi id="S3.SS1.p2.1.m1.3.3.4" xref="S3.SS1.p2.1.m1.3.3.4.cmml">σ</mi><mo id="S3.SS1.p2.1.m1.3.3.3" xref="S3.SS1.p2.1.m1.3.3.3.cmml">=</mo><mrow id="S3.SS1.p2.1.m1.3.3.2.2" xref="S3.SS1.p2.1.m1.3.3.2.3.cmml"><mo id="S3.SS1.p2.1.m1.3.3.2.2.3" stretchy="false" xref="S3.SS1.p2.1.m1.3.3.2.3.cmml">(</mo><msub id="S3.SS1.p2.1.m1.2.2.1.1.1" xref="S3.SS1.p2.1.m1.2.2.1.1.1.cmml"><mi id="S3.SS1.p2.1.m1.2.2.1.1.1.2" xref="S3.SS1.p2.1.m1.2.2.1.1.1.2.cmml">σ</mi><mn id="S3.SS1.p2.1.m1.2.2.1.1.1.3" xref="S3.SS1.p2.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.p2.1.m1.3.3.2.2.4" xref="S3.SS1.p2.1.m1.3.3.2.3.cmml">,</mo><mi id="S3.SS1.p2.1.m1.1.1" mathvariant="normal" xref="S3.SS1.p2.1.m1.1.1.cmml">…</mi><mo id="S3.SS1.p2.1.m1.3.3.2.2.5" xref="S3.SS1.p2.1.m1.3.3.2.3.cmml">,</mo><msub id="S3.SS1.p2.1.m1.3.3.2.2.2" xref="S3.SS1.p2.1.m1.3.3.2.2.2.cmml"><mi id="S3.SS1.p2.1.m1.3.3.2.2.2.2" xref="S3.SS1.p2.1.m1.3.3.2.2.2.2.cmml">σ</mi><mi id="S3.SS1.p2.1.m1.3.3.2.2.2.3" xref="S3.SS1.p2.1.m1.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S3.SS1.p2.1.m1.3.3.2.2.6" stretchy="false" xref="S3.SS1.p2.1.m1.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.3b"><apply id="S3.SS1.p2.1.m1.3.3.cmml" xref="S3.SS1.p2.1.m1.3.3"><eq id="S3.SS1.p2.1.m1.3.3.3.cmml" xref="S3.SS1.p2.1.m1.3.3.3"></eq><ci id="S3.SS1.p2.1.m1.3.3.4.cmml" xref="S3.SS1.p2.1.m1.3.3.4">𝜎</ci><vector id="S3.SS1.p2.1.m1.3.3.2.3.cmml" xref="S3.SS1.p2.1.m1.3.3.2.2"><apply id="S3.SS1.p2.1.m1.2.2.1.1.1.cmml" xref="S3.SS1.p2.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.2.2.1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.p2.1.m1.2.2.1.1.1.2.cmml" xref="S3.SS1.p2.1.m1.2.2.1.1.1.2">𝜎</ci><cn id="S3.SS1.p2.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.p2.1.m1.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">…</ci><apply id="S3.SS1.p2.1.m1.3.3.2.2.2.cmml" xref="S3.SS1.p2.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.3.3.2.2.2.1.cmml" xref="S3.SS1.p2.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S3.SS1.p2.1.m1.3.3.2.2.2.2.cmml" xref="S3.SS1.p2.1.m1.3.3.2.2.2.2">𝜎</ci><ci id="S3.SS1.p2.1.m1.3.3.2.2.2.3.cmml" xref="S3.SS1.p2.1.m1.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.3c">\sigma=(\sigma_{1},...,\sigma_{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.3d">italic_σ = ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> as vectors <math alttext="\sigma\in\mathbb{R}^{mn}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml">σ</mi><mo id="S3.SS1.p2.2.m2.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.2" xref="S3.SS1.p2.2.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS1.p2.2.m2.1.1.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.3.2" xref="S3.SS1.p2.2.m2.1.1.3.3.2.cmml">m</mi><mo id="S3.SS1.p2.2.m2.1.1.3.3.1" xref="S3.SS1.p2.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.2.m2.1.1.3.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><in id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1"></in><ci id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2">𝜎</ci><apply id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3">superscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2">ℝ</ci><apply id="S3.SS1.p2.2.m2.1.1.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3"><times id="S3.SS1.p2.2.m2.1.1.3.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.1"></times><ci id="S3.SS1.p2.2.m2.1.1.3.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.2">𝑚</ci><ci id="S3.SS1.p2.2.m2.1.1.3.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">\sigma\in\mathbb{R}^{mn}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_σ ∈ blackboard_R start_POSTSUPERSCRIPT italic_m italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Now, for the continuity purposes above we use the <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><msub id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2" mathvariant="normal" xref="S3.SS1.p2.3.m3.1.1.2.cmml">ℓ</mi><mn id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">ℓ</ci><cn id="S3.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-norm to measure distances between these distributions.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>The model</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.32"><span class="ltx_text ltx_font_bold" id="S3.SS2.p1.32.1">The graph </span> In the game there is a directed <math alttext="s-t" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mrow id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.1.1.2" xref="S3.SS2.p1.1.m1.1.1.2.cmml">s</mi><mo id="S3.SS2.p1.1.m1.1.1.1" xref="S3.SS2.p1.1.m1.1.1.1.cmml">−</mo><mi id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><minus id="S3.SS2.p1.1.m1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1.1"></minus><ci id="S3.SS2.p1.1.m1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.1.1.2">𝑠</ci><ci id="S3.SS2.p1.1.m1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_s - italic_t</annotation></semantics></math> graph <math alttext="G=(V,E,s,t)" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.4"><semantics id="S3.SS2.p1.2.m2.4a"><mrow id="S3.SS2.p1.2.m2.4.5" xref="S3.SS2.p1.2.m2.4.5.cmml"><mi id="S3.SS2.p1.2.m2.4.5.2" xref="S3.SS2.p1.2.m2.4.5.2.cmml">G</mi><mo id="S3.SS2.p1.2.m2.4.5.1" xref="S3.SS2.p1.2.m2.4.5.1.cmml">=</mo><mrow id="S3.SS2.p1.2.m2.4.5.3.2" xref="S3.SS2.p1.2.m2.4.5.3.1.cmml"><mo id="S3.SS2.p1.2.m2.4.5.3.2.1" stretchy="false" xref="S3.SS2.p1.2.m2.4.5.3.1.cmml">(</mo><mi id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml">V</mi><mo id="S3.SS2.p1.2.m2.4.5.3.2.2" xref="S3.SS2.p1.2.m2.4.5.3.1.cmml">,</mo><mi id="S3.SS2.p1.2.m2.2.2" xref="S3.SS2.p1.2.m2.2.2.cmml">E</mi><mo id="S3.SS2.p1.2.m2.4.5.3.2.3" xref="S3.SS2.p1.2.m2.4.5.3.1.cmml">,</mo><mi id="S3.SS2.p1.2.m2.3.3" xref="S3.SS2.p1.2.m2.3.3.cmml">s</mi><mo id="S3.SS2.p1.2.m2.4.5.3.2.4" xref="S3.SS2.p1.2.m2.4.5.3.1.cmml">,</mo><mi id="S3.SS2.p1.2.m2.4.4" xref="S3.SS2.p1.2.m2.4.4.cmml">t</mi><mo id="S3.SS2.p1.2.m2.4.5.3.2.5" stretchy="false" xref="S3.SS2.p1.2.m2.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.4b"><apply id="S3.SS2.p1.2.m2.4.5.cmml" xref="S3.SS2.p1.2.m2.4.5"><eq id="S3.SS2.p1.2.m2.4.5.1.cmml" xref="S3.SS2.p1.2.m2.4.5.1"></eq><ci id="S3.SS2.p1.2.m2.4.5.2.cmml" xref="S3.SS2.p1.2.m2.4.5.2">𝐺</ci><vector id="S3.SS2.p1.2.m2.4.5.3.1.cmml" xref="S3.SS2.p1.2.m2.4.5.3.2"><ci id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1">𝑉</ci><ci id="S3.SS2.p1.2.m2.2.2.cmml" xref="S3.SS2.p1.2.m2.2.2">𝐸</ci><ci id="S3.SS2.p1.2.m2.3.3.cmml" xref="S3.SS2.p1.2.m2.3.3">𝑠</ci><ci id="S3.SS2.p1.2.m2.4.4.cmml" xref="S3.SS2.p1.2.m2.4.4">𝑡</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.4c">G=(V,E,s,t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.4d">italic_G = ( italic_V , italic_E , italic_s , italic_t )</annotation></semantics></math>, where <math alttext="V" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mi id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><ci id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_V</annotation></semantics></math> is the set of vertices, <math alttext="s,t\in V" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.2"><semantics id="S3.SS2.p1.4.m4.2a"><mrow id="S3.SS2.p1.4.m4.2.3" xref="S3.SS2.p1.4.m4.2.3.cmml"><mrow id="S3.SS2.p1.4.m4.2.3.2.2" xref="S3.SS2.p1.4.m4.2.3.2.1.cmml"><mi id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml">s</mi><mo id="S3.SS2.p1.4.m4.2.3.2.2.1" xref="S3.SS2.p1.4.m4.2.3.2.1.cmml">,</mo><mi id="S3.SS2.p1.4.m4.2.2" xref="S3.SS2.p1.4.m4.2.2.cmml">t</mi></mrow><mo id="S3.SS2.p1.4.m4.2.3.1" xref="S3.SS2.p1.4.m4.2.3.1.cmml">∈</mo><mi id="S3.SS2.p1.4.m4.2.3.3" xref="S3.SS2.p1.4.m4.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.2b"><apply id="S3.SS2.p1.4.m4.2.3.cmml" xref="S3.SS2.p1.4.m4.2.3"><in id="S3.SS2.p1.4.m4.2.3.1.cmml" xref="S3.SS2.p1.4.m4.2.3.1"></in><list id="S3.SS2.p1.4.m4.2.3.2.1.cmml" xref="S3.SS2.p1.4.m4.2.3.2.2"><ci id="S3.SS2.p1.4.m4.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1">𝑠</ci><ci id="S3.SS2.p1.4.m4.2.2.cmml" xref="S3.SS2.p1.4.m4.2.2">𝑡</ci></list><ci id="S3.SS2.p1.4.m4.2.3.3.cmml" xref="S3.SS2.p1.4.m4.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.2c">s,t\in V</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.2d">italic_s , italic_t ∈ italic_V</annotation></semantics></math> are the source and the sink of <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.1"><semantics id="S3.SS2.p1.5.m5.1a"><mi id="S3.SS2.p1.5.m5.1.1" xref="S3.SS2.p1.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.1b"><ci id="S3.SS2.p1.5.m5.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m5.1d">italic_G</annotation></semantics></math> and <math alttext="E" class="ltx_Math" display="inline" id="S3.SS2.p1.6.m6.1"><semantics id="S3.SS2.p1.6.m6.1a"><mi id="S3.SS2.p1.6.m6.1.1" xref="S3.SS2.p1.6.m6.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.6.m6.1b"><ci id="S3.SS2.p1.6.m6.1.1.cmml" xref="S3.SS2.p1.6.m6.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.6.m6.1c">E</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.6.m6.1d">italic_E</annotation></semantics></math> is the <span class="ltx_text ltx_font_italic" id="S3.SS2.p1.32.2">multiset</span> of edges. We assume that <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p1.7.m7.1"><semantics id="S3.SS2.p1.7.m7.1a"><mi id="S3.SS2.p1.7.m7.1.1" xref="S3.SS2.p1.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.7.m7.1b"><ci id="S3.SS2.p1.7.m7.1.1.cmml" xref="S3.SS2.p1.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.7.m7.1d">italic_G</annotation></semantics></math> is acyclic and that all edges are oriented from <math alttext="s" class="ltx_Math" display="inline" id="S3.SS2.p1.8.m8.1"><semantics id="S3.SS2.p1.8.m8.1a"><mi id="S3.SS2.p1.8.m8.1.1" xref="S3.SS2.p1.8.m8.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.8.m8.1b"><ci id="S3.SS2.p1.8.m8.1.1.cmml" xref="S3.SS2.p1.8.m8.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.8.m8.1c">s</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.8.m8.1d">italic_s</annotation></semantics></math> to <math alttext="t" class="ltx_Math" display="inline" id="S3.SS2.p1.9.m9.1"><semantics id="S3.SS2.p1.9.m9.1a"><mi id="S3.SS2.p1.9.m9.1.1" xref="S3.SS2.p1.9.m9.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.9.m9.1b"><ci id="S3.SS2.p1.9.m9.1.1.cmml" xref="S3.SS2.p1.9.m9.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.9.m9.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.9.m9.1d">italic_t</annotation></semantics></math>, i.e., every edge <math alttext="e\in E" class="ltx_Math" display="inline" id="S3.SS2.p1.10.m10.1"><semantics id="S3.SS2.p1.10.m10.1a"><mrow id="S3.SS2.p1.10.m10.1.1" xref="S3.SS2.p1.10.m10.1.1.cmml"><mi id="S3.SS2.p1.10.m10.1.1.2" xref="S3.SS2.p1.10.m10.1.1.2.cmml">e</mi><mo id="S3.SS2.p1.10.m10.1.1.1" xref="S3.SS2.p1.10.m10.1.1.1.cmml">∈</mo><mi id="S3.SS2.p1.10.m10.1.1.3" xref="S3.SS2.p1.10.m10.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.10.m10.1b"><apply id="S3.SS2.p1.10.m10.1.1.cmml" xref="S3.SS2.p1.10.m10.1.1"><in id="S3.SS2.p1.10.m10.1.1.1.cmml" xref="S3.SS2.p1.10.m10.1.1.1"></in><ci id="S3.SS2.p1.10.m10.1.1.2.cmml" xref="S3.SS2.p1.10.m10.1.1.2">𝑒</ci><ci id="S3.SS2.p1.10.m10.1.1.3.cmml" xref="S3.SS2.p1.10.m10.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.10.m10.1c">e\in E</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.10.m10.1d">italic_e ∈ italic_E</annotation></semantics></math> is contained in some directed <math alttext="s-t" class="ltx_Math" display="inline" id="S3.SS2.p1.11.m11.1"><semantics id="S3.SS2.p1.11.m11.1a"><mrow id="S3.SS2.p1.11.m11.1.1" xref="S3.SS2.p1.11.m11.1.1.cmml"><mi id="S3.SS2.p1.11.m11.1.1.2" xref="S3.SS2.p1.11.m11.1.1.2.cmml">s</mi><mo id="S3.SS2.p1.11.m11.1.1.1" xref="S3.SS2.p1.11.m11.1.1.1.cmml">−</mo><mi id="S3.SS2.p1.11.m11.1.1.3" xref="S3.SS2.p1.11.m11.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.11.m11.1b"><apply id="S3.SS2.p1.11.m11.1.1.cmml" xref="S3.SS2.p1.11.m11.1.1"><minus id="S3.SS2.p1.11.m11.1.1.1.cmml" xref="S3.SS2.p1.11.m11.1.1.1"></minus><ci id="S3.SS2.p1.11.m11.1.1.2.cmml" xref="S3.SS2.p1.11.m11.1.1.2">𝑠</ci><ci id="S3.SS2.p1.11.m11.1.1.3.cmml" xref="S3.SS2.p1.11.m11.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.11.m11.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.11.m11.1d">italic_s - italic_t</annotation></semantics></math> path. Let <math alttext="P_{st}=\left\{p_{1},...,p_{m}\right\}" class="ltx_Math" display="inline" id="S3.SS2.p1.12.m12.3"><semantics id="S3.SS2.p1.12.m12.3a"><mrow id="S3.SS2.p1.12.m12.3.3" xref="S3.SS2.p1.12.m12.3.3.cmml"><msub id="S3.SS2.p1.12.m12.3.3.4" xref="S3.SS2.p1.12.m12.3.3.4.cmml"><mi id="S3.SS2.p1.12.m12.3.3.4.2" xref="S3.SS2.p1.12.m12.3.3.4.2.cmml">P</mi><mrow id="S3.SS2.p1.12.m12.3.3.4.3" xref="S3.SS2.p1.12.m12.3.3.4.3.cmml"><mi id="S3.SS2.p1.12.m12.3.3.4.3.2" xref="S3.SS2.p1.12.m12.3.3.4.3.2.cmml">s</mi><mo id="S3.SS2.p1.12.m12.3.3.4.3.1" xref="S3.SS2.p1.12.m12.3.3.4.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.12.m12.3.3.4.3.3" xref="S3.SS2.p1.12.m12.3.3.4.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.p1.12.m12.3.3.3" xref="S3.SS2.p1.12.m12.3.3.3.cmml">=</mo><mrow id="S3.SS2.p1.12.m12.3.3.2.2" xref="S3.SS2.p1.12.m12.3.3.2.3.cmml"><mo id="S3.SS2.p1.12.m12.3.3.2.2.3" xref="S3.SS2.p1.12.m12.3.3.2.3.cmml">{</mo><msub id="S3.SS2.p1.12.m12.2.2.1.1.1" xref="S3.SS2.p1.12.m12.2.2.1.1.1.cmml"><mi id="S3.SS2.p1.12.m12.2.2.1.1.1.2" xref="S3.SS2.p1.12.m12.2.2.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p1.12.m12.2.2.1.1.1.3" xref="S3.SS2.p1.12.m12.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p1.12.m12.3.3.2.2.4" xref="S3.SS2.p1.12.m12.3.3.2.3.cmml">,</mo><mi id="S3.SS2.p1.12.m12.1.1" mathvariant="normal" xref="S3.SS2.p1.12.m12.1.1.cmml">…</mi><mo id="S3.SS2.p1.12.m12.3.3.2.2.5" xref="S3.SS2.p1.12.m12.3.3.2.3.cmml">,</mo><msub id="S3.SS2.p1.12.m12.3.3.2.2.2" xref="S3.SS2.p1.12.m12.3.3.2.2.2.cmml"><mi id="S3.SS2.p1.12.m12.3.3.2.2.2.2" xref="S3.SS2.p1.12.m12.3.3.2.2.2.2.cmml">p</mi><mi id="S3.SS2.p1.12.m12.3.3.2.2.2.3" xref="S3.SS2.p1.12.m12.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S3.SS2.p1.12.m12.3.3.2.2.6" xref="S3.SS2.p1.12.m12.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.12.m12.3b"><apply id="S3.SS2.p1.12.m12.3.3.cmml" xref="S3.SS2.p1.12.m12.3.3"><eq id="S3.SS2.p1.12.m12.3.3.3.cmml" xref="S3.SS2.p1.12.m12.3.3.3"></eq><apply id="S3.SS2.p1.12.m12.3.3.4.cmml" xref="S3.SS2.p1.12.m12.3.3.4"><csymbol cd="ambiguous" id="S3.SS2.p1.12.m12.3.3.4.1.cmml" xref="S3.SS2.p1.12.m12.3.3.4">subscript</csymbol><ci id="S3.SS2.p1.12.m12.3.3.4.2.cmml" xref="S3.SS2.p1.12.m12.3.3.4.2">𝑃</ci><apply id="S3.SS2.p1.12.m12.3.3.4.3.cmml" xref="S3.SS2.p1.12.m12.3.3.4.3"><times id="S3.SS2.p1.12.m12.3.3.4.3.1.cmml" xref="S3.SS2.p1.12.m12.3.3.4.3.1"></times><ci id="S3.SS2.p1.12.m12.3.3.4.3.2.cmml" xref="S3.SS2.p1.12.m12.3.3.4.3.2">𝑠</ci><ci id="S3.SS2.p1.12.m12.3.3.4.3.3.cmml" xref="S3.SS2.p1.12.m12.3.3.4.3.3">𝑡</ci></apply></apply><set id="S3.SS2.p1.12.m12.3.3.2.3.cmml" xref="S3.SS2.p1.12.m12.3.3.2.2"><apply id="S3.SS2.p1.12.m12.2.2.1.1.1.cmml" xref="S3.SS2.p1.12.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.12.m12.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.12.m12.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.12.m12.2.2.1.1.1.2.cmml" xref="S3.SS2.p1.12.m12.2.2.1.1.1.2">𝑝</ci><cn id="S3.SS2.p1.12.m12.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.12.m12.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS2.p1.12.m12.1.1.cmml" xref="S3.SS2.p1.12.m12.1.1">…</ci><apply id="S3.SS2.p1.12.m12.3.3.2.2.2.cmml" xref="S3.SS2.p1.12.m12.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.12.m12.3.3.2.2.2.1.cmml" xref="S3.SS2.p1.12.m12.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.12.m12.3.3.2.2.2.2.cmml" xref="S3.SS2.p1.12.m12.3.3.2.2.2.2">𝑝</ci><ci id="S3.SS2.p1.12.m12.3.3.2.2.2.3.cmml" xref="S3.SS2.p1.12.m12.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.12.m12.3c">P_{st}=\left\{p_{1},...,p_{m}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.12.m12.3d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT = { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math> be the set of directed <math alttext="s-t" class="ltx_Math" display="inline" id="S3.SS2.p1.13.m13.1"><semantics id="S3.SS2.p1.13.m13.1a"><mrow id="S3.SS2.p1.13.m13.1.1" xref="S3.SS2.p1.13.m13.1.1.cmml"><mi id="S3.SS2.p1.13.m13.1.1.2" xref="S3.SS2.p1.13.m13.1.1.2.cmml">s</mi><mo id="S3.SS2.p1.13.m13.1.1.1" xref="S3.SS2.p1.13.m13.1.1.1.cmml">−</mo><mi id="S3.SS2.p1.13.m13.1.1.3" xref="S3.SS2.p1.13.m13.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.13.m13.1b"><apply id="S3.SS2.p1.13.m13.1.1.cmml" xref="S3.SS2.p1.13.m13.1.1"><minus id="S3.SS2.p1.13.m13.1.1.1.cmml" xref="S3.SS2.p1.13.m13.1.1.1"></minus><ci id="S3.SS2.p1.13.m13.1.1.2.cmml" xref="S3.SS2.p1.13.m13.1.1.2">𝑠</ci><ci id="S3.SS2.p1.13.m13.1.1.3.cmml" xref="S3.SS2.p1.13.m13.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.13.m13.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.13.m13.1d">italic_s - italic_t</annotation></semantics></math> paths in <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p1.14.m14.1"><semantics id="S3.SS2.p1.14.m14.1a"><mi id="S3.SS2.p1.14.m14.1.1" xref="S3.SS2.p1.14.m14.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.14.m14.1b"><ci id="S3.SS2.p1.14.m14.1.1.cmml" xref="S3.SS2.p1.14.m14.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.14.m14.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.14.m14.1d">italic_G</annotation></semantics></math>. Paths will be treated as sets of edges, which can be disjoint. Here, edge-disjointness and vertex-disjointness are the same because if 2 paths share a vertex but no edges, then they imply the existence of paths which are not edge-disjoint. On the other hand, vertex-disjointness implies edge-disjointness. If the paths are disjoint, we will assume that <math alttext="|p_{1}|\leq...\leq|p_{m}|" class="ltx_Math" display="inline" id="S3.SS2.p1.15.m15.2"><semantics id="S3.SS2.p1.15.m15.2a"><mrow id="S3.SS2.p1.15.m15.2.2" xref="S3.SS2.p1.15.m15.2.2.cmml"><mrow id="S3.SS2.p1.15.m15.1.1.1.1" xref="S3.SS2.p1.15.m15.1.1.1.2.cmml"><mo id="S3.SS2.p1.15.m15.1.1.1.1.2" stretchy="false" xref="S3.SS2.p1.15.m15.1.1.1.2.1.cmml">|</mo><msub id="S3.SS2.p1.15.m15.1.1.1.1.1" xref="S3.SS2.p1.15.m15.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.15.m15.1.1.1.1.1.2" xref="S3.SS2.p1.15.m15.1.1.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p1.15.m15.1.1.1.1.1.3" xref="S3.SS2.p1.15.m15.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p1.15.m15.1.1.1.1.3" stretchy="false" xref="S3.SS2.p1.15.m15.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS2.p1.15.m15.2.2.4" xref="S3.SS2.p1.15.m15.2.2.4.cmml">≤</mo><mi id="S3.SS2.p1.15.m15.2.2.5" mathvariant="normal" xref="S3.SS2.p1.15.m15.2.2.5.cmml">…</mi><mo id="S3.SS2.p1.15.m15.2.2.6" xref="S3.SS2.p1.15.m15.2.2.6.cmml">≤</mo><mrow id="S3.SS2.p1.15.m15.2.2.2.1" xref="S3.SS2.p1.15.m15.2.2.2.2.cmml"><mo id="S3.SS2.p1.15.m15.2.2.2.1.2" stretchy="false" xref="S3.SS2.p1.15.m15.2.2.2.2.1.cmml">|</mo><msub id="S3.SS2.p1.15.m15.2.2.2.1.1" xref="S3.SS2.p1.15.m15.2.2.2.1.1.cmml"><mi id="S3.SS2.p1.15.m15.2.2.2.1.1.2" xref="S3.SS2.p1.15.m15.2.2.2.1.1.2.cmml">p</mi><mi id="S3.SS2.p1.15.m15.2.2.2.1.1.3" xref="S3.SS2.p1.15.m15.2.2.2.1.1.3.cmml">m</mi></msub><mo id="S3.SS2.p1.15.m15.2.2.2.1.3" stretchy="false" xref="S3.SS2.p1.15.m15.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.15.m15.2b"><apply id="S3.SS2.p1.15.m15.2.2.cmml" xref="S3.SS2.p1.15.m15.2.2"><and id="S3.SS2.p1.15.m15.2.2a.cmml" xref="S3.SS2.p1.15.m15.2.2"></and><apply id="S3.SS2.p1.15.m15.2.2b.cmml" xref="S3.SS2.p1.15.m15.2.2"><leq id="S3.SS2.p1.15.m15.2.2.4.cmml" xref="S3.SS2.p1.15.m15.2.2.4"></leq><apply id="S3.SS2.p1.15.m15.1.1.1.2.cmml" xref="S3.SS2.p1.15.m15.1.1.1.1"><abs id="S3.SS2.p1.15.m15.1.1.1.2.1.cmml" xref="S3.SS2.p1.15.m15.1.1.1.1.2"></abs><apply id="S3.SS2.p1.15.m15.1.1.1.1.1.cmml" xref="S3.SS2.p1.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.15.m15.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.15.m15.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.15.m15.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.15.m15.1.1.1.1.1.2">𝑝</ci><cn id="S3.SS2.p1.15.m15.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.15.m15.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.SS2.p1.15.m15.2.2.5.cmml" xref="S3.SS2.p1.15.m15.2.2.5">…</ci></apply><apply id="S3.SS2.p1.15.m15.2.2c.cmml" xref="S3.SS2.p1.15.m15.2.2"><leq id="S3.SS2.p1.15.m15.2.2.6.cmml" xref="S3.SS2.p1.15.m15.2.2.6"></leq><share href="https://arxiv.org/html/2308.08047v3#S3.SS2.p1.15.m15.2.2.5.cmml" id="S3.SS2.p1.15.m15.2.2d.cmml" xref="S3.SS2.p1.15.m15.2.2"></share><apply id="S3.SS2.p1.15.m15.2.2.2.2.cmml" xref="S3.SS2.p1.15.m15.2.2.2.1"><abs id="S3.SS2.p1.15.m15.2.2.2.2.1.cmml" xref="S3.SS2.p1.15.m15.2.2.2.1.2"></abs><apply id="S3.SS2.p1.15.m15.2.2.2.1.1.cmml" xref="S3.SS2.p1.15.m15.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.15.m15.2.2.2.1.1.1.cmml" xref="S3.SS2.p1.15.m15.2.2.2.1.1">subscript</csymbol><ci id="S3.SS2.p1.15.m15.2.2.2.1.1.2.cmml" xref="S3.SS2.p1.15.m15.2.2.2.1.1.2">𝑝</ci><ci id="S3.SS2.p1.15.m15.2.2.2.1.1.3.cmml" xref="S3.SS2.p1.15.m15.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.15.m15.2c">|p_{1}|\leq...\leq|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.15.m15.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ … ≤ | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math>, and denote <math alttext="P_{i}=\left\{p_{1},...,p_{i}\right\}" class="ltx_Math" display="inline" id="S3.SS2.p1.16.m16.3"><semantics id="S3.SS2.p1.16.m16.3a"><mrow id="S3.SS2.p1.16.m16.3.3" xref="S3.SS2.p1.16.m16.3.3.cmml"><msub id="S3.SS2.p1.16.m16.3.3.4" xref="S3.SS2.p1.16.m16.3.3.4.cmml"><mi id="S3.SS2.p1.16.m16.3.3.4.2" xref="S3.SS2.p1.16.m16.3.3.4.2.cmml">P</mi><mi id="S3.SS2.p1.16.m16.3.3.4.3" xref="S3.SS2.p1.16.m16.3.3.4.3.cmml">i</mi></msub><mo id="S3.SS2.p1.16.m16.3.3.3" xref="S3.SS2.p1.16.m16.3.3.3.cmml">=</mo><mrow id="S3.SS2.p1.16.m16.3.3.2.2" xref="S3.SS2.p1.16.m16.3.3.2.3.cmml"><mo id="S3.SS2.p1.16.m16.3.3.2.2.3" xref="S3.SS2.p1.16.m16.3.3.2.3.cmml">{</mo><msub id="S3.SS2.p1.16.m16.2.2.1.1.1" xref="S3.SS2.p1.16.m16.2.2.1.1.1.cmml"><mi id="S3.SS2.p1.16.m16.2.2.1.1.1.2" xref="S3.SS2.p1.16.m16.2.2.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p1.16.m16.2.2.1.1.1.3" xref="S3.SS2.p1.16.m16.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p1.16.m16.3.3.2.2.4" xref="S3.SS2.p1.16.m16.3.3.2.3.cmml">,</mo><mi id="S3.SS2.p1.16.m16.1.1" mathvariant="normal" xref="S3.SS2.p1.16.m16.1.1.cmml">…</mi><mo id="S3.SS2.p1.16.m16.3.3.2.2.5" xref="S3.SS2.p1.16.m16.3.3.2.3.cmml">,</mo><msub id="S3.SS2.p1.16.m16.3.3.2.2.2" xref="S3.SS2.p1.16.m16.3.3.2.2.2.cmml"><mi id="S3.SS2.p1.16.m16.3.3.2.2.2.2" xref="S3.SS2.p1.16.m16.3.3.2.2.2.2.cmml">p</mi><mi id="S3.SS2.p1.16.m16.3.3.2.2.2.3" xref="S3.SS2.p1.16.m16.3.3.2.2.2.3.cmml">i</mi></msub><mo id="S3.SS2.p1.16.m16.3.3.2.2.6" xref="S3.SS2.p1.16.m16.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.16.m16.3b"><apply id="S3.SS2.p1.16.m16.3.3.cmml" xref="S3.SS2.p1.16.m16.3.3"><eq id="S3.SS2.p1.16.m16.3.3.3.cmml" xref="S3.SS2.p1.16.m16.3.3.3"></eq><apply id="S3.SS2.p1.16.m16.3.3.4.cmml" xref="S3.SS2.p1.16.m16.3.3.4"><csymbol cd="ambiguous" id="S3.SS2.p1.16.m16.3.3.4.1.cmml" xref="S3.SS2.p1.16.m16.3.3.4">subscript</csymbol><ci id="S3.SS2.p1.16.m16.3.3.4.2.cmml" xref="S3.SS2.p1.16.m16.3.3.4.2">𝑃</ci><ci id="S3.SS2.p1.16.m16.3.3.4.3.cmml" xref="S3.SS2.p1.16.m16.3.3.4.3">𝑖</ci></apply><set id="S3.SS2.p1.16.m16.3.3.2.3.cmml" xref="S3.SS2.p1.16.m16.3.3.2.2"><apply id="S3.SS2.p1.16.m16.2.2.1.1.1.cmml" xref="S3.SS2.p1.16.m16.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.16.m16.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.16.m16.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.16.m16.2.2.1.1.1.2.cmml" xref="S3.SS2.p1.16.m16.2.2.1.1.1.2">𝑝</ci><cn id="S3.SS2.p1.16.m16.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.16.m16.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS2.p1.16.m16.1.1.cmml" xref="S3.SS2.p1.16.m16.1.1">…</ci><apply id="S3.SS2.p1.16.m16.3.3.2.2.2.cmml" xref="S3.SS2.p1.16.m16.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.16.m16.3.3.2.2.2.1.cmml" xref="S3.SS2.p1.16.m16.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.16.m16.3.3.2.2.2.2.cmml" xref="S3.SS2.p1.16.m16.3.3.2.2.2.2">𝑝</ci><ci id="S3.SS2.p1.16.m16.3.3.2.2.2.3.cmml" xref="S3.SS2.p1.16.m16.3.3.2.2.2.3">𝑖</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.16.m16.3c">P_{i}=\left\{p_{1},...,p_{i}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.16.m16.3d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math>. In general we denote with <math alttext="p_{max},p_{min}" class="ltx_Math" display="inline" id="S3.SS2.p1.17.m17.2"><semantics id="S3.SS2.p1.17.m17.2a"><mrow id="S3.SS2.p1.17.m17.2.2.2" xref="S3.SS2.p1.17.m17.2.2.3.cmml"><msub id="S3.SS2.p1.17.m17.1.1.1.1" xref="S3.SS2.p1.17.m17.1.1.1.1.cmml"><mi id="S3.SS2.p1.17.m17.1.1.1.1.2" xref="S3.SS2.p1.17.m17.1.1.1.1.2.cmml">p</mi><mrow id="S3.SS2.p1.17.m17.1.1.1.1.3" xref="S3.SS2.p1.17.m17.1.1.1.1.3.cmml"><mi id="S3.SS2.p1.17.m17.1.1.1.1.3.2" xref="S3.SS2.p1.17.m17.1.1.1.1.3.2.cmml">m</mi><mo id="S3.SS2.p1.17.m17.1.1.1.1.3.1" xref="S3.SS2.p1.17.m17.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.17.m17.1.1.1.1.3.3" xref="S3.SS2.p1.17.m17.1.1.1.1.3.3.cmml">a</mi><mo id="S3.SS2.p1.17.m17.1.1.1.1.3.1a" xref="S3.SS2.p1.17.m17.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.17.m17.1.1.1.1.3.4" xref="S3.SS2.p1.17.m17.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="S3.SS2.p1.17.m17.2.2.2.3" xref="S3.SS2.p1.17.m17.2.2.3.cmml">,</mo><msub id="S3.SS2.p1.17.m17.2.2.2.2" xref="S3.SS2.p1.17.m17.2.2.2.2.cmml"><mi id="S3.SS2.p1.17.m17.2.2.2.2.2" xref="S3.SS2.p1.17.m17.2.2.2.2.2.cmml">p</mi><mrow id="S3.SS2.p1.17.m17.2.2.2.2.3" xref="S3.SS2.p1.17.m17.2.2.2.2.3.cmml"><mi id="S3.SS2.p1.17.m17.2.2.2.2.3.2" xref="S3.SS2.p1.17.m17.2.2.2.2.3.2.cmml">m</mi><mo id="S3.SS2.p1.17.m17.2.2.2.2.3.1" xref="S3.SS2.p1.17.m17.2.2.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.17.m17.2.2.2.2.3.3" xref="S3.SS2.p1.17.m17.2.2.2.2.3.3.cmml">i</mi><mo id="S3.SS2.p1.17.m17.2.2.2.2.3.1a" xref="S3.SS2.p1.17.m17.2.2.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.17.m17.2.2.2.2.3.4" xref="S3.SS2.p1.17.m17.2.2.2.2.3.4.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.17.m17.2b"><list id="S3.SS2.p1.17.m17.2.2.3.cmml" xref="S3.SS2.p1.17.m17.2.2.2"><apply id="S3.SS2.p1.17.m17.1.1.1.1.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.17.m17.1.1.1.1.1.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.17.m17.1.1.1.1.2.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.2">𝑝</ci><apply id="S3.SS2.p1.17.m17.1.1.1.1.3.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.3"><times id="S3.SS2.p1.17.m17.1.1.1.1.3.1.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.3.1"></times><ci id="S3.SS2.p1.17.m17.1.1.1.1.3.2.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.3.2">𝑚</ci><ci id="S3.SS2.p1.17.m17.1.1.1.1.3.3.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.3.3">𝑎</ci><ci id="S3.SS2.p1.17.m17.1.1.1.1.3.4.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.3.4">𝑥</ci></apply></apply><apply id="S3.SS2.p1.17.m17.2.2.2.2.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.17.m17.2.2.2.2.1.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.17.m17.2.2.2.2.2.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.2">𝑝</ci><apply id="S3.SS2.p1.17.m17.2.2.2.2.3.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.3"><times id="S3.SS2.p1.17.m17.2.2.2.2.3.1.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.3.1"></times><ci id="S3.SS2.p1.17.m17.2.2.2.2.3.2.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.3.2">𝑚</ci><ci id="S3.SS2.p1.17.m17.2.2.2.2.3.3.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.3.3">𝑖</ci><ci id="S3.SS2.p1.17.m17.2.2.2.2.3.4.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.3.4">𝑛</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.17.m17.2c">p_{max},p_{min}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.17.m17.2d">italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math> an arbitrary pair of longest and shortest paths in <math alttext="P_{st}" class="ltx_Math" display="inline" id="S3.SS2.p1.18.m18.1"><semantics id="S3.SS2.p1.18.m18.1a"><msub id="S3.SS2.p1.18.m18.1.1" xref="S3.SS2.p1.18.m18.1.1.cmml"><mi id="S3.SS2.p1.18.m18.1.1.2" xref="S3.SS2.p1.18.m18.1.1.2.cmml">P</mi><mrow id="S3.SS2.p1.18.m18.1.1.3" xref="S3.SS2.p1.18.m18.1.1.3.cmml"><mi id="S3.SS2.p1.18.m18.1.1.3.2" xref="S3.SS2.p1.18.m18.1.1.3.2.cmml">s</mi><mo id="S3.SS2.p1.18.m18.1.1.3.1" xref="S3.SS2.p1.18.m18.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.18.m18.1.1.3.3" xref="S3.SS2.p1.18.m18.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.18.m18.1b"><apply id="S3.SS2.p1.18.m18.1.1.cmml" xref="S3.SS2.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.18.m18.1.1.1.cmml" xref="S3.SS2.p1.18.m18.1.1">subscript</csymbol><ci id="S3.SS2.p1.18.m18.1.1.2.cmml" xref="S3.SS2.p1.18.m18.1.1.2">𝑃</ci><apply id="S3.SS2.p1.18.m18.1.1.3.cmml" xref="S3.SS2.p1.18.m18.1.1.3"><times id="S3.SS2.p1.18.m18.1.1.3.1.cmml" xref="S3.SS2.p1.18.m18.1.1.3.1"></times><ci id="S3.SS2.p1.18.m18.1.1.3.2.cmml" xref="S3.SS2.p1.18.m18.1.1.3.2">𝑠</ci><ci id="S3.SS2.p1.18.m18.1.1.3.3.cmml" xref="S3.SS2.p1.18.m18.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.18.m18.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.18.m18.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. We also define <math alttext="|p_{0}|=0" class="ltx_Math" display="inline" id="S3.SS2.p1.19.m19.1"><semantics id="S3.SS2.p1.19.m19.1a"><mrow id="S3.SS2.p1.19.m19.1.1" xref="S3.SS2.p1.19.m19.1.1.cmml"><mrow id="S3.SS2.p1.19.m19.1.1.1.1" xref="S3.SS2.p1.19.m19.1.1.1.2.cmml"><mo id="S3.SS2.p1.19.m19.1.1.1.1.2" stretchy="false" xref="S3.SS2.p1.19.m19.1.1.1.2.1.cmml">|</mo><msub id="S3.SS2.p1.19.m19.1.1.1.1.1" xref="S3.SS2.p1.19.m19.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.19.m19.1.1.1.1.1.2" xref="S3.SS2.p1.19.m19.1.1.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p1.19.m19.1.1.1.1.1.3" xref="S3.SS2.p1.19.m19.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S3.SS2.p1.19.m19.1.1.1.1.3" stretchy="false" xref="S3.SS2.p1.19.m19.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS2.p1.19.m19.1.1.2" xref="S3.SS2.p1.19.m19.1.1.2.cmml">=</mo><mn id="S3.SS2.p1.19.m19.1.1.3" xref="S3.SS2.p1.19.m19.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.19.m19.1b"><apply id="S3.SS2.p1.19.m19.1.1.cmml" xref="S3.SS2.p1.19.m19.1.1"><eq id="S3.SS2.p1.19.m19.1.1.2.cmml" xref="S3.SS2.p1.19.m19.1.1.2"></eq><apply id="S3.SS2.p1.19.m19.1.1.1.2.cmml" xref="S3.SS2.p1.19.m19.1.1.1.1"><abs id="S3.SS2.p1.19.m19.1.1.1.2.1.cmml" xref="S3.SS2.p1.19.m19.1.1.1.1.2"></abs><apply id="S3.SS2.p1.19.m19.1.1.1.1.1.cmml" xref="S3.SS2.p1.19.m19.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.19.m19.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.19.m19.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.19.m19.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.19.m19.1.1.1.1.1.2">𝑝</ci><cn id="S3.SS2.p1.19.m19.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.19.m19.1.1.1.1.1.3">0</cn></apply></apply><cn id="S3.SS2.p1.19.m19.1.1.3.cmml" type="integer" xref="S3.SS2.p1.19.m19.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.19.m19.1c">|p_{0}|=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.19.m19.1d">| italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | = 0</annotation></semantics></math>. For an edge <math alttext="e\in E" class="ltx_Math" display="inline" id="S3.SS2.p1.20.m20.1"><semantics id="S3.SS2.p1.20.m20.1a"><mrow id="S3.SS2.p1.20.m20.1.1" xref="S3.SS2.p1.20.m20.1.1.cmml"><mi id="S3.SS2.p1.20.m20.1.1.2" xref="S3.SS2.p1.20.m20.1.1.2.cmml">e</mi><mo id="S3.SS2.p1.20.m20.1.1.1" xref="S3.SS2.p1.20.m20.1.1.1.cmml">∈</mo><mi id="S3.SS2.p1.20.m20.1.1.3" xref="S3.SS2.p1.20.m20.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.20.m20.1b"><apply id="S3.SS2.p1.20.m20.1.1.cmml" xref="S3.SS2.p1.20.m20.1.1"><in id="S3.SS2.p1.20.m20.1.1.1.cmml" xref="S3.SS2.p1.20.m20.1.1.1"></in><ci id="S3.SS2.p1.20.m20.1.1.2.cmml" xref="S3.SS2.p1.20.m20.1.1.2">𝑒</ci><ci id="S3.SS2.p1.20.m20.1.1.3.cmml" xref="S3.SS2.p1.20.m20.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.20.m20.1c">e\in E</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.20.m20.1d">italic_e ∈ italic_E</annotation></semantics></math>, let <math alttext="P_{st,e}=\left\{p\in P_{st}\mid e\in p\right\}" class="ltx_Math" display="inline" id="S3.SS2.p1.21.m21.4"><semantics id="S3.SS2.p1.21.m21.4a"><mrow id="S3.SS2.p1.21.m21.4.4" xref="S3.SS2.p1.21.m21.4.4.cmml"><msub id="S3.SS2.p1.21.m21.4.4.4" xref="S3.SS2.p1.21.m21.4.4.4.cmml"><mi id="S3.SS2.p1.21.m21.4.4.4.2" xref="S3.SS2.p1.21.m21.4.4.4.2.cmml">P</mi><mrow id="S3.SS2.p1.21.m21.2.2.2.2" xref="S3.SS2.p1.21.m21.2.2.2.3.cmml"><mrow id="S3.SS2.p1.21.m21.2.2.2.2.1" xref="S3.SS2.p1.21.m21.2.2.2.2.1.cmml"><mi id="S3.SS2.p1.21.m21.2.2.2.2.1.2" xref="S3.SS2.p1.21.m21.2.2.2.2.1.2.cmml">s</mi><mo id="S3.SS2.p1.21.m21.2.2.2.2.1.1" xref="S3.SS2.p1.21.m21.2.2.2.2.1.1.cmml">⁢</mo><mi id="S3.SS2.p1.21.m21.2.2.2.2.1.3" xref="S3.SS2.p1.21.m21.2.2.2.2.1.3.cmml">t</mi></mrow><mo id="S3.SS2.p1.21.m21.2.2.2.2.2" xref="S3.SS2.p1.21.m21.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p1.21.m21.1.1.1.1" xref="S3.SS2.p1.21.m21.1.1.1.1.cmml">e</mi></mrow></msub><mo id="S3.SS2.p1.21.m21.4.4.3" xref="S3.SS2.p1.21.m21.4.4.3.cmml">=</mo><mrow id="S3.SS2.p1.21.m21.4.4.2.2" xref="S3.SS2.p1.21.m21.4.4.2.3.cmml"><mo id="S3.SS2.p1.21.m21.4.4.2.2.3" xref="S3.SS2.p1.21.m21.4.4.2.3.1.cmml">{</mo><mrow id="S3.SS2.p1.21.m21.3.3.1.1.1" xref="S3.SS2.p1.21.m21.3.3.1.1.1.cmml"><mi id="S3.SS2.p1.21.m21.3.3.1.1.1.2" xref="S3.SS2.p1.21.m21.3.3.1.1.1.2.cmml">p</mi><mo id="S3.SS2.p1.21.m21.3.3.1.1.1.1" xref="S3.SS2.p1.21.m21.3.3.1.1.1.1.cmml">∈</mo><msub id="S3.SS2.p1.21.m21.3.3.1.1.1.3" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.cmml"><mi id="S3.SS2.p1.21.m21.3.3.1.1.1.3.2" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.2.cmml">P</mi><mrow id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.cmml"><mi id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.2" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.2.cmml">s</mi><mo id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.1" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.3" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.3.cmml">t</mi></mrow></msub></mrow><mo fence="true" id="S3.SS2.p1.21.m21.4.4.2.2.4" lspace="0em" rspace="0em" xref="S3.SS2.p1.21.m21.4.4.2.3.1.cmml">∣</mo><mrow id="S3.SS2.p1.21.m21.4.4.2.2.2" xref="S3.SS2.p1.21.m21.4.4.2.2.2.cmml"><mi id="S3.SS2.p1.21.m21.4.4.2.2.2.2" xref="S3.SS2.p1.21.m21.4.4.2.2.2.2.cmml">e</mi><mo id="S3.SS2.p1.21.m21.4.4.2.2.2.1" xref="S3.SS2.p1.21.m21.4.4.2.2.2.1.cmml">∈</mo><mi id="S3.SS2.p1.21.m21.4.4.2.2.2.3" xref="S3.SS2.p1.21.m21.4.4.2.2.2.3.cmml">p</mi></mrow><mo id="S3.SS2.p1.21.m21.4.4.2.2.5" xref="S3.SS2.p1.21.m21.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.21.m21.4b"><apply id="S3.SS2.p1.21.m21.4.4.cmml" xref="S3.SS2.p1.21.m21.4.4"><eq id="S3.SS2.p1.21.m21.4.4.3.cmml" xref="S3.SS2.p1.21.m21.4.4.3"></eq><apply id="S3.SS2.p1.21.m21.4.4.4.cmml" xref="S3.SS2.p1.21.m21.4.4.4"><csymbol cd="ambiguous" id="S3.SS2.p1.21.m21.4.4.4.1.cmml" xref="S3.SS2.p1.21.m21.4.4.4">subscript</csymbol><ci id="S3.SS2.p1.21.m21.4.4.4.2.cmml" xref="S3.SS2.p1.21.m21.4.4.4.2">𝑃</ci><list id="S3.SS2.p1.21.m21.2.2.2.3.cmml" xref="S3.SS2.p1.21.m21.2.2.2.2"><apply id="S3.SS2.p1.21.m21.2.2.2.2.1.cmml" xref="S3.SS2.p1.21.m21.2.2.2.2.1"><times id="S3.SS2.p1.21.m21.2.2.2.2.1.1.cmml" xref="S3.SS2.p1.21.m21.2.2.2.2.1.1"></times><ci id="S3.SS2.p1.21.m21.2.2.2.2.1.2.cmml" xref="S3.SS2.p1.21.m21.2.2.2.2.1.2">𝑠</ci><ci id="S3.SS2.p1.21.m21.2.2.2.2.1.3.cmml" xref="S3.SS2.p1.21.m21.2.2.2.2.1.3">𝑡</ci></apply><ci id="S3.SS2.p1.21.m21.1.1.1.1.cmml" xref="S3.SS2.p1.21.m21.1.1.1.1">𝑒</ci></list></apply><apply id="S3.SS2.p1.21.m21.4.4.2.3.cmml" xref="S3.SS2.p1.21.m21.4.4.2.2"><csymbol cd="latexml" id="S3.SS2.p1.21.m21.4.4.2.3.1.cmml" xref="S3.SS2.p1.21.m21.4.4.2.2.3">conditional-set</csymbol><apply id="S3.SS2.p1.21.m21.3.3.1.1.1.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1"><in id="S3.SS2.p1.21.m21.3.3.1.1.1.1.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.1"></in><ci id="S3.SS2.p1.21.m21.3.3.1.1.1.2.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.2">𝑝</ci><apply id="S3.SS2.p1.21.m21.3.3.1.1.1.3.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p1.21.m21.3.3.1.1.1.3.1.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3">subscript</csymbol><ci id="S3.SS2.p1.21.m21.3.3.1.1.1.3.2.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.2">𝑃</ci><apply id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3"><times id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.1.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.1"></times><ci id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.2.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.2">𝑠</ci><ci id="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.3.cmml" xref="S3.SS2.p1.21.m21.3.3.1.1.1.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.SS2.p1.21.m21.4.4.2.2.2.cmml" xref="S3.SS2.p1.21.m21.4.4.2.2.2"><in id="S3.SS2.p1.21.m21.4.4.2.2.2.1.cmml" xref="S3.SS2.p1.21.m21.4.4.2.2.2.1"></in><ci id="S3.SS2.p1.21.m21.4.4.2.2.2.2.cmml" xref="S3.SS2.p1.21.m21.4.4.2.2.2.2">𝑒</ci><ci id="S3.SS2.p1.21.m21.4.4.2.2.2.3.cmml" xref="S3.SS2.p1.21.m21.4.4.2.2.2.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.21.m21.4c">P_{st,e}=\left\{p\in P_{st}\mid e\in p\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.21.m21.4d">italic_P start_POSTSUBSCRIPT italic_s italic_t , italic_e end_POSTSUBSCRIPT = { italic_p ∈ italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ∣ italic_e ∈ italic_p }</annotation></semantics></math>. <br class="ltx_break"/> <br class="ltx_break"/><span class="ltx_text ltx_font_bold" id="S3.SS2.p1.32.3">The strategies </span> There are <math alttext="n" class="ltx_Math" display="inline" id="S3.SS2.p1.22.m22.1"><semantics id="S3.SS2.p1.22.m22.1a"><mi id="S3.SS2.p1.22.m22.1.1" xref="S3.SS2.p1.22.m22.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.22.m22.1b"><ci id="S3.SS2.p1.22.m22.1.1.cmml" xref="S3.SS2.p1.22.m22.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.22.m22.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.22.m22.1d">italic_n</annotation></semantics></math> agents and <math alttext="k" class="ltx_Math" display="inline" id="S3.SS2.p1.23.m23.1"><semantics id="S3.SS2.p1.23.m23.1a"><mi id="S3.SS2.p1.23.m23.1.1" xref="S3.SS2.p1.23.m23.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.23.m23.1b"><ci id="S3.SS2.p1.23.m23.1.1.cmml" xref="S3.SS2.p1.23.m23.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.23.m23.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.23.m23.1d">italic_k</annotation></semantics></math> interceptors. The set of pure strategies of each agent is defined to be <math alttext="P_{st}" class="ltx_Math" display="inline" id="S3.SS2.p1.24.m24.1"><semantics id="S3.SS2.p1.24.m24.1a"><msub id="S3.SS2.p1.24.m24.1.1" xref="S3.SS2.p1.24.m24.1.1.cmml"><mi id="S3.SS2.p1.24.m24.1.1.2" xref="S3.SS2.p1.24.m24.1.1.2.cmml">P</mi><mrow id="S3.SS2.p1.24.m24.1.1.3" xref="S3.SS2.p1.24.m24.1.1.3.cmml"><mi id="S3.SS2.p1.24.m24.1.1.3.2" xref="S3.SS2.p1.24.m24.1.1.3.2.cmml">s</mi><mo id="S3.SS2.p1.24.m24.1.1.3.1" xref="S3.SS2.p1.24.m24.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.24.m24.1.1.3.3" xref="S3.SS2.p1.24.m24.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.24.m24.1b"><apply id="S3.SS2.p1.24.m24.1.1.cmml" xref="S3.SS2.p1.24.m24.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.24.m24.1.1.1.cmml" xref="S3.SS2.p1.24.m24.1.1">subscript</csymbol><ci id="S3.SS2.p1.24.m24.1.1.2.cmml" xref="S3.SS2.p1.24.m24.1.1.2">𝑃</ci><apply id="S3.SS2.p1.24.m24.1.1.3.cmml" xref="S3.SS2.p1.24.m24.1.1.3"><times id="S3.SS2.p1.24.m24.1.1.3.1.cmml" xref="S3.SS2.p1.24.m24.1.1.3.1"></times><ci id="S3.SS2.p1.24.m24.1.1.3.2.cmml" xref="S3.SS2.p1.24.m24.1.1.3.2">𝑠</ci><ci id="S3.SS2.p1.24.m24.1.1.3.3.cmml" xref="S3.SS2.p1.24.m24.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.24.m24.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.24.m24.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, and that of each interceptor is <math alttext="E" class="ltx_Math" display="inline" id="S3.SS2.p1.25.m25.1"><semantics id="S3.SS2.p1.25.m25.1a"><mi id="S3.SS2.p1.25.m25.1.1" xref="S3.SS2.p1.25.m25.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.25.m25.1b"><ci id="S3.SS2.p1.25.m25.1.1.cmml" xref="S3.SS2.p1.25.m25.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.25.m25.1c">E</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.25.m25.1d">italic_E</annotation></semantics></math>. Paths picked by the agents will be denoted by <math alttext="a=(a_{1},...,a_{n})" class="ltx_Math" display="inline" id="S3.SS2.p1.26.m26.3"><semantics id="S3.SS2.p1.26.m26.3a"><mrow id="S3.SS2.p1.26.m26.3.3" xref="S3.SS2.p1.26.m26.3.3.cmml"><mi id="S3.SS2.p1.26.m26.3.3.4" xref="S3.SS2.p1.26.m26.3.3.4.cmml">a</mi><mo id="S3.SS2.p1.26.m26.3.3.3" xref="S3.SS2.p1.26.m26.3.3.3.cmml">=</mo><mrow id="S3.SS2.p1.26.m26.3.3.2.2" xref="S3.SS2.p1.26.m26.3.3.2.3.cmml"><mo id="S3.SS2.p1.26.m26.3.3.2.2.3" stretchy="false" xref="S3.SS2.p1.26.m26.3.3.2.3.cmml">(</mo><msub id="S3.SS2.p1.26.m26.2.2.1.1.1" xref="S3.SS2.p1.26.m26.2.2.1.1.1.cmml"><mi id="S3.SS2.p1.26.m26.2.2.1.1.1.2" xref="S3.SS2.p1.26.m26.2.2.1.1.1.2.cmml">a</mi><mn id="S3.SS2.p1.26.m26.2.2.1.1.1.3" xref="S3.SS2.p1.26.m26.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p1.26.m26.3.3.2.2.4" xref="S3.SS2.p1.26.m26.3.3.2.3.cmml">,</mo><mi id="S3.SS2.p1.26.m26.1.1" mathvariant="normal" xref="S3.SS2.p1.26.m26.1.1.cmml">…</mi><mo id="S3.SS2.p1.26.m26.3.3.2.2.5" xref="S3.SS2.p1.26.m26.3.3.2.3.cmml">,</mo><msub id="S3.SS2.p1.26.m26.3.3.2.2.2" xref="S3.SS2.p1.26.m26.3.3.2.2.2.cmml"><mi id="S3.SS2.p1.26.m26.3.3.2.2.2.2" xref="S3.SS2.p1.26.m26.3.3.2.2.2.2.cmml">a</mi><mi id="S3.SS2.p1.26.m26.3.3.2.2.2.3" xref="S3.SS2.p1.26.m26.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S3.SS2.p1.26.m26.3.3.2.2.6" stretchy="false" xref="S3.SS2.p1.26.m26.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.26.m26.3b"><apply id="S3.SS2.p1.26.m26.3.3.cmml" xref="S3.SS2.p1.26.m26.3.3"><eq id="S3.SS2.p1.26.m26.3.3.3.cmml" xref="S3.SS2.p1.26.m26.3.3.3"></eq><ci id="S3.SS2.p1.26.m26.3.3.4.cmml" xref="S3.SS2.p1.26.m26.3.3.4">𝑎</ci><vector id="S3.SS2.p1.26.m26.3.3.2.3.cmml" xref="S3.SS2.p1.26.m26.3.3.2.2"><apply id="S3.SS2.p1.26.m26.2.2.1.1.1.cmml" xref="S3.SS2.p1.26.m26.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.26.m26.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.26.m26.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.26.m26.2.2.1.1.1.2.cmml" xref="S3.SS2.p1.26.m26.2.2.1.1.1.2">𝑎</ci><cn id="S3.SS2.p1.26.m26.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.26.m26.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS2.p1.26.m26.1.1.cmml" xref="S3.SS2.p1.26.m26.1.1">…</ci><apply id="S3.SS2.p1.26.m26.3.3.2.2.2.cmml" xref="S3.SS2.p1.26.m26.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.26.m26.3.3.2.2.2.1.cmml" xref="S3.SS2.p1.26.m26.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.26.m26.3.3.2.2.2.2.cmml" xref="S3.SS2.p1.26.m26.3.3.2.2.2.2">𝑎</ci><ci id="S3.SS2.p1.26.m26.3.3.2.2.2.3.cmml" xref="S3.SS2.p1.26.m26.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.26.m26.3c">a=(a_{1},...,a_{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.26.m26.3d">italic_a = ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>, and edges picked by interceptors will be denoted by <math alttext="d=(d_{1},...,d_{k})" class="ltx_Math" display="inline" id="S3.SS2.p1.27.m27.3"><semantics id="S3.SS2.p1.27.m27.3a"><mrow id="S3.SS2.p1.27.m27.3.3" xref="S3.SS2.p1.27.m27.3.3.cmml"><mi id="S3.SS2.p1.27.m27.3.3.4" xref="S3.SS2.p1.27.m27.3.3.4.cmml">d</mi><mo id="S3.SS2.p1.27.m27.3.3.3" xref="S3.SS2.p1.27.m27.3.3.3.cmml">=</mo><mrow id="S3.SS2.p1.27.m27.3.3.2.2" xref="S3.SS2.p1.27.m27.3.3.2.3.cmml"><mo id="S3.SS2.p1.27.m27.3.3.2.2.3" stretchy="false" xref="S3.SS2.p1.27.m27.3.3.2.3.cmml">(</mo><msub id="S3.SS2.p1.27.m27.2.2.1.1.1" xref="S3.SS2.p1.27.m27.2.2.1.1.1.cmml"><mi id="S3.SS2.p1.27.m27.2.2.1.1.1.2" xref="S3.SS2.p1.27.m27.2.2.1.1.1.2.cmml">d</mi><mn id="S3.SS2.p1.27.m27.2.2.1.1.1.3" xref="S3.SS2.p1.27.m27.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p1.27.m27.3.3.2.2.4" xref="S3.SS2.p1.27.m27.3.3.2.3.cmml">,</mo><mi id="S3.SS2.p1.27.m27.1.1" mathvariant="normal" xref="S3.SS2.p1.27.m27.1.1.cmml">…</mi><mo id="S3.SS2.p1.27.m27.3.3.2.2.5" xref="S3.SS2.p1.27.m27.3.3.2.3.cmml">,</mo><msub id="S3.SS2.p1.27.m27.3.3.2.2.2" xref="S3.SS2.p1.27.m27.3.3.2.2.2.cmml"><mi id="S3.SS2.p1.27.m27.3.3.2.2.2.2" xref="S3.SS2.p1.27.m27.3.3.2.2.2.2.cmml">d</mi><mi id="S3.SS2.p1.27.m27.3.3.2.2.2.3" xref="S3.SS2.p1.27.m27.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S3.SS2.p1.27.m27.3.3.2.2.6" stretchy="false" xref="S3.SS2.p1.27.m27.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.27.m27.3b"><apply id="S3.SS2.p1.27.m27.3.3.cmml" xref="S3.SS2.p1.27.m27.3.3"><eq id="S3.SS2.p1.27.m27.3.3.3.cmml" xref="S3.SS2.p1.27.m27.3.3.3"></eq><ci id="S3.SS2.p1.27.m27.3.3.4.cmml" xref="S3.SS2.p1.27.m27.3.3.4">𝑑</ci><vector id="S3.SS2.p1.27.m27.3.3.2.3.cmml" xref="S3.SS2.p1.27.m27.3.3.2.2"><apply id="S3.SS2.p1.27.m27.2.2.1.1.1.cmml" xref="S3.SS2.p1.27.m27.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.27.m27.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.27.m27.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.27.m27.2.2.1.1.1.2.cmml" xref="S3.SS2.p1.27.m27.2.2.1.1.1.2">𝑑</ci><cn id="S3.SS2.p1.27.m27.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.27.m27.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS2.p1.27.m27.1.1.cmml" xref="S3.SS2.p1.27.m27.1.1">…</ci><apply id="S3.SS2.p1.27.m27.3.3.2.2.2.cmml" xref="S3.SS2.p1.27.m27.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.27.m27.3.3.2.2.2.1.cmml" xref="S3.SS2.p1.27.m27.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.27.m27.3.3.2.2.2.2.cmml" xref="S3.SS2.p1.27.m27.3.3.2.2.2.2">𝑑</ci><ci id="S3.SS2.p1.27.m27.3.3.2.2.2.3.cmml" xref="S3.SS2.p1.27.m27.3.3.2.2.2.3">𝑘</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.27.m27.3c">d=(d_{1},...,d_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.27.m27.3d">italic_d = ( italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. We will sometimes treat <math alttext="a" class="ltx_Math" display="inline" id="S3.SS2.p1.28.m28.1"><semantics id="S3.SS2.p1.28.m28.1a"><mi id="S3.SS2.p1.28.m28.1.1" xref="S3.SS2.p1.28.m28.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.28.m28.1b"><ci id="S3.SS2.p1.28.m28.1.1.cmml" xref="S3.SS2.p1.28.m28.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.28.m28.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.28.m28.1d">italic_a</annotation></semantics></math> as an integer vector <math alttext="(|a_{i}|)_{i}\in\mathbb{Z}^{n}\subseteq\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S3.SS2.p1.29.m29.1"><semantics id="S3.SS2.p1.29.m29.1a"><mrow id="S3.SS2.p1.29.m29.1.1" xref="S3.SS2.p1.29.m29.1.1.cmml"><msub id="S3.SS2.p1.29.m29.1.1.1" xref="S3.SS2.p1.29.m29.1.1.1.cmml"><mrow id="S3.SS2.p1.29.m29.1.1.1.1.1" xref="S3.SS2.p1.29.m29.1.1.1.cmml"><mo id="S3.SS2.p1.29.m29.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p1.29.m29.1.1.1.cmml">(</mo><mrow id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.2.cmml"><mo id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.2.1.cmml">|</mo><msub id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.2" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.3" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS2.p1.29.m29.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p1.29.m29.1.1.1.cmml">)</mo></mrow><mi id="S3.SS2.p1.29.m29.1.1.1.3" xref="S3.SS2.p1.29.m29.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS2.p1.29.m29.1.1.3" xref="S3.SS2.p1.29.m29.1.1.3.cmml">∈</mo><msup id="S3.SS2.p1.29.m29.1.1.4" xref="S3.SS2.p1.29.m29.1.1.4.cmml"><mi id="S3.SS2.p1.29.m29.1.1.4.2" xref="S3.SS2.p1.29.m29.1.1.4.2.cmml">ℤ</mi><mi id="S3.SS2.p1.29.m29.1.1.4.3" xref="S3.SS2.p1.29.m29.1.1.4.3.cmml">n</mi></msup><mo id="S3.SS2.p1.29.m29.1.1.5" xref="S3.SS2.p1.29.m29.1.1.5.cmml">⊆</mo><msup id="S3.SS2.p1.29.m29.1.1.6" xref="S3.SS2.p1.29.m29.1.1.6.cmml"><mi id="S3.SS2.p1.29.m29.1.1.6.2" xref="S3.SS2.p1.29.m29.1.1.6.2.cmml">ℝ</mi><mi id="S3.SS2.p1.29.m29.1.1.6.3" xref="S3.SS2.p1.29.m29.1.1.6.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.29.m29.1b"><apply id="S3.SS2.p1.29.m29.1.1.cmml" xref="S3.SS2.p1.29.m29.1.1"><and id="S3.SS2.p1.29.m29.1.1a.cmml" xref="S3.SS2.p1.29.m29.1.1"></and><apply id="S3.SS2.p1.29.m29.1.1b.cmml" xref="S3.SS2.p1.29.m29.1.1"><in id="S3.SS2.p1.29.m29.1.1.3.cmml" xref="S3.SS2.p1.29.m29.1.1.3"></in><apply id="S3.SS2.p1.29.m29.1.1.1.cmml" xref="S3.SS2.p1.29.m29.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.29.m29.1.1.1.2.cmml" xref="S3.SS2.p1.29.m29.1.1.1">subscript</csymbol><apply id="S3.SS2.p1.29.m29.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1"><abs id="S3.SS2.p1.29.m29.1.1.1.1.1.1.2.1.cmml" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.2"></abs><apply id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p1.29.m29.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><ci id="S3.SS2.p1.29.m29.1.1.1.3.cmml" xref="S3.SS2.p1.29.m29.1.1.1.3">𝑖</ci></apply><apply id="S3.SS2.p1.29.m29.1.1.4.cmml" xref="S3.SS2.p1.29.m29.1.1.4"><csymbol cd="ambiguous" id="S3.SS2.p1.29.m29.1.1.4.1.cmml" xref="S3.SS2.p1.29.m29.1.1.4">superscript</csymbol><ci id="S3.SS2.p1.29.m29.1.1.4.2.cmml" xref="S3.SS2.p1.29.m29.1.1.4.2">ℤ</ci><ci id="S3.SS2.p1.29.m29.1.1.4.3.cmml" xref="S3.SS2.p1.29.m29.1.1.4.3">𝑛</ci></apply></apply><apply id="S3.SS2.p1.29.m29.1.1c.cmml" xref="S3.SS2.p1.29.m29.1.1"><subset id="S3.SS2.p1.29.m29.1.1.5.cmml" xref="S3.SS2.p1.29.m29.1.1.5"></subset><share href="https://arxiv.org/html/2308.08047v3#S3.SS2.p1.29.m29.1.1.4.cmml" id="S3.SS2.p1.29.m29.1.1d.cmml" xref="S3.SS2.p1.29.m29.1.1"></share><apply id="S3.SS2.p1.29.m29.1.1.6.cmml" xref="S3.SS2.p1.29.m29.1.1.6"><csymbol cd="ambiguous" id="S3.SS2.p1.29.m29.1.1.6.1.cmml" xref="S3.SS2.p1.29.m29.1.1.6">superscript</csymbol><ci id="S3.SS2.p1.29.m29.1.1.6.2.cmml" xref="S3.SS2.p1.29.m29.1.1.6.2">ℝ</ci><ci id="S3.SS2.p1.29.m29.1.1.6.3.cmml" xref="S3.SS2.p1.29.m29.1.1.6.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.29.m29.1c">(|a_{i}|)_{i}\in\mathbb{Z}^{n}\subseteq\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.29.m29.1d">( | italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ⊆ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, particularly when computing its norm <math alttext="\left\lVert a\right\rVert" class="ltx_Math" display="inline" id="S3.SS2.p1.30.m30.1"><semantics id="S3.SS2.p1.30.m30.1a"><mrow id="S3.SS2.p1.30.m30.1.2.2" xref="S3.SS2.p1.30.m30.1.2.1.cmml"><mo fence="true" id="S3.SS2.p1.30.m30.1.2.2.1" rspace="0em" stretchy="true" xref="S3.SS2.p1.30.m30.1.2.1.1.cmml">∥</mo><mi id="S3.SS2.p1.30.m30.1.1" xref="S3.SS2.p1.30.m30.1.1.cmml">a</mi><mo fence="true" id="S3.SS2.p1.30.m30.1.2.2.2" lspace="0em" stretchy="true" xref="S3.SS2.p1.30.m30.1.2.1.1.cmml">∥</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.30.m30.1b"><apply id="S3.SS2.p1.30.m30.1.2.1.cmml" xref="S3.SS2.p1.30.m30.1.2.2"><csymbol cd="latexml" id="S3.SS2.p1.30.m30.1.2.1.1.cmml" xref="S3.SS2.p1.30.m30.1.2.2.1">delimited-∥∥</csymbol><ci id="S3.SS2.p1.30.m30.1.1.cmml" xref="S3.SS2.p1.30.m30.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.30.m30.1c">\left\lVert a\right\rVert</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.30.m30.1d">∥ italic_a ∥</annotation></semantics></math>. Here we will consider only the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S3.SS2.p1.31.m31.1"><semantics id="S3.SS2.p1.31.m31.1a"><msub id="S3.SS2.p1.31.m31.1.1" xref="S3.SS2.p1.31.m31.1.1.cmml"><mi id="S3.SS2.p1.31.m31.1.1.2" mathvariant="normal" xref="S3.SS2.p1.31.m31.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p1.31.m31.1.1.3" mathvariant="normal" xref="S3.SS2.p1.31.m31.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.31.m31.1b"><apply id="S3.SS2.p1.31.m31.1.1.cmml" xref="S3.SS2.p1.31.m31.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.31.m31.1.1.1.cmml" xref="S3.SS2.p1.31.m31.1.1">subscript</csymbol><ci id="S3.SS2.p1.31.m31.1.1.2.cmml" xref="S3.SS2.p1.31.m31.1.1.2">ℓ</ci><infinity id="S3.SS2.p1.31.m31.1.1.3.cmml" xref="S3.SS2.p1.31.m31.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.31.m31.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.31.m31.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-norm: <math alttext="\left\lVert a\right\rVert_{\infty}:=\max_{i\in[n]}|a_{i}|" class="ltx_Math" display="inline" id="S3.SS2.p1.32.m32.3"><semantics id="S3.SS2.p1.32.m32.3a"><mrow id="S3.SS2.p1.32.m32.3.3" xref="S3.SS2.p1.32.m32.3.3.cmml"><msub id="S3.SS2.p1.32.m32.3.3.3" xref="S3.SS2.p1.32.m32.3.3.3.cmml"><mrow id="S3.SS2.p1.32.m32.3.3.3.2.2" xref="S3.SS2.p1.32.m32.3.3.3.2.1.cmml"><mo fence="true" id="S3.SS2.p1.32.m32.3.3.3.2.2.1" rspace="0em" stretchy="true" xref="S3.SS2.p1.32.m32.3.3.3.2.1.1.cmml">∥</mo><mi id="S3.SS2.p1.32.m32.2.2" xref="S3.SS2.p1.32.m32.2.2.cmml">a</mi><mo fence="true" id="S3.SS2.p1.32.m32.3.3.3.2.2.2" lspace="0em" stretchy="true" xref="S3.SS2.p1.32.m32.3.3.3.2.1.1.cmml">∥</mo></mrow><mi id="S3.SS2.p1.32.m32.3.3.3.3" mathvariant="normal" xref="S3.SS2.p1.32.m32.3.3.3.3.cmml">∞</mi></msub><mo id="S3.SS2.p1.32.m32.3.3.2" lspace="0.278em" rspace="0.278em" xref="S3.SS2.p1.32.m32.3.3.2.cmml">:=</mo><mrow id="S3.SS2.p1.32.m32.3.3.1" xref="S3.SS2.p1.32.m32.3.3.1.cmml"><msub id="S3.SS2.p1.32.m32.3.3.1.2" xref="S3.SS2.p1.32.m32.3.3.1.2.cmml"><mi id="S3.SS2.p1.32.m32.3.3.1.2.2" xref="S3.SS2.p1.32.m32.3.3.1.2.2.cmml">max</mi><mrow id="S3.SS2.p1.32.m32.1.1.1" xref="S3.SS2.p1.32.m32.1.1.1.cmml"><mi id="S3.SS2.p1.32.m32.1.1.1.3" xref="S3.SS2.p1.32.m32.1.1.1.3.cmml">i</mi><mo id="S3.SS2.p1.32.m32.1.1.1.2" xref="S3.SS2.p1.32.m32.1.1.1.2.cmml">∈</mo><mrow id="S3.SS2.p1.32.m32.1.1.1.4.2" xref="S3.SS2.p1.32.m32.1.1.1.4.1.cmml"><mo id="S3.SS2.p1.32.m32.1.1.1.4.2.1" stretchy="false" xref="S3.SS2.p1.32.m32.1.1.1.4.1.1.cmml">[</mo><mi id="S3.SS2.p1.32.m32.1.1.1.1" xref="S3.SS2.p1.32.m32.1.1.1.1.cmml">n</mi><mo id="S3.SS2.p1.32.m32.1.1.1.4.2.2" stretchy="false" xref="S3.SS2.p1.32.m32.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mo id="S3.SS2.p1.32.m32.3.3.1a" xref="S3.SS2.p1.32.m32.3.3.1.cmml">⁡</mo><mrow id="S3.SS2.p1.32.m32.3.3.1.1.1" xref="S3.SS2.p1.32.m32.3.3.1.1.2.cmml"><mo id="S3.SS2.p1.32.m32.3.3.1.1.1.2" stretchy="false" xref="S3.SS2.p1.32.m32.3.3.1.1.2.1.cmml">|</mo><msub id="S3.SS2.p1.32.m32.3.3.1.1.1.1" xref="S3.SS2.p1.32.m32.3.3.1.1.1.1.cmml"><mi id="S3.SS2.p1.32.m32.3.3.1.1.1.1.2" xref="S3.SS2.p1.32.m32.3.3.1.1.1.1.2.cmml">a</mi><mi id="S3.SS2.p1.32.m32.3.3.1.1.1.1.3" xref="S3.SS2.p1.32.m32.3.3.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS2.p1.32.m32.3.3.1.1.1.3" stretchy="false" xref="S3.SS2.p1.32.m32.3.3.1.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.32.m32.3b"><apply id="S3.SS2.p1.32.m32.3.3.cmml" xref="S3.SS2.p1.32.m32.3.3"><csymbol cd="latexml" id="S3.SS2.p1.32.m32.3.3.2.cmml" xref="S3.SS2.p1.32.m32.3.3.2">assign</csymbol><apply id="S3.SS2.p1.32.m32.3.3.3.cmml" xref="S3.SS2.p1.32.m32.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p1.32.m32.3.3.3.1.cmml" xref="S3.SS2.p1.32.m32.3.3.3">subscript</csymbol><apply id="S3.SS2.p1.32.m32.3.3.3.2.1.cmml" xref="S3.SS2.p1.32.m32.3.3.3.2.2"><csymbol cd="latexml" id="S3.SS2.p1.32.m32.3.3.3.2.1.1.cmml" xref="S3.SS2.p1.32.m32.3.3.3.2.2.1">delimited-∥∥</csymbol><ci id="S3.SS2.p1.32.m32.2.2.cmml" xref="S3.SS2.p1.32.m32.2.2">𝑎</ci></apply><infinity id="S3.SS2.p1.32.m32.3.3.3.3.cmml" xref="S3.SS2.p1.32.m32.3.3.3.3"></infinity></apply><apply id="S3.SS2.p1.32.m32.3.3.1.cmml" xref="S3.SS2.p1.32.m32.3.3.1"><apply id="S3.SS2.p1.32.m32.3.3.1.2.cmml" xref="S3.SS2.p1.32.m32.3.3.1.2"><csymbol cd="ambiguous" id="S3.SS2.p1.32.m32.3.3.1.2.1.cmml" xref="S3.SS2.p1.32.m32.3.3.1.2">subscript</csymbol><max id="S3.SS2.p1.32.m32.3.3.1.2.2.cmml" xref="S3.SS2.p1.32.m32.3.3.1.2.2"></max><apply id="S3.SS2.p1.32.m32.1.1.1.cmml" xref="S3.SS2.p1.32.m32.1.1.1"><in id="S3.SS2.p1.32.m32.1.1.1.2.cmml" xref="S3.SS2.p1.32.m32.1.1.1.2"></in><ci id="S3.SS2.p1.32.m32.1.1.1.3.cmml" xref="S3.SS2.p1.32.m32.1.1.1.3">𝑖</ci><apply id="S3.SS2.p1.32.m32.1.1.1.4.1.cmml" xref="S3.SS2.p1.32.m32.1.1.1.4.2"><csymbol cd="latexml" id="S3.SS2.p1.32.m32.1.1.1.4.1.1.cmml" xref="S3.SS2.p1.32.m32.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S3.SS2.p1.32.m32.1.1.1.1.cmml" xref="S3.SS2.p1.32.m32.1.1.1.1">𝑛</ci></apply></apply></apply><apply id="S3.SS2.p1.32.m32.3.3.1.1.2.cmml" xref="S3.SS2.p1.32.m32.3.3.1.1.1"><abs id="S3.SS2.p1.32.m32.3.3.1.1.2.1.cmml" xref="S3.SS2.p1.32.m32.3.3.1.1.1.2"></abs><apply id="S3.SS2.p1.32.m32.3.3.1.1.1.1.cmml" xref="S3.SS2.p1.32.m32.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.32.m32.3.3.1.1.1.1.1.cmml" xref="S3.SS2.p1.32.m32.3.3.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.32.m32.3.3.1.1.1.1.2.cmml" xref="S3.SS2.p1.32.m32.3.3.1.1.1.1.2">𝑎</ci><ci id="S3.SS2.p1.32.m32.3.3.1.1.1.1.3.cmml" xref="S3.SS2.p1.32.m32.3.3.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.32.m32.3c">\left\lVert a\right\rVert_{\infty}:=\max_{i\in[n]}|a_{i}|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.32.m32.3d">∥ italic_a ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT := roman_max start_POSTSUBSCRIPT italic_i ∈ [ italic_n ] end_POSTSUBSCRIPT | italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.14">Joint mixed strategies of agents and interceptors are denoted by <math alttext="\sigma\in\Delta((P_{st})^{n})" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mrow id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml">σ</mi><mo id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">∈</mo><mrow id="S3.SS2.p2.1.m1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.1.3" mathvariant="normal" xref="S3.SS2.p2.1.m1.1.1.1.3.cmml">Δ</mi><mo id="S3.SS2.p2.1.m1.1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p2.1.m1.1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS2.p2.1.m1.1.1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml"><mrow id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><in id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2"></in><ci id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3">𝜎</ci><apply id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1"><times id="S3.SS2.p2.1.m1.1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.1.2"></times><ci id="S3.SS2.p2.1.m1.1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.3">Δ</ci><apply id="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1">superscript</csymbol><apply id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3"><times id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><ci id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">\sigma\in\Delta((P_{st})^{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\tau\in\Delta(E^{k})" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mrow id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml"><mi id="S3.SS2.p2.2.m2.1.1.3" xref="S3.SS2.p2.2.m2.1.1.3.cmml">τ</mi><mo id="S3.SS2.p2.2.m2.1.1.2" xref="S3.SS2.p2.2.m2.1.1.2.cmml">∈</mo><mrow id="S3.SS2.p2.2.m2.1.1.1" xref="S3.SS2.p2.2.m2.1.1.1.cmml"><mi id="S3.SS2.p2.2.m2.1.1.1.3" mathvariant="normal" xref="S3.SS2.p2.2.m2.1.1.1.3.cmml">Δ</mi><mo id="S3.SS2.p2.2.m2.1.1.1.2" xref="S3.SS2.p2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p2.2.m2.1.1.1.1.1" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS2.p2.2.m2.1.1.1.1.1.1" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p2.2.m2.1.1.1.1.1.1.2" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.2.cmml">E</mi><mi id="S3.SS2.p2.2.m2.1.1.1.1.1.1.3" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.3.cmml">k</mi></msup><mo id="S3.SS2.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><apply id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1"><in id="S3.SS2.p2.2.m2.1.1.2.cmml" xref="S3.SS2.p2.2.m2.1.1.2"></in><ci id="S3.SS2.p2.2.m2.1.1.3.cmml" xref="S3.SS2.p2.2.m2.1.1.3">𝜏</ci><apply id="S3.SS2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1.1"><times id="S3.SS2.p2.2.m2.1.1.1.2.cmml" xref="S3.SS2.p2.2.m2.1.1.1.2"></times><ci id="S3.SS2.p2.2.m2.1.1.1.3.cmml" xref="S3.SS2.p2.2.m2.1.1.1.3">Δ</ci><apply id="S3.SS2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S3.SS2.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.2">𝐸</ci><ci id="S3.SS2.p2.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.2.m2.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">\tau\in\Delta(E^{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_τ ∈ roman_Δ ( italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> respectively, and <math alttext="\sigma_{j},\tau_{j}" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.2"><semantics id="S3.SS2.p2.3.m3.2a"><mrow id="S3.SS2.p2.3.m3.2.2.2" xref="S3.SS2.p2.3.m3.2.2.3.cmml"><msub id="S3.SS2.p2.3.m3.1.1.1.1" xref="S3.SS2.p2.3.m3.1.1.1.1.cmml"><mi id="S3.SS2.p2.3.m3.1.1.1.1.2" xref="S3.SS2.p2.3.m3.1.1.1.1.2.cmml">σ</mi><mi id="S3.SS2.p2.3.m3.1.1.1.1.3" xref="S3.SS2.p2.3.m3.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.SS2.p2.3.m3.2.2.2.3" xref="S3.SS2.p2.3.m3.2.2.3.cmml">,</mo><msub id="S3.SS2.p2.3.m3.2.2.2.2" xref="S3.SS2.p2.3.m3.2.2.2.2.cmml"><mi id="S3.SS2.p2.3.m3.2.2.2.2.2" xref="S3.SS2.p2.3.m3.2.2.2.2.2.cmml">τ</mi><mi id="S3.SS2.p2.3.m3.2.2.2.2.3" xref="S3.SS2.p2.3.m3.2.2.2.2.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.2b"><list id="S3.SS2.p2.3.m3.2.2.3.cmml" xref="S3.SS2.p2.3.m3.2.2.2"><apply id="S3.SS2.p2.3.m3.1.1.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.3.m3.1.1.1.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.3.m3.1.1.1.1.2.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.2">𝜎</ci><ci id="S3.SS2.p2.3.m3.1.1.1.1.3.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.3">𝑗</ci></apply><apply id="S3.SS2.p2.3.m3.2.2.2.2.cmml" xref="S3.SS2.p2.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p2.3.m3.2.2.2.2.1.cmml" xref="S3.SS2.p2.3.m3.2.2.2.2">subscript</csymbol><ci id="S3.SS2.p2.3.m3.2.2.2.2.2.cmml" xref="S3.SS2.p2.3.m3.2.2.2.2.2">𝜏</ci><ci id="S3.SS2.p2.3.m3.2.2.2.2.3.cmml" xref="S3.SS2.p2.3.m3.2.2.2.2.3">𝑗</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.2c">\sigma_{j},\tau_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.2d">italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are the marginal distributions of single agents and interceptors. If <math alttext="P_{st}=\left\{p_{1},...,p_{m}\right\}" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.3"><semantics id="S3.SS2.p2.4.m4.3a"><mrow id="S3.SS2.p2.4.m4.3.3" xref="S3.SS2.p2.4.m4.3.3.cmml"><msub id="S3.SS2.p2.4.m4.3.3.4" xref="S3.SS2.p2.4.m4.3.3.4.cmml"><mi id="S3.SS2.p2.4.m4.3.3.4.2" xref="S3.SS2.p2.4.m4.3.3.4.2.cmml">P</mi><mrow id="S3.SS2.p2.4.m4.3.3.4.3" xref="S3.SS2.p2.4.m4.3.3.4.3.cmml"><mi id="S3.SS2.p2.4.m4.3.3.4.3.2" xref="S3.SS2.p2.4.m4.3.3.4.3.2.cmml">s</mi><mo id="S3.SS2.p2.4.m4.3.3.4.3.1" xref="S3.SS2.p2.4.m4.3.3.4.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.4.m4.3.3.4.3.3" xref="S3.SS2.p2.4.m4.3.3.4.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.p2.4.m4.3.3.3" xref="S3.SS2.p2.4.m4.3.3.3.cmml">=</mo><mrow id="S3.SS2.p2.4.m4.3.3.2.2" xref="S3.SS2.p2.4.m4.3.3.2.3.cmml"><mo id="S3.SS2.p2.4.m4.3.3.2.2.3" xref="S3.SS2.p2.4.m4.3.3.2.3.cmml">{</mo><msub id="S3.SS2.p2.4.m4.2.2.1.1.1" xref="S3.SS2.p2.4.m4.2.2.1.1.1.cmml"><mi id="S3.SS2.p2.4.m4.2.2.1.1.1.2" xref="S3.SS2.p2.4.m4.2.2.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p2.4.m4.2.2.1.1.1.3" xref="S3.SS2.p2.4.m4.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p2.4.m4.3.3.2.2.4" xref="S3.SS2.p2.4.m4.3.3.2.3.cmml">,</mo><mi id="S3.SS2.p2.4.m4.1.1" mathvariant="normal" xref="S3.SS2.p2.4.m4.1.1.cmml">…</mi><mo id="S3.SS2.p2.4.m4.3.3.2.2.5" xref="S3.SS2.p2.4.m4.3.3.2.3.cmml">,</mo><msub id="S3.SS2.p2.4.m4.3.3.2.2.2" xref="S3.SS2.p2.4.m4.3.3.2.2.2.cmml"><mi id="S3.SS2.p2.4.m4.3.3.2.2.2.2" xref="S3.SS2.p2.4.m4.3.3.2.2.2.2.cmml">p</mi><mi id="S3.SS2.p2.4.m4.3.3.2.2.2.3" xref="S3.SS2.p2.4.m4.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S3.SS2.p2.4.m4.3.3.2.2.6" xref="S3.SS2.p2.4.m4.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.3b"><apply id="S3.SS2.p2.4.m4.3.3.cmml" xref="S3.SS2.p2.4.m4.3.3"><eq id="S3.SS2.p2.4.m4.3.3.3.cmml" xref="S3.SS2.p2.4.m4.3.3.3"></eq><apply id="S3.SS2.p2.4.m4.3.3.4.cmml" xref="S3.SS2.p2.4.m4.3.3.4"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.3.3.4.1.cmml" xref="S3.SS2.p2.4.m4.3.3.4">subscript</csymbol><ci id="S3.SS2.p2.4.m4.3.3.4.2.cmml" xref="S3.SS2.p2.4.m4.3.3.4.2">𝑃</ci><apply id="S3.SS2.p2.4.m4.3.3.4.3.cmml" xref="S3.SS2.p2.4.m4.3.3.4.3"><times id="S3.SS2.p2.4.m4.3.3.4.3.1.cmml" xref="S3.SS2.p2.4.m4.3.3.4.3.1"></times><ci id="S3.SS2.p2.4.m4.3.3.4.3.2.cmml" xref="S3.SS2.p2.4.m4.3.3.4.3.2">𝑠</ci><ci id="S3.SS2.p2.4.m4.3.3.4.3.3.cmml" xref="S3.SS2.p2.4.m4.3.3.4.3.3">𝑡</ci></apply></apply><set id="S3.SS2.p2.4.m4.3.3.2.3.cmml" xref="S3.SS2.p2.4.m4.3.3.2.2"><apply id="S3.SS2.p2.4.m4.2.2.1.1.1.cmml" xref="S3.SS2.p2.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS2.p2.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.4.m4.2.2.1.1.1.2.cmml" xref="S3.SS2.p2.4.m4.2.2.1.1.1.2">𝑝</ci><cn id="S3.SS2.p2.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS2.p2.4.m4.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1">…</ci><apply id="S3.SS2.p2.4.m4.3.3.2.2.2.cmml" xref="S3.SS2.p2.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.3.3.2.2.2.1.cmml" xref="S3.SS2.p2.4.m4.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.p2.4.m4.3.3.2.2.2.2.cmml" xref="S3.SS2.p2.4.m4.3.3.2.2.2.2">𝑝</ci><ci id="S3.SS2.p2.4.m4.3.3.2.2.2.3.cmml" xref="S3.SS2.p2.4.m4.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.3c">P_{st}=\left\{p_{1},...,p_{m}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.3d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT = { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math>, we will also write sometimes <math alttext="\sigma_{ij}=\sigma_{j}(p_{i})" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m5.1"><semantics id="S3.SS2.p2.5.m5.1a"><mrow id="S3.SS2.p2.5.m5.1.1" xref="S3.SS2.p2.5.m5.1.1.cmml"><msub id="S3.SS2.p2.5.m5.1.1.3" xref="S3.SS2.p2.5.m5.1.1.3.cmml"><mi id="S3.SS2.p2.5.m5.1.1.3.2" xref="S3.SS2.p2.5.m5.1.1.3.2.cmml">σ</mi><mrow id="S3.SS2.p2.5.m5.1.1.3.3" xref="S3.SS2.p2.5.m5.1.1.3.3.cmml"><mi id="S3.SS2.p2.5.m5.1.1.3.3.2" xref="S3.SS2.p2.5.m5.1.1.3.3.2.cmml">i</mi><mo id="S3.SS2.p2.5.m5.1.1.3.3.1" xref="S3.SS2.p2.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.5.m5.1.1.3.3.3" xref="S3.SS2.p2.5.m5.1.1.3.3.3.cmml">j</mi></mrow></msub><mo id="S3.SS2.p2.5.m5.1.1.2" xref="S3.SS2.p2.5.m5.1.1.2.cmml">=</mo><mrow id="S3.SS2.p2.5.m5.1.1.1" xref="S3.SS2.p2.5.m5.1.1.1.cmml"><msub id="S3.SS2.p2.5.m5.1.1.1.3" xref="S3.SS2.p2.5.m5.1.1.1.3.cmml"><mi id="S3.SS2.p2.5.m5.1.1.1.3.2" xref="S3.SS2.p2.5.m5.1.1.1.3.2.cmml">σ</mi><mi id="S3.SS2.p2.5.m5.1.1.1.3.3" xref="S3.SS2.p2.5.m5.1.1.1.3.3.cmml">j</mi></msub><mo id="S3.SS2.p2.5.m5.1.1.1.2" xref="S3.SS2.p2.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p2.5.m5.1.1.1.1.1" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p2.5.m5.1.1.1.1.1.1" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p2.5.m5.1.1.1.1.1.1.2" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.2.cmml">p</mi><mi id="S3.SS2.p2.5.m5.1.1.1.1.1.1.3" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS2.p2.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m5.1b"><apply id="S3.SS2.p2.5.m5.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1"><eq id="S3.SS2.p2.5.m5.1.1.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2"></eq><apply id="S3.SS2.p2.5.m5.1.1.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m5.1.1.3.1.cmml" xref="S3.SS2.p2.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS2.p2.5.m5.1.1.3.2.cmml" xref="S3.SS2.p2.5.m5.1.1.3.2">𝜎</ci><apply id="S3.SS2.p2.5.m5.1.1.3.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3"><times id="S3.SS2.p2.5.m5.1.1.3.3.1.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3.1"></times><ci id="S3.SS2.p2.5.m5.1.1.3.3.2.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3.2">𝑖</ci><ci id="S3.SS2.p2.5.m5.1.1.3.3.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3.3">𝑗</ci></apply></apply><apply id="S3.SS2.p2.5.m5.1.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1.1"><times id="S3.SS2.p2.5.m5.1.1.1.2.cmml" xref="S3.SS2.p2.5.m5.1.1.1.2"></times><apply id="S3.SS2.p2.5.m5.1.1.1.3.cmml" xref="S3.SS2.p2.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m5.1.1.1.3.1.cmml" xref="S3.SS2.p2.5.m5.1.1.1.3">subscript</csymbol><ci id="S3.SS2.p2.5.m5.1.1.1.3.2.cmml" xref="S3.SS2.p2.5.m5.1.1.1.3.2">𝜎</ci><ci id="S3.SS2.p2.5.m5.1.1.1.3.3.cmml" xref="S3.SS2.p2.5.m5.1.1.1.3.3">𝑗</ci></apply><apply id="S3.SS2.p2.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.2">𝑝</ci><ci id="S3.SS2.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m5.1c">\sigma_{ij}=\sigma_{j}(p_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m5.1d">italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. For <math alttext="\sigma\in\Delta((P_{st})^{n})" class="ltx_Math" display="inline" id="S3.SS2.p2.6.m6.1"><semantics id="S3.SS2.p2.6.m6.1a"><mrow id="S3.SS2.p2.6.m6.1.1" xref="S3.SS2.p2.6.m6.1.1.cmml"><mi id="S3.SS2.p2.6.m6.1.1.3" xref="S3.SS2.p2.6.m6.1.1.3.cmml">σ</mi><mo id="S3.SS2.p2.6.m6.1.1.2" xref="S3.SS2.p2.6.m6.1.1.2.cmml">∈</mo><mrow id="S3.SS2.p2.6.m6.1.1.1" xref="S3.SS2.p2.6.m6.1.1.1.cmml"><mi id="S3.SS2.p2.6.m6.1.1.1.3" mathvariant="normal" xref="S3.SS2.p2.6.m6.1.1.1.3.cmml">Δ</mi><mo id="S3.SS2.p2.6.m6.1.1.1.2" xref="S3.SS2.p2.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p2.6.m6.1.1.1.1.1" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.6.m6.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS2.p2.6.m6.1.1.1.1.1.1" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.cmml"><mrow id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.2" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.2" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.1" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.3" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.SS2.p2.6.m6.1.1.1.1.1.1.3" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="S3.SS2.p2.6.m6.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.6.m6.1b"><apply id="S3.SS2.p2.6.m6.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1"><in id="S3.SS2.p2.6.m6.1.1.2.cmml" xref="S3.SS2.p2.6.m6.1.1.2"></in><ci id="S3.SS2.p2.6.m6.1.1.3.cmml" xref="S3.SS2.p2.6.m6.1.1.3">𝜎</ci><apply id="S3.SS2.p2.6.m6.1.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1.1"><times id="S3.SS2.p2.6.m6.1.1.1.2.cmml" xref="S3.SS2.p2.6.m6.1.1.1.2"></times><ci id="S3.SS2.p2.6.m6.1.1.1.3.cmml" xref="S3.SS2.p2.6.m6.1.1.1.3">Δ</ci><apply id="S3.SS2.p2.6.m6.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1">superscript</csymbol><apply id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3"><times id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><ci id="S3.SS2.p2.6.m6.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.6.m6.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.6.m6.1c">\sigma\in\Delta((P_{st})^{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.6.m6.1d">italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="p\in P_{st}" class="ltx_Math" display="inline" id="S3.SS2.p2.7.m7.1"><semantics id="S3.SS2.p2.7.m7.1a"><mrow id="S3.SS2.p2.7.m7.1.1" xref="S3.SS2.p2.7.m7.1.1.cmml"><mi id="S3.SS2.p2.7.m7.1.1.2" xref="S3.SS2.p2.7.m7.1.1.2.cmml">p</mi><mo id="S3.SS2.p2.7.m7.1.1.1" xref="S3.SS2.p2.7.m7.1.1.1.cmml">∈</mo><msub id="S3.SS2.p2.7.m7.1.1.3" xref="S3.SS2.p2.7.m7.1.1.3.cmml"><mi id="S3.SS2.p2.7.m7.1.1.3.2" xref="S3.SS2.p2.7.m7.1.1.3.2.cmml">P</mi><mrow id="S3.SS2.p2.7.m7.1.1.3.3" xref="S3.SS2.p2.7.m7.1.1.3.3.cmml"><mi id="S3.SS2.p2.7.m7.1.1.3.3.2" xref="S3.SS2.p2.7.m7.1.1.3.3.2.cmml">s</mi><mo id="S3.SS2.p2.7.m7.1.1.3.3.1" xref="S3.SS2.p2.7.m7.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.7.m7.1.1.3.3.3" xref="S3.SS2.p2.7.m7.1.1.3.3.3.cmml">t</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.7.m7.1b"><apply id="S3.SS2.p2.7.m7.1.1.cmml" xref="S3.SS2.p2.7.m7.1.1"><in id="S3.SS2.p2.7.m7.1.1.1.cmml" xref="S3.SS2.p2.7.m7.1.1.1"></in><ci id="S3.SS2.p2.7.m7.1.1.2.cmml" xref="S3.SS2.p2.7.m7.1.1.2">𝑝</ci><apply id="S3.SS2.p2.7.m7.1.1.3.cmml" xref="S3.SS2.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.7.m7.1.1.3.1.cmml" xref="S3.SS2.p2.7.m7.1.1.3">subscript</csymbol><ci id="S3.SS2.p2.7.m7.1.1.3.2.cmml" xref="S3.SS2.p2.7.m7.1.1.3.2">𝑃</ci><apply id="S3.SS2.p2.7.m7.1.1.3.3.cmml" xref="S3.SS2.p2.7.m7.1.1.3.3"><times id="S3.SS2.p2.7.m7.1.1.3.3.1.cmml" xref="S3.SS2.p2.7.m7.1.1.3.3.1"></times><ci id="S3.SS2.p2.7.m7.1.1.3.3.2.cmml" xref="S3.SS2.p2.7.m7.1.1.3.3.2">𝑠</ci><ci id="S3.SS2.p2.7.m7.1.1.3.3.3.cmml" xref="S3.SS2.p2.7.m7.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.7.m7.1c">p\in P_{st}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.7.m7.1d">italic_p ∈ italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="\sigma(p)=\Pr_{a\sim\sigma}(p\in a)" class="ltx_Math" display="inline" id="S3.SS2.p2.8.m8.3"><semantics id="S3.SS2.p2.8.m8.3a"><mrow id="S3.SS2.p2.8.m8.3.3" xref="S3.SS2.p2.8.m8.3.3.cmml"><mrow id="S3.SS2.p2.8.m8.3.3.4" xref="S3.SS2.p2.8.m8.3.3.4.cmml"><mi id="S3.SS2.p2.8.m8.3.3.4.2" xref="S3.SS2.p2.8.m8.3.3.4.2.cmml">σ</mi><mo id="S3.SS2.p2.8.m8.3.3.4.1" xref="S3.SS2.p2.8.m8.3.3.4.1.cmml">⁢</mo><mrow id="S3.SS2.p2.8.m8.3.3.4.3.2" xref="S3.SS2.p2.8.m8.3.3.4.cmml"><mo id="S3.SS2.p2.8.m8.3.3.4.3.2.1" stretchy="false" xref="S3.SS2.p2.8.m8.3.3.4.cmml">(</mo><mi id="S3.SS2.p2.8.m8.1.1" xref="S3.SS2.p2.8.m8.1.1.cmml">p</mi><mo id="S3.SS2.p2.8.m8.3.3.4.3.2.2" stretchy="false" xref="S3.SS2.p2.8.m8.3.3.4.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p2.8.m8.3.3.3" xref="S3.SS2.p2.8.m8.3.3.3.cmml">=</mo><mrow id="S3.SS2.p2.8.m8.3.3.2.2" xref="S3.SS2.p2.8.m8.3.3.2.3.cmml"><msub id="S3.SS2.p2.8.m8.2.2.1.1.1" xref="S3.SS2.p2.8.m8.2.2.1.1.1.cmml"><mi id="S3.SS2.p2.8.m8.2.2.1.1.1.2" xref="S3.SS2.p2.8.m8.2.2.1.1.1.2.cmml">Pr</mi><mrow id="S3.SS2.p2.8.m8.2.2.1.1.1.3" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3.cmml"><mi id="S3.SS2.p2.8.m8.2.2.1.1.1.3.2" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3.2.cmml">a</mi><mo id="S3.SS2.p2.8.m8.2.2.1.1.1.3.1" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3.1.cmml">∼</mo><mi id="S3.SS2.p2.8.m8.2.2.1.1.1.3.3" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3.3.cmml">σ</mi></mrow></msub><mo id="S3.SS2.p2.8.m8.3.3.2.2a" xref="S3.SS2.p2.8.m8.3.3.2.3.cmml">⁡</mo><mrow id="S3.SS2.p2.8.m8.3.3.2.2.2" xref="S3.SS2.p2.8.m8.3.3.2.3.cmml"><mo id="S3.SS2.p2.8.m8.3.3.2.2.2.2" stretchy="false" xref="S3.SS2.p2.8.m8.3.3.2.3.cmml">(</mo><mrow id="S3.SS2.p2.8.m8.3.3.2.2.2.1" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1.cmml"><mi id="S3.SS2.p2.8.m8.3.3.2.2.2.1.2" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1.2.cmml">p</mi><mo id="S3.SS2.p2.8.m8.3.3.2.2.2.1.1" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1.1.cmml">∈</mo><mi id="S3.SS2.p2.8.m8.3.3.2.2.2.1.3" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1.3.cmml">a</mi></mrow><mo id="S3.SS2.p2.8.m8.3.3.2.2.2.3" stretchy="false" xref="S3.SS2.p2.8.m8.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.8.m8.3b"><apply id="S3.SS2.p2.8.m8.3.3.cmml" xref="S3.SS2.p2.8.m8.3.3"><eq id="S3.SS2.p2.8.m8.3.3.3.cmml" xref="S3.SS2.p2.8.m8.3.3.3"></eq><apply id="S3.SS2.p2.8.m8.3.3.4.cmml" xref="S3.SS2.p2.8.m8.3.3.4"><times id="S3.SS2.p2.8.m8.3.3.4.1.cmml" xref="S3.SS2.p2.8.m8.3.3.4.1"></times><ci id="S3.SS2.p2.8.m8.3.3.4.2.cmml" xref="S3.SS2.p2.8.m8.3.3.4.2">𝜎</ci><ci id="S3.SS2.p2.8.m8.1.1.cmml" xref="S3.SS2.p2.8.m8.1.1">𝑝</ci></apply><apply id="S3.SS2.p2.8.m8.3.3.2.3.cmml" xref="S3.SS2.p2.8.m8.3.3.2.2"><apply id="S3.SS2.p2.8.m8.2.2.1.1.1.cmml" xref="S3.SS2.p2.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.8.m8.2.2.1.1.1.1.cmml" xref="S3.SS2.p2.8.m8.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.8.m8.2.2.1.1.1.2.cmml" xref="S3.SS2.p2.8.m8.2.2.1.1.1.2">Pr</ci><apply id="S3.SS2.p2.8.m8.2.2.1.1.1.3.cmml" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3"><csymbol cd="latexml" id="S3.SS2.p2.8.m8.2.2.1.1.1.3.1.cmml" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3.1">similar-to</csymbol><ci id="S3.SS2.p2.8.m8.2.2.1.1.1.3.2.cmml" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3.2">𝑎</ci><ci id="S3.SS2.p2.8.m8.2.2.1.1.1.3.3.cmml" xref="S3.SS2.p2.8.m8.2.2.1.1.1.3.3">𝜎</ci></apply></apply><apply id="S3.SS2.p2.8.m8.3.3.2.2.2.1.cmml" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1"><in id="S3.SS2.p2.8.m8.3.3.2.2.2.1.1.cmml" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1.1"></in><ci id="S3.SS2.p2.8.m8.3.3.2.2.2.1.2.cmml" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1.2">𝑝</ci><ci id="S3.SS2.p2.8.m8.3.3.2.2.2.1.3.cmml" xref="S3.SS2.p2.8.m8.3.3.2.2.2.1.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.8.m8.3c">\sigma(p)=\Pr_{a\sim\sigma}(p\in a)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.8.m8.3d">italic_σ ( italic_p ) = roman_Pr start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( italic_p ∈ italic_a )</annotation></semantics></math>. Note that for <math alttext="n=1" class="ltx_Math" display="inline" id="S3.SS2.p2.9.m9.1"><semantics id="S3.SS2.p2.9.m9.1a"><mrow id="S3.SS2.p2.9.m9.1.1" xref="S3.SS2.p2.9.m9.1.1.cmml"><mi id="S3.SS2.p2.9.m9.1.1.2" xref="S3.SS2.p2.9.m9.1.1.2.cmml">n</mi><mo id="S3.SS2.p2.9.m9.1.1.1" xref="S3.SS2.p2.9.m9.1.1.1.cmml">=</mo><mn id="S3.SS2.p2.9.m9.1.1.3" xref="S3.SS2.p2.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.9.m9.1b"><apply id="S3.SS2.p2.9.m9.1.1.cmml" xref="S3.SS2.p2.9.m9.1.1"><eq id="S3.SS2.p2.9.m9.1.1.1.cmml" xref="S3.SS2.p2.9.m9.1.1.1"></eq><ci id="S3.SS2.p2.9.m9.1.1.2.cmml" xref="S3.SS2.p2.9.m9.1.1.2">𝑛</ci><cn id="S3.SS2.p2.9.m9.1.1.3.cmml" type="integer" xref="S3.SS2.p2.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.9.m9.1c">n=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.9.m9.1d">italic_n = 1</annotation></semantics></math> this definition and the original definition of <math alttext="\sigma(p)" class="ltx_Math" display="inline" id="S3.SS2.p2.10.m10.1"><semantics id="S3.SS2.p2.10.m10.1a"><mrow id="S3.SS2.p2.10.m10.1.2" xref="S3.SS2.p2.10.m10.1.2.cmml"><mi id="S3.SS2.p2.10.m10.1.2.2" xref="S3.SS2.p2.10.m10.1.2.2.cmml">σ</mi><mo id="S3.SS2.p2.10.m10.1.2.1" xref="S3.SS2.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p2.10.m10.1.2.3.2" xref="S3.SS2.p2.10.m10.1.2.cmml"><mo id="S3.SS2.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S3.SS2.p2.10.m10.1.2.cmml">(</mo><mi id="S3.SS2.p2.10.m10.1.1" xref="S3.SS2.p2.10.m10.1.1.cmml">p</mi><mo id="S3.SS2.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S3.SS2.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.10.m10.1b"><apply id="S3.SS2.p2.10.m10.1.2.cmml" xref="S3.SS2.p2.10.m10.1.2"><times id="S3.SS2.p2.10.m10.1.2.1.cmml" xref="S3.SS2.p2.10.m10.1.2.1"></times><ci id="S3.SS2.p2.10.m10.1.2.2.cmml" xref="S3.SS2.p2.10.m10.1.2.2">𝜎</ci><ci id="S3.SS2.p2.10.m10.1.1.cmml" xref="S3.SS2.p2.10.m10.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.10.m10.1c">\sigma(p)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.10.m10.1d">italic_σ ( italic_p )</annotation></semantics></math> coincide. Let <math alttext="\sigma^{E}" class="ltx_Math" display="inline" id="S3.SS2.p2.11.m11.1"><semantics id="S3.SS2.p2.11.m11.1a"><msup id="S3.SS2.p2.11.m11.1.1" xref="S3.SS2.p2.11.m11.1.1.cmml"><mi id="S3.SS2.p2.11.m11.1.1.2" xref="S3.SS2.p2.11.m11.1.1.2.cmml">σ</mi><mi id="S3.SS2.p2.11.m11.1.1.3" xref="S3.SS2.p2.11.m11.1.1.3.cmml">E</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.11.m11.1b"><apply id="S3.SS2.p2.11.m11.1.1.cmml" xref="S3.SS2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.11.m11.1.1.1.cmml" xref="S3.SS2.p2.11.m11.1.1">superscript</csymbol><ci id="S3.SS2.p2.11.m11.1.1.2.cmml" xref="S3.SS2.p2.11.m11.1.1.2">𝜎</ci><ci id="S3.SS2.p2.11.m11.1.1.3.cmml" xref="S3.SS2.p2.11.m11.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.11.m11.1c">\sigma^{E}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.11.m11.1d">italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT</annotation></semantics></math> be a distribution on subsets of <math alttext="E" class="ltx_Math" display="inline" id="S3.SS2.p2.12.m12.1"><semantics id="S3.SS2.p2.12.m12.1a"><mi id="S3.SS2.p2.12.m12.1.1" xref="S3.SS2.p2.12.m12.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.12.m12.1b"><ci id="S3.SS2.p2.12.m12.1.1.cmml" xref="S3.SS2.p2.12.m12.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.12.m12.1c">E</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.12.m12.1d">italic_E</annotation></semantics></math>, where <math alttext="\sigma^{E}(e)=\Pr_{a\sim\sigma}(e\in\bigcup a)" class="ltx_Math" display="inline" id="S3.SS2.p2.13.m13.3"><semantics id="S3.SS2.p2.13.m13.3a"><mrow id="S3.SS2.p2.13.m13.3.3" xref="S3.SS2.p2.13.m13.3.3.cmml"><mrow id="S3.SS2.p2.13.m13.3.3.4" xref="S3.SS2.p2.13.m13.3.3.4.cmml"><msup id="S3.SS2.p2.13.m13.3.3.4.2" xref="S3.SS2.p2.13.m13.3.3.4.2.cmml"><mi id="S3.SS2.p2.13.m13.3.3.4.2.2" xref="S3.SS2.p2.13.m13.3.3.4.2.2.cmml">σ</mi><mi id="S3.SS2.p2.13.m13.3.3.4.2.3" xref="S3.SS2.p2.13.m13.3.3.4.2.3.cmml">E</mi></msup><mo id="S3.SS2.p2.13.m13.3.3.4.1" xref="S3.SS2.p2.13.m13.3.3.4.1.cmml">⁢</mo><mrow id="S3.SS2.p2.13.m13.3.3.4.3.2" xref="S3.SS2.p2.13.m13.3.3.4.cmml"><mo id="S3.SS2.p2.13.m13.3.3.4.3.2.1" stretchy="false" xref="S3.SS2.p2.13.m13.3.3.4.cmml">(</mo><mi id="S3.SS2.p2.13.m13.1.1" xref="S3.SS2.p2.13.m13.1.1.cmml">e</mi><mo id="S3.SS2.p2.13.m13.3.3.4.3.2.2" stretchy="false" xref="S3.SS2.p2.13.m13.3.3.4.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p2.13.m13.3.3.3" xref="S3.SS2.p2.13.m13.3.3.3.cmml">=</mo><mrow id="S3.SS2.p2.13.m13.3.3.2.2" xref="S3.SS2.p2.13.m13.3.3.2.3.cmml"><msub id="S3.SS2.p2.13.m13.2.2.1.1.1" xref="S3.SS2.p2.13.m13.2.2.1.1.1.cmml"><mi id="S3.SS2.p2.13.m13.2.2.1.1.1.2" xref="S3.SS2.p2.13.m13.2.2.1.1.1.2.cmml">Pr</mi><mrow id="S3.SS2.p2.13.m13.2.2.1.1.1.3" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3.cmml"><mi id="S3.SS2.p2.13.m13.2.2.1.1.1.3.2" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3.2.cmml">a</mi><mo id="S3.SS2.p2.13.m13.2.2.1.1.1.3.1" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3.1.cmml">∼</mo><mi id="S3.SS2.p2.13.m13.2.2.1.1.1.3.3" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3.3.cmml">σ</mi></mrow></msub><mo id="S3.SS2.p2.13.m13.3.3.2.2a" xref="S3.SS2.p2.13.m13.3.3.2.3.cmml">⁡</mo><mrow id="S3.SS2.p2.13.m13.3.3.2.2.2" xref="S3.SS2.p2.13.m13.3.3.2.3.cmml"><mo id="S3.SS2.p2.13.m13.3.3.2.2.2.2" stretchy="false" xref="S3.SS2.p2.13.m13.3.3.2.3.cmml">(</mo><mrow id="S3.SS2.p2.13.m13.3.3.2.2.2.1" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.cmml"><mi id="S3.SS2.p2.13.m13.3.3.2.2.2.1.2" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.2.cmml">e</mi><mo id="S3.SS2.p2.13.m13.3.3.2.2.2.1.1" rspace="0.111em" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.1.cmml">∈</mo><mrow id="S3.SS2.p2.13.m13.3.3.2.2.2.1.3" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.cmml"><mo id="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.1" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.1.cmml">⋃</mo><mi id="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.2" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.2.cmml">a</mi></mrow></mrow><mo id="S3.SS2.p2.13.m13.3.3.2.2.2.3" stretchy="false" xref="S3.SS2.p2.13.m13.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.13.m13.3b"><apply id="S3.SS2.p2.13.m13.3.3.cmml" xref="S3.SS2.p2.13.m13.3.3"><eq id="S3.SS2.p2.13.m13.3.3.3.cmml" xref="S3.SS2.p2.13.m13.3.3.3"></eq><apply id="S3.SS2.p2.13.m13.3.3.4.cmml" xref="S3.SS2.p2.13.m13.3.3.4"><times id="S3.SS2.p2.13.m13.3.3.4.1.cmml" xref="S3.SS2.p2.13.m13.3.3.4.1"></times><apply id="S3.SS2.p2.13.m13.3.3.4.2.cmml" xref="S3.SS2.p2.13.m13.3.3.4.2"><csymbol cd="ambiguous" id="S3.SS2.p2.13.m13.3.3.4.2.1.cmml" xref="S3.SS2.p2.13.m13.3.3.4.2">superscript</csymbol><ci id="S3.SS2.p2.13.m13.3.3.4.2.2.cmml" xref="S3.SS2.p2.13.m13.3.3.4.2.2">𝜎</ci><ci id="S3.SS2.p2.13.m13.3.3.4.2.3.cmml" xref="S3.SS2.p2.13.m13.3.3.4.2.3">𝐸</ci></apply><ci id="S3.SS2.p2.13.m13.1.1.cmml" xref="S3.SS2.p2.13.m13.1.1">𝑒</ci></apply><apply id="S3.SS2.p2.13.m13.3.3.2.3.cmml" xref="S3.SS2.p2.13.m13.3.3.2.2"><apply id="S3.SS2.p2.13.m13.2.2.1.1.1.cmml" xref="S3.SS2.p2.13.m13.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.13.m13.2.2.1.1.1.1.cmml" xref="S3.SS2.p2.13.m13.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.13.m13.2.2.1.1.1.2.cmml" xref="S3.SS2.p2.13.m13.2.2.1.1.1.2">Pr</ci><apply id="S3.SS2.p2.13.m13.2.2.1.1.1.3.cmml" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3"><csymbol cd="latexml" id="S3.SS2.p2.13.m13.2.2.1.1.1.3.1.cmml" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3.1">similar-to</csymbol><ci id="S3.SS2.p2.13.m13.2.2.1.1.1.3.2.cmml" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3.2">𝑎</ci><ci id="S3.SS2.p2.13.m13.2.2.1.1.1.3.3.cmml" xref="S3.SS2.p2.13.m13.2.2.1.1.1.3.3">𝜎</ci></apply></apply><apply id="S3.SS2.p2.13.m13.3.3.2.2.2.1.cmml" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1"><in id="S3.SS2.p2.13.m13.3.3.2.2.2.1.1.cmml" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.1"></in><ci id="S3.SS2.p2.13.m13.3.3.2.2.2.1.2.cmml" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.2">𝑒</ci><apply id="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.cmml" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.3"><union id="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.1.cmml" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.1"></union><ci id="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.2.cmml" xref="S3.SS2.p2.13.m13.3.3.2.2.2.1.3.2">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.13.m13.3c">\sigma^{E}(e)=\Pr_{a\sim\sigma}(e\in\bigcup a)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.13.m13.3d">italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e ) = roman_Pr start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( italic_e ∈ ⋃ italic_a )</annotation></semantics></math>. Then, let <math alttext="\beta(\sigma)=\max_{e\in E}\sigma^{E}(e)" class="ltx_Math" display="inline" id="S3.SS2.p2.14.m14.2"><semantics id="S3.SS2.p2.14.m14.2a"><mrow id="S3.SS2.p2.14.m14.2.3" xref="S3.SS2.p2.14.m14.2.3.cmml"><mrow id="S3.SS2.p2.14.m14.2.3.2" xref="S3.SS2.p2.14.m14.2.3.2.cmml"><mi id="S3.SS2.p2.14.m14.2.3.2.2" xref="S3.SS2.p2.14.m14.2.3.2.2.cmml">β</mi><mo id="S3.SS2.p2.14.m14.2.3.2.1" xref="S3.SS2.p2.14.m14.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS2.p2.14.m14.2.3.2.3.2" xref="S3.SS2.p2.14.m14.2.3.2.cmml"><mo id="S3.SS2.p2.14.m14.2.3.2.3.2.1" stretchy="false" xref="S3.SS2.p2.14.m14.2.3.2.cmml">(</mo><mi id="S3.SS2.p2.14.m14.1.1" xref="S3.SS2.p2.14.m14.1.1.cmml">σ</mi><mo id="S3.SS2.p2.14.m14.2.3.2.3.2.2" stretchy="false" xref="S3.SS2.p2.14.m14.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p2.14.m14.2.3.1" xref="S3.SS2.p2.14.m14.2.3.1.cmml">=</mo><mrow id="S3.SS2.p2.14.m14.2.3.3" xref="S3.SS2.p2.14.m14.2.3.3.cmml"><mrow id="S3.SS2.p2.14.m14.2.3.3.2" xref="S3.SS2.p2.14.m14.2.3.3.2.cmml"><msub id="S3.SS2.p2.14.m14.2.3.3.2.1" xref="S3.SS2.p2.14.m14.2.3.3.2.1.cmml"><mi id="S3.SS2.p2.14.m14.2.3.3.2.1.2" xref="S3.SS2.p2.14.m14.2.3.3.2.1.2.cmml">max</mi><mrow id="S3.SS2.p2.14.m14.2.3.3.2.1.3" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3.cmml"><mi id="S3.SS2.p2.14.m14.2.3.3.2.1.3.2" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3.2.cmml">e</mi><mo id="S3.SS2.p2.14.m14.2.3.3.2.1.3.1" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3.1.cmml">∈</mo><mi id="S3.SS2.p2.14.m14.2.3.3.2.1.3.3" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3.3.cmml">E</mi></mrow></msub><mo id="S3.SS2.p2.14.m14.2.3.3.2a" lspace="0.167em" xref="S3.SS2.p2.14.m14.2.3.3.2.cmml">⁡</mo><msup id="S3.SS2.p2.14.m14.2.3.3.2.2" xref="S3.SS2.p2.14.m14.2.3.3.2.2.cmml"><mi id="S3.SS2.p2.14.m14.2.3.3.2.2.2" xref="S3.SS2.p2.14.m14.2.3.3.2.2.2.cmml">σ</mi><mi id="S3.SS2.p2.14.m14.2.3.3.2.2.3" xref="S3.SS2.p2.14.m14.2.3.3.2.2.3.cmml">E</mi></msup></mrow><mo id="S3.SS2.p2.14.m14.2.3.3.1" xref="S3.SS2.p2.14.m14.2.3.3.1.cmml">⁢</mo><mrow id="S3.SS2.p2.14.m14.2.3.3.3.2" xref="S3.SS2.p2.14.m14.2.3.3.cmml"><mo id="S3.SS2.p2.14.m14.2.3.3.3.2.1" stretchy="false" xref="S3.SS2.p2.14.m14.2.3.3.cmml">(</mo><mi id="S3.SS2.p2.14.m14.2.2" xref="S3.SS2.p2.14.m14.2.2.cmml">e</mi><mo id="S3.SS2.p2.14.m14.2.3.3.3.2.2" stretchy="false" xref="S3.SS2.p2.14.m14.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.14.m14.2b"><apply id="S3.SS2.p2.14.m14.2.3.cmml" xref="S3.SS2.p2.14.m14.2.3"><eq id="S3.SS2.p2.14.m14.2.3.1.cmml" xref="S3.SS2.p2.14.m14.2.3.1"></eq><apply id="S3.SS2.p2.14.m14.2.3.2.cmml" xref="S3.SS2.p2.14.m14.2.3.2"><times id="S3.SS2.p2.14.m14.2.3.2.1.cmml" xref="S3.SS2.p2.14.m14.2.3.2.1"></times><ci id="S3.SS2.p2.14.m14.2.3.2.2.cmml" xref="S3.SS2.p2.14.m14.2.3.2.2">𝛽</ci><ci id="S3.SS2.p2.14.m14.1.1.cmml" xref="S3.SS2.p2.14.m14.1.1">𝜎</ci></apply><apply id="S3.SS2.p2.14.m14.2.3.3.cmml" xref="S3.SS2.p2.14.m14.2.3.3"><times id="S3.SS2.p2.14.m14.2.3.3.1.cmml" xref="S3.SS2.p2.14.m14.2.3.3.1"></times><apply id="S3.SS2.p2.14.m14.2.3.3.2.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2"><apply id="S3.SS2.p2.14.m14.2.3.3.2.1.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.1"><csymbol cd="ambiguous" id="S3.SS2.p2.14.m14.2.3.3.2.1.1.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.1">subscript</csymbol><max id="S3.SS2.p2.14.m14.2.3.3.2.1.2.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.1.2"></max><apply id="S3.SS2.p2.14.m14.2.3.3.2.1.3.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3"><in id="S3.SS2.p2.14.m14.2.3.3.2.1.3.1.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3.1"></in><ci id="S3.SS2.p2.14.m14.2.3.3.2.1.3.2.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3.2">𝑒</ci><ci id="S3.SS2.p2.14.m14.2.3.3.2.1.3.3.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.1.3.3">𝐸</ci></apply></apply><apply id="S3.SS2.p2.14.m14.2.3.3.2.2.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS2.p2.14.m14.2.3.3.2.2.1.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.2">superscript</csymbol><ci id="S3.SS2.p2.14.m14.2.3.3.2.2.2.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.2.2">𝜎</ci><ci id="S3.SS2.p2.14.m14.2.3.3.2.2.3.cmml" xref="S3.SS2.p2.14.m14.2.3.3.2.2.3">𝐸</ci></apply></apply><ci id="S3.SS2.p2.14.m14.2.2.cmml" xref="S3.SS2.p2.14.m14.2.2">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.14.m14.2c">\beta(\sigma)=\max_{e\in E}\sigma^{E}(e)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.14.m14.2d">italic_β ( italic_σ ) = roman_max start_POSTSUBSCRIPT italic_e ∈ italic_E end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.7">When the agents are correlated, their set of mixed strategies is <math alttext="\Delta((P_{st})^{n})" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mrow id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.1.1.3" mathvariant="normal" xref="S3.SS2.p3.1.m1.1.1.3.cmml">Δ</mi><mo id="S3.SS2.p3.1.m1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p3.1.m1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.1.m1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p3.1.m1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS2.p3.1.m1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.cmml"><mrow id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3.3" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.SS2.p3.1.m1.1.1.1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="S3.SS2.p3.1.m1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1"><times id="S3.SS2.p3.1.m1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.2"></times><ci id="S3.SS2.p3.1.m1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3">Δ</ci><apply id="S3.SS2.p3.1.m1.1.1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1">superscript</csymbol><apply 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start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>. When they don’t, it is only <math alttext="(\Delta(P_{st}))^{n}" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><msup id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><mrow id="S3.SS2.p3.2.m2.1.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p3.2.m2.1.1.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS2.p3.2.m2.1.1.1.1.1.3" mathvariant="normal" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3.cmml">Δ</mi><mo id="S3.SS2.p3.2.m2.1.1.1.1.1.2" xref="S3.SS2.p3.2.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.2" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.2" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.3" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m2.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1">superscript</csymbol><apply id="S3.SS2.p3.2.m2.1.1.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1"><times id="S3.SS2.p3.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.2"></times><ci id="S3.SS2.p3.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3">Δ</ci><apply id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3"><times id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.1"></times><ci id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="S3.SS2.p3.2.m2.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">(\Delta(P_{st}))^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. The interceptors can always share their randomness. <br class="ltx_break"/> <br class="ltx_break"/><span class="ltx_text ltx_font_bold" id="S3.SS2.p3.7.1">The cost functions </span> The cost function of the game is <math alttext="C:(P_{st})^{*}\times E^{*}\times(0,\infty)\to[0,\infty]" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.5"><semantics id="S3.SS2.p3.3.m3.5a"><mrow id="S3.SS2.p3.3.m3.5.5" xref="S3.SS2.p3.3.m3.5.5.cmml"><mi id="S3.SS2.p3.3.m3.5.5.3" xref="S3.SS2.p3.3.m3.5.5.3.cmml">C</mi><mo id="S3.SS2.p3.3.m3.5.5.2" lspace="0.278em" rspace="0.278em" xref="S3.SS2.p3.3.m3.5.5.2.cmml">:</mo><mrow id="S3.SS2.p3.3.m3.5.5.1" xref="S3.SS2.p3.3.m3.5.5.1.cmml"><mrow id="S3.SS2.p3.3.m3.5.5.1.1" xref="S3.SS2.p3.3.m3.5.5.1.1.cmml"><msup id="S3.SS2.p3.3.m3.5.5.1.1.1" xref="S3.SS2.p3.3.m3.5.5.1.1.1.cmml"><mrow id="S3.SS2.p3.3.m3.5.5.1.1.1.1.1" xref="S3.SS2.p3.3.m3.5.5.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.3.m3.5.5.1.1.1.1.1.2" stretchy="false" 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encoding="application/x-tex" id="S3.SS2.p3.3.m3.5c">C:(P_{st})^{*}\times E^{*}\times(0,\infty)\to[0,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.5d">italic_C : ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT × italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT × ( 0 , ∞ ) → [ 0 , ∞ ]</annotation></semantics></math>, where <math alttext="C(a,d,\alpha)" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.3"><semantics id="S3.SS2.p3.4.m4.3a"><mrow id="S3.SS2.p3.4.m4.3.4" xref="S3.SS2.p3.4.m4.3.4.cmml"><mi id="S3.SS2.p3.4.m4.3.4.2" xref="S3.SS2.p3.4.m4.3.4.2.cmml">C</mi><mo id="S3.SS2.p3.4.m4.3.4.1" xref="S3.SS2.p3.4.m4.3.4.1.cmml">⁢</mo><mrow id="S3.SS2.p3.4.m4.3.4.3.2" xref="S3.SS2.p3.4.m4.3.4.3.1.cmml"><mo id="S3.SS2.p3.4.m4.3.4.3.2.1" stretchy="false" xref="S3.SS2.p3.4.m4.3.4.3.1.cmml">(</mo><mi id="S3.SS2.p3.4.m4.1.1" xref="S3.SS2.p3.4.m4.1.1.cmml">a</mi><mo id="S3.SS2.p3.4.m4.3.4.3.2.2" xref="S3.SS2.p3.4.m4.3.4.3.1.cmml">,</mo><mi id="S3.SS2.p3.4.m4.2.2" xref="S3.SS2.p3.4.m4.2.2.cmml">d</mi><mo id="S3.SS2.p3.4.m4.3.4.3.2.3" xref="S3.SS2.p3.4.m4.3.4.3.1.cmml">,</mo><mi id="S3.SS2.p3.4.m4.3.3" xref="S3.SS2.p3.4.m4.3.3.cmml">α</mi><mo id="S3.SS2.p3.4.m4.3.4.3.2.4" stretchy="false" xref="S3.SS2.p3.4.m4.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.3b"><apply id="S3.SS2.p3.4.m4.3.4.cmml" xref="S3.SS2.p3.4.m4.3.4"><times id="S3.SS2.p3.4.m4.3.4.1.cmml" xref="S3.SS2.p3.4.m4.3.4.1"></times><ci id="S3.SS2.p3.4.m4.3.4.2.cmml" xref="S3.SS2.p3.4.m4.3.4.2">𝐶</ci><vector id="S3.SS2.p3.4.m4.3.4.3.1.cmml" xref="S3.SS2.p3.4.m4.3.4.3.2"><ci id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1">𝑎</ci><ci id="S3.SS2.p3.4.m4.2.2.cmml" xref="S3.SS2.p3.4.m4.2.2">𝑑</ci><ci id="S3.SS2.p3.4.m4.3.3.cmml" xref="S3.SS2.p3.4.m4.3.3">𝛼</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.3c">C(a,d,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.3d">italic_C ( italic_a , italic_d , italic_α )</annotation></semantics></math> is the cost to the agents in the corresponding outcome. More formally, <math alttext="-C" class="ltx_Math" display="inline" id="S3.SS2.p3.5.m5.1"><semantics id="S3.SS2.p3.5.m5.1a"><mrow id="S3.SS2.p3.5.m5.1.1" xref="S3.SS2.p3.5.m5.1.1.cmml"><mo id="S3.SS2.p3.5.m5.1.1a" xref="S3.SS2.p3.5.m5.1.1.cmml">−</mo><mi id="S3.SS2.p3.5.m5.1.1.2" xref="S3.SS2.p3.5.m5.1.1.2.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.5.m5.1b"><apply id="S3.SS2.p3.5.m5.1.1.cmml" xref="S3.SS2.p3.5.m5.1.1"><minus id="S3.SS2.p3.5.m5.1.1.1.cmml" xref="S3.SS2.p3.5.m5.1.1"></minus><ci id="S3.SS2.p3.5.m5.1.1.2.cmml" xref="S3.SS2.p3.5.m5.1.1.2">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.5.m5.1c">-C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.5.m5.1d">- italic_C</annotation></semantics></math> is the utility of every agent, while <math alttext="C" class="ltx_Math" display="inline" id="S3.SS2.p3.6.m6.1"><semantics id="S3.SS2.p3.6.m6.1a"><mi id="S3.SS2.p3.6.m6.1.1" xref="S3.SS2.p3.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.6.m6.1b"><ci id="S3.SS2.p3.6.m6.1.1.cmml" xref="S3.SS2.p3.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.6.m6.1d">italic_C</annotation></semantics></math> is the utility of every interceptor. <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS2.p3.7.m7.1"><semantics id="S3.SS2.p3.7.m7.1a"><mi id="S3.SS2.p3.7.m7.1.1" xref="S3.SS2.p3.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.7.m7.1b"><ci id="S3.SS2.p3.7.m7.1.1.cmml" xref="S3.SS2.p3.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.7.m7.1d">italic_α</annotation></semantics></math> will be referred to as either additional cost or delay imposed on edges by interceptors. The two cost functions we will consider for a large part of the paper are the SUM cost:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx1"> <tbody id="S3.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C(a,d,\alpha)=\left|\bigcup a\right|+\alpha\sum_{j\leq k}1\left% \{d_{j}\in\bigcup a\right\}" class="ltx_Math" display="inline" id="S3.Ex2.m1.5"><semantics id="S3.Ex2.m1.5a"><mrow id="S3.Ex2.m1.5.5" xref="S3.Ex2.m1.5.5.cmml"><mrow id="S3.Ex2.m1.5.5.4" xref="S3.Ex2.m1.5.5.4.cmml"><mi id="S3.Ex2.m1.5.5.4.2" xref="S3.Ex2.m1.5.5.4.2.cmml">C</mi><mo id="S3.Ex2.m1.5.5.4.1" xref="S3.Ex2.m1.5.5.4.1.cmml">⁢</mo><mrow id="S3.Ex2.m1.5.5.4.3.2" xref="S3.Ex2.m1.5.5.4.3.1.cmml"><mo id="S3.Ex2.m1.5.5.4.3.2.1" stretchy="false" xref="S3.Ex2.m1.5.5.4.3.1.cmml">(</mo><mi id="S3.Ex2.m1.1.1" xref="S3.Ex2.m1.1.1.cmml">a</mi><mo 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start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ ⋃ italic_a }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(SUM)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p3.9">and the MAX cost:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx2"> <tbody id="S3.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C(a,d,\alpha)" class="ltx_Math" display="inline" id="S3.Ex3.m1.3"><semantics id="S3.Ex3.m1.3a"><mrow id="S3.Ex3.m1.3.4" xref="S3.Ex3.m1.3.4.cmml"><mi id="S3.Ex3.m1.3.4.2" xref="S3.Ex3.m1.3.4.2.cmml">C</mi><mo id="S3.Ex3.m1.3.4.1" xref="S3.Ex3.m1.3.4.1.cmml">⁢</mo><mrow id="S3.Ex3.m1.3.4.3.2" 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class="ltx_tag ltx_tag_equation ltx_align_right">(MAX)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p3.8"><math alttext="C" class="ltx_Math" display="inline" id="S3.SS2.p3.8.m1.1"><semantics id="S3.SS2.p3.8.m1.1a"><mi id="S3.SS2.p3.8.m1.1.1" xref="S3.SS2.p3.8.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.8.m1.1b"><ci id="S3.SS2.p3.8.m1.1.1.cmml" xref="S3.SS2.p3.8.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.8.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.8.m1.1d">italic_C</annotation></semantics></math> is extended to mixed strategies in the usual way, as an expectation over the outcomes of the mixed strategies. <br class="ltx_break"/> <br class="ltx_break"/><span class="ltx_text ltx_font_bold" id="S3.SS2.p3.8.1">Price of uncorrelation </span> We measure the gap in the performances of correlated and uncorrelated agents with the PoU. It is defined by the costs at TMEs and TMECors of the game, as defined in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">von1997team</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>]</cite>.<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Technically, the PoU is defined by payoff-maximizing agents. See Appendix <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A2" title="Appendix 0.B Connections of Previous Results to Our Model ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.B</span></a> to see why we can define it with cost-minimizing agents without changing it.</span></span></span></p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Definition 1</span></h6> <div class="ltx_para" id="Thmdefinition1.p1"> <p class="ltx_p" id="Thmdefinition1.p1.5">For a game with graph <math alttext="G" class="ltx_Math" display="inline" id="Thmdefinition1.p1.1.m1.1"><semantics id="Thmdefinition1.p1.1.m1.1a"><mi id="Thmdefinition1.p1.1.m1.1.1" xref="Thmdefinition1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.1.m1.1b"><ci id="Thmdefinition1.p1.1.m1.1.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.1.m1.1d">italic_G</annotation></semantics></math>, <math alttext="n" class="ltx_Math" display="inline" id="Thmdefinition1.p1.2.m2.1"><semantics id="Thmdefinition1.p1.2.m2.1a"><mi id="Thmdefinition1.p1.2.m2.1.1" xref="Thmdefinition1.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.2.m2.1b"><ci id="Thmdefinition1.p1.2.m2.1.1.cmml" xref="Thmdefinition1.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.2.m2.1d">italic_n</annotation></semantics></math> agents, <math alttext="k" class="ltx_Math" display="inline" id="Thmdefinition1.p1.3.m3.1"><semantics id="Thmdefinition1.p1.3.m3.1a"><mi id="Thmdefinition1.p1.3.m3.1.1" xref="Thmdefinition1.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.3.m3.1b"><ci id="Thmdefinition1.p1.3.m3.1.1.cmml" xref="Thmdefinition1.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.3.m3.1d">italic_k</annotation></semantics></math> (correlated) interceptors, a parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmdefinition1.p1.4.m4.1"><semantics id="Thmdefinition1.p1.4.m4.1a"><mi id="Thmdefinition1.p1.4.m4.1.1" xref="Thmdefinition1.p1.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.4.m4.1b"><ci id="Thmdefinition1.p1.4.m4.1.1.cmml" xref="Thmdefinition1.p1.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.4.m4.1d">italic_α</annotation></semantics></math> and a cost function <math alttext="C" class="ltx_Math" display="inline" id="Thmdefinition1.p1.5.m5.1"><semantics id="Thmdefinition1.p1.5.m5.1a"><mi id="Thmdefinition1.p1.5.m5.1.1" xref="Thmdefinition1.p1.5.m5.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.5.m5.1b"><ci id="Thmdefinition1.p1.5.m5.1.1.cmml" xref="Thmdefinition1.p1.5.m5.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.5.m5.1c">C</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.5.m5.1d">italic_C</annotation></semantics></math>, the PoU of the game is defined by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx3"> <tbody id="S3.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(G,n,k,C,\alpha):=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}% \max_{d\in E^{k}}C(\sigma,\tau,\alpha)}{\min_{\sigma\in\Delta((P_{st})^{n})}% \max_{d\in E^{k}}C(\sigma,\tau,\alpha)}" class="ltx_Math" display="inline" id="S3.E1.m1.13"><semantics id="S3.E1.m1.13a"><mrow id="S3.E1.m1.13.14" xref="S3.E1.m1.13.14.cmml"><mrow id="S3.E1.m1.13.14.2" xref="S3.E1.m1.13.14.2.cmml"><mi id="S3.E1.m1.13.14.2.2" xref="S3.E1.m1.13.14.2.2.cmml">r</mi><mo id="S3.E1.m1.13.14.2.1" xref="S3.E1.m1.13.14.2.1.cmml">⁢</mo><mrow id="S3.E1.m1.13.14.2.3.2" xref="S3.E1.m1.13.14.2.3.1.cmml"><mo id="S3.E1.m1.13.14.2.3.2.1" stretchy="false" xref="S3.E1.m1.13.14.2.3.1.cmml">(</mo><mi id="S3.E1.m1.9.9" xref="S3.E1.m1.9.9.cmml">G</mi><mo id="S3.E1.m1.13.14.2.3.2.2" xref="S3.E1.m1.13.14.2.3.1.cmml">,</mo><mi id="S3.E1.m1.10.10" xref="S3.E1.m1.10.10.cmml">n</mi><mo id="S3.E1.m1.13.14.2.3.2.3" xref="S3.E1.m1.13.14.2.3.1.cmml">,</mo><mi id="S3.E1.m1.11.11" xref="S3.E1.m1.11.11.cmml">k</mi><mo id="S3.E1.m1.13.14.2.3.2.4" xref="S3.E1.m1.13.14.2.3.1.cmml">,</mo><mi id="S3.E1.m1.12.12" xref="S3.E1.m1.12.12.cmml">C</mi><mo id="S3.E1.m1.13.14.2.3.2.5" xref="S3.E1.m1.13.14.2.3.1.cmml">,</mo><mi id="S3.E1.m1.13.13" xref="S3.E1.m1.13.13.cmml">α</mi><mo id="S3.E1.m1.13.14.2.3.2.6" rspace="0.278em" stretchy="false" xref="S3.E1.m1.13.14.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.E1.m1.13.14.1" rspace="0.278em" xref="S3.E1.m1.13.14.1.cmml">:=</mo><mstyle displaystyle="true" id="S3.E1.m1.8.8" xref="S3.E1.m1.8.8.cmml"><mfrac id="S3.E1.m1.8.8a" xref="S3.E1.m1.8.8.cmml"><mrow id="S3.E1.m1.4.4.4" xref="S3.E1.m1.4.4.4.cmml"><mrow id="S3.E1.m1.4.4.4.6" xref="S3.E1.m1.4.4.4.6.cmml"><msub id="S3.E1.m1.4.4.4.6.1" xref="S3.E1.m1.4.4.4.6.1.cmml"><mi id="S3.E1.m1.4.4.4.6.1.2" xref="S3.E1.m1.4.4.4.6.1.2.cmml">min</mi><mrow id="S3.E1.m1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.cmml"><mi id="S3.E1.m1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.3.cmml">σ</mi><mo id="S3.E1.m1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.2.cmml">∈</mo><msup id="S3.E1.m1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.cmml"><mrow id="S3.E1.m1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="S3.E1.m1.1.1.1.1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.E1.m1.1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.3.cmml">n</mi></msup></mrow></msub><mo id="S3.E1.m1.4.4.4.6a" lspace="0.167em" xref="S3.E1.m1.4.4.4.6.cmml">⁡</mo><mrow id="S3.E1.m1.4.4.4.6.2" xref="S3.E1.m1.4.4.4.6.2.cmml"><msub id="S3.E1.m1.4.4.4.6.2.1" xref="S3.E1.m1.4.4.4.6.2.1.cmml"><mi id="S3.E1.m1.4.4.4.6.2.1.2" xref="S3.E1.m1.4.4.4.6.2.1.2.cmml">max</mi><mrow id="S3.E1.m1.4.4.4.6.2.1.3" xref="S3.E1.m1.4.4.4.6.2.1.3.cmml"><mi id="S3.E1.m1.4.4.4.6.2.1.3.2" xref="S3.E1.m1.4.4.4.6.2.1.3.2.cmml">d</mi><mo id="S3.E1.m1.4.4.4.6.2.1.3.1" xref="S3.E1.m1.4.4.4.6.2.1.3.1.cmml">∈</mo><msup id="S3.E1.m1.4.4.4.6.2.1.3.3" xref="S3.E1.m1.4.4.4.6.2.1.3.3.cmml"><mi id="S3.E1.m1.4.4.4.6.2.1.3.3.2" xref="S3.E1.m1.4.4.4.6.2.1.3.3.2.cmml">E</mi><mi id="S3.E1.m1.4.4.4.6.2.1.3.3.3" xref="S3.E1.m1.4.4.4.6.2.1.3.3.3.cmml">k</mi></msup></mrow></msub><mo id="S3.E1.m1.4.4.4.6.2a" lspace="0.167em" xref="S3.E1.m1.4.4.4.6.2.cmml">⁡</mo><mi id="S3.E1.m1.4.4.4.6.2.2" xref="S3.E1.m1.4.4.4.6.2.2.cmml">C</mi></mrow></mrow><mo id="S3.E1.m1.4.4.4.5" xref="S3.E1.m1.4.4.4.5.cmml">⁢</mo><mrow id="S3.E1.m1.4.4.4.7.2" xref="S3.E1.m1.4.4.4.7.1.cmml"><mo id="S3.E1.m1.4.4.4.7.2.1" stretchy="false" xref="S3.E1.m1.4.4.4.7.1.cmml">(</mo><mi id="S3.E1.m1.2.2.2.2" xref="S3.E1.m1.2.2.2.2.cmml">σ</mi><mo id="S3.E1.m1.4.4.4.7.2.2" xref="S3.E1.m1.4.4.4.7.1.cmml">,</mo><mi id="S3.E1.m1.3.3.3.3" xref="S3.E1.m1.3.3.3.3.cmml">τ</mi><mo id="S3.E1.m1.4.4.4.7.2.3" xref="S3.E1.m1.4.4.4.7.1.cmml">,</mo><mi id="S3.E1.m1.4.4.4.4" xref="S3.E1.m1.4.4.4.4.cmml">α</mi><mo id="S3.E1.m1.4.4.4.7.2.4" stretchy="false" xref="S3.E1.m1.4.4.4.7.1.cmml">)</mo></mrow></mrow><mrow id="S3.E1.m1.8.8.8" xref="S3.E1.m1.8.8.8.cmml"><mrow id="S3.E1.m1.8.8.8.6" xref="S3.E1.m1.8.8.8.6.cmml"><msub id="S3.E1.m1.8.8.8.6.1" xref="S3.E1.m1.8.8.8.6.1.cmml"><mi id="S3.E1.m1.8.8.8.6.1.2" xref="S3.E1.m1.8.8.8.6.1.2.cmml">min</mi><mrow id="S3.E1.m1.5.5.5.1.1" xref="S3.E1.m1.5.5.5.1.1.cmml"><mi id="S3.E1.m1.5.5.5.1.1.3" xref="S3.E1.m1.5.5.5.1.1.3.cmml">σ</mi><mo id="S3.E1.m1.5.5.5.1.1.2" xref="S3.E1.m1.5.5.5.1.1.2.cmml">∈</mo><mrow id="S3.E1.m1.5.5.5.1.1.1" xref="S3.E1.m1.5.5.5.1.1.1.cmml"><mi id="S3.E1.m1.5.5.5.1.1.1.3" mathvariant="normal" xref="S3.E1.m1.5.5.5.1.1.1.3.cmml">Δ</mi><mo id="S3.E1.m1.5.5.5.1.1.1.2" xref="S3.E1.m1.5.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.5.5.5.1.1.1.1.1" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.5.5.5.1.1.1.1.1.2" stretchy="false" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.cmml">(</mo><msup id="S3.E1.m1.5.5.5.1.1.1.1.1.1" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.cmml"><mrow id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.2" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.2" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.1" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.3" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.E1.m1.5.5.5.1.1.1.1.1.1.3" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="S3.E1.m1.5.5.5.1.1.1.1.1.3" stretchy="false" xref="S3.E1.m1.5.5.5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mo id="S3.E1.m1.8.8.8.6a" lspace="0.167em" xref="S3.E1.m1.8.8.8.6.cmml">⁡</mo><mrow id="S3.E1.m1.8.8.8.6.2" xref="S3.E1.m1.8.8.8.6.2.cmml"><msub id="S3.E1.m1.8.8.8.6.2.1" xref="S3.E1.m1.8.8.8.6.2.1.cmml"><mi id="S3.E1.m1.8.8.8.6.2.1.2" xref="S3.E1.m1.8.8.8.6.2.1.2.cmml">max</mi><mrow id="S3.E1.m1.8.8.8.6.2.1.3" xref="S3.E1.m1.8.8.8.6.2.1.3.cmml"><mi id="S3.E1.m1.8.8.8.6.2.1.3.2" xref="S3.E1.m1.8.8.8.6.2.1.3.2.cmml">d</mi><mo id="S3.E1.m1.8.8.8.6.2.1.3.1" xref="S3.E1.m1.8.8.8.6.2.1.3.1.cmml">∈</mo><msup id="S3.E1.m1.8.8.8.6.2.1.3.3" xref="S3.E1.m1.8.8.8.6.2.1.3.3.cmml"><mi 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id="S3.E1.m1.8.8.8.6.2.2.cmml" xref="S3.E1.m1.8.8.8.6.2.2">𝐶</ci></apply></apply><vector id="S3.E1.m1.8.8.8.7.1.cmml" xref="S3.E1.m1.8.8.8.7.2"><ci id="S3.E1.m1.6.6.6.2.cmml" xref="S3.E1.m1.6.6.6.2">𝜎</ci><ci id="S3.E1.m1.7.7.7.3.cmml" xref="S3.E1.m1.7.7.7.3">𝜏</ci><ci id="S3.E1.m1.8.8.8.4.cmml" xref="S3.E1.m1.8.8.8.4">𝛼</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1.m1.13c">\displaystyle r(G,n,k,C,\alpha):=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}% \max_{d\in E^{k}}C(\sigma,\tau,\alpha)}{\min_{\sigma\in\Delta((P_{st})^{n})}% \max_{d\in E^{k}}C(\sigma,\tau,\alpha)}</annotation><annotation encoding="application/x-llamapun" id="S3.E1.m1.13d">italic_r ( italic_G , italic_n , italic_k , italic_C , italic_α ) := divide start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_C ( italic_σ , italic_τ , italic_α ) end_ARG start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_C ( italic_σ , italic_τ , italic_α ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.7">We omit <math alttext="C" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><mi id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><ci id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">italic_C</annotation></semantics></math> when it is clear from context. Observe that <math alttext="r(G,n,k,C,\alpha)\geq 1" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.5"><semantics id="S3.SS2.p4.2.m2.5a"><mrow id="S3.SS2.p4.2.m2.5.6" xref="S3.SS2.p4.2.m2.5.6.cmml"><mrow id="S3.SS2.p4.2.m2.5.6.2" xref="S3.SS2.p4.2.m2.5.6.2.cmml"><mi id="S3.SS2.p4.2.m2.5.6.2.2" xref="S3.SS2.p4.2.m2.5.6.2.2.cmml">r</mi><mo id="S3.SS2.p4.2.m2.5.6.2.1" xref="S3.SS2.p4.2.m2.5.6.2.1.cmml">⁢</mo><mrow id="S3.SS2.p4.2.m2.5.6.2.3.2" xref="S3.SS2.p4.2.m2.5.6.2.3.1.cmml"><mo id="S3.SS2.p4.2.m2.5.6.2.3.2.1" stretchy="false" xref="S3.SS2.p4.2.m2.5.6.2.3.1.cmml">(</mo><mi id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml">G</mi><mo id="S3.SS2.p4.2.m2.5.6.2.3.2.2" xref="S3.SS2.p4.2.m2.5.6.2.3.1.cmml">,</mo><mi id="S3.SS2.p4.2.m2.2.2" xref="S3.SS2.p4.2.m2.2.2.cmml">n</mi><mo id="S3.SS2.p4.2.m2.5.6.2.3.2.3" xref="S3.SS2.p4.2.m2.5.6.2.3.1.cmml">,</mo><mi id="S3.SS2.p4.2.m2.3.3" xref="S3.SS2.p4.2.m2.3.3.cmml">k</mi><mo id="S3.SS2.p4.2.m2.5.6.2.3.2.4" xref="S3.SS2.p4.2.m2.5.6.2.3.1.cmml">,</mo><mi id="S3.SS2.p4.2.m2.4.4" xref="S3.SS2.p4.2.m2.4.4.cmml">C</mi><mo id="S3.SS2.p4.2.m2.5.6.2.3.2.5" xref="S3.SS2.p4.2.m2.5.6.2.3.1.cmml">,</mo><mi id="S3.SS2.p4.2.m2.5.5" xref="S3.SS2.p4.2.m2.5.5.cmml">α</mi><mo id="S3.SS2.p4.2.m2.5.6.2.3.2.6" stretchy="false" xref="S3.SS2.p4.2.m2.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p4.2.m2.5.6.1" xref="S3.SS2.p4.2.m2.5.6.1.cmml">≥</mo><mn id="S3.SS2.p4.2.m2.5.6.3" xref="S3.SS2.p4.2.m2.5.6.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.5b"><apply id="S3.SS2.p4.2.m2.5.6.cmml" xref="S3.SS2.p4.2.m2.5.6"><geq id="S3.SS2.p4.2.m2.5.6.1.cmml" xref="S3.SS2.p4.2.m2.5.6.1"></geq><apply id="S3.SS2.p4.2.m2.5.6.2.cmml" xref="S3.SS2.p4.2.m2.5.6.2"><times id="S3.SS2.p4.2.m2.5.6.2.1.cmml" xref="S3.SS2.p4.2.m2.5.6.2.1"></times><ci id="S3.SS2.p4.2.m2.5.6.2.2.cmml" xref="S3.SS2.p4.2.m2.5.6.2.2">𝑟</ci><vector id="S3.SS2.p4.2.m2.5.6.2.3.1.cmml" xref="S3.SS2.p4.2.m2.5.6.2.3.2"><ci id="S3.SS2.p4.2.m2.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1">𝐺</ci><ci id="S3.SS2.p4.2.m2.2.2.cmml" xref="S3.SS2.p4.2.m2.2.2">𝑛</ci><ci id="S3.SS2.p4.2.m2.3.3.cmml" xref="S3.SS2.p4.2.m2.3.3">𝑘</ci><ci id="S3.SS2.p4.2.m2.4.4.cmml" xref="S3.SS2.p4.2.m2.4.4">𝐶</ci><ci id="S3.SS2.p4.2.m2.5.5.cmml" xref="S3.SS2.p4.2.m2.5.5">𝛼</ci></vector></apply><cn id="S3.SS2.p4.2.m2.5.6.3.cmml" type="integer" xref="S3.SS2.p4.2.m2.5.6.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.5c">r(G,n,k,C,\alpha)\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.5d">italic_r ( italic_G , italic_n , italic_k , italic_C , italic_α ) ≥ 1</annotation></semantics></math> as long as <math alttext="C>0" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.1"><semantics id="S3.SS2.p4.3.m3.1a"><mrow id="S3.SS2.p4.3.m3.1.1" xref="S3.SS2.p4.3.m3.1.1.cmml"><mi id="S3.SS2.p4.3.m3.1.1.2" xref="S3.SS2.p4.3.m3.1.1.2.cmml">C</mi><mo id="S3.SS2.p4.3.m3.1.1.1" xref="S3.SS2.p4.3.m3.1.1.1.cmml">></mo><mn id="S3.SS2.p4.3.m3.1.1.3" xref="S3.SS2.p4.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.1b"><apply id="S3.SS2.p4.3.m3.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1"><gt id="S3.SS2.p4.3.m3.1.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1.1"></gt><ci id="S3.SS2.p4.3.m3.1.1.2.cmml" xref="S3.SS2.p4.3.m3.1.1.2">𝐶</ci><cn id="S3.SS2.p4.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.1c">C>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.1d">italic_C > 0</annotation></semantics></math>. Note that in the PoU definition, the interceptor pick edges deterministically. This is wlog for the purposes of computing the costs at TMEs and TMECors of the game (or TME and TMECor values). As shown in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">von1997team</span>]</cite>, for every <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS2.p4.4.m4.1"><semantics id="S3.SS2.p4.4.m4.1a"><mi id="S3.SS2.p4.4.m4.1.1" xref="S3.SS2.p4.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.4.m4.1b"><ci id="S3.SS2.p4.4.m4.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.4.m4.1d">italic_σ</annotation></semantics></math> that attains the TME value (subject to a best response <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="S3.SS2.p4.5.m5.1"><semantics id="S3.SS2.p4.5.m5.1a"><mrow id="S3.SS2.p4.5.m5.1.1" xref="S3.SS2.p4.5.m5.1.1.cmml"><mi id="S3.SS2.p4.5.m5.1.1.2" xref="S3.SS2.p4.5.m5.1.1.2.cmml">d</mi><mo id="S3.SS2.p4.5.m5.1.1.1" xref="S3.SS2.p4.5.m5.1.1.1.cmml">∈</mo><msup id="S3.SS2.p4.5.m5.1.1.3" xref="S3.SS2.p4.5.m5.1.1.3.cmml"><mi id="S3.SS2.p4.5.m5.1.1.3.2" xref="S3.SS2.p4.5.m5.1.1.3.2.cmml">E</mi><mi id="S3.SS2.p4.5.m5.1.1.3.3" xref="S3.SS2.p4.5.m5.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.5.m5.1b"><apply id="S3.SS2.p4.5.m5.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1"><in id="S3.SS2.p4.5.m5.1.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1.1"></in><ci id="S3.SS2.p4.5.m5.1.1.2.cmml" xref="S3.SS2.p4.5.m5.1.1.2">𝑑</ci><apply id="S3.SS2.p4.5.m5.1.1.3.cmml" xref="S3.SS2.p4.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p4.5.m5.1.1.3.1.cmml" xref="S3.SS2.p4.5.m5.1.1.3">superscript</csymbol><ci id="S3.SS2.p4.5.m5.1.1.3.2.cmml" xref="S3.SS2.p4.5.m5.1.1.3.2">𝐸</ci><ci id="S3.SS2.p4.5.m5.1.1.3.3.cmml" xref="S3.SS2.p4.5.m5.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.5.m5.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.5.m5.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>), there exists a <math alttext="\tau\in\Delta(E^{k})" class="ltx_Math" display="inline" id="S3.SS2.p4.6.m6.1"><semantics id="S3.SS2.p4.6.m6.1a"><mrow id="S3.SS2.p4.6.m6.1.1" xref="S3.SS2.p4.6.m6.1.1.cmml"><mi id="S3.SS2.p4.6.m6.1.1.3" xref="S3.SS2.p4.6.m6.1.1.3.cmml">τ</mi><mo id="S3.SS2.p4.6.m6.1.1.2" xref="S3.SS2.p4.6.m6.1.1.2.cmml">∈</mo><mrow id="S3.SS2.p4.6.m6.1.1.1" xref="S3.SS2.p4.6.m6.1.1.1.cmml"><mi id="S3.SS2.p4.6.m6.1.1.1.3" mathvariant="normal" xref="S3.SS2.p4.6.m6.1.1.1.3.cmml">Δ</mi><mo id="S3.SS2.p4.6.m6.1.1.1.2" xref="S3.SS2.p4.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p4.6.m6.1.1.1.1.1" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p4.6.m6.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS2.p4.6.m6.1.1.1.1.1.1" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p4.6.m6.1.1.1.1.1.1.2" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.2.cmml">E</mi><mi id="S3.SS2.p4.6.m6.1.1.1.1.1.1.3" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.3.cmml">k</mi></msup><mo id="S3.SS2.p4.6.m6.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.6.m6.1b"><apply id="S3.SS2.p4.6.m6.1.1.cmml" xref="S3.SS2.p4.6.m6.1.1"><in id="S3.SS2.p4.6.m6.1.1.2.cmml" xref="S3.SS2.p4.6.m6.1.1.2"></in><ci id="S3.SS2.p4.6.m6.1.1.3.cmml" xref="S3.SS2.p4.6.m6.1.1.3">𝜏</ci><apply id="S3.SS2.p4.6.m6.1.1.1.cmml" xref="S3.SS2.p4.6.m6.1.1.1"><times id="S3.SS2.p4.6.m6.1.1.1.2.cmml" xref="S3.SS2.p4.6.m6.1.1.1.2"></times><ci id="S3.SS2.p4.6.m6.1.1.1.3.cmml" xref="S3.SS2.p4.6.m6.1.1.1.3">Δ</ci><apply id="S3.SS2.p4.6.m6.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.6.m6.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S3.SS2.p4.6.m6.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.2">𝐸</ci><ci id="S3.SS2.p4.6.m6.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p4.6.m6.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.6.m6.1c">\tau\in\Delta(E^{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.6.m6.1d">italic_τ ∈ roman_Δ ( italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> such that <math alttext="(\sigma,\tau)" class="ltx_Math" display="inline" id="S3.SS2.p4.7.m7.2"><semantics id="S3.SS2.p4.7.m7.2a"><mrow id="S3.SS2.p4.7.m7.2.3.2" xref="S3.SS2.p4.7.m7.2.3.1.cmml"><mo id="S3.SS2.p4.7.m7.2.3.2.1" stretchy="false" xref="S3.SS2.p4.7.m7.2.3.1.cmml">(</mo><mi id="S3.SS2.p4.7.m7.1.1" xref="S3.SS2.p4.7.m7.1.1.cmml">σ</mi><mo id="S3.SS2.p4.7.m7.2.3.2.2" xref="S3.SS2.p4.7.m7.2.3.1.cmml">,</mo><mi id="S3.SS2.p4.7.m7.2.2" xref="S3.SS2.p4.7.m7.2.2.cmml">τ</mi><mo id="S3.SS2.p4.7.m7.2.3.2.3" stretchy="false" xref="S3.SS2.p4.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.7.m7.2b"><interval closure="open" id="S3.SS2.p4.7.m7.2.3.1.cmml" xref="S3.SS2.p4.7.m7.2.3.2"><ci id="S3.SS2.p4.7.m7.1.1.cmml" xref="S3.SS2.p4.7.m7.1.1">𝜎</ci><ci id="S3.SS2.p4.7.m7.2.2.cmml" xref="S3.SS2.p4.7.m7.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.7.m7.2c">(\sigma,\tau)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.7.m7.2d">( italic_σ , italic_τ )</annotation></semantics></math> is a TME of the game. A similar result holds for the TMECor value.</p> </div> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.9">For <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="S3.SS2.p5.1.m1.1"><semantics id="S3.SS2.p5.1.m1.1a"><mrow id="S3.SS2.p5.1.m1.1.1" xref="S3.SS2.p5.1.m1.1.1.cmml"><mi id="S3.SS2.p5.1.m1.1.1.2" xref="S3.SS2.p5.1.m1.1.1.2.cmml">d</mi><mo id="S3.SS2.p5.1.m1.1.1.1" xref="S3.SS2.p5.1.m1.1.1.1.cmml">∈</mo><msup id="S3.SS2.p5.1.m1.1.1.3" xref="S3.SS2.p5.1.m1.1.1.3.cmml"><mi id="S3.SS2.p5.1.m1.1.1.3.2" xref="S3.SS2.p5.1.m1.1.1.3.2.cmml">E</mi><mi id="S3.SS2.p5.1.m1.1.1.3.3" xref="S3.SS2.p5.1.m1.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.1.m1.1b"><apply id="S3.SS2.p5.1.m1.1.1.cmml" xref="S3.SS2.p5.1.m1.1.1"><in id="S3.SS2.p5.1.m1.1.1.1.cmml" xref="S3.SS2.p5.1.m1.1.1.1"></in><ci id="S3.SS2.p5.1.m1.1.1.2.cmml" xref="S3.SS2.p5.1.m1.1.1.2">𝑑</ci><apply id="S3.SS2.p5.1.m1.1.1.3.cmml" xref="S3.SS2.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p5.1.m1.1.1.3.1.cmml" xref="S3.SS2.p5.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS2.p5.1.m1.1.1.3.2.cmml" xref="S3.SS2.p5.1.m1.1.1.3.2">𝐸</ci><ci id="S3.SS2.p5.1.m1.1.1.3.3.cmml" xref="S3.SS2.p5.1.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.1.m1.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.1.m1.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> let <math alttext="f_{d}(\sigma,\alpha)=C(\sigma,d,\alpha)" class="ltx_Math" display="inline" id="S3.SS2.p5.2.m2.5"><semantics id="S3.SS2.p5.2.m2.5a"><mrow id="S3.SS2.p5.2.m2.5.6" xref="S3.SS2.p5.2.m2.5.6.cmml"><mrow id="S3.SS2.p5.2.m2.5.6.2" xref="S3.SS2.p5.2.m2.5.6.2.cmml"><msub id="S3.SS2.p5.2.m2.5.6.2.2" xref="S3.SS2.p5.2.m2.5.6.2.2.cmml"><mi id="S3.SS2.p5.2.m2.5.6.2.2.2" xref="S3.SS2.p5.2.m2.5.6.2.2.2.cmml">f</mi><mi id="S3.SS2.p5.2.m2.5.6.2.2.3" xref="S3.SS2.p5.2.m2.5.6.2.2.3.cmml">d</mi></msub><mo id="S3.SS2.p5.2.m2.5.6.2.1" xref="S3.SS2.p5.2.m2.5.6.2.1.cmml">⁢</mo><mrow id="S3.SS2.p5.2.m2.5.6.2.3.2" xref="S3.SS2.p5.2.m2.5.6.2.3.1.cmml"><mo id="S3.SS2.p5.2.m2.5.6.2.3.2.1" stretchy="false" xref="S3.SS2.p5.2.m2.5.6.2.3.1.cmml">(</mo><mi id="S3.SS2.p5.2.m2.1.1" xref="S3.SS2.p5.2.m2.1.1.cmml">σ</mi><mo id="S3.SS2.p5.2.m2.5.6.2.3.2.2" xref="S3.SS2.p5.2.m2.5.6.2.3.1.cmml">,</mo><mi id="S3.SS2.p5.2.m2.2.2" xref="S3.SS2.p5.2.m2.2.2.cmml">α</mi><mo id="S3.SS2.p5.2.m2.5.6.2.3.2.3" stretchy="false" xref="S3.SS2.p5.2.m2.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p5.2.m2.5.6.1" xref="S3.SS2.p5.2.m2.5.6.1.cmml">=</mo><mrow id="S3.SS2.p5.2.m2.5.6.3" xref="S3.SS2.p5.2.m2.5.6.3.cmml"><mi id="S3.SS2.p5.2.m2.5.6.3.2" xref="S3.SS2.p5.2.m2.5.6.3.2.cmml">C</mi><mo id="S3.SS2.p5.2.m2.5.6.3.1" xref="S3.SS2.p5.2.m2.5.6.3.1.cmml">⁢</mo><mrow id="S3.SS2.p5.2.m2.5.6.3.3.2" xref="S3.SS2.p5.2.m2.5.6.3.3.1.cmml"><mo id="S3.SS2.p5.2.m2.5.6.3.3.2.1" stretchy="false" xref="S3.SS2.p5.2.m2.5.6.3.3.1.cmml">(</mo><mi id="S3.SS2.p5.2.m2.3.3" xref="S3.SS2.p5.2.m2.3.3.cmml">σ</mi><mo id="S3.SS2.p5.2.m2.5.6.3.3.2.2" xref="S3.SS2.p5.2.m2.5.6.3.3.1.cmml">,</mo><mi id="S3.SS2.p5.2.m2.4.4" xref="S3.SS2.p5.2.m2.4.4.cmml">d</mi><mo id="S3.SS2.p5.2.m2.5.6.3.3.2.3" xref="S3.SS2.p5.2.m2.5.6.3.3.1.cmml">,</mo><mi id="S3.SS2.p5.2.m2.5.5" xref="S3.SS2.p5.2.m2.5.5.cmml">α</mi><mo id="S3.SS2.p5.2.m2.5.6.3.3.2.4" stretchy="false" xref="S3.SS2.p5.2.m2.5.6.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.2.m2.5b"><apply id="S3.SS2.p5.2.m2.5.6.cmml" xref="S3.SS2.p5.2.m2.5.6"><eq id="S3.SS2.p5.2.m2.5.6.1.cmml" xref="S3.SS2.p5.2.m2.5.6.1"></eq><apply id="S3.SS2.p5.2.m2.5.6.2.cmml" xref="S3.SS2.p5.2.m2.5.6.2"><times id="S3.SS2.p5.2.m2.5.6.2.1.cmml" xref="S3.SS2.p5.2.m2.5.6.2.1"></times><apply id="S3.SS2.p5.2.m2.5.6.2.2.cmml" xref="S3.SS2.p5.2.m2.5.6.2.2"><csymbol cd="ambiguous" id="S3.SS2.p5.2.m2.5.6.2.2.1.cmml" xref="S3.SS2.p5.2.m2.5.6.2.2">subscript</csymbol><ci id="S3.SS2.p5.2.m2.5.6.2.2.2.cmml" xref="S3.SS2.p5.2.m2.5.6.2.2.2">𝑓</ci><ci id="S3.SS2.p5.2.m2.5.6.2.2.3.cmml" xref="S3.SS2.p5.2.m2.5.6.2.2.3">𝑑</ci></apply><interval closure="open" id="S3.SS2.p5.2.m2.5.6.2.3.1.cmml" xref="S3.SS2.p5.2.m2.5.6.2.3.2"><ci id="S3.SS2.p5.2.m2.1.1.cmml" xref="S3.SS2.p5.2.m2.1.1">𝜎</ci><ci id="S3.SS2.p5.2.m2.2.2.cmml" xref="S3.SS2.p5.2.m2.2.2">𝛼</ci></interval></apply><apply id="S3.SS2.p5.2.m2.5.6.3.cmml" xref="S3.SS2.p5.2.m2.5.6.3"><times id="S3.SS2.p5.2.m2.5.6.3.1.cmml" xref="S3.SS2.p5.2.m2.5.6.3.1"></times><ci id="S3.SS2.p5.2.m2.5.6.3.2.cmml" xref="S3.SS2.p5.2.m2.5.6.3.2">𝐶</ci><vector id="S3.SS2.p5.2.m2.5.6.3.3.1.cmml" xref="S3.SS2.p5.2.m2.5.6.3.3.2"><ci id="S3.SS2.p5.2.m2.3.3.cmml" xref="S3.SS2.p5.2.m2.3.3">𝜎</ci><ci id="S3.SS2.p5.2.m2.4.4.cmml" xref="S3.SS2.p5.2.m2.4.4">𝑑</ci><ci id="S3.SS2.p5.2.m2.5.5.cmml" xref="S3.SS2.p5.2.m2.5.5">𝛼</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.2.m2.5c">f_{d}(\sigma,\alpha)=C(\sigma,d,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.2.m2.5d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_σ , italic_α ) = italic_C ( italic_σ , italic_d , italic_α )</annotation></semantics></math>, and let <math alttext="F_{n}(\sigma,\alpha)=\max_{d\in E^{k}}f_{d}(\sigma,\alpha)" class="ltx_Math" display="inline" id="S3.SS2.p5.3.m3.4"><semantics id="S3.SS2.p5.3.m3.4a"><mrow id="S3.SS2.p5.3.m3.4.5" xref="S3.SS2.p5.3.m3.4.5.cmml"><mrow id="S3.SS2.p5.3.m3.4.5.2" xref="S3.SS2.p5.3.m3.4.5.2.cmml"><msub id="S3.SS2.p5.3.m3.4.5.2.2" xref="S3.SS2.p5.3.m3.4.5.2.2.cmml"><mi id="S3.SS2.p5.3.m3.4.5.2.2.2" xref="S3.SS2.p5.3.m3.4.5.2.2.2.cmml">F</mi><mi id="S3.SS2.p5.3.m3.4.5.2.2.3" xref="S3.SS2.p5.3.m3.4.5.2.2.3.cmml">n</mi></msub><mo id="S3.SS2.p5.3.m3.4.5.2.1" xref="S3.SS2.p5.3.m3.4.5.2.1.cmml">⁢</mo><mrow id="S3.SS2.p5.3.m3.4.5.2.3.2" xref="S3.SS2.p5.3.m3.4.5.2.3.1.cmml"><mo id="S3.SS2.p5.3.m3.4.5.2.3.2.1" stretchy="false" xref="S3.SS2.p5.3.m3.4.5.2.3.1.cmml">(</mo><mi id="S3.SS2.p5.3.m3.1.1" xref="S3.SS2.p5.3.m3.1.1.cmml">σ</mi><mo id="S3.SS2.p5.3.m3.4.5.2.3.2.2" xref="S3.SS2.p5.3.m3.4.5.2.3.1.cmml">,</mo><mi id="S3.SS2.p5.3.m3.2.2" xref="S3.SS2.p5.3.m3.2.2.cmml">α</mi><mo id="S3.SS2.p5.3.m3.4.5.2.3.2.3" stretchy="false" xref="S3.SS2.p5.3.m3.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p5.3.m3.4.5.1" xref="S3.SS2.p5.3.m3.4.5.1.cmml">=</mo><mrow id="S3.SS2.p5.3.m3.4.5.3" xref="S3.SS2.p5.3.m3.4.5.3.cmml"><mrow id="S3.SS2.p5.3.m3.4.5.3.2" xref="S3.SS2.p5.3.m3.4.5.3.2.cmml"><msub id="S3.SS2.p5.3.m3.4.5.3.2.1" xref="S3.SS2.p5.3.m3.4.5.3.2.1.cmml"><mi id="S3.SS2.p5.3.m3.4.5.3.2.1.2" xref="S3.SS2.p5.3.m3.4.5.3.2.1.2.cmml">max</mi><mrow id="S3.SS2.p5.3.m3.4.5.3.2.1.3" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.cmml"><mi id="S3.SS2.p5.3.m3.4.5.3.2.1.3.2" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.2.cmml">d</mi><mo id="S3.SS2.p5.3.m3.4.5.3.2.1.3.1" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.1.cmml">∈</mo><msup id="S3.SS2.p5.3.m3.4.5.3.2.1.3.3" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.cmml"><mi id="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.2" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.2.cmml">E</mi><mi id="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.3" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.3.cmml">k</mi></msup></mrow></msub><mo id="S3.SS2.p5.3.m3.4.5.3.2a" lspace="0.167em" xref="S3.SS2.p5.3.m3.4.5.3.2.cmml">⁡</mo><msub id="S3.SS2.p5.3.m3.4.5.3.2.2" xref="S3.SS2.p5.3.m3.4.5.3.2.2.cmml"><mi id="S3.SS2.p5.3.m3.4.5.3.2.2.2" xref="S3.SS2.p5.3.m3.4.5.3.2.2.2.cmml">f</mi><mi id="S3.SS2.p5.3.m3.4.5.3.2.2.3" xref="S3.SS2.p5.3.m3.4.5.3.2.2.3.cmml">d</mi></msub></mrow><mo id="S3.SS2.p5.3.m3.4.5.3.1" xref="S3.SS2.p5.3.m3.4.5.3.1.cmml">⁢</mo><mrow id="S3.SS2.p5.3.m3.4.5.3.3.2" xref="S3.SS2.p5.3.m3.4.5.3.3.1.cmml"><mo id="S3.SS2.p5.3.m3.4.5.3.3.2.1" stretchy="false" xref="S3.SS2.p5.3.m3.4.5.3.3.1.cmml">(</mo><mi id="S3.SS2.p5.3.m3.3.3" xref="S3.SS2.p5.3.m3.3.3.cmml">σ</mi><mo id="S3.SS2.p5.3.m3.4.5.3.3.2.2" xref="S3.SS2.p5.3.m3.4.5.3.3.1.cmml">,</mo><mi id="S3.SS2.p5.3.m3.4.4" xref="S3.SS2.p5.3.m3.4.4.cmml">α</mi><mo id="S3.SS2.p5.3.m3.4.5.3.3.2.3" stretchy="false" xref="S3.SS2.p5.3.m3.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.3.m3.4b"><apply id="S3.SS2.p5.3.m3.4.5.cmml" xref="S3.SS2.p5.3.m3.4.5"><eq id="S3.SS2.p5.3.m3.4.5.1.cmml" xref="S3.SS2.p5.3.m3.4.5.1"></eq><apply id="S3.SS2.p5.3.m3.4.5.2.cmml" xref="S3.SS2.p5.3.m3.4.5.2"><times id="S3.SS2.p5.3.m3.4.5.2.1.cmml" xref="S3.SS2.p5.3.m3.4.5.2.1"></times><apply id="S3.SS2.p5.3.m3.4.5.2.2.cmml" xref="S3.SS2.p5.3.m3.4.5.2.2"><csymbol cd="ambiguous" id="S3.SS2.p5.3.m3.4.5.2.2.1.cmml" xref="S3.SS2.p5.3.m3.4.5.2.2">subscript</csymbol><ci id="S3.SS2.p5.3.m3.4.5.2.2.2.cmml" xref="S3.SS2.p5.3.m3.4.5.2.2.2">𝐹</ci><ci id="S3.SS2.p5.3.m3.4.5.2.2.3.cmml" xref="S3.SS2.p5.3.m3.4.5.2.2.3">𝑛</ci></apply><interval closure="open" id="S3.SS2.p5.3.m3.4.5.2.3.1.cmml" xref="S3.SS2.p5.3.m3.4.5.2.3.2"><ci id="S3.SS2.p5.3.m3.1.1.cmml" xref="S3.SS2.p5.3.m3.1.1">𝜎</ci><ci id="S3.SS2.p5.3.m3.2.2.cmml" xref="S3.SS2.p5.3.m3.2.2">𝛼</ci></interval></apply><apply id="S3.SS2.p5.3.m3.4.5.3.cmml" xref="S3.SS2.p5.3.m3.4.5.3"><times id="S3.SS2.p5.3.m3.4.5.3.1.cmml" xref="S3.SS2.p5.3.m3.4.5.3.1"></times><apply id="S3.SS2.p5.3.m3.4.5.3.2.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2"><apply id="S3.SS2.p5.3.m3.4.5.3.2.1.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1"><csymbol cd="ambiguous" id="S3.SS2.p5.3.m3.4.5.3.2.1.1.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1">subscript</csymbol><max id="S3.SS2.p5.3.m3.4.5.3.2.1.2.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.2"></max><apply id="S3.SS2.p5.3.m3.4.5.3.2.1.3.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3"><in id="S3.SS2.p5.3.m3.4.5.3.2.1.3.1.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.1"></in><ci id="S3.SS2.p5.3.m3.4.5.3.2.1.3.2.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.2">𝑑</ci><apply id="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.1.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.3">superscript</csymbol><ci id="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.2.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.2">𝐸</ci><ci id="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.3.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.1.3.3.3">𝑘</ci></apply></apply></apply><apply id="S3.SS2.p5.3.m3.4.5.3.2.2.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.2"><csymbol cd="ambiguous" id="S3.SS2.p5.3.m3.4.5.3.2.2.1.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.2">subscript</csymbol><ci id="S3.SS2.p5.3.m3.4.5.3.2.2.2.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.2.2">𝑓</ci><ci id="S3.SS2.p5.3.m3.4.5.3.2.2.3.cmml" xref="S3.SS2.p5.3.m3.4.5.3.2.2.3">𝑑</ci></apply></apply><interval closure="open" id="S3.SS2.p5.3.m3.4.5.3.3.1.cmml" xref="S3.SS2.p5.3.m3.4.5.3.3.2"><ci id="S3.SS2.p5.3.m3.3.3.cmml" xref="S3.SS2.p5.3.m3.3.3">𝜎</ci><ci id="S3.SS2.p5.3.m3.4.4.cmml" xref="S3.SS2.p5.3.m3.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.3.m3.4c">F_{n}(\sigma,\alpha)=\max_{d\in E^{k}}f_{d}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.3.m3.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = roman_max start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math>, where <math alttext="\sigma\in\Delta((P_{st})^{n})" class="ltx_Math" display="inline" id="S3.SS2.p5.4.m4.1"><semantics id="S3.SS2.p5.4.m4.1a"><mrow id="S3.SS2.p5.4.m4.1.1" xref="S3.SS2.p5.4.m4.1.1.cmml"><mi id="S3.SS2.p5.4.m4.1.1.3" xref="S3.SS2.p5.4.m4.1.1.3.cmml">σ</mi><mo id="S3.SS2.p5.4.m4.1.1.2" 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id="S3.SS2.p5.4.m4.1.1.1.2.cmml" xref="S3.SS2.p5.4.m4.1.1.1.2"></times><ci id="S3.SS2.p5.4.m4.1.1.1.3.cmml" xref="S3.SS2.p5.4.m4.1.1.1.3">Δ</ci><apply id="S3.SS2.p5.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1">superscript</csymbol><apply id="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3"><times id="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><ci id="S3.SS2.p5.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p5.4.m4.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.4.m4.1c">\sigma\in\Delta((P_{st})^{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.4.m4.1d">italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>. In words, <math alttext="F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="S3.SS2.p5.5.m5.2"><semantics id="S3.SS2.p5.5.m5.2a"><mrow id="S3.SS2.p5.5.m5.2.3" xref="S3.SS2.p5.5.m5.2.3.cmml"><msub id="S3.SS2.p5.5.m5.2.3.2" xref="S3.SS2.p5.5.m5.2.3.2.cmml"><mi id="S3.SS2.p5.5.m5.2.3.2.2" xref="S3.SS2.p5.5.m5.2.3.2.2.cmml">F</mi><mi id="S3.SS2.p5.5.m5.2.3.2.3" xref="S3.SS2.p5.5.m5.2.3.2.3.cmml">n</mi></msub><mo id="S3.SS2.p5.5.m5.2.3.1" xref="S3.SS2.p5.5.m5.2.3.1.cmml">⁢</mo><mrow id="S3.SS2.p5.5.m5.2.3.3.2" xref="S3.SS2.p5.5.m5.2.3.3.1.cmml"><mo id="S3.SS2.p5.5.m5.2.3.3.2.1" stretchy="false" xref="S3.SS2.p5.5.m5.2.3.3.1.cmml">(</mo><mi id="S3.SS2.p5.5.m5.1.1" xref="S3.SS2.p5.5.m5.1.1.cmml">σ</mi><mo id="S3.SS2.p5.5.m5.2.3.3.2.2" xref="S3.SS2.p5.5.m5.2.3.3.1.cmml">,</mo><mi id="S3.SS2.p5.5.m5.2.2" xref="S3.SS2.p5.5.m5.2.2.cmml">α</mi><mo id="S3.SS2.p5.5.m5.2.3.3.2.3" stretchy="false" xref="S3.SS2.p5.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.5.m5.2b"><apply id="S3.SS2.p5.5.m5.2.3.cmml" xref="S3.SS2.p5.5.m5.2.3"><times id="S3.SS2.p5.5.m5.2.3.1.cmml" xref="S3.SS2.p5.5.m5.2.3.1"></times><apply id="S3.SS2.p5.5.m5.2.3.2.cmml" xref="S3.SS2.p5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.p5.5.m5.2.3.2.1.cmml" xref="S3.SS2.p5.5.m5.2.3.2">subscript</csymbol><ci id="S3.SS2.p5.5.m5.2.3.2.2.cmml" xref="S3.SS2.p5.5.m5.2.3.2.2">𝐹</ci><ci id="S3.SS2.p5.5.m5.2.3.2.3.cmml" xref="S3.SS2.p5.5.m5.2.3.2.3">𝑛</ci></apply><interval closure="open" id="S3.SS2.p5.5.m5.2.3.3.1.cmml" xref="S3.SS2.p5.5.m5.2.3.3.2"><ci id="S3.SS2.p5.5.m5.1.1.cmml" xref="S3.SS2.p5.5.m5.1.1">𝜎</ci><ci id="S3.SS2.p5.5.m5.2.2.cmml" xref="S3.SS2.p5.5.m5.2.2">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.5.m5.2c">F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.5.m5.2d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math> is the cost that <math alttext="n" class="ltx_Math" display="inline" id="S3.SS2.p5.6.m6.1"><semantics id="S3.SS2.p5.6.m6.1a"><mi id="S3.SS2.p5.6.m6.1.1" xref="S3.SS2.p5.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.6.m6.1b"><ci id="S3.SS2.p5.6.m6.1.1.cmml" xref="S3.SS2.p5.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.6.m6.1d">italic_n</annotation></semantics></math> agents will incur when playing the joint strategy <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS2.p5.7.m7.1"><semantics id="S3.SS2.p5.7.m7.1a"><mi id="S3.SS2.p5.7.m7.1.1" xref="S3.SS2.p5.7.m7.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.7.m7.1b"><ci id="S3.SS2.p5.7.m7.1.1.cmml" xref="S3.SS2.p5.7.m7.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.7.m7.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.7.m7.1d">italic_σ</annotation></semantics></math> against <math alttext="k" class="ltx_Math" display="inline" id="S3.SS2.p5.8.m8.1"><semantics id="S3.SS2.p5.8.m8.1a"><mi id="S3.SS2.p5.8.m8.1.1" xref="S3.SS2.p5.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.8.m8.1b"><ci id="S3.SS2.p5.8.m8.1.1.cmml" xref="S3.SS2.p5.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.8.m8.1d">italic_k</annotation></semantics></math> interceptors with strength <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS2.p5.9.m9.1"><semantics id="S3.SS2.p5.9.m9.1a"><mi id="S3.SS2.p5.9.m9.1.1" xref="S3.SS2.p5.9.m9.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.9.m9.1b"><ci id="S3.SS2.p5.9.m9.1.1.cmml" xref="S3.SS2.p5.9.m9.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.9.m9.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.9.m9.1d">italic_α</annotation></semantics></math>. Further observe that for SUM and MAX, when the agents minimize their cost and are correlated, they will always pick the same path. So Equation <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S3.E1" title="In Definition 1 ‣ 3.2 The model ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1</span></a> simplifies to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx4"> <tbody id="S3.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(G,n,k,C,\alpha)=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n% }(\sigma,\alpha)}{\min_{\sigma\in\Delta(P_{st})}F_{1}(\sigma,\alpha)}" class="ltx_Math" display="inline" id="S3.E2.m1.11"><semantics id="S3.E2.m1.11a"><mrow id="S3.E2.m1.11.12" xref="S3.E2.m1.11.12.cmml"><mrow id="S3.E2.m1.11.12.2" xref="S3.E2.m1.11.12.2.cmml"><mi id="S3.E2.m1.11.12.2.2" xref="S3.E2.m1.11.12.2.2.cmml">r</mi><mo id="S3.E2.m1.11.12.2.1" 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id="S3.E2.m1.4.4.4.1.1.1.1.1.1.3.3.cmml" xref="S3.E2.m1.4.4.4.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></apply><apply id="S3.E2.m1.6.6.6.5.2.cmml" xref="S3.E2.m1.6.6.6.5.2"><csymbol cd="ambiguous" id="S3.E2.m1.6.6.6.5.2.1.cmml" xref="S3.E2.m1.6.6.6.5.2">subscript</csymbol><ci id="S3.E2.m1.6.6.6.5.2.2.cmml" xref="S3.E2.m1.6.6.6.5.2.2">𝐹</ci><cn id="S3.E2.m1.6.6.6.5.2.3.cmml" type="integer" xref="S3.E2.m1.6.6.6.5.2.3">1</cn></apply></apply><interval closure="open" id="S3.E2.m1.6.6.6.6.1.cmml" xref="S3.E2.m1.6.6.6.6.2"><ci id="S3.E2.m1.5.5.5.2.cmml" xref="S3.E2.m1.5.5.5.2">𝜎</ci><ci id="S3.E2.m1.6.6.6.3.cmml" xref="S3.E2.m1.6.6.6.3">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.11c">\displaystyle r(G,n,k,C,\alpha)=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n% }(\sigma,\alpha)}{\min_{\sigma\in\Delta(P_{st})}F_{1}(\sigma,\alpha)}</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.11d">italic_r ( italic_G , italic_n , italic_k , italic_C , italic_α ) = divide start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) end_ARG start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_σ , italic_α ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS2.p6"> <p class="ltx_p" id="S3.SS2.p6.13">We now define precisely the uniform strategies and the uniform PoU, using terminology from flow networks. <br class="ltx_break"/> <br class="ltx_break"/><math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p6.1.m1.1"><semantics id="S3.SS2.p6.1.m1.1a"><mi id="S3.SS2.p6.1.m1.1.1" xref="S3.SS2.p6.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.1.m1.1b"><ci id="S3.SS2.p6.1.m1.1.1.cmml" xref="S3.SS2.p6.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.1.m1.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_bold" id="S3.SS2.p6.13.1"> as a flow network </span> We recall the notion of mincuts. Endow every edge in <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p6.2.m2.1"><semantics id="S3.SS2.p6.2.m2.1a"><mi id="S3.SS2.p6.2.m2.1.1" xref="S3.SS2.p6.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.2.m2.1b"><ci id="S3.SS2.p6.2.m2.1.1.cmml" xref="S3.SS2.p6.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.2.m2.1d">italic_G</annotation></semantics></math> with capacity 1. An <math alttext="s-t" class="ltx_Math" display="inline" id="S3.SS2.p6.3.m3.1"><semantics id="S3.SS2.p6.3.m3.1a"><mrow id="S3.SS2.p6.3.m3.1.1" xref="S3.SS2.p6.3.m3.1.1.cmml"><mi id="S3.SS2.p6.3.m3.1.1.2" xref="S3.SS2.p6.3.m3.1.1.2.cmml">s</mi><mo id="S3.SS2.p6.3.m3.1.1.1" xref="S3.SS2.p6.3.m3.1.1.1.cmml">−</mo><mi id="S3.SS2.p6.3.m3.1.1.3" xref="S3.SS2.p6.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.3.m3.1b"><apply id="S3.SS2.p6.3.m3.1.1.cmml" xref="S3.SS2.p6.3.m3.1.1"><minus id="S3.SS2.p6.3.m3.1.1.1.cmml" xref="S3.SS2.p6.3.m3.1.1.1"></minus><ci id="S3.SS2.p6.3.m3.1.1.2.cmml" xref="S3.SS2.p6.3.m3.1.1.2">𝑠</ci><ci id="S3.SS2.p6.3.m3.1.1.3.cmml" xref="S3.SS2.p6.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.3.m3.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.3.m3.1d">italic_s - italic_t</annotation></semantics></math> cut in <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p6.4.m4.1"><semantics id="S3.SS2.p6.4.m4.1a"><mi id="S3.SS2.p6.4.m4.1.1" xref="S3.SS2.p6.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.4.m4.1b"><ci id="S3.SS2.p6.4.m4.1.1.cmml" xref="S3.SS2.p6.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.4.m4.1d">italic_G</annotation></semantics></math> is a partition of <math alttext="V" class="ltx_Math" display="inline" id="S3.SS2.p6.5.m5.1"><semantics id="S3.SS2.p6.5.m5.1a"><mi id="S3.SS2.p6.5.m5.1.1" xref="S3.SS2.p6.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.5.m5.1b"><ci id="S3.SS2.p6.5.m5.1.1.cmml" xref="S3.SS2.p6.5.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.5.m5.1d">italic_V</annotation></semantics></math> into <math alttext="S\sqcup T" class="ltx_Math" display="inline" id="S3.SS2.p6.6.m6.1"><semantics id="S3.SS2.p6.6.m6.1a"><mrow id="S3.SS2.p6.6.m6.1.1" xref="S3.SS2.p6.6.m6.1.1.cmml"><mi id="S3.SS2.p6.6.m6.1.1.2" xref="S3.SS2.p6.6.m6.1.1.2.cmml">S</mi><mo id="S3.SS2.p6.6.m6.1.1.1" xref="S3.SS2.p6.6.m6.1.1.1.cmml">⊔</mo><mi id="S3.SS2.p6.6.m6.1.1.3" xref="S3.SS2.p6.6.m6.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.6.m6.1b"><apply id="S3.SS2.p6.6.m6.1.1.cmml" xref="S3.SS2.p6.6.m6.1.1"><csymbol cd="latexml" id="S3.SS2.p6.6.m6.1.1.1.cmml" xref="S3.SS2.p6.6.m6.1.1.1">square-union</csymbol><ci id="S3.SS2.p6.6.m6.1.1.2.cmml" xref="S3.SS2.p6.6.m6.1.1.2">𝑆</ci><ci id="S3.SS2.p6.6.m6.1.1.3.cmml" xref="S3.SS2.p6.6.m6.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.6.m6.1c">S\sqcup T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.6.m6.1d">italic_S ⊔ italic_T</annotation></semantics></math> such that <math alttext="s\in S,~{}t\in T" class="ltx_Math" display="inline" id="S3.SS2.p6.7.m7.2"><semantics id="S3.SS2.p6.7.m7.2a"><mrow id="S3.SS2.p6.7.m7.2.2.2" xref="S3.SS2.p6.7.m7.2.2.3.cmml"><mrow id="S3.SS2.p6.7.m7.1.1.1.1" xref="S3.SS2.p6.7.m7.1.1.1.1.cmml"><mi id="S3.SS2.p6.7.m7.1.1.1.1.2" xref="S3.SS2.p6.7.m7.1.1.1.1.2.cmml">s</mi><mo id="S3.SS2.p6.7.m7.1.1.1.1.1" xref="S3.SS2.p6.7.m7.1.1.1.1.1.cmml">∈</mo><mi id="S3.SS2.p6.7.m7.1.1.1.1.3" xref="S3.SS2.p6.7.m7.1.1.1.1.3.cmml">S</mi></mrow><mo id="S3.SS2.p6.7.m7.2.2.2.3" rspace="0.497em" xref="S3.SS2.p6.7.m7.2.2.3a.cmml">,</mo><mrow id="S3.SS2.p6.7.m7.2.2.2.2" xref="S3.SS2.p6.7.m7.2.2.2.2.cmml"><mi id="S3.SS2.p6.7.m7.2.2.2.2.2" xref="S3.SS2.p6.7.m7.2.2.2.2.2.cmml">t</mi><mo id="S3.SS2.p6.7.m7.2.2.2.2.1" xref="S3.SS2.p6.7.m7.2.2.2.2.1.cmml">∈</mo><mi id="S3.SS2.p6.7.m7.2.2.2.2.3" xref="S3.SS2.p6.7.m7.2.2.2.2.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.7.m7.2b"><apply id="S3.SS2.p6.7.m7.2.2.3.cmml" xref="S3.SS2.p6.7.m7.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p6.7.m7.2.2.3a.cmml" xref="S3.SS2.p6.7.m7.2.2.2.3">formulae-sequence</csymbol><apply id="S3.SS2.p6.7.m7.1.1.1.1.cmml" xref="S3.SS2.p6.7.m7.1.1.1.1"><in id="S3.SS2.p6.7.m7.1.1.1.1.1.cmml" xref="S3.SS2.p6.7.m7.1.1.1.1.1"></in><ci id="S3.SS2.p6.7.m7.1.1.1.1.2.cmml" xref="S3.SS2.p6.7.m7.1.1.1.1.2">𝑠</ci><ci id="S3.SS2.p6.7.m7.1.1.1.1.3.cmml" xref="S3.SS2.p6.7.m7.1.1.1.1.3">𝑆</ci></apply><apply id="S3.SS2.p6.7.m7.2.2.2.2.cmml" xref="S3.SS2.p6.7.m7.2.2.2.2"><in id="S3.SS2.p6.7.m7.2.2.2.2.1.cmml" xref="S3.SS2.p6.7.m7.2.2.2.2.1"></in><ci id="S3.SS2.p6.7.m7.2.2.2.2.2.cmml" xref="S3.SS2.p6.7.m7.2.2.2.2.2">𝑡</ci><ci id="S3.SS2.p6.7.m7.2.2.2.2.3.cmml" xref="S3.SS2.p6.7.m7.2.2.2.2.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.7.m7.2c">s\in S,~{}t\in T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.7.m7.2d">italic_s ∈ italic_S , italic_t ∈ italic_T</annotation></semantics></math>. Let <math alttext="E(S,T):=E\cap(S\times T)" class="ltx_Math" display="inline" id="S3.SS2.p6.8.m8.3"><semantics id="S3.SS2.p6.8.m8.3a"><mrow id="S3.SS2.p6.8.m8.3.3" xref="S3.SS2.p6.8.m8.3.3.cmml"><mrow id="S3.SS2.p6.8.m8.3.3.3" xref="S3.SS2.p6.8.m8.3.3.3.cmml"><mi id="S3.SS2.p6.8.m8.3.3.3.2" xref="S3.SS2.p6.8.m8.3.3.3.2.cmml">E</mi><mo id="S3.SS2.p6.8.m8.3.3.3.1" xref="S3.SS2.p6.8.m8.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS2.p6.8.m8.3.3.3.3.2" xref="S3.SS2.p6.8.m8.3.3.3.3.1.cmml"><mo id="S3.SS2.p6.8.m8.3.3.3.3.2.1" stretchy="false" xref="S3.SS2.p6.8.m8.3.3.3.3.1.cmml">(</mo><mi id="S3.SS2.p6.8.m8.1.1" xref="S3.SS2.p6.8.m8.1.1.cmml">S</mi><mo id="S3.SS2.p6.8.m8.3.3.3.3.2.2" xref="S3.SS2.p6.8.m8.3.3.3.3.1.cmml">,</mo><mi id="S3.SS2.p6.8.m8.2.2" xref="S3.SS2.p6.8.m8.2.2.cmml">T</mi><mo id="S3.SS2.p6.8.m8.3.3.3.3.2.3" rspace="0.278em" stretchy="false" xref="S3.SS2.p6.8.m8.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p6.8.m8.3.3.2" rspace="0.278em" xref="S3.SS2.p6.8.m8.3.3.2.cmml">:=</mo><mrow id="S3.SS2.p6.8.m8.3.3.1" xref="S3.SS2.p6.8.m8.3.3.1.cmml"><mi id="S3.SS2.p6.8.m8.3.3.1.3" xref="S3.SS2.p6.8.m8.3.3.1.3.cmml">E</mi><mo id="S3.SS2.p6.8.m8.3.3.1.2" xref="S3.SS2.p6.8.m8.3.3.1.2.cmml">∩</mo><mrow id="S3.SS2.p6.8.m8.3.3.1.1.1" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.cmml"><mo id="S3.SS2.p6.8.m8.3.3.1.1.1.2" stretchy="false" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p6.8.m8.3.3.1.1.1.1" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.cmml"><mi id="S3.SS2.p6.8.m8.3.3.1.1.1.1.2" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.2.cmml">S</mi><mo id="S3.SS2.p6.8.m8.3.3.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.1.cmml">×</mo><mi id="S3.SS2.p6.8.m8.3.3.1.1.1.1.3" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.3.cmml">T</mi></mrow><mo id="S3.SS2.p6.8.m8.3.3.1.1.1.3" stretchy="false" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.8.m8.3b"><apply id="S3.SS2.p6.8.m8.3.3.cmml" xref="S3.SS2.p6.8.m8.3.3"><csymbol cd="latexml" id="S3.SS2.p6.8.m8.3.3.2.cmml" xref="S3.SS2.p6.8.m8.3.3.2">assign</csymbol><apply id="S3.SS2.p6.8.m8.3.3.3.cmml" xref="S3.SS2.p6.8.m8.3.3.3"><times id="S3.SS2.p6.8.m8.3.3.3.1.cmml" xref="S3.SS2.p6.8.m8.3.3.3.1"></times><ci id="S3.SS2.p6.8.m8.3.3.3.2.cmml" xref="S3.SS2.p6.8.m8.3.3.3.2">𝐸</ci><interval closure="open" id="S3.SS2.p6.8.m8.3.3.3.3.1.cmml" xref="S3.SS2.p6.8.m8.3.3.3.3.2"><ci id="S3.SS2.p6.8.m8.1.1.cmml" xref="S3.SS2.p6.8.m8.1.1">𝑆</ci><ci id="S3.SS2.p6.8.m8.2.2.cmml" xref="S3.SS2.p6.8.m8.2.2">𝑇</ci></interval></apply><apply id="S3.SS2.p6.8.m8.3.3.1.cmml" xref="S3.SS2.p6.8.m8.3.3.1"><intersect id="S3.SS2.p6.8.m8.3.3.1.2.cmml" xref="S3.SS2.p6.8.m8.3.3.1.2"></intersect><ci id="S3.SS2.p6.8.m8.3.3.1.3.cmml" xref="S3.SS2.p6.8.m8.3.3.1.3">𝐸</ci><apply id="S3.SS2.p6.8.m8.3.3.1.1.1.1.cmml" xref="S3.SS2.p6.8.m8.3.3.1.1.1"><times id="S3.SS2.p6.8.m8.3.3.1.1.1.1.1.cmml" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.1"></times><ci id="S3.SS2.p6.8.m8.3.3.1.1.1.1.2.cmml" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.2">𝑆</ci><ci id="S3.SS2.p6.8.m8.3.3.1.1.1.1.3.cmml" xref="S3.SS2.p6.8.m8.3.3.1.1.1.1.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.8.m8.3c">E(S,T):=E\cap(S\times T)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.8.m8.3d">italic_E ( italic_S , italic_T ) := italic_E ∩ ( italic_S × italic_T )</annotation></semantics></math> and <math alttext="|(S,T)|=|E(S\times T)|" class="ltx_Math" display="inline" id="S3.SS2.p6.9.m9.4"><semantics id="S3.SS2.p6.9.m9.4a"><mrow id="S3.SS2.p6.9.m9.4.4" xref="S3.SS2.p6.9.m9.4.4.cmml"><mrow id="S3.SS2.p6.9.m9.3.3.1.1" xref="S3.SS2.p6.9.m9.3.3.1.2.cmml"><mo id="S3.SS2.p6.9.m9.3.3.1.1.2" stretchy="false" xref="S3.SS2.p6.9.m9.3.3.1.2.1.cmml">|</mo><mrow id="S3.SS2.p6.9.m9.3.3.1.1.1.2" xref="S3.SS2.p6.9.m9.3.3.1.1.1.1.cmml"><mo id="S3.SS2.p6.9.m9.3.3.1.1.1.2.1" stretchy="false" xref="S3.SS2.p6.9.m9.3.3.1.1.1.1.cmml">(</mo><mi id="S3.SS2.p6.9.m9.1.1" xref="S3.SS2.p6.9.m9.1.1.cmml">S</mi><mo id="S3.SS2.p6.9.m9.3.3.1.1.1.2.2" xref="S3.SS2.p6.9.m9.3.3.1.1.1.1.cmml">,</mo><mi id="S3.SS2.p6.9.m9.2.2" xref="S3.SS2.p6.9.m9.2.2.cmml">T</mi><mo id="S3.SS2.p6.9.m9.3.3.1.1.1.2.3" stretchy="false" xref="S3.SS2.p6.9.m9.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS2.p6.9.m9.3.3.1.1.3" stretchy="false" xref="S3.SS2.p6.9.m9.3.3.1.2.1.cmml">|</mo></mrow><mo id="S3.SS2.p6.9.m9.4.4.3" xref="S3.SS2.p6.9.m9.4.4.3.cmml">=</mo><mrow id="S3.SS2.p6.9.m9.4.4.2.1" xref="S3.SS2.p6.9.m9.4.4.2.2.cmml"><mo id="S3.SS2.p6.9.m9.4.4.2.1.2" stretchy="false" xref="S3.SS2.p6.9.m9.4.4.2.2.1.cmml">|</mo><mrow id="S3.SS2.p6.9.m9.4.4.2.1.1" xref="S3.SS2.p6.9.m9.4.4.2.1.1.cmml"><mi id="S3.SS2.p6.9.m9.4.4.2.1.1.3" xref="S3.SS2.p6.9.m9.4.4.2.1.1.3.cmml">E</mi><mo id="S3.SS2.p6.9.m9.4.4.2.1.1.2" xref="S3.SS2.p6.9.m9.4.4.2.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.cmml"><mo id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.2" stretchy="false" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.cmml"><mi id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.2" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.2.cmml">S</mi><mo id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.1.cmml">×</mo><mi id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.3" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.3.cmml">T</mi></mrow><mo id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.3" stretchy="false" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p6.9.m9.4.4.2.1.3" stretchy="false" xref="S3.SS2.p6.9.m9.4.4.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.9.m9.4b"><apply id="S3.SS2.p6.9.m9.4.4.cmml" xref="S3.SS2.p6.9.m9.4.4"><eq id="S3.SS2.p6.9.m9.4.4.3.cmml" xref="S3.SS2.p6.9.m9.4.4.3"></eq><apply id="S3.SS2.p6.9.m9.3.3.1.2.cmml" xref="S3.SS2.p6.9.m9.3.3.1.1"><abs id="S3.SS2.p6.9.m9.3.3.1.2.1.cmml" xref="S3.SS2.p6.9.m9.3.3.1.1.2"></abs><interval closure="open" id="S3.SS2.p6.9.m9.3.3.1.1.1.1.cmml" xref="S3.SS2.p6.9.m9.3.3.1.1.1.2"><ci id="S3.SS2.p6.9.m9.1.1.cmml" xref="S3.SS2.p6.9.m9.1.1">𝑆</ci><ci id="S3.SS2.p6.9.m9.2.2.cmml" xref="S3.SS2.p6.9.m9.2.2">𝑇</ci></interval></apply><apply id="S3.SS2.p6.9.m9.4.4.2.2.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1"><abs id="S3.SS2.p6.9.m9.4.4.2.2.1.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.2"></abs><apply id="S3.SS2.p6.9.m9.4.4.2.1.1.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.1"><times id="S3.SS2.p6.9.m9.4.4.2.1.1.2.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.1.2"></times><ci id="S3.SS2.p6.9.m9.4.4.2.1.1.3.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.1.3">𝐸</ci><apply id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1"><times id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.1.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.1"></times><ci id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.2.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.2">𝑆</ci><ci id="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.3.cmml" xref="S3.SS2.p6.9.m9.4.4.2.1.1.1.1.1.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.9.m9.4c">|(S,T)|=|E(S\times T)|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.9.m9.4d">| ( italic_S , italic_T ) | = | italic_E ( italic_S × italic_T ) |</annotation></semantics></math>, and let <math alttext="m_{c}(G)=\min_{S,T}|(S,T)|" class="ltx_Math" display="inline" id="S3.SS2.p6.10.m10.6"><semantics id="S3.SS2.p6.10.m10.6a"><mrow id="S3.SS2.p6.10.m10.6.6" xref="S3.SS2.p6.10.m10.6.6.cmml"><mrow id="S3.SS2.p6.10.m10.6.6.3" xref="S3.SS2.p6.10.m10.6.6.3.cmml"><msub id="S3.SS2.p6.10.m10.6.6.3.2" xref="S3.SS2.p6.10.m10.6.6.3.2.cmml"><mi id="S3.SS2.p6.10.m10.6.6.3.2.2" xref="S3.SS2.p6.10.m10.6.6.3.2.2.cmml">m</mi><mi id="S3.SS2.p6.10.m10.6.6.3.2.3" xref="S3.SS2.p6.10.m10.6.6.3.2.3.cmml">c</mi></msub><mo id="S3.SS2.p6.10.m10.6.6.3.1" xref="S3.SS2.p6.10.m10.6.6.3.1.cmml">⁢</mo><mrow id="S3.SS2.p6.10.m10.6.6.3.3.2" xref="S3.SS2.p6.10.m10.6.6.3.cmml"><mo id="S3.SS2.p6.10.m10.6.6.3.3.2.1" stretchy="false" xref="S3.SS2.p6.10.m10.6.6.3.cmml">(</mo><mi id="S3.SS2.p6.10.m10.3.3" xref="S3.SS2.p6.10.m10.3.3.cmml">G</mi><mo id="S3.SS2.p6.10.m10.6.6.3.3.2.2" stretchy="false" xref="S3.SS2.p6.10.m10.6.6.3.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p6.10.m10.6.6.2" xref="S3.SS2.p6.10.m10.6.6.2.cmml">=</mo><mrow id="S3.SS2.p6.10.m10.6.6.1" xref="S3.SS2.p6.10.m10.6.6.1.cmml"><msub id="S3.SS2.p6.10.m10.6.6.1.2" xref="S3.SS2.p6.10.m10.6.6.1.2.cmml"><mi id="S3.SS2.p6.10.m10.6.6.1.2.2" xref="S3.SS2.p6.10.m10.6.6.1.2.2.cmml">min</mi><mrow id="S3.SS2.p6.10.m10.2.2.2.4" xref="S3.SS2.p6.10.m10.2.2.2.3.cmml"><mi id="S3.SS2.p6.10.m10.1.1.1.1" xref="S3.SS2.p6.10.m10.1.1.1.1.cmml">S</mi><mo id="S3.SS2.p6.10.m10.2.2.2.4.1" xref="S3.SS2.p6.10.m10.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p6.10.m10.2.2.2.2" xref="S3.SS2.p6.10.m10.2.2.2.2.cmml">T</mi></mrow></msub><mo id="S3.SS2.p6.10.m10.6.6.1a" xref="S3.SS2.p6.10.m10.6.6.1.cmml">⁡</mo><mrow id="S3.SS2.p6.10.m10.6.6.1.1.1" xref="S3.SS2.p6.10.m10.6.6.1.1.2.cmml"><mo id="S3.SS2.p6.10.m10.6.6.1.1.1.2" stretchy="false" xref="S3.SS2.p6.10.m10.6.6.1.1.2.1.cmml">|</mo><mrow id="S3.SS2.p6.10.m10.6.6.1.1.1.1.2" xref="S3.SS2.p6.10.m10.6.6.1.1.1.1.1.cmml"><mo id="S3.SS2.p6.10.m10.6.6.1.1.1.1.2.1" stretchy="false" xref="S3.SS2.p6.10.m10.6.6.1.1.1.1.1.cmml">(</mo><mi id="S3.SS2.p6.10.m10.4.4" xref="S3.SS2.p6.10.m10.4.4.cmml">S</mi><mo id="S3.SS2.p6.10.m10.6.6.1.1.1.1.2.2" xref="S3.SS2.p6.10.m10.6.6.1.1.1.1.1.cmml">,</mo><mi id="S3.SS2.p6.10.m10.5.5" xref="S3.SS2.p6.10.m10.5.5.cmml">T</mi><mo id="S3.SS2.p6.10.m10.6.6.1.1.1.1.2.3" stretchy="false" xref="S3.SS2.p6.10.m10.6.6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS2.p6.10.m10.6.6.1.1.1.3" stretchy="false" xref="S3.SS2.p6.10.m10.6.6.1.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.10.m10.6b"><apply id="S3.SS2.p6.10.m10.6.6.cmml" xref="S3.SS2.p6.10.m10.6.6"><eq id="S3.SS2.p6.10.m10.6.6.2.cmml" xref="S3.SS2.p6.10.m10.6.6.2"></eq><apply id="S3.SS2.p6.10.m10.6.6.3.cmml" xref="S3.SS2.p6.10.m10.6.6.3"><times id="S3.SS2.p6.10.m10.6.6.3.1.cmml" xref="S3.SS2.p6.10.m10.6.6.3.1"></times><apply id="S3.SS2.p6.10.m10.6.6.3.2.cmml" xref="S3.SS2.p6.10.m10.6.6.3.2"><csymbol cd="ambiguous" id="S3.SS2.p6.10.m10.6.6.3.2.1.cmml" xref="S3.SS2.p6.10.m10.6.6.3.2">subscript</csymbol><ci id="S3.SS2.p6.10.m10.6.6.3.2.2.cmml" xref="S3.SS2.p6.10.m10.6.6.3.2.2">𝑚</ci><ci id="S3.SS2.p6.10.m10.6.6.3.2.3.cmml" xref="S3.SS2.p6.10.m10.6.6.3.2.3">𝑐</ci></apply><ci id="S3.SS2.p6.10.m10.3.3.cmml" xref="S3.SS2.p6.10.m10.3.3">𝐺</ci></apply><apply id="S3.SS2.p6.10.m10.6.6.1.cmml" xref="S3.SS2.p6.10.m10.6.6.1"><apply id="S3.SS2.p6.10.m10.6.6.1.2.cmml" xref="S3.SS2.p6.10.m10.6.6.1.2"><csymbol cd="ambiguous" id="S3.SS2.p6.10.m10.6.6.1.2.1.cmml" xref="S3.SS2.p6.10.m10.6.6.1.2">subscript</csymbol><min id="S3.SS2.p6.10.m10.6.6.1.2.2.cmml" xref="S3.SS2.p6.10.m10.6.6.1.2.2"></min><list id="S3.SS2.p6.10.m10.2.2.2.3.cmml" xref="S3.SS2.p6.10.m10.2.2.2.4"><ci id="S3.SS2.p6.10.m10.1.1.1.1.cmml" xref="S3.SS2.p6.10.m10.1.1.1.1">𝑆</ci><ci id="S3.SS2.p6.10.m10.2.2.2.2.cmml" xref="S3.SS2.p6.10.m10.2.2.2.2">𝑇</ci></list></apply><apply id="S3.SS2.p6.10.m10.6.6.1.1.2.cmml" xref="S3.SS2.p6.10.m10.6.6.1.1.1"><abs id="S3.SS2.p6.10.m10.6.6.1.1.2.1.cmml" xref="S3.SS2.p6.10.m10.6.6.1.1.1.2"></abs><interval closure="open" id="S3.SS2.p6.10.m10.6.6.1.1.1.1.1.cmml" xref="S3.SS2.p6.10.m10.6.6.1.1.1.1.2"><ci id="S3.SS2.p6.10.m10.4.4.cmml" xref="S3.SS2.p6.10.m10.4.4">𝑆</ci><ci id="S3.SS2.p6.10.m10.5.5.cmml" xref="S3.SS2.p6.10.m10.5.5">𝑇</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.10.m10.6c">m_{c}(G)=\min_{S,T}|(S,T)|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.10.m10.6d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G ) = roman_min start_POSTSUBSCRIPT italic_S , italic_T end_POSTSUBSCRIPT | ( italic_S , italic_T ) |</annotation></semantics></math>, the size of a mincut of <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p6.11.m11.1"><semantics id="S3.SS2.p6.11.m11.1a"><mi id="S3.SS2.p6.11.m11.1.1" xref="S3.SS2.p6.11.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.11.m11.1b"><ci id="S3.SS2.p6.11.m11.1.1.cmml" xref="S3.SS2.p6.11.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.11.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.11.m11.1d">italic_G</annotation></semantics></math>. We write <math alttext="m_{c}" class="ltx_Math" display="inline" id="S3.SS2.p6.12.m12.1"><semantics id="S3.SS2.p6.12.m12.1a"><msub id="S3.SS2.p6.12.m12.1.1" xref="S3.SS2.p6.12.m12.1.1.cmml"><mi id="S3.SS2.p6.12.m12.1.1.2" xref="S3.SS2.p6.12.m12.1.1.2.cmml">m</mi><mi id="S3.SS2.p6.12.m12.1.1.3" xref="S3.SS2.p6.12.m12.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.12.m12.1b"><apply id="S3.SS2.p6.12.m12.1.1.cmml" xref="S3.SS2.p6.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS2.p6.12.m12.1.1.1.cmml" xref="S3.SS2.p6.12.m12.1.1">subscript</csymbol><ci id="S3.SS2.p6.12.m12.1.1.2.cmml" xref="S3.SS2.p6.12.m12.1.1.2">𝑚</ci><ci id="S3.SS2.p6.12.m12.1.1.3.cmml" xref="S3.SS2.p6.12.m12.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.12.m12.1c">m_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.12.m12.1d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> in short when <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p6.13.m13.1"><semantics id="S3.SS2.p6.13.m13.1a"><mi id="S3.SS2.p6.13.m13.1.1" xref="S3.SS2.p6.13.m13.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.13.m13.1b"><ci id="S3.SS2.p6.13.m13.1.1.cmml" xref="S3.SS2.p6.13.m13.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.13.m13.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.13.m13.1d">italic_G</annotation></semantics></math> is known from context.</p> </div> <div class="ltx_para" id="S3.SS2.p7"> <p class="ltx_p" id="S3.SS2.p7.10">We rephrase <math alttext="m_{c}" class="ltx_Math" display="inline" id="S3.SS2.p7.1.m1.1"><semantics id="S3.SS2.p7.1.m1.1a"><msub id="S3.SS2.p7.1.m1.1.1" xref="S3.SS2.p7.1.m1.1.1.cmml"><mi id="S3.SS2.p7.1.m1.1.1.2" xref="S3.SS2.p7.1.m1.1.1.2.cmml">m</mi><mi id="S3.SS2.p7.1.m1.1.1.3" xref="S3.SS2.p7.1.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.1.m1.1b"><apply id="S3.SS2.p7.1.m1.1.1.cmml" xref="S3.SS2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p7.1.m1.1.1.1.cmml" xref="S3.SS2.p7.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p7.1.m1.1.1.2.cmml" xref="S3.SS2.p7.1.m1.1.1.2">𝑚</ci><ci id="S3.SS2.p7.1.m1.1.1.3.cmml" xref="S3.SS2.p7.1.m1.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.1.m1.1c">m_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> in terms of maximal integer flows (integer flows which cannot be extended). Let <math alttext="MF(G)" class="ltx_Math" display="inline" id="S3.SS2.p7.2.m2.1"><semantics id="S3.SS2.p7.2.m2.1a"><mrow id="S3.SS2.p7.2.m2.1.2" xref="S3.SS2.p7.2.m2.1.2.cmml"><mi id="S3.SS2.p7.2.m2.1.2.2" xref="S3.SS2.p7.2.m2.1.2.2.cmml">M</mi><mo id="S3.SS2.p7.2.m2.1.2.1" xref="S3.SS2.p7.2.m2.1.2.1.cmml">⁢</mo><mi id="S3.SS2.p7.2.m2.1.2.3" xref="S3.SS2.p7.2.m2.1.2.3.cmml">F</mi><mo id="S3.SS2.p7.2.m2.1.2.1a" xref="S3.SS2.p7.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p7.2.m2.1.2.4.2" xref="S3.SS2.p7.2.m2.1.2.cmml"><mo id="S3.SS2.p7.2.m2.1.2.4.2.1" stretchy="false" xref="S3.SS2.p7.2.m2.1.2.cmml">(</mo><mi id="S3.SS2.p7.2.m2.1.1" xref="S3.SS2.p7.2.m2.1.1.cmml">G</mi><mo id="S3.SS2.p7.2.m2.1.2.4.2.2" stretchy="false" xref="S3.SS2.p7.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.2.m2.1b"><apply id="S3.SS2.p7.2.m2.1.2.cmml" xref="S3.SS2.p7.2.m2.1.2"><times id="S3.SS2.p7.2.m2.1.2.1.cmml" xref="S3.SS2.p7.2.m2.1.2.1"></times><ci id="S3.SS2.p7.2.m2.1.2.2.cmml" xref="S3.SS2.p7.2.m2.1.2.2">𝑀</ci><ci id="S3.SS2.p7.2.m2.1.2.3.cmml" xref="S3.SS2.p7.2.m2.1.2.3">𝐹</ci><ci id="S3.SS2.p7.2.m2.1.1.cmml" xref="S3.SS2.p7.2.m2.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.2.m2.1c">MF(G)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.2.m2.1d">italic_M italic_F ( italic_G )</annotation></semantics></math> be the set of maximal integer (here, binary) <math alttext="s-t" class="ltx_Math" display="inline" id="S3.SS2.p7.3.m3.1"><semantics id="S3.SS2.p7.3.m3.1a"><mrow id="S3.SS2.p7.3.m3.1.1" xref="S3.SS2.p7.3.m3.1.1.cmml"><mi id="S3.SS2.p7.3.m3.1.1.2" xref="S3.SS2.p7.3.m3.1.1.2.cmml">s</mi><mo id="S3.SS2.p7.3.m3.1.1.1" xref="S3.SS2.p7.3.m3.1.1.1.cmml">−</mo><mi id="S3.SS2.p7.3.m3.1.1.3" xref="S3.SS2.p7.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.3.m3.1b"><apply id="S3.SS2.p7.3.m3.1.1.cmml" xref="S3.SS2.p7.3.m3.1.1"><minus id="S3.SS2.p7.3.m3.1.1.1.cmml" xref="S3.SS2.p7.3.m3.1.1.1"></minus><ci id="S3.SS2.p7.3.m3.1.1.2.cmml" xref="S3.SS2.p7.3.m3.1.1.2">𝑠</ci><ci id="S3.SS2.p7.3.m3.1.1.3.cmml" xref="S3.SS2.p7.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.3.m3.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.3.m3.1d">italic_s - italic_t</annotation></semantics></math> flows on <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p7.4.m4.1"><semantics id="S3.SS2.p7.4.m4.1a"><mi id="S3.SS2.p7.4.m4.1.1" xref="S3.SS2.p7.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.4.m4.1b"><ci id="S3.SS2.p7.4.m4.1.1.cmml" xref="S3.SS2.p7.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.4.m4.1d">italic_G</annotation></semantics></math>. Again we write <math alttext="MF" class="ltx_Math" display="inline" id="S3.SS2.p7.5.m5.1"><semantics id="S3.SS2.p7.5.m5.1a"><mrow id="S3.SS2.p7.5.m5.1.1" xref="S3.SS2.p7.5.m5.1.1.cmml"><mi id="S3.SS2.p7.5.m5.1.1.2" xref="S3.SS2.p7.5.m5.1.1.2.cmml">M</mi><mo id="S3.SS2.p7.5.m5.1.1.1" xref="S3.SS2.p7.5.m5.1.1.1.cmml">⁢</mo><mi id="S3.SS2.p7.5.m5.1.1.3" xref="S3.SS2.p7.5.m5.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.5.m5.1b"><apply id="S3.SS2.p7.5.m5.1.1.cmml" xref="S3.SS2.p7.5.m5.1.1"><times id="S3.SS2.p7.5.m5.1.1.1.cmml" xref="S3.SS2.p7.5.m5.1.1.1"></times><ci id="S3.SS2.p7.5.m5.1.1.2.cmml" xref="S3.SS2.p7.5.m5.1.1.2">𝑀</ci><ci id="S3.SS2.p7.5.m5.1.1.3.cmml" xref="S3.SS2.p7.5.m5.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.5.m5.1c">MF</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.5.m5.1d">italic_M italic_F</annotation></semantics></math> when <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p7.6.m6.1"><semantics id="S3.SS2.p7.6.m6.1a"><mi id="S3.SS2.p7.6.m6.1.1" xref="S3.SS2.p7.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.6.m6.1b"><ci id="S3.SS2.p7.6.m6.1.1.cmml" xref="S3.SS2.p7.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.6.m6.1d">italic_G</annotation></semantics></math> is known from context. By the maxflow-mincut theorem, <math alttext="m_{c}=\max_{f\in MF}|f|" class="ltx_Math" display="inline" id="S3.SS2.p7.7.m7.1"><semantics id="S3.SS2.p7.7.m7.1a"><mrow id="S3.SS2.p7.7.m7.1.2" xref="S3.SS2.p7.7.m7.1.2.cmml"><msub id="S3.SS2.p7.7.m7.1.2.2" xref="S3.SS2.p7.7.m7.1.2.2.cmml"><mi id="S3.SS2.p7.7.m7.1.2.2.2" xref="S3.SS2.p7.7.m7.1.2.2.2.cmml">m</mi><mi id="S3.SS2.p7.7.m7.1.2.2.3" xref="S3.SS2.p7.7.m7.1.2.2.3.cmml">c</mi></msub><mo id="S3.SS2.p7.7.m7.1.2.1" xref="S3.SS2.p7.7.m7.1.2.1.cmml">=</mo><mrow id="S3.SS2.p7.7.m7.1.2.3" xref="S3.SS2.p7.7.m7.1.2.3.cmml"><msub id="S3.SS2.p7.7.m7.1.2.3.1" xref="S3.SS2.p7.7.m7.1.2.3.1.cmml"><mi id="S3.SS2.p7.7.m7.1.2.3.1.2" xref="S3.SS2.p7.7.m7.1.2.3.1.2.cmml">max</mi><mrow id="S3.SS2.p7.7.m7.1.2.3.1.3" xref="S3.SS2.p7.7.m7.1.2.3.1.3.cmml"><mi id="S3.SS2.p7.7.m7.1.2.3.1.3.2" xref="S3.SS2.p7.7.m7.1.2.3.1.3.2.cmml">f</mi><mo id="S3.SS2.p7.7.m7.1.2.3.1.3.1" xref="S3.SS2.p7.7.m7.1.2.3.1.3.1.cmml">∈</mo><mrow id="S3.SS2.p7.7.m7.1.2.3.1.3.3" xref="S3.SS2.p7.7.m7.1.2.3.1.3.3.cmml"><mi id="S3.SS2.p7.7.m7.1.2.3.1.3.3.2" xref="S3.SS2.p7.7.m7.1.2.3.1.3.3.2.cmml">M</mi><mo id="S3.SS2.p7.7.m7.1.2.3.1.3.3.1" xref="S3.SS2.p7.7.m7.1.2.3.1.3.3.1.cmml">⁢</mo><mi id="S3.SS2.p7.7.m7.1.2.3.1.3.3.3" xref="S3.SS2.p7.7.m7.1.2.3.1.3.3.3.cmml">F</mi></mrow></mrow></msub><mo id="S3.SS2.p7.7.m7.1.2.3a" xref="S3.SS2.p7.7.m7.1.2.3.cmml">⁡</mo><mrow id="S3.SS2.p7.7.m7.1.2.3.2.2" xref="S3.SS2.p7.7.m7.1.2.3.2.1.cmml"><mo id="S3.SS2.p7.7.m7.1.2.3.2.2.1" stretchy="false" xref="S3.SS2.p7.7.m7.1.2.3.2.1.1.cmml">|</mo><mi id="S3.SS2.p7.7.m7.1.1" xref="S3.SS2.p7.7.m7.1.1.cmml">f</mi><mo id="S3.SS2.p7.7.m7.1.2.3.2.2.2" stretchy="false" xref="S3.SS2.p7.7.m7.1.2.3.2.1.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.7.m7.1b"><apply id="S3.SS2.p7.7.m7.1.2.cmml" xref="S3.SS2.p7.7.m7.1.2"><eq id="S3.SS2.p7.7.m7.1.2.1.cmml" xref="S3.SS2.p7.7.m7.1.2.1"></eq><apply id="S3.SS2.p7.7.m7.1.2.2.cmml" 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xref="S3.SS2.p7.7.m7.1.2.3.1.3.3.1"></times><ci id="S3.SS2.p7.7.m7.1.2.3.1.3.3.2.cmml" xref="S3.SS2.p7.7.m7.1.2.3.1.3.3.2">𝑀</ci><ci id="S3.SS2.p7.7.m7.1.2.3.1.3.3.3.cmml" xref="S3.SS2.p7.7.m7.1.2.3.1.3.3.3">𝐹</ci></apply></apply></apply><apply id="S3.SS2.p7.7.m7.1.2.3.2.1.cmml" xref="S3.SS2.p7.7.m7.1.2.3.2.2"><abs id="S3.SS2.p7.7.m7.1.2.3.2.1.1.cmml" xref="S3.SS2.p7.7.m7.1.2.3.2.2.1"></abs><ci id="S3.SS2.p7.7.m7.1.1.cmml" xref="S3.SS2.p7.7.m7.1.1">𝑓</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.7.m7.1c">m_{c}=\max_{f\in MF}|f|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.7.m7.1d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = roman_max start_POSTSUBSCRIPT italic_f ∈ italic_M italic_F end_POSTSUBSCRIPT | italic_f |</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span><math alttext="G" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">G</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_G</annotation></semantics></math> can be converted to a simple graph (where <math alttext="E" class="ltx_Math" display="inline" id="footnote2.m2.1"><semantics id="footnote2.m2.1b"><mi id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><ci id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">E</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.1e">italic_E</annotation></semantics></math> is not a multiset), and then the theorem holds, and when going back to the multigraph the fluxes and the cut sizes remain the same.</span></span></span> We distinguish between maximal flows so that we have a mapping from each such flow to a unique maximal subset of edge-disjoint paths. <math alttext="MF" class="ltx_Math" display="inline" id="S3.SS2.p7.8.m8.1"><semantics id="S3.SS2.p7.8.m8.1a"><mrow id="S3.SS2.p7.8.m8.1.1" xref="S3.SS2.p7.8.m8.1.1.cmml"><mi id="S3.SS2.p7.8.m8.1.1.2" xref="S3.SS2.p7.8.m8.1.1.2.cmml">M</mi><mo id="S3.SS2.p7.8.m8.1.1.1" xref="S3.SS2.p7.8.m8.1.1.1.cmml">⁢</mo><mi id="S3.SS2.p7.8.m8.1.1.3" xref="S3.SS2.p7.8.m8.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.8.m8.1b"><apply id="S3.SS2.p7.8.m8.1.1.cmml" xref="S3.SS2.p7.8.m8.1.1"><times id="S3.SS2.p7.8.m8.1.1.1.cmml" xref="S3.SS2.p7.8.m8.1.1.1"></times><ci id="S3.SS2.p7.8.m8.1.1.2.cmml" xref="S3.SS2.p7.8.m8.1.1.2">𝑀</ci><ci id="S3.SS2.p7.8.m8.1.1.3.cmml" xref="S3.SS2.p7.8.m8.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.8.m8.1c">MF</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.8.m8.1d">italic_M italic_F</annotation></semantics></math> changes but <math alttext="m_{c}" class="ltx_Math" display="inline" id="S3.SS2.p7.9.m9.1"><semantics id="S3.SS2.p7.9.m9.1a"><msub id="S3.SS2.p7.9.m9.1.1" xref="S3.SS2.p7.9.m9.1.1.cmml"><mi id="S3.SS2.p7.9.m9.1.1.2" xref="S3.SS2.p7.9.m9.1.1.2.cmml">m</mi><mi id="S3.SS2.p7.9.m9.1.1.3" xref="S3.SS2.p7.9.m9.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.9.m9.1b"><apply id="S3.SS2.p7.9.m9.1.1.cmml" xref="S3.SS2.p7.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS2.p7.9.m9.1.1.1.cmml" xref="S3.SS2.p7.9.m9.1.1">subscript</csymbol><ci id="S3.SS2.p7.9.m9.1.1.2.cmml" xref="S3.SS2.p7.9.m9.1.1.2">𝑚</ci><ci id="S3.SS2.p7.9.m9.1.1.3.cmml" xref="S3.SS2.p7.9.m9.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.9.m9.1c">m_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.9.m9.1d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\max_{f\in MF}|f|" class="ltx_Math" display="inline" id="S3.SS2.p7.10.m10.1"><semantics id="S3.SS2.p7.10.m10.1a"><mrow id="S3.SS2.p7.10.m10.1.2" xref="S3.SS2.p7.10.m10.1.2.cmml"><msub id="S3.SS2.p7.10.m10.1.2.1" xref="S3.SS2.p7.10.m10.1.2.1.cmml"><mi id="S3.SS2.p7.10.m10.1.2.1.2" xref="S3.SS2.p7.10.m10.1.2.1.2.cmml">max</mi><mrow id="S3.SS2.p7.10.m10.1.2.1.3" xref="S3.SS2.p7.10.m10.1.2.1.3.cmml"><mi id="S3.SS2.p7.10.m10.1.2.1.3.2" xref="S3.SS2.p7.10.m10.1.2.1.3.2.cmml">f</mi><mo id="S3.SS2.p7.10.m10.1.2.1.3.1" xref="S3.SS2.p7.10.m10.1.2.1.3.1.cmml">∈</mo><mrow id="S3.SS2.p7.10.m10.1.2.1.3.3" xref="S3.SS2.p7.10.m10.1.2.1.3.3.cmml"><mi id="S3.SS2.p7.10.m10.1.2.1.3.3.2" xref="S3.SS2.p7.10.m10.1.2.1.3.3.2.cmml">M</mi><mo id="S3.SS2.p7.10.m10.1.2.1.3.3.1" xref="S3.SS2.p7.10.m10.1.2.1.3.3.1.cmml">⁢</mo><mi id="S3.SS2.p7.10.m10.1.2.1.3.3.3" xref="S3.SS2.p7.10.m10.1.2.1.3.3.3.cmml">F</mi></mrow></mrow></msub><mo id="S3.SS2.p7.10.m10.1.2a" xref="S3.SS2.p7.10.m10.1.2.cmml">⁡</mo><mrow id="S3.SS2.p7.10.m10.1.2.2.2" xref="S3.SS2.p7.10.m10.1.2.2.1.cmml"><mo id="S3.SS2.p7.10.m10.1.2.2.2.1" stretchy="false" xref="S3.SS2.p7.10.m10.1.2.2.1.1.cmml">|</mo><mi id="S3.SS2.p7.10.m10.1.1" xref="S3.SS2.p7.10.m10.1.1.cmml">f</mi><mo id="S3.SS2.p7.10.m10.1.2.2.2.2" stretchy="false" xref="S3.SS2.p7.10.m10.1.2.2.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.10.m10.1b"><apply id="S3.SS2.p7.10.m10.1.2.cmml" xref="S3.SS2.p7.10.m10.1.2"><apply id="S3.SS2.p7.10.m10.1.2.1.cmml" xref="S3.SS2.p7.10.m10.1.2.1"><csymbol cd="ambiguous" id="S3.SS2.p7.10.m10.1.2.1.1.cmml" xref="S3.SS2.p7.10.m10.1.2.1">subscript</csymbol><max id="S3.SS2.p7.10.m10.1.2.1.2.cmml" xref="S3.SS2.p7.10.m10.1.2.1.2"></max><apply id="S3.SS2.p7.10.m10.1.2.1.3.cmml" xref="S3.SS2.p7.10.m10.1.2.1.3"><in id="S3.SS2.p7.10.m10.1.2.1.3.1.cmml" xref="S3.SS2.p7.10.m10.1.2.1.3.1"></in><ci id="S3.SS2.p7.10.m10.1.2.1.3.2.cmml" xref="S3.SS2.p7.10.m10.1.2.1.3.2">𝑓</ci><apply id="S3.SS2.p7.10.m10.1.2.1.3.3.cmml" xref="S3.SS2.p7.10.m10.1.2.1.3.3"><times id="S3.SS2.p7.10.m10.1.2.1.3.3.1.cmml" xref="S3.SS2.p7.10.m10.1.2.1.3.3.1"></times><ci id="S3.SS2.p7.10.m10.1.2.1.3.3.2.cmml" xref="S3.SS2.p7.10.m10.1.2.1.3.3.2">𝑀</ci><ci id="S3.SS2.p7.10.m10.1.2.1.3.3.3.cmml" xref="S3.SS2.p7.10.m10.1.2.1.3.3.3">𝐹</ci></apply></apply></apply><apply id="S3.SS2.p7.10.m10.1.2.2.1.cmml" xref="S3.SS2.p7.10.m10.1.2.2.2"><abs id="S3.SS2.p7.10.m10.1.2.2.1.1.cmml" xref="S3.SS2.p7.10.m10.1.2.2.2.1"></abs><ci id="S3.SS2.p7.10.m10.1.1.cmml" xref="S3.SS2.p7.10.m10.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.10.m10.1c">\max_{f\in MF}|f|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.10.m10.1d">roman_max start_POSTSUBSCRIPT italic_f ∈ italic_M italic_F end_POSTSUBSCRIPT | italic_f |</annotation></semantics></math> remain the same.</p> </div> <div class="ltx_para" id="S3.SS2.p8"> <p class="ltx_p" id="S3.SS2.p8.8">For an <math alttext="s-t" class="ltx_Math" display="inline" id="S3.SS2.p8.1.m1.1"><semantics id="S3.SS2.p8.1.m1.1a"><mrow id="S3.SS2.p8.1.m1.1.1" xref="S3.SS2.p8.1.m1.1.1.cmml"><mi id="S3.SS2.p8.1.m1.1.1.2" xref="S3.SS2.p8.1.m1.1.1.2.cmml">s</mi><mo id="S3.SS2.p8.1.m1.1.1.1" xref="S3.SS2.p8.1.m1.1.1.1.cmml">−</mo><mi id="S3.SS2.p8.1.m1.1.1.3" xref="S3.SS2.p8.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.1.m1.1b"><apply id="S3.SS2.p8.1.m1.1.1.cmml" xref="S3.SS2.p8.1.m1.1.1"><minus id="S3.SS2.p8.1.m1.1.1.1.cmml" xref="S3.SS2.p8.1.m1.1.1.1"></minus><ci id="S3.SS2.p8.1.m1.1.1.2.cmml" xref="S3.SS2.p8.1.m1.1.1.2">𝑠</ci><ci id="S3.SS2.p8.1.m1.1.1.3.cmml" xref="S3.SS2.p8.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.1.m1.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.1.m1.1d">italic_s - italic_t</annotation></semantics></math> flow <math alttext="f" class="ltx_Math" display="inline" id="S3.SS2.p8.2.m2.1"><semantics id="S3.SS2.p8.2.m2.1a"><mi id="S3.SS2.p8.2.m2.1.1" xref="S3.SS2.p8.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.2.m2.1b"><ci id="S3.SS2.p8.2.m2.1.1.cmml" xref="S3.SS2.p8.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.2.m2.1d">italic_f</annotation></semantics></math>, let <math alttext="P_{f}=\left\{p_{1},...,p_{m^{\prime}}\right\}" class="ltx_Math" display="inline" id="S3.SS2.p8.3.m3.3"><semantics id="S3.SS2.p8.3.m3.3a"><mrow id="S3.SS2.p8.3.m3.3.3" xref="S3.SS2.p8.3.m3.3.3.cmml"><msub id="S3.SS2.p8.3.m3.3.3.4" xref="S3.SS2.p8.3.m3.3.3.4.cmml"><mi id="S3.SS2.p8.3.m3.3.3.4.2" xref="S3.SS2.p8.3.m3.3.3.4.2.cmml">P</mi><mi id="S3.SS2.p8.3.m3.3.3.4.3" xref="S3.SS2.p8.3.m3.3.3.4.3.cmml">f</mi></msub><mo id="S3.SS2.p8.3.m3.3.3.3" xref="S3.SS2.p8.3.m3.3.3.3.cmml">=</mo><mrow id="S3.SS2.p8.3.m3.3.3.2.2" xref="S3.SS2.p8.3.m3.3.3.2.3.cmml"><mo id="S3.SS2.p8.3.m3.3.3.2.2.3" xref="S3.SS2.p8.3.m3.3.3.2.3.cmml">{</mo><msub id="S3.SS2.p8.3.m3.2.2.1.1.1" xref="S3.SS2.p8.3.m3.2.2.1.1.1.cmml"><mi id="S3.SS2.p8.3.m3.2.2.1.1.1.2" xref="S3.SS2.p8.3.m3.2.2.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p8.3.m3.2.2.1.1.1.3" xref="S3.SS2.p8.3.m3.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p8.3.m3.3.3.2.2.4" xref="S3.SS2.p8.3.m3.3.3.2.3.cmml">,</mo><mi id="S3.SS2.p8.3.m3.1.1" mathvariant="normal" xref="S3.SS2.p8.3.m3.1.1.cmml">…</mi><mo id="S3.SS2.p8.3.m3.3.3.2.2.5" xref="S3.SS2.p8.3.m3.3.3.2.3.cmml">,</mo><msub id="S3.SS2.p8.3.m3.3.3.2.2.2" xref="S3.SS2.p8.3.m3.3.3.2.2.2.cmml"><mi id="S3.SS2.p8.3.m3.3.3.2.2.2.2" xref="S3.SS2.p8.3.m3.3.3.2.2.2.2.cmml">p</mi><msup id="S3.SS2.p8.3.m3.3.3.2.2.2.3" xref="S3.SS2.p8.3.m3.3.3.2.2.2.3.cmml"><mi id="S3.SS2.p8.3.m3.3.3.2.2.2.3.2" xref="S3.SS2.p8.3.m3.3.3.2.2.2.3.2.cmml">m</mi><mo id="S3.SS2.p8.3.m3.3.3.2.2.2.3.3" xref="S3.SS2.p8.3.m3.3.3.2.2.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS2.p8.3.m3.3.3.2.2.6" xref="S3.SS2.p8.3.m3.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.3.m3.3b"><apply id="S3.SS2.p8.3.m3.3.3.cmml" xref="S3.SS2.p8.3.m3.3.3"><eq id="S3.SS2.p8.3.m3.3.3.3.cmml" xref="S3.SS2.p8.3.m3.3.3.3"></eq><apply id="S3.SS2.p8.3.m3.3.3.4.cmml" xref="S3.SS2.p8.3.m3.3.3.4"><csymbol cd="ambiguous" id="S3.SS2.p8.3.m3.3.3.4.1.cmml" xref="S3.SS2.p8.3.m3.3.3.4">subscript</csymbol><ci id="S3.SS2.p8.3.m3.3.3.4.2.cmml" xref="S3.SS2.p8.3.m3.3.3.4.2">𝑃</ci><ci id="S3.SS2.p8.3.m3.3.3.4.3.cmml" xref="S3.SS2.p8.3.m3.3.3.4.3">𝑓</ci></apply><set id="S3.SS2.p8.3.m3.3.3.2.3.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2"><apply id="S3.SS2.p8.3.m3.2.2.1.1.1.cmml" xref="S3.SS2.p8.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p8.3.m3.2.2.1.1.1.1.cmml" xref="S3.SS2.p8.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p8.3.m3.2.2.1.1.1.2.cmml" xref="S3.SS2.p8.3.m3.2.2.1.1.1.2">𝑝</ci><cn id="S3.SS2.p8.3.m3.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS2.p8.3.m3.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS2.p8.3.m3.1.1.cmml" xref="S3.SS2.p8.3.m3.1.1">…</ci><apply id="S3.SS2.p8.3.m3.3.3.2.2.2.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p8.3.m3.3.3.2.2.2.1.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.p8.3.m3.3.3.2.2.2.2.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2.2.2">𝑝</ci><apply id="S3.SS2.p8.3.m3.3.3.2.2.2.3.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p8.3.m3.3.3.2.2.2.3.1.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2.2.3">superscript</csymbol><ci id="S3.SS2.p8.3.m3.3.3.2.2.2.3.2.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2.2.3.2">𝑚</ci><ci id="S3.SS2.p8.3.m3.3.3.2.2.2.3.3.cmml" xref="S3.SS2.p8.3.m3.3.3.2.2.2.3.3">′</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.3.m3.3c">P_{f}=\left\{p_{1},...,p_{m^{\prime}}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.3.m3.3d">italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math> be the set of edge-disjoint paths on which a flow of 1 is transferred. Suppose that <math alttext="|p_{1}|\leq...\leq|p_{m^{\prime}}|" class="ltx_Math" display="inline" id="S3.SS2.p8.4.m4.2"><semantics id="S3.SS2.p8.4.m4.2a"><mrow id="S3.SS2.p8.4.m4.2.2" xref="S3.SS2.p8.4.m4.2.2.cmml"><mrow id="S3.SS2.p8.4.m4.1.1.1.1" xref="S3.SS2.p8.4.m4.1.1.1.2.cmml"><mo id="S3.SS2.p8.4.m4.1.1.1.1.2" stretchy="false" xref="S3.SS2.p8.4.m4.1.1.1.2.1.cmml">|</mo><msub id="S3.SS2.p8.4.m4.1.1.1.1.1" xref="S3.SS2.p8.4.m4.1.1.1.1.1.cmml"><mi id="S3.SS2.p8.4.m4.1.1.1.1.1.2" xref="S3.SS2.p8.4.m4.1.1.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p8.4.m4.1.1.1.1.1.3" xref="S3.SS2.p8.4.m4.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p8.4.m4.1.1.1.1.3" stretchy="false" xref="S3.SS2.p8.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS2.p8.4.m4.2.2.4" xref="S3.SS2.p8.4.m4.2.2.4.cmml">≤</mo><mi id="S3.SS2.p8.4.m4.2.2.5" mathvariant="normal" xref="S3.SS2.p8.4.m4.2.2.5.cmml">…</mi><mo id="S3.SS2.p8.4.m4.2.2.6" xref="S3.SS2.p8.4.m4.2.2.6.cmml">≤</mo><mrow id="S3.SS2.p8.4.m4.2.2.2.1" xref="S3.SS2.p8.4.m4.2.2.2.2.cmml"><mo id="S3.SS2.p8.4.m4.2.2.2.1.2" stretchy="false" xref="S3.SS2.p8.4.m4.2.2.2.2.1.cmml">|</mo><msub id="S3.SS2.p8.4.m4.2.2.2.1.1" xref="S3.SS2.p8.4.m4.2.2.2.1.1.cmml"><mi id="S3.SS2.p8.4.m4.2.2.2.1.1.2" xref="S3.SS2.p8.4.m4.2.2.2.1.1.2.cmml">p</mi><msup id="S3.SS2.p8.4.m4.2.2.2.1.1.3" xref="S3.SS2.p8.4.m4.2.2.2.1.1.3.cmml"><mi id="S3.SS2.p8.4.m4.2.2.2.1.1.3.2" xref="S3.SS2.p8.4.m4.2.2.2.1.1.3.2.cmml">m</mi><mo id="S3.SS2.p8.4.m4.2.2.2.1.1.3.3" xref="S3.SS2.p8.4.m4.2.2.2.1.1.3.3.cmml">′</mo></msup></msub><mo id="S3.SS2.p8.4.m4.2.2.2.1.3" stretchy="false" xref="S3.SS2.p8.4.m4.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.4.m4.2b"><apply id="S3.SS2.p8.4.m4.2.2.cmml" xref="S3.SS2.p8.4.m4.2.2"><and id="S3.SS2.p8.4.m4.2.2a.cmml" xref="S3.SS2.p8.4.m4.2.2"></and><apply id="S3.SS2.p8.4.m4.2.2b.cmml" xref="S3.SS2.p8.4.m4.2.2"><leq id="S3.SS2.p8.4.m4.2.2.4.cmml" xref="S3.SS2.p8.4.m4.2.2.4"></leq><apply id="S3.SS2.p8.4.m4.1.1.1.2.cmml" xref="S3.SS2.p8.4.m4.1.1.1.1"><abs id="S3.SS2.p8.4.m4.1.1.1.2.1.cmml" xref="S3.SS2.p8.4.m4.1.1.1.1.2"></abs><apply id="S3.SS2.p8.4.m4.1.1.1.1.1.cmml" xref="S3.SS2.p8.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p8.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS2.p8.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p8.4.m4.1.1.1.1.1.2.cmml" xref="S3.SS2.p8.4.m4.1.1.1.1.1.2">𝑝</ci><cn id="S3.SS2.p8.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.p8.4.m4.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.SS2.p8.4.m4.2.2.5.cmml" xref="S3.SS2.p8.4.m4.2.2.5">…</ci></apply><apply id="S3.SS2.p8.4.m4.2.2c.cmml" xref="S3.SS2.p8.4.m4.2.2"><leq id="S3.SS2.p8.4.m4.2.2.6.cmml" xref="S3.SS2.p8.4.m4.2.2.6"></leq><share href="https://arxiv.org/html/2308.08047v3#S3.SS2.p8.4.m4.2.2.5.cmml" id="S3.SS2.p8.4.m4.2.2d.cmml" xref="S3.SS2.p8.4.m4.2.2"></share><apply id="S3.SS2.p8.4.m4.2.2.2.2.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1"><abs id="S3.SS2.p8.4.m4.2.2.2.2.1.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.2"></abs><apply id="S3.SS2.p8.4.m4.2.2.2.1.1.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p8.4.m4.2.2.2.1.1.1.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.1">subscript</csymbol><ci id="S3.SS2.p8.4.m4.2.2.2.1.1.2.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.1.2">𝑝</ci><apply id="S3.SS2.p8.4.m4.2.2.2.1.1.3.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p8.4.m4.2.2.2.1.1.3.1.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.1.3">superscript</csymbol><ci id="S3.SS2.p8.4.m4.2.2.2.1.1.3.2.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.1.3.2">𝑚</ci><ci id="S3.SS2.p8.4.m4.2.2.2.1.1.3.3.cmml" xref="S3.SS2.p8.4.m4.2.2.2.1.1.3.3">′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.4.m4.2c">|p_{1}|\leq...\leq|p_{m^{\prime}}|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.4.m4.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ … ≤ | italic_p start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math>, and let <math alttext="\operatorname{\mathcal{S}}(P_{f})=\left\{\left\{p_{1},...,p_{i}\right\}\mid i% \leq m^{\prime}\right\}" class="ltx_Math" display="inline" id="S3.SS2.p8.5.m5.5"><semantics id="S3.SS2.p8.5.m5.5a"><mrow id="S3.SS2.p8.5.m5.5.5" xref="S3.SS2.p8.5.m5.5.5.cmml"><mrow id="S3.SS2.p8.5.m5.3.3.1.1" xref="S3.SS2.p8.5.m5.3.3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p8.5.m5.1.1" xref="S3.SS2.p8.5.m5.1.1.cmml">𝒮</mi><mo id="S3.SS2.p8.5.m5.3.3.1.1a" xref="S3.SS2.p8.5.m5.3.3.1.2.cmml">⁡</mo><mrow id="S3.SS2.p8.5.m5.3.3.1.1.1" xref="S3.SS2.p8.5.m5.3.3.1.2.cmml"><mo id="S3.SS2.p8.5.m5.3.3.1.1.1.2" stretchy="false" xref="S3.SS2.p8.5.m5.3.3.1.2.cmml">(</mo><msub id="S3.SS2.p8.5.m5.3.3.1.1.1.1" xref="S3.SS2.p8.5.m5.3.3.1.1.1.1.cmml"><mi id="S3.SS2.p8.5.m5.3.3.1.1.1.1.2" xref="S3.SS2.p8.5.m5.3.3.1.1.1.1.2.cmml">P</mi><mi id="S3.SS2.p8.5.m5.3.3.1.1.1.1.3" xref="S3.SS2.p8.5.m5.3.3.1.1.1.1.3.cmml">f</mi></msub><mo id="S3.SS2.p8.5.m5.3.3.1.1.1.3" stretchy="false" xref="S3.SS2.p8.5.m5.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p8.5.m5.5.5.4" xref="S3.SS2.p8.5.m5.5.5.4.cmml">=</mo><mrow id="S3.SS2.p8.5.m5.5.5.3.2" xref="S3.SS2.p8.5.m5.5.5.3.3.cmml"><mo id="S3.SS2.p8.5.m5.5.5.3.2.3" xref="S3.SS2.p8.5.m5.5.5.3.3.1.cmml">{</mo><mrow id="S3.SS2.p8.5.m5.4.4.2.1.1.2" xref="S3.SS2.p8.5.m5.4.4.2.1.1.3.cmml"><mo id="S3.SS2.p8.5.m5.4.4.2.1.1.2.3" xref="S3.SS2.p8.5.m5.4.4.2.1.1.3.cmml">{</mo><msub id="S3.SS2.p8.5.m5.4.4.2.1.1.1.1" xref="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.cmml"><mi id="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.2" xref="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.2.cmml">p</mi><mn id="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.3" xref="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p8.5.m5.4.4.2.1.1.2.4" xref="S3.SS2.p8.5.m5.4.4.2.1.1.3.cmml">,</mo><mi id="S3.SS2.p8.5.m5.2.2" mathvariant="normal" xref="S3.SS2.p8.5.m5.2.2.cmml">…</mi><mo id="S3.SS2.p8.5.m5.4.4.2.1.1.2.5" xref="S3.SS2.p8.5.m5.4.4.2.1.1.3.cmml">,</mo><msub id="S3.SS2.p8.5.m5.4.4.2.1.1.2.2" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.cmml"><mi id="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.2" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.2.cmml">p</mi><mi id="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.3" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS2.p8.5.m5.4.4.2.1.1.2.6" xref="S3.SS2.p8.5.m5.4.4.2.1.1.3.cmml">}</mo></mrow><mo fence="true" id="S3.SS2.p8.5.m5.5.5.3.2.4" lspace="0em" rspace="0em" xref="S3.SS2.p8.5.m5.5.5.3.3.1.cmml">∣</mo><mrow id="S3.SS2.p8.5.m5.5.5.3.2.2" xref="S3.SS2.p8.5.m5.5.5.3.2.2.cmml"><mi id="S3.SS2.p8.5.m5.5.5.3.2.2.2" xref="S3.SS2.p8.5.m5.5.5.3.2.2.2.cmml">i</mi><mo id="S3.SS2.p8.5.m5.5.5.3.2.2.1" xref="S3.SS2.p8.5.m5.5.5.3.2.2.1.cmml">≤</mo><msup id="S3.SS2.p8.5.m5.5.5.3.2.2.3" xref="S3.SS2.p8.5.m5.5.5.3.2.2.3.cmml"><mi id="S3.SS2.p8.5.m5.5.5.3.2.2.3.2" xref="S3.SS2.p8.5.m5.5.5.3.2.2.3.2.cmml">m</mi><mo id="S3.SS2.p8.5.m5.5.5.3.2.2.3.3" xref="S3.SS2.p8.5.m5.5.5.3.2.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.SS2.p8.5.m5.5.5.3.2.5" xref="S3.SS2.p8.5.m5.5.5.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.5.m5.5b"><apply id="S3.SS2.p8.5.m5.5.5.cmml" xref="S3.SS2.p8.5.m5.5.5"><eq id="S3.SS2.p8.5.m5.5.5.4.cmml" xref="S3.SS2.p8.5.m5.5.5.4"></eq><apply id="S3.SS2.p8.5.m5.3.3.1.2.cmml" xref="S3.SS2.p8.5.m5.3.3.1.1"><ci id="S3.SS2.p8.5.m5.1.1.cmml" xref="S3.SS2.p8.5.m5.1.1">𝒮</ci><apply id="S3.SS2.p8.5.m5.3.3.1.1.1.1.cmml" xref="S3.SS2.p8.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p8.5.m5.3.3.1.1.1.1.1.cmml" xref="S3.SS2.p8.5.m5.3.3.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p8.5.m5.3.3.1.1.1.1.2.cmml" xref="S3.SS2.p8.5.m5.3.3.1.1.1.1.2">𝑃</ci><ci id="S3.SS2.p8.5.m5.3.3.1.1.1.1.3.cmml" xref="S3.SS2.p8.5.m5.3.3.1.1.1.1.3">𝑓</ci></apply></apply><apply id="S3.SS2.p8.5.m5.5.5.3.3.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2"><csymbol cd="latexml" id="S3.SS2.p8.5.m5.5.5.3.3.1.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.3">conditional-set</csymbol><set id="S3.SS2.p8.5.m5.4.4.2.1.1.3.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2"><apply id="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.1.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.2.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.2">𝑝</ci><cn id="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.p8.5.m5.4.4.2.1.1.1.1.3">1</cn></apply><ci id="S3.SS2.p8.5.m5.2.2.cmml" xref="S3.SS2.p8.5.m5.2.2">…</ci><apply id="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.1.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2.2">subscript</csymbol><ci id="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.2.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.2">𝑝</ci><ci id="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.3.cmml" xref="S3.SS2.p8.5.m5.4.4.2.1.1.2.2.3">𝑖</ci></apply></set><apply id="S3.SS2.p8.5.m5.5.5.3.2.2.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.2"><leq id="S3.SS2.p8.5.m5.5.5.3.2.2.1.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.2.1"></leq><ci id="S3.SS2.p8.5.m5.5.5.3.2.2.2.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.2.2">𝑖</ci><apply id="S3.SS2.p8.5.m5.5.5.3.2.2.3.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p8.5.m5.5.5.3.2.2.3.1.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.2.3">superscript</csymbol><ci id="S3.SS2.p8.5.m5.5.5.3.2.2.3.2.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.2.3.2">𝑚</ci><ci id="S3.SS2.p8.5.m5.5.5.3.2.2.3.3.cmml" xref="S3.SS2.p8.5.m5.5.5.3.2.2.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.5.m5.5c">\operatorname{\mathcal{S}}(P_{f})=\left\{\left\{p_{1},...,p_{i}\right\}\mid i% \leq m^{\prime}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.5.m5.5d">caligraphic_S ( italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) = { { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } ∣ italic_i ≤ italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math>. For <math alttext="P" class="ltx_Math" display="inline" id="S3.SS2.p8.6.m6.1"><semantics id="S3.SS2.p8.6.m6.1a"><mi id="S3.SS2.p8.6.m6.1.1" xref="S3.SS2.p8.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.6.m6.1b"><ci id="S3.SS2.p8.6.m6.1.1.cmml" xref="S3.SS2.p8.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.6.m6.1d">italic_P</annotation></semantics></math> let <math alttext="U_{P}" class="ltx_Math" display="inline" id="S3.SS2.p8.7.m7.1"><semantics id="S3.SS2.p8.7.m7.1a"><msub id="S3.SS2.p8.7.m7.1.1" xref="S3.SS2.p8.7.m7.1.1.cmml"><mi id="S3.SS2.p8.7.m7.1.1.2" xref="S3.SS2.p8.7.m7.1.1.2.cmml">U</mi><mi id="S3.SS2.p8.7.m7.1.1.3" xref="S3.SS2.p8.7.m7.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.7.m7.1b"><apply id="S3.SS2.p8.7.m7.1.1.cmml" xref="S3.SS2.p8.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS2.p8.7.m7.1.1.1.cmml" xref="S3.SS2.p8.7.m7.1.1">subscript</csymbol><ci id="S3.SS2.p8.7.m7.1.1.2.cmml" xref="S3.SS2.p8.7.m7.1.1.2">𝑈</ci><ci id="S3.SS2.p8.7.m7.1.1.3.cmml" xref="S3.SS2.p8.7.m7.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.7.m7.1c">U_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.7.m7.1d">italic_U start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> be the uniform distribution on <math alttext="P" class="ltx_Math" display="inline" id="S3.SS2.p8.8.m8.1"><semantics id="S3.SS2.p8.8.m8.1a"><mi id="S3.SS2.p8.8.m8.1.1" xref="S3.SS2.p8.8.m8.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.8.m8.1b"><ci id="S3.SS2.p8.8.m8.1.1.cmml" xref="S3.SS2.p8.8.m8.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.8.m8.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.8.m8.1d">italic_P</annotation></semantics></math>. We now formally define the uniform strategies from earlier.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Definition 2</span></h6> <div class="ltx_para" id="Thmdefinition2.p1"> <p class="ltx_p" id="Thmdefinition2.p1.1">For <math alttext="n" class="ltx_Math" display="inline" id="Thmdefinition2.p1.1.m1.1"><semantics id="Thmdefinition2.p1.1.m1.1a"><mi id="Thmdefinition2.p1.1.m1.1.1" xref="Thmdefinition2.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.1.m1.1b"><ci id="Thmdefinition2.p1.1.m1.1.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.1.m1.1d">italic_n</annotation></semantics></math> uncorrelated agents, the set of uniform agent strategies is defined as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx5"> <tbody id="S3.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{U}_{n}(G)=\left\{U_{P}^{n}\mid\exists f\in MF,~{}P\in% \operatorname{\mathcal{S}}(P_{f})\right\}" class="ltx_Math" display="inline" id="S3.E3.m1.4"><semantics id="S3.E3.m1.4a"><mrow id="S3.E3.m1.4.4" xref="S3.E3.m1.4.4.cmml"><mrow id="S3.E3.m1.4.4.4" xref="S3.E3.m1.4.4.4.cmml"><msub id="S3.E3.m1.4.4.4.2" xref="S3.E3.m1.4.4.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E3.m1.4.4.4.2.2" xref="S3.E3.m1.4.4.4.2.2.cmml">𝒰</mi><mi id="S3.E3.m1.4.4.4.2.3" xref="S3.E3.m1.4.4.4.2.3.cmml">n</mi></msub><mo id="S3.E3.m1.4.4.4.1" xref="S3.E3.m1.4.4.4.1.cmml">⁢</mo><mrow id="S3.E3.m1.4.4.4.3.2" xref="S3.E3.m1.4.4.4.cmml"><mo id="S3.E3.m1.4.4.4.3.2.1" stretchy="false" 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rspace="0.167em" xref="S3.E3.m1.4.4.2.2.2.1.1.2.1.cmml">∃</mo><mi id="S3.E3.m1.4.4.2.2.2.1.1.2.2" xref="S3.E3.m1.4.4.2.2.2.1.1.2.2.cmml">f</mi></mrow><mo id="S3.E3.m1.4.4.2.2.2.1.1.1" xref="S3.E3.m1.4.4.2.2.2.1.1.1.cmml">∈</mo><mrow id="S3.E3.m1.4.4.2.2.2.1.1.3" xref="S3.E3.m1.4.4.2.2.2.1.1.3.cmml"><mi id="S3.E3.m1.4.4.2.2.2.1.1.3.2" xref="S3.E3.m1.4.4.2.2.2.1.1.3.2.cmml">M</mi><mo id="S3.E3.m1.4.4.2.2.2.1.1.3.1" xref="S3.E3.m1.4.4.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S3.E3.m1.4.4.2.2.2.1.1.3.3" xref="S3.E3.m1.4.4.2.2.2.1.1.3.3.cmml">F</mi></mrow></mrow><mo id="S3.E3.m1.4.4.2.2.2.2.3" rspace="0.497em" xref="S3.E3.m1.4.4.2.2.2.3a.cmml">,</mo><mrow id="S3.E3.m1.4.4.2.2.2.2.2" xref="S3.E3.m1.4.4.2.2.2.2.2.cmml"><mi id="S3.E3.m1.4.4.2.2.2.2.2.3" xref="S3.E3.m1.4.4.2.2.2.2.2.3.cmml">P</mi><mo id="S3.E3.m1.4.4.2.2.2.2.2.2" xref="S3.E3.m1.4.4.2.2.2.2.2.2.cmml">∈</mo><mrow id="S3.E3.m1.4.4.2.2.2.2.2.1.1" xref="S3.E3.m1.4.4.2.2.2.2.2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E3.m1.2.2" 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id="S3.E3.m1.4c">\displaystyle\mathcal{U}_{n}(G)=\left\{U_{P}^{n}\mid\exists f\in MF,~{}P\in% \operatorname{\mathcal{S}}(P_{f})\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.4d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_G ) = { italic_U start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∣ ∃ italic_f ∈ italic_M italic_F , italic_P ∈ caligraphic_S ( italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS2.p9"> <p class="ltx_p" id="S3.SS2.p9.4">Each distribution <math alttext="U_{P}^{n}" class="ltx_Math" display="inline" id="S3.SS2.p9.1.m1.1"><semantics id="S3.SS2.p9.1.m1.1a"><msubsup id="S3.SS2.p9.1.m1.1.1" xref="S3.SS2.p9.1.m1.1.1.cmml"><mi id="S3.SS2.p9.1.m1.1.1.2.2" xref="S3.SS2.p9.1.m1.1.1.2.2.cmml">U</mi><mi id="S3.SS2.p9.1.m1.1.1.2.3" xref="S3.SS2.p9.1.m1.1.1.2.3.cmml">P</mi><mi id="S3.SS2.p9.1.m1.1.1.3" xref="S3.SS2.p9.1.m1.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.1.m1.1b"><apply id="S3.SS2.p9.1.m1.1.1.cmml" xref="S3.SS2.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.1.m1.1.1.1.cmml" xref="S3.SS2.p9.1.m1.1.1">superscript</csymbol><apply id="S3.SS2.p9.1.m1.1.1.2.cmml" xref="S3.SS2.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.1.m1.1.1.2.1.cmml" xref="S3.SS2.p9.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p9.1.m1.1.1.2.2.cmml" xref="S3.SS2.p9.1.m1.1.1.2.2">𝑈</ci><ci id="S3.SS2.p9.1.m1.1.1.2.3.cmml" xref="S3.SS2.p9.1.m1.1.1.2.3">𝑃</ci></apply><ci id="S3.SS2.p9.1.m1.1.1.3.cmml" xref="S3.SS2.p9.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.1.m1.1c">U_{P}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.1.m1.1d">italic_U start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is made up of <math alttext="n" class="ltx_Math" display="inline" id="S3.SS2.p9.2.m2.1"><semantics id="S3.SS2.p9.2.m2.1a"><mi id="S3.SS2.p9.2.m2.1.1" xref="S3.SS2.p9.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.2.m2.1b"><ci id="S3.SS2.p9.2.m2.1.1.cmml" xref="S3.SS2.p9.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.2.m2.1d">italic_n</annotation></semantics></math> independent copies of <math alttext="U_{P}" class="ltx_Math" display="inline" id="S3.SS2.p9.3.m3.1"><semantics id="S3.SS2.p9.3.m3.1a"><msub id="S3.SS2.p9.3.m3.1.1" xref="S3.SS2.p9.3.m3.1.1.cmml"><mi id="S3.SS2.p9.3.m3.1.1.2" xref="S3.SS2.p9.3.m3.1.1.2.cmml">U</mi><mi id="S3.SS2.p9.3.m3.1.1.3" xref="S3.SS2.p9.3.m3.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.3.m3.1b"><apply id="S3.SS2.p9.3.m3.1.1.cmml" xref="S3.SS2.p9.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.3.m3.1.1.1.cmml" xref="S3.SS2.p9.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.p9.3.m3.1.1.2.cmml" xref="S3.SS2.p9.3.m3.1.1.2">𝑈</ci><ci id="S3.SS2.p9.3.m3.1.1.3.cmml" xref="S3.SS2.p9.3.m3.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.3.m3.1c">U_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.3.m3.1d">italic_U start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>, the uniform distribution on the set of paths <math alttext="P" class="ltx_Math" display="inline" id="S3.SS2.p9.4.m4.1"><semantics id="S3.SS2.p9.4.m4.1a"><mi id="S3.SS2.p9.4.m4.1.1" xref="S3.SS2.p9.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.4.m4.1b"><ci id="S3.SS2.p9.4.m4.1.1.cmml" xref="S3.SS2.p9.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.4.m4.1d">italic_P</annotation></semantics></math>. Now we can formally define the uniform PoU:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Definition 3</span></h6> <div class="ltx_para" id="Thmdefinition3.p1"> <p class="ltx_p" id="Thmdefinition3.p1.5">For a game with graph <math alttext="G" class="ltx_Math" display="inline" id="Thmdefinition3.p1.1.m1.1"><semantics id="Thmdefinition3.p1.1.m1.1a"><mi id="Thmdefinition3.p1.1.m1.1.1" xref="Thmdefinition3.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.1.m1.1b"><ci id="Thmdefinition3.p1.1.m1.1.1.cmml" xref="Thmdefinition3.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.1.m1.1d">italic_G</annotation></semantics></math>, <math alttext="n" class="ltx_Math" display="inline" id="Thmdefinition3.p1.2.m2.1"><semantics id="Thmdefinition3.p1.2.m2.1a"><mi id="Thmdefinition3.p1.2.m2.1.1" xref="Thmdefinition3.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.2.m2.1b"><ci id="Thmdefinition3.p1.2.m2.1.1.cmml" xref="Thmdefinition3.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.2.m2.1d">italic_n</annotation></semantics></math> agents, <math alttext="k" class="ltx_Math" display="inline" id="Thmdefinition3.p1.3.m3.1"><semantics id="Thmdefinition3.p1.3.m3.1a"><mi id="Thmdefinition3.p1.3.m3.1.1" xref="Thmdefinition3.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.3.m3.1b"><ci id="Thmdefinition3.p1.3.m3.1.1.cmml" xref="Thmdefinition3.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.3.m3.1d">italic_k</annotation></semantics></math> (correlated) interceptors, a parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmdefinition3.p1.4.m4.1"><semantics id="Thmdefinition3.p1.4.m4.1a"><mi id="Thmdefinition3.p1.4.m4.1.1" xref="Thmdefinition3.p1.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.4.m4.1b"><ci id="Thmdefinition3.p1.4.m4.1.1.cmml" xref="Thmdefinition3.p1.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.4.m4.1d">italic_α</annotation></semantics></math> and a cost function <math alttext="C" class="ltx_Math" display="inline" id="Thmdefinition3.p1.5.m5.1"><semantics id="Thmdefinition3.p1.5.m5.1a"><mi id="Thmdefinition3.p1.5.m5.1.1" xref="Thmdefinition3.p1.5.m5.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.5.m5.1b"><ci id="Thmdefinition3.p1.5.m5.1.1.cmml" xref="Thmdefinition3.p1.5.m5.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.5.m5.1c">C</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.5.m5.1d">italic_C</annotation></semantics></math>, the uniform PoU of the game is defined by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx6"> <tbody id="S3.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r_{U}(G,n,k,C,\alpha):=\frac{\min_{\sigma\in\mathcal{U}_{n}}F_{n% }(\sigma,\alpha)}{\min_{\sigma\in\mathcal{U}_{1}}F_{n}(\sigma,\alpha)}" class="ltx_Math" display="inline" id="S3.E4.m1.9"><semantics id="S3.E4.m1.9a"><mrow id="S3.E4.m1.9.10" 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id="S3.E4.m1.9.9" xref="S3.E4.m1.9.9.cmml">α</mi><mo id="S3.E4.m1.9.10.2.3.2.6" rspace="0.278em" stretchy="false" xref="S3.E4.m1.9.10.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.E4.m1.9.10.1" rspace="0.278em" xref="S3.E4.m1.9.10.1.cmml">:=</mo><mstyle displaystyle="true" id="S3.E4.m1.4.4" xref="S3.E4.m1.4.4.cmml"><mfrac id="S3.E4.m1.4.4a" xref="S3.E4.m1.4.4.cmml"><mrow id="S3.E4.m1.2.2.2" xref="S3.E4.m1.2.2.2.cmml"><mrow id="S3.E4.m1.2.2.2.4" xref="S3.E4.m1.2.2.2.4.cmml"><msub id="S3.E4.m1.2.2.2.4.1" xref="S3.E4.m1.2.2.2.4.1.cmml"><mi id="S3.E4.m1.2.2.2.4.1.2" xref="S3.E4.m1.2.2.2.4.1.2.cmml">min</mi><mrow id="S3.E4.m1.2.2.2.4.1.3" xref="S3.E4.m1.2.2.2.4.1.3.cmml"><mi id="S3.E4.m1.2.2.2.4.1.3.2" xref="S3.E4.m1.2.2.2.4.1.3.2.cmml">σ</mi><mo id="S3.E4.m1.2.2.2.4.1.3.1" xref="S3.E4.m1.2.2.2.4.1.3.1.cmml">∈</mo><msub id="S3.E4.m1.2.2.2.4.1.3.3" xref="S3.E4.m1.2.2.2.4.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E4.m1.2.2.2.4.1.3.3.2" xref="S3.E4.m1.2.2.2.4.1.3.3.2.cmml">𝒰</mi><mi 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id="S3.E4.m1.4.4.4.4.1" xref="S3.E4.m1.4.4.4.4.1.cmml"><mi id="S3.E4.m1.4.4.4.4.1.2" xref="S3.E4.m1.4.4.4.4.1.2.cmml">min</mi><mrow id="S3.E4.m1.4.4.4.4.1.3" xref="S3.E4.m1.4.4.4.4.1.3.cmml"><mi id="S3.E4.m1.4.4.4.4.1.3.2" xref="S3.E4.m1.4.4.4.4.1.3.2.cmml">σ</mi><mo id="S3.E4.m1.4.4.4.4.1.3.1" xref="S3.E4.m1.4.4.4.4.1.3.1.cmml">∈</mo><msub id="S3.E4.m1.4.4.4.4.1.3.3" xref="S3.E4.m1.4.4.4.4.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E4.m1.4.4.4.4.1.3.3.2" xref="S3.E4.m1.4.4.4.4.1.3.3.2.cmml">𝒰</mi><mn id="S3.E4.m1.4.4.4.4.1.3.3.3" xref="S3.E4.m1.4.4.4.4.1.3.3.3.cmml">1</mn></msub></mrow></msub><mo id="S3.E4.m1.4.4.4.4a" lspace="0.167em" xref="S3.E4.m1.4.4.4.4.cmml">⁡</mo><msub id="S3.E4.m1.4.4.4.4.2" xref="S3.E4.m1.4.4.4.4.2.cmml"><mi id="S3.E4.m1.4.4.4.4.2.2" xref="S3.E4.m1.4.4.4.4.2.2.cmml">F</mi><mi id="S3.E4.m1.4.4.4.4.2.3" xref="S3.E4.m1.4.4.4.4.2.3.cmml">n</mi></msub></mrow><mo id="S3.E4.m1.4.4.4.3" xref="S3.E4.m1.4.4.4.3.cmml">⁢</mo><mrow id="S3.E4.m1.4.4.4.5.2" 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type="integer" xref="S3.E4.m1.4.4.4.4.1.3.3.3">1</cn></apply></apply></apply><apply id="S3.E4.m1.4.4.4.4.2.cmml" xref="S3.E4.m1.4.4.4.4.2"><csymbol cd="ambiguous" id="S3.E4.m1.4.4.4.4.2.1.cmml" xref="S3.E4.m1.4.4.4.4.2">subscript</csymbol><ci id="S3.E4.m1.4.4.4.4.2.2.cmml" xref="S3.E4.m1.4.4.4.4.2.2">𝐹</ci><ci id="S3.E4.m1.4.4.4.4.2.3.cmml" xref="S3.E4.m1.4.4.4.4.2.3">𝑛</ci></apply></apply><interval closure="open" id="S3.E4.m1.4.4.4.5.1.cmml" xref="S3.E4.m1.4.4.4.5.2"><ci id="S3.E4.m1.3.3.3.1.cmml" xref="S3.E4.m1.3.3.3.1">𝜎</ci><ci id="S3.E4.m1.4.4.4.2.cmml" xref="S3.E4.m1.4.4.4.2">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E4.m1.9c">\displaystyle r_{U}(G,n,k,C,\alpha):=\frac{\min_{\sigma\in\mathcal{U}_{n}}F_{n% }(\sigma,\alpha)}{\min_{\sigma\in\mathcal{U}_{1}}F_{n}(\sigma,\alpha)}</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.9d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_C , italic_α ) := divide start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) end_ARG start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS2.p10"> <p class="ltx_p" id="S3.SS2.p10.4">We omit <math alttext="C" class="ltx_Math" display="inline" id="S3.SS2.p10.1.m1.1"><semantics id="S3.SS2.p10.1.m1.1a"><mi id="S3.SS2.p10.1.m1.1.1" xref="S3.SS2.p10.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.1.m1.1b"><ci id="S3.SS2.p10.1.m1.1.1.cmml" xref="S3.SS2.p10.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.1.m1.1d">italic_C</annotation></semantics></math> when clear from context. Note that <math alttext="r_{U}\geq 1" class="ltx_Math" display="inline" id="S3.SS2.p10.2.m2.1"><semantics id="S3.SS2.p10.2.m2.1a"><mrow id="S3.SS2.p10.2.m2.1.1" xref="S3.SS2.p10.2.m2.1.1.cmml"><msub id="S3.SS2.p10.2.m2.1.1.2" xref="S3.SS2.p10.2.m2.1.1.2.cmml"><mi id="S3.SS2.p10.2.m2.1.1.2.2" xref="S3.SS2.p10.2.m2.1.1.2.2.cmml">r</mi><mi id="S3.SS2.p10.2.m2.1.1.2.3" xref="S3.SS2.p10.2.m2.1.1.2.3.cmml">U</mi></msub><mo id="S3.SS2.p10.2.m2.1.1.1" xref="S3.SS2.p10.2.m2.1.1.1.cmml">≥</mo><mn id="S3.SS2.p10.2.m2.1.1.3" xref="S3.SS2.p10.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.2.m2.1b"><apply id="S3.SS2.p10.2.m2.1.1.cmml" xref="S3.SS2.p10.2.m2.1.1"><geq id="S3.SS2.p10.2.m2.1.1.1.cmml" xref="S3.SS2.p10.2.m2.1.1.1"></geq><apply id="S3.SS2.p10.2.m2.1.1.2.cmml" xref="S3.SS2.p10.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p10.2.m2.1.1.2.1.cmml" xref="S3.SS2.p10.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS2.p10.2.m2.1.1.2.2.cmml" xref="S3.SS2.p10.2.m2.1.1.2.2">𝑟</ci><ci id="S3.SS2.p10.2.m2.1.1.2.3.cmml" xref="S3.SS2.p10.2.m2.1.1.2.3">𝑈</ci></apply><cn id="S3.SS2.p10.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.p10.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.2.m2.1c">r_{U}\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.2.m2.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ≥ 1</annotation></semantics></math> is not generally true, however it is true in our problem instances, because the agents are correlated, even <math alttext="\mathcal{U}_{1}" class="ltx_Math" display="inline" id="S3.SS2.p10.3.m3.1"><semantics id="S3.SS2.p10.3.m3.1a"><msub id="S3.SS2.p10.3.m3.1.1" xref="S3.SS2.p10.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.3.m3.1.1.2" xref="S3.SS2.p10.3.m3.1.1.2.cmml">𝒰</mi><mn id="S3.SS2.p10.3.m3.1.1.3" xref="S3.SS2.p10.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.3.m3.1b"><apply id="S3.SS2.p10.3.m3.1.1.cmml" xref="S3.SS2.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.3.m3.1.1.1.cmml" xref="S3.SS2.p10.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.p10.3.m3.1.1.2.cmml" xref="S3.SS2.p10.3.m3.1.1.2">𝒰</ci><cn id="S3.SS2.p10.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p10.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.3.m3.1c">\mathcal{U}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.3.m3.1d">caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> will strictly dominate <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S3.SS2.p10.4.m4.1"><semantics id="S3.SS2.p10.4.m4.1a"><msub id="S3.SS2.p10.4.m4.1.1" xref="S3.SS2.p10.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.4.m4.1.1.2" xref="S3.SS2.p10.4.m4.1.1.2.cmml">𝒰</mi><mi id="S3.SS2.p10.4.m4.1.1.3" xref="S3.SS2.p10.4.m4.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.4.m4.1b"><apply id="S3.SS2.p10.4.m4.1.1.cmml" xref="S3.SS2.p10.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.4.m4.1.1.1.cmml" xref="S3.SS2.p10.4.m4.1.1">subscript</csymbol><ci id="S3.SS2.p10.4.m4.1.1.2.cmml" xref="S3.SS2.p10.4.m4.1.1.2">𝒰</ci><ci id="S3.SS2.p10.4.m4.1.1.3.cmml" xref="S3.SS2.p10.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.4.m4.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.4.m4.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> in our problem instances.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>General lemmas</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">Here we state a few general lemmas that are used in the proofs that appear here. For the full list of lemmas and their proofs, see Appendix <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A1" title="Appendix 0.A Proofs of General Lemmas ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.A</span></a>.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.5">For <math alttext="\ell,n\geq 1" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.2"><semantics id="S3.SS3.p2.1.m1.2a"><mrow id="S3.SS3.p2.1.m1.2.3" xref="S3.SS3.p2.1.m1.2.3.cmml"><mrow id="S3.SS3.p2.1.m1.2.3.2.2" xref="S3.SS3.p2.1.m1.2.3.2.1.cmml"><mi id="S3.SS3.p2.1.m1.1.1" mathvariant="normal" xref="S3.SS3.p2.1.m1.1.1.cmml">ℓ</mi><mo id="S3.SS3.p2.1.m1.2.3.2.2.1" xref="S3.SS3.p2.1.m1.2.3.2.1.cmml">,</mo><mi id="S3.SS3.p2.1.m1.2.2" xref="S3.SS3.p2.1.m1.2.2.cmml">n</mi></mrow><mo id="S3.SS3.p2.1.m1.2.3.1" xref="S3.SS3.p2.1.m1.2.3.1.cmml">≥</mo><mn id="S3.SS3.p2.1.m1.2.3.3" xref="S3.SS3.p2.1.m1.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.2b"><apply id="S3.SS3.p2.1.m1.2.3.cmml" xref="S3.SS3.p2.1.m1.2.3"><geq id="S3.SS3.p2.1.m1.2.3.1.cmml" xref="S3.SS3.p2.1.m1.2.3.1"></geq><list id="S3.SS3.p2.1.m1.2.3.2.1.cmml" xref="S3.SS3.p2.1.m1.2.3.2.2"><ci id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">ℓ</ci><ci id="S3.SS3.p2.1.m1.2.2.cmml" xref="S3.SS3.p2.1.m1.2.2">𝑛</ci></list><cn id="S3.SS3.p2.1.m1.2.3.3.cmml" type="integer" xref="S3.SS3.p2.1.m1.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.2c">\ell,n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.2d">roman_ℓ , italic_n ≥ 1</annotation></semantics></math>, let <math alttext="\mathcal{E}_{\ell,n}=\ell\left(1-\left(1-\frac{1}{\ell}\right)^{n}\right)" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.3"><semantics id="S3.SS3.p2.2.m2.3a"><mrow id="S3.SS3.p2.2.m2.3.3" 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id="S3.SS3.p2.2.m2.3.3.1.1.1.1" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.cmml"><mn id="S3.SS3.p2.2.m2.3.3.1.1.1.1.3" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.3.cmml">1</mn><mo id="S3.SS3.p2.2.m2.3.3.1.1.1.1.2" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.2.cmml">−</mo><msup id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.cmml"><mrow id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.2" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.cmml"><mn id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.2" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.1" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.cmml"><mn id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.2" 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id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.1"></minus><cn id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.2">1</cn><apply id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3"><divide id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3"></divide><cn id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.1.1.1.3.3">ℓ</ci></apply></apply><ci id="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.2.m2.3.3.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.3c">\mathcal{E}_{\ell,n}=\ell\left(1-\left(1-\frac{1}{\ell}\right)^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.3d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT = roman_ℓ ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG roman_ℓ end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>. In the appendix we show that when <math alttext="\ell,n" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m3.2"><semantics id="S3.SS3.p2.3.m3.2a"><mrow id="S3.SS3.p2.3.m3.2.3.2" xref="S3.SS3.p2.3.m3.2.3.1.cmml"><mi id="S3.SS3.p2.3.m3.1.1" mathvariant="normal" xref="S3.SS3.p2.3.m3.1.1.cmml">ℓ</mi><mo id="S3.SS3.p2.3.m3.2.3.2.1" xref="S3.SS3.p2.3.m3.2.3.1.cmml">,</mo><mi id="S3.SS3.p2.3.m3.2.2" xref="S3.SS3.p2.3.m3.2.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.3.m3.2b"><list id="S3.SS3.p2.3.m3.2.3.1.cmml" xref="S3.SS3.p2.3.m3.2.3.2"><ci id="S3.SS3.p2.3.m3.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1">ℓ</ci><ci id="S3.SS3.p2.3.m3.2.2.cmml" xref="S3.SS3.p2.3.m3.2.2">𝑛</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.3.m3.2c">\ell,n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.3.m3.2d">roman_ℓ , italic_n</annotation></semantics></math> are large, <math alttext="\mathcal{E}_{\ell,n}=\Theta(\min(\ell,n))" class="ltx_Math" display="inline" id="S3.SS3.p2.4.m4.6"><semantics id="S3.SS3.p2.4.m4.6a"><mrow id="S3.SS3.p2.4.m4.6.6" xref="S3.SS3.p2.4.m4.6.6.cmml"><msub id="S3.SS3.p2.4.m4.6.6.3" xref="S3.SS3.p2.4.m4.6.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.4.m4.6.6.3.2" xref="S3.SS3.p2.4.m4.6.6.3.2.cmml">ℰ</mi><mrow id="S3.SS3.p2.4.m4.2.2.2.4" xref="S3.SS3.p2.4.m4.2.2.2.3.cmml"><mi id="S3.SS3.p2.4.m4.1.1.1.1" mathvariant="normal" xref="S3.SS3.p2.4.m4.1.1.1.1.cmml">ℓ</mi><mo id="S3.SS3.p2.4.m4.2.2.2.4.1" xref="S3.SS3.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p2.4.m4.2.2.2.2" xref="S3.SS3.p2.4.m4.2.2.2.2.cmml">n</mi></mrow></msub><mo id="S3.SS3.p2.4.m4.6.6.2" xref="S3.SS3.p2.4.m4.6.6.2.cmml">=</mo><mrow id="S3.SS3.p2.4.m4.6.6.1" xref="S3.SS3.p2.4.m4.6.6.1.cmml"><mi id="S3.SS3.p2.4.m4.6.6.1.3" mathvariant="normal" xref="S3.SS3.p2.4.m4.6.6.1.3.cmml">Θ</mi><mo id="S3.SS3.p2.4.m4.6.6.1.2" xref="S3.SS3.p2.4.m4.6.6.1.2.cmml">⁢</mo><mrow id="S3.SS3.p2.4.m4.6.6.1.1.1" xref="S3.SS3.p2.4.m4.6.6.1.cmml"><mo id="S3.SS3.p2.4.m4.6.6.1.1.1.2" stretchy="false" xref="S3.SS3.p2.4.m4.6.6.1.cmml">(</mo><mrow id="S3.SS3.p2.4.m4.6.6.1.1.1.1.2" xref="S3.SS3.p2.4.m4.6.6.1.1.1.1.1.cmml"><mi id="S3.SS3.p2.4.m4.3.3" xref="S3.SS3.p2.4.m4.3.3.cmml">min</mi><mo id="S3.SS3.p2.4.m4.6.6.1.1.1.1.2a" xref="S3.SS3.p2.4.m4.6.6.1.1.1.1.1.cmml">⁡</mo><mrow id="S3.SS3.p2.4.m4.6.6.1.1.1.1.2.1" xref="S3.SS3.p2.4.m4.6.6.1.1.1.1.1.cmml"><mo id="S3.SS3.p2.4.m4.6.6.1.1.1.1.2.1.1" stretchy="false" xref="S3.SS3.p2.4.m4.6.6.1.1.1.1.1.cmml">(</mo><mi id="S3.SS3.p2.4.m4.4.4" mathvariant="normal" xref="S3.SS3.p2.4.m4.4.4.cmml">ℓ</mi><mo id="S3.SS3.p2.4.m4.6.6.1.1.1.1.2.1.2" xref="S3.SS3.p2.4.m4.6.6.1.1.1.1.1.cmml">,</mo><mi id="S3.SS3.p2.4.m4.5.5" xref="S3.SS3.p2.4.m4.5.5.cmml">n</mi><mo id="S3.SS3.p2.4.m4.6.6.1.1.1.1.2.1.3" stretchy="false" xref="S3.SS3.p2.4.m4.6.6.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p2.4.m4.6.6.1.1.1.3" stretchy="false" xref="S3.SS3.p2.4.m4.6.6.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.4.m4.6b"><apply id="S3.SS3.p2.4.m4.6.6.cmml" xref="S3.SS3.p2.4.m4.6.6"><eq id="S3.SS3.p2.4.m4.6.6.2.cmml" xref="S3.SS3.p2.4.m4.6.6.2"></eq><apply id="S3.SS3.p2.4.m4.6.6.3.cmml" xref="S3.SS3.p2.4.m4.6.6.3"><csymbol cd="ambiguous" id="S3.SS3.p2.4.m4.6.6.3.1.cmml" xref="S3.SS3.p2.4.m4.6.6.3">subscript</csymbol><ci id="S3.SS3.p2.4.m4.6.6.3.2.cmml" xref="S3.SS3.p2.4.m4.6.6.3.2">ℰ</ci><list id="S3.SS3.p2.4.m4.2.2.2.3.cmml" xref="S3.SS3.p2.4.m4.2.2.2.4"><ci id="S3.SS3.p2.4.m4.1.1.1.1.cmml" xref="S3.SS3.p2.4.m4.1.1.1.1">ℓ</ci><ci id="S3.SS3.p2.4.m4.2.2.2.2.cmml" xref="S3.SS3.p2.4.m4.2.2.2.2">𝑛</ci></list></apply><apply id="S3.SS3.p2.4.m4.6.6.1.cmml" xref="S3.SS3.p2.4.m4.6.6.1"><times id="S3.SS3.p2.4.m4.6.6.1.2.cmml" xref="S3.SS3.p2.4.m4.6.6.1.2"></times><ci id="S3.SS3.p2.4.m4.6.6.1.3.cmml" xref="S3.SS3.p2.4.m4.6.6.1.3">Θ</ci><apply id="S3.SS3.p2.4.m4.6.6.1.1.1.1.1.cmml" xref="S3.SS3.p2.4.m4.6.6.1.1.1.1.2"><min id="S3.SS3.p2.4.m4.3.3.cmml" xref="S3.SS3.p2.4.m4.3.3"></min><ci id="S3.SS3.p2.4.m4.4.4.cmml" xref="S3.SS3.p2.4.m4.4.4">ℓ</ci><ci id="S3.SS3.p2.4.m4.5.5.cmml" xref="S3.SS3.p2.4.m4.5.5">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.4.m4.6c">\mathcal{E}_{\ell,n}=\Theta(\min(\ell,n))</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.4.m4.6d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT = roman_Θ ( roman_min ( roman_ℓ , italic_n ) )</annotation></semantics></math>, with an <math alttext="\leq" class="ltx_Math" display="inline" id="S3.SS3.p2.5.m5.1"><semantics id="S3.SS3.p2.5.m5.1a"><mo id="S3.SS3.p2.5.m5.1.1" xref="S3.SS3.p2.5.m5.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.5.m5.1b"><leq id="S3.SS3.p2.5.m5.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.5.m5.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.5.m5.1d">≤</annotation></semantics></math> inequality in general.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma4"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 4</span></h6> <div class="ltx_para" id="Thmlemma4.p1"> <p class="ltx_p" id="Thmlemma4.p1.7"><span class="ltx_text ltx_font_italic" id="Thmlemma4.p1.7.7">Let <math alttext="\sigma\in(\Delta(P_{st}))^{n}" class="ltx_Math" display="inline" id="Thmlemma4.p1.1.1.m1.1"><semantics id="Thmlemma4.p1.1.1.m1.1a"><mrow id="Thmlemma4.p1.1.1.m1.1.1" xref="Thmlemma4.p1.1.1.m1.1.1.cmml"><mi id="Thmlemma4.p1.1.1.m1.1.1.3" xref="Thmlemma4.p1.1.1.m1.1.1.3.cmml">σ</mi><mo id="Thmlemma4.p1.1.1.m1.1.1.2" xref="Thmlemma4.p1.1.1.m1.1.1.2.cmml">∈</mo><msup id="Thmlemma4.p1.1.1.m1.1.1.1" xref="Thmlemma4.p1.1.1.m1.1.1.1.cmml"><mrow id="Thmlemma4.p1.1.1.m1.1.1.1.1.1" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.3" mathvariant="normal" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.2" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmlemma4.p1.1.1.m1.1.1.1.3" xref="Thmlemma4.p1.1.1.m1.1.1.1.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma4.p1.1.1.m1.1b"><apply id="Thmlemma4.p1.1.1.m1.1.1.cmml" xref="Thmlemma4.p1.1.1.m1.1.1"><in id="Thmlemma4.p1.1.1.m1.1.1.2.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.2"></in><ci id="Thmlemma4.p1.1.1.m1.1.1.3.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.3">𝜎</ci><apply id="Thmlemma4.p1.1.1.m1.1.1.1.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma4.p1.1.1.m1.1.1.1.2.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1">superscript</csymbol><apply id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1"><times id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.2"></times><ci id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.3">Δ</ci><apply id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3"><times id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="Thmlemma4.p1.1.1.m1.1.1.1.3.cmml" xref="Thmlemma4.p1.1.1.m1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma4.p1.1.1.m1.1c">\sigma\in(\Delta(P_{st}))^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma4.p1.1.1.m1.1d">italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="Thmlemma4.p1.2.2.m2.1"><semantics id="Thmlemma4.p1.2.2.m2.1a"><msup id="Thmlemma4.p1.2.2.m2.1.1" xref="Thmlemma4.p1.2.2.m2.1.1.cmml"><mi id="Thmlemma4.p1.2.2.m2.1.1.2" xref="Thmlemma4.p1.2.2.m2.1.1.2.cmml">G</mi><mo id="Thmlemma4.p1.2.2.m2.1.1.3" xref="Thmlemma4.p1.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmlemma4.p1.2.2.m2.1b"><apply id="Thmlemma4.p1.2.2.m2.1.1.cmml" xref="Thmlemma4.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmlemma4.p1.2.2.m2.1.1.1.cmml" xref="Thmlemma4.p1.2.2.m2.1.1">superscript</csymbol><ci id="Thmlemma4.p1.2.2.m2.1.1.2.cmml" xref="Thmlemma4.p1.2.2.m2.1.1.2">𝐺</ci><ci id="Thmlemma4.p1.2.2.m2.1.1.3.cmml" xref="Thmlemma4.p1.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma4.p1.2.2.m2.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma4.p1.2.2.m2.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> result from <math alttext="G" class="ltx_Math" display="inline" id="Thmlemma4.p1.3.3.m3.1"><semantics id="Thmlemma4.p1.3.3.m3.1a"><mi id="Thmlemma4.p1.3.3.m3.1.1" xref="Thmlemma4.p1.3.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmlemma4.p1.3.3.m3.1b"><ci id="Thmlemma4.p1.3.3.m3.1.1.cmml" xref="Thmlemma4.p1.3.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma4.p1.3.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmlemma4.p1.3.3.m3.1d">italic_G</annotation></semantics></math> by keeping only vertices and edges on paths <math alttext="p" class="ltx_Math" display="inline" id="Thmlemma4.p1.4.4.m4.1"><semantics id="Thmlemma4.p1.4.4.m4.1a"><mi id="Thmlemma4.p1.4.4.m4.1.1" xref="Thmlemma4.p1.4.4.m4.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="Thmlemma4.p1.4.4.m4.1b"><ci id="Thmlemma4.p1.4.4.m4.1.1.cmml" xref="Thmlemma4.p1.4.4.m4.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma4.p1.4.4.m4.1c">p</annotation><annotation encoding="application/x-llamapun" id="Thmlemma4.p1.4.4.m4.1d">italic_p</annotation></semantics></math> for which <math alttext="\sigma(p)>0" class="ltx_Math" display="inline" id="Thmlemma4.p1.5.5.m5.1"><semantics id="Thmlemma4.p1.5.5.m5.1a"><mrow id="Thmlemma4.p1.5.5.m5.1.2" xref="Thmlemma4.p1.5.5.m5.1.2.cmml"><mrow id="Thmlemma4.p1.5.5.m5.1.2.2" xref="Thmlemma4.p1.5.5.m5.1.2.2.cmml"><mi id="Thmlemma4.p1.5.5.m5.1.2.2.2" xref="Thmlemma4.p1.5.5.m5.1.2.2.2.cmml">σ</mi><mo id="Thmlemma4.p1.5.5.m5.1.2.2.1" xref="Thmlemma4.p1.5.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="Thmlemma4.p1.5.5.m5.1.2.2.3.2" xref="Thmlemma4.p1.5.5.m5.1.2.2.cmml"><mo id="Thmlemma4.p1.5.5.m5.1.2.2.3.2.1" stretchy="false" xref="Thmlemma4.p1.5.5.m5.1.2.2.cmml">(</mo><mi id="Thmlemma4.p1.5.5.m5.1.1" xref="Thmlemma4.p1.5.5.m5.1.1.cmml">p</mi><mo id="Thmlemma4.p1.5.5.m5.1.2.2.3.2.2" stretchy="false" xref="Thmlemma4.p1.5.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma4.p1.5.5.m5.1.2.1" xref="Thmlemma4.p1.5.5.m5.1.2.1.cmml">></mo><mn id="Thmlemma4.p1.5.5.m5.1.2.3" xref="Thmlemma4.p1.5.5.m5.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma4.p1.5.5.m5.1b"><apply id="Thmlemma4.p1.5.5.m5.1.2.cmml" xref="Thmlemma4.p1.5.5.m5.1.2"><gt id="Thmlemma4.p1.5.5.m5.1.2.1.cmml" xref="Thmlemma4.p1.5.5.m5.1.2.1"></gt><apply id="Thmlemma4.p1.5.5.m5.1.2.2.cmml" xref="Thmlemma4.p1.5.5.m5.1.2.2"><times id="Thmlemma4.p1.5.5.m5.1.2.2.1.cmml" xref="Thmlemma4.p1.5.5.m5.1.2.2.1"></times><ci id="Thmlemma4.p1.5.5.m5.1.2.2.2.cmml" xref="Thmlemma4.p1.5.5.m5.1.2.2.2">𝜎</ci><ci id="Thmlemma4.p1.5.5.m5.1.1.cmml" xref="Thmlemma4.p1.5.5.m5.1.1">𝑝</ci></apply><cn id="Thmlemma4.p1.5.5.m5.1.2.3.cmml" type="integer" xref="Thmlemma4.p1.5.5.m5.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma4.p1.5.5.m5.1c">\sigma(p)>0</annotation><annotation encoding="application/x-llamapun" id="Thmlemma4.p1.5.5.m5.1d">italic_σ ( italic_p ) > 0</annotation></semantics></math>. Let <math alttext="m^{\prime}=m_{c}(G^{\prime})" class="ltx_Math" display="inline" id="Thmlemma4.p1.6.6.m6.1"><semantics id="Thmlemma4.p1.6.6.m6.1a"><mrow id="Thmlemma4.p1.6.6.m6.1.1" xref="Thmlemma4.p1.6.6.m6.1.1.cmml"><msup id="Thmlemma4.p1.6.6.m6.1.1.3" xref="Thmlemma4.p1.6.6.m6.1.1.3.cmml"><mi id="Thmlemma4.p1.6.6.m6.1.1.3.2" xref="Thmlemma4.p1.6.6.m6.1.1.3.2.cmml">m</mi><mo id="Thmlemma4.p1.6.6.m6.1.1.3.3" xref="Thmlemma4.p1.6.6.m6.1.1.3.3.cmml">′</mo></msup><mo id="Thmlemma4.p1.6.6.m6.1.1.2" xref="Thmlemma4.p1.6.6.m6.1.1.2.cmml">=</mo><mrow id="Thmlemma4.p1.6.6.m6.1.1.1" xref="Thmlemma4.p1.6.6.m6.1.1.1.cmml"><msub id="Thmlemma4.p1.6.6.m6.1.1.1.3" xref="Thmlemma4.p1.6.6.m6.1.1.1.3.cmml"><mi id="Thmlemma4.p1.6.6.m6.1.1.1.3.2" xref="Thmlemma4.p1.6.6.m6.1.1.1.3.2.cmml">m</mi><mi id="Thmlemma4.p1.6.6.m6.1.1.1.3.3" xref="Thmlemma4.p1.6.6.m6.1.1.1.3.3.cmml">c</mi></msub><mo id="Thmlemma4.p1.6.6.m6.1.1.1.2" xref="Thmlemma4.p1.6.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma4.p1.6.6.m6.1.1.1.1.1" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.2" stretchy="false" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.cmml"><mi id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.2" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.2.cmml">G</mi><mo id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.3" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.3" stretchy="false" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma4.p1.6.6.m6.1b"><apply id="Thmlemma4.p1.6.6.m6.1.1.cmml" xref="Thmlemma4.p1.6.6.m6.1.1"><eq id="Thmlemma4.p1.6.6.m6.1.1.2.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.2"></eq><apply id="Thmlemma4.p1.6.6.m6.1.1.3.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="Thmlemma4.p1.6.6.m6.1.1.3.1.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.3">superscript</csymbol><ci id="Thmlemma4.p1.6.6.m6.1.1.3.2.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.3.2">𝑚</ci><ci id="Thmlemma4.p1.6.6.m6.1.1.3.3.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.3.3">′</ci></apply><apply id="Thmlemma4.p1.6.6.m6.1.1.1.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1"><times id="Thmlemma4.p1.6.6.m6.1.1.1.2.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.2"></times><apply id="Thmlemma4.p1.6.6.m6.1.1.1.3.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma4.p1.6.6.m6.1.1.1.3.1.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.3">subscript</csymbol><ci id="Thmlemma4.p1.6.6.m6.1.1.1.3.2.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.3.2">𝑚</ci><ci id="Thmlemma4.p1.6.6.m6.1.1.1.3.3.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.3.3">𝑐</ci></apply><apply id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1">superscript</csymbol><ci id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.2.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.2">𝐺</ci><ci id="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.3.cmml" xref="Thmlemma4.p1.6.6.m6.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma4.p1.6.6.m6.1c">m^{\prime}=m_{c}(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma4.p1.6.6.m6.1d">italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then <math alttext="\beta(\sigma)\geq 1-\left(1-\frac{1}{m^{\prime}}\right)^{n}=\frac{\mathcal{E}_% {m^{\prime},n}}{m^{\prime}}" class="ltx_Math" display="inline" id="Thmlemma4.p1.7.7.m7.4"><semantics id="Thmlemma4.p1.7.7.m7.4a"><mrow id="Thmlemma4.p1.7.7.m7.4.4" xref="Thmlemma4.p1.7.7.m7.4.4.cmml"><mrow id="Thmlemma4.p1.7.7.m7.4.4.3" xref="Thmlemma4.p1.7.7.m7.4.4.3.cmml"><mi id="Thmlemma4.p1.7.7.m7.4.4.3.2" xref="Thmlemma4.p1.7.7.m7.4.4.3.2.cmml">β</mi><mo id="Thmlemma4.p1.7.7.m7.4.4.3.1" xref="Thmlemma4.p1.7.7.m7.4.4.3.1.cmml">⁢</mo><mrow id="Thmlemma4.p1.7.7.m7.4.4.3.3.2" xref="Thmlemma4.p1.7.7.m7.4.4.3.cmml"><mo id="Thmlemma4.p1.7.7.m7.4.4.3.3.2.1" stretchy="false" xref="Thmlemma4.p1.7.7.m7.4.4.3.cmml">(</mo><mi id="Thmlemma4.p1.7.7.m7.3.3" xref="Thmlemma4.p1.7.7.m7.3.3.cmml">σ</mi><mo id="Thmlemma4.p1.7.7.m7.4.4.3.3.2.2" stretchy="false" xref="Thmlemma4.p1.7.7.m7.4.4.3.cmml">)</mo></mrow></mrow><mo id="Thmlemma4.p1.7.7.m7.4.4.4" xref="Thmlemma4.p1.7.7.m7.4.4.4.cmml">≥</mo><mrow id="Thmlemma4.p1.7.7.m7.4.4.1" xref="Thmlemma4.p1.7.7.m7.4.4.1.cmml"><mn id="Thmlemma4.p1.7.7.m7.4.4.1.3" xref="Thmlemma4.p1.7.7.m7.4.4.1.3.cmml">1</mn><mo id="Thmlemma4.p1.7.7.m7.4.4.1.2" xref="Thmlemma4.p1.7.7.m7.4.4.1.2.cmml">−</mo><msup id="Thmlemma4.p1.7.7.m7.4.4.1.1" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.cmml"><mrow id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.cmml"><mo id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.2" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.cmml">(</mo><mrow id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.cmml"><mn id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.2" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.2.cmml">1</mn><mo id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.1" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.cmml"><mn id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.2" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.2.cmml">1</mn><msup id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.3" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.3.cmml"><mi id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.3.2" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.3.2.cmml">m</mi><mo id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.3.3" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.3.3.cmml">′</mo></msup></mfrac></mrow><mo id="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.3" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmlemma4.p1.7.7.m7.4.4.1.1.3" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.3.cmml">n</mi></msup></mrow><mo id="Thmlemma4.p1.7.7.m7.4.4.5" xref="Thmlemma4.p1.7.7.m7.4.4.5.cmml">=</mo><mfrac id="Thmlemma4.p1.7.7.m7.2.2" xref="Thmlemma4.p1.7.7.m7.2.2.cmml"><msub id="Thmlemma4.p1.7.7.m7.2.2.2" xref="Thmlemma4.p1.7.7.m7.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma4.p1.7.7.m7.2.2.2.4" xref="Thmlemma4.p1.7.7.m7.2.2.2.4.cmml">ℰ</mi><mrow id="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2" xref="Thmlemma4.p1.7.7.m7.2.2.2.2.2.3.cmml"><msup 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xref="Thmlemma4.p1.7.7.m7.4.4.1.1.1.1.1.3.3.3">′</ci></apply></apply></apply><ci id="Thmlemma4.p1.7.7.m7.4.4.1.1.3.cmml" xref="Thmlemma4.p1.7.7.m7.4.4.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmlemma4.p1.7.7.m7.4.4c.cmml" xref="Thmlemma4.p1.7.7.m7.4.4"><eq id="Thmlemma4.p1.7.7.m7.4.4.5.cmml" xref="Thmlemma4.p1.7.7.m7.4.4.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmlemma4.p1.7.7.m7.4.4.1.cmml" id="Thmlemma4.p1.7.7.m7.4.4d.cmml" xref="Thmlemma4.p1.7.7.m7.4.4"></share><apply id="Thmlemma4.p1.7.7.m7.2.2.cmml" xref="Thmlemma4.p1.7.7.m7.2.2"><divide id="Thmlemma4.p1.7.7.m7.2.2.3.cmml" xref="Thmlemma4.p1.7.7.m7.2.2"></divide><apply id="Thmlemma4.p1.7.7.m7.2.2.2.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2"><csymbol cd="ambiguous" id="Thmlemma4.p1.7.7.m7.2.2.2.3.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2">subscript</csymbol><ci id="Thmlemma4.p1.7.7.m7.2.2.2.4.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2.4">ℰ</ci><list id="Thmlemma4.p1.7.7.m7.2.2.2.2.2.3.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2"><apply id="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1.1.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1">superscript</csymbol><ci id="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1.2.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1.2">𝑚</ci><ci id="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1.3.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.2.2.2.2.1.3">′</ci></apply><ci id="Thmlemma4.p1.7.7.m7.1.1.1.1.1.1.cmml" xref="Thmlemma4.p1.7.7.m7.1.1.1.1.1.1">𝑛</ci></list></apply><apply id="Thmlemma4.p1.7.7.m7.2.2.4.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.4"><csymbol cd="ambiguous" id="Thmlemma4.p1.7.7.m7.2.2.4.1.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.4">superscript</csymbol><ci id="Thmlemma4.p1.7.7.m7.2.2.4.2.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.4.2">𝑚</ci><ci id="Thmlemma4.p1.7.7.m7.2.2.4.3.cmml" xref="Thmlemma4.p1.7.7.m7.2.2.4.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma4.p1.7.7.m7.4c">\beta(\sigma)\geq 1-\left(1-\frac{1}{m^{\prime}}\right)^{n}=\frac{\mathcal{E}_% {m^{\prime},n}}{m^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma4.p1.7.7.m7.4d">italic_β ( italic_σ ) ≥ 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>, and this bound is tight.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma5"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 5</span></h6> <div class="ltx_para" id="Thmlemma5.p1"> <p class="ltx_p" id="Thmlemma5.p1.3"><span class="ltx_text ltx_font_italic" id="Thmlemma5.p1.3.3">For every <math alttext="\sigma\in\Delta(P_{st})" class="ltx_Math" display="inline" id="Thmlemma5.p1.1.1.m1.1"><semantics id="Thmlemma5.p1.1.1.m1.1a"><mrow id="Thmlemma5.p1.1.1.m1.1.1" xref="Thmlemma5.p1.1.1.m1.1.1.cmml"><mi id="Thmlemma5.p1.1.1.m1.1.1.3" xref="Thmlemma5.p1.1.1.m1.1.1.3.cmml">σ</mi><mo id="Thmlemma5.p1.1.1.m1.1.1.2" xref="Thmlemma5.p1.1.1.m1.1.1.2.cmml">∈</mo><mrow id="Thmlemma5.p1.1.1.m1.1.1.1" xref="Thmlemma5.p1.1.1.m1.1.1.1.cmml"><mi id="Thmlemma5.p1.1.1.m1.1.1.1.3" mathvariant="normal" xref="Thmlemma5.p1.1.1.m1.1.1.1.3.cmml">Δ</mi><mo id="Thmlemma5.p1.1.1.m1.1.1.1.2" xref="Thmlemma5.p1.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma5.p1.1.1.m1.1.1.1.1.1" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.cmml"><mi 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xref="Thmlemma5.p1.1.1.m1.1.1.1"><times id="Thmlemma5.p1.1.1.m1.1.1.1.2.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.2"></times><ci id="Thmlemma5.p1.1.1.m1.1.1.1.3.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.3">Δ</ci><apply id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3"><times id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3.1"></times><ci id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="Thmlemma5.p1.1.1.m1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma5.p1.1.1.m1.1c">\sigma\in\Delta(P_{st})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma5.p1.1.1.m1.1d">italic_σ ∈ roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="Thmlemma5.p1.2.2.m2.1"><semantics id="Thmlemma5.p1.2.2.m2.1a"><mrow id="Thmlemma5.p1.2.2.m2.1.1" xref="Thmlemma5.p1.2.2.m2.1.1.cmml"><mi id="Thmlemma5.p1.2.2.m2.1.1.2" xref="Thmlemma5.p1.2.2.m2.1.1.2.cmml">n</mi><mo id="Thmlemma5.p1.2.2.m2.1.1.1" xref="Thmlemma5.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="Thmlemma5.p1.2.2.m2.1.1.3" xref="Thmlemma5.p1.2.2.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma5.p1.2.2.m2.1b"><apply id="Thmlemma5.p1.2.2.m2.1.1.cmml" xref="Thmlemma5.p1.2.2.m2.1.1"><in id="Thmlemma5.p1.2.2.m2.1.1.1.cmml" xref="Thmlemma5.p1.2.2.m2.1.1.1"></in><ci id="Thmlemma5.p1.2.2.m2.1.1.2.cmml" xref="Thmlemma5.p1.2.2.m2.1.1.2">𝑛</ci><ci id="Thmlemma5.p1.2.2.m2.1.1.3.cmml" xref="Thmlemma5.p1.2.2.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma5.p1.2.2.m2.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma5.p1.2.2.m2.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, <math alttext="\beta(\sigma)\leq\beta(\sigma^{n})\leq\min(m_{c},n)\beta(\sigma)" class="ltx_Math" display="inline" id="Thmlemma5.p1.3.3.m3.6"><semantics id="Thmlemma5.p1.3.3.m3.6a"><mrow id="Thmlemma5.p1.3.3.m3.6.6" xref="Thmlemma5.p1.3.3.m3.6.6.cmml"><mrow id="Thmlemma5.p1.3.3.m3.6.6.4" xref="Thmlemma5.p1.3.3.m3.6.6.4.cmml"><mi id="Thmlemma5.p1.3.3.m3.6.6.4.2" xref="Thmlemma5.p1.3.3.m3.6.6.4.2.cmml">β</mi><mo id="Thmlemma5.p1.3.3.m3.6.6.4.1" xref="Thmlemma5.p1.3.3.m3.6.6.4.1.cmml">⁢</mo><mrow id="Thmlemma5.p1.3.3.m3.6.6.4.3.2" xref="Thmlemma5.p1.3.3.m3.6.6.4.cmml"><mo id="Thmlemma5.p1.3.3.m3.6.6.4.3.2.1" stretchy="false" xref="Thmlemma5.p1.3.3.m3.6.6.4.cmml">(</mo><mi id="Thmlemma5.p1.3.3.m3.1.1" xref="Thmlemma5.p1.3.3.m3.1.1.cmml">σ</mi><mo id="Thmlemma5.p1.3.3.m3.6.6.4.3.2.2" stretchy="false" xref="Thmlemma5.p1.3.3.m3.6.6.4.cmml">)</mo></mrow></mrow><mo id="Thmlemma5.p1.3.3.m3.6.6.5" xref="Thmlemma5.p1.3.3.m3.6.6.5.cmml">≤</mo><mrow id="Thmlemma5.p1.3.3.m3.5.5.1" xref="Thmlemma5.p1.3.3.m3.5.5.1.cmml"><mi id="Thmlemma5.p1.3.3.m3.5.5.1.3" xref="Thmlemma5.p1.3.3.m3.5.5.1.3.cmml">β</mi><mo id="Thmlemma5.p1.3.3.m3.5.5.1.2" xref="Thmlemma5.p1.3.3.m3.5.5.1.2.cmml">⁢</mo><mrow id="Thmlemma5.p1.3.3.m3.5.5.1.1.1" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.cmml"><mo id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.2" stretchy="false" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.cmml">(</mo><msup id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.cmml"><mi id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.2" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.2.cmml">σ</mi><mi id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.3" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.3" stretchy="false" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma5.p1.3.3.m3.6.6.6" xref="Thmlemma5.p1.3.3.m3.6.6.6.cmml">≤</mo><mrow id="Thmlemma5.p1.3.3.m3.6.6.2" xref="Thmlemma5.p1.3.3.m3.6.6.2.cmml"><mrow id="Thmlemma5.p1.3.3.m3.6.6.2.1.1" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.2.cmml"><mi id="Thmlemma5.p1.3.3.m3.2.2" xref="Thmlemma5.p1.3.3.m3.2.2.cmml">min</mi><mo id="Thmlemma5.p1.3.3.m3.6.6.2.1.1a" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.2.cmml">⁡</mo><mrow id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.2.cmml"><mo id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.2" stretchy="false" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.2.cmml">(</mo><msub id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.cmml"><mi id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.2" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.2.cmml">m</mi><mi id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.3" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.3.cmml">c</mi></msub><mo id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.3" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.2.cmml">,</mo><mi id="Thmlemma5.p1.3.3.m3.3.3" xref="Thmlemma5.p1.3.3.m3.3.3.cmml">n</mi><mo id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.4" stretchy="false" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma5.p1.3.3.m3.6.6.2.2" xref="Thmlemma5.p1.3.3.m3.6.6.2.2.cmml">⁢</mo><mi id="Thmlemma5.p1.3.3.m3.6.6.2.3" xref="Thmlemma5.p1.3.3.m3.6.6.2.3.cmml">β</mi><mo id="Thmlemma5.p1.3.3.m3.6.6.2.2a" xref="Thmlemma5.p1.3.3.m3.6.6.2.2.cmml">⁢</mo><mrow id="Thmlemma5.p1.3.3.m3.6.6.2.4.2" xref="Thmlemma5.p1.3.3.m3.6.6.2.cmml"><mo id="Thmlemma5.p1.3.3.m3.6.6.2.4.2.1" stretchy="false" xref="Thmlemma5.p1.3.3.m3.6.6.2.cmml">(</mo><mi id="Thmlemma5.p1.3.3.m3.4.4" xref="Thmlemma5.p1.3.3.m3.4.4.cmml">σ</mi><mo id="Thmlemma5.p1.3.3.m3.6.6.2.4.2.2" stretchy="false" xref="Thmlemma5.p1.3.3.m3.6.6.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma5.p1.3.3.m3.6b"><apply id="Thmlemma5.p1.3.3.m3.6.6.cmml" xref="Thmlemma5.p1.3.3.m3.6.6"><and id="Thmlemma5.p1.3.3.m3.6.6a.cmml" xref="Thmlemma5.p1.3.3.m3.6.6"></and><apply id="Thmlemma5.p1.3.3.m3.6.6b.cmml" xref="Thmlemma5.p1.3.3.m3.6.6"><leq id="Thmlemma5.p1.3.3.m3.6.6.5.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.5"></leq><apply id="Thmlemma5.p1.3.3.m3.6.6.4.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.4"><times id="Thmlemma5.p1.3.3.m3.6.6.4.1.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.4.1"></times><ci id="Thmlemma5.p1.3.3.m3.6.6.4.2.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.4.2">𝛽</ci><ci id="Thmlemma5.p1.3.3.m3.1.1.cmml" xref="Thmlemma5.p1.3.3.m3.1.1">𝜎</ci></apply><apply id="Thmlemma5.p1.3.3.m3.5.5.1.cmml" xref="Thmlemma5.p1.3.3.m3.5.5.1"><times id="Thmlemma5.p1.3.3.m3.5.5.1.2.cmml" xref="Thmlemma5.p1.3.3.m3.5.5.1.2"></times><ci id="Thmlemma5.p1.3.3.m3.5.5.1.3.cmml" xref="Thmlemma5.p1.3.3.m3.5.5.1.3">𝛽</ci><apply id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.cmml" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1"><csymbol cd="ambiguous" id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.1.cmml" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1">superscript</csymbol><ci id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.2.cmml" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.2">𝜎</ci><ci id="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.3.cmml" xref="Thmlemma5.p1.3.3.m3.5.5.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmlemma5.p1.3.3.m3.6.6c.cmml" xref="Thmlemma5.p1.3.3.m3.6.6"><leq id="Thmlemma5.p1.3.3.m3.6.6.6.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.6"></leq><share href="https://arxiv.org/html/2308.08047v3#Thmlemma5.p1.3.3.m3.5.5.1.cmml" id="Thmlemma5.p1.3.3.m3.6.6d.cmml" xref="Thmlemma5.p1.3.3.m3.6.6"></share><apply id="Thmlemma5.p1.3.3.m3.6.6.2.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2"><times id="Thmlemma5.p1.3.3.m3.6.6.2.2.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2.2"></times><apply id="Thmlemma5.p1.3.3.m3.6.6.2.1.2.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1"><min id="Thmlemma5.p1.3.3.m3.2.2.cmml" xref="Thmlemma5.p1.3.3.m3.2.2"></min><apply id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.1.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1">subscript</csymbol><ci id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.2.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.2">𝑚</ci><ci id="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.3.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2.1.1.1.1.3">𝑐</ci></apply><ci id="Thmlemma5.p1.3.3.m3.3.3.cmml" xref="Thmlemma5.p1.3.3.m3.3.3">𝑛</ci></apply><ci id="Thmlemma5.p1.3.3.m3.6.6.2.3.cmml" xref="Thmlemma5.p1.3.3.m3.6.6.2.3">𝛽</ci><ci id="Thmlemma5.p1.3.3.m3.4.4.cmml" xref="Thmlemma5.p1.3.3.m3.4.4">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma5.p1.3.3.m3.6c">\beta(\sigma)\leq\beta(\sigma^{n})\leq\min(m_{c},n)\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma5.p1.3.3.m3.6d">italic_β ( italic_σ ) ≤ italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ≤ roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n ) italic_β ( italic_σ )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma6"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 6</span></h6> <div class="ltx_para" id="Thmlemma6.p1"> <p class="ltx_p" id="Thmlemma6.p1.6"><span class="ltx_text ltx_font_italic" id="Thmlemma6.p1.6.6">Let <math alttext="x\in\mathbb{R}" class="ltx_Math" display="inline" id="Thmlemma6.p1.1.1.m1.1"><semantics id="Thmlemma6.p1.1.1.m1.1a"><mrow id="Thmlemma6.p1.1.1.m1.1.1" xref="Thmlemma6.p1.1.1.m1.1.1.cmml"><mi id="Thmlemma6.p1.1.1.m1.1.1.2" xref="Thmlemma6.p1.1.1.m1.1.1.2.cmml">x</mi><mo id="Thmlemma6.p1.1.1.m1.1.1.1" xref="Thmlemma6.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="Thmlemma6.p1.1.1.m1.1.1.3" xref="Thmlemma6.p1.1.1.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma6.p1.1.1.m1.1b"><apply id="Thmlemma6.p1.1.1.m1.1.1.cmml" xref="Thmlemma6.p1.1.1.m1.1.1"><in id="Thmlemma6.p1.1.1.m1.1.1.1.cmml" xref="Thmlemma6.p1.1.1.m1.1.1.1"></in><ci id="Thmlemma6.p1.1.1.m1.1.1.2.cmml" xref="Thmlemma6.p1.1.1.m1.1.1.2">𝑥</ci><ci id="Thmlemma6.p1.1.1.m1.1.1.3.cmml" xref="Thmlemma6.p1.1.1.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma6.p1.1.1.m1.1c">x\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma6.p1.1.1.m1.1d">italic_x ∈ blackboard_R</annotation></semantics></math>. Suppose that <math alttext="C" class="ltx_Math" display="inline" id="Thmlemma6.p1.2.2.m2.1"><semantics id="Thmlemma6.p1.2.2.m2.1a"><mi id="Thmlemma6.p1.2.2.m2.1.1" xref="Thmlemma6.p1.2.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="Thmlemma6.p1.2.2.m2.1b"><ci id="Thmlemma6.p1.2.2.m2.1.1.cmml" xref="Thmlemma6.p1.2.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma6.p1.2.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="Thmlemma6.p1.2.2.m2.1d">italic_C</annotation></semantics></math> is <math alttext="F" class="ltx_Math" display="inline" id="Thmlemma6.p1.3.3.m3.1"><semantics id="Thmlemma6.p1.3.3.m3.1a"><mi id="Thmlemma6.p1.3.3.m3.1.1" xref="Thmlemma6.p1.3.3.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmlemma6.p1.3.3.m3.1b"><ci id="Thmlemma6.p1.3.3.m3.1.1.cmml" xref="Thmlemma6.p1.3.3.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma6.p1.3.3.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmlemma6.p1.3.3.m3.1d">italic_F</annotation></semantics></math>-nice and for every <math alttext="\sigma\in\Delta(P_{st})" class="ltx_Math" display="inline" id="Thmlemma6.p1.4.4.m4.1"><semantics id="Thmlemma6.p1.4.4.m4.1a"><mrow id="Thmlemma6.p1.4.4.m4.1.1" xref="Thmlemma6.p1.4.4.m4.1.1.cmml"><mi id="Thmlemma6.p1.4.4.m4.1.1.3" xref="Thmlemma6.p1.4.4.m4.1.1.3.cmml">σ</mi><mo id="Thmlemma6.p1.4.4.m4.1.1.2" xref="Thmlemma6.p1.4.4.m4.1.1.2.cmml">∈</mo><mrow id="Thmlemma6.p1.4.4.m4.1.1.1" xref="Thmlemma6.p1.4.4.m4.1.1.1.cmml"><mi id="Thmlemma6.p1.4.4.m4.1.1.1.3" mathvariant="normal" xref="Thmlemma6.p1.4.4.m4.1.1.1.3.cmml">Δ</mi><mo id="Thmlemma6.p1.4.4.m4.1.1.1.2" xref="Thmlemma6.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma6.p1.4.4.m4.1.1.1.1.1" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.2" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.cmml"><mi id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.2" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.1" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.3" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma6.p1.4.4.m4.1b"><apply id="Thmlemma6.p1.4.4.m4.1.1.cmml" xref="Thmlemma6.p1.4.4.m4.1.1"><in id="Thmlemma6.p1.4.4.m4.1.1.2.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.2"></in><ci id="Thmlemma6.p1.4.4.m4.1.1.3.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.3">𝜎</ci><apply id="Thmlemma6.p1.4.4.m4.1.1.1.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1"><times id="Thmlemma6.p1.4.4.m4.1.1.1.2.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.2"></times><ci id="Thmlemma6.p1.4.4.m4.1.1.1.3.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.3">Δ</ci><apply id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3"><times id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.1.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.1"></times><ci id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.2.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.3.cmml" xref="Thmlemma6.p1.4.4.m4.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma6.p1.4.4.m4.1c">\sigma\in\Delta(P_{st})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma6.p1.4.4.m4.1d">italic_σ ∈ roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> there holds <math alttext="F_{n}(\sigma^{n})\leq xF_{1}(\sigma)" class="ltx_Math" display="inline" id="Thmlemma6.p1.5.5.m5.2"><semantics id="Thmlemma6.p1.5.5.m5.2a"><mrow id="Thmlemma6.p1.5.5.m5.2.2" xref="Thmlemma6.p1.5.5.m5.2.2.cmml"><mrow id="Thmlemma6.p1.5.5.m5.2.2.1" xref="Thmlemma6.p1.5.5.m5.2.2.1.cmml"><msub id="Thmlemma6.p1.5.5.m5.2.2.1.3" xref="Thmlemma6.p1.5.5.m5.2.2.1.3.cmml"><mi id="Thmlemma6.p1.5.5.m5.2.2.1.3.2" xref="Thmlemma6.p1.5.5.m5.2.2.1.3.2.cmml">F</mi><mi id="Thmlemma6.p1.5.5.m5.2.2.1.3.3" xref="Thmlemma6.p1.5.5.m5.2.2.1.3.3.cmml">n</mi></msub><mo id="Thmlemma6.p1.5.5.m5.2.2.1.2" xref="Thmlemma6.p1.5.5.m5.2.2.1.2.cmml">⁢</mo><mrow id="Thmlemma6.p1.5.5.m5.2.2.1.1.1" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.cmml"><mo id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.2" stretchy="false" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.cmml">(</mo><msup id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.cmml"><mi id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.2" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.2.cmml">σ</mi><mi id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.3" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.3" stretchy="false" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma6.p1.5.5.m5.2.2.2" xref="Thmlemma6.p1.5.5.m5.2.2.2.cmml">≤</mo><mrow id="Thmlemma6.p1.5.5.m5.2.2.3" xref="Thmlemma6.p1.5.5.m5.2.2.3.cmml"><mi id="Thmlemma6.p1.5.5.m5.2.2.3.2" xref="Thmlemma6.p1.5.5.m5.2.2.3.2.cmml">x</mi><mo id="Thmlemma6.p1.5.5.m5.2.2.3.1" xref="Thmlemma6.p1.5.5.m5.2.2.3.1.cmml">⁢</mo><msub id="Thmlemma6.p1.5.5.m5.2.2.3.3" xref="Thmlemma6.p1.5.5.m5.2.2.3.3.cmml"><mi id="Thmlemma6.p1.5.5.m5.2.2.3.3.2" xref="Thmlemma6.p1.5.5.m5.2.2.3.3.2.cmml">F</mi><mn id="Thmlemma6.p1.5.5.m5.2.2.3.3.3" xref="Thmlemma6.p1.5.5.m5.2.2.3.3.3.cmml">1</mn></msub><mo id="Thmlemma6.p1.5.5.m5.2.2.3.1a" xref="Thmlemma6.p1.5.5.m5.2.2.3.1.cmml">⁢</mo><mrow id="Thmlemma6.p1.5.5.m5.2.2.3.4.2" xref="Thmlemma6.p1.5.5.m5.2.2.3.cmml"><mo id="Thmlemma6.p1.5.5.m5.2.2.3.4.2.1" stretchy="false" xref="Thmlemma6.p1.5.5.m5.2.2.3.cmml">(</mo><mi id="Thmlemma6.p1.5.5.m5.1.1" xref="Thmlemma6.p1.5.5.m5.1.1.cmml">σ</mi><mo id="Thmlemma6.p1.5.5.m5.2.2.3.4.2.2" stretchy="false" xref="Thmlemma6.p1.5.5.m5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma6.p1.5.5.m5.2b"><apply id="Thmlemma6.p1.5.5.m5.2.2.cmml" xref="Thmlemma6.p1.5.5.m5.2.2"><leq id="Thmlemma6.p1.5.5.m5.2.2.2.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.2"></leq><apply id="Thmlemma6.p1.5.5.m5.2.2.1.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1"><times id="Thmlemma6.p1.5.5.m5.2.2.1.2.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.2"></times><apply id="Thmlemma6.p1.5.5.m5.2.2.1.3.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.3"><csymbol cd="ambiguous" id="Thmlemma6.p1.5.5.m5.2.2.1.3.1.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.3">subscript</csymbol><ci id="Thmlemma6.p1.5.5.m5.2.2.1.3.2.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.3.2">𝐹</ci><ci id="Thmlemma6.p1.5.5.m5.2.2.1.3.3.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.3.3">𝑛</ci></apply><apply id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.1.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1">superscript</csymbol><ci id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.2.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.2">𝜎</ci><ci id="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.3.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmlemma6.p1.5.5.m5.2.2.3.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.3"><times id="Thmlemma6.p1.5.5.m5.2.2.3.1.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.3.1"></times><ci id="Thmlemma6.p1.5.5.m5.2.2.3.2.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.3.2">𝑥</ci><apply id="Thmlemma6.p1.5.5.m5.2.2.3.3.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.3.3"><csymbol cd="ambiguous" id="Thmlemma6.p1.5.5.m5.2.2.3.3.1.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.3.3">subscript</csymbol><ci id="Thmlemma6.p1.5.5.m5.2.2.3.3.2.cmml" xref="Thmlemma6.p1.5.5.m5.2.2.3.3.2">𝐹</ci><cn id="Thmlemma6.p1.5.5.m5.2.2.3.3.3.cmml" type="integer" xref="Thmlemma6.p1.5.5.m5.2.2.3.3.3">1</cn></apply><ci id="Thmlemma6.p1.5.5.m5.1.1.cmml" xref="Thmlemma6.p1.5.5.m5.1.1">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma6.p1.5.5.m5.2c">F_{n}(\sigma^{n})\leq xF_{1}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma6.p1.5.5.m5.2d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ≤ italic_x italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_σ )</annotation></semantics></math>. Then <math alttext="r(G,n,k,\alpha)\leq x" class="ltx_Math" display="inline" id="Thmlemma6.p1.6.6.m6.4"><semantics id="Thmlemma6.p1.6.6.m6.4a"><mrow id="Thmlemma6.p1.6.6.m6.4.5" xref="Thmlemma6.p1.6.6.m6.4.5.cmml"><mrow id="Thmlemma6.p1.6.6.m6.4.5.2" xref="Thmlemma6.p1.6.6.m6.4.5.2.cmml"><mi id="Thmlemma6.p1.6.6.m6.4.5.2.2" xref="Thmlemma6.p1.6.6.m6.4.5.2.2.cmml">r</mi><mo id="Thmlemma6.p1.6.6.m6.4.5.2.1" xref="Thmlemma6.p1.6.6.m6.4.5.2.1.cmml">⁢</mo><mrow id="Thmlemma6.p1.6.6.m6.4.5.2.3.2" xref="Thmlemma6.p1.6.6.m6.4.5.2.3.1.cmml"><mo id="Thmlemma6.p1.6.6.m6.4.5.2.3.2.1" stretchy="false" xref="Thmlemma6.p1.6.6.m6.4.5.2.3.1.cmml">(</mo><mi id="Thmlemma6.p1.6.6.m6.1.1" xref="Thmlemma6.p1.6.6.m6.1.1.cmml">G</mi><mo id="Thmlemma6.p1.6.6.m6.4.5.2.3.2.2" xref="Thmlemma6.p1.6.6.m6.4.5.2.3.1.cmml">,</mo><mi id="Thmlemma6.p1.6.6.m6.2.2" xref="Thmlemma6.p1.6.6.m6.2.2.cmml">n</mi><mo id="Thmlemma6.p1.6.6.m6.4.5.2.3.2.3" xref="Thmlemma6.p1.6.6.m6.4.5.2.3.1.cmml">,</mo><mi id="Thmlemma6.p1.6.6.m6.3.3" xref="Thmlemma6.p1.6.6.m6.3.3.cmml">k</mi><mo id="Thmlemma6.p1.6.6.m6.4.5.2.3.2.4" xref="Thmlemma6.p1.6.6.m6.4.5.2.3.1.cmml">,</mo><mi id="Thmlemma6.p1.6.6.m6.4.4" xref="Thmlemma6.p1.6.6.m6.4.4.cmml">α</mi><mo id="Thmlemma6.p1.6.6.m6.4.5.2.3.2.5" stretchy="false" xref="Thmlemma6.p1.6.6.m6.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma6.p1.6.6.m6.4.5.1" xref="Thmlemma6.p1.6.6.m6.4.5.1.cmml">≤</mo><mi id="Thmlemma6.p1.6.6.m6.4.5.3" xref="Thmlemma6.p1.6.6.m6.4.5.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma6.p1.6.6.m6.4b"><apply id="Thmlemma6.p1.6.6.m6.4.5.cmml" xref="Thmlemma6.p1.6.6.m6.4.5"><leq id="Thmlemma6.p1.6.6.m6.4.5.1.cmml" xref="Thmlemma6.p1.6.6.m6.4.5.1"></leq><apply id="Thmlemma6.p1.6.6.m6.4.5.2.cmml" xref="Thmlemma6.p1.6.6.m6.4.5.2"><times id="Thmlemma6.p1.6.6.m6.4.5.2.1.cmml" xref="Thmlemma6.p1.6.6.m6.4.5.2.1"></times><ci id="Thmlemma6.p1.6.6.m6.4.5.2.2.cmml" xref="Thmlemma6.p1.6.6.m6.4.5.2.2">𝑟</ci><vector id="Thmlemma6.p1.6.6.m6.4.5.2.3.1.cmml" xref="Thmlemma6.p1.6.6.m6.4.5.2.3.2"><ci id="Thmlemma6.p1.6.6.m6.1.1.cmml" xref="Thmlemma6.p1.6.6.m6.1.1">𝐺</ci><ci id="Thmlemma6.p1.6.6.m6.2.2.cmml" xref="Thmlemma6.p1.6.6.m6.2.2">𝑛</ci><ci id="Thmlemma6.p1.6.6.m6.3.3.cmml" xref="Thmlemma6.p1.6.6.m6.3.3">𝑘</ci><ci id="Thmlemma6.p1.6.6.m6.4.4.cmml" xref="Thmlemma6.p1.6.6.m6.4.4">𝛼</ci></vector></apply><ci id="Thmlemma6.p1.6.6.m6.4.5.3.cmml" xref="Thmlemma6.p1.6.6.m6.4.5.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma6.p1.6.6.m6.4c">r(G,n,k,\alpha)\leq x</annotation><annotation encoding="application/x-llamapun" id="Thmlemma6.p1.6.6.m6.4d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≤ italic_x</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.4">Suppose that <math alttext="P_{st}=\left\{p_{1},...,p_{m}\right\}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.3"><semantics id="S3.SS3.p3.1.m1.3a"><mrow id="S3.SS3.p3.1.m1.3.3" xref="S3.SS3.p3.1.m1.3.3.cmml"><msub id="S3.SS3.p3.1.m1.3.3.4" xref="S3.SS3.p3.1.m1.3.3.4.cmml"><mi id="S3.SS3.p3.1.m1.3.3.4.2" xref="S3.SS3.p3.1.m1.3.3.4.2.cmml">P</mi><mrow id="S3.SS3.p3.1.m1.3.3.4.3" xref="S3.SS3.p3.1.m1.3.3.4.3.cmml"><mi id="S3.SS3.p3.1.m1.3.3.4.3.2" xref="S3.SS3.p3.1.m1.3.3.4.3.2.cmml">s</mi><mo id="S3.SS3.p3.1.m1.3.3.4.3.1" xref="S3.SS3.p3.1.m1.3.3.4.3.1.cmml">⁢</mo><mi id="S3.SS3.p3.1.m1.3.3.4.3.3" xref="S3.SS3.p3.1.m1.3.3.4.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS3.p3.1.m1.3.3.3" xref="S3.SS3.p3.1.m1.3.3.3.cmml">=</mo><mrow id="S3.SS3.p3.1.m1.3.3.2.2" xref="S3.SS3.p3.1.m1.3.3.2.3.cmml"><mo id="S3.SS3.p3.1.m1.3.3.2.2.3" xref="S3.SS3.p3.1.m1.3.3.2.3.cmml">{</mo><msub id="S3.SS3.p3.1.m1.2.2.1.1.1" xref="S3.SS3.p3.1.m1.2.2.1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.2.2.1.1.1.2" xref="S3.SS3.p3.1.m1.2.2.1.1.1.2.cmml">p</mi><mn id="S3.SS3.p3.1.m1.2.2.1.1.1.3" xref="S3.SS3.p3.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p3.1.m1.3.3.2.2.4" xref="S3.SS3.p3.1.m1.3.3.2.3.cmml">,</mo><mi id="S3.SS3.p3.1.m1.1.1" mathvariant="normal" xref="S3.SS3.p3.1.m1.1.1.cmml">…</mi><mo id="S3.SS3.p3.1.m1.3.3.2.2.5" xref="S3.SS3.p3.1.m1.3.3.2.3.cmml">,</mo><msub id="S3.SS3.p3.1.m1.3.3.2.2.2" xref="S3.SS3.p3.1.m1.3.3.2.2.2.cmml"><mi id="S3.SS3.p3.1.m1.3.3.2.2.2.2" xref="S3.SS3.p3.1.m1.3.3.2.2.2.2.cmml">p</mi><mi id="S3.SS3.p3.1.m1.3.3.2.2.2.3" xref="S3.SS3.p3.1.m1.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S3.SS3.p3.1.m1.3.3.2.2.6" xref="S3.SS3.p3.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.3b"><apply id="S3.SS3.p3.1.m1.3.3.cmml" xref="S3.SS3.p3.1.m1.3.3"><eq id="S3.SS3.p3.1.m1.3.3.3.cmml" xref="S3.SS3.p3.1.m1.3.3.3"></eq><apply id="S3.SS3.p3.1.m1.3.3.4.cmml" xref="S3.SS3.p3.1.m1.3.3.4"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.3.3.4.1.cmml" xref="S3.SS3.p3.1.m1.3.3.4">subscript</csymbol><ci id="S3.SS3.p3.1.m1.3.3.4.2.cmml" xref="S3.SS3.p3.1.m1.3.3.4.2">𝑃</ci><apply id="S3.SS3.p3.1.m1.3.3.4.3.cmml" xref="S3.SS3.p3.1.m1.3.3.4.3"><times id="S3.SS3.p3.1.m1.3.3.4.3.1.cmml" xref="S3.SS3.p3.1.m1.3.3.4.3.1"></times><ci id="S3.SS3.p3.1.m1.3.3.4.3.2.cmml" xref="S3.SS3.p3.1.m1.3.3.4.3.2">𝑠</ci><ci id="S3.SS3.p3.1.m1.3.3.4.3.3.cmml" xref="S3.SS3.p3.1.m1.3.3.4.3.3">𝑡</ci></apply></apply><set id="S3.SS3.p3.1.m1.3.3.2.3.cmml" xref="S3.SS3.p3.1.m1.3.3.2.2"><apply id="S3.SS3.p3.1.m1.2.2.1.1.1.cmml" xref="S3.SS3.p3.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.2.2.1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S3.SS3.p3.1.m1.2.2.1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.2.2.1.1.1.2">𝑝</ci><cn id="S3.SS3.p3.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS3.p3.1.m1.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1">…</ci><apply id="S3.SS3.p3.1.m1.3.3.2.2.2.cmml" xref="S3.SS3.p3.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.3.3.2.2.2.1.cmml" xref="S3.SS3.p3.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S3.SS3.p3.1.m1.3.3.2.2.2.2.cmml" xref="S3.SS3.p3.1.m1.3.3.2.2.2.2">𝑝</ci><ci id="S3.SS3.p3.1.m1.3.3.2.2.2.3.cmml" xref="S3.SS3.p3.1.m1.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.3c">P_{st}=\left\{p_{1},...,p_{m}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.3d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT = { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math> is a set of disjoint paths. Let <math alttext="S=\{X\in\Delta(P_{st})\mid\forall j<j^{\prime},~{}~{}X(p_{i})\geq X(p_{i^{% \prime}})\}" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.2"><semantics id="S3.SS3.p3.2.m2.2a"><mrow id="S3.SS3.p3.2.m2.2.2" xref="S3.SS3.p3.2.m2.2.2.cmml"><mi id="S3.SS3.p3.2.m2.2.2.4" xref="S3.SS3.p3.2.m2.2.2.4.cmml">S</mi><mo id="S3.SS3.p3.2.m2.2.2.3" xref="S3.SS3.p3.2.m2.2.2.3.cmml">=</mo><mrow id="S3.SS3.p3.2.m2.2.2.2.2" xref="S3.SS3.p3.2.m2.2.2.2.3.cmml"><mo id="S3.SS3.p3.2.m2.2.2.2.2.3" stretchy="false" xref="S3.SS3.p3.2.m2.2.2.2.3.1.cmml">{</mo><mrow id="S3.SS3.p3.2.m2.1.1.1.1.1" xref="S3.SS3.p3.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS3.p3.2.m2.1.1.1.1.1.3" xref="S3.SS3.p3.2.m2.1.1.1.1.1.3.cmml">X</mi><mo id="S3.SS3.p3.2.m2.1.1.1.1.1.2" xref="S3.SS3.p3.2.m2.1.1.1.1.1.2.cmml">∈</mo><mrow id="S3.SS3.p3.2.m2.1.1.1.1.1.1" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p3.2.m2.1.1.1.1.1.1.3" mathvariant="normal" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="S3.SS3.p3.2.m2.1.1.1.1.1.1.2" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.2" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3.2" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3.1" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3.3" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p3.2.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S3.SS3.p3.2.m2.2.2.2.2.4" lspace="0em" rspace="0.167em" xref="S3.SS3.p3.2.m2.2.2.2.3.1.cmml">∣</mo><mrow id="S3.SS3.p3.2.m2.2.2.2.2.2.2" xref="S3.SS3.p3.2.m2.2.2.2.2.2.3.cmml"><mrow id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1" xref="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.cmml"><mrow id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.2" xref="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.2.cmml"><mo id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.2.1" rspace="0.167em" xref="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.2.1.cmml">∀</mo><mi id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.2.2" xref="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.2.2.cmml">j</mi></mrow><mo id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.1" xref="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.1.cmml"><</mo><msup id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.3" xref="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.3.cmml"><mi id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.3.2" xref="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.3.2.cmml">j</mi><mo id="S3.SS3.p3.2.m2.2.2.2.2.2.1.1.3.3" 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( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) }</annotation></semantics></math>, the set of decreasing distributions with respect to the indexing of <math alttext="P_{st}" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><msub id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml"><mi id="S3.SS3.p3.3.m3.1.1.2" xref="S3.SS3.p3.3.m3.1.1.2.cmml">P</mi><mrow id="S3.SS3.p3.3.m3.1.1.3" xref="S3.SS3.p3.3.m3.1.1.3.cmml"><mi id="S3.SS3.p3.3.m3.1.1.3.2" xref="S3.SS3.p3.3.m3.1.1.3.2.cmml">s</mi><mo id="S3.SS3.p3.3.m3.1.1.3.1" xref="S3.SS3.p3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.SS3.p3.3.m3.1.1.3.3" xref="S3.SS3.p3.3.m3.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><apply id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.3.m3.1.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.p3.3.m3.1.1.2.cmml" xref="S3.SS3.p3.3.m3.1.1.2">𝑃</ci><apply id="S3.SS3.p3.3.m3.1.1.3.cmml" xref="S3.SS3.p3.3.m3.1.1.3"><times id="S3.SS3.p3.3.m3.1.1.3.1.cmml" xref="S3.SS3.p3.3.m3.1.1.3.1"></times><ci id="S3.SS3.p3.3.m3.1.1.3.2.cmml" xref="S3.SS3.p3.3.m3.1.1.3.2">𝑠</ci><ci id="S3.SS3.p3.3.m3.1.1.3.3.cmml" xref="S3.SS3.p3.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. In our analysis, <math alttext="S" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mi id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><ci id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">italic_S</annotation></semantics></math> will come up in the case of disjoint paths, where path lengths are increasing with respect to indexing.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma7"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 7</span></h6> <div class="ltx_para" id="Thmlemma7.p1"> <p class="ltx_p" id="Thmlemma7.p1.3"><math alttext="S=\operatorname{conv}(\mathcal{U}_{1})" class="ltx_Math" display="inline" 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id="Thmlemma7.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="Thmlemma7.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma7.p1.1.m1.2b"><apply id="Thmlemma7.p1.1.m1.2.2.cmml" xref="Thmlemma7.p1.1.m1.2.2"><eq id="Thmlemma7.p1.1.m1.2.2.2.cmml" xref="Thmlemma7.p1.1.m1.2.2.2"></eq><ci id="Thmlemma7.p1.1.m1.2.2.3.cmml" xref="Thmlemma7.p1.1.m1.2.2.3">𝑆</ci><apply id="Thmlemma7.p1.1.m1.2.2.1.2.cmml" xref="Thmlemma7.p1.1.m1.2.2.1.1"><ci id="Thmlemma7.p1.1.m1.1.1.cmml" xref="Thmlemma7.p1.1.m1.1.1">conv</ci><apply id="Thmlemma7.p1.1.m1.2.2.1.1.1.1.cmml" xref="Thmlemma7.p1.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma7.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="Thmlemma7.p1.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="Thmlemma7.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="Thmlemma7.p1.1.m1.2.2.1.1.1.1.2">𝒰</ci><cn id="Thmlemma7.p1.1.m1.2.2.1.1.1.1.3.cmml" type="integer" xref="Thmlemma7.p1.1.m1.2.2.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma7.p1.1.m1.2c">S=\operatorname{conv}(\mathcal{U}_{1})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma7.p1.1.m1.2d">italic_S = roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Thmlemma7.p1.3.2">, and for every <math alttext="n," class="ltx_Math" display="inline" id="Thmlemma7.p1.2.1.m1.1"><semantics id="Thmlemma7.p1.2.1.m1.1a"><mrow id="Thmlemma7.p1.2.1.m1.1.2.2"><mi id="Thmlemma7.p1.2.1.m1.1.1" xref="Thmlemma7.p1.2.1.m1.1.1.cmml">n</mi><mo id="Thmlemma7.p1.2.1.m1.1.2.2.1">,</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma7.p1.2.1.m1.1b"><ci id="Thmlemma7.p1.2.1.m1.1.1.cmml" xref="Thmlemma7.p1.2.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma7.p1.2.1.m1.1c">n,</annotation><annotation encoding="application/x-llamapun" id="Thmlemma7.p1.2.1.m1.1d">italic_n ,</annotation></semantics></math> <math alttext="\operatorname{conv}(\mathcal{U}_{n})=\left\{X^{n}\mid X\in\operatorname{conv}(% \mathcal{U}_{1})\right\}" class="ltx_Math" display="inline" id="Thmlemma7.p1.3.2.m2.5"><semantics id="Thmlemma7.p1.3.2.m2.5a"><mrow id="Thmlemma7.p1.3.2.m2.5.5" xref="Thmlemma7.p1.3.2.m2.5.5.cmml"><mrow id="Thmlemma7.p1.3.2.m2.3.3.1.1" xref="Thmlemma7.p1.3.2.m2.3.3.1.2.cmml"><mi id="Thmlemma7.p1.3.2.m2.1.1" xref="Thmlemma7.p1.3.2.m2.1.1.cmml">conv</mi><mo id="Thmlemma7.p1.3.2.m2.3.3.1.1a" xref="Thmlemma7.p1.3.2.m2.3.3.1.2.cmml">⁡</mo><mrow id="Thmlemma7.p1.3.2.m2.3.3.1.1.1" xref="Thmlemma7.p1.3.2.m2.3.3.1.2.cmml"><mo id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.2" stretchy="false" xref="Thmlemma7.p1.3.2.m2.3.3.1.2.cmml">(</mo><msub id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1" xref="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.2" xref="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.2.cmml">𝒰</mi><mi id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.3" xref="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.3.cmml">n</mi></msub><mo id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.3" stretchy="false" xref="Thmlemma7.p1.3.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma7.p1.3.2.m2.5.5.4" xref="Thmlemma7.p1.3.2.m2.5.5.4.cmml">=</mo><mrow id="Thmlemma7.p1.3.2.m2.5.5.3.2" xref="Thmlemma7.p1.3.2.m2.5.5.3.3.cmml"><mo id="Thmlemma7.p1.3.2.m2.5.5.3.2.3" xref="Thmlemma7.p1.3.2.m2.5.5.3.3.1.cmml">{</mo><msup id="Thmlemma7.p1.3.2.m2.4.4.2.1.1" xref="Thmlemma7.p1.3.2.m2.4.4.2.1.1.cmml"><mi id="Thmlemma7.p1.3.2.m2.4.4.2.1.1.2" xref="Thmlemma7.p1.3.2.m2.4.4.2.1.1.2.cmml">X</mi><mi id="Thmlemma7.p1.3.2.m2.4.4.2.1.1.3" xref="Thmlemma7.p1.3.2.m2.4.4.2.1.1.3.cmml">n</mi></msup><mo fence="true" id="Thmlemma7.p1.3.2.m2.5.5.3.2.4" lspace="0em" rspace="0em" xref="Thmlemma7.p1.3.2.m2.5.5.3.3.1.cmml">∣</mo><mrow id="Thmlemma7.p1.3.2.m2.5.5.3.2.2" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.cmml"><mi id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.3" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.3.cmml">X</mi><mo id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.2" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.2.cmml">∈</mo><mrow id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.2.cmml"><mi id="Thmlemma7.p1.3.2.m2.2.2" xref="Thmlemma7.p1.3.2.m2.2.2.cmml">conv</mi><mo id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1a" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.2.cmml">⁡</mo><mrow id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.2.cmml"><mo id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.2" stretchy="false" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.2.cmml">(</mo><msub id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.2" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.2.cmml">𝒰</mi><mn id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.3" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.3" stretchy="false" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="Thmlemma7.p1.3.2.m2.5.5.3.2.5" xref="Thmlemma7.p1.3.2.m2.5.5.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma7.p1.3.2.m2.5b"><apply id="Thmlemma7.p1.3.2.m2.5.5.cmml" xref="Thmlemma7.p1.3.2.m2.5.5"><eq id="Thmlemma7.p1.3.2.m2.5.5.4.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.4"></eq><apply id="Thmlemma7.p1.3.2.m2.3.3.1.2.cmml" xref="Thmlemma7.p1.3.2.m2.3.3.1.1"><ci id="Thmlemma7.p1.3.2.m2.1.1.cmml" xref="Thmlemma7.p1.3.2.m2.1.1">conv</ci><apply id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.cmml" xref="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.1.cmml" xref="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1">subscript</csymbol><ci id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.2.cmml" xref="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.2">𝒰</ci><ci id="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.3.cmml" xref="Thmlemma7.p1.3.2.m2.3.3.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmlemma7.p1.3.2.m2.5.5.3.3.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2"><csymbol cd="latexml" id="Thmlemma7.p1.3.2.m2.5.5.3.3.1.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.3">conditional-set</csymbol><apply id="Thmlemma7.p1.3.2.m2.4.4.2.1.1.cmml" xref="Thmlemma7.p1.3.2.m2.4.4.2.1.1"><csymbol cd="ambiguous" id="Thmlemma7.p1.3.2.m2.4.4.2.1.1.1.cmml" xref="Thmlemma7.p1.3.2.m2.4.4.2.1.1">superscript</csymbol><ci id="Thmlemma7.p1.3.2.m2.4.4.2.1.1.2.cmml" xref="Thmlemma7.p1.3.2.m2.4.4.2.1.1.2">𝑋</ci><ci id="Thmlemma7.p1.3.2.m2.4.4.2.1.1.3.cmml" xref="Thmlemma7.p1.3.2.m2.4.4.2.1.1.3">𝑛</ci></apply><apply id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2"><in id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.2.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.2"></in><ci id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.3.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.3">𝑋</ci><apply id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.2.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1"><ci id="Thmlemma7.p1.3.2.m2.2.2.cmml" xref="Thmlemma7.p1.3.2.m2.2.2">conv</ci><apply id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.1.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1">subscript</csymbol><ci id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.2.cmml" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.2">𝒰</ci><cn id="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.3.cmml" type="integer" xref="Thmlemma7.p1.3.2.m2.5.5.3.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma7.p1.3.2.m2.5c">\operatorname{conv}(\mathcal{U}_{n})=\left\{X^{n}\mid X\in\operatorname{conv}(% \mathcal{U}_{1})\right\}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma7.p1.3.2.m2.5d">roman_conv ( caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = { italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∣ italic_X ∈ roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) }</annotation></semantics></math>.</span></p> </div> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>SUM Cost</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">The full proofs of this section can be found in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A3" title="Appendix 0.C Omitted Proofs in Section 4 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.C</span></a>.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>General <math alttext="s-t" class="ltx_Math" display="inline" id="S4.SS1.1.m1.1"><semantics id="S4.SS1.1.m1.1b"><mrow id="S4.SS1.1.m1.1.1" xref="S4.SS1.1.m1.1.1.cmml"><mi id="S4.SS1.1.m1.1.1.2" xref="S4.SS1.1.m1.1.1.2.cmml">s</mi><mo id="S4.SS1.1.m1.1.1.1" xref="S4.SS1.1.m1.1.1.1.cmml">−</mo><mi id="S4.SS1.1.m1.1.1.3" xref="S4.SS1.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.1.m1.1c"><apply id="S4.SS1.1.m1.1.1.cmml" xref="S4.SS1.1.m1.1.1"><minus id="S4.SS1.1.m1.1.1.1.cmml" xref="S4.SS1.1.m1.1.1.1"></minus><ci id="S4.SS1.1.m1.1.1.2.cmml" xref="S4.SS1.1.m1.1.1.2">𝑠</ci><ci id="S4.SS1.1.m1.1.1.3.cmml" xref="S4.SS1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.m1.1d">s-t</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.m1.1e">italic_s - italic_t</annotation></semantics></math> graphs</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">We begin with Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.Thmtheorem1" title="Theorem 1.1 (Informal version of Theorem 4.3 and Corollary 2) ‣ 1.1 Our results ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1.1</span></a> for the SUM cost. We begin with the following technical lemma. Essentially, it observes that the SUM cost is a sum of two factors. The first is the cost incurred by the interceptors, which is proportional to the maximum probability mass placed on an edge by the joint strategy of the agents. The second factor is simply the expected number of edges used by the agents.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma8"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 8</span></h6> <div class="ltx_para" id="Thmlemma8.p1"> <p class="ltx_p" id="Thmlemma8.p1.3"><math alttext="F_{n}(\sigma,\alpha)=k\alpha\beta(\sigma)+\operatorname*{\mathbb{E}}_{a\sim% \sigma}\left(\left|\bigcup a\right|\right)" class="ltx_Math" display="inline" id="Thmlemma8.p1.1.m1.5"><semantics id="Thmlemma8.p1.1.m1.5a"><mrow id="Thmlemma8.p1.1.m1.5.5" xref="Thmlemma8.p1.1.m1.5.5.cmml"><mrow id="Thmlemma8.p1.1.m1.5.5.4" xref="Thmlemma8.p1.1.m1.5.5.4.cmml"><msub id="Thmlemma8.p1.1.m1.5.5.4.2" xref="Thmlemma8.p1.1.m1.5.5.4.2.cmml"><mi id="Thmlemma8.p1.1.m1.5.5.4.2.2" xref="Thmlemma8.p1.1.m1.5.5.4.2.2.cmml">F</mi><mi id="Thmlemma8.p1.1.m1.5.5.4.2.3" xref="Thmlemma8.p1.1.m1.5.5.4.2.3.cmml">n</mi></msub><mo id="Thmlemma8.p1.1.m1.5.5.4.1" xref="Thmlemma8.p1.1.m1.5.5.4.1.cmml">⁢</mo><mrow id="Thmlemma8.p1.1.m1.5.5.4.3.2" xref="Thmlemma8.p1.1.m1.5.5.4.3.1.cmml"><mo id="Thmlemma8.p1.1.m1.5.5.4.3.2.1" stretchy="false" xref="Thmlemma8.p1.1.m1.5.5.4.3.1.cmml">(</mo><mi id="Thmlemma8.p1.1.m1.1.1" xref="Thmlemma8.p1.1.m1.1.1.cmml">σ</mi><mo id="Thmlemma8.p1.1.m1.5.5.4.3.2.2" xref="Thmlemma8.p1.1.m1.5.5.4.3.1.cmml">,</mo><mi id="Thmlemma8.p1.1.m1.2.2" xref="Thmlemma8.p1.1.m1.2.2.cmml">α</mi><mo id="Thmlemma8.p1.1.m1.5.5.4.3.2.3" stretchy="false" xref="Thmlemma8.p1.1.m1.5.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma8.p1.1.m1.5.5.3" xref="Thmlemma8.p1.1.m1.5.5.3.cmml">=</mo><mrow id="Thmlemma8.p1.1.m1.5.5.2" xref="Thmlemma8.p1.1.m1.5.5.2.cmml"><mrow id="Thmlemma8.p1.1.m1.5.5.2.4" xref="Thmlemma8.p1.1.m1.5.5.2.4.cmml"><mi id="Thmlemma8.p1.1.m1.5.5.2.4.2" xref="Thmlemma8.p1.1.m1.5.5.2.4.2.cmml">k</mi><mo id="Thmlemma8.p1.1.m1.5.5.2.4.1" xref="Thmlemma8.p1.1.m1.5.5.2.4.1.cmml">⁢</mo><mi id="Thmlemma8.p1.1.m1.5.5.2.4.3" xref="Thmlemma8.p1.1.m1.5.5.2.4.3.cmml">α</mi><mo id="Thmlemma8.p1.1.m1.5.5.2.4.1a" xref="Thmlemma8.p1.1.m1.5.5.2.4.1.cmml">⁢</mo><mi id="Thmlemma8.p1.1.m1.5.5.2.4.4" xref="Thmlemma8.p1.1.m1.5.5.2.4.4.cmml">β</mi><mo id="Thmlemma8.p1.1.m1.5.5.2.4.1b" xref="Thmlemma8.p1.1.m1.5.5.2.4.1.cmml">⁢</mo><mrow id="Thmlemma8.p1.1.m1.5.5.2.4.5.2" xref="Thmlemma8.p1.1.m1.5.5.2.4.cmml"><mo id="Thmlemma8.p1.1.m1.5.5.2.4.5.2.1" stretchy="false" xref="Thmlemma8.p1.1.m1.5.5.2.4.cmml">(</mo><mi id="Thmlemma8.p1.1.m1.3.3" xref="Thmlemma8.p1.1.m1.3.3.cmml">σ</mi><mo id="Thmlemma8.p1.1.m1.5.5.2.4.5.2.2" stretchy="false" xref="Thmlemma8.p1.1.m1.5.5.2.4.cmml">)</mo></mrow></mrow><mo id="Thmlemma8.p1.1.m1.5.5.2.3" xref="Thmlemma8.p1.1.m1.5.5.2.3.cmml">+</mo><mrow id="Thmlemma8.p1.1.m1.5.5.2.2.2" xref="Thmlemma8.p1.1.m1.5.5.2.2.3.cmml"><msub id="Thmlemma8.p1.1.m1.4.4.1.1.1.1" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.cmml"><mo id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.2" lspace="0em" rspace="0em" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.2.cmml">𝔼</mo><mrow id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.cmml"><mi id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.2" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.2.cmml">a</mi><mo id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.1" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.1.cmml">∼</mo><mi id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.3" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.3.cmml">σ</mi></mrow></msub><mrow id="Thmlemma8.p1.1.m1.5.5.2.2.2.2" xref="Thmlemma8.p1.1.m1.5.5.2.2.3.cmml"><mo id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.2" xref="Thmlemma8.p1.1.m1.5.5.2.2.3.cmml">(</mo><mrow id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.2.cmml"><mo id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.2" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.2.1.cmml">|</mo><mrow id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.cmml"><mo id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.1" lspace="0em" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.1.cmml">⋃</mo><mi id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.2" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.2.cmml">a</mi></mrow><mo id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.3" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.3" xref="Thmlemma8.p1.1.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma8.p1.1.m1.5b"><apply id="Thmlemma8.p1.1.m1.5.5.cmml" xref="Thmlemma8.p1.1.m1.5.5"><eq id="Thmlemma8.p1.1.m1.5.5.3.cmml" xref="Thmlemma8.p1.1.m1.5.5.3"></eq><apply id="Thmlemma8.p1.1.m1.5.5.4.cmml" xref="Thmlemma8.p1.1.m1.5.5.4"><times id="Thmlemma8.p1.1.m1.5.5.4.1.cmml" xref="Thmlemma8.p1.1.m1.5.5.4.1"></times><apply id="Thmlemma8.p1.1.m1.5.5.4.2.cmml" xref="Thmlemma8.p1.1.m1.5.5.4.2"><csymbol cd="ambiguous" id="Thmlemma8.p1.1.m1.5.5.4.2.1.cmml" xref="Thmlemma8.p1.1.m1.5.5.4.2">subscript</csymbol><ci id="Thmlemma8.p1.1.m1.5.5.4.2.2.cmml" xref="Thmlemma8.p1.1.m1.5.5.4.2.2">𝐹</ci><ci id="Thmlemma8.p1.1.m1.5.5.4.2.3.cmml" xref="Thmlemma8.p1.1.m1.5.5.4.2.3">𝑛</ci></apply><interval closure="open" id="Thmlemma8.p1.1.m1.5.5.4.3.1.cmml" xref="Thmlemma8.p1.1.m1.5.5.4.3.2"><ci id="Thmlemma8.p1.1.m1.1.1.cmml" xref="Thmlemma8.p1.1.m1.1.1">𝜎</ci><ci id="Thmlemma8.p1.1.m1.2.2.cmml" xref="Thmlemma8.p1.1.m1.2.2">𝛼</ci></interval></apply><apply id="Thmlemma8.p1.1.m1.5.5.2.cmml" xref="Thmlemma8.p1.1.m1.5.5.2"><plus id="Thmlemma8.p1.1.m1.5.5.2.3.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.3"></plus><apply id="Thmlemma8.p1.1.m1.5.5.2.4.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.4"><times id="Thmlemma8.p1.1.m1.5.5.2.4.1.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.4.1"></times><ci id="Thmlemma8.p1.1.m1.5.5.2.4.2.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.4.2">𝑘</ci><ci id="Thmlemma8.p1.1.m1.5.5.2.4.3.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.4.3">𝛼</ci><ci id="Thmlemma8.p1.1.m1.5.5.2.4.4.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.4.4">𝛽</ci><ci id="Thmlemma8.p1.1.m1.3.3.cmml" xref="Thmlemma8.p1.1.m1.3.3">𝜎</ci></apply><apply id="Thmlemma8.p1.1.m1.5.5.2.2.3.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.2.2"><apply id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.cmml" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.1.cmml" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1">subscript</csymbol><ci id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.2.cmml" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.2">𝔼</ci><apply id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.cmml" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3"><csymbol cd="latexml" id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.1.cmml" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.1">similar-to</csymbol><ci id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.2.cmml" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.2">𝑎</ci><ci id="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.3.cmml" xref="Thmlemma8.p1.1.m1.4.4.1.1.1.1.3.3">𝜎</ci></apply></apply><apply id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.2.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1"><abs id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.2.1.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.2"></abs><apply id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1"><union id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.1.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.1"></union><ci id="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.2.cmml" xref="Thmlemma8.p1.1.m1.5.5.2.2.2.2.1.1.1.2">𝑎</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma8.p1.1.m1.5c">F_{n}(\sigma,\alpha)=k\alpha\beta(\sigma)+\operatorname*{\mathbb{E}}_{a\sim% \sigma}\left(\left|\bigcup a\right|\right)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma8.p1.1.m1.5d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = italic_k italic_α italic_β ( italic_σ ) + blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( | ⋃ italic_a | )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Thmlemma8.p1.3.2">, and when <math alttext="P:=\bigcup_{j}\operatorname{supp}\sigma_{j}" class="ltx_Math" display="inline" id="Thmlemma8.p1.2.1.m1.1"><semantics id="Thmlemma8.p1.2.1.m1.1a"><mrow id="Thmlemma8.p1.2.1.m1.1.1" xref="Thmlemma8.p1.2.1.m1.1.1.cmml"><mi id="Thmlemma8.p1.2.1.m1.1.1.2" xref="Thmlemma8.p1.2.1.m1.1.1.2.cmml">P</mi><mo id="Thmlemma8.p1.2.1.m1.1.1.1" lspace="0.278em" rspace="0.111em" xref="Thmlemma8.p1.2.1.m1.1.1.1.cmml">:=</mo><mrow id="Thmlemma8.p1.2.1.m1.1.1.3" xref="Thmlemma8.p1.2.1.m1.1.1.3.cmml"><msub id="Thmlemma8.p1.2.1.m1.1.1.3.1" xref="Thmlemma8.p1.2.1.m1.1.1.3.1.cmml"><mo id="Thmlemma8.p1.2.1.m1.1.1.3.1.2" xref="Thmlemma8.p1.2.1.m1.1.1.3.1.2.cmml">⋃</mo><mi id="Thmlemma8.p1.2.1.m1.1.1.3.1.3" xref="Thmlemma8.p1.2.1.m1.1.1.3.1.3.cmml">j</mi></msub><mrow id="Thmlemma8.p1.2.1.m1.1.1.3.2" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.cmml"><mi id="Thmlemma8.p1.2.1.m1.1.1.3.2.1" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.1.cmml">supp</mi><mo id="Thmlemma8.p1.2.1.m1.1.1.3.2a" lspace="0.167em" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.cmml">⁡</mo><msub id="Thmlemma8.p1.2.1.m1.1.1.3.2.2" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.2.cmml"><mi id="Thmlemma8.p1.2.1.m1.1.1.3.2.2.2" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.2.2.cmml">σ</mi><mi id="Thmlemma8.p1.2.1.m1.1.1.3.2.2.3" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.2.3.cmml">j</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma8.p1.2.1.m1.1b"><apply id="Thmlemma8.p1.2.1.m1.1.1.cmml" xref="Thmlemma8.p1.2.1.m1.1.1"><csymbol cd="latexml" id="Thmlemma8.p1.2.1.m1.1.1.1.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.1">assign</csymbol><ci id="Thmlemma8.p1.2.1.m1.1.1.2.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.2">𝑃</ci><apply id="Thmlemma8.p1.2.1.m1.1.1.3.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3"><apply id="Thmlemma8.p1.2.1.m1.1.1.3.1.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.1"><csymbol cd="ambiguous" id="Thmlemma8.p1.2.1.m1.1.1.3.1.1.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.1">subscript</csymbol><union id="Thmlemma8.p1.2.1.m1.1.1.3.1.2.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.1.2"></union><ci id="Thmlemma8.p1.2.1.m1.1.1.3.1.3.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.1.3">𝑗</ci></apply><apply id="Thmlemma8.p1.2.1.m1.1.1.3.2.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.2"><ci id="Thmlemma8.p1.2.1.m1.1.1.3.2.1.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.1">supp</ci><apply id="Thmlemma8.p1.2.1.m1.1.1.3.2.2.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="Thmlemma8.p1.2.1.m1.1.1.3.2.2.1.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.2">subscript</csymbol><ci id="Thmlemma8.p1.2.1.m1.1.1.3.2.2.2.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.2.2">𝜎</ci><ci id="Thmlemma8.p1.2.1.m1.1.1.3.2.2.3.cmml" xref="Thmlemma8.p1.2.1.m1.1.1.3.2.2.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma8.p1.2.1.m1.1c">P:=\bigcup_{j}\operatorname{supp}\sigma_{j}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma8.p1.2.1.m1.1d">italic_P := ⋃ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_supp italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is a set of disjoint paths, <math alttext="F_{n}(\sigma,\alpha)=k\alpha\beta(\sigma)+\sum_{p\in P}\sigma(p)|p|" class="ltx_Math" display="inline" id="Thmlemma8.p1.3.2.m2.5"><semantics id="Thmlemma8.p1.3.2.m2.5a"><mrow id="Thmlemma8.p1.3.2.m2.5.6" xref="Thmlemma8.p1.3.2.m2.5.6.cmml"><mrow id="Thmlemma8.p1.3.2.m2.5.6.2" xref="Thmlemma8.p1.3.2.m2.5.6.2.cmml"><msub id="Thmlemma8.p1.3.2.m2.5.6.2.2" xref="Thmlemma8.p1.3.2.m2.5.6.2.2.cmml"><mi id="Thmlemma8.p1.3.2.m2.5.6.2.2.2" xref="Thmlemma8.p1.3.2.m2.5.6.2.2.2.cmml">F</mi><mi id="Thmlemma8.p1.3.2.m2.5.6.2.2.3" xref="Thmlemma8.p1.3.2.m2.5.6.2.2.3.cmml">n</mi></msub><mo id="Thmlemma8.p1.3.2.m2.5.6.2.1" xref="Thmlemma8.p1.3.2.m2.5.6.2.1.cmml">⁢</mo><mrow id="Thmlemma8.p1.3.2.m2.5.6.2.3.2" xref="Thmlemma8.p1.3.2.m2.5.6.2.3.1.cmml"><mo id="Thmlemma8.p1.3.2.m2.5.6.2.3.2.1" stretchy="false" 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xref="Thmlemma8.p1.3.2.m2.5.6.3.3.1.3.3">𝑃</ci></apply></apply><apply id="Thmlemma8.p1.3.2.m2.5.6.3.3.2.cmml" xref="Thmlemma8.p1.3.2.m2.5.6.3.3.2"><times id="Thmlemma8.p1.3.2.m2.5.6.3.3.2.1.cmml" xref="Thmlemma8.p1.3.2.m2.5.6.3.3.2.1"></times><ci id="Thmlemma8.p1.3.2.m2.5.6.3.3.2.2.cmml" xref="Thmlemma8.p1.3.2.m2.5.6.3.3.2.2">𝜎</ci><ci id="Thmlemma8.p1.3.2.m2.4.4.cmml" xref="Thmlemma8.p1.3.2.m2.4.4">𝑝</ci><apply id="Thmlemma8.p1.3.2.m2.5.6.3.3.2.4.1.cmml" xref="Thmlemma8.p1.3.2.m2.5.6.3.3.2.4.2"><abs id="Thmlemma8.p1.3.2.m2.5.6.3.3.2.4.1.1.cmml" xref="Thmlemma8.p1.3.2.m2.5.6.3.3.2.4.2.1"></abs><ci id="Thmlemma8.p1.3.2.m2.5.5.cmml" xref="Thmlemma8.p1.3.2.m2.5.5">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma8.p1.3.2.m2.5c">F_{n}(\sigma,\alpha)=k\alpha\beta(\sigma)+\sum_{p\in P}\sigma(p)|p|</annotation><annotation encoding="application/x-llamapun" id="Thmlemma8.p1.3.2.m2.5d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = italic_k italic_α italic_β ( italic_σ ) + ∑ start_POSTSUBSCRIPT italic_p ∈ italic_P end_POSTSUBSCRIPT italic_σ ( italic_p ) | italic_p |</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.1">We now state the following theorem about uniform strategies, which shows that they are approximately optimal agent strategies.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 4.1</span></h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.2.2">Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.1.m1.1"><semantics id="S4.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem1.p1.1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">ε</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">></mo><mn id="S4.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.1.m1.1b"><apply id="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1"><gt id="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1"></gt><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2">𝜀</ci><cn id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.1.m1.1d">italic_ε > 0</annotation></semantics></math>, and <math alttext="\alpha\geq\frac{m_{c}}{k\varepsilon\mathcal{E}_{m_{c},n}}\left(n\left|p_{max}% \right|-(1+\varepsilon)p_{min}\right)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.2.m2.3"><semantics id="S4.Thmtheorem1.p1.2.2.m2.3a"><mrow id="S4.Thmtheorem1.p1.2.2.m2.3.3" xref="S4.Thmtheorem1.p1.2.2.m2.3.3.cmml"><mi id="S4.Thmtheorem1.p1.2.2.m2.3.3.3" xref="S4.Thmtheorem1.p1.2.2.m2.3.3.3.cmml">α</mi><mo id="S4.Thmtheorem1.p1.2.2.m2.3.3.2" xref="S4.Thmtheorem1.p1.2.2.m2.3.3.2.cmml">≥</mo><mrow id="S4.Thmtheorem1.p1.2.2.m2.3.3.1" xref="S4.Thmtheorem1.p1.2.2.m2.3.3.1.cmml"><mfrac id="S4.Thmtheorem1.p1.2.2.m2.2.2" xref="S4.Thmtheorem1.p1.2.2.m2.2.2.cmml"><msub id="S4.Thmtheorem1.p1.2.2.m2.2.2.4" xref="S4.Thmtheorem1.p1.2.2.m2.2.2.4.cmml"><mi id="S4.Thmtheorem1.p1.2.2.m2.2.2.4.2" xref="S4.Thmtheorem1.p1.2.2.m2.2.2.4.2.cmml">m</mi><mi id="S4.Thmtheorem1.p1.2.2.m2.2.2.4.3" xref="S4.Thmtheorem1.p1.2.2.m2.2.2.4.3.cmml">c</mi></msub><mrow 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id="S4.Thmtheorem1.p1.2.2.m2.3c">\alpha\geq\frac{m_{c}}{k\varepsilon\mathcal{E}_{m_{c},n}}\left(n\left|p_{max}% \right|-(1+\varepsilon)p_{min}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.3d">italic_α ≥ divide start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG start_ARG italic_k italic_ε caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG ( italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - ( 1 + italic_ε ) italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>. Then</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx7"> <tbody id="S4.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min_{\sigma\in\mathcal{U}_{n}}F_{n}(\sigma,\alpha)\leq(1+% \varepsilon)\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="S4.Ex4.m1.6"><semantics id="S4.Ex4.m1.6a"><mrow id="S4.Ex4.m1.6.6" xref="S4.Ex4.m1.6.6.cmml"><mrow id="S4.Ex4.m1.6.6.3" xref="S4.Ex4.m1.6.6.3.cmml"><mrow id="S4.Ex4.m1.6.6.3.2" xref="S4.Ex4.m1.6.6.3.2.cmml"><munder id="S4.Ex4.m1.6.6.3.2.1" xref="S4.Ex4.m1.6.6.3.2.1.cmml"><mi id="S4.Ex4.m1.6.6.3.2.1.2" xref="S4.Ex4.m1.6.6.3.2.1.2.cmml">min</mi><mrow id="S4.Ex4.m1.6.6.3.2.1.3" xref="S4.Ex4.m1.6.6.3.2.1.3.cmml"><mi id="S4.Ex4.m1.6.6.3.2.1.3.2" xref="S4.Ex4.m1.6.6.3.2.1.3.2.cmml">σ</mi><mo id="S4.Ex4.m1.6.6.3.2.1.3.1" xref="S4.Ex4.m1.6.6.3.2.1.3.1.cmml">∈</mo><msub id="S4.Ex4.m1.6.6.3.2.1.3.3" xref="S4.Ex4.m1.6.6.3.2.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex4.m1.6.6.3.2.1.3.3.2" xref="S4.Ex4.m1.6.6.3.2.1.3.3.2.cmml">𝒰</mi><mi id="S4.Ex4.m1.6.6.3.2.1.3.3.3" xref="S4.Ex4.m1.6.6.3.2.1.3.3.3.cmml">n</mi></msub></mrow></munder><mo id="S4.Ex4.m1.6.6.3.2a" lspace="0.167em" xref="S4.Ex4.m1.6.6.3.2.cmml">⁡</mo><msub id="S4.Ex4.m1.6.6.3.2.2" xref="S4.Ex4.m1.6.6.3.2.2.cmml"><mi id="S4.Ex4.m1.6.6.3.2.2.2" xref="S4.Ex4.m1.6.6.3.2.2.2.cmml">F</mi><mi id="S4.Ex4.m1.6.6.3.2.2.3" xref="S4.Ex4.m1.6.6.3.2.2.3.cmml">n</mi></msub></mrow><mo id="S4.Ex4.m1.6.6.3.1" xref="S4.Ex4.m1.6.6.3.1.cmml">⁢</mo><mrow id="S4.Ex4.m1.6.6.3.3.2" xref="S4.Ex4.m1.6.6.3.3.1.cmml"><mo id="S4.Ex4.m1.6.6.3.3.2.1" stretchy="false" xref="S4.Ex4.m1.6.6.3.3.1.cmml">(</mo><mi id="S4.Ex4.m1.2.2" xref="S4.Ex4.m1.2.2.cmml">σ</mi><mo id="S4.Ex4.m1.6.6.3.3.2.2" xref="S4.Ex4.m1.6.6.3.3.1.cmml">,</mo><mi id="S4.Ex4.m1.3.3" 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xref="S4.Ex4.m1.5.5">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m1.6c">\displaystyle\min_{\sigma\in\mathcal{U}_{n}}F_{n}(\sigma,\alpha)\leq(1+% \varepsilon)\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m1.6d">roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≤ ( 1 + italic_ε ) roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.6">Intuitively, as <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mi id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">italic_α</annotation></semantics></math> grows large, the paths’ inherent lengths (namely, ignoring <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mi id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><ci id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">italic_α</annotation></semantics></math>) become smaller than <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">italic_α</annotation></semantics></math>. Therefore, the agents will get increasingly concerned with avoiding encounters with interceptors over the base cost of sending traffic over long paths. To address this concern, the agents will prefer strategies <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.1"><semantics id="S4.SS1.p3.4.m4.1a"><mi id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.1b"><ci id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">italic_σ</annotation></semantics></math> that have small <math alttext="\beta(\sigma)" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mrow id="S4.SS1.p3.5.m5.1.2" xref="S4.SS1.p3.5.m5.1.2.cmml"><mi id="S4.SS1.p3.5.m5.1.2.2" xref="S4.SS1.p3.5.m5.1.2.2.cmml">β</mi><mo id="S4.SS1.p3.5.m5.1.2.1" xref="S4.SS1.p3.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.5.m5.1.2.3.2" xref="S4.SS1.p3.5.m5.1.2.cmml"><mo id="S4.SS1.p3.5.m5.1.2.3.2.1" stretchy="false" xref="S4.SS1.p3.5.m5.1.2.cmml">(</mo><mi id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml">σ</mi><mo id="S4.SS1.p3.5.m5.1.2.3.2.2" stretchy="false" xref="S4.SS1.p3.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><apply id="S4.SS1.p3.5.m5.1.2.cmml" xref="S4.SS1.p3.5.m5.1.2"><times id="S4.SS1.p3.5.m5.1.2.1.cmml" xref="S4.SS1.p3.5.m5.1.2.1"></times><ci id="S4.SS1.p3.5.m5.1.2.2.cmml" xref="S4.SS1.p3.5.m5.1.2.2">𝛽</ci><ci id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">italic_β ( italic_σ )</annotation></semantics></math>. This is because the additional cost caused by the interceptors will be <math alttext="k\alpha\beta(\sigma)" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.1"><semantics id="S4.SS1.p3.6.m6.1a"><mrow id="S4.SS1.p3.6.m6.1.2" xref="S4.SS1.p3.6.m6.1.2.cmml"><mi id="S4.SS1.p3.6.m6.1.2.2" xref="S4.SS1.p3.6.m6.1.2.2.cmml">k</mi><mo id="S4.SS1.p3.6.m6.1.2.1" xref="S4.SS1.p3.6.m6.1.2.1.cmml">⁢</mo><mi id="S4.SS1.p3.6.m6.1.2.3" xref="S4.SS1.p3.6.m6.1.2.3.cmml">α</mi><mo id="S4.SS1.p3.6.m6.1.2.1a" xref="S4.SS1.p3.6.m6.1.2.1.cmml">⁢</mo><mi id="S4.SS1.p3.6.m6.1.2.4" xref="S4.SS1.p3.6.m6.1.2.4.cmml">β</mi><mo id="S4.SS1.p3.6.m6.1.2.1b" xref="S4.SS1.p3.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.6.m6.1.2.5.2" xref="S4.SS1.p3.6.m6.1.2.cmml"><mo id="S4.SS1.p3.6.m6.1.2.5.2.1" stretchy="false" xref="S4.SS1.p3.6.m6.1.2.cmml">(</mo><mi id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml">σ</mi><mo id="S4.SS1.p3.6.m6.1.2.5.2.2" stretchy="false" xref="S4.SS1.p3.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.1b"><apply id="S4.SS1.p3.6.m6.1.2.cmml" xref="S4.SS1.p3.6.m6.1.2"><times id="S4.SS1.p3.6.m6.1.2.1.cmml" xref="S4.SS1.p3.6.m6.1.2.1"></times><ci id="S4.SS1.p3.6.m6.1.2.2.cmml" xref="S4.SS1.p3.6.m6.1.2.2">𝑘</ci><ci id="S4.SS1.p3.6.m6.1.2.3.cmml" xref="S4.SS1.p3.6.m6.1.2.3">𝛼</ci><ci id="S4.SS1.p3.6.m6.1.2.4.cmml" xref="S4.SS1.p3.6.m6.1.2.4">𝛽</ci><ci id="S4.SS1.p3.6.m6.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.1c">k\alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.1d">italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math>, and the agents can only control the rightmost factor.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.1">The theorem above implies our two main results. The first is the approximation of the PoU with the uniform PoU.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 4.2</span></h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.2.2">Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem2.p1.1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">ε</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">></mo><mn id="S4.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.1.m1.1b"><apply id="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1"><gt id="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1"></gt><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.2">𝜀</ci><cn id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.1d">italic_ε > 0</annotation></semantics></math>, and <math alttext="\alpha\geq\frac{m_{c}}{k\varepsilon}\left(n\left|p_{max}\right|-(1+\varepsilon% )\left|p_{min}\right|\right)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem2.p1.2.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1.3" 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xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.2.1.1.1.3">𝜀</ci></apply><apply id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1"><abs id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.2.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.2"></abs><apply id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.2">𝑝</ci><apply id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3"><times id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.1"></times><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.2">𝑚</ci><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.3">𝑖</ci><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.4.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.3.2.1.1.3.4">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.1c">\alpha\geq\frac{m_{c}}{k\varepsilon}\left(n\left|p_{max}\right|-(1+\varepsilon% )\left|p_{min}\right|\right)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.1d">italic_α ≥ divide start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG start_ARG italic_k italic_ε end_ARG ( italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - ( 1 + italic_ε ) | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | )</annotation></semantics></math>. Then</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx8"> <tbody id="S4.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{r(G,n,k,\alpha)}{r_{U}(G,n,k,\alpha)}\in\left[\frac{1}{1+% \varepsilon},1+\varepsilon\right]" class="ltx_Math" display="inline" id="S4.Ex5.m1.10"><semantics id="S4.Ex5.m1.10a"><mrow id="S4.Ex5.m1.10.10" xref="S4.Ex5.m1.10.10.cmml"><mstyle displaystyle="true" id="S4.Ex5.m1.8.8" xref="S4.Ex5.m1.8.8.cmml"><mfrac id="S4.Ex5.m1.8.8a" xref="S4.Ex5.m1.8.8.cmml"><mrow id="S4.Ex5.m1.4.4.4" xref="S4.Ex5.m1.4.4.4.cmml"><mi id="S4.Ex5.m1.4.4.4.6" xref="S4.Ex5.m1.4.4.4.6.cmml">r</mi><mo id="S4.Ex5.m1.4.4.4.5" xref="S4.Ex5.m1.4.4.4.5.cmml">⁢</mo><mrow id="S4.Ex5.m1.4.4.4.7.2" xref="S4.Ex5.m1.4.4.4.7.1.cmml"><mo id="S4.Ex5.m1.4.4.4.7.2.1" stretchy="false" xref="S4.Ex5.m1.4.4.4.7.1.cmml">(</mo><mi id="S4.Ex5.m1.1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.cmml">G</mi><mo id="S4.Ex5.m1.4.4.4.7.2.2" xref="S4.Ex5.m1.4.4.4.7.1.cmml">,</mo><mi id="S4.Ex5.m1.2.2.2.2" xref="S4.Ex5.m1.2.2.2.2.cmml">n</mi><mo id="S4.Ex5.m1.4.4.4.7.2.3" xref="S4.Ex5.m1.4.4.4.7.1.cmml">,</mo><mi id="S4.Ex5.m1.3.3.3.3" xref="S4.Ex5.m1.3.3.3.3.cmml">k</mi><mo id="S4.Ex5.m1.4.4.4.7.2.4" xref="S4.Ex5.m1.4.4.4.7.1.cmml">,</mo><mi id="S4.Ex5.m1.4.4.4.4" xref="S4.Ex5.m1.4.4.4.4.cmml">α</mi><mo id="S4.Ex5.m1.4.4.4.7.2.5" stretchy="false" xref="S4.Ex5.m1.4.4.4.7.1.cmml">)</mo></mrow></mrow><mrow id="S4.Ex5.m1.8.8.8" xref="S4.Ex5.m1.8.8.8.cmml"><msub id="S4.Ex5.m1.8.8.8.6" xref="S4.Ex5.m1.8.8.8.6.cmml"><mi id="S4.Ex5.m1.8.8.8.6.2" xref="S4.Ex5.m1.8.8.8.6.2.cmml">r</mi><mi id="S4.Ex5.m1.8.8.8.6.3" xref="S4.Ex5.m1.8.8.8.6.3.cmml">U</mi></msub><mo id="S4.Ex5.m1.8.8.8.5" xref="S4.Ex5.m1.8.8.8.5.cmml">⁢</mo><mrow id="S4.Ex5.m1.8.8.8.7.2" xref="S4.Ex5.m1.8.8.8.7.1.cmml"><mo id="S4.Ex5.m1.8.8.8.7.2.1" stretchy="false" xref="S4.Ex5.m1.8.8.8.7.1.cmml">(</mo><mi id="S4.Ex5.m1.5.5.5.1" xref="S4.Ex5.m1.5.5.5.1.cmml">G</mi><mo id="S4.Ex5.m1.8.8.8.7.2.2" xref="S4.Ex5.m1.8.8.8.7.1.cmml">,</mo><mi id="S4.Ex5.m1.6.6.6.2" xref="S4.Ex5.m1.6.6.6.2.cmml">n</mi><mo id="S4.Ex5.m1.8.8.8.7.2.3" xref="S4.Ex5.m1.8.8.8.7.1.cmml">,</mo><mi id="S4.Ex5.m1.7.7.7.3" xref="S4.Ex5.m1.7.7.7.3.cmml">k</mi><mo id="S4.Ex5.m1.8.8.8.7.2.4" xref="S4.Ex5.m1.8.8.8.7.1.cmml">,</mo><mi id="S4.Ex5.m1.8.8.8.4" xref="S4.Ex5.m1.8.8.8.4.cmml">α</mi><mo id="S4.Ex5.m1.8.8.8.7.2.5" stretchy="false" xref="S4.Ex5.m1.8.8.8.7.1.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="S4.Ex5.m1.10.10.2" xref="S4.Ex5.m1.10.10.2.cmml">∈</mo><mrow id="S4.Ex5.m1.10.10.1.1" xref="S4.Ex5.m1.10.10.1.2.cmml"><mo id="S4.Ex5.m1.10.10.1.1.2" xref="S4.Ex5.m1.10.10.1.2.cmml">[</mo><mstyle displaystyle="true" id="S4.Ex5.m1.9.9" xref="S4.Ex5.m1.9.9.cmml"><mfrac id="S4.Ex5.m1.9.9a" xref="S4.Ex5.m1.9.9.cmml"><mn id="S4.Ex5.m1.9.9.2" xref="S4.Ex5.m1.9.9.2.cmml">1</mn><mrow id="S4.Ex5.m1.9.9.3" xref="S4.Ex5.m1.9.9.3.cmml"><mn id="S4.Ex5.m1.9.9.3.2" xref="S4.Ex5.m1.9.9.3.2.cmml">1</mn><mo id="S4.Ex5.m1.9.9.3.1" xref="S4.Ex5.m1.9.9.3.1.cmml">+</mo><mi id="S4.Ex5.m1.9.9.3.3" xref="S4.Ex5.m1.9.9.3.3.cmml">ε</mi></mrow></mfrac></mstyle><mo id="S4.Ex5.m1.10.10.1.1.3" xref="S4.Ex5.m1.10.10.1.2.cmml">,</mo><mrow id="S4.Ex5.m1.10.10.1.1.1" xref="S4.Ex5.m1.10.10.1.1.1.cmml"><mn id="S4.Ex5.m1.10.10.1.1.1.2" xref="S4.Ex5.m1.10.10.1.1.1.2.cmml">1</mn><mo id="S4.Ex5.m1.10.10.1.1.1.1" xref="S4.Ex5.m1.10.10.1.1.1.1.cmml">+</mo><mi id="S4.Ex5.m1.10.10.1.1.1.3" xref="S4.Ex5.m1.10.10.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.Ex5.m1.10.10.1.1.4" xref="S4.Ex5.m1.10.10.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex5.m1.10b"><apply id="S4.Ex5.m1.10.10.cmml" xref="S4.Ex5.m1.10.10"><in id="S4.Ex5.m1.10.10.2.cmml" xref="S4.Ex5.m1.10.10.2"></in><apply id="S4.Ex5.m1.8.8.cmml" xref="S4.Ex5.m1.8.8"><divide id="S4.Ex5.m1.8.8.9.cmml" xref="S4.Ex5.m1.8.8"></divide><apply id="S4.Ex5.m1.4.4.4.cmml" xref="S4.Ex5.m1.4.4.4"><times id="S4.Ex5.m1.4.4.4.5.cmml" xref="S4.Ex5.m1.4.4.4.5"></times><ci id="S4.Ex5.m1.4.4.4.6.cmml" xref="S4.Ex5.m1.4.4.4.6">𝑟</ci><vector id="S4.Ex5.m1.4.4.4.7.1.cmml" xref="S4.Ex5.m1.4.4.4.7.2"><ci id="S4.Ex5.m1.1.1.1.1.cmml" xref="S4.Ex5.m1.1.1.1.1">𝐺</ci><ci id="S4.Ex5.m1.2.2.2.2.cmml" xref="S4.Ex5.m1.2.2.2.2">𝑛</ci><ci id="S4.Ex5.m1.3.3.3.3.cmml" xref="S4.Ex5.m1.3.3.3.3">𝑘</ci><ci id="S4.Ex5.m1.4.4.4.4.cmml" xref="S4.Ex5.m1.4.4.4.4">𝛼</ci></vector></apply><apply id="S4.Ex5.m1.8.8.8.cmml" xref="S4.Ex5.m1.8.8.8"><times id="S4.Ex5.m1.8.8.8.5.cmml" xref="S4.Ex5.m1.8.8.8.5"></times><apply id="S4.Ex5.m1.8.8.8.6.cmml" xref="S4.Ex5.m1.8.8.8.6"><csymbol cd="ambiguous" id="S4.Ex5.m1.8.8.8.6.1.cmml" xref="S4.Ex5.m1.8.8.8.6">subscript</csymbol><ci id="S4.Ex5.m1.8.8.8.6.2.cmml" xref="S4.Ex5.m1.8.8.8.6.2">𝑟</ci><ci id="S4.Ex5.m1.8.8.8.6.3.cmml" xref="S4.Ex5.m1.8.8.8.6.3">𝑈</ci></apply><vector 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xref="S4.Ex5.m1.10.10.1.1.1.1"></plus><cn id="S4.Ex5.m1.10.10.1.1.1.2.cmml" type="integer" xref="S4.Ex5.m1.10.10.1.1.1.2">1</cn><ci id="S4.Ex5.m1.10.10.1.1.1.3.cmml" xref="S4.Ex5.m1.10.10.1.1.1.3">𝜀</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex5.m1.10c">\displaystyle\frac{r(G,n,k,\alpha)}{r_{U}(G,n,k,\alpha)}\in\left[\frac{1}{1+% \varepsilon},1+\varepsilon\right]</annotation><annotation encoding="application/x-llamapun" id="S4.Ex5.m1.10d">divide start_ARG italic_r ( italic_G , italic_n , italic_k , italic_α ) end_ARG start_ARG italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α ) end_ARG ∈ [ divide start_ARG 1 end_ARG start_ARG 1 + italic_ε end_ARG , 1 + italic_ε ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.1">The second main result of this section is our bounds on the PoU. We provide both asymptotic and non-asymptotic bounds.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 4.3</span></h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.2.2">Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">ε</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.cmml">></mo><mn id="S4.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.1b"><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1"><gt id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1"></gt><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2">𝜀</ci><cn id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.1d">italic_ε > 0</annotation></semantics></math>. We have the following bounds on <math alttext="r" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mi id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">italic_r</annotation></semantics></math>:</span></p> <ol class="ltx_enumerate" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.1.1">(Simple upper bound) </span><math alttext="r(G,n,k,\alpha)\leq\min(m_{c},n)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.7"><semantics id="S4.I1.i1.p1.1.m1.7a"><mrow id="S4.I1.i1.p1.1.m1.7.7" xref="S4.I1.i1.p1.1.m1.7.7.cmml"><mrow id="S4.I1.i1.p1.1.m1.7.7.3" xref="S4.I1.i1.p1.1.m1.7.7.3.cmml"><mi id="S4.I1.i1.p1.1.m1.7.7.3.2" xref="S4.I1.i1.p1.1.m1.7.7.3.2.cmml">r</mi><mo id="S4.I1.i1.p1.1.m1.7.7.3.1" xref="S4.I1.i1.p1.1.m1.7.7.3.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.1.m1.7.7.3.3.2" xref="S4.I1.i1.p1.1.m1.7.7.3.3.1.cmml"><mo id="S4.I1.i1.p1.1.m1.7.7.3.3.2.1" stretchy="false" xref="S4.I1.i1.p1.1.m1.7.7.3.3.1.cmml">(</mo><mi id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml">G</mi><mo id="S4.I1.i1.p1.1.m1.7.7.3.3.2.2" xref="S4.I1.i1.p1.1.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.I1.i1.p1.1.m1.2.2" xref="S4.I1.i1.p1.1.m1.2.2.cmml">n</mi><mo id="S4.I1.i1.p1.1.m1.7.7.3.3.2.3" xref="S4.I1.i1.p1.1.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.I1.i1.p1.1.m1.3.3" xref="S4.I1.i1.p1.1.m1.3.3.cmml">k</mi><mo id="S4.I1.i1.p1.1.m1.7.7.3.3.2.4" xref="S4.I1.i1.p1.1.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.I1.i1.p1.1.m1.4.4" xref="S4.I1.i1.p1.1.m1.4.4.cmml">α</mi><mo id="S4.I1.i1.p1.1.m1.7.7.3.3.2.5" stretchy="false" xref="S4.I1.i1.p1.1.m1.7.7.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i1.p1.1.m1.7.7.2" xref="S4.I1.i1.p1.1.m1.7.7.2.cmml">≤</mo><mrow id="S4.I1.i1.p1.1.m1.7.7.1.1" xref="S4.I1.i1.p1.1.m1.7.7.1.2.cmml"><mi id="S4.I1.i1.p1.1.m1.5.5" xref="S4.I1.i1.p1.1.m1.5.5.cmml">min</mi><mo id="S4.I1.i1.p1.1.m1.7.7.1.1a" xref="S4.I1.i1.p1.1.m1.7.7.1.2.cmml">⁡</mo><mrow id="S4.I1.i1.p1.1.m1.7.7.1.1.1" xref="S4.I1.i1.p1.1.m1.7.7.1.2.cmml"><mo id="S4.I1.i1.p1.1.m1.7.7.1.1.1.2" stretchy="false" xref="S4.I1.i1.p1.1.m1.7.7.1.2.cmml">(</mo><msub id="S4.I1.i1.p1.1.m1.7.7.1.1.1.1" xref="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.cmml"><mi id="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.2" xref="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.2.cmml">m</mi><mi id="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.3" xref="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.3.cmml">c</mi></msub><mo id="S4.I1.i1.p1.1.m1.7.7.1.1.1.3" xref="S4.I1.i1.p1.1.m1.7.7.1.2.cmml">,</mo><mi id="S4.I1.i1.p1.1.m1.6.6" xref="S4.I1.i1.p1.1.m1.6.6.cmml">n</mi><mo id="S4.I1.i1.p1.1.m1.7.7.1.1.1.4" stretchy="false" xref="S4.I1.i1.p1.1.m1.7.7.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.7b"><apply id="S4.I1.i1.p1.1.m1.7.7.cmml" xref="S4.I1.i1.p1.1.m1.7.7"><leq id="S4.I1.i1.p1.1.m1.7.7.2.cmml" xref="S4.I1.i1.p1.1.m1.7.7.2"></leq><apply id="S4.I1.i1.p1.1.m1.7.7.3.cmml" xref="S4.I1.i1.p1.1.m1.7.7.3"><times id="S4.I1.i1.p1.1.m1.7.7.3.1.cmml" xref="S4.I1.i1.p1.1.m1.7.7.3.1"></times><ci id="S4.I1.i1.p1.1.m1.7.7.3.2.cmml" xref="S4.I1.i1.p1.1.m1.7.7.3.2">𝑟</ci><vector id="S4.I1.i1.p1.1.m1.7.7.3.3.1.cmml" xref="S4.I1.i1.p1.1.m1.7.7.3.3.2"><ci id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1">𝐺</ci><ci id="S4.I1.i1.p1.1.m1.2.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2">𝑛</ci><ci id="S4.I1.i1.p1.1.m1.3.3.cmml" xref="S4.I1.i1.p1.1.m1.3.3">𝑘</ci><ci id="S4.I1.i1.p1.1.m1.4.4.cmml" xref="S4.I1.i1.p1.1.m1.4.4">𝛼</ci></vector></apply><apply id="S4.I1.i1.p1.1.m1.7.7.1.2.cmml" xref="S4.I1.i1.p1.1.m1.7.7.1.1"><min id="S4.I1.i1.p1.1.m1.5.5.cmml" xref="S4.I1.i1.p1.1.m1.5.5"></min><apply id="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.7.7.1.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.7.7.1.1.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.2.cmml" xref="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.2">𝑚</ci><ci id="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.3.cmml" xref="S4.I1.i1.p1.1.m1.7.7.1.1.1.1.3">𝑐</ci></apply><ci id="S4.I1.i1.p1.1.m1.6.6.cmml" xref="S4.I1.i1.p1.1.m1.6.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.7c">r(G,n,k,\alpha)\leq\min(m_{c},n)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.7d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≤ roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.1.2">.</span></p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.1"><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.1.1">(Upper bound) Let </span><math alttext="\alpha\geq\frac{m_{c}}{k\mathcal{E}_{m_{c},n}\varepsilon}\left(n\left|p_{max}% \right|-(1+\varepsilon)\mathcal{E}_{m_{c},n}\left|p_{min}\right|\right)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.5"><semantics id="S4.I1.i2.p1.1.m1.5a"><mrow id="S4.I1.i2.p1.1.m1.5.5" xref="S4.I1.i2.p1.1.m1.5.5.cmml"><mi id="S4.I1.i2.p1.1.m1.5.5.3" xref="S4.I1.i2.p1.1.m1.5.5.3.cmml">α</mi><mo id="S4.I1.i2.p1.1.m1.5.5.2" xref="S4.I1.i2.p1.1.m1.5.5.2.cmml">≥</mo><mrow id="S4.I1.i2.p1.1.m1.5.5.1" xref="S4.I1.i2.p1.1.m1.5.5.1.cmml"><mfrac 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xref="S4.I1.i2.p1.1.m1.2.2.2.2.2.2.1.3.cmml">c</mi></msub><mo id="S4.I1.i2.p1.1.m1.2.2.2.2.2.2.2" xref="S4.I1.i2.p1.1.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.I1.i2.p1.1.m1.1.1.1.1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml">n</mi></mrow></msub><mo id="S4.I1.i2.p1.1.m1.2.2.2.3a" xref="S4.I1.i2.p1.1.m1.2.2.2.3.cmml">⁢</mo><mi id="S4.I1.i2.p1.1.m1.2.2.2.6" xref="S4.I1.i2.p1.1.m1.2.2.2.6.cmml">ε</mi></mrow></mfrac><mo id="S4.I1.i2.p1.1.m1.5.5.1.2" xref="S4.I1.i2.p1.1.m1.5.5.1.2.cmml">⁢</mo><mrow id="S4.I1.i2.p1.1.m1.5.5.1.1.1" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.cmml"><mo id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.cmml"><mrow id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.1" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.1.cmml"><mi id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.1.3" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.1.3.cmml">n</mi><mo id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.1.2" 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cd="ambiguous" id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1">subscript</csymbol><ci id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.2">𝑝</ci><apply id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3"><times id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.1.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.1"></times><ci id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.2">𝑚</ci><ci id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.3.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.3">𝑖</ci><ci id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.4.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.3.2.1.1.3.4">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.5c">\alpha\geq\frac{m_{c}}{k\mathcal{E}_{m_{c},n}\varepsilon}\left(n\left|p_{max}% \right|-(1+\varepsilon)\mathcal{E}_{m_{c},n}\left|p_{min}\right|\right)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.5d">italic_α ≥ divide start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG start_ARG italic_k caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT italic_ε end_ARG ( italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - ( 1 + italic_ε ) caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.1.2">. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="r(G,n,k,\alpha)\leq(1+\varepsilon)\mathcal{E}_{m_{c},n}" class="ltx_Math" display="block" id="S4.Ex6.m1.7"><semantics id="S4.Ex6.m1.7a"><mrow id="S4.Ex6.m1.7.7" xref="S4.Ex6.m1.7.7.cmml"><mrow id="S4.Ex6.m1.7.7.3" xref="S4.Ex6.m1.7.7.3.cmml"><mi id="S4.Ex6.m1.7.7.3.2" xref="S4.Ex6.m1.7.7.3.2.cmml">r</mi><mo id="S4.Ex6.m1.7.7.3.1" xref="S4.Ex6.m1.7.7.3.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.7.7.3.3.2" xref="S4.Ex6.m1.7.7.3.3.1.cmml"><mo id="S4.Ex6.m1.7.7.3.3.2.1" stretchy="false" xref="S4.Ex6.m1.7.7.3.3.1.cmml">(</mo><mi id="S4.Ex6.m1.3.3" xref="S4.Ex6.m1.3.3.cmml">G</mi><mo id="S4.Ex6.m1.7.7.3.3.2.2" xref="S4.Ex6.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.Ex6.m1.4.4" xref="S4.Ex6.m1.4.4.cmml">n</mi><mo id="S4.Ex6.m1.7.7.3.3.2.3" xref="S4.Ex6.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.Ex6.m1.5.5" xref="S4.Ex6.m1.5.5.cmml">k</mi><mo id="S4.Ex6.m1.7.7.3.3.2.4" xref="S4.Ex6.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.Ex6.m1.6.6" xref="S4.Ex6.m1.6.6.cmml">α</mi><mo id="S4.Ex6.m1.7.7.3.3.2.5" stretchy="false" xref="S4.Ex6.m1.7.7.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex6.m1.7.7.2" xref="S4.Ex6.m1.7.7.2.cmml">≤</mo><mrow id="S4.Ex6.m1.7.7.1" xref="S4.Ex6.m1.7.7.1.cmml"><mrow id="S4.Ex6.m1.7.7.1.1.1" xref="S4.Ex6.m1.7.7.1.1.1.1.cmml"><mo id="S4.Ex6.m1.7.7.1.1.1.2" stretchy="false" xref="S4.Ex6.m1.7.7.1.1.1.1.cmml">(</mo><mrow id="S4.Ex6.m1.7.7.1.1.1.1" xref="S4.Ex6.m1.7.7.1.1.1.1.cmml"><mn id="S4.Ex6.m1.7.7.1.1.1.1.2" xref="S4.Ex6.m1.7.7.1.1.1.1.2.cmml">1</mn><mo id="S4.Ex6.m1.7.7.1.1.1.1.1" xref="S4.Ex6.m1.7.7.1.1.1.1.1.cmml">+</mo><mi id="S4.Ex6.m1.7.7.1.1.1.1.3" xref="S4.Ex6.m1.7.7.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.Ex6.m1.7.7.1.1.1.3" stretchy="false" xref="S4.Ex6.m1.7.7.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex6.m1.7.7.1.2" xref="S4.Ex6.m1.7.7.1.2.cmml">⁢</mo><msub id="S4.Ex6.m1.7.7.1.3" xref="S4.Ex6.m1.7.7.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex6.m1.7.7.1.3.2" xref="S4.Ex6.m1.7.7.1.3.2.cmml">ℰ</mi><mrow id="S4.Ex6.m1.2.2.2.2" xref="S4.Ex6.m1.2.2.2.3.cmml"><msub id="S4.Ex6.m1.2.2.2.2.1" xref="S4.Ex6.m1.2.2.2.2.1.cmml"><mi id="S4.Ex6.m1.2.2.2.2.1.2" xref="S4.Ex6.m1.2.2.2.2.1.2.cmml">m</mi><mi id="S4.Ex6.m1.2.2.2.2.1.3" xref="S4.Ex6.m1.2.2.2.2.1.3.cmml">c</mi></msub><mo id="S4.Ex6.m1.2.2.2.2.2" xref="S4.Ex6.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex6.m1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.cmml">n</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex6.m1.7b"><apply id="S4.Ex6.m1.7.7.cmml" xref="S4.Ex6.m1.7.7"><leq id="S4.Ex6.m1.7.7.2.cmml" xref="S4.Ex6.m1.7.7.2"></leq><apply id="S4.Ex6.m1.7.7.3.cmml" xref="S4.Ex6.m1.7.7.3"><times id="S4.Ex6.m1.7.7.3.1.cmml" xref="S4.Ex6.m1.7.7.3.1"></times><ci id="S4.Ex6.m1.7.7.3.2.cmml" xref="S4.Ex6.m1.7.7.3.2">𝑟</ci><vector id="S4.Ex6.m1.7.7.3.3.1.cmml" xref="S4.Ex6.m1.7.7.3.3.2"><ci id="S4.Ex6.m1.3.3.cmml" xref="S4.Ex6.m1.3.3">𝐺</ci><ci id="S4.Ex6.m1.4.4.cmml" xref="S4.Ex6.m1.4.4">𝑛</ci><ci id="S4.Ex6.m1.5.5.cmml" xref="S4.Ex6.m1.5.5">𝑘</ci><ci id="S4.Ex6.m1.6.6.cmml" xref="S4.Ex6.m1.6.6">𝛼</ci></vector></apply><apply id="S4.Ex6.m1.7.7.1.cmml" xref="S4.Ex6.m1.7.7.1"><times id="S4.Ex6.m1.7.7.1.2.cmml" xref="S4.Ex6.m1.7.7.1.2"></times><apply id="S4.Ex6.m1.7.7.1.1.1.1.cmml" xref="S4.Ex6.m1.7.7.1.1.1"><plus id="S4.Ex6.m1.7.7.1.1.1.1.1.cmml" xref="S4.Ex6.m1.7.7.1.1.1.1.1"></plus><cn id="S4.Ex6.m1.7.7.1.1.1.1.2.cmml" type="integer" xref="S4.Ex6.m1.7.7.1.1.1.1.2">1</cn><ci id="S4.Ex6.m1.7.7.1.1.1.1.3.cmml" xref="S4.Ex6.m1.7.7.1.1.1.1.3">𝜀</ci></apply><apply id="S4.Ex6.m1.7.7.1.3.cmml" xref="S4.Ex6.m1.7.7.1.3"><csymbol cd="ambiguous" id="S4.Ex6.m1.7.7.1.3.1.cmml" xref="S4.Ex6.m1.7.7.1.3">subscript</csymbol><ci id="S4.Ex6.m1.7.7.1.3.2.cmml" xref="S4.Ex6.m1.7.7.1.3.2">ℰ</ci><list id="S4.Ex6.m1.2.2.2.3.cmml" xref="S4.Ex6.m1.2.2.2.2"><apply id="S4.Ex6.m1.2.2.2.2.1.cmml" xref="S4.Ex6.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.Ex6.m1.2.2.2.2.1.1.cmml" xref="S4.Ex6.m1.2.2.2.2.1">subscript</csymbol><ci id="S4.Ex6.m1.2.2.2.2.1.2.cmml" xref="S4.Ex6.m1.2.2.2.2.1.2">𝑚</ci><ci id="S4.Ex6.m1.2.2.2.2.1.3.cmml" xref="S4.Ex6.m1.2.2.2.2.1.3">𝑐</ci></apply><ci id="S4.Ex6.m1.1.1.1.1.cmml" xref="S4.Ex6.m1.1.1.1.1">𝑛</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex6.m1.7c">r(G,n,k,\alpha)\leq(1+\varepsilon)\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex6.m1.7d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≤ ( 1 + italic_ε ) caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S4.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S4.I1.i3.p1"> <p class="ltx_p" id="S4.I1.i3.p1.1"><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.1.1">(Lower bound) Let </span><math alttext="\alpha\geq\frac{m_{c}}{k\mathcal{E}_{m_{c},n}\varepsilon}\left((1-\varepsilon)% \mathcal{E}_{m_{c},n}n\left|p_{max}\right|-\left|p_{min}\right|\right)" class="ltx_Math" display="inline" id="S4.I1.i3.p1.1.m1.5"><semantics id="S4.I1.i3.p1.1.m1.5a"><mrow id="S4.I1.i3.p1.1.m1.5.5" xref="S4.I1.i3.p1.1.m1.5.5.cmml"><mi id="S4.I1.i3.p1.1.m1.5.5.3" xref="S4.I1.i3.p1.1.m1.5.5.3.cmml">α</mi><mo id="S4.I1.i3.p1.1.m1.5.5.2" xref="S4.I1.i3.p1.1.m1.5.5.2.cmml">≥</mo><mrow id="S4.I1.i3.p1.1.m1.5.5.1" xref="S4.I1.i3.p1.1.m1.5.5.1.cmml"><mfrac id="S4.I1.i3.p1.1.m1.2.2" xref="S4.I1.i3.p1.1.m1.2.2.cmml"><msub id="S4.I1.i3.p1.1.m1.2.2.4" xref="S4.I1.i3.p1.1.m1.2.2.4.cmml"><mi 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xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.5">𝑛</ci><apply id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.2.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1"><abs id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.2.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.2"></abs><apply id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1">subscript</csymbol><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.2.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.2">𝑝</ci><apply id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3"><times id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.1"></times><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.2.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.2">𝑚</ci><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.3.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.3">𝑎</ci><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.4.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.2.2.1.1.3.4">𝑥</ci></apply></apply></apply></apply><apply id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.2.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1"><abs id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.2.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.2"></abs><apply id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1"><csymbol cd="ambiguous" id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1">subscript</csymbol><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.2.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.2">𝑝</ci><apply id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3"><times id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.1.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.1"></times><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.2.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.2">𝑚</ci><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.3.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.3">𝑖</ci><ci id="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.4.cmml" xref="S4.I1.i3.p1.1.m1.5.5.1.1.1.1.3.1.1.3.4">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.1.m1.5c">\alpha\geq\frac{m_{c}}{k\mathcal{E}_{m_{c},n}\varepsilon}\left((1-\varepsilon)% \mathcal{E}_{m_{c},n}n\left|p_{max}\right|-\left|p_{min}\right|\right)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.1.m1.5d">italic_α ≥ divide start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG start_ARG italic_k caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT italic_ε end_ARG ( ( 1 - italic_ε ) caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.1.2">. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="r(G,n,k,\alpha)\geq(1-\varepsilon)\mathcal{E}_{m_{c},n}" class="ltx_Math" display="block" id="S4.Ex7.m1.7"><semantics id="S4.Ex7.m1.7a"><mrow id="S4.Ex7.m1.7.7" xref="S4.Ex7.m1.7.7.cmml"><mrow id="S4.Ex7.m1.7.7.3" xref="S4.Ex7.m1.7.7.3.cmml"><mi id="S4.Ex7.m1.7.7.3.2" xref="S4.Ex7.m1.7.7.3.2.cmml">r</mi><mo id="S4.Ex7.m1.7.7.3.1" xref="S4.Ex7.m1.7.7.3.1.cmml">⁢</mo><mrow id="S4.Ex7.m1.7.7.3.3.2" xref="S4.Ex7.m1.7.7.3.3.1.cmml"><mo id="S4.Ex7.m1.7.7.3.3.2.1" stretchy="false" xref="S4.Ex7.m1.7.7.3.3.1.cmml">(</mo><mi id="S4.Ex7.m1.3.3" xref="S4.Ex7.m1.3.3.cmml">G</mi><mo id="S4.Ex7.m1.7.7.3.3.2.2" xref="S4.Ex7.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.Ex7.m1.4.4" xref="S4.Ex7.m1.4.4.cmml">n</mi><mo id="S4.Ex7.m1.7.7.3.3.2.3" xref="S4.Ex7.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.Ex7.m1.5.5" xref="S4.Ex7.m1.5.5.cmml">k</mi><mo id="S4.Ex7.m1.7.7.3.3.2.4" xref="S4.Ex7.m1.7.7.3.3.1.cmml">,</mo><mi id="S4.Ex7.m1.6.6" xref="S4.Ex7.m1.6.6.cmml">α</mi><mo id="S4.Ex7.m1.7.7.3.3.2.5" stretchy="false" xref="S4.Ex7.m1.7.7.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.7.7.2" xref="S4.Ex7.m1.7.7.2.cmml">≥</mo><mrow id="S4.Ex7.m1.7.7.1" xref="S4.Ex7.m1.7.7.1.cmml"><mrow id="S4.Ex7.m1.7.7.1.1.1" xref="S4.Ex7.m1.7.7.1.1.1.1.cmml"><mo id="S4.Ex7.m1.7.7.1.1.1.2" stretchy="false" xref="S4.Ex7.m1.7.7.1.1.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.7.7.1.1.1.1" xref="S4.Ex7.m1.7.7.1.1.1.1.cmml"><mn id="S4.Ex7.m1.7.7.1.1.1.1.2" xref="S4.Ex7.m1.7.7.1.1.1.1.2.cmml">1</mn><mo id="S4.Ex7.m1.7.7.1.1.1.1.1" xref="S4.Ex7.m1.7.7.1.1.1.1.1.cmml">−</mo><mi id="S4.Ex7.m1.7.7.1.1.1.1.3" xref="S4.Ex7.m1.7.7.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.Ex7.m1.7.7.1.1.1.3" stretchy="false" xref="S4.Ex7.m1.7.7.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex7.m1.7.7.1.2" xref="S4.Ex7.m1.7.7.1.2.cmml">⁢</mo><msub id="S4.Ex7.m1.7.7.1.3" xref="S4.Ex7.m1.7.7.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex7.m1.7.7.1.3.2" xref="S4.Ex7.m1.7.7.1.3.2.cmml">ℰ</mi><mrow id="S4.Ex7.m1.2.2.2.2" xref="S4.Ex7.m1.2.2.2.3.cmml"><msub id="S4.Ex7.m1.2.2.2.2.1" xref="S4.Ex7.m1.2.2.2.2.1.cmml"><mi id="S4.Ex7.m1.2.2.2.2.1.2" xref="S4.Ex7.m1.2.2.2.2.1.2.cmml">m</mi><mi id="S4.Ex7.m1.2.2.2.2.1.3" xref="S4.Ex7.m1.2.2.2.2.1.3.cmml">c</mi></msub><mo id="S4.Ex7.m1.2.2.2.2.2" xref="S4.Ex7.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex7.m1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.cmml">n</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex7.m1.7b"><apply id="S4.Ex7.m1.7.7.cmml" xref="S4.Ex7.m1.7.7"><geq id="S4.Ex7.m1.7.7.2.cmml" xref="S4.Ex7.m1.7.7.2"></geq><apply id="S4.Ex7.m1.7.7.3.cmml" xref="S4.Ex7.m1.7.7.3"><times id="S4.Ex7.m1.7.7.3.1.cmml" xref="S4.Ex7.m1.7.7.3.1"></times><ci id="S4.Ex7.m1.7.7.3.2.cmml" xref="S4.Ex7.m1.7.7.3.2">𝑟</ci><vector id="S4.Ex7.m1.7.7.3.3.1.cmml" xref="S4.Ex7.m1.7.7.3.3.2"><ci id="S4.Ex7.m1.3.3.cmml" xref="S4.Ex7.m1.3.3">𝐺</ci><ci id="S4.Ex7.m1.4.4.cmml" xref="S4.Ex7.m1.4.4">𝑛</ci><ci id="S4.Ex7.m1.5.5.cmml" xref="S4.Ex7.m1.5.5">𝑘</ci><ci id="S4.Ex7.m1.6.6.cmml" xref="S4.Ex7.m1.6.6">𝛼</ci></vector></apply><apply id="S4.Ex7.m1.7.7.1.cmml" xref="S4.Ex7.m1.7.7.1"><times id="S4.Ex7.m1.7.7.1.2.cmml" xref="S4.Ex7.m1.7.7.1.2"></times><apply id="S4.Ex7.m1.7.7.1.1.1.1.cmml" xref="S4.Ex7.m1.7.7.1.1.1"><minus id="S4.Ex7.m1.7.7.1.1.1.1.1.cmml" xref="S4.Ex7.m1.7.7.1.1.1.1.1"></minus><cn id="S4.Ex7.m1.7.7.1.1.1.1.2.cmml" type="integer" xref="S4.Ex7.m1.7.7.1.1.1.1.2">1</cn><ci id="S4.Ex7.m1.7.7.1.1.1.1.3.cmml" xref="S4.Ex7.m1.7.7.1.1.1.1.3">𝜀</ci></apply><apply id="S4.Ex7.m1.7.7.1.3.cmml" xref="S4.Ex7.m1.7.7.1.3"><csymbol cd="ambiguous" id="S4.Ex7.m1.7.7.1.3.1.cmml" xref="S4.Ex7.m1.7.7.1.3">subscript</csymbol><ci id="S4.Ex7.m1.7.7.1.3.2.cmml" xref="S4.Ex7.m1.7.7.1.3.2">ℰ</ci><list id="S4.Ex7.m1.2.2.2.3.cmml" xref="S4.Ex7.m1.2.2.2.2"><apply id="S4.Ex7.m1.2.2.2.2.1.cmml" xref="S4.Ex7.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.Ex7.m1.2.2.2.2.1.1.cmml" xref="S4.Ex7.m1.2.2.2.2.1">subscript</csymbol><ci id="S4.Ex7.m1.2.2.2.2.1.2.cmml" xref="S4.Ex7.m1.2.2.2.2.1.2">𝑚</ci><ci id="S4.Ex7.m1.2.2.2.2.1.3.cmml" xref="S4.Ex7.m1.2.2.2.2.1.3">𝑐</ci></apply><ci id="S4.Ex7.m1.1.1.1.1.cmml" xref="S4.Ex7.m1.1.1.1.1">𝑛</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.7c">r(G,n,k,\alpha)\geq(1-\varepsilon)\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.7d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≥ ( 1 - italic_ε ) caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S4.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S4.I1.i4.p1"> <p class="ltx_p" id="S4.I1.i4.p1.1"><span class="ltx_text ltx_font_italic" id="S4.I1.i4.p1.1.1">(Limit) </span><math alttext="\lim_{\alpha\to\infty}r(G,n,k,\alpha)=\mathcal{E}_{m_{c},n}" class="ltx_Math" display="inline" id="S4.I1.i4.p1.1.m1.6"><semantics id="S4.I1.i4.p1.1.m1.6a"><mrow id="S4.I1.i4.p1.1.m1.6.7" xref="S4.I1.i4.p1.1.m1.6.7.cmml"><mrow id="S4.I1.i4.p1.1.m1.6.7.2" xref="S4.I1.i4.p1.1.m1.6.7.2.cmml"><msub id="S4.I1.i4.p1.1.m1.6.7.2.1" xref="S4.I1.i4.p1.1.m1.6.7.2.1.cmml"><mo id="S4.I1.i4.p1.1.m1.6.7.2.1.2" xref="S4.I1.i4.p1.1.m1.6.7.2.1.2.cmml">lim</mo><mrow id="S4.I1.i4.p1.1.m1.6.7.2.1.3" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3.cmml"><mi id="S4.I1.i4.p1.1.m1.6.7.2.1.3.2" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3.2.cmml">α</mi><mo id="S4.I1.i4.p1.1.m1.6.7.2.1.3.1" stretchy="false" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3.1.cmml">→</mo><mi id="S4.I1.i4.p1.1.m1.6.7.2.1.3.3" mathvariant="normal" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="S4.I1.i4.p1.1.m1.6.7.2.2" xref="S4.I1.i4.p1.1.m1.6.7.2.2.cmml"><mi id="S4.I1.i4.p1.1.m1.6.7.2.2.2" xref="S4.I1.i4.p1.1.m1.6.7.2.2.2.cmml">r</mi><mo id="S4.I1.i4.p1.1.m1.6.7.2.2.1" xref="S4.I1.i4.p1.1.m1.6.7.2.2.1.cmml">⁢</mo><mrow id="S4.I1.i4.p1.1.m1.6.7.2.2.3.2" xref="S4.I1.i4.p1.1.m1.6.7.2.2.3.1.cmml"><mo id="S4.I1.i4.p1.1.m1.6.7.2.2.3.2.1" stretchy="false" xref="S4.I1.i4.p1.1.m1.6.7.2.2.3.1.cmml">(</mo><mi id="S4.I1.i4.p1.1.m1.3.3" xref="S4.I1.i4.p1.1.m1.3.3.cmml">G</mi><mo id="S4.I1.i4.p1.1.m1.6.7.2.2.3.2.2" xref="S4.I1.i4.p1.1.m1.6.7.2.2.3.1.cmml">,</mo><mi id="S4.I1.i4.p1.1.m1.4.4" xref="S4.I1.i4.p1.1.m1.4.4.cmml">n</mi><mo id="S4.I1.i4.p1.1.m1.6.7.2.2.3.2.3" xref="S4.I1.i4.p1.1.m1.6.7.2.2.3.1.cmml">,</mo><mi id="S4.I1.i4.p1.1.m1.5.5" xref="S4.I1.i4.p1.1.m1.5.5.cmml">k</mi><mo 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xref="S4.I1.i4.p1.1.m1.1.1.1.1.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i4.p1.1.m1.6b"><apply id="S4.I1.i4.p1.1.m1.6.7.cmml" xref="S4.I1.i4.p1.1.m1.6.7"><eq id="S4.I1.i4.p1.1.m1.6.7.1.cmml" xref="S4.I1.i4.p1.1.m1.6.7.1"></eq><apply id="S4.I1.i4.p1.1.m1.6.7.2.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2"><apply id="S4.I1.i4.p1.1.m1.6.7.2.1.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2.1"><csymbol cd="ambiguous" id="S4.I1.i4.p1.1.m1.6.7.2.1.1.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2.1">subscript</csymbol><limit id="S4.I1.i4.p1.1.m1.6.7.2.1.2.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2.1.2"></limit><apply id="S4.I1.i4.p1.1.m1.6.7.2.1.3.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3"><ci id="S4.I1.i4.p1.1.m1.6.7.2.1.3.1.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3.1">→</ci><ci id="S4.I1.i4.p1.1.m1.6.7.2.1.3.2.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3.2">𝛼</ci><infinity id="S4.I1.i4.p1.1.m1.6.7.2.1.3.3.cmml" xref="S4.I1.i4.p1.1.m1.6.7.2.1.3.3"></infinity></apply></apply><apply 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</div> </li> </ol> </div> </div> <div class="ltx_para" id="S4.SS1.p6"> <p class="ltx_p" id="S4.SS1.p6.1">We conjecture that for general <math alttext="s-t" class="ltx_Math" display="inline" id="S4.SS1.p6.1.m1.1"><semantics id="S4.SS1.p6.1.m1.1a"><mrow id="S4.SS1.p6.1.m1.1.1" xref="S4.SS1.p6.1.m1.1.1.cmml"><mi id="S4.SS1.p6.1.m1.1.1.2" xref="S4.SS1.p6.1.m1.1.1.2.cmml">s</mi><mo id="S4.SS1.p6.1.m1.1.1.1" xref="S4.SS1.p6.1.m1.1.1.1.cmml">−</mo><mi id="S4.SS1.p6.1.m1.1.1.3" xref="S4.SS1.p6.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.1.m1.1b"><apply id="S4.SS1.p6.1.m1.1.1.cmml" xref="S4.SS1.p6.1.m1.1.1"><minus id="S4.SS1.p6.1.m1.1.1.1.cmml" xref="S4.SS1.p6.1.m1.1.1.1"></minus><ci id="S4.SS1.p6.1.m1.1.1.2.cmml" xref="S4.SS1.p6.1.m1.1.1.2">𝑠</ci><ci id="S4.SS1.p6.1.m1.1.1.3.cmml" xref="S4.SS1.p6.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.1.m1.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.1.m1.1d">italic_s - italic_t</annotation></semantics></math> graphs, there holds</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx9"> <tbody id="S4.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)=\min_{% \sigma\in\mathcal{U}_{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="S4.E5.m1.5"><semantics id="S4.E5.m1.5a"><mrow id="S4.E5.m1.5.6" xref="S4.E5.m1.5.6.cmml"><mrow id="S4.E5.m1.5.6.2" xref="S4.E5.m1.5.6.2.cmml"><mrow id="S4.E5.m1.5.6.2.2" xref="S4.E5.m1.5.6.2.2.cmml"><munder id="S4.E5.m1.5.6.2.2.1" xref="S4.E5.m1.5.6.2.2.1.cmml"><mi id="S4.E5.m1.5.6.2.2.1.2" xref="S4.E5.m1.5.6.2.2.1.2.cmml">min</mi><mrow id="S4.E5.m1.1.1.1" xref="S4.E5.m1.1.1.1.cmml"><mi id="S4.E5.m1.1.1.1.3" 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encoding="application/x-tex" id="S4.E5.m1.5c">\displaystyle\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)=\min_{% \sigma\in\mathcal{U}_{n}}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.5d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p6.9">and as a consequence, there holds <math alttext="r(G,n,k,\text{SUM},\alpha)=r_{U}(G,n,k,\text{SUM},\alpha)" class="ltx_Math" display="inline" id="S4.SS1.p6.2.m1.10"><semantics id="S4.SS1.p6.2.m1.10a"><mrow id="S4.SS1.p6.2.m1.10.11" xref="S4.SS1.p6.2.m1.10.11.cmml"><mrow id="S4.SS1.p6.2.m1.10.11.2" xref="S4.SS1.p6.2.m1.10.11.2.cmml"><mi id="S4.SS1.p6.2.m1.10.11.2.2" xref="S4.SS1.p6.2.m1.10.11.2.2.cmml">r</mi><mo id="S4.SS1.p6.2.m1.10.11.2.1" xref="S4.SS1.p6.2.m1.10.11.2.1.cmml">⁢</mo><mrow id="S4.SS1.p6.2.m1.10.11.2.3.2" xref="S4.SS1.p6.2.m1.10.11.2.3.1.cmml"><mo id="S4.SS1.p6.2.m1.10.11.2.3.2.1" stretchy="false" xref="S4.SS1.p6.2.m1.10.11.2.3.1.cmml">(</mo><mi id="S4.SS1.p6.2.m1.1.1" xref="S4.SS1.p6.2.m1.1.1.cmml">G</mi><mo id="S4.SS1.p6.2.m1.10.11.2.3.2.2" xref="S4.SS1.p6.2.m1.10.11.2.3.1.cmml">,</mo><mi id="S4.SS1.p6.2.m1.2.2" xref="S4.SS1.p6.2.m1.2.2.cmml">n</mi><mo id="S4.SS1.p6.2.m1.10.11.2.3.2.3" xref="S4.SS1.p6.2.m1.10.11.2.3.1.cmml">,</mo><mi 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xref="S4.SS1.p6.2.m1.10.11.3.3.1.cmml"><mo id="S4.SS1.p6.2.m1.10.11.3.3.2.1" stretchy="false" xref="S4.SS1.p6.2.m1.10.11.3.3.1.cmml">(</mo><mi id="S4.SS1.p6.2.m1.6.6" xref="S4.SS1.p6.2.m1.6.6.cmml">G</mi><mo id="S4.SS1.p6.2.m1.10.11.3.3.2.2" xref="S4.SS1.p6.2.m1.10.11.3.3.1.cmml">,</mo><mi id="S4.SS1.p6.2.m1.7.7" xref="S4.SS1.p6.2.m1.7.7.cmml">n</mi><mo id="S4.SS1.p6.2.m1.10.11.3.3.2.3" xref="S4.SS1.p6.2.m1.10.11.3.3.1.cmml">,</mo><mi id="S4.SS1.p6.2.m1.8.8" xref="S4.SS1.p6.2.m1.8.8.cmml">k</mi><mo id="S4.SS1.p6.2.m1.10.11.3.3.2.4" xref="S4.SS1.p6.2.m1.10.11.3.3.1.cmml">,</mo><mtext id="S4.SS1.p6.2.m1.9.9" xref="S4.SS1.p6.2.m1.9.9a.cmml">SUM</mtext><mo id="S4.SS1.p6.2.m1.10.11.3.3.2.5" xref="S4.SS1.p6.2.m1.10.11.3.3.1.cmml">,</mo><mi id="S4.SS1.p6.2.m1.10.10" xref="S4.SS1.p6.2.m1.10.10.cmml">α</mi><mo id="S4.SS1.p6.2.m1.10.11.3.3.2.6" stretchy="false" xref="S4.SS1.p6.2.m1.10.11.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" 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xref="S4.SS1.p6.2.m1.10.11.3.1"></times><apply id="S4.SS1.p6.2.m1.10.11.3.2.cmml" xref="S4.SS1.p6.2.m1.10.11.3.2"><csymbol cd="ambiguous" id="S4.SS1.p6.2.m1.10.11.3.2.1.cmml" xref="S4.SS1.p6.2.m1.10.11.3.2">subscript</csymbol><ci id="S4.SS1.p6.2.m1.10.11.3.2.2.cmml" xref="S4.SS1.p6.2.m1.10.11.3.2.2">𝑟</ci><ci id="S4.SS1.p6.2.m1.10.11.3.2.3.cmml" xref="S4.SS1.p6.2.m1.10.11.3.2.3">𝑈</ci></apply><vector id="S4.SS1.p6.2.m1.10.11.3.3.1.cmml" xref="S4.SS1.p6.2.m1.10.11.3.3.2"><ci id="S4.SS1.p6.2.m1.6.6.cmml" xref="S4.SS1.p6.2.m1.6.6">𝐺</ci><ci id="S4.SS1.p6.2.m1.7.7.cmml" xref="S4.SS1.p6.2.m1.7.7">𝑛</ci><ci id="S4.SS1.p6.2.m1.8.8.cmml" xref="S4.SS1.p6.2.m1.8.8">𝑘</ci><ci id="S4.SS1.p6.2.m1.9.9a.cmml" xref="S4.SS1.p6.2.m1.9.9"><mtext id="S4.SS1.p6.2.m1.9.9.cmml" xref="S4.SS1.p6.2.m1.9.9">SUM</mtext></ci><ci id="S4.SS1.p6.2.m1.10.10.cmml" xref="S4.SS1.p6.2.m1.10.10">𝛼</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.2.m1.10c">r(G,n,k,\text{SUM},\alpha)=r_{U}(G,n,k,\text{SUM},\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.2.m1.10d">italic_r ( italic_G , italic_n , italic_k , SUM , italic_α ) = italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , SUM , italic_α )</annotation></semantics></math>. This is a stronger result about SUM than Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem3" title="Theorem 4.3 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.3</span></a>, for which we do not have a proof at this time. However, we do prove the conjecture in the case where <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p6.3.m2.1"><semantics id="S4.SS1.p6.3.m2.1a"><mi id="S4.SS1.p6.3.m2.1.1" xref="S4.SS1.p6.3.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.3.m2.1b"><ci id="S4.SS1.p6.3.m2.1.1.cmml" xref="S4.SS1.p6.3.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.3.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.3.m2.1d">italic_G</annotation></semantics></math> has only disjoint paths, which seems to support the validity of the conjecture. If the conjecture is true, then <math alttext="r" class="ltx_Math" display="inline" id="S4.SS1.p6.4.m3.1"><semantics id="S4.SS1.p6.4.m3.1a"><mi id="S4.SS1.p6.4.m3.1.1" xref="S4.SS1.p6.4.m3.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.4.m3.1b"><ci id="S4.SS1.p6.4.m3.1.1.cmml" xref="S4.SS1.p6.4.m3.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.4.m3.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.4.m3.1d">italic_r</annotation></semantics></math> will satisfy all the properties that <math alttext="r_{U}" class="ltx_Math" display="inline" id="S4.SS1.p6.5.m4.1"><semantics id="S4.SS1.p6.5.m4.1a"><msub id="S4.SS1.p6.5.m4.1.1" xref="S4.SS1.p6.5.m4.1.1.cmml"><mi id="S4.SS1.p6.5.m4.1.1.2" xref="S4.SS1.p6.5.m4.1.1.2.cmml">r</mi><mi id="S4.SS1.p6.5.m4.1.1.3" xref="S4.SS1.p6.5.m4.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.5.m4.1b"><apply id="S4.SS1.p6.5.m4.1.1.cmml" xref="S4.SS1.p6.5.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.5.m4.1.1.1.cmml" xref="S4.SS1.p6.5.m4.1.1">subscript</csymbol><ci id="S4.SS1.p6.5.m4.1.1.2.cmml" xref="S4.SS1.p6.5.m4.1.1.2">𝑟</ci><ci id="S4.SS1.p6.5.m4.1.1.3.cmml" xref="S4.SS1.p6.5.m4.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.5.m4.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.5.m4.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> satisfies. Because SUM has convenient properties (e.g. Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma8" title="Lemma 8 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">8</span></a>), <math alttext="r_{U}" class="ltx_Math" display="inline" id="S4.SS1.p6.6.m5.1"><semantics id="S4.SS1.p6.6.m5.1a"><msub id="S4.SS1.p6.6.m5.1.1" xref="S4.SS1.p6.6.m5.1.1.cmml"><mi id="S4.SS1.p6.6.m5.1.1.2" xref="S4.SS1.p6.6.m5.1.1.2.cmml">r</mi><mi id="S4.SS1.p6.6.m5.1.1.3" xref="S4.SS1.p6.6.m5.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.6.m5.1b"><apply id="S4.SS1.p6.6.m5.1.1.cmml" xref="S4.SS1.p6.6.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.6.m5.1.1.1.cmml" xref="S4.SS1.p6.6.m5.1.1">subscript</csymbol><ci id="S4.SS1.p6.6.m5.1.1.2.cmml" xref="S4.SS1.p6.6.m5.1.1.2">𝑟</ci><ci id="S4.SS1.p6.6.m5.1.1.3.cmml" xref="S4.SS1.p6.6.m5.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.6.m5.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.6.m5.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> also has several appealing properties: it is smooth a.s. (almost surely), continuous and strictly increasing in <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.p6.7.m6.1"><semantics id="S4.SS1.p6.7.m6.1a"><mi id="S4.SS1.p6.7.m6.1.1" xref="S4.SS1.p6.7.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.7.m6.1b"><ci id="S4.SS1.p6.7.m6.1.1.cmml" xref="S4.SS1.p6.7.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.7.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.7.m6.1d">italic_α</annotation></semantics></math> until it hits <math alttext="\mathcal{E}_{m_{c},n}" class="ltx_Math" display="inline" id="S4.SS1.p6.8.m7.2"><semantics id="S4.SS1.p6.8.m7.2a"><msub id="S4.SS1.p6.8.m7.2.3" xref="S4.SS1.p6.8.m7.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.8.m7.2.3.2" xref="S4.SS1.p6.8.m7.2.3.2.cmml">ℰ</mi><mrow id="S4.SS1.p6.8.m7.2.2.2.2" xref="S4.SS1.p6.8.m7.2.2.2.3.cmml"><msub id="S4.SS1.p6.8.m7.2.2.2.2.1" xref="S4.SS1.p6.8.m7.2.2.2.2.1.cmml"><mi id="S4.SS1.p6.8.m7.2.2.2.2.1.2" xref="S4.SS1.p6.8.m7.2.2.2.2.1.2.cmml">m</mi><mi id="S4.SS1.p6.8.m7.2.2.2.2.1.3" xref="S4.SS1.p6.8.m7.2.2.2.2.1.3.cmml">c</mi></msub><mo id="S4.SS1.p6.8.m7.2.2.2.2.2" xref="S4.SS1.p6.8.m7.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p6.8.m7.1.1.1.1" xref="S4.SS1.p6.8.m7.1.1.1.1.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.8.m7.2b"><apply id="S4.SS1.p6.8.m7.2.3.cmml" xref="S4.SS1.p6.8.m7.2.3"><csymbol cd="ambiguous" id="S4.SS1.p6.8.m7.2.3.1.cmml" xref="S4.SS1.p6.8.m7.2.3">subscript</csymbol><ci id="S4.SS1.p6.8.m7.2.3.2.cmml" xref="S4.SS1.p6.8.m7.2.3.2">ℰ</ci><list id="S4.SS1.p6.8.m7.2.2.2.3.cmml" xref="S4.SS1.p6.8.m7.2.2.2.2"><apply id="S4.SS1.p6.8.m7.2.2.2.2.1.cmml" xref="S4.SS1.p6.8.m7.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p6.8.m7.2.2.2.2.1.1.cmml" xref="S4.SS1.p6.8.m7.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.p6.8.m7.2.2.2.2.1.2.cmml" xref="S4.SS1.p6.8.m7.2.2.2.2.1.2">𝑚</ci><ci id="S4.SS1.p6.8.m7.2.2.2.2.1.3.cmml" xref="S4.SS1.p6.8.m7.2.2.2.2.1.3">𝑐</ci></apply><ci id="S4.SS1.p6.8.m7.1.1.1.1.cmml" xref="S4.SS1.p6.8.m7.1.1.1.1">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.8.m7.2c">\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.8.m7.2d">caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, where it plateaus. In addition, <math alttext="r_{U}" class="ltx_Math" display="inline" id="S4.SS1.p6.9.m8.1"><semantics id="S4.SS1.p6.9.m8.1a"><msub id="S4.SS1.p6.9.m8.1.1" xref="S4.SS1.p6.9.m8.1.1.cmml"><mi id="S4.SS1.p6.9.m8.1.1.2" xref="S4.SS1.p6.9.m8.1.1.2.cmml">r</mi><mi id="S4.SS1.p6.9.m8.1.1.3" xref="S4.SS1.p6.9.m8.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.9.m8.1b"><apply id="S4.SS1.p6.9.m8.1.1.cmml" xref="S4.SS1.p6.9.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.9.m8.1.1.1.cmml" xref="S4.SS1.p6.9.m8.1.1">subscript</csymbol><ci id="S4.SS1.p6.9.m8.1.1.2.cmml" xref="S4.SS1.p6.9.m8.1.1.2">𝑟</ci><ci id="S4.SS1.p6.9.m8.1.1.3.cmml" xref="S4.SS1.p6.9.m8.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.9.m8.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.9.m8.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> can be computed in polynomial time in the game size.</p> </div> <section class="ltx_subsubsection" id="S4.SS1.SSSx1"> <h4 class="ltx_title ltx_title_subsubsection">Properties of <math alttext="r_{U}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.1.m1.1"><semantics id="S4.SS1.SSSx1.1.m1.1b"><msub id="S4.SS1.SSSx1.1.m1.1.1" xref="S4.SS1.SSSx1.1.m1.1.1.cmml"><mi id="S4.SS1.SSSx1.1.m1.1.1.2" xref="S4.SS1.SSSx1.1.m1.1.1.2.cmml">r</mi><mi id="S4.SS1.SSSx1.1.m1.1.1.3" xref="S4.SS1.SSSx1.1.m1.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.1.m1.1c"><apply id="S4.SS1.SSSx1.1.m1.1.1.cmml" xref="S4.SS1.SSSx1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.1.m1.1.1.1.cmml" xref="S4.SS1.SSSx1.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.SSSx1.1.m1.1.1.2.cmml" xref="S4.SS1.SSSx1.1.m1.1.1.2">𝑟</ci><ci id="S4.SS1.SSSx1.1.m1.1.1.3.cmml" xref="S4.SS1.SSSx1.1.m1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.1.m1.1d">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.1.m1.1e">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> </h4> <div class="ltx_para" id="S4.SS1.SSSx1.p1"> <p class="ltx_p" id="S4.SS1.SSSx1.p1.1">For every <math alttext="i\in[m_{c}]" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p1.1.m1.1"><semantics id="S4.SS1.SSSx1.p1.1.m1.1a"><mrow id="S4.SS1.SSSx1.p1.1.m1.1.1" xref="S4.SS1.SSSx1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.SSSx1.p1.1.m1.1.1.3" xref="S4.SS1.SSSx1.p1.1.m1.1.1.3.cmml">i</mi><mo id="S4.SS1.SSSx1.p1.1.m1.1.1.2" xref="S4.SS1.SSSx1.p1.1.m1.1.1.2.cmml">∈</mo><mrow id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.2.cmml"><mo id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.2.1.cmml">[</mo><msub id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.cmml"><mi id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.2" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.2.cmml">m</mi><mi id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.3" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p1.1.m1.1b"><apply id="S4.SS1.SSSx1.p1.1.m1.1.1.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1"><in id="S4.SS1.SSSx1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.2"></in><ci id="S4.SS1.SSSx1.p1.1.m1.1.1.3.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.3">𝑖</ci><apply id="S4.SS1.SSSx1.p1.1.m1.1.1.1.2.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1"><csymbol cd="latexml" id="S4.SS1.SSSx1.p1.1.m1.1.1.1.2.1.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.2">𝑚</ci><ci id="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S4.SS1.SSSx1.p1.1.m1.1.1.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p1.1.m1.1c">i\in[m_{c}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p1.1.m1.1d">italic_i ∈ [ italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ]</annotation></semantics></math> pick a subset</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P^{*}_{i}\in\operatorname*{arg\,min}_{f\in MF,P\in\operatorname{\mathcal{S}}(P% _{f}),|P|=i}\sum_{p\in P}|p|" class="ltx_Math" display="block" id="S4.Ex8.m1.5"><semantics id="S4.Ex8.m1.5a"><mrow id="S4.Ex8.m1.5.6" xref="S4.Ex8.m1.5.6.cmml"><msubsup id="S4.Ex8.m1.5.6.2" xref="S4.Ex8.m1.5.6.2.cmml"><mi id="S4.Ex8.m1.5.6.2.2.2" xref="S4.Ex8.m1.5.6.2.2.2.cmml">P</mi><mi id="S4.Ex8.m1.5.6.2.3" xref="S4.Ex8.m1.5.6.2.3.cmml">i</mi><mo id="S4.Ex8.m1.5.6.2.2.3" xref="S4.Ex8.m1.5.6.2.2.3.cmml">∗</mo></msubsup><mo id="S4.Ex8.m1.5.6.1" xref="S4.Ex8.m1.5.6.1.cmml">∈</mo><mrow id="S4.Ex8.m1.5.6.3" xref="S4.Ex8.m1.5.6.3.cmml"><munder id="S4.Ex8.m1.5.6.3.2" xref="S4.Ex8.m1.5.6.3.2.cmml"><mrow id="S4.Ex8.m1.5.6.3.2.2" xref="S4.Ex8.m1.5.6.3.2.2.cmml"><mi id="S4.Ex8.m1.5.6.3.2.2.2" xref="S4.Ex8.m1.5.6.3.2.2.2.cmml">arg</mi><mo id="S4.Ex8.m1.5.6.3.2.2.1" lspace="0.170em" xref="S4.Ex8.m1.5.6.3.2.2.1.cmml">⁢</mo><mi id="S4.Ex8.m1.5.6.3.2.2.3" xref="S4.Ex8.m1.5.6.3.2.2.3.cmml">min</mi></mrow><mrow id="S4.Ex8.m1.4.4.4.4" xref="S4.Ex8.m1.4.4.4.5.cmml"><mrow id="S4.Ex8.m1.3.3.3.3.1" xref="S4.Ex8.m1.3.3.3.3.1.cmml"><mi id="S4.Ex8.m1.3.3.3.3.1.2" xref="S4.Ex8.m1.3.3.3.3.1.2.cmml">f</mi><mo id="S4.Ex8.m1.3.3.3.3.1.1" xref="S4.Ex8.m1.3.3.3.3.1.1.cmml">∈</mo><mrow id="S4.Ex8.m1.3.3.3.3.1.3" xref="S4.Ex8.m1.3.3.3.3.1.3.cmml"><mi id="S4.Ex8.m1.3.3.3.3.1.3.2" xref="S4.Ex8.m1.3.3.3.3.1.3.2.cmml">M</mi><mo id="S4.Ex8.m1.3.3.3.3.1.3.1" xref="S4.Ex8.m1.3.3.3.3.1.3.1.cmml">⁢</mo><mi id="S4.Ex8.m1.3.3.3.3.1.3.3" xref="S4.Ex8.m1.3.3.3.3.1.3.3.cmml">F</mi></mrow></mrow><mo id="S4.Ex8.m1.4.4.4.4.3" xref="S4.Ex8.m1.4.4.4.5a.cmml">,</mo><mrow id="S4.Ex8.m1.4.4.4.4.2.2" xref="S4.Ex8.m1.4.4.4.4.2.3.cmml"><mrow id="S4.Ex8.m1.4.4.4.4.2.1.1" xref="S4.Ex8.m1.4.4.4.4.2.1.1.cmml"><mi id="S4.Ex8.m1.4.4.4.4.2.1.1.3" xref="S4.Ex8.m1.4.4.4.4.2.1.1.3.cmml">P</mi><mo id="S4.Ex8.m1.4.4.4.4.2.1.1.2" xref="S4.Ex8.m1.4.4.4.4.2.1.1.2.cmml">∈</mo><mrow id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex8.m1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.cmml">𝒮</mi><mo id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1a" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.2.cmml">⁡</mo><mrow id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.2.cmml"><mo id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.2" stretchy="false" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.2.cmml">(</mo><msub id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.1" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.1.cmml"><mi id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.1.2" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.1.2.cmml">P</mi><mi id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.1.3" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.1.3.cmml">f</mi></msub><mo id="S4.Ex8.m1.4.4.4.4.2.1.1.1.1.1.3" stretchy="false" xref="S4.Ex8.m1.4.4.4.4.2.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex8.m1.4.4.4.4.2.2.3" xref="S4.Ex8.m1.4.4.4.4.2.3a.cmml">,</mo><mrow id="S4.Ex8.m1.4.4.4.4.2.2.2" xref="S4.Ex8.m1.4.4.4.4.2.2.2.cmml"><mrow id="S4.Ex8.m1.4.4.4.4.2.2.2.2.2" xref="S4.Ex8.m1.4.4.4.4.2.2.2.2.1.cmml"><mo id="S4.Ex8.m1.4.4.4.4.2.2.2.2.2.1" stretchy="false" xref="S4.Ex8.m1.4.4.4.4.2.2.2.2.1.1.cmml">|</mo><mi id="S4.Ex8.m1.2.2.2.2" xref="S4.Ex8.m1.2.2.2.2.cmml">P</mi><mo id="S4.Ex8.m1.4.4.4.4.2.2.2.2.2.2" stretchy="false" 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xref="S4.Ex8.m1.5.6.3.3.1.2"></sum><apply id="S4.Ex8.m1.5.6.3.3.1.3.cmml" xref="S4.Ex8.m1.5.6.3.3.1.3"><in id="S4.Ex8.m1.5.6.3.3.1.3.1.cmml" xref="S4.Ex8.m1.5.6.3.3.1.3.1"></in><ci id="S4.Ex8.m1.5.6.3.3.1.3.2.cmml" xref="S4.Ex8.m1.5.6.3.3.1.3.2">𝑝</ci><ci id="S4.Ex8.m1.5.6.3.3.1.3.3.cmml" xref="S4.Ex8.m1.5.6.3.3.1.3.3">𝑃</ci></apply></apply><apply id="S4.Ex8.m1.5.6.3.3.2.1.cmml" xref="S4.Ex8.m1.5.6.3.3.2.2"><abs id="S4.Ex8.m1.5.6.3.3.2.1.1.cmml" xref="S4.Ex8.m1.5.6.3.3.2.2.1"></abs><ci id="S4.Ex8.m1.5.5.cmml" xref="S4.Ex8.m1.5.5">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.5c">P^{*}_{i}\in\operatorname*{arg\,min}_{f\in MF,P\in\operatorname{\mathcal{S}}(P% _{f}),|P|=i}\sum_{p\in P}|p|</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.5d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_f ∈ italic_M italic_F , italic_P ∈ caligraphic_S ( italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) , | italic_P | = italic_i end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_p ∈ italic_P end_POSTSUBSCRIPT | italic_p |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.SSSx1.p1.2">By definition, for every <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p1.2.m1.1"><semantics id="S4.SS1.SSSx1.p1.2.m1.1a"><mrow id="S4.SS1.SSSx1.p1.2.m1.1.1" xref="S4.SS1.SSSx1.p1.2.m1.1.1.cmml"><mi id="S4.SS1.SSSx1.p1.2.m1.1.1.2" xref="S4.SS1.SSSx1.p1.2.m1.1.1.2.cmml">n</mi><mo id="S4.SS1.SSSx1.p1.2.m1.1.1.1" xref="S4.SS1.SSSx1.p1.2.m1.1.1.1.cmml">∈</mo><mi id="S4.SS1.SSSx1.p1.2.m1.1.1.3" xref="S4.SS1.SSSx1.p1.2.m1.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p1.2.m1.1b"><apply id="S4.SS1.SSSx1.p1.2.m1.1.1.cmml" xref="S4.SS1.SSSx1.p1.2.m1.1.1"><in id="S4.SS1.SSSx1.p1.2.m1.1.1.1.cmml" xref="S4.SS1.SSSx1.p1.2.m1.1.1.1"></in><ci id="S4.SS1.SSSx1.p1.2.m1.1.1.2.cmml" xref="S4.SS1.SSSx1.p1.2.m1.1.1.2">𝑛</ci><ci id="S4.SS1.SSSx1.p1.2.m1.1.1.3.cmml" xref="S4.SS1.SSSx1.p1.2.m1.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p1.2.m1.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p1.2.m1.1d">italic_n ∈ blackboard_N</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx10"> <tbody id="S4.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min_{U^{n}\in\mathcal{U}_{n}}F_{n}(U^{n},\alpha)=\min_{i}F_{n}(U% _{P^{*}_{i}}^{n},\alpha)=\min_{i}\left(\frac{\mathcal{E}_{i,n}}{i}\left(k% \alpha+\sum_{p\in P^{*}_{i}}|p|\right)\right)" class="ltx_Math" display="inline" id="S4.Ex9.m1.9"><semantics id="S4.Ex9.m1.9a"><mrow id="S4.Ex9.m1.9.9" xref="S4.Ex9.m1.9.9.cmml"><mrow id="S4.Ex9.m1.6.6.1" xref="S4.Ex9.m1.6.6.1.cmml"><mrow id="S4.Ex9.m1.6.6.1.3" xref="S4.Ex9.m1.6.6.1.3.cmml"><munder id="S4.Ex9.m1.6.6.1.3.1" xref="S4.Ex9.m1.6.6.1.3.1.cmml"><mi id="S4.Ex9.m1.6.6.1.3.1.2" xref="S4.Ex9.m1.6.6.1.3.1.2.cmml">min</mi><mrow id="S4.Ex9.m1.6.6.1.3.1.3" xref="S4.Ex9.m1.6.6.1.3.1.3.cmml"><msup id="S4.Ex9.m1.6.6.1.3.1.3.2" xref="S4.Ex9.m1.6.6.1.3.1.3.2.cmml"><mi id="S4.Ex9.m1.6.6.1.3.1.3.2.2" xref="S4.Ex9.m1.6.6.1.3.1.3.2.2.cmml">U</mi><mi id="S4.Ex9.m1.6.6.1.3.1.3.2.3" xref="S4.Ex9.m1.6.6.1.3.1.3.2.3.cmml">n</mi></msup><mo id="S4.Ex9.m1.6.6.1.3.1.3.1" xref="S4.Ex9.m1.6.6.1.3.1.3.1.cmml">∈</mo><msub id="S4.Ex9.m1.6.6.1.3.1.3.3" xref="S4.Ex9.m1.6.6.1.3.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex9.m1.6.6.1.3.1.3.3.2" xref="S4.Ex9.m1.6.6.1.3.1.3.3.2.cmml">𝒰</mi><mi 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xref="S4.Ex9.m1.5.5">𝑝</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.9c">\displaystyle\min_{U^{n}\in\mathcal{U}_{n}}F_{n}(U^{n},\alpha)=\min_{i}F_{n}(U% _{P^{*}_{i}}^{n},\alpha)=\min_{i}\left(\frac{\mathcal{E}_{i,n}}{i}\left(k% \alpha+\sum_{p\in P^{*}_{i}}|p|\right)\right)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.9d">roman_min start_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG ( italic_k italic_α + ∑ start_POSTSUBSCRIPT italic_p ∈ italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_p | ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.SSSx1.p1.6">This defines a more explicit form for <math alttext="r_{U}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p1.3.m1.1"><semantics id="S4.SS1.SSSx1.p1.3.m1.1a"><msub id="S4.SS1.SSSx1.p1.3.m1.1.1" xref="S4.SS1.SSSx1.p1.3.m1.1.1.cmml"><mi id="S4.SS1.SSSx1.p1.3.m1.1.1.2" xref="S4.SS1.SSSx1.p1.3.m1.1.1.2.cmml">r</mi><mi id="S4.SS1.SSSx1.p1.3.m1.1.1.3" xref="S4.SS1.SSSx1.p1.3.m1.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p1.3.m1.1b"><apply id="S4.SS1.SSSx1.p1.3.m1.1.1.cmml" xref="S4.SS1.SSSx1.p1.3.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p1.3.m1.1.1.1.cmml" xref="S4.SS1.SSSx1.p1.3.m1.1.1">subscript</csymbol><ci id="S4.SS1.SSSx1.p1.3.m1.1.1.2.cmml" xref="S4.SS1.SSSx1.p1.3.m1.1.1.2">𝑟</ci><ci id="S4.SS1.SSSx1.p1.3.m1.1.1.3.cmml" xref="S4.SS1.SSSx1.p1.3.m1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p1.3.m1.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p1.3.m1.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>. By the minimality of each <math alttext="P^{*}_{i}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p1.4.m2.1"><semantics id="S4.SS1.SSSx1.p1.4.m2.1a"><msubsup id="S4.SS1.SSSx1.p1.4.m2.1.1" xref="S4.SS1.SSSx1.p1.4.m2.1.1.cmml"><mi id="S4.SS1.SSSx1.p1.4.m2.1.1.2.2" xref="S4.SS1.SSSx1.p1.4.m2.1.1.2.2.cmml">P</mi><mi id="S4.SS1.SSSx1.p1.4.m2.1.1.3" xref="S4.SS1.SSSx1.p1.4.m2.1.1.3.cmml">i</mi><mo id="S4.SS1.SSSx1.p1.4.m2.1.1.2.3" xref="S4.SS1.SSSx1.p1.4.m2.1.1.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p1.4.m2.1b"><apply id="S4.SS1.SSSx1.p1.4.m2.1.1.cmml" xref="S4.SS1.SSSx1.p1.4.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p1.4.m2.1.1.1.cmml" xref="S4.SS1.SSSx1.p1.4.m2.1.1">subscript</csymbol><apply id="S4.SS1.SSSx1.p1.4.m2.1.1.2.cmml" xref="S4.SS1.SSSx1.p1.4.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p1.4.m2.1.1.2.1.cmml" xref="S4.SS1.SSSx1.p1.4.m2.1.1">superscript</csymbol><ci id="S4.SS1.SSSx1.p1.4.m2.1.1.2.2.cmml" xref="S4.SS1.SSSx1.p1.4.m2.1.1.2.2">𝑃</ci><times id="S4.SS1.SSSx1.p1.4.m2.1.1.2.3.cmml" xref="S4.SS1.SSSx1.p1.4.m2.1.1.2.3"></times></apply><ci id="S4.SS1.SSSx1.p1.4.m2.1.1.3.cmml" xref="S4.SS1.SSSx1.p1.4.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p1.4.m2.1c">P^{*}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p1.4.m2.1d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the sums <math alttext="S_{i}:=\sum_{p\in P^{*}_{i}}|p|" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p1.5.m3.1"><semantics id="S4.SS1.SSSx1.p1.5.m3.1a"><mrow id="S4.SS1.SSSx1.p1.5.m3.1.2" xref="S4.SS1.SSSx1.p1.5.m3.1.2.cmml"><msub id="S4.SS1.SSSx1.p1.5.m3.1.2.2" xref="S4.SS1.SSSx1.p1.5.m3.1.2.2.cmml"><mi id="S4.SS1.SSSx1.p1.5.m3.1.2.2.2" xref="S4.SS1.SSSx1.p1.5.m3.1.2.2.2.cmml">S</mi><mi id="S4.SS1.SSSx1.p1.5.m3.1.2.2.3" 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id="S4.SS1.SSSx1.p1.5.m3.1c">S_{i}:=\sum_{p\in P^{*}_{i}}|p|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p1.5.m3.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := ∑ start_POSTSUBSCRIPT italic_p ∈ italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_p |</annotation></semantics></math> form a strictly increasing sequence. We now formally state the properties of <math alttext="r_{U}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p1.6.m4.1"><semantics id="S4.SS1.SSSx1.p1.6.m4.1a"><msub id="S4.SS1.SSSx1.p1.6.m4.1.1" xref="S4.SS1.SSSx1.p1.6.m4.1.1.cmml"><mi id="S4.SS1.SSSx1.p1.6.m4.1.1.2" xref="S4.SS1.SSSx1.p1.6.m4.1.1.2.cmml">r</mi><mi id="S4.SS1.SSSx1.p1.6.m4.1.1.3" xref="S4.SS1.SSSx1.p1.6.m4.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p1.6.m4.1b"><apply id="S4.SS1.SSSx1.p1.6.m4.1.1.cmml" xref="S4.SS1.SSSx1.p1.6.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p1.6.m4.1.1.1.cmml" xref="S4.SS1.SSSx1.p1.6.m4.1.1">subscript</csymbol><ci id="S4.SS1.SSSx1.p1.6.m4.1.1.2.cmml" xref="S4.SS1.SSSx1.p1.6.m4.1.1.2">𝑟</ci><ci id="S4.SS1.SSSx1.p1.6.m4.1.1.3.cmml" xref="S4.SS1.SSSx1.p1.6.m4.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p1.6.m4.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p1.6.m4.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> described above.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 4.4</span></h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.5"><math alttext="r_{U}(G,n,k,\alpha)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.m1.4"><semantics id="S4.Thmtheorem4.p1.1.m1.4a"><mrow id="S4.Thmtheorem4.p1.1.m1.4.5" xref="S4.Thmtheorem4.p1.1.m1.4.5.cmml"><msub id="S4.Thmtheorem4.p1.1.m1.4.5.2" xref="S4.Thmtheorem4.p1.1.m1.4.5.2.cmml"><mi id="S4.Thmtheorem4.p1.1.m1.4.5.2.2" xref="S4.Thmtheorem4.p1.1.m1.4.5.2.2.cmml">r</mi><mi id="S4.Thmtheorem4.p1.1.m1.4.5.2.3" xref="S4.Thmtheorem4.p1.1.m1.4.5.2.3.cmml">U</mi></msub><mo id="S4.Thmtheorem4.p1.1.m1.4.5.1" xref="S4.Thmtheorem4.p1.1.m1.4.5.1.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.1.m1.4.5.3.2" xref="S4.Thmtheorem4.p1.1.m1.4.5.3.1.cmml"><mo id="S4.Thmtheorem4.p1.1.m1.4.5.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.1.m1.4.5.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.m1.1.1.cmml">G</mi><mo id="S4.Thmtheorem4.p1.1.m1.4.5.3.2.2" xref="S4.Thmtheorem4.p1.1.m1.4.5.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.1.m1.2.2" xref="S4.Thmtheorem4.p1.1.m1.2.2.cmml">n</mi><mo id="S4.Thmtheorem4.p1.1.m1.4.5.3.2.3" xref="S4.Thmtheorem4.p1.1.m1.4.5.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.1.m1.3.3" xref="S4.Thmtheorem4.p1.1.m1.3.3.cmml">k</mi><mo id="S4.Thmtheorem4.p1.1.m1.4.5.3.2.4" xref="S4.Thmtheorem4.p1.1.m1.4.5.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.1.m1.4.4" xref="S4.Thmtheorem4.p1.1.m1.4.4.cmml">α</mi><mo id="S4.Thmtheorem4.p1.1.m1.4.5.3.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.1.m1.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.m1.4b"><apply id="S4.Thmtheorem4.p1.1.m1.4.5.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.5"><times id="S4.Thmtheorem4.p1.1.m1.4.5.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.5.1"></times><apply id="S4.Thmtheorem4.p1.1.m1.4.5.2.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.5.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.1.m1.4.5.2.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.5.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.1.m1.4.5.2.2.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.5.2.2">𝑟</ci><ci id="S4.Thmtheorem4.p1.1.m1.4.5.2.3.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.5.2.3">𝑈</ci></apply><vector id="S4.Thmtheorem4.p1.1.m1.4.5.3.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.5.3.2"><ci id="S4.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1">𝐺</ci><ci id="S4.Thmtheorem4.p1.1.m1.2.2.cmml" xref="S4.Thmtheorem4.p1.1.m1.2.2">𝑛</ci><ci id="S4.Thmtheorem4.p1.1.m1.3.3.cmml" xref="S4.Thmtheorem4.p1.1.m1.3.3">𝑘</ci><ci id="S4.Thmtheorem4.p1.1.m1.4.4.cmml" xref="S4.Thmtheorem4.p1.1.m1.4.4">𝛼</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.m1.4c">r_{U}(G,n,k,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.m1.4d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.5.4"> is a function that is smooth a.s., continuous and strictly increasing in <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.1.m1.1"><semantics id="S4.Thmtheorem4.p1.2.1.m1.1a"><mi id="S4.Thmtheorem4.p1.2.1.m1.1.1" xref="S4.Thmtheorem4.p1.2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.1.m1.1b"><ci id="S4.Thmtheorem4.p1.2.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.1.m1.1d">italic_α</annotation></semantics></math>. 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xref="S4.Thmtheorem4.p1.3.2.m2.3.3.cmml">G</mi><mo id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.2.2" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.3.2.m2.4.4" xref="S4.Thmtheorem4.p1.3.2.m2.4.4.cmml">n</mi><mo id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.2.3" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.3.2.m2.5.5" xref="S4.Thmtheorem4.p1.3.2.m2.5.5.cmml">k</mi><mo id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.2.4" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.3.2.m2.6.6" xref="S4.Thmtheorem4.p1.3.2.m2.6.6.cmml">α</mi><mo id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem4.p1.3.2.m2.6.7.1" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.1.cmml">≤</mo><msub id="S4.Thmtheorem4.p1.3.2.m2.6.7.3" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.3.2.m2.6.7.3.2" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.3.2.cmml">ℰ</mi><mrow id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.3.cmml"><msub id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.cmml"><mi id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.2" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.2.cmml">m</mi><mi id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.3" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.3.cmml">c</mi></msub><mo id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem4.p1.3.2.m2.1.1.1.1" xref="S4.Thmtheorem4.p1.3.2.m2.1.1.1.1.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.3.2.m2.6b"><apply id="S4.Thmtheorem4.p1.3.2.m2.6.7.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7"><leq id="S4.Thmtheorem4.p1.3.2.m2.6.7.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.1"></leq><apply id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2"><times id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.1"></times><apply id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2.2">𝑟</ci><ci id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2.3.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.2.3">𝑈</ci></apply><vector id="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.2.3.2"><ci id="S4.Thmtheorem4.p1.3.2.m2.3.3.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.3.3">𝐺</ci><ci id="S4.Thmtheorem4.p1.3.2.m2.4.4.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.4.4">𝑛</ci><ci id="S4.Thmtheorem4.p1.3.2.m2.5.5.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.5.5">𝑘</ci><ci id="S4.Thmtheorem4.p1.3.2.m2.6.6.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.6">𝛼</ci></vector></apply><apply id="S4.Thmtheorem4.p1.3.2.m2.6.7.3.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.2.m2.6.7.3.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.3">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.2.m2.6.7.3.2.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.6.7.3.2">ℰ</ci><list id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2"><apply id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.2.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.2">𝑚</ci><ci id="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.3.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.2.2.2.2.1.3">𝑐</ci></apply><ci id="S4.Thmtheorem4.p1.3.2.m2.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.2.m2.1.1.1.1">𝑛</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.2.m2.6c">r_{U}(G,n,k,\alpha)\leq\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.2.m2.6d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α ) ≤ caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. There is a function <math alttext="A(G,n,k)\in[0,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.3.m3.5"><semantics id="S4.Thmtheorem4.p1.4.3.m3.5a"><mrow id="S4.Thmtheorem4.p1.4.3.m3.5.6" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.cmml"><mrow id="S4.Thmtheorem4.p1.4.3.m3.5.6.2" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.cmml"><mi id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.2" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.2.cmml">A</mi><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.1" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.2" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.1.cmml"><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.4.3.m3.1.1" xref="S4.Thmtheorem4.p1.4.3.m3.1.1.cmml">G</mi><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.2.2" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.4.3.m3.2.2" xref="S4.Thmtheorem4.p1.4.3.m3.2.2.cmml">n</mi><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.2.3" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.4.3.m3.3.3" xref="S4.Thmtheorem4.p1.4.3.m3.3.3.cmml">k</mi><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.2.4" stretchy="false" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.1" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.1.cmml">∈</mo><mrow id="S4.Thmtheorem4.p1.4.3.m3.5.6.3.2" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.3.1.cmml"><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.3.1.cmml">[</mo><mn id="S4.Thmtheorem4.p1.4.3.m3.4.4" xref="S4.Thmtheorem4.p1.4.3.m3.4.4.cmml">0</mn><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.3.2.2" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.4.3.m3.5.5" mathvariant="normal" xref="S4.Thmtheorem4.p1.4.3.m3.5.5.cmml">∞</mi><mo id="S4.Thmtheorem4.p1.4.3.m3.5.6.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.3.m3.5b"><apply id="S4.Thmtheorem4.p1.4.3.m3.5.6.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.6"><in id="S4.Thmtheorem4.p1.4.3.m3.5.6.1.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.1"></in><apply id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2"><times id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.1.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.1"></times><ci id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.2.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.2">𝐴</ci><vector id="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.1.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.2.3.2"><ci id="S4.Thmtheorem4.p1.4.3.m3.1.1.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.1.1">𝐺</ci><ci id="S4.Thmtheorem4.p1.4.3.m3.2.2.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.2.2">𝑛</ci><ci id="S4.Thmtheorem4.p1.4.3.m3.3.3.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.3.3">𝑘</ci></vector></apply><interval closure="closed-open" id="S4.Thmtheorem4.p1.4.3.m3.5.6.3.1.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.6.3.2"><cn id="S4.Thmtheorem4.p1.4.3.m3.4.4.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.3.m3.4.4">0</cn><infinity id="S4.Thmtheorem4.p1.4.3.m3.5.5.cmml" xref="S4.Thmtheorem4.p1.4.3.m3.5.5"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.3.m3.5c">A(G,n,k)\in[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.3.m3.5d">italic_A ( italic_G , italic_n , italic_k ) ∈ [ 0 , ∞ )</annotation></semantics></math> such that equality holds for every <math alttext="\alpha\geq A(G,n,k)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.5.4.m4.3"><semantics id="S4.Thmtheorem4.p1.5.4.m4.3a"><mrow id="S4.Thmtheorem4.p1.5.4.m4.3.4" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.cmml"><mi id="S4.Thmtheorem4.p1.5.4.m4.3.4.2" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.2.cmml">α</mi><mo id="S4.Thmtheorem4.p1.5.4.m4.3.4.1" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.1.cmml">≥</mo><mrow id="S4.Thmtheorem4.p1.5.4.m4.3.4.3" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.cmml"><mi id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.2" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.2.cmml">A</mi><mo id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.1" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.2" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.1.cmml"><mo id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.5.4.m4.1.1" xref="S4.Thmtheorem4.p1.5.4.m4.1.1.cmml">G</mi><mo id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.2.2" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.5.4.m4.2.2" xref="S4.Thmtheorem4.p1.5.4.m4.2.2.cmml">n</mi><mo id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.2.3" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.5.4.m4.3.3" xref="S4.Thmtheorem4.p1.5.4.m4.3.3.cmml">k</mi><mo id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.2.4" stretchy="false" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.5.4.m4.3b"><apply id="S4.Thmtheorem4.p1.5.4.m4.3.4.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.4"><geq id="S4.Thmtheorem4.p1.5.4.m4.3.4.1.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.1"></geq><ci id="S4.Thmtheorem4.p1.5.4.m4.3.4.2.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.2">𝛼</ci><apply id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3"><times id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.1.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.1"></times><ci id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.2.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.2">𝐴</ci><vector id="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.1.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.4.3.3.2"><ci id="S4.Thmtheorem4.p1.5.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.1.1">𝐺</ci><ci id="S4.Thmtheorem4.p1.5.4.m4.2.2.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.2.2">𝑛</ci><ci id="S4.Thmtheorem4.p1.5.4.m4.3.3.cmml" xref="S4.Thmtheorem4.p1.5.4.m4.3.3">𝑘</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.5.4.m4.3c">\alpha\geq A(G,n,k)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.5.4.m4.3d">italic_α ≥ italic_A ( italic_G , italic_n , italic_k )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS1.SSSx1.p2"> <p class="ltx_p" id="S4.SS1.SSSx1.p2.4">We remark that since <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p2.1.m1.1"><semantics id="S4.SS1.SSSx1.p2.1.m1.1a"><mi id="S4.SS1.SSSx1.p2.1.m1.1.1" xref="S4.SS1.SSSx1.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p2.1.m1.1b"><ci id="S4.SS1.SSSx1.p2.1.m1.1.1.cmml" xref="S4.SS1.SSSx1.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p2.1.m1.1d">italic_k</annotation></semantics></math> has the same effect on <math alttext="F_{n}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p2.2.m2.1"><semantics id="S4.SS1.SSSx1.p2.2.m2.1a"><msub id="S4.SS1.SSSx1.p2.2.m2.1.1" xref="S4.SS1.SSSx1.p2.2.m2.1.1.cmml"><mi id="S4.SS1.SSSx1.p2.2.m2.1.1.2" xref="S4.SS1.SSSx1.p2.2.m2.1.1.2.cmml">F</mi><mi id="S4.SS1.SSSx1.p2.2.m2.1.1.3" xref="S4.SS1.SSSx1.p2.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p2.2.m2.1b"><apply id="S4.SS1.SSSx1.p2.2.m2.1.1.cmml" xref="S4.SS1.SSSx1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.SSSx1.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.SSSx1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.SSSx1.p2.2.m2.1.1.2">𝐹</ci><ci id="S4.SS1.SSSx1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.SSSx1.p2.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p2.2.m2.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p2.2.m2.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p2.3.m3.1"><semantics id="S4.SS1.SSSx1.p2.3.m3.1a"><mi id="S4.SS1.SSSx1.p2.3.m3.1.1" xref="S4.SS1.SSSx1.p2.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p2.3.m3.1b"><ci id="S4.SS1.SSSx1.p2.3.m3.1.1.cmml" xref="S4.SS1.SSSx1.p2.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p2.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p2.3.m3.1d">italic_α</annotation></semantics></math>, an analogous claim to the above holds as a function of <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p2.4.m4.1"><semantics id="S4.SS1.SSSx1.p2.4.m4.1a"><mi id="S4.SS1.SSSx1.p2.4.m4.1.1" xref="S4.SS1.SSSx1.p2.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p2.4.m4.1b"><ci id="S4.SS1.SSSx1.p2.4.m4.1.1.cmml" xref="S4.SS1.SSSx1.p2.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p2.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p2.4.m4.1d">italic_k</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.SSSx1.p3"> <p class="ltx_p" id="S4.SS1.SSSx1.p3.1">Next, we have a polynomial-time algorithm for computing <math alttext="r_{U}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p3.1.m1.1"><semantics id="S4.SS1.SSSx1.p3.1.m1.1a"><msub id="S4.SS1.SSSx1.p3.1.m1.1.1" xref="S4.SS1.SSSx1.p3.1.m1.1.1.cmml"><mi id="S4.SS1.SSSx1.p3.1.m1.1.1.2" xref="S4.SS1.SSSx1.p3.1.m1.1.1.2.cmml">r</mi><mi id="S4.SS1.SSSx1.p3.1.m1.1.1.3" xref="S4.SS1.SSSx1.p3.1.m1.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p3.1.m1.1b"><apply id="S4.SS1.SSSx1.p3.1.m1.1.1.cmml" xref="S4.SS1.SSSx1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p3.1.m1.1.1.1.cmml" xref="S4.SS1.SSSx1.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.SSSx1.p3.1.m1.1.1.2.cmml" xref="S4.SS1.SSSx1.p3.1.m1.1.1.2">𝑟</ci><ci id="S4.SS1.SSSx1.p3.1.m1.1.1.3.cmml" xref="S4.SS1.SSSx1.p3.1.m1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p3.1.m1.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p3.1.m1.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>, using mincost flow algorithms (see <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">ahuja1988network</span>]</cite> for reference).</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Corollary 1</span></h6> <div class="ltx_para" id="Thmcorollary1.p1"> <p class="ltx_p" id="Thmcorollary1.p1.3"><math alttext="r_{U}(G,n,k,\alpha)" class="ltx_Math" display="inline" id="Thmcorollary1.p1.1.m1.4"><semantics id="Thmcorollary1.p1.1.m1.4a"><mrow id="Thmcorollary1.p1.1.m1.4.5" xref="Thmcorollary1.p1.1.m1.4.5.cmml"><msub id="Thmcorollary1.p1.1.m1.4.5.2" xref="Thmcorollary1.p1.1.m1.4.5.2.cmml"><mi id="Thmcorollary1.p1.1.m1.4.5.2.2" xref="Thmcorollary1.p1.1.m1.4.5.2.2.cmml">r</mi><mi id="Thmcorollary1.p1.1.m1.4.5.2.3" xref="Thmcorollary1.p1.1.m1.4.5.2.3.cmml">U</mi></msub><mo id="Thmcorollary1.p1.1.m1.4.5.1" xref="Thmcorollary1.p1.1.m1.4.5.1.cmml">⁢</mo><mrow id="Thmcorollary1.p1.1.m1.4.5.3.2" xref="Thmcorollary1.p1.1.m1.4.5.3.1.cmml"><mo id="Thmcorollary1.p1.1.m1.4.5.3.2.1" stretchy="false" xref="Thmcorollary1.p1.1.m1.4.5.3.1.cmml">(</mo><mi id="Thmcorollary1.p1.1.m1.1.1" xref="Thmcorollary1.p1.1.m1.1.1.cmml">G</mi><mo id="Thmcorollary1.p1.1.m1.4.5.3.2.2" xref="Thmcorollary1.p1.1.m1.4.5.3.1.cmml">,</mo><mi id="Thmcorollary1.p1.1.m1.2.2" xref="Thmcorollary1.p1.1.m1.2.2.cmml">n</mi><mo id="Thmcorollary1.p1.1.m1.4.5.3.2.3" xref="Thmcorollary1.p1.1.m1.4.5.3.1.cmml">,</mo><mi id="Thmcorollary1.p1.1.m1.3.3" xref="Thmcorollary1.p1.1.m1.3.3.cmml">k</mi><mo id="Thmcorollary1.p1.1.m1.4.5.3.2.4" xref="Thmcorollary1.p1.1.m1.4.5.3.1.cmml">,</mo><mi id="Thmcorollary1.p1.1.m1.4.4" xref="Thmcorollary1.p1.1.m1.4.4.cmml">α</mi><mo id="Thmcorollary1.p1.1.m1.4.5.3.2.5" stretchy="false" xref="Thmcorollary1.p1.1.m1.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary1.p1.1.m1.4b"><apply id="Thmcorollary1.p1.1.m1.4.5.cmml" xref="Thmcorollary1.p1.1.m1.4.5"><times id="Thmcorollary1.p1.1.m1.4.5.1.cmml" xref="Thmcorollary1.p1.1.m1.4.5.1"></times><apply id="Thmcorollary1.p1.1.m1.4.5.2.cmml" xref="Thmcorollary1.p1.1.m1.4.5.2"><csymbol cd="ambiguous" id="Thmcorollary1.p1.1.m1.4.5.2.1.cmml" xref="Thmcorollary1.p1.1.m1.4.5.2">subscript</csymbol><ci id="Thmcorollary1.p1.1.m1.4.5.2.2.cmml" xref="Thmcorollary1.p1.1.m1.4.5.2.2">𝑟</ci><ci id="Thmcorollary1.p1.1.m1.4.5.2.3.cmml" xref="Thmcorollary1.p1.1.m1.4.5.2.3">𝑈</ci></apply><vector id="Thmcorollary1.p1.1.m1.4.5.3.1.cmml" xref="Thmcorollary1.p1.1.m1.4.5.3.2"><ci id="Thmcorollary1.p1.1.m1.1.1.cmml" xref="Thmcorollary1.p1.1.m1.1.1">𝐺</ci><ci id="Thmcorollary1.p1.1.m1.2.2.cmml" xref="Thmcorollary1.p1.1.m1.2.2">𝑛</ci><ci id="Thmcorollary1.p1.1.m1.3.3.cmml" xref="Thmcorollary1.p1.1.m1.3.3">𝑘</ci><ci id="Thmcorollary1.p1.1.m1.4.4.cmml" xref="Thmcorollary1.p1.1.m1.4.4">𝛼</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary1.p1.1.m1.4c">r_{U}(G,n,k,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary1.p1.1.m1.4d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Thmcorollary1.p1.3.2"> can be computed in time <math alttext="O\left(|E|^{1+o(1)}m_{c}\right)" class="ltx_Math" display="inline" id="Thmcorollary1.p1.2.1.m1.3"><semantics id="Thmcorollary1.p1.2.1.m1.3a"><mrow id="Thmcorollary1.p1.2.1.m1.3.3" xref="Thmcorollary1.p1.2.1.m1.3.3.cmml"><mi id="Thmcorollary1.p1.2.1.m1.3.3.3" xref="Thmcorollary1.p1.2.1.m1.3.3.3.cmml">O</mi><mo id="Thmcorollary1.p1.2.1.m1.3.3.2" xref="Thmcorollary1.p1.2.1.m1.3.3.2.cmml">⁢</mo><mrow id="Thmcorollary1.p1.2.1.m1.3.3.1.1" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.cmml"><mo id="Thmcorollary1.p1.2.1.m1.3.3.1.1.2" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.cmml">(</mo><mrow id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.cmml"><msup id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.cmml"><mrow id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.2" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.1.cmml"><mo id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.2.1" stretchy="false" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.1.1.cmml">|</mo><mi id="Thmcorollary1.p1.2.1.m1.2.2" xref="Thmcorollary1.p1.2.1.m1.2.2.cmml">E</mi><mo id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.2.2" stretchy="false" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.1.1.cmml">|</mo></mrow><mrow id="Thmcorollary1.p1.2.1.m1.1.1.1" xref="Thmcorollary1.p1.2.1.m1.1.1.1.cmml"><mn id="Thmcorollary1.p1.2.1.m1.1.1.1.3" xref="Thmcorollary1.p1.2.1.m1.1.1.1.3.cmml">1</mn><mo id="Thmcorollary1.p1.2.1.m1.1.1.1.2" xref="Thmcorollary1.p1.2.1.m1.1.1.1.2.cmml">+</mo><mrow id="Thmcorollary1.p1.2.1.m1.1.1.1.4" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.cmml"><mi id="Thmcorollary1.p1.2.1.m1.1.1.1.4.2" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.2.cmml">o</mi><mo id="Thmcorollary1.p1.2.1.m1.1.1.1.4.1" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="Thmcorollary1.p1.2.1.m1.1.1.1.4.3.2" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.cmml"><mo id="Thmcorollary1.p1.2.1.m1.1.1.1.4.3.2.1" stretchy="false" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.cmml">(</mo><mn id="Thmcorollary1.p1.2.1.m1.1.1.1.1" xref="Thmcorollary1.p1.2.1.m1.1.1.1.1.cmml">1</mn><mo id="Thmcorollary1.p1.2.1.m1.1.1.1.4.3.2.2" stretchy="false" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msup><mo id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.1" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.1.cmml">⁢</mo><msub id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.cmml"><mi id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.2" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.2.cmml">m</mi><mi id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.3" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.3.cmml">c</mi></msub></mrow><mo id="Thmcorollary1.p1.2.1.m1.3.3.1.1.3" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary1.p1.2.1.m1.3b"><apply id="Thmcorollary1.p1.2.1.m1.3.3.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3"><times id="Thmcorollary1.p1.2.1.m1.3.3.2.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.2"></times><ci id="Thmcorollary1.p1.2.1.m1.3.3.3.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.3">𝑂</ci><apply id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1"><times id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.1.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.1"></times><apply id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.1.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2">superscript</csymbol><apply id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.1.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.2"><abs id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.1.1.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.2.2.2.1"></abs><ci id="Thmcorollary1.p1.2.1.m1.2.2.cmml" xref="Thmcorollary1.p1.2.1.m1.2.2">𝐸</ci></apply><apply id="Thmcorollary1.p1.2.1.m1.1.1.1.cmml" xref="Thmcorollary1.p1.2.1.m1.1.1.1"><plus id="Thmcorollary1.p1.2.1.m1.1.1.1.2.cmml" xref="Thmcorollary1.p1.2.1.m1.1.1.1.2"></plus><cn id="Thmcorollary1.p1.2.1.m1.1.1.1.3.cmml" type="integer" xref="Thmcorollary1.p1.2.1.m1.1.1.1.3">1</cn><apply id="Thmcorollary1.p1.2.1.m1.1.1.1.4.cmml" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4"><times id="Thmcorollary1.p1.2.1.m1.1.1.1.4.1.cmml" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.1"></times><ci id="Thmcorollary1.p1.2.1.m1.1.1.1.4.2.cmml" xref="Thmcorollary1.p1.2.1.m1.1.1.1.4.2">𝑜</ci><cn id="Thmcorollary1.p1.2.1.m1.1.1.1.1.cmml" type="integer" xref="Thmcorollary1.p1.2.1.m1.1.1.1.1">1</cn></apply></apply></apply><apply id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.1.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3">subscript</csymbol><ci id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.2.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.2">𝑚</ci><ci id="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.3.cmml" xref="Thmcorollary1.p1.2.1.m1.3.3.1.1.1.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary1.p1.2.1.m1.3c">O\left(|E|^{1+o(1)}m_{c}\right)</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary1.p1.2.1.m1.3d">italic_O ( | italic_E | start_POSTSUPERSCRIPT 1 + italic_o ( 1 ) end_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math> whp, and can be computed in time <math alttext="O\left(|E|m_{c}|V|\log|V|\right)" class="ltx_Math" display="inline" id="Thmcorollary1.p1.3.2.m2.4"><semantics id="Thmcorollary1.p1.3.2.m2.4a"><mrow id="Thmcorollary1.p1.3.2.m2.4.4" xref="Thmcorollary1.p1.3.2.m2.4.4.cmml"><mi id="Thmcorollary1.p1.3.2.m2.4.4.3" xref="Thmcorollary1.p1.3.2.m2.4.4.3.cmml">O</mi><mo id="Thmcorollary1.p1.3.2.m2.4.4.2" xref="Thmcorollary1.p1.3.2.m2.4.4.2.cmml">⁢</mo><mrow id="Thmcorollary1.p1.3.2.m2.4.4.1.1" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.cmml"><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.2" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.cmml">(</mo><mrow id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.cmml"><mrow id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.2" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.1.cmml"><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.2.1" stretchy="false" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.1.1.cmml">|</mo><mi id="Thmcorollary1.p1.3.2.m2.1.1" xref="Thmcorollary1.p1.3.2.m2.1.1.cmml">E</mi><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.2.2" stretchy="false" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.1.1.cmml">|</mo></mrow><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1.cmml">⁢</mo><msub id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.cmml"><mi id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.2" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.2.cmml">m</mi><mi id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.3" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.3.cmml">c</mi></msub><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1a" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1.cmml">⁢</mo><mrow id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.2" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.1.cmml"><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.2.1" stretchy="false" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.1.1.cmml">|</mo><mi id="Thmcorollary1.p1.3.2.m2.2.2" xref="Thmcorollary1.p1.3.2.m2.2.2.cmml">V</mi><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.2.2" stretchy="false" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.1.1.cmml">|</mo></mrow><mo id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1b" lspace="0.167em" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1.cmml">⁢</mo><mrow id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.cmml"><mi 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xref="Thmcorollary1.p1.3.2.m2.4.4.2"></times><ci id="Thmcorollary1.p1.3.2.m2.4.4.3.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.3">𝑂</ci><apply id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1"><times id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.1"></times><apply id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.2"><abs id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.1.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.2.2.1"></abs><ci id="Thmcorollary1.p1.3.2.m2.1.1.cmml" xref="Thmcorollary1.p1.3.2.m2.1.1">𝐸</ci></apply><apply id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3"><csymbol cd="ambiguous" id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3">subscript</csymbol><ci id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.2.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.2">𝑚</ci><ci id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.3.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.3.3">𝑐</ci></apply><apply id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.2"><abs id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.1.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.4.2.1"></abs><ci id="Thmcorollary1.p1.3.2.m2.2.2.cmml" xref="Thmcorollary1.p1.3.2.m2.2.2">𝑉</ci></apply><apply id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5"><log id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.1"></log><apply id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.2.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.2.2"><abs id="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.2.1.1.cmml" xref="Thmcorollary1.p1.3.2.m2.4.4.1.1.1.5.2.2.1"></abs><ci id="Thmcorollary1.p1.3.2.m2.3.3.cmml" xref="Thmcorollary1.p1.3.2.m2.3.3">𝑉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary1.p1.3.2.m2.4c">O\left(|E|m_{c}|V|\log|V|\right)</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary1.p1.3.2.m2.4d">italic_O ( | italic_E | italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT | italic_V | roman_log | italic_V | )</annotation></semantics></math> deterministically.</span></p> </div> </div> <div class="ltx_para" id="S4.SS1.SSSx1.p4"> <p class="ltx_p" id="S4.SS1.SSSx1.p4.4">We also find bounds on the minimal <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p4.1.m1.1"><semantics id="S4.SS1.SSSx1.p4.1.m1.1a"><mi id="S4.SS1.SSSx1.p4.1.m1.1.1" xref="S4.SS1.SSSx1.p4.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p4.1.m1.1b"><ci id="S4.SS1.SSSx1.p4.1.m1.1.1.cmml" xref="S4.SS1.SSSx1.p4.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p4.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p4.1.m1.1d">italic_α</annotation></semantics></math> for which <math alttext="r_{U}(G,n,k,\alpha)=\mathcal{E}_{m_{c},n}" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p4.2.m2.6"><semantics id="S4.SS1.SSSx1.p4.2.m2.6a"><mrow id="S4.SS1.SSSx1.p4.2.m2.6.7" xref="S4.SS1.SSSx1.p4.2.m2.6.7.cmml"><mrow id="S4.SS1.SSSx1.p4.2.m2.6.7.2" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.cmml"><msub id="S4.SS1.SSSx1.p4.2.m2.6.7.2.2" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.cmml"><mi id="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.2" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.2.cmml">r</mi><mi id="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.3" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.3.cmml">U</mi></msub><mo id="S4.SS1.SSSx1.p4.2.m2.6.7.2.1" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.2" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.1.cmml"><mo id="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.2.1" stretchy="false" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.1.cmml">(</mo><mi id="S4.SS1.SSSx1.p4.2.m2.3.3" xref="S4.SS1.SSSx1.p4.2.m2.3.3.cmml">G</mi><mo id="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.2.2" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S4.SS1.SSSx1.p4.2.m2.4.4" xref="S4.SS1.SSSx1.p4.2.m2.4.4.cmml">n</mi><mo id="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.2.3" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S4.SS1.SSSx1.p4.2.m2.5.5" xref="S4.SS1.SSSx1.p4.2.m2.5.5.cmml">k</mi><mo id="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.2.4" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S4.SS1.SSSx1.p4.2.m2.6.6" xref="S4.SS1.SSSx1.p4.2.m2.6.6.cmml">α</mi><mo id="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.2.5" stretchy="false" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSSx1.p4.2.m2.6.7.1" xref="S4.SS1.SSSx1.p4.2.m2.6.7.1.cmml">=</mo><msub id="S4.SS1.SSSx1.p4.2.m2.6.7.3" xref="S4.SS1.SSSx1.p4.2.m2.6.7.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSSx1.p4.2.m2.6.7.3.2" xref="S4.SS1.SSSx1.p4.2.m2.6.7.3.2.cmml">ℰ</mi><mrow id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.3.cmml"><msub id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.cmml"><mi id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.2" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.2.cmml">m</mi><mi id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.3" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.3.cmml">c</mi></msub><mo id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.2" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.SSSx1.p4.2.m2.1.1.1.1" xref="S4.SS1.SSSx1.p4.2.m2.1.1.1.1.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p4.2.m2.6b"><apply id="S4.SS1.SSSx1.p4.2.m2.6.7.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7"><eq id="S4.SS1.SSSx1.p4.2.m2.6.7.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.1"></eq><apply id="S4.SS1.SSSx1.p4.2.m2.6.7.2.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2"><times id="S4.SS1.SSSx1.p4.2.m2.6.7.2.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.1"></times><apply id="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.2">subscript</csymbol><ci id="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.2.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.2">𝑟</ci><ci id="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.3.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.2.3">𝑈</ci></apply><vector id="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.2.3.2"><ci id="S4.SS1.SSSx1.p4.2.m2.3.3.cmml" xref="S4.SS1.SSSx1.p4.2.m2.3.3">𝐺</ci><ci id="S4.SS1.SSSx1.p4.2.m2.4.4.cmml" xref="S4.SS1.SSSx1.p4.2.m2.4.4">𝑛</ci><ci id="S4.SS1.SSSx1.p4.2.m2.5.5.cmml" xref="S4.SS1.SSSx1.p4.2.m2.5.5">𝑘</ci><ci id="S4.SS1.SSSx1.p4.2.m2.6.6.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.6">𝛼</ci></vector></apply><apply id="S4.SS1.SSSx1.p4.2.m2.6.7.3.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.3"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p4.2.m2.6.7.3.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.3">subscript</csymbol><ci id="S4.SS1.SSSx1.p4.2.m2.6.7.3.2.cmml" xref="S4.SS1.SSSx1.p4.2.m2.6.7.3.2">ℰ</ci><list id="S4.SS1.SSSx1.p4.2.m2.2.2.2.3.cmml" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2"><apply id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.2.cmml" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.2">𝑚</ci><ci id="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.3.cmml" xref="S4.SS1.SSSx1.p4.2.m2.2.2.2.2.1.3">𝑐</ci></apply><ci id="S4.SS1.SSSx1.p4.2.m2.1.1.1.1.cmml" xref="S4.SS1.SSSx1.p4.2.m2.1.1.1.1">𝑛</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p4.2.m2.6c">r_{U}(G,n,k,\alpha)=\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p4.2.m2.6d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α ) = caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. 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xref="S4.SS1.SSSx1.p4.3.m3.2.2.1.1">subscript</csymbol><ci id="S4.SS1.SSSx1.p4.3.m3.2.2.1.1.2.cmml" xref="S4.SS1.SSSx1.p4.3.m3.2.2.1.1.2">𝑚</ci><ci id="S4.SS1.SSSx1.p4.3.m3.2.2.1.1.3.cmml" xref="S4.SS1.SSSx1.p4.3.m3.2.2.1.1.3">𝑐</ci></apply><ci id="S4.SS1.SSSx1.p4.3.m3.1.1.cmml" xref="S4.SS1.SSSx1.p4.3.m3.1.1">𝑛</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSSx1.p4.3.m3.2c">m_{c},n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSSx1.p4.3.m3.2d">italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n</annotation></semantics></math> and the paths lengths in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.SSSx1.p4.4.m4.1"><semantics id="S4.SS1.SSSx1.p4.4.m4.1a"><mi id="S4.SS1.SSSx1.p4.4.m4.1.1" xref="S4.SS1.SSSx1.p4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSSx1.p4.4.m4.1b"><ci id="S4.SS1.SSSx1.p4.4.m4.1.1.cmml" 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xref="S4.Thmtheorem5.p1.1.m1.2.2.2.4">subscript</csymbol><ci id="S4.Thmtheorem5.p1.1.m1.2.2.2.4.2.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.4.2">𝑆</ci><apply id="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3.1.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3.2.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3.2">𝑚</ci><ci id="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3.3.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.4.3.3">𝑐</ci></apply></apply></apply><ci id="S4.Thmtheorem5.p1.1.m1.2.2.4.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.m1.5c">A(G,n,k)\geq\frac{e^{\Theta(n/m_{c})}\left(S_{m_{c}}-S_{m_{c}-1}\right)-S_{m_{% c}}}{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.m1.5d">italic_A ( italic_G , italic_n , italic_k ) ≥ divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Θ ( italic_n / italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.1.1">.</span></p> </div> </div> </section> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Disjoint paths</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.7">As stated earlier, our main result for the SUM cost and disjoint paths is that <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">𝒰</mi><mi id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝒰</ci><ci id="S4.SS2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> always admits an optimal agents strategy. Hence <math alttext="r=r_{U}" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><mrow id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2" xref="S4.SS2.p1.2.m2.1.1.2.cmml">r</mi><mo id="S4.SS2.p1.2.m2.1.1.1" xref="S4.SS2.p1.2.m2.1.1.1.cmml">=</mo><msub id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml"><mi id="S4.SS2.p1.2.m2.1.1.3.2" xref="S4.SS2.p1.2.m2.1.1.3.2.cmml">r</mi><mi id="S4.SS2.p1.2.m2.1.1.3.3" xref="S4.SS2.p1.2.m2.1.1.3.3.cmml">U</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><eq id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1.1"></eq><ci id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2">𝑟</ci><apply id="S4.SS2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.p1.2.m2.1.1.3.2">𝑟</ci><ci id="S4.SS2.p1.2.m2.1.1.3.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">r=r_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_r = italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> in these instances. Intuitively, for fixed <math alttext="G,k,\alpha" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.3"><semantics id="S4.SS2.p1.3.m3.3a"><mrow id="S4.SS2.p1.3.m3.3.4.2" xref="S4.SS2.p1.3.m3.3.4.1.cmml"><mi id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml">G</mi><mo id="S4.SS2.p1.3.m3.3.4.2.1" xref="S4.SS2.p1.3.m3.3.4.1.cmml">,</mo><mi id="S4.SS2.p1.3.m3.2.2" xref="S4.SS2.p1.3.m3.2.2.cmml">k</mi><mo id="S4.SS2.p1.3.m3.3.4.2.2" xref="S4.SS2.p1.3.m3.3.4.1.cmml">,</mo><mi id="S4.SS2.p1.3.m3.3.3" xref="S4.SS2.p1.3.m3.3.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.3b"><list id="S4.SS2.p1.3.m3.3.4.1.cmml" xref="S4.SS2.p1.3.m3.3.4.2"><ci id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">𝐺</ci><ci id="S4.SS2.p1.3.m3.2.2.cmml" xref="S4.SS2.p1.3.m3.2.2">𝑘</ci><ci id="S4.SS2.p1.3.m3.3.3.cmml" xref="S4.SS2.p1.3.m3.3.3">𝛼</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.3c">G,k,\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.3d">italic_G , italic_k , italic_α</annotation></semantics></math>, some paths in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><mi id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><ci id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.1d">italic_G</annotation></semantics></math> are too expensive for the agents to consider – they would rather incur <math alttext="k\alpha" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.1"><semantics id="S4.SS2.p1.5.m5.1a"><mrow id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml"><mi id="S4.SS2.p1.5.m5.1.1.2" xref="S4.SS2.p1.5.m5.1.1.2.cmml">k</mi><mo id="S4.SS2.p1.5.m5.1.1.1" xref="S4.SS2.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p1.5.m5.1.1.3" xref="S4.SS2.p1.5.m5.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.1b"><apply id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1"><times id="S4.SS2.p1.5.m5.1.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1.1"></times><ci id="S4.SS2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.p1.5.m5.1.1.2">𝑘</ci><ci id="S4.SS2.p1.5.m5.1.1.3.cmml" xref="S4.SS2.p1.5.m5.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">k\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">italic_k italic_α</annotation></semantics></math> additional cost than pick these long paths, so they ignore them. Then, <math alttext="k\alpha" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m6.1"><semantics id="S4.SS2.p1.6.m6.1a"><mrow id="S4.SS2.p1.6.m6.1.1" xref="S4.SS2.p1.6.m6.1.1.cmml"><mi id="S4.SS2.p1.6.m6.1.1.2" xref="S4.SS2.p1.6.m6.1.1.2.cmml">k</mi><mo id="S4.SS2.p1.6.m6.1.1.1" xref="S4.SS2.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p1.6.m6.1.1.3" xref="S4.SS2.p1.6.m6.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m6.1b"><apply id="S4.SS2.p1.6.m6.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1"><times id="S4.SS2.p1.6.m6.1.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1.1"></times><ci id="S4.SS2.p1.6.m6.1.1.2.cmml" xref="S4.SS2.p1.6.m6.1.1.2">𝑘</ci><ci id="S4.SS2.p1.6.m6.1.1.3.cmml" xref="S4.SS2.p1.6.m6.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m6.1c">k\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m6.1d">italic_k italic_α</annotation></semantics></math> becomes very large, relative to the remaining paths. Hence, the agents will pick a joint strategy that minimize the probability that agents collide with interceptors, and <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S4.SS2.p1.7.m7.1"><semantics id="S4.SS2.p1.7.m7.1a"><msub id="S4.SS2.p1.7.m7.1.1" xref="S4.SS2.p1.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p1.7.m7.1.1.2" xref="S4.SS2.p1.7.m7.1.1.2.cmml">𝒰</mi><mi id="S4.SS2.p1.7.m7.1.1.3" xref="S4.SS2.p1.7.m7.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m7.1b"><apply id="S4.SS2.p1.7.m7.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m7.1.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.p1.7.m7.1.1.2">𝒰</ci><ci id="S4.SS2.p1.7.m7.1.1.3.cmml" xref="S4.SS2.p1.7.m7.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m7.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.7.m7.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> offers one.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">To prove the result, we reduce the strategy space that we need to consider. Next we will establish Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma11" title="Lemma 11 ‣ 4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">11</span></a> in two steps. The first reduces the strategy space to distributions whose mass on paths decreases as the path gets longer.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma9"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 9</span></h6> <div class="ltx_para" id="Thmlemma9.p1"> <p class="ltx_p" id="Thmlemma9.p1.4"><span class="ltx_text ltx_font_italic" id="Thmlemma9.p1.4.4">When minimizing <math alttext="F_{n}(\cdot,\alpha)" class="ltx_Math" display="inline" id="Thmlemma9.p1.1.1.m1.2"><semantics id="Thmlemma9.p1.1.1.m1.2a"><mrow id="Thmlemma9.p1.1.1.m1.2.3" xref="Thmlemma9.p1.1.1.m1.2.3.cmml"><msub id="Thmlemma9.p1.1.1.m1.2.3.2" xref="Thmlemma9.p1.1.1.m1.2.3.2.cmml"><mi id="Thmlemma9.p1.1.1.m1.2.3.2.2" xref="Thmlemma9.p1.1.1.m1.2.3.2.2.cmml">F</mi><mi id="Thmlemma9.p1.1.1.m1.2.3.2.3" xref="Thmlemma9.p1.1.1.m1.2.3.2.3.cmml">n</mi></msub><mo id="Thmlemma9.p1.1.1.m1.2.3.1" xref="Thmlemma9.p1.1.1.m1.2.3.1.cmml">⁢</mo><mrow 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id="Thmlemma9.p1.1.1.m1.2.3.2.2.cmml" xref="Thmlemma9.p1.1.1.m1.2.3.2.2">𝐹</ci><ci id="Thmlemma9.p1.1.1.m1.2.3.2.3.cmml" xref="Thmlemma9.p1.1.1.m1.2.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmlemma9.p1.1.1.m1.2.3.3.1.cmml" xref="Thmlemma9.p1.1.1.m1.2.3.3.2"><ci id="Thmlemma9.p1.1.1.m1.1.1.cmml" xref="Thmlemma9.p1.1.1.m1.1.1">⋅</ci><ci id="Thmlemma9.p1.1.1.m1.2.2.cmml" xref="Thmlemma9.p1.1.1.m1.2.2">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma9.p1.1.1.m1.2c">F_{n}(\cdot,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma9.p1.1.1.m1.2d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ⋅ , italic_α )</annotation></semantics></math> we can restrict ourselves to the set of decreasing distributions: all <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmlemma9.p1.2.2.m2.1"><semantics id="Thmlemma9.p1.2.2.m2.1a"><mi id="Thmlemma9.p1.2.2.m2.1.1" xref="Thmlemma9.p1.2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmlemma9.p1.2.2.m2.1b"><ci id="Thmlemma9.p1.2.2.m2.1.1.cmml" xref="Thmlemma9.p1.2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma9.p1.2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmlemma9.p1.2.2.m2.1d">italic_σ</annotation></semantics></math> such that for <math alttext="i<i^{\prime}" class="ltx_Math" display="inline" id="Thmlemma9.p1.3.3.m3.1"><semantics id="Thmlemma9.p1.3.3.m3.1a"><mrow id="Thmlemma9.p1.3.3.m3.1.1" xref="Thmlemma9.p1.3.3.m3.1.1.cmml"><mi id="Thmlemma9.p1.3.3.m3.1.1.2" xref="Thmlemma9.p1.3.3.m3.1.1.2.cmml">i</mi><mo id="Thmlemma9.p1.3.3.m3.1.1.1" xref="Thmlemma9.p1.3.3.m3.1.1.1.cmml"><</mo><msup id="Thmlemma9.p1.3.3.m3.1.1.3" xref="Thmlemma9.p1.3.3.m3.1.1.3.cmml"><mi id="Thmlemma9.p1.3.3.m3.1.1.3.2" xref="Thmlemma9.p1.3.3.m3.1.1.3.2.cmml">i</mi><mo id="Thmlemma9.p1.3.3.m3.1.1.3.3" 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end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\sigma(p_{i})\geq\sigma(p_{i^{\prime}})" class="ltx_Math" display="inline" id="Thmlemma9.p1.4.4.m4.2"><semantics id="Thmlemma9.p1.4.4.m4.2a"><mrow id="Thmlemma9.p1.4.4.m4.2.2" xref="Thmlemma9.p1.4.4.m4.2.2.cmml"><mrow id="Thmlemma9.p1.4.4.m4.1.1.1" xref="Thmlemma9.p1.4.4.m4.1.1.1.cmml"><mi id="Thmlemma9.p1.4.4.m4.1.1.1.3" xref="Thmlemma9.p1.4.4.m4.1.1.1.3.cmml">σ</mi><mo id="Thmlemma9.p1.4.4.m4.1.1.1.2" xref="Thmlemma9.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma9.p1.4.4.m4.1.1.1.1.1" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.2" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.3" 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id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.2" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.3" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.3" stretchy="false" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma9.p1.4.4.m4.2b"><apply id="Thmlemma9.p1.4.4.m4.2.2.cmml" xref="Thmlemma9.p1.4.4.m4.2.2"><geq id="Thmlemma9.p1.4.4.m4.2.2.3.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.3"></geq><apply id="Thmlemma9.p1.4.4.m4.1.1.1.cmml" xref="Thmlemma9.p1.4.4.m4.1.1.1"><times id="Thmlemma9.p1.4.4.m4.1.1.1.2.cmml" xref="Thmlemma9.p1.4.4.m4.1.1.1.2"></times><ci id="Thmlemma9.p1.4.4.m4.1.1.1.3.cmml" xref="Thmlemma9.p1.4.4.m4.1.1.1.3">𝜎</ci><apply id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmlemma9.p1.4.4.m4.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmlemma9.p1.4.4.m4.2.2.2.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2"><times id="Thmlemma9.p1.4.4.m4.2.2.2.2.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.2"></times><ci id="Thmlemma9.p1.4.4.m4.2.2.2.3.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.3">𝜎</ci><apply id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.1.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1">subscript</csymbol><ci id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.2.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.2">𝑝</ci><apply id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.1.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3">superscript</csymbol><ci id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.2.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.2">𝑖</ci><ci id="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.3.cmml" xref="Thmlemma9.p1.4.4.m4.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma9.p1.4.4.m4.2c">\sigma(p_{i})\geq\sigma(p_{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma9.p1.4.4.m4.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ italic_σ ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.2">For such <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mi id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><ci id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_σ</annotation></semantics></math>, <math alttext="F_{n}(\sigma,\alpha)=\sum_{p_{i}\in P_{st}}\sigma(p)+\alpha\sigma(p_{1})" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.4"><semantics id="S4.SS2.p3.2.m2.4a"><mrow id="S4.SS2.p3.2.m2.4.4" xref="S4.SS2.p3.2.m2.4.4.cmml"><mrow id="S4.SS2.p3.2.m2.4.4.3" xref="S4.SS2.p3.2.m2.4.4.3.cmml"><msub id="S4.SS2.p3.2.m2.4.4.3.2" xref="S4.SS2.p3.2.m2.4.4.3.2.cmml"><mi id="S4.SS2.p3.2.m2.4.4.3.2.2" xref="S4.SS2.p3.2.m2.4.4.3.2.2.cmml">F</mi><mi id="S4.SS2.p3.2.m2.4.4.3.2.3" xref="S4.SS2.p3.2.m2.4.4.3.2.3.cmml">n</mi></msub><mo id="S4.SS2.p3.2.m2.4.4.3.1" xref="S4.SS2.p3.2.m2.4.4.3.1.cmml">⁢</mo><mrow 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Our second reduction is to iid agent distributions.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma10"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 10</span></h6> <div class="ltx_para" id="Thmlemma10.p1"> <p class="ltx_p" id="Thmlemma10.p1.4"><span class="ltx_text ltx_font_italic" id="Thmlemma10.p1.4.4">When minimizing <math alttext="F_{n}(\cdot,\alpha)" class="ltx_Math" display="inline" id="Thmlemma10.p1.1.1.m1.2"><semantics id="Thmlemma10.p1.1.1.m1.2a"><mrow id="Thmlemma10.p1.1.1.m1.2.3" xref="Thmlemma10.p1.1.1.m1.2.3.cmml"><msub id="Thmlemma10.p1.1.1.m1.2.3.2" xref="Thmlemma10.p1.1.1.m1.2.3.2.cmml"><mi id="Thmlemma10.p1.1.1.m1.2.3.2.2" xref="Thmlemma10.p1.1.1.m1.2.3.2.2.cmml">F</mi><mi id="Thmlemma10.p1.1.1.m1.2.3.2.3" xref="Thmlemma10.p1.1.1.m1.2.3.2.3.cmml">n</mi></msub><mo id="Thmlemma10.p1.1.1.m1.2.3.1" xref="Thmlemma10.p1.1.1.m1.2.3.1.cmml">⁢</mo><mrow id="Thmlemma10.p1.1.1.m1.2.3.3.2" xref="Thmlemma10.p1.1.1.m1.2.3.3.1.cmml"><mo id="Thmlemma10.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="Thmlemma10.p1.1.1.m1.2.3.3.1.cmml">(</mo><mo id="Thmlemma10.p1.1.1.m1.1.1" lspace="0em" rspace="0em" xref="Thmlemma10.p1.1.1.m1.1.1.cmml">⋅</mo><mo id="Thmlemma10.p1.1.1.m1.2.3.3.2.2" xref="Thmlemma10.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="Thmlemma10.p1.1.1.m1.2.2" xref="Thmlemma10.p1.1.1.m1.2.2.cmml">α</mi><mo id="Thmlemma10.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="Thmlemma10.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma10.p1.1.1.m1.2b"><apply id="Thmlemma10.p1.1.1.m1.2.3.cmml" xref="Thmlemma10.p1.1.1.m1.2.3"><times id="Thmlemma10.p1.1.1.m1.2.3.1.cmml" xref="Thmlemma10.p1.1.1.m1.2.3.1"></times><apply id="Thmlemma10.p1.1.1.m1.2.3.2.cmml" xref="Thmlemma10.p1.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="Thmlemma10.p1.1.1.m1.2.3.2.1.cmml" xref="Thmlemma10.p1.1.1.m1.2.3.2">subscript</csymbol><ci id="Thmlemma10.p1.1.1.m1.2.3.2.2.cmml" xref="Thmlemma10.p1.1.1.m1.2.3.2.2">𝐹</ci><ci id="Thmlemma10.p1.1.1.m1.2.3.2.3.cmml" xref="Thmlemma10.p1.1.1.m1.2.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmlemma10.p1.1.1.m1.2.3.3.1.cmml" xref="Thmlemma10.p1.1.1.m1.2.3.3.2"><ci id="Thmlemma10.p1.1.1.m1.1.1.cmml" xref="Thmlemma10.p1.1.1.m1.1.1">⋅</ci><ci id="Thmlemma10.p1.1.1.m1.2.2.cmml" xref="Thmlemma10.p1.1.1.m1.2.2">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma10.p1.1.1.m1.2c">F_{n}(\cdot,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma10.p1.1.1.m1.2d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ⋅ , italic_α )</annotation></semantics></math> we can restrict ourselves to the set of iid distributions: all <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmlemma10.p1.2.2.m2.1"><semantics id="Thmlemma10.p1.2.2.m2.1a"><mi id="Thmlemma10.p1.2.2.m2.1.1" xref="Thmlemma10.p1.2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmlemma10.p1.2.2.m2.1b"><ci id="Thmlemma10.p1.2.2.m2.1.1.cmml" xref="Thmlemma10.p1.2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma10.p1.2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmlemma10.p1.2.2.m2.1d">italic_σ</annotation></semantics></math> which are the product distribution of <math alttext="n" class="ltx_Math" display="inline" id="Thmlemma10.p1.3.3.m3.1"><semantics id="Thmlemma10.p1.3.3.m3.1a"><mi id="Thmlemma10.p1.3.3.m3.1.1" xref="Thmlemma10.p1.3.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmlemma10.p1.3.3.m3.1b"><ci id="Thmlemma10.p1.3.3.m3.1.1.cmml" xref="Thmlemma10.p1.3.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma10.p1.3.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmlemma10.p1.3.3.m3.1d">italic_n</annotation></semantics></math> copies of a distribution on <math alttext="P_{st}" class="ltx_Math" display="inline" id="Thmlemma10.p1.4.4.m4.1"><semantics id="Thmlemma10.p1.4.4.m4.1a"><msub id="Thmlemma10.p1.4.4.m4.1.1" xref="Thmlemma10.p1.4.4.m4.1.1.cmml"><mi id="Thmlemma10.p1.4.4.m4.1.1.2" xref="Thmlemma10.p1.4.4.m4.1.1.2.cmml">P</mi><mrow id="Thmlemma10.p1.4.4.m4.1.1.3" xref="Thmlemma10.p1.4.4.m4.1.1.3.cmml"><mi id="Thmlemma10.p1.4.4.m4.1.1.3.2" xref="Thmlemma10.p1.4.4.m4.1.1.3.2.cmml">s</mi><mo id="Thmlemma10.p1.4.4.m4.1.1.3.1" xref="Thmlemma10.p1.4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="Thmlemma10.p1.4.4.m4.1.1.3.3" xref="Thmlemma10.p1.4.4.m4.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmlemma10.p1.4.4.m4.1b"><apply id="Thmlemma10.p1.4.4.m4.1.1.cmml" xref="Thmlemma10.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="Thmlemma10.p1.4.4.m4.1.1.1.cmml" xref="Thmlemma10.p1.4.4.m4.1.1">subscript</csymbol><ci id="Thmlemma10.p1.4.4.m4.1.1.2.cmml" xref="Thmlemma10.p1.4.4.m4.1.1.2">𝑃</ci><apply id="Thmlemma10.p1.4.4.m4.1.1.3.cmml" xref="Thmlemma10.p1.4.4.m4.1.1.3"><times id="Thmlemma10.p1.4.4.m4.1.1.3.1.cmml" xref="Thmlemma10.p1.4.4.m4.1.1.3.1"></times><ci id="Thmlemma10.p1.4.4.m4.1.1.3.2.cmml" xref="Thmlemma10.p1.4.4.m4.1.1.3.2">𝑠</ci><ci id="Thmlemma10.p1.4.4.m4.1.1.3.3.cmml" xref="Thmlemma10.p1.4.4.m4.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma10.p1.4.4.m4.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma10.p1.4.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.1">These two lemmas and Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma7" title="Lemma 7 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">7</span></a> imply the following lemma (which was discussed in Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS2" title="1.2 Our techniques ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1.2</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma11"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 11</span></h6> <div class="ltx_para" id="Thmlemma11.p1"> <p class="ltx_p" id="Thmlemma11.p1.1"><math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)=\min_{\sigma\in% \operatorname{conv}(\mathcal{U}_{n})}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Thmlemma11.p1.1.m1.7"><semantics id="Thmlemma11.p1.1.m1.7a"><mrow id="Thmlemma11.p1.1.m1.7.8" xref="Thmlemma11.p1.1.m1.7.8.cmml"><mrow id="Thmlemma11.p1.1.m1.7.8.2" xref="Thmlemma11.p1.1.m1.7.8.2.cmml"><mrow id="Thmlemma11.p1.1.m1.7.8.2.2" 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xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.2" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmlemma11.p1.1.m1.1.1.1.1.3" xref="Thmlemma11.p1.1.m1.1.1.1.1.3.cmml">n</mi></msup></mrow></msub><mo id="Thmlemma11.p1.1.m1.7.8.2.2a" lspace="0.167em" xref="Thmlemma11.p1.1.m1.7.8.2.2.cmml">⁡</mo><msub id="Thmlemma11.p1.1.m1.7.8.2.2.2" xref="Thmlemma11.p1.1.m1.7.8.2.2.2.cmml"><mi id="Thmlemma11.p1.1.m1.7.8.2.2.2.2" xref="Thmlemma11.p1.1.m1.7.8.2.2.2.2.cmml">F</mi><mi id="Thmlemma11.p1.1.m1.7.8.2.2.2.3" xref="Thmlemma11.p1.1.m1.7.8.2.2.2.3.cmml">n</mi></msub></mrow><mo id="Thmlemma11.p1.1.m1.7.8.2.1" xref="Thmlemma11.p1.1.m1.7.8.2.1.cmml">⁢</mo><mrow id="Thmlemma11.p1.1.m1.7.8.2.3.2" xref="Thmlemma11.p1.1.m1.7.8.2.3.1.cmml"><mo id="Thmlemma11.p1.1.m1.7.8.2.3.2.1" stretchy="false" xref="Thmlemma11.p1.1.m1.7.8.2.3.1.cmml">(</mo><mi id="Thmlemma11.p1.1.m1.4.4" xref="Thmlemma11.p1.1.m1.4.4.cmml">σ</mi><mo id="Thmlemma11.p1.1.m1.7.8.2.3.2.2" xref="Thmlemma11.p1.1.m1.7.8.2.3.1.cmml">,</mo><mi id="Thmlemma11.p1.1.m1.5.5" xref="Thmlemma11.p1.1.m1.5.5.cmml">α</mi><mo id="Thmlemma11.p1.1.m1.7.8.2.3.2.3" stretchy="false" xref="Thmlemma11.p1.1.m1.7.8.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma11.p1.1.m1.7.8.1" xref="Thmlemma11.p1.1.m1.7.8.1.cmml">=</mo><mrow id="Thmlemma11.p1.1.m1.7.8.3" xref="Thmlemma11.p1.1.m1.7.8.3.cmml"><mrow id="Thmlemma11.p1.1.m1.7.8.3.2" xref="Thmlemma11.p1.1.m1.7.8.3.2.cmml"><msub id="Thmlemma11.p1.1.m1.7.8.3.2.1" xref="Thmlemma11.p1.1.m1.7.8.3.2.1.cmml"><mi id="Thmlemma11.p1.1.m1.7.8.3.2.1.2" xref="Thmlemma11.p1.1.m1.7.8.3.2.1.2.cmml">min</mi><mrow id="Thmlemma11.p1.1.m1.3.3.2" xref="Thmlemma11.p1.1.m1.3.3.2.cmml"><mi id="Thmlemma11.p1.1.m1.3.3.2.4" xref="Thmlemma11.p1.1.m1.3.3.2.4.cmml">σ</mi><mo id="Thmlemma11.p1.1.m1.3.3.2.3" xref="Thmlemma11.p1.1.m1.3.3.2.3.cmml">∈</mo><mrow id="Thmlemma11.p1.1.m1.3.3.2.2.1" xref="Thmlemma11.p1.1.m1.3.3.2.2.2.cmml"><mi id="Thmlemma11.p1.1.m1.2.2.1.1" xref="Thmlemma11.p1.1.m1.2.2.1.1.cmml">conv</mi><mo id="Thmlemma11.p1.1.m1.3.3.2.2.1a" xref="Thmlemma11.p1.1.m1.3.3.2.2.2.cmml">⁡</mo><mrow id="Thmlemma11.p1.1.m1.3.3.2.2.1.1" xref="Thmlemma11.p1.1.m1.3.3.2.2.2.cmml"><mo id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.2" stretchy="false" xref="Thmlemma11.p1.1.m1.3.3.2.2.2.cmml">(</mo><msub id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1" xref="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.2" xref="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.2.cmml">𝒰</mi><mi id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.3" xref="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.3.cmml">n</mi></msub><mo id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.3" stretchy="false" xref="Thmlemma11.p1.1.m1.3.3.2.2.2.cmml">)</mo></mrow></mrow></mrow></msub><mo id="Thmlemma11.p1.1.m1.7.8.3.2a" lspace="0.167em" xref="Thmlemma11.p1.1.m1.7.8.3.2.cmml">⁡</mo><msub id="Thmlemma11.p1.1.m1.7.8.3.2.2" xref="Thmlemma11.p1.1.m1.7.8.3.2.2.cmml"><mi id="Thmlemma11.p1.1.m1.7.8.3.2.2.2" xref="Thmlemma11.p1.1.m1.7.8.3.2.2.2.cmml">F</mi><mi id="Thmlemma11.p1.1.m1.7.8.3.2.2.3" xref="Thmlemma11.p1.1.m1.7.8.3.2.2.3.cmml">n</mi></msub></mrow><mo id="Thmlemma11.p1.1.m1.7.8.3.1" xref="Thmlemma11.p1.1.m1.7.8.3.1.cmml">⁢</mo><mrow id="Thmlemma11.p1.1.m1.7.8.3.3.2" xref="Thmlemma11.p1.1.m1.7.8.3.3.1.cmml"><mo id="Thmlemma11.p1.1.m1.7.8.3.3.2.1" stretchy="false" xref="Thmlemma11.p1.1.m1.7.8.3.3.1.cmml">(</mo><mi id="Thmlemma11.p1.1.m1.6.6" xref="Thmlemma11.p1.1.m1.6.6.cmml">σ</mi><mo id="Thmlemma11.p1.1.m1.7.8.3.3.2.2" xref="Thmlemma11.p1.1.m1.7.8.3.3.1.cmml">,</mo><mi id="Thmlemma11.p1.1.m1.7.7" xref="Thmlemma11.p1.1.m1.7.7.cmml">α</mi><mo id="Thmlemma11.p1.1.m1.7.8.3.3.2.3" stretchy="false" xref="Thmlemma11.p1.1.m1.7.8.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma11.p1.1.m1.7b"><apply id="Thmlemma11.p1.1.m1.7.8.cmml" xref="Thmlemma11.p1.1.m1.7.8"><eq id="Thmlemma11.p1.1.m1.7.8.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.1"></eq><apply id="Thmlemma11.p1.1.m1.7.8.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.2"><times id="Thmlemma11.p1.1.m1.7.8.2.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.1"></times><apply id="Thmlemma11.p1.1.m1.7.8.2.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2"><apply id="Thmlemma11.p1.1.m1.7.8.2.2.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2.1"><csymbol cd="ambiguous" id="Thmlemma11.p1.1.m1.7.8.2.2.1.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2.1">subscript</csymbol><min id="Thmlemma11.p1.1.m1.7.8.2.2.1.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2.1.2"></min><apply id="Thmlemma11.p1.1.m1.1.1.1.cmml" xref="Thmlemma11.p1.1.m1.1.1.1"><in id="Thmlemma11.p1.1.m1.1.1.1.2.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.2"></in><ci id="Thmlemma11.p1.1.m1.1.1.1.3.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.3">𝜎</ci><apply id="Thmlemma11.p1.1.m1.1.1.1.1.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma11.p1.1.m1.1.1.1.1.2.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1">superscript</csymbol><apply id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1"><times id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.2"></times><ci id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.3">Δ</ci><apply id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3"><times id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="Thmlemma11.p1.1.m1.1.1.1.1.3.cmml" xref="Thmlemma11.p1.1.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmlemma11.p1.1.m1.7.8.2.2.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2.2"><csymbol cd="ambiguous" id="Thmlemma11.p1.1.m1.7.8.2.2.2.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2.2">subscript</csymbol><ci id="Thmlemma11.p1.1.m1.7.8.2.2.2.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2.2.2">𝐹</ci><ci id="Thmlemma11.p1.1.m1.7.8.2.2.2.3.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmlemma11.p1.1.m1.7.8.2.3.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.2.3.2"><ci id="Thmlemma11.p1.1.m1.4.4.cmml" xref="Thmlemma11.p1.1.m1.4.4">𝜎</ci><ci id="Thmlemma11.p1.1.m1.5.5.cmml" xref="Thmlemma11.p1.1.m1.5.5">𝛼</ci></interval></apply><apply id="Thmlemma11.p1.1.m1.7.8.3.cmml" xref="Thmlemma11.p1.1.m1.7.8.3"><times id="Thmlemma11.p1.1.m1.7.8.3.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.1"></times><apply id="Thmlemma11.p1.1.m1.7.8.3.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2"><apply id="Thmlemma11.p1.1.m1.7.8.3.2.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2.1"><csymbol cd="ambiguous" id="Thmlemma11.p1.1.m1.7.8.3.2.1.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2.1">subscript</csymbol><min id="Thmlemma11.p1.1.m1.7.8.3.2.1.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2.1.2"></min><apply id="Thmlemma11.p1.1.m1.3.3.2.cmml" xref="Thmlemma11.p1.1.m1.3.3.2"><in id="Thmlemma11.p1.1.m1.3.3.2.3.cmml" xref="Thmlemma11.p1.1.m1.3.3.2.3"></in><ci id="Thmlemma11.p1.1.m1.3.3.2.4.cmml" xref="Thmlemma11.p1.1.m1.3.3.2.4">𝜎</ci><apply id="Thmlemma11.p1.1.m1.3.3.2.2.2.cmml" xref="Thmlemma11.p1.1.m1.3.3.2.2.1"><ci id="Thmlemma11.p1.1.m1.2.2.1.1.cmml" xref="Thmlemma11.p1.1.m1.2.2.1.1">conv</ci><apply id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.cmml" xref="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.1.cmml" xref="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1">subscript</csymbol><ci id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.2.cmml" xref="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.2">𝒰</ci><ci id="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.3.cmml" xref="Thmlemma11.p1.1.m1.3.3.2.2.1.1.1.3">𝑛</ci></apply></apply></apply></apply><apply id="Thmlemma11.p1.1.m1.7.8.3.2.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2.2"><csymbol cd="ambiguous" id="Thmlemma11.p1.1.m1.7.8.3.2.2.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2.2">subscript</csymbol><ci id="Thmlemma11.p1.1.m1.7.8.3.2.2.2.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2.2.2">𝐹</ci><ci id="Thmlemma11.p1.1.m1.7.8.3.2.2.3.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmlemma11.p1.1.m1.7.8.3.3.1.cmml" xref="Thmlemma11.p1.1.m1.7.8.3.3.2"><ci id="Thmlemma11.p1.1.m1.6.6.cmml" xref="Thmlemma11.p1.1.m1.6.6">𝜎</ci><ci id="Thmlemma11.p1.1.m1.7.7.cmml" xref="Thmlemma11.p1.1.m1.7.7">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma11.p1.1.m1.7c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)=\min_{\sigma\in% \operatorname{conv}(\mathcal{U}_{n})}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma11.p1.1.m1.7d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = roman_min start_POSTSUBSCRIPT italic_σ ∈ roman_conv ( caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Thmlemma11.p1.1.1">.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.8">Now, define <math alttext="F^{1}_{n}(X)=F_{n}(X^{n},\alpha)" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.3"><semantics id="S4.SS2.p5.1.m1.3a"><mrow id="S4.SS2.p5.1.m1.3.3" xref="S4.SS2.p5.1.m1.3.3.cmml"><mrow id="S4.SS2.p5.1.m1.3.3.3" xref="S4.SS2.p5.1.m1.3.3.3.cmml"><msubsup id="S4.SS2.p5.1.m1.3.3.3.2" xref="S4.SS2.p5.1.m1.3.3.3.2.cmml"><mi id="S4.SS2.p5.1.m1.3.3.3.2.2.2" xref="S4.SS2.p5.1.m1.3.3.3.2.2.2.cmml">F</mi><mi id="S4.SS2.p5.1.m1.3.3.3.2.3" xref="S4.SS2.p5.1.m1.3.3.3.2.3.cmml">n</mi><mn id="S4.SS2.p5.1.m1.3.3.3.2.2.3" xref="S4.SS2.p5.1.m1.3.3.3.2.2.3.cmml">1</mn></msubsup><mo id="S4.SS2.p5.1.m1.3.3.3.1" xref="S4.SS2.p5.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.p5.1.m1.3.3.3.3.2" xref="S4.SS2.p5.1.m1.3.3.3.cmml"><mo id="S4.SS2.p5.1.m1.3.3.3.3.2.1" stretchy="false" xref="S4.SS2.p5.1.m1.3.3.3.cmml">(</mo><mi id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml">X</mi><mo id="S4.SS2.p5.1.m1.3.3.3.3.2.2" stretchy="false" xref="S4.SS2.p5.1.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p5.1.m1.3.3.2" xref="S4.SS2.p5.1.m1.3.3.2.cmml">=</mo><mrow id="S4.SS2.p5.1.m1.3.3.1" xref="S4.SS2.p5.1.m1.3.3.1.cmml"><msub id="S4.SS2.p5.1.m1.3.3.1.3" xref="S4.SS2.p5.1.m1.3.3.1.3.cmml"><mi id="S4.SS2.p5.1.m1.3.3.1.3.2" xref="S4.SS2.p5.1.m1.3.3.1.3.2.cmml">F</mi><mi id="S4.SS2.p5.1.m1.3.3.1.3.3" xref="S4.SS2.p5.1.m1.3.3.1.3.3.cmml">n</mi></msub><mo id="S4.SS2.p5.1.m1.3.3.1.2" xref="S4.SS2.p5.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS2.p5.1.m1.3.3.1.1.1" xref="S4.SS2.p5.1.m1.3.3.1.1.2.cmml"><mo id="S4.SS2.p5.1.m1.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.p5.1.m1.3.3.1.1.2.cmml">(</mo><msup id="S4.SS2.p5.1.m1.3.3.1.1.1.1" xref="S4.SS2.p5.1.m1.3.3.1.1.1.1.cmml"><mi id="S4.SS2.p5.1.m1.3.3.1.1.1.1.2" xref="S4.SS2.p5.1.m1.3.3.1.1.1.1.2.cmml">X</mi><mi id="S4.SS2.p5.1.m1.3.3.1.1.1.1.3" xref="S4.SS2.p5.1.m1.3.3.1.1.1.1.3.cmml">n</mi></msup><mo id="S4.SS2.p5.1.m1.3.3.1.1.1.3" xref="S4.SS2.p5.1.m1.3.3.1.1.2.cmml">,</mo><mi id="S4.SS2.p5.1.m1.2.2" xref="S4.SS2.p5.1.m1.2.2.cmml">α</mi><mo id="S4.SS2.p5.1.m1.3.3.1.1.1.4" stretchy="false" xref="S4.SS2.p5.1.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.3b"><apply id="S4.SS2.p5.1.m1.3.3.cmml" xref="S4.SS2.p5.1.m1.3.3"><eq id="S4.SS2.p5.1.m1.3.3.2.cmml" xref="S4.SS2.p5.1.m1.3.3.2"></eq><apply id="S4.SS2.p5.1.m1.3.3.3.cmml" xref="S4.SS2.p5.1.m1.3.3.3"><times id="S4.SS2.p5.1.m1.3.3.3.1.cmml" xref="S4.SS2.p5.1.m1.3.3.3.1"></times><apply id="S4.SS2.p5.1.m1.3.3.3.2.cmml" xref="S4.SS2.p5.1.m1.3.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.3.3.3.2.1.cmml" xref="S4.SS2.p5.1.m1.3.3.3.2">subscript</csymbol><apply id="S4.SS2.p5.1.m1.3.3.3.2.2.cmml" xref="S4.SS2.p5.1.m1.3.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.3.3.3.2.2.1.cmml" xref="S4.SS2.p5.1.m1.3.3.3.2">superscript</csymbol><ci id="S4.SS2.p5.1.m1.3.3.3.2.2.2.cmml" xref="S4.SS2.p5.1.m1.3.3.3.2.2.2">𝐹</ci><cn id="S4.SS2.p5.1.m1.3.3.3.2.2.3.cmml" type="integer" xref="S4.SS2.p5.1.m1.3.3.3.2.2.3">1</cn></apply><ci id="S4.SS2.p5.1.m1.3.3.3.2.3.cmml" xref="S4.SS2.p5.1.m1.3.3.3.2.3">𝑛</ci></apply><ci id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">𝑋</ci></apply><apply id="S4.SS2.p5.1.m1.3.3.1.cmml" xref="S4.SS2.p5.1.m1.3.3.1"><times id="S4.SS2.p5.1.m1.3.3.1.2.cmml" xref="S4.SS2.p5.1.m1.3.3.1.2"></times><apply id="S4.SS2.p5.1.m1.3.3.1.3.cmml" xref="S4.SS2.p5.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.3.3.1.3.1.cmml" xref="S4.SS2.p5.1.m1.3.3.1.3">subscript</csymbol><ci id="S4.SS2.p5.1.m1.3.3.1.3.2.cmml" xref="S4.SS2.p5.1.m1.3.3.1.3.2">𝐹</ci><ci id="S4.SS2.p5.1.m1.3.3.1.3.3.cmml" xref="S4.SS2.p5.1.m1.3.3.1.3.3">𝑛</ci></apply><interval closure="open" id="S4.SS2.p5.1.m1.3.3.1.1.2.cmml" xref="S4.SS2.p5.1.m1.3.3.1.1.1"><apply id="S4.SS2.p5.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS2.p5.1.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.3.3.1.1.1.1.1.cmml" xref="S4.SS2.p5.1.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S4.SS2.p5.1.m1.3.3.1.1.1.1.2.cmml" xref="S4.SS2.p5.1.m1.3.3.1.1.1.1.2">𝑋</ci><ci id="S4.SS2.p5.1.m1.3.3.1.1.1.1.3.cmml" xref="S4.SS2.p5.1.m1.3.3.1.1.1.1.3">𝑛</ci></apply><ci id="S4.SS2.p5.1.m1.2.2.cmml" xref="S4.SS2.p5.1.m1.2.2">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.3c">F^{1}_{n}(X)=F_{n}(X^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.3d">italic_F start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_X ) = italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math> for <math alttext="X\in\operatorname{conv}(\mathcal{U}_{1})" class="ltx_Math" display="inline" id="S4.SS2.p5.2.m2.2"><semantics id="S4.SS2.p5.2.m2.2a"><mrow id="S4.SS2.p5.2.m2.2.2" xref="S4.SS2.p5.2.m2.2.2.cmml"><mi id="S4.SS2.p5.2.m2.2.2.3" xref="S4.SS2.p5.2.m2.2.2.3.cmml">X</mi><mo id="S4.SS2.p5.2.m2.2.2.2" xref="S4.SS2.p5.2.m2.2.2.2.cmml">∈</mo><mrow id="S4.SS2.p5.2.m2.2.2.1.1" xref="S4.SS2.p5.2.m2.2.2.1.2.cmml"><mi id="S4.SS2.p5.2.m2.1.1" xref="S4.SS2.p5.2.m2.1.1.cmml">conv</mi><mo id="S4.SS2.p5.2.m2.2.2.1.1a" xref="S4.SS2.p5.2.m2.2.2.1.2.cmml">⁡</mo><mrow id="S4.SS2.p5.2.m2.2.2.1.1.1" xref="S4.SS2.p5.2.m2.2.2.1.2.cmml"><mo id="S4.SS2.p5.2.m2.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.p5.2.m2.2.2.1.2.cmml">(</mo><msub id="S4.SS2.p5.2.m2.2.2.1.1.1.1" xref="S4.SS2.p5.2.m2.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p5.2.m2.2.2.1.1.1.1.2" xref="S4.SS2.p5.2.m2.2.2.1.1.1.1.2.cmml">𝒰</mi><mn id="S4.SS2.p5.2.m2.2.2.1.1.1.1.3" xref="S4.SS2.p5.2.m2.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.p5.2.m2.2.2.1.1.1.3" stretchy="false" xref="S4.SS2.p5.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.2.m2.2b"><apply id="S4.SS2.p5.2.m2.2.2.cmml" xref="S4.SS2.p5.2.m2.2.2"><in id="S4.SS2.p5.2.m2.2.2.2.cmml" xref="S4.SS2.p5.2.m2.2.2.2"></in><ci id="S4.SS2.p5.2.m2.2.2.3.cmml" xref="S4.SS2.p5.2.m2.2.2.3">𝑋</ci><apply id="S4.SS2.p5.2.m2.2.2.1.2.cmml" xref="S4.SS2.p5.2.m2.2.2.1.1"><ci id="S4.SS2.p5.2.m2.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1">conv</ci><apply id="S4.SS2.p5.2.m2.2.2.1.1.1.1.cmml" xref="S4.SS2.p5.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.2.m2.2.2.1.1.1.1.1.cmml" xref="S4.SS2.p5.2.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.2.m2.2.2.1.1.1.1.2.cmml" xref="S4.SS2.p5.2.m2.2.2.1.1.1.1.2">𝒰</ci><cn id="S4.SS2.p5.2.m2.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p5.2.m2.2.2.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.2.m2.2c">X\in\operatorname{conv}(\mathcal{U}_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.2.m2.2d">italic_X ∈ roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Our goal is to compute <math alttext="\min_{X\in\operatorname{conv}(\mathcal{U}_{1})}F^{1}_{n}(X)" class="ltx_Math" display="inline" id="S4.SS2.p5.3.m3.3"><semantics id="S4.SS2.p5.3.m3.3a"><mrow id="S4.SS2.p5.3.m3.3.4" xref="S4.SS2.p5.3.m3.3.4.cmml"><mrow id="S4.SS2.p5.3.m3.3.4.2" xref="S4.SS2.p5.3.m3.3.4.2.cmml"><msub id="S4.SS2.p5.3.m3.3.4.2.1" xref="S4.SS2.p5.3.m3.3.4.2.1.cmml"><mi id="S4.SS2.p5.3.m3.3.4.2.1.2" xref="S4.SS2.p5.3.m3.3.4.2.1.2.cmml">min</mi><mrow id="S4.SS2.p5.3.m3.2.2.2" xref="S4.SS2.p5.3.m3.2.2.2.cmml"><mi id="S4.SS2.p5.3.m3.2.2.2.4" xref="S4.SS2.p5.3.m3.2.2.2.4.cmml">X</mi><mo id="S4.SS2.p5.3.m3.2.2.2.3" xref="S4.SS2.p5.3.m3.2.2.2.3.cmml">∈</mo><mrow id="S4.SS2.p5.3.m3.2.2.2.2.1" xref="S4.SS2.p5.3.m3.2.2.2.2.2.cmml"><mi id="S4.SS2.p5.3.m3.1.1.1.1" xref="S4.SS2.p5.3.m3.1.1.1.1.cmml">conv</mi><mo id="S4.SS2.p5.3.m3.2.2.2.2.1a" xref="S4.SS2.p5.3.m3.2.2.2.2.2.cmml">⁡</mo><mrow id="S4.SS2.p5.3.m3.2.2.2.2.1.1" xref="S4.SS2.p5.3.m3.2.2.2.2.2.cmml"><mo 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xref="S4.SS2.p5.3.m3.3.4.1.cmml">⁢</mo><mrow id="S4.SS2.p5.3.m3.3.4.3.2" xref="S4.SS2.p5.3.m3.3.4.cmml"><mo id="S4.SS2.p5.3.m3.3.4.3.2.1" stretchy="false" xref="S4.SS2.p5.3.m3.3.4.cmml">(</mo><mi id="S4.SS2.p5.3.m3.3.3" xref="S4.SS2.p5.3.m3.3.3.cmml">X</mi><mo id="S4.SS2.p5.3.m3.3.4.3.2.2" stretchy="false" xref="S4.SS2.p5.3.m3.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.3.m3.3b"><apply id="S4.SS2.p5.3.m3.3.4.cmml" xref="S4.SS2.p5.3.m3.3.4"><times id="S4.SS2.p5.3.m3.3.4.1.cmml" xref="S4.SS2.p5.3.m3.3.4.1"></times><apply id="S4.SS2.p5.3.m3.3.4.2.cmml" xref="S4.SS2.p5.3.m3.3.4.2"><apply id="S4.SS2.p5.3.m3.3.4.2.1.cmml" xref="S4.SS2.p5.3.m3.3.4.2.1"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.3.4.2.1.1.cmml" xref="S4.SS2.p5.3.m3.3.4.2.1">subscript</csymbol><min id="S4.SS2.p5.3.m3.3.4.2.1.2.cmml" xref="S4.SS2.p5.3.m3.3.4.2.1.2"></min><apply id="S4.SS2.p5.3.m3.2.2.2.cmml" xref="S4.SS2.p5.3.m3.2.2.2"><in id="S4.SS2.p5.3.m3.2.2.2.3.cmml" xref="S4.SS2.p5.3.m3.2.2.2.3"></in><ci id="S4.SS2.p5.3.m3.2.2.2.4.cmml" xref="S4.SS2.p5.3.m3.2.2.2.4">𝑋</ci><apply id="S4.SS2.p5.3.m3.2.2.2.2.2.cmml" xref="S4.SS2.p5.3.m3.2.2.2.2.1"><ci id="S4.SS2.p5.3.m3.1.1.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1.1.1">conv</ci><apply id="S4.SS2.p5.3.m3.2.2.2.2.1.1.1.cmml" xref="S4.SS2.p5.3.m3.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.2.2.2.2.1.1.1.1.cmml" xref="S4.SS2.p5.3.m3.2.2.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.3.m3.2.2.2.2.1.1.1.2.cmml" xref="S4.SS2.p5.3.m3.2.2.2.2.1.1.1.2">𝒰</ci><cn id="S4.SS2.p5.3.m3.2.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS2.p5.3.m3.2.2.2.2.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S4.SS2.p5.3.m3.3.4.2.2.cmml" xref="S4.SS2.p5.3.m3.3.4.2.2"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.3.4.2.2.1.cmml" xref="S4.SS2.p5.3.m3.3.4.2.2">subscript</csymbol><apply id="S4.SS2.p5.3.m3.3.4.2.2.2.cmml" xref="S4.SS2.p5.3.m3.3.4.2.2"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.3.4.2.2.2.1.cmml" xref="S4.SS2.p5.3.m3.3.4.2.2">superscript</csymbol><ci id="S4.SS2.p5.3.m3.3.4.2.2.2.2.cmml" xref="S4.SS2.p5.3.m3.3.4.2.2.2.2">𝐹</ci><cn id="S4.SS2.p5.3.m3.3.4.2.2.2.3.cmml" type="integer" xref="S4.SS2.p5.3.m3.3.4.2.2.2.3">1</cn></apply><ci id="S4.SS2.p5.3.m3.3.4.2.2.3.cmml" xref="S4.SS2.p5.3.m3.3.4.2.2.3">𝑛</ci></apply></apply><ci id="S4.SS2.p5.3.m3.3.3.cmml" xref="S4.SS2.p5.3.m3.3.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.3.m3.3c">\min_{X\in\operatorname{conv}(\mathcal{U}_{1})}F^{1}_{n}(X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.3.m3.3d">roman_min start_POSTSUBSCRIPT italic_X ∈ roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math>. <math alttext="F^{1}_{n}" class="ltx_Math" display="inline" id="S4.SS2.p5.4.m4.1"><semantics id="S4.SS2.p5.4.m4.1a"><msubsup id="S4.SS2.p5.4.m4.1.1" xref="S4.SS2.p5.4.m4.1.1.cmml"><mi id="S4.SS2.p5.4.m4.1.1.2.2" xref="S4.SS2.p5.4.m4.1.1.2.2.cmml">F</mi><mi id="S4.SS2.p5.4.m4.1.1.3" xref="S4.SS2.p5.4.m4.1.1.3.cmml">n</mi><mn id="S4.SS2.p5.4.m4.1.1.2.3" xref="S4.SS2.p5.4.m4.1.1.2.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.4.m4.1b"><apply id="S4.SS2.p5.4.m4.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.1.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1">subscript</csymbol><apply id="S4.SS2.p5.4.m4.1.1.2.cmml" xref="S4.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.1.1.2.1.cmml" xref="S4.SS2.p5.4.m4.1.1">superscript</csymbol><ci id="S4.SS2.p5.4.m4.1.1.2.2.cmml" xref="S4.SS2.p5.4.m4.1.1.2.2">𝐹</ci><cn id="S4.SS2.p5.4.m4.1.1.2.3.cmml" type="integer" xref="S4.SS2.p5.4.m4.1.1.2.3">1</cn></apply><ci id="S4.SS2.p5.4.m4.1.1.3.cmml" xref="S4.SS2.p5.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.4.m4.1c">F^{1}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.4.m4.1d">italic_F start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is concave on <math alttext="\operatorname{conv}(\mathcal{U}_{1})" class="ltx_Math" display="inline" id="S4.SS2.p5.5.m5.2"><semantics id="S4.SS2.p5.5.m5.2a"><mrow id="S4.SS2.p5.5.m5.2.2.1" xref="S4.SS2.p5.5.m5.2.2.2.cmml"><mi id="S4.SS2.p5.5.m5.1.1" xref="S4.SS2.p5.5.m5.1.1.cmml">conv</mi><mo id="S4.SS2.p5.5.m5.2.2.1a" xref="S4.SS2.p5.5.m5.2.2.2.cmml">⁡</mo><mrow id="S4.SS2.p5.5.m5.2.2.1.1" xref="S4.SS2.p5.5.m5.2.2.2.cmml"><mo id="S4.SS2.p5.5.m5.2.2.1.1.2" stretchy="false" xref="S4.SS2.p5.5.m5.2.2.2.cmml">(</mo><msub id="S4.SS2.p5.5.m5.2.2.1.1.1" xref="S4.SS2.p5.5.m5.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p5.5.m5.2.2.1.1.1.2" xref="S4.SS2.p5.5.m5.2.2.1.1.1.2.cmml">𝒰</mi><mn id="S4.SS2.p5.5.m5.2.2.1.1.1.3" xref="S4.SS2.p5.5.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.p5.5.m5.2.2.1.1.3" stretchy="false" xref="S4.SS2.p5.5.m5.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.5.m5.2b"><apply id="S4.SS2.p5.5.m5.2.2.2.cmml" xref="S4.SS2.p5.5.m5.2.2.1"><ci id="S4.SS2.p5.5.m5.1.1.cmml" xref="S4.SS2.p5.5.m5.1.1">conv</ci><apply id="S4.SS2.p5.5.m5.2.2.1.1.1.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m5.2.2.1.1.1.1.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.5.m5.2.2.1.1.1.2.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1.1.2">𝒰</ci><cn id="S4.SS2.p5.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS2.p5.5.m5.2.2.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.5.m5.2c">\operatorname{conv}(\mathcal{U}_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.5.m5.2d">roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> as a positive sum of concave functions: this is because each <math alttext="\sigma(p_{i})" class="ltx_Math" display="inline" id="S4.SS2.p5.6.m6.1"><semantics id="S4.SS2.p5.6.m6.1a"><mrow id="S4.SS2.p5.6.m6.1.1" xref="S4.SS2.p5.6.m6.1.1.cmml"><mi id="S4.SS2.p5.6.m6.1.1.3" xref="S4.SS2.p5.6.m6.1.1.3.cmml">σ</mi><mo id="S4.SS2.p5.6.m6.1.1.2" xref="S4.SS2.p5.6.m6.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p5.6.m6.1.1.1.1" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml"><mo id="S4.SS2.p5.6.m6.1.1.1.1.2" stretchy="false" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p5.6.m6.1.1.1.1.1" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml"><mi id="S4.SS2.p5.6.m6.1.1.1.1.1.2" xref="S4.SS2.p5.6.m6.1.1.1.1.1.2.cmml">p</mi><mi id="S4.SS2.p5.6.m6.1.1.1.1.1.3" xref="S4.SS2.p5.6.m6.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS2.p5.6.m6.1.1.1.1.3" stretchy="false" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.6.m6.1b"><apply id="S4.SS2.p5.6.m6.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1"><times id="S4.SS2.p5.6.m6.1.1.2.cmml" xref="S4.SS2.p5.6.m6.1.1.2"></times><ci id="S4.SS2.p5.6.m6.1.1.3.cmml" xref="S4.SS2.p5.6.m6.1.1.3">𝜎</ci><apply id="S4.SS2.p5.6.m6.1.1.1.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.6.m6.1.1.1.1.1.2.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1.1.2">𝑝</ci><ci id="S4.SS2.p5.6.m6.1.1.1.1.1.3.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.6.m6.1c">\sigma(p_{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.6.m6.1d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is a function of the form <math alttext="x\mapsto 1-(1-x)^{n}" class="ltx_Math" display="inline" id="S4.SS2.p5.7.m7.1"><semantics id="S4.SS2.p5.7.m7.1a"><mrow id="S4.SS2.p5.7.m7.1.1" xref="S4.SS2.p5.7.m7.1.1.cmml"><mi id="S4.SS2.p5.7.m7.1.1.3" xref="S4.SS2.p5.7.m7.1.1.3.cmml">x</mi><mo id="S4.SS2.p5.7.m7.1.1.2" stretchy="false" xref="S4.SS2.p5.7.m7.1.1.2.cmml">↦</mo><mrow id="S4.SS2.p5.7.m7.1.1.1" xref="S4.SS2.p5.7.m7.1.1.1.cmml"><mn id="S4.SS2.p5.7.m7.1.1.1.3" xref="S4.SS2.p5.7.m7.1.1.1.3.cmml">1</mn><mo id="S4.SS2.p5.7.m7.1.1.1.2" xref="S4.SS2.p5.7.m7.1.1.1.2.cmml">−</mo><msup id="S4.SS2.p5.7.m7.1.1.1.1" xref="S4.SS2.p5.7.m7.1.1.1.1.cmml"><mrow id="S4.SS2.p5.7.m7.1.1.1.1.1.1" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p5.7.m7.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.cmml"><mn id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.2" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.1" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.3" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.SS2.p5.7.m7.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S4.SS2.p5.7.m7.1.1.1.1.3" xref="S4.SS2.p5.7.m7.1.1.1.1.3.cmml">n</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.7.m7.1b"><apply id="S4.SS2.p5.7.m7.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1"><csymbol cd="latexml" id="S4.SS2.p5.7.m7.1.1.2.cmml" xref="S4.SS2.p5.7.m7.1.1.2">maps-to</csymbol><ci id="S4.SS2.p5.7.m7.1.1.3.cmml" xref="S4.SS2.p5.7.m7.1.1.3">𝑥</ci><apply id="S4.SS2.p5.7.m7.1.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1.1"><minus id="S4.SS2.p5.7.m7.1.1.1.2.cmml" xref="S4.SS2.p5.7.m7.1.1.1.2"></minus><cn id="S4.SS2.p5.7.m7.1.1.1.3.cmml" type="integer" xref="S4.SS2.p5.7.m7.1.1.1.3">1</cn><apply id="S4.SS2.p5.7.m7.1.1.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.7.m7.1.1.1.1.2.cmml" xref="S4.SS2.p5.7.m7.1.1.1.1">superscript</csymbol><apply id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1"><minus id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.1"></minus><cn id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.2">1</cn><ci id="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p5.7.m7.1.1.1.1.1.1.1.3">𝑥</ci></apply><ci id="S4.SS2.p5.7.m7.1.1.1.1.3.cmml" xref="S4.SS2.p5.7.m7.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.7.m7.1c">x\mapsto 1-(1-x)^{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.7.m7.1d">italic_x ↦ 1 - ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. We now obtain the optimality of <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S4.SS2.p5.8.m8.1"><semantics id="S4.SS2.p5.8.m8.1a"><msub id="S4.SS2.p5.8.m8.1.1" xref="S4.SS2.p5.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p5.8.m8.1.1.2" xref="S4.SS2.p5.8.m8.1.1.2.cmml">𝒰</mi><mi id="S4.SS2.p5.8.m8.1.1.3" xref="S4.SS2.p5.8.m8.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.8.m8.1b"><apply id="S4.SS2.p5.8.m8.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.8.m8.1.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1">subscript</csymbol><ci id="S4.SS2.p5.8.m8.1.1.2.cmml" xref="S4.SS2.p5.8.m8.1.1.2">𝒰</ci><ci id="S4.SS2.p5.8.m8.1.1.3.cmml" xref="S4.SS2.p5.8.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.8.m8.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.8.m8.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> with respect to the SUM cost, as was discussed in the Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.SS1" title="1.1 Our results ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1.1</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 4.6</span></h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem6.p1.3.3">Let <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.1.1.m1.1"><semantics id="S4.Thmtheorem6.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem6.p1.1.1.m1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem6.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.2.cmml">α</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.1.cmml">></mo><mn id="S4.Thmtheorem6.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.1.1.m1.1b"><apply id="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1"><gt id="S4.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.1"></gt><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.2">𝛼</ci><cn id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.1.1.m1.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.1.1.m1.1d">italic_α > 0</annotation></semantics></math>. Then <math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)=\min_{\sigma\in% \mathcal{U}_{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.2.m2.5"><semantics id="S4.Thmtheorem6.p1.2.2.m2.5a"><mrow id="S4.Thmtheorem6.p1.2.2.m2.5.6" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.cmml"><mrow id="S4.Thmtheorem6.p1.2.2.m2.5.6.2" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.2.cmml"><mrow id="S4.Thmtheorem6.p1.2.2.m2.5.6.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.2.2.cmml"><msub id="S4.Thmtheorem6.p1.2.2.m2.5.6.2.2.1" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.2.2.1.cmml"><mi id="S4.Thmtheorem6.p1.2.2.m2.5.6.2.2.1.2" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.2.2.1.2.cmml">min</mi><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.1.cmml"><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1.1.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.1.3.cmml">σ</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.1.1.1.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.1.2.cmml">∈</mo><msup 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id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.2">𝜎</ci><apply id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3.2">𝒰</ci><ci id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3.3.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.1.3.3.3">𝑛</ci></apply></apply></apply><apply id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2">subscript</csymbol><ci id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2.2">𝐹</ci><ci id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2.3.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="S4.Thmtheorem6.p1.2.2.m2.5.6.3.3.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.6.3.3.2"><ci id="S4.Thmtheorem6.p1.2.2.m2.4.4.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.4.4">𝜎</ci><ci id="S4.Thmtheorem6.p1.2.2.m2.5.5.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.5.5">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.2.m2.5c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)=\min_{\sigma\in% \mathcal{U}_{n}}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.2.m2.5d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math>. 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id="S4.Thmtheorem6.p1.3.3.m3.3.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.3">𝑘</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.4.4.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.4.4">𝛼</ci></vector></apply><apply id="S4.Thmtheorem6.p1.3.3.m3.8.9.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.9.3"><times id="S4.Thmtheorem6.p1.3.3.m3.8.9.3.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.9.3.1"></times><apply id="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2">subscript</csymbol><ci id="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2.2">𝑟</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.9.3.2.3">𝑈</ci></apply><vector id="S4.Thmtheorem6.p1.3.3.m3.8.9.3.3.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.9.3.3.2"><ci id="S4.Thmtheorem6.p1.3.3.m3.5.5.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.5.5">𝐺</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.6.6.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.6.6">𝑛</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.7.7.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.7.7">𝑘</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.8.8.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.8.8">𝛼</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.3.3.m3.8c">r(G,n,k,\alpha)=r_{U}(G,n,k,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.3.3.m3.8d">italic_r ( italic_G , italic_n , italic_k , italic_α ) = italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α )</annotation></semantics></math>.</span></p> </div> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>MAX Cost</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.6">The full details of this section can be found in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4" title="Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.D</span></a>. Throughout the rest of the paper, we will assume that <math alttext="\alpha>\frac{\left|p_{max}\right|-\left|p_{min}\right|}{k}" class="ltx_Math" display="inline" id="S5.p1.1.m1.2"><semantics id="S5.p1.1.m1.2a"><mrow id="S5.p1.1.m1.2.3" xref="S5.p1.1.m1.2.3.cmml"><mi id="S5.p1.1.m1.2.3.2" xref="S5.p1.1.m1.2.3.2.cmml">α</mi><mo id="S5.p1.1.m1.2.3.1" xref="S5.p1.1.m1.2.3.1.cmml">></mo><mfrac id="S5.p1.1.m1.2.2" xref="S5.p1.1.m1.2.2.cmml"><mrow id="S5.p1.1.m1.2.2.2" xref="S5.p1.1.m1.2.2.2.cmml"><mrow id="S5.p1.1.m1.1.1.1.1.1" xref="S5.p1.1.m1.1.1.1.1.2.cmml"><mo id="S5.p1.1.m1.1.1.1.1.1.2" xref="S5.p1.1.m1.1.1.1.1.2.1.cmml">|</mo><msub id="S5.p1.1.m1.1.1.1.1.1.1" xref="S5.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.1.1.1.1.2" xref="S5.p1.1.m1.1.1.1.1.1.1.2.cmml">p</mi><mrow id="S5.p1.1.m1.1.1.1.1.1.1.3" xref="S5.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S5.p1.1.m1.1.1.1.1.1.1.3.2" xref="S5.p1.1.m1.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="S5.p1.1.m1.1.1.1.1.1.1.3.1" xref="S5.p1.1.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.1.m1.1.1.1.1.1.1.3.3" xref="S5.p1.1.m1.1.1.1.1.1.1.3.3.cmml">a</mi><mo id="S5.p1.1.m1.1.1.1.1.1.1.3.1a" xref="S5.p1.1.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.1.m1.1.1.1.1.1.1.3.4" xref="S5.p1.1.m1.1.1.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="S5.p1.1.m1.1.1.1.1.1.3" xref="S5.p1.1.m1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.p1.1.m1.2.2.2.3" xref="S5.p1.1.m1.2.2.2.3.cmml">−</mo><mrow id="S5.p1.1.m1.2.2.2.2.1" xref="S5.p1.1.m1.2.2.2.2.2.cmml"><mo id="S5.p1.1.m1.2.2.2.2.1.2" xref="S5.p1.1.m1.2.2.2.2.2.1.cmml">|</mo><msub id="S5.p1.1.m1.2.2.2.2.1.1" xref="S5.p1.1.m1.2.2.2.2.1.1.cmml"><mi id="S5.p1.1.m1.2.2.2.2.1.1.2" xref="S5.p1.1.m1.2.2.2.2.1.1.2.cmml">p</mi><mrow id="S5.p1.1.m1.2.2.2.2.1.1.3" xref="S5.p1.1.m1.2.2.2.2.1.1.3.cmml"><mi id="S5.p1.1.m1.2.2.2.2.1.1.3.2" xref="S5.p1.1.m1.2.2.2.2.1.1.3.2.cmml">m</mi><mo id="S5.p1.1.m1.2.2.2.2.1.1.3.1" xref="S5.p1.1.m1.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.1.m1.2.2.2.2.1.1.3.3" xref="S5.p1.1.m1.2.2.2.2.1.1.3.3.cmml">i</mi><mo id="S5.p1.1.m1.2.2.2.2.1.1.3.1a" xref="S5.p1.1.m1.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.1.m1.2.2.2.2.1.1.3.4" xref="S5.p1.1.m1.2.2.2.2.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S5.p1.1.m1.2.2.2.2.1.3" xref="S5.p1.1.m1.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mi id="S5.p1.1.m1.2.2.4" xref="S5.p1.1.m1.2.2.4.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.2b"><apply id="S5.p1.1.m1.2.3.cmml" xref="S5.p1.1.m1.2.3"><gt id="S5.p1.1.m1.2.3.1.cmml" xref="S5.p1.1.m1.2.3.1"></gt><ci id="S5.p1.1.m1.2.3.2.cmml" xref="S5.p1.1.m1.2.3.2">𝛼</ci><apply id="S5.p1.1.m1.2.2.cmml" xref="S5.p1.1.m1.2.2"><divide id="S5.p1.1.m1.2.2.3.cmml" xref="S5.p1.1.m1.2.2"></divide><apply id="S5.p1.1.m1.2.2.2.cmml" xref="S5.p1.1.m1.2.2.2"><minus id="S5.p1.1.m1.2.2.2.3.cmml" xref="S5.p1.1.m1.2.2.2.3"></minus><apply id="S5.p1.1.m1.1.1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.1.1.1"><abs id="S5.p1.1.m1.1.1.1.1.2.1.cmml" xref="S5.p1.1.m1.1.1.1.1.1.2"></abs><apply id="S5.p1.1.m1.1.1.1.1.1.1.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1.2">𝑝</ci><apply id="S5.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1.3"><times id="S5.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1.3.1"></times><ci id="S5.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1.3.2">𝑚</ci><ci id="S5.p1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1.3.3">𝑎</ci><ci id="S5.p1.1.m1.1.1.1.1.1.1.3.4.cmml" xref="S5.p1.1.m1.1.1.1.1.1.1.3.4">𝑥</ci></apply></apply></apply><apply id="S5.p1.1.m1.2.2.2.2.2.cmml" xref="S5.p1.1.m1.2.2.2.2.1"><abs id="S5.p1.1.m1.2.2.2.2.2.1.cmml" xref="S5.p1.1.m1.2.2.2.2.1.2"></abs><apply id="S5.p1.1.m1.2.2.2.2.1.1.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.p1.1.m1.2.2.2.2.1.1.1.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1">subscript</csymbol><ci id="S5.p1.1.m1.2.2.2.2.1.1.2.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1.2">𝑝</ci><apply id="S5.p1.1.m1.2.2.2.2.1.1.3.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1.3"><times id="S5.p1.1.m1.2.2.2.2.1.1.3.1.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1.3.1"></times><ci id="S5.p1.1.m1.2.2.2.2.1.1.3.2.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1.3.2">𝑚</ci><ci id="S5.p1.1.m1.2.2.2.2.1.1.3.3.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1.3.3">𝑖</ci><ci id="S5.p1.1.m1.2.2.2.2.1.1.3.4.cmml" xref="S5.p1.1.m1.2.2.2.2.1.1.3.4">𝑛</ci></apply></apply></apply></apply><ci id="S5.p1.1.m1.2.2.4.cmml" xref="S5.p1.1.m1.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.2c">\alpha>\frac{\left|p_{max}\right|-\left|p_{min}\right|}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.2d">italic_α > divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>. If <math alttext="\alpha<\frac{\left|p_{max}\right|-\left|p_{min}\right|}{k}" class="ltx_Math" display="inline" id="S5.p1.2.m2.2"><semantics id="S5.p1.2.m2.2a"><mrow id="S5.p1.2.m2.2.3" xref="S5.p1.2.m2.2.3.cmml"><mi id="S5.p1.2.m2.2.3.2" xref="S5.p1.2.m2.2.3.2.cmml">α</mi><mo id="S5.p1.2.m2.2.3.1" xref="S5.p1.2.m2.2.3.1.cmml"><</mo><mfrac id="S5.p1.2.m2.2.2" xref="S5.p1.2.m2.2.2.cmml"><mrow id="S5.p1.2.m2.2.2.2" xref="S5.p1.2.m2.2.2.2.cmml"><mrow id="S5.p1.2.m2.1.1.1.1.1" xref="S5.p1.2.m2.1.1.1.1.2.cmml"><mo id="S5.p1.2.m2.1.1.1.1.1.2" xref="S5.p1.2.m2.1.1.1.1.2.1.cmml">|</mo><msub id="S5.p1.2.m2.1.1.1.1.1.1" xref="S5.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.p1.2.m2.1.1.1.1.1.1.2" xref="S5.p1.2.m2.1.1.1.1.1.1.2.cmml">p</mi><mrow id="S5.p1.2.m2.1.1.1.1.1.1.3" xref="S5.p1.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S5.p1.2.m2.1.1.1.1.1.1.3.2" xref="S5.p1.2.m2.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="S5.p1.2.m2.1.1.1.1.1.1.3.1" xref="S5.p1.2.m2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.2.m2.1.1.1.1.1.1.3.3" xref="S5.p1.2.m2.1.1.1.1.1.1.3.3.cmml">a</mi><mo id="S5.p1.2.m2.1.1.1.1.1.1.3.1a" xref="S5.p1.2.m2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.2.m2.1.1.1.1.1.1.3.4" xref="S5.p1.2.m2.1.1.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="S5.p1.2.m2.1.1.1.1.1.3" xref="S5.p1.2.m2.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.p1.2.m2.2.2.2.3" xref="S5.p1.2.m2.2.2.2.3.cmml">−</mo><mrow id="S5.p1.2.m2.2.2.2.2.1" xref="S5.p1.2.m2.2.2.2.2.2.cmml"><mo id="S5.p1.2.m2.2.2.2.2.1.2" xref="S5.p1.2.m2.2.2.2.2.2.1.cmml">|</mo><msub id="S5.p1.2.m2.2.2.2.2.1.1" xref="S5.p1.2.m2.2.2.2.2.1.1.cmml"><mi id="S5.p1.2.m2.2.2.2.2.1.1.2" xref="S5.p1.2.m2.2.2.2.2.1.1.2.cmml">p</mi><mrow id="S5.p1.2.m2.2.2.2.2.1.1.3" xref="S5.p1.2.m2.2.2.2.2.1.1.3.cmml"><mi id="S5.p1.2.m2.2.2.2.2.1.1.3.2" xref="S5.p1.2.m2.2.2.2.2.1.1.3.2.cmml">m</mi><mo id="S5.p1.2.m2.2.2.2.2.1.1.3.1" xref="S5.p1.2.m2.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.2.m2.2.2.2.2.1.1.3.3" xref="S5.p1.2.m2.2.2.2.2.1.1.3.3.cmml">i</mi><mo id="S5.p1.2.m2.2.2.2.2.1.1.3.1a" xref="S5.p1.2.m2.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.2.m2.2.2.2.2.1.1.3.4" xref="S5.p1.2.m2.2.2.2.2.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S5.p1.2.m2.2.2.2.2.1.3" xref="S5.p1.2.m2.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mi id="S5.p1.2.m2.2.2.4" xref="S5.p1.2.m2.2.2.4.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.2b"><apply id="S5.p1.2.m2.2.3.cmml" xref="S5.p1.2.m2.2.3"><lt id="S5.p1.2.m2.2.3.1.cmml" xref="S5.p1.2.m2.2.3.1"></lt><ci id="S5.p1.2.m2.2.3.2.cmml" xref="S5.p1.2.m2.2.3.2">𝛼</ci><apply id="S5.p1.2.m2.2.2.cmml" xref="S5.p1.2.m2.2.2"><divide id="S5.p1.2.m2.2.2.3.cmml" xref="S5.p1.2.m2.2.2"></divide><apply id="S5.p1.2.m2.2.2.2.cmml" xref="S5.p1.2.m2.2.2.2"><minus id="S5.p1.2.m2.2.2.2.3.cmml" xref="S5.p1.2.m2.2.2.2.3"></minus><apply id="S5.p1.2.m2.1.1.1.1.2.cmml" xref="S5.p1.2.m2.1.1.1.1.1"><abs id="S5.p1.2.m2.1.1.1.1.2.1.cmml" xref="S5.p1.2.m2.1.1.1.1.1.2"></abs><apply id="S5.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1.2">𝑝</ci><apply id="S5.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1.3"><times id="S5.p1.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1.3.1"></times><ci id="S5.p1.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1.3.2">𝑚</ci><ci id="S5.p1.2.m2.1.1.1.1.1.1.3.3.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1.3.3">𝑎</ci><ci id="S5.p1.2.m2.1.1.1.1.1.1.3.4.cmml" xref="S5.p1.2.m2.1.1.1.1.1.1.3.4">𝑥</ci></apply></apply></apply><apply id="S5.p1.2.m2.2.2.2.2.2.cmml" xref="S5.p1.2.m2.2.2.2.2.1"><abs id="S5.p1.2.m2.2.2.2.2.2.1.cmml" xref="S5.p1.2.m2.2.2.2.2.1.2"></abs><apply id="S5.p1.2.m2.2.2.2.2.1.1.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.p1.2.m2.2.2.2.2.1.1.1.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1">subscript</csymbol><ci id="S5.p1.2.m2.2.2.2.2.1.1.2.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1.2">𝑝</ci><apply id="S5.p1.2.m2.2.2.2.2.1.1.3.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1.3"><times id="S5.p1.2.m2.2.2.2.2.1.1.3.1.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1.3.1"></times><ci id="S5.p1.2.m2.2.2.2.2.1.1.3.2.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1.3.2">𝑚</ci><ci id="S5.p1.2.m2.2.2.2.2.1.1.3.3.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1.3.3">𝑖</ci><ci id="S5.p1.2.m2.2.2.2.2.1.1.3.4.cmml" xref="S5.p1.2.m2.2.2.2.2.1.1.3.4">𝑛</ci></apply></apply></apply></apply><ci id="S5.p1.2.m2.2.2.4.cmml" xref="S5.p1.2.m2.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.2c">\alpha<\frac{\left|p_{max}\right|-\left|p_{min}\right|}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.2d">italic_α < divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>, then the agents can strictly decrease their worst-case cost by never picking <math alttext="p_{max}" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><msub id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2" xref="S5.p1.3.m3.1.1.2.cmml">p</mi><mrow id="S5.p1.3.m3.1.1.3" xref="S5.p1.3.m3.1.1.3.cmml"><mi id="S5.p1.3.m3.1.1.3.2" xref="S5.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="S5.p1.3.m3.1.1.3.1" xref="S5.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.3.m3.1.1.3.3" xref="S5.p1.3.m3.1.1.3.3.cmml">a</mi><mo id="S5.p1.3.m3.1.1.3.1a" xref="S5.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.3.m3.1.1.3.4" xref="S5.p1.3.m3.1.1.3.4.cmml">x</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1">subscript</csymbol><ci id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1.2">𝑝</ci><apply id="S5.p1.3.m3.1.1.3.cmml" xref="S5.p1.3.m3.1.1.3"><times id="S5.p1.3.m3.1.1.3.1.cmml" xref="S5.p1.3.m3.1.1.3.1"></times><ci id="S5.p1.3.m3.1.1.3.2.cmml" xref="S5.p1.3.m3.1.1.3.2">𝑚</ci><ci id="S5.p1.3.m3.1.1.3.3.cmml" xref="S5.p1.3.m3.1.1.3.3">𝑎</ci><ci id="S5.p1.3.m3.1.1.3.4.cmml" xref="S5.p1.3.m3.1.1.3.4">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">p_{max}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Then, <math alttext="P_{st}" class="ltx_Math" display="inline" id="S5.p1.4.m4.1"><semantics id="S5.p1.4.m4.1a"><msub id="S5.p1.4.m4.1.1" xref="S5.p1.4.m4.1.1.cmml"><mi id="S5.p1.4.m4.1.1.2" xref="S5.p1.4.m4.1.1.2.cmml">P</mi><mrow id="S5.p1.4.m4.1.1.3" xref="S5.p1.4.m4.1.1.3.cmml"><mi id="S5.p1.4.m4.1.1.3.2" xref="S5.p1.4.m4.1.1.3.2.cmml">s</mi><mo id="S5.p1.4.m4.1.1.3.1" xref="S5.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.4.m4.1.1.3.3" xref="S5.p1.4.m4.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.p1.4.m4.1b"><apply id="S5.p1.4.m4.1.1.cmml" xref="S5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.p1.4.m4.1.1.1.cmml" xref="S5.p1.4.m4.1.1">subscript</csymbol><ci id="S5.p1.4.m4.1.1.2.cmml" xref="S5.p1.4.m4.1.1.2">𝑃</ci><apply id="S5.p1.4.m4.1.1.3.cmml" xref="S5.p1.4.m4.1.1.3"><times id="S5.p1.4.m4.1.1.3.1.cmml" xref="S5.p1.4.m4.1.1.3.1"></times><ci id="S5.p1.4.m4.1.1.3.2.cmml" xref="S5.p1.4.m4.1.1.3.2">𝑠</ci><ci id="S5.p1.4.m4.1.1.3.3.cmml" xref="S5.p1.4.m4.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math> becomes effectively <math alttext="P_{st}\setminus\{p_{max}\}" class="ltx_Math" display="inline" id="S5.p1.5.m5.1"><semantics id="S5.p1.5.m5.1a"><mrow id="S5.p1.5.m5.1.1" xref="S5.p1.5.m5.1.1.cmml"><msub id="S5.p1.5.m5.1.1.3" xref="S5.p1.5.m5.1.1.3.cmml"><mi id="S5.p1.5.m5.1.1.3.2" xref="S5.p1.5.m5.1.1.3.2.cmml">P</mi><mrow id="S5.p1.5.m5.1.1.3.3" xref="S5.p1.5.m5.1.1.3.3.cmml"><mi id="S5.p1.5.m5.1.1.3.3.2" xref="S5.p1.5.m5.1.1.3.3.2.cmml">s</mi><mo id="S5.p1.5.m5.1.1.3.3.1" xref="S5.p1.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="S5.p1.5.m5.1.1.3.3.3" xref="S5.p1.5.m5.1.1.3.3.3.cmml">t</mi></mrow></msub><mo id="S5.p1.5.m5.1.1.2" xref="S5.p1.5.m5.1.1.2.cmml">∖</mo><mrow id="S5.p1.5.m5.1.1.1.1" xref="S5.p1.5.m5.1.1.1.2.cmml"><mo id="S5.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S5.p1.5.m5.1.1.1.2.cmml">{</mo><msub id="S5.p1.5.m5.1.1.1.1.1" xref="S5.p1.5.m5.1.1.1.1.1.cmml"><mi id="S5.p1.5.m5.1.1.1.1.1.2" xref="S5.p1.5.m5.1.1.1.1.1.2.cmml">p</mi><mrow id="S5.p1.5.m5.1.1.1.1.1.3" xref="S5.p1.5.m5.1.1.1.1.1.3.cmml"><mi id="S5.p1.5.m5.1.1.1.1.1.3.2" xref="S5.p1.5.m5.1.1.1.1.1.3.2.cmml">m</mi><mo id="S5.p1.5.m5.1.1.1.1.1.3.1" xref="S5.p1.5.m5.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.5.m5.1.1.1.1.1.3.3" xref="S5.p1.5.m5.1.1.1.1.1.3.3.cmml">a</mi><mo id="S5.p1.5.m5.1.1.1.1.1.3.1a" xref="S5.p1.5.m5.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.5.m5.1.1.1.1.1.3.4" xref="S5.p1.5.m5.1.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="S5.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S5.p1.5.m5.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.5.m5.1b"><apply id="S5.p1.5.m5.1.1.cmml" xref="S5.p1.5.m5.1.1"><setdiff id="S5.p1.5.m5.1.1.2.cmml" xref="S5.p1.5.m5.1.1.2"></setdiff><apply id="S5.p1.5.m5.1.1.3.cmml" xref="S5.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.p1.5.m5.1.1.3.1.cmml" xref="S5.p1.5.m5.1.1.3">subscript</csymbol><ci id="S5.p1.5.m5.1.1.3.2.cmml" xref="S5.p1.5.m5.1.1.3.2">𝑃</ci><apply id="S5.p1.5.m5.1.1.3.3.cmml" xref="S5.p1.5.m5.1.1.3.3"><times id="S5.p1.5.m5.1.1.3.3.1.cmml" xref="S5.p1.5.m5.1.1.3.3.1"></times><ci id="S5.p1.5.m5.1.1.3.3.2.cmml" xref="S5.p1.5.m5.1.1.3.3.2">𝑠</ci><ci id="S5.p1.5.m5.1.1.3.3.3.cmml" xref="S5.p1.5.m5.1.1.3.3.3">𝑡</ci></apply></apply><set id="S5.p1.5.m5.1.1.1.2.cmml" xref="S5.p1.5.m5.1.1.1.1"><apply id="S5.p1.5.m5.1.1.1.1.1.cmml" xref="S5.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p1.5.m5.1.1.1.1.1.1.cmml" xref="S5.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S5.p1.5.m5.1.1.1.1.1.2.cmml" xref="S5.p1.5.m5.1.1.1.1.1.2">𝑝</ci><apply id="S5.p1.5.m5.1.1.1.1.1.3.cmml" xref="S5.p1.5.m5.1.1.1.1.1.3"><times id="S5.p1.5.m5.1.1.1.1.1.3.1.cmml" xref="S5.p1.5.m5.1.1.1.1.1.3.1"></times><ci id="S5.p1.5.m5.1.1.1.1.1.3.2.cmml" xref="S5.p1.5.m5.1.1.1.1.1.3.2">𝑚</ci><ci id="S5.p1.5.m5.1.1.1.1.1.3.3.cmml" xref="S5.p1.5.m5.1.1.1.1.1.3.3">𝑎</ci><ci id="S5.p1.5.m5.1.1.1.1.1.3.4.cmml" xref="S5.p1.5.m5.1.1.1.1.1.3.4">𝑥</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.5.m5.1c">P_{st}\setminus\{p_{max}\}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.5.m5.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ∖ { italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT }</annotation></semantics></math>. For simplicity, we do not consider the case <math alttext="\alpha=\frac{\left|p_{max}\right|-\left|p_{min}\right|}{k}" class="ltx_Math" display="inline" id="S5.p1.6.m6.2"><semantics id="S5.p1.6.m6.2a"><mrow id="S5.p1.6.m6.2.3" xref="S5.p1.6.m6.2.3.cmml"><mi id="S5.p1.6.m6.2.3.2" xref="S5.p1.6.m6.2.3.2.cmml">α</mi><mo id="S5.p1.6.m6.2.3.1" xref="S5.p1.6.m6.2.3.1.cmml">=</mo><mfrac id="S5.p1.6.m6.2.2" xref="S5.p1.6.m6.2.2.cmml"><mrow id="S5.p1.6.m6.2.2.2" xref="S5.p1.6.m6.2.2.2.cmml"><mrow id="S5.p1.6.m6.1.1.1.1.1" xref="S5.p1.6.m6.1.1.1.1.2.cmml"><mo id="S5.p1.6.m6.1.1.1.1.1.2" xref="S5.p1.6.m6.1.1.1.1.2.1.cmml">|</mo><msub id="S5.p1.6.m6.1.1.1.1.1.1" xref="S5.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.p1.6.m6.1.1.1.1.1.1.2" xref="S5.p1.6.m6.1.1.1.1.1.1.2.cmml">p</mi><mrow id="S5.p1.6.m6.1.1.1.1.1.1.3" xref="S5.p1.6.m6.1.1.1.1.1.1.3.cmml"><mi id="S5.p1.6.m6.1.1.1.1.1.1.3.2" xref="S5.p1.6.m6.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="S5.p1.6.m6.1.1.1.1.1.1.3.1" xref="S5.p1.6.m6.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.6.m6.1.1.1.1.1.1.3.3" xref="S5.p1.6.m6.1.1.1.1.1.1.3.3.cmml">a</mi><mo id="S5.p1.6.m6.1.1.1.1.1.1.3.1a" xref="S5.p1.6.m6.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.6.m6.1.1.1.1.1.1.3.4" xref="S5.p1.6.m6.1.1.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="S5.p1.6.m6.1.1.1.1.1.3" xref="S5.p1.6.m6.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.p1.6.m6.2.2.2.3" xref="S5.p1.6.m6.2.2.2.3.cmml">−</mo><mrow id="S5.p1.6.m6.2.2.2.2.1" xref="S5.p1.6.m6.2.2.2.2.2.cmml"><mo id="S5.p1.6.m6.2.2.2.2.1.2" xref="S5.p1.6.m6.2.2.2.2.2.1.cmml">|</mo><msub id="S5.p1.6.m6.2.2.2.2.1.1" xref="S5.p1.6.m6.2.2.2.2.1.1.cmml"><mi id="S5.p1.6.m6.2.2.2.2.1.1.2" xref="S5.p1.6.m6.2.2.2.2.1.1.2.cmml">p</mi><mrow id="S5.p1.6.m6.2.2.2.2.1.1.3" xref="S5.p1.6.m6.2.2.2.2.1.1.3.cmml"><mi id="S5.p1.6.m6.2.2.2.2.1.1.3.2" xref="S5.p1.6.m6.2.2.2.2.1.1.3.2.cmml">m</mi><mo id="S5.p1.6.m6.2.2.2.2.1.1.3.1" xref="S5.p1.6.m6.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.6.m6.2.2.2.2.1.1.3.3" xref="S5.p1.6.m6.2.2.2.2.1.1.3.3.cmml">i</mi><mo id="S5.p1.6.m6.2.2.2.2.1.1.3.1a" xref="S5.p1.6.m6.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S5.p1.6.m6.2.2.2.2.1.1.3.4" xref="S5.p1.6.m6.2.2.2.2.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S5.p1.6.m6.2.2.2.2.1.3" xref="S5.p1.6.m6.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mi id="S5.p1.6.m6.2.2.4" xref="S5.p1.6.m6.2.2.4.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.6.m6.2b"><apply id="S5.p1.6.m6.2.3.cmml" xref="S5.p1.6.m6.2.3"><eq id="S5.p1.6.m6.2.3.1.cmml" xref="S5.p1.6.m6.2.3.1"></eq><ci id="S5.p1.6.m6.2.3.2.cmml" xref="S5.p1.6.m6.2.3.2">𝛼</ci><apply id="S5.p1.6.m6.2.2.cmml" xref="S5.p1.6.m6.2.2"><divide id="S5.p1.6.m6.2.2.3.cmml" xref="S5.p1.6.m6.2.2"></divide><apply id="S5.p1.6.m6.2.2.2.cmml" xref="S5.p1.6.m6.2.2.2"><minus id="S5.p1.6.m6.2.2.2.3.cmml" xref="S5.p1.6.m6.2.2.2.3"></minus><apply id="S5.p1.6.m6.1.1.1.1.2.cmml" xref="S5.p1.6.m6.1.1.1.1.1"><abs id="S5.p1.6.m6.1.1.1.1.2.1.cmml" xref="S5.p1.6.m6.1.1.1.1.1.2"></abs><apply id="S5.p1.6.m6.1.1.1.1.1.1.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1.2">𝑝</ci><apply id="S5.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1.3"><times id="S5.p1.6.m6.1.1.1.1.1.1.3.1.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1.3.1"></times><ci id="S5.p1.6.m6.1.1.1.1.1.1.3.2.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1.3.2">𝑚</ci><ci id="S5.p1.6.m6.1.1.1.1.1.1.3.3.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1.3.3">𝑎</ci><ci id="S5.p1.6.m6.1.1.1.1.1.1.3.4.cmml" xref="S5.p1.6.m6.1.1.1.1.1.1.3.4">𝑥</ci></apply></apply></apply><apply id="S5.p1.6.m6.2.2.2.2.2.cmml" xref="S5.p1.6.m6.2.2.2.2.1"><abs id="S5.p1.6.m6.2.2.2.2.2.1.cmml" xref="S5.p1.6.m6.2.2.2.2.1.2"></abs><apply id="S5.p1.6.m6.2.2.2.2.1.1.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.p1.6.m6.2.2.2.2.1.1.1.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1">subscript</csymbol><ci id="S5.p1.6.m6.2.2.2.2.1.1.2.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1.2">𝑝</ci><apply id="S5.p1.6.m6.2.2.2.2.1.1.3.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1.3"><times id="S5.p1.6.m6.2.2.2.2.1.1.3.1.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1.3.1"></times><ci id="S5.p1.6.m6.2.2.2.2.1.1.3.2.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1.3.2">𝑚</ci><ci id="S5.p1.6.m6.2.2.2.2.1.1.3.3.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1.3.3">𝑖</ci><ci id="S5.p1.6.m6.2.2.2.2.1.1.3.4.cmml" xref="S5.p1.6.m6.2.2.2.2.1.1.3.4">𝑛</ci></apply></apply></apply></apply><ci id="S5.p1.6.m6.2.2.4.cmml" xref="S5.p1.6.m6.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.6.m6.2c">\alpha=\frac{\left|p_{max}\right|-\left|p_{min}\right|}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.6.m6.2d">italic_α = divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> here.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">In short, the main result here is that in nontrivial game instances with the MAX cost, <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><msub id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p2.1.m1.1.1.2" xref="S5.p2.1.m1.1.1.2.cmml">𝒰</mi><mi id="S5.p2.1.m1.1.1.3" xref="S5.p2.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><apply id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.1.m1.1.1.1.cmml" xref="S5.p2.1.m1.1.1">subscript</csymbol><ci id="S5.p2.1.m1.1.1.2.cmml" xref="S5.p2.1.m1.1.1.2">𝒰</ci><ci id="S5.p2.1.m1.1.1.3.cmml" xref="S5.p2.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> does not admit an optimal agent strategy (Theorems <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem1" title="Theorem 5.1 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem2" title="Theorem 5.2 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.2</span></a>).</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">In Appendix <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.SS1" title="0.D.1 Illustrating example ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.D.1</span></a> we illustrate this with an example.</p> </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.1 </span>General <math alttext="s-t" class="ltx_Math" display="inline" id="S5.SS1.1.m1.1"><semantics id="S5.SS1.1.m1.1b"><mrow id="S5.SS1.1.m1.1.1" xref="S5.SS1.1.m1.1.1.cmml"><mi id="S5.SS1.1.m1.1.1.2" xref="S5.SS1.1.m1.1.1.2.cmml">s</mi><mo id="S5.SS1.1.m1.1.1.1" xref="S5.SS1.1.m1.1.1.1.cmml">−</mo><mi id="S5.SS1.1.m1.1.1.3" xref="S5.SS1.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.m1.1c"><apply id="S5.SS1.1.m1.1.1.cmml" xref="S5.SS1.1.m1.1.1"><minus id="S5.SS1.1.m1.1.1.1.cmml" xref="S5.SS1.1.m1.1.1.1"></minus><ci id="S5.SS1.1.m1.1.1.2.cmml" xref="S5.SS1.1.m1.1.1.2">𝑠</ci><ci id="S5.SS1.1.m1.1.1.3.cmml" xref="S5.SS1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.m1.1d">s-t</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.m1.1e">italic_s - italic_t</annotation></semantics></math> graphs</h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.7">Again, we have the function <math alttext="F_{n}(\sigma,\alpha)=\max_{d\in E^{k}}f_{d}(\sigma,\alpha)" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.4"><semantics id="S5.SS1.p1.1.m1.4a"><mrow id="S5.SS1.p1.1.m1.4.5" xref="S5.SS1.p1.1.m1.4.5.cmml"><mrow id="S5.SS1.p1.1.m1.4.5.2" xref="S5.SS1.p1.1.m1.4.5.2.cmml"><msub id="S5.SS1.p1.1.m1.4.5.2.2" xref="S5.SS1.p1.1.m1.4.5.2.2.cmml"><mi id="S5.SS1.p1.1.m1.4.5.2.2.2" xref="S5.SS1.p1.1.m1.4.5.2.2.2.cmml">F</mi><mi id="S5.SS1.p1.1.m1.4.5.2.2.3" xref="S5.SS1.p1.1.m1.4.5.2.2.3.cmml">n</mi></msub><mo id="S5.SS1.p1.1.m1.4.5.2.1" xref="S5.SS1.p1.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="S5.SS1.p1.1.m1.4.5.2.3.2" xref="S5.SS1.p1.1.m1.4.5.2.3.1.cmml"><mo id="S5.SS1.p1.1.m1.4.5.2.3.2.1" stretchy="false" xref="S5.SS1.p1.1.m1.4.5.2.3.1.cmml">(</mo><mi id="S5.SS1.p1.1.m1.1.1" xref="S5.SS1.p1.1.m1.1.1.cmml">σ</mi><mo id="S5.SS1.p1.1.m1.4.5.2.3.2.2" xref="S5.SS1.p1.1.m1.4.5.2.3.1.cmml">,</mo><mi id="S5.SS1.p1.1.m1.2.2" xref="S5.SS1.p1.1.m1.2.2.cmml">α</mi><mo id="S5.SS1.p1.1.m1.4.5.2.3.2.3" stretchy="false" xref="S5.SS1.p1.1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.p1.1.m1.4.5.1" xref="S5.SS1.p1.1.m1.4.5.1.cmml">=</mo><mrow id="S5.SS1.p1.1.m1.4.5.3" xref="S5.SS1.p1.1.m1.4.5.3.cmml"><mrow id="S5.SS1.p1.1.m1.4.5.3.2" xref="S5.SS1.p1.1.m1.4.5.3.2.cmml"><msub id="S5.SS1.p1.1.m1.4.5.3.2.1" xref="S5.SS1.p1.1.m1.4.5.3.2.1.cmml"><mi id="S5.SS1.p1.1.m1.4.5.3.2.1.2" xref="S5.SS1.p1.1.m1.4.5.3.2.1.2.cmml">max</mi><mrow 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xref="S5.SS1.p1.1.m1.4.5.3.3.2"><ci id="S5.SS1.p1.1.m1.3.3.cmml" xref="S5.SS1.p1.1.m1.3.3">𝜎</ci><ci id="S5.SS1.p1.1.m1.4.4.cmml" xref="S5.SS1.p1.1.m1.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.4c">F_{n}(\sigma,\alpha)=\max_{d\in E^{k}}f_{d}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = roman_max start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math> for every <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="S5.SS1.p1.2.m2.1"><semantics id="S5.SS1.p1.2.m2.1a"><mrow id="S5.SS1.p1.2.m2.1.1" xref="S5.SS1.p1.2.m2.1.1.cmml"><mi id="S5.SS1.p1.2.m2.1.1.2" xref="S5.SS1.p1.2.m2.1.1.2.cmml">n</mi><mo id="S5.SS1.p1.2.m2.1.1.1" xref="S5.SS1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S5.SS1.p1.2.m2.1.1.3" xref="S5.SS1.p1.2.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.2.m2.1b"><apply id="S5.SS1.p1.2.m2.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1"><in id="S5.SS1.p1.2.m2.1.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1.1"></in><ci id="S5.SS1.p1.2.m2.1.1.2.cmml" xref="S5.SS1.p1.2.m2.1.1.2">𝑛</ci><ci id="S5.SS1.p1.2.m2.1.1.3.cmml" xref="S5.SS1.p1.2.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.2.m2.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.2.m2.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, however <math alttext="F_{n}" class="ltx_Math" display="inline" id="S5.SS1.p1.3.m3.1"><semantics id="S5.SS1.p1.3.m3.1a"><msub id="S5.SS1.p1.3.m3.1.1" xref="S5.SS1.p1.3.m3.1.1.cmml"><mi id="S5.SS1.p1.3.m3.1.1.2" xref="S5.SS1.p1.3.m3.1.1.2.cmml">F</mi><mi id="S5.SS1.p1.3.m3.1.1.3" xref="S5.SS1.p1.3.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.3.m3.1b"><apply id="S5.SS1.p1.3.m3.1.1.cmml" xref="S5.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.1.1.1.cmml" xref="S5.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S5.SS1.p1.3.m3.1.1.2.cmml" xref="S5.SS1.p1.3.m3.1.1.2">𝐹</ci><ci id="S5.SS1.p1.3.m3.1.1.3.cmml" xref="S5.SS1.p1.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.3.m3.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.3.m3.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> doesn’t have the same form as in Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a> for <math alttext="n>1" class="ltx_Math" display="inline" id="S5.SS1.p1.4.m4.1"><semantics id="S5.SS1.p1.4.m4.1a"><mrow id="S5.SS1.p1.4.m4.1.1" xref="S5.SS1.p1.4.m4.1.1.cmml"><mi id="S5.SS1.p1.4.m4.1.1.2" xref="S5.SS1.p1.4.m4.1.1.2.cmml">n</mi><mo id="S5.SS1.p1.4.m4.1.1.1" xref="S5.SS1.p1.4.m4.1.1.1.cmml">></mo><mn id="S5.SS1.p1.4.m4.1.1.3" xref="S5.SS1.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.4.m4.1b"><apply id="S5.SS1.p1.4.m4.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1"><gt id="S5.SS1.p1.4.m4.1.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1"></gt><ci id="S5.SS1.p1.4.m4.1.1.2.cmml" xref="S5.SS1.p1.4.m4.1.1.2">𝑛</ci><cn id="S5.SS1.p1.4.m4.1.1.3.cmml" type="integer" xref="S5.SS1.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.4.m4.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.4.m4.1d">italic_n > 1</annotation></semantics></math>. However, <math alttext="F_{1}" class="ltx_Math" display="inline" id="S5.SS1.p1.5.m5.1"><semantics id="S5.SS1.p1.5.m5.1a"><msub id="S5.SS1.p1.5.m5.1.1" xref="S5.SS1.p1.5.m5.1.1.cmml"><mi id="S5.SS1.p1.5.m5.1.1.2" xref="S5.SS1.p1.5.m5.1.1.2.cmml">F</mi><mn id="S5.SS1.p1.5.m5.1.1.3" xref="S5.SS1.p1.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.5.m5.1b"><apply id="S5.SS1.p1.5.m5.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.5.m5.1.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S5.SS1.p1.5.m5.1.1.2.cmml" xref="S5.SS1.p1.5.m5.1.1.2">𝐹</ci><cn id="S5.SS1.p1.5.m5.1.1.3.cmml" type="integer" xref="S5.SS1.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.5.m5.1c">F_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.5.m5.1d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> does collide with the definition that results from taking SUM cost, which we have analyzed. Therefore, we only have <math alttext="F_{n}" class="ltx_Math" display="inline" id="S5.SS1.p1.6.m6.1"><semantics id="S5.SS1.p1.6.m6.1a"><msub id="S5.SS1.p1.6.m6.1.1" xref="S5.SS1.p1.6.m6.1.1.cmml"><mi id="S5.SS1.p1.6.m6.1.1.2" xref="S5.SS1.p1.6.m6.1.1.2.cmml">F</mi><mi id="S5.SS1.p1.6.m6.1.1.3" xref="S5.SS1.p1.6.m6.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.6.m6.1b"><apply id="S5.SS1.p1.6.m6.1.1.cmml" xref="S5.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.1.1.1.cmml" xref="S5.SS1.p1.6.m6.1.1">subscript</csymbol><ci id="S5.SS1.p1.6.m6.1.1.2.cmml" xref="S5.SS1.p1.6.m6.1.1.2">𝐹</ci><ci id="S5.SS1.p1.6.m6.1.1.3.cmml" xref="S5.SS1.p1.6.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.6.m6.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.6.m6.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="n>1" class="ltx_Math" display="inline" id="S5.SS1.p1.7.m7.1"><semantics id="S5.SS1.p1.7.m7.1a"><mrow id="S5.SS1.p1.7.m7.1.1" xref="S5.SS1.p1.7.m7.1.1.cmml"><mi id="S5.SS1.p1.7.m7.1.1.2" xref="S5.SS1.p1.7.m7.1.1.2.cmml">n</mi><mo id="S5.SS1.p1.7.m7.1.1.1" xref="S5.SS1.p1.7.m7.1.1.1.cmml">></mo><mn id="S5.SS1.p1.7.m7.1.1.3" xref="S5.SS1.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.7.m7.1b"><apply id="S5.SS1.p1.7.m7.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1"><gt id="S5.SS1.p1.7.m7.1.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1.1"></gt><ci id="S5.SS1.p1.7.m7.1.1.2.cmml" xref="S5.SS1.p1.7.m7.1.1.2">𝑛</ci><cn id="S5.SS1.p1.7.m7.1.1.3.cmml" type="integer" xref="S5.SS1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.7.m7.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.7.m7.1d">italic_n > 1</annotation></semantics></math> to analyze.</p> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.1">We begin by establishing Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.1</span></a>, Corollary <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem2" title="Theorem 4.2 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.2</span></a> and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem3" title="Theorem 4.3 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.3</span></a> from Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a> but for the MAX cost. The proofs are almost identical, regardless of the cost function. In order for these proofs to go through, it suffices to prove the following lemma:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma12"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 12</span></h6> <div class="ltx_para" id="Thmlemma12.p1"> <p class="ltx_p" id="Thmlemma12.p1.1"><span class="ltx_text ltx_font_italic" id="Thmlemma12.p1.1.1">For every <math alttext="\sigma\in\Delta(P_{st})" class="ltx_Math" display="inline" id="Thmlemma12.p1.1.1.m1.1"><semantics id="Thmlemma12.p1.1.1.m1.1a"><mrow id="Thmlemma12.p1.1.1.m1.1.1" xref="Thmlemma12.p1.1.1.m1.1.1.cmml"><mi id="Thmlemma12.p1.1.1.m1.1.1.3" xref="Thmlemma12.p1.1.1.m1.1.1.3.cmml">σ</mi><mo id="Thmlemma12.p1.1.1.m1.1.1.2" xref="Thmlemma12.p1.1.1.m1.1.1.2.cmml">∈</mo><mrow id="Thmlemma12.p1.1.1.m1.1.1.1" xref="Thmlemma12.p1.1.1.m1.1.1.1.cmml"><mi id="Thmlemma12.p1.1.1.m1.1.1.1.3" mathvariant="normal" xref="Thmlemma12.p1.1.1.m1.1.1.1.3.cmml">Δ</mi><mo id="Thmlemma12.p1.1.1.m1.1.1.1.2" xref="Thmlemma12.p1.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma12.p1.1.1.m1.1.1.1.1.1" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.2" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.2" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.1" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.3" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma12.p1.1.1.m1.1b"><apply id="Thmlemma12.p1.1.1.m1.1.1.cmml" xref="Thmlemma12.p1.1.1.m1.1.1"><in id="Thmlemma12.p1.1.1.m1.1.1.2.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.2"></in><ci id="Thmlemma12.p1.1.1.m1.1.1.3.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.3">𝜎</ci><apply id="Thmlemma12.p1.1.1.m1.1.1.1.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1"><times id="Thmlemma12.p1.1.1.m1.1.1.1.2.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.2"></times><ci id="Thmlemma12.p1.1.1.m1.1.1.1.3.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.3">Δ</ci><apply id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3"><times id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.1"></times><ci id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="Thmlemma12.p1.1.1.m1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma12.p1.1.1.m1.1c">\sigma\in\Delta(P_{st})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma12.p1.1.1.m1.1d">italic_σ ∈ roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT )</annotation></semantics></math>,</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx11"> <tbody id="S5.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left|p_{min}\right|+k\alpha\leq F_{n}(\sigma,\alpha)\leq\left|p_% {max}\right|+k\alpha\leq n\left|p_{max}\right|+k\alpha" class="ltx_Math" display="inline" id="S5.Ex10.m1.5"><semantics id="S5.Ex10.m1.5a"><mrow id="S5.Ex10.m1.5.5" xref="S5.Ex10.m1.5.5.cmml"><mrow id="S5.Ex10.m1.3.3.1" xref="S5.Ex10.m1.3.3.1.cmml"><mrow id="S5.Ex10.m1.3.3.1.1.1" xref="S5.Ex10.m1.3.3.1.1.2.cmml"><mo id="S5.Ex10.m1.3.3.1.1.1.2" xref="S5.Ex10.m1.3.3.1.1.2.1.cmml">|</mo><msub id="S5.Ex10.m1.3.3.1.1.1.1" xref="S5.Ex10.m1.3.3.1.1.1.1.cmml"><mi id="S5.Ex10.m1.3.3.1.1.1.1.2" xref="S5.Ex10.m1.3.3.1.1.1.1.2.cmml">p</mi><mrow id="S5.Ex10.m1.3.3.1.1.1.1.3" xref="S5.Ex10.m1.3.3.1.1.1.1.3.cmml"><mi id="S5.Ex10.m1.3.3.1.1.1.1.3.2" xref="S5.Ex10.m1.3.3.1.1.1.1.3.2.cmml">m</mi><mo id="S5.Ex10.m1.3.3.1.1.1.1.3.1" xref="S5.Ex10.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.Ex10.m1.3.3.1.1.1.1.3.3" xref="S5.Ex10.m1.3.3.1.1.1.1.3.3.cmml">i</mi><mo id="S5.Ex10.m1.3.3.1.1.1.1.3.1a" xref="S5.Ex10.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.Ex10.m1.3.3.1.1.1.1.3.4" xref="S5.Ex10.m1.3.3.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S5.Ex10.m1.3.3.1.1.1.3" xref="S5.Ex10.m1.3.3.1.1.2.1.cmml">|</mo></mrow><mo id="S5.Ex10.m1.3.3.1.2" xref="S5.Ex10.m1.3.3.1.2.cmml">+</mo><mrow id="S5.Ex10.m1.3.3.1.3" xref="S5.Ex10.m1.3.3.1.3.cmml"><mi id="S5.Ex10.m1.3.3.1.3.2" xref="S5.Ex10.m1.3.3.1.3.2.cmml">k</mi><mo id="S5.Ex10.m1.3.3.1.3.1" xref="S5.Ex10.m1.3.3.1.3.1.cmml">⁢</mo><mi id="S5.Ex10.m1.3.3.1.3.3" xref="S5.Ex10.m1.3.3.1.3.3.cmml">α</mi></mrow></mrow><mo id="S5.Ex10.m1.5.5.5" xref="S5.Ex10.m1.5.5.5.cmml">≤</mo><mrow id="S5.Ex10.m1.5.5.6" xref="S5.Ex10.m1.5.5.6.cmml"><msub id="S5.Ex10.m1.5.5.6.2" xref="S5.Ex10.m1.5.5.6.2.cmml"><mi id="S5.Ex10.m1.5.5.6.2.2" xref="S5.Ex10.m1.5.5.6.2.2.cmml">F</mi><mi id="S5.Ex10.m1.5.5.6.2.3" xref="S5.Ex10.m1.5.5.6.2.3.cmml">n</mi></msub><mo id="S5.Ex10.m1.5.5.6.1" xref="S5.Ex10.m1.5.5.6.1.cmml">⁢</mo><mrow id="S5.Ex10.m1.5.5.6.3.2" 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id="S5.Ex10.m1.5c">\displaystyle\left|p_{min}\right|+k\alpha\leq F_{n}(\sigma,\alpha)\leq\left|p_% {max}\right|+k\alpha\leq n\left|p_{max}\right|+k\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.Ex10.m1.5d">| italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + italic_k italic_α ≤ italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≤ | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + italic_k italic_α ≤ italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Corollary 2</span></h6> <div class="ltx_para" id="Thmcorollary2.p1"> <p class="ltx_p" id="Thmcorollary2.p1.1"><span class="ltx_text ltx_font_italic" id="Thmcorollary2.p1.1.1">Theorems <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.1</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem2" title="Theorem 4.2 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.2</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem3" title="Theorem 4.3 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.3</span></a> all hold for the MAX cost verbatim.</span></p> </div> </div> <div class="ltx_para" id="S5.SS1.p3"> <p class="ltx_p" id="S5.SS1.p3.1">An analogue to Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma10" title="Lemma 10 ‣ 4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">10</span></a> holds for MAX as well, but for general <math alttext="s-t" class="ltx_Math" display="inline" id="S5.SS1.p3.1.m1.1"><semantics id="S5.SS1.p3.1.m1.1a"><mrow id="S5.SS1.p3.1.m1.1.1" xref="S5.SS1.p3.1.m1.1.1.cmml"><mi id="S5.SS1.p3.1.m1.1.1.2" xref="S5.SS1.p3.1.m1.1.1.2.cmml">s</mi><mo id="S5.SS1.p3.1.m1.1.1.1" xref="S5.SS1.p3.1.m1.1.1.1.cmml">−</mo><mi id="S5.SS1.p3.1.m1.1.1.3" xref="S5.SS1.p3.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.1.m1.1b"><apply id="S5.SS1.p3.1.m1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1"><minus id="S5.SS1.p3.1.m1.1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1.1"></minus><ci id="S5.SS1.p3.1.m1.1.1.2.cmml" xref="S5.SS1.p3.1.m1.1.1.2">𝑠</ci><ci id="S5.SS1.p3.1.m1.1.1.3.cmml" xref="S5.SS1.p3.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.1.m1.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.1.m1.1d">italic_s - italic_t</annotation></semantics></math> graphs:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma13"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 13</span></h6> <div class="ltx_para" id="Thmlemma13.p1"> <p class="ltx_p" id="Thmlemma13.p1.4"><span class="ltx_text ltx_font_italic" id="Thmlemma13.p1.4.4">When minimizing <math alttext="F_{n}(\cdot,\alpha)" class="ltx_Math" display="inline" id="Thmlemma13.p1.1.1.m1.2"><semantics id="Thmlemma13.p1.1.1.m1.2a"><mrow id="Thmlemma13.p1.1.1.m1.2.3" xref="Thmlemma13.p1.1.1.m1.2.3.cmml"><msub id="Thmlemma13.p1.1.1.m1.2.3.2" xref="Thmlemma13.p1.1.1.m1.2.3.2.cmml"><mi id="Thmlemma13.p1.1.1.m1.2.3.2.2" xref="Thmlemma13.p1.1.1.m1.2.3.2.2.cmml">F</mi><mi id="Thmlemma13.p1.1.1.m1.2.3.2.3" xref="Thmlemma13.p1.1.1.m1.2.3.2.3.cmml">n</mi></msub><mo id="Thmlemma13.p1.1.1.m1.2.3.1" xref="Thmlemma13.p1.1.1.m1.2.3.1.cmml">⁢</mo><mrow id="Thmlemma13.p1.1.1.m1.2.3.3.2" xref="Thmlemma13.p1.1.1.m1.2.3.3.1.cmml"><mo id="Thmlemma13.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="Thmlemma13.p1.1.1.m1.2.3.3.1.cmml">(</mo><mo id="Thmlemma13.p1.1.1.m1.1.1" lspace="0em" rspace="0em" xref="Thmlemma13.p1.1.1.m1.1.1.cmml">⋅</mo><mo id="Thmlemma13.p1.1.1.m1.2.3.3.2.2" xref="Thmlemma13.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="Thmlemma13.p1.1.1.m1.2.2" xref="Thmlemma13.p1.1.1.m1.2.2.cmml">α</mi><mo id="Thmlemma13.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="Thmlemma13.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma13.p1.1.1.m1.2b"><apply id="Thmlemma13.p1.1.1.m1.2.3.cmml" xref="Thmlemma13.p1.1.1.m1.2.3"><times id="Thmlemma13.p1.1.1.m1.2.3.1.cmml" xref="Thmlemma13.p1.1.1.m1.2.3.1"></times><apply id="Thmlemma13.p1.1.1.m1.2.3.2.cmml" xref="Thmlemma13.p1.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="Thmlemma13.p1.1.1.m1.2.3.2.1.cmml" xref="Thmlemma13.p1.1.1.m1.2.3.2">subscript</csymbol><ci id="Thmlemma13.p1.1.1.m1.2.3.2.2.cmml" xref="Thmlemma13.p1.1.1.m1.2.3.2.2">𝐹</ci><ci id="Thmlemma13.p1.1.1.m1.2.3.2.3.cmml" xref="Thmlemma13.p1.1.1.m1.2.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmlemma13.p1.1.1.m1.2.3.3.1.cmml" xref="Thmlemma13.p1.1.1.m1.2.3.3.2"><ci id="Thmlemma13.p1.1.1.m1.1.1.cmml" xref="Thmlemma13.p1.1.1.m1.1.1">⋅</ci><ci id="Thmlemma13.p1.1.1.m1.2.2.cmml" xref="Thmlemma13.p1.1.1.m1.2.2">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma13.p1.1.1.m1.2c">F_{n}(\cdot,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma13.p1.1.1.m1.2d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ⋅ , italic_α )</annotation></semantics></math> we can restrict ourselves to the set of iid distributions: all <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmlemma13.p1.2.2.m2.1"><semantics id="Thmlemma13.p1.2.2.m2.1a"><mi id="Thmlemma13.p1.2.2.m2.1.1" xref="Thmlemma13.p1.2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmlemma13.p1.2.2.m2.1b"><ci id="Thmlemma13.p1.2.2.m2.1.1.cmml" xref="Thmlemma13.p1.2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma13.p1.2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmlemma13.p1.2.2.m2.1d">italic_σ</annotation></semantics></math> which are the product distribution of <math alttext="n" class="ltx_Math" display="inline" id="Thmlemma13.p1.3.3.m3.1"><semantics id="Thmlemma13.p1.3.3.m3.1a"><mi id="Thmlemma13.p1.3.3.m3.1.1" xref="Thmlemma13.p1.3.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmlemma13.p1.3.3.m3.1b"><ci id="Thmlemma13.p1.3.3.m3.1.1.cmml" xref="Thmlemma13.p1.3.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma13.p1.3.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmlemma13.p1.3.3.m3.1d">italic_n</annotation></semantics></math> copies of a distribution on <math alttext="P_{st}" class="ltx_Math" display="inline" id="Thmlemma13.p1.4.4.m4.1"><semantics id="Thmlemma13.p1.4.4.m4.1a"><msub id="Thmlemma13.p1.4.4.m4.1.1" xref="Thmlemma13.p1.4.4.m4.1.1.cmml"><mi id="Thmlemma13.p1.4.4.m4.1.1.2" xref="Thmlemma13.p1.4.4.m4.1.1.2.cmml">P</mi><mrow id="Thmlemma13.p1.4.4.m4.1.1.3" xref="Thmlemma13.p1.4.4.m4.1.1.3.cmml"><mi id="Thmlemma13.p1.4.4.m4.1.1.3.2" xref="Thmlemma13.p1.4.4.m4.1.1.3.2.cmml">s</mi><mo id="Thmlemma13.p1.4.4.m4.1.1.3.1" xref="Thmlemma13.p1.4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="Thmlemma13.p1.4.4.m4.1.1.3.3" xref="Thmlemma13.p1.4.4.m4.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmlemma13.p1.4.4.m4.1b"><apply id="Thmlemma13.p1.4.4.m4.1.1.cmml" xref="Thmlemma13.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="Thmlemma13.p1.4.4.m4.1.1.1.cmml" xref="Thmlemma13.p1.4.4.m4.1.1">subscript</csymbol><ci id="Thmlemma13.p1.4.4.m4.1.1.2.cmml" xref="Thmlemma13.p1.4.4.m4.1.1.2">𝑃</ci><apply id="Thmlemma13.p1.4.4.m4.1.1.3.cmml" xref="Thmlemma13.p1.4.4.m4.1.1.3"><times id="Thmlemma13.p1.4.4.m4.1.1.3.1.cmml" xref="Thmlemma13.p1.4.4.m4.1.1.3.1"></times><ci id="Thmlemma13.p1.4.4.m4.1.1.3.2.cmml" xref="Thmlemma13.p1.4.4.m4.1.1.3.2">𝑠</ci><ci id="Thmlemma13.p1.4.4.m4.1.1.3.3.cmml" xref="Thmlemma13.p1.4.4.m4.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma13.p1.4.4.m4.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma13.p1.4.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS1.p4"> <p class="ltx_p" id="S5.SS1.p4.1">Before focusing on graphs of disjoint paths, we end with stating the second part of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.Thmtheorem3" title="Theorem 1.3 (Informal version of Theorems 5.1 and 5.3) ‣ 1.1 Our results ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1.3</span></a>:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 5.1</span></h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.9"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p1.9.9">Let <math alttext="G" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.1.m1.1"><semantics id="S5.Thmtheorem1.p1.1.1.m1.1a"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.1.m1.1b"><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be an <math alttext="s-t" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.2.2.m2.1"><semantics id="S5.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem1.p1.2.2.m2.1.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">s</mi><mo id="S5.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">−</mo><mi id="S5.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.2.2.m2.1b"><apply id="S5.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1"><minus id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1"></minus><ci id="S5.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.2">𝑠</ci><ci id="S5.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.2.m2.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.2.m2.1d">italic_s - italic_t</annotation></semantics></math> graph, and let <math alttext="n>1,k\geq 1" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.3.3.m3.2"><semantics id="S5.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.3.cmml"><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.2.cmml">n</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml">></mo><mn id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.3a.cmml">,</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.cmml"><mi id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.cmml">k</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.1.cmml">≥</mo><mn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.3.m3.2b"><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.3.3.m3.2.2.3a.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3">formulae-sequence</csymbol><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1"><gt id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"></gt><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.2">𝑛</ci><cn id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.3">1</cn></apply><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2"><geq id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.1"></geq><ci id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.2">𝑘</ci><cn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.3.m3.2c">n>1,k\geq 1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.3.m3.2d">italic_n > 1 , italic_k ≥ 1</annotation></semantics></math> such that the following holds: For every <math alttext="f\in MF" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.4.4.m4.1"><semantics id="S5.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S5.Thmtheorem1.p1.4.4.m4.1.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.cmml"><mi id="S5.Thmtheorem1.p1.4.4.m4.1.1.2" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.2.cmml">f</mi><mo id="S5.Thmtheorem1.p1.4.4.m4.1.1.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.1.cmml">∈</mo><mrow id="S5.Thmtheorem1.p1.4.4.m4.1.1.3" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.cmml"><mi id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.2" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.2.cmml">M</mi><mo id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.3" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.3.cmml">F</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.4.4.m4.1b"><apply id="S5.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1"><in id="S5.Thmtheorem1.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.1"></in><ci id="S5.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.2">𝑓</ci><apply id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3"><times id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.1"></times><ci id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.2">𝑀</ci><ci id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.3">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.4.4.m4.1c">f\in MF</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.4.4.m4.1d">italic_f ∈ italic_M italic_F</annotation></semantics></math>, and for <math alttext="P_{f}:=\left\{p_{1},...,p_{m^{\prime}}\right\}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.5.5.m5.3"><semantics id="S5.Thmtheorem1.p1.5.5.m5.3a"><mrow id="S5.Thmtheorem1.p1.5.5.m5.3.3" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.cmml"><msub id="S5.Thmtheorem1.p1.5.5.m5.3.3.4" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.4.cmml"><mi id="S5.Thmtheorem1.p1.5.5.m5.3.3.4.2" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.4.2.cmml">P</mi><mi id="S5.Thmtheorem1.p1.5.5.m5.3.3.4.3" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.4.3.cmml">f</mi></msub><mo id="S5.Thmtheorem1.p1.5.5.m5.3.3.3" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.3.cmml">:=</mo><mrow id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml"><mo id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.3" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml">{</mo><msub id="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1" xref="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2" xref="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2.cmml">p</mi><mn id="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3" xref="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.4" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml">,</mo><mi id="S5.Thmtheorem1.p1.5.5.m5.1.1" mathvariant="normal" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.cmml">…</mi><mo id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.5" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml">,</mo><msub id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.cmml"><mi id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.cmml">p</mi><msup id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.cmml"><mi id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.2" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.2.cmml">m</mi><mo id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.3" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.3.cmml">′</mo></msup></msub><mo id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.6" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.5.5.m5.3b"><apply id="S5.Thmtheorem1.p1.5.5.m5.3.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3"><csymbol cd="latexml" id="S5.Thmtheorem1.p1.5.5.m5.3.3.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.3">assign</csymbol><apply id="S5.Thmtheorem1.p1.5.5.m5.3.3.4.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.4"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.5.5.m5.3.3.4.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.4">subscript</csymbol><ci id="S5.Thmtheorem1.p1.5.5.m5.3.3.4.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.4.2">𝑃</ci><ci id="S5.Thmtheorem1.p1.5.5.m5.3.3.4.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.4.3">𝑓</ci></apply><set id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2"><apply id="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2">𝑝</ci><cn id="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3">1</cn></apply><ci id="S5.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.1.1">…</ci><apply id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2">𝑝</ci><apply id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3">superscript</csymbol><ci id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.2">𝑚</ci><ci id="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.3">′</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.5.5.m5.3c">P_{f}:=\left\{p_{1},...,p_{m^{\prime}}\right\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.5.5.m5.3d">italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT := { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math>, there holds <math alttext="|p_{1}|<|p_{m^{\prime}}|" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.6.6.m6.2"><semantics id="S5.Thmtheorem1.p1.6.6.m6.2a"><mrow id="S5.Thmtheorem1.p1.6.6.m6.2.2" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.cmml"><mrow id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.2.cmml"><mo id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.2.1.cmml">|</mo><msub id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.2" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.2.cmml">p</mi><mn id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.3" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.Thmtheorem1.p1.6.6.m6.2.2.3" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.3.cmml"><</mo><mrow id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.2.cmml"><mo id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.2.1.cmml">|</mo><msub id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.cmml"><mi id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.2" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.2.cmml">p</mi><msup id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.cmml"><mi id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.2" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.2.cmml">m</mi><mo id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.3" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.3.cmml">′</mo></msup></msub><mo id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.6.6.m6.2b"><apply id="S5.Thmtheorem1.p1.6.6.m6.2.2.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2"><lt id="S5.Thmtheorem1.p1.6.6.m6.2.2.3.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.3"></lt><apply id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1"><abs id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.2.1.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.2"></abs><apply id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.2">𝑝</ci><cn id="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.6.6.m6.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1"><abs id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.2.1.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.2"></abs><apply id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.2">𝑝</ci><apply id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.2">𝑚</ci><ci id="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.6.6.m6.2.2.2.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.6.6.m6.2c">|p_{1}|<|p_{m^{\prime}}|</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.6.6.m6.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math>. Also, <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.7.7.m7.1"><semantics id="S5.Thmtheorem1.p1.7.7.m7.1a"><mi id="S5.Thmtheorem1.p1.7.7.m7.1.1" xref="S5.Thmtheorem1.p1.7.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.7.7.m7.1b"><ci id="S5.Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="S5.Thmtheorem1.p1.7.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.7.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.7.7.m7.1d">italic_α</annotation></semantics></math> is large enough such that <math alttext="\min_{\sigma\in(\Delta(P_{f}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{\mathcal% {E}_{m^{\prime},n}}{m^{\prime}}k\alpha" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.8.8.m8.6"><semantics id="S5.Thmtheorem1.p1.8.8.m8.6a"><mrow id="S5.Thmtheorem1.p1.8.8.m8.6.6" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.cmml"><mrow 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id="S5.Thmtheorem1.p1.8.8.m8.6.6.3.3.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.3.3.2"><ci id="S5.Thmtheorem1.p1.8.8.m8.4.4.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.4.4">𝜎</ci><ci id="S5.Thmtheorem1.p1.8.8.m8.5.5.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.5.5">𝛼</ci></interval></apply><apply id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1"><plus id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.2.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.2"></plus><apply id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.2.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1"><abs id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.2.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.2"></abs><apply id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1.2">𝑝</ci><cn id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.1.1.1.3">1</cn></apply></apply><apply id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3"><times id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3.1"></times><apply id="S5.Thmtheorem1.p1.8.8.m8.3.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3"><divide id="S5.Thmtheorem1.p1.8.8.m8.3.3.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3"></divide><apply id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2">subscript</csymbol><ci id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.4.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2.4">ℰ</ci><list id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2"><apply id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1">superscript</csymbol><ci id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1.2">𝑚</ci><ci id="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.2.2.2.2.1.3">′</ci></apply><ci id="S5.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.1">𝑛</ci></list></apply><apply id="S5.Thmtheorem1.p1.8.8.m8.3.3.4.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.4"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.8.8.m8.3.3.4.1.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.4">superscript</csymbol><ci id="S5.Thmtheorem1.p1.8.8.m8.3.3.4.2.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.4.2">𝑚</ci><ci id="S5.Thmtheorem1.p1.8.8.m8.3.3.4.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.3.3.4.3">′</ci></apply></apply><ci id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3.2.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3.2">𝑘</ci><ci id="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3.3.cmml" xref="S5.Thmtheorem1.p1.8.8.m8.6.6.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.8.8.m8.6c">\min_{\sigma\in(\Delta(P_{f}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{\mathcal% {E}_{m^{\prime},n}}{m^{\prime}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.8.8.m8.6d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG italic_k italic_α</annotation></semantics></math> for every such <math alttext="f,P_{f}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.9.9.m9.2"><semantics id="S5.Thmtheorem1.p1.9.9.m9.2a"><mrow id="S5.Thmtheorem1.p1.9.9.m9.2.2.1" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.2.cmml"><mi id="S5.Thmtheorem1.p1.9.9.m9.1.1" xref="S5.Thmtheorem1.p1.9.9.m9.1.1.cmml">f</mi><mo id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.2" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.2.cmml">,</mo><msub id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.cmml"><mi id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.2" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.2.cmml">P</mi><mi id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.3" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.3.cmml">f</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.9.9.m9.2b"><list id="S5.Thmtheorem1.p1.9.9.m9.2.2.2.cmml" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1"><ci id="S5.Thmtheorem1.p1.9.9.m9.1.1.cmml" xref="S5.Thmtheorem1.p1.9.9.m9.1.1">𝑓</ci><apply id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.2">𝑃</ci><ci id="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.3.cmml" xref="S5.Thmtheorem1.p1.9.9.m9.2.2.1.1.3">𝑓</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.9.9.m9.2c">f,P_{f}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.9.9.m9.2d">italic_f , italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem1.p2"> <p class="ltx_p" id="S5.Thmtheorem1.p2.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p2.1.1">Then</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx12"> <tbody id="S5.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<\min_{% \sigma\in\mathcal{U}_{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="S5.Ex11.m1.5"><semantics id="S5.Ex11.m1.5a"><mrow id="S5.Ex11.m1.5.6" xref="S5.Ex11.m1.5.6.cmml"><mrow id="S5.Ex11.m1.5.6.2" xref="S5.Ex11.m1.5.6.2.cmml"><mrow id="S5.Ex11.m1.5.6.2.2" xref="S5.Ex11.m1.5.6.2.2.cmml"><munder id="S5.Ex11.m1.5.6.2.2.1" xref="S5.Ex11.m1.5.6.2.2.1.cmml"><mi id="S5.Ex11.m1.5.6.2.2.1.2" xref="S5.Ex11.m1.5.6.2.2.1.2.cmml">min</mi><mrow id="S5.Ex11.m1.1.1.1" xref="S5.Ex11.m1.1.1.1.cmml"><mi id="S5.Ex11.m1.1.1.1.3" xref="S5.Ex11.m1.1.1.1.3.cmml">σ</mi><mo id="S5.Ex11.m1.1.1.1.2" xref="S5.Ex11.m1.1.1.1.2.cmml">∈</mo><msup id="S5.Ex11.m1.1.1.1.1" xref="S5.Ex11.m1.1.1.1.1.cmml"><mrow id="S5.Ex11.m1.1.1.1.1.1.1" xref="S5.Ex11.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex11.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex11.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex11.m1.1.1.1.1.1.1.1" xref="S5.Ex11.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex11.m1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="S5.Ex11.m1.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="S5.Ex11.m1.1.1.1.1.1.1.1.2" xref="S5.Ex11.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex11.m1.1.1.1.1.1.1.1.1.1" xref="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex11.m1.1.1.1.1.1.1.1.1.1.1.3.2" 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xref="S5.Ex11.m1.5.6.3.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="S5.Ex11.m1.5.6.3.3.1.cmml" xref="S5.Ex11.m1.5.6.3.3.2"><ci id="S5.Ex11.m1.4.4.cmml" xref="S5.Ex11.m1.4.4">𝜎</ci><ci id="S5.Ex11.m1.5.5.cmml" xref="S5.Ex11.m1.5.5">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex11.m1.5c">\displaystyle\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<\min_{% \sigma\in\mathcal{U}_{n}}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex11.m1.5d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S5.SS1.p5"> <p class="ltx_p" id="S5.SS1.p5.2">When <math alttext="G" class="ltx_Math" display="inline" id="S5.SS1.p5.1.m1.1"><semantics id="S5.SS1.p5.1.m1.1a"><mi id="S5.SS1.p5.1.m1.1.1" xref="S5.SS1.p5.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.1.m1.1b"><ci id="S5.SS1.p5.1.m1.1.1.cmml" xref="S5.SS1.p5.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.1.m1.1d">italic_G</annotation></semantics></math> has only disjoint paths, the theorem has a simpler phrasing. Also, the proof of the general case follows immediately from the disjoint-paths case (using the definition of <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S5.SS1.p5.2.m2.1"><semantics id="S5.SS1.p5.2.m2.1a"><msub id="S5.SS1.p5.2.m2.1.1" xref="S5.SS1.p5.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.p5.2.m2.1.1.2" xref="S5.SS1.p5.2.m2.1.1.2.cmml">𝒰</mi><mi id="S5.SS1.p5.2.m2.1.1.3" xref="S5.SS1.p5.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.2.m2.1b"><apply id="S5.SS1.p5.2.m2.1.1.cmml" xref="S5.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.p5.2.m2.1.1.1.cmml" xref="S5.SS1.p5.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.p5.2.m2.1.1.2.cmml" xref="S5.SS1.p5.2.m2.1.1.2">𝒰</ci><ci id="S5.SS1.p5.2.m2.1.1.3.cmml" xref="S5.SS1.p5.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.2.m2.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.2.m2.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>), so we only prove the latter (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem2" title="Theorem 5.2 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.2</span></a>).</p> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.2 </span>Disjoint paths</h3> <section class="ltx_subsubsection" id="S5.SS2.SSSx1"> <h4 class="ltx_title ltx_title_subsubsection">The general <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.1.m1.1"><semantics id="S5.SS2.SSSx1.1.m1.1b"><mi id="S5.SS2.SSSx1.1.m1.1.1" xref="S5.SS2.SSSx1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.1.m1.1c"><ci id="S5.SS2.SSSx1.1.m1.1.1.cmml" xref="S5.SS2.SSSx1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.1.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.1.m1.1e">italic_k</annotation></semantics></math> case</h4> <div class="ltx_para" id="S5.SS2.SSSx1.p1"> <p class="ltx_p" id="S5.SS2.SSSx1.p1.2">As the paths are disjoint, we sometimes abuse notation by identifying each <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p1.1.m1.1"><semantics id="S5.SS2.SSSx1.p1.1.m1.1a"><mrow id="S5.SS2.SSSx1.p1.1.m1.1.1" xref="S5.SS2.SSSx1.p1.1.m1.1.1.cmml"><mi id="S5.SS2.SSSx1.p1.1.m1.1.1.2" xref="S5.SS2.SSSx1.p1.1.m1.1.1.2.cmml">d</mi><mo id="S5.SS2.SSSx1.p1.1.m1.1.1.1" xref="S5.SS2.SSSx1.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S5.SS2.SSSx1.p1.1.m1.1.1.3" xref="S5.SS2.SSSx1.p1.1.m1.1.1.3.cmml"><mi id="S5.SS2.SSSx1.p1.1.m1.1.1.3.2" xref="S5.SS2.SSSx1.p1.1.m1.1.1.3.2.cmml">E</mi><mi id="S5.SS2.SSSx1.p1.1.m1.1.1.3.3" xref="S5.SS2.SSSx1.p1.1.m1.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p1.1.m1.1b"><apply id="S5.SS2.SSSx1.p1.1.m1.1.1.cmml" xref="S5.SS2.SSSx1.p1.1.m1.1.1"><in id="S5.SS2.SSSx1.p1.1.m1.1.1.1.cmml" xref="S5.SS2.SSSx1.p1.1.m1.1.1.1"></in><ci id="S5.SS2.SSSx1.p1.1.m1.1.1.2.cmml" xref="S5.SS2.SSSx1.p1.1.m1.1.1.2">𝑑</ci><apply id="S5.SS2.SSSx1.p1.1.m1.1.1.3.cmml" xref="S5.SS2.SSSx1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p1.1.m1.1.1.3.1.cmml" xref="S5.SS2.SSSx1.p1.1.m1.1.1.3">superscript</csymbol><ci id="S5.SS2.SSSx1.p1.1.m1.1.1.3.2.cmml" xref="S5.SS2.SSSx1.p1.1.m1.1.1.3.2">𝐸</ci><ci id="S5.SS2.SSSx1.p1.1.m1.1.1.3.3.cmml" xref="S5.SS2.SSSx1.p1.1.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p1.1.m1.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p1.1.m1.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> with the corresponding tuple in <math alttext="[m]^{k}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p1.2.m2.1"><semantics id="S5.SS2.SSSx1.p1.2.m2.1a"><msup id="S5.SS2.SSSx1.p1.2.m2.1.2" xref="S5.SS2.SSSx1.p1.2.m2.1.2.cmml"><mrow id="S5.SS2.SSSx1.p1.2.m2.1.2.2.2" xref="S5.SS2.SSSx1.p1.2.m2.1.2.2.1.cmml"><mo id="S5.SS2.SSSx1.p1.2.m2.1.2.2.2.1" stretchy="false" xref="S5.SS2.SSSx1.p1.2.m2.1.2.2.1.1.cmml">[</mo><mi id="S5.SS2.SSSx1.p1.2.m2.1.1" xref="S5.SS2.SSSx1.p1.2.m2.1.1.cmml">m</mi><mo id="S5.SS2.SSSx1.p1.2.m2.1.2.2.2.2" stretchy="false" xref="S5.SS2.SSSx1.p1.2.m2.1.2.2.1.1.cmml">]</mo></mrow><mi id="S5.SS2.SSSx1.p1.2.m2.1.2.3" xref="S5.SS2.SSSx1.p1.2.m2.1.2.3.cmml">k</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p1.2.m2.1b"><apply id="S5.SS2.SSSx1.p1.2.m2.1.2.cmml" xref="S5.SS2.SSSx1.p1.2.m2.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p1.2.m2.1.2.1.cmml" xref="S5.SS2.SSSx1.p1.2.m2.1.2">superscript</csymbol><apply id="S5.SS2.SSSx1.p1.2.m2.1.2.2.1.cmml" xref="S5.SS2.SSSx1.p1.2.m2.1.2.2.2"><csymbol cd="latexml" id="S5.SS2.SSSx1.p1.2.m2.1.2.2.1.1.cmml" xref="S5.SS2.SSSx1.p1.2.m2.1.2.2.2.1">delimited-[]</csymbol><ci id="S5.SS2.SSSx1.p1.2.m2.1.1.cmml" xref="S5.SS2.SSSx1.p1.2.m2.1.1">𝑚</ci></apply><ci id="S5.SS2.SSSx1.p1.2.m2.1.2.3.cmml" xref="S5.SS2.SSSx1.p1.2.m2.1.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p1.2.m2.1c">[m]^{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p1.2.m2.1d">[ italic_m ] start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.SSSx1.p2"> <p class="ltx_p" id="S5.SS2.SSSx1.p2.15">We begin with proving Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem2" title="Theorem 5.2 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.2</span></a>. Fix <math alttext="G" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.1.m1.1"><semantics id="S5.SS2.SSSx1.p2.1.m1.1a"><mi id="S5.SS2.SSSx1.p2.1.m1.1.1" xref="S5.SS2.SSSx1.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.1.m1.1b"><ci id="S5.SS2.SSSx1.p2.1.m1.1.1.cmml" xref="S5.SS2.SSSx1.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.1.m1.1d">italic_G</annotation></semantics></math> with disjoint paths <math alttext="P_{st}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.2.m2.1"><semantics id="S5.SS2.SSSx1.p2.2.m2.1a"><msub id="S5.SS2.SSSx1.p2.2.m2.1.1" xref="S5.SS2.SSSx1.p2.2.m2.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.2.m2.1.1.2" xref="S5.SS2.SSSx1.p2.2.m2.1.1.2.cmml">P</mi><mrow id="S5.SS2.SSSx1.p2.2.m2.1.1.3" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3.cmml"><mi id="S5.SS2.SSSx1.p2.2.m2.1.1.3.2" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3.2.cmml">s</mi><mo id="S5.SS2.SSSx1.p2.2.m2.1.1.3.1" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S5.SS2.SSSx1.p2.2.m2.1.1.3.3" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.2.m2.1b"><apply id="S5.SS2.SSSx1.p2.2.m2.1.1.cmml" xref="S5.SS2.SSSx1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.2.m2.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.2.m2.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.2.m2.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.2.m2.1.1.2">𝑃</ci><apply id="S5.SS2.SSSx1.p2.2.m2.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3"><times id="S5.SS2.SSSx1.p2.2.m2.1.1.3.1.cmml" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3.1"></times><ci id="S5.SS2.SSSx1.p2.2.m2.1.1.3.2.cmml" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3.2">𝑠</ci><ci id="S5.SS2.SSSx1.p2.2.m2.1.1.3.3.cmml" xref="S5.SS2.SSSx1.p2.2.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.2.m2.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, such that <math alttext="|p_{1}|<|p_{m}|" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.3.m3.2"><semantics id="S5.SS2.SSSx1.p2.3.m3.2a"><mrow id="S5.SS2.SSSx1.p2.3.m3.2.2" xref="S5.SS2.SSSx1.p2.3.m3.2.2.cmml"><mrow id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.2.cmml"><mo id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.2.1.cmml">|</mo><msub id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.2" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.2.cmml">p</mi><mn id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.3" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.SSSx1.p2.3.m3.2.2.3" xref="S5.SS2.SSSx1.p2.3.m3.2.2.3.cmml"><</mo><mrow id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.2.cmml"><mo id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.2.1.cmml">|</mo><msub id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.2" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.2.cmml">p</mi><mi id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.3" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.3.cmml">m</mi></msub><mo id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.3.m3.2b"><apply id="S5.SS2.SSSx1.p2.3.m3.2.2.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2"><lt id="S5.SS2.SSSx1.p2.3.m3.2.2.3.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2.3"></lt><apply id="S5.SS2.SSSx1.p2.3.m3.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1"><abs id="S5.SS2.SSSx1.p2.3.m3.1.1.1.2.1.cmml" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.2"></abs><apply id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.2">𝑝</ci><cn id="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSSx1.p2.3.m3.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.SS2.SSSx1.p2.3.m3.2.2.2.2.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1"><abs id="S5.SS2.SSSx1.p2.3.m3.2.2.2.2.1.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.2"></abs><apply id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.2">𝑝</ci><ci id="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.3.m3.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.3.m3.2c">|p_{1}|<|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.3.m3.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math>. We will sometimes consider <math alttext="G" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.4.m4.1"><semantics id="S5.SS2.SSSx1.p2.4.m4.1a"><mi id="S5.SS2.SSSx1.p2.4.m4.1.1" xref="S5.SS2.SSSx1.p2.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.4.m4.1b"><ci id="S5.SS2.SSSx1.p2.4.m4.1.1.cmml" xref="S5.SS2.SSSx1.p2.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.4.m4.1d">italic_G</annotation></semantics></math> with only the first <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.5.m5.1"><semantics id="S5.SS2.SSSx1.p2.5.m5.1a"><mi id="S5.SS2.SSSx1.p2.5.m5.1.1" xref="S5.SS2.SSSx1.p2.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.5.m5.1b"><ci id="S5.SS2.SSSx1.p2.5.m5.1.1.cmml" xref="S5.SS2.SSSx1.p2.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.5.m5.1d">italic_i</annotation></semantics></math> paths for <math alttext="i\leq m" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.6.m6.1"><semantics id="S5.SS2.SSSx1.p2.6.m6.1a"><mrow id="S5.SS2.SSSx1.p2.6.m6.1.1" xref="S5.SS2.SSSx1.p2.6.m6.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.6.m6.1.1.2" xref="S5.SS2.SSSx1.p2.6.m6.1.1.2.cmml">i</mi><mo id="S5.SS2.SSSx1.p2.6.m6.1.1.1" xref="S5.SS2.SSSx1.p2.6.m6.1.1.1.cmml">≤</mo><mi id="S5.SS2.SSSx1.p2.6.m6.1.1.3" xref="S5.SS2.SSSx1.p2.6.m6.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.6.m6.1b"><apply id="S5.SS2.SSSx1.p2.6.m6.1.1.cmml" xref="S5.SS2.SSSx1.p2.6.m6.1.1"><leq id="S5.SS2.SSSx1.p2.6.m6.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.6.m6.1.1.1"></leq><ci id="S5.SS2.SSSx1.p2.6.m6.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.6.m6.1.1.2">𝑖</ci><ci id="S5.SS2.SSSx1.p2.6.m6.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.6.m6.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.6.m6.1c">i\leq m</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.6.m6.1d">italic_i ≤ italic_m</annotation></semantics></math>, denote it as <math alttext="G_{i}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.7.m7.1"><semantics id="S5.SS2.SSSx1.p2.7.m7.1a"><msub id="S5.SS2.SSSx1.p2.7.m7.1.1" xref="S5.SS2.SSSx1.p2.7.m7.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.7.m7.1.1.2" xref="S5.SS2.SSSx1.p2.7.m7.1.1.2.cmml">G</mi><mi id="S5.SS2.SSSx1.p2.7.m7.1.1.3" xref="S5.SS2.SSSx1.p2.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.7.m7.1b"><apply id="S5.SS2.SSSx1.p2.7.m7.1.1.cmml" xref="S5.SS2.SSSx1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.7.m7.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.7.m7.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.7.m7.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.7.m7.1.1.2">𝐺</ci><ci id="S5.SS2.SSSx1.p2.7.m7.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.7.m7.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.7.m7.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Then, the <math alttext="F_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.8.m8.1"><semantics id="S5.SS2.SSSx1.p2.8.m8.1a"><msub id="S5.SS2.SSSx1.p2.8.m8.1.1" xref="S5.SS2.SSSx1.p2.8.m8.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.8.m8.1.1.2" xref="S5.SS2.SSSx1.p2.8.m8.1.1.2.cmml">F</mi><mi id="S5.SS2.SSSx1.p2.8.m8.1.1.3" xref="S5.SS2.SSSx1.p2.8.m8.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.8.m8.1b"><apply id="S5.SS2.SSSx1.p2.8.m8.1.1.cmml" xref="S5.SS2.SSSx1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.8.m8.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.8.m8.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.8.m8.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.8.m8.1.1.2">𝐹</ci><ci id="S5.SS2.SSSx1.p2.8.m8.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.8.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.8.m8.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.8.m8.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> function defined on <math alttext="G_{i}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.9.m9.1"><semantics id="S5.SS2.SSSx1.p2.9.m9.1a"><msub id="S5.SS2.SSSx1.p2.9.m9.1.1" xref="S5.SS2.SSSx1.p2.9.m9.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.9.m9.1.1.2" xref="S5.SS2.SSSx1.p2.9.m9.1.1.2.cmml">G</mi><mi id="S5.SS2.SSSx1.p2.9.m9.1.1.3" xref="S5.SS2.SSSx1.p2.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.9.m9.1b"><apply id="S5.SS2.SSSx1.p2.9.m9.1.1.cmml" xref="S5.SS2.SSSx1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.9.m9.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.9.m9.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.9.m9.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.9.m9.1.1.2">𝐺</ci><ci id="S5.SS2.SSSx1.p2.9.m9.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.9.m9.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.9.m9.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is simply <math alttext="F_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.10.m10.1"><semantics id="S5.SS2.SSSx1.p2.10.m10.1a"><msub id="S5.SS2.SSSx1.p2.10.m10.1.1" xref="S5.SS2.SSSx1.p2.10.m10.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.10.m10.1.1.2" xref="S5.SS2.SSSx1.p2.10.m10.1.1.2.cmml">F</mi><mi id="S5.SS2.SSSx1.p2.10.m10.1.1.3" xref="S5.SS2.SSSx1.p2.10.m10.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.10.m10.1b"><apply id="S5.SS2.SSSx1.p2.10.m10.1.1.cmml" xref="S5.SS2.SSSx1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.10.m10.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.10.m10.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.10.m10.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.10.m10.1.1.2">𝐹</ci><ci id="S5.SS2.SSSx1.p2.10.m10.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.10.m10.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.10.m10.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.10.m10.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> when the domain of the distributions is <math alttext="(\Delta(P_{i}))^{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.11.m11.1"><semantics id="S5.SS2.SSSx1.p2.11.m11.1a"><msup id="S5.SS2.SSSx1.p2.11.m11.1.1" xref="S5.SS2.SSSx1.p2.11.m11.1.1.cmml"><mrow id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.cmml"><mo id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.3" mathvariant="normal" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.3.cmml">Δ</mi><mo id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.2" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.2" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.2.cmml">P</mi><mi id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.3" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.SS2.SSSx1.p2.11.m11.1.1.3" xref="S5.SS2.SSSx1.p2.11.m11.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.11.m11.1b"><apply id="S5.SS2.SSSx1.p2.11.m11.1.1.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.11.m11.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1">superscript</csymbol><apply id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1"><times id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.2"></times><ci id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.3">Δ</ci><apply id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.2">𝑃</ci><ci id="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><ci id="S5.SS2.SSSx1.p2.11.m11.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.11.m11.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.11.m11.1c">(\Delta(P_{i}))^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.11.m11.1d">( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> instead of <math alttext="(\Delta(P_{st}))^{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.12.m12.1"><semantics id="S5.SS2.SSSx1.p2.12.m12.1a"><msup id="S5.SS2.SSSx1.p2.12.m12.1.1" xref="S5.SS2.SSSx1.p2.12.m12.1.1.cmml"><mrow id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.cmml"><mo id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.3" mathvariant="normal" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.3.cmml">Δ</mi><mo id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.2" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.2" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3.2" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3.1" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3.3" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.SS2.SSSx1.p2.12.m12.1.1.3" xref="S5.SS2.SSSx1.p2.12.m12.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.12.m12.1b"><apply id="S5.SS2.SSSx1.p2.12.m12.1.1.cmml" xref="S5.SS2.SSSx1.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.12.m12.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.12.m12.1.1">superscript</csymbol><apply id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1"><times id="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.12.m12.1.1.1.1.1.2"></times><ci 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xref="S5.SS2.SSSx1.p2.12.m12.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.12.m12.1c">(\Delta(P_{st}))^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.12.m12.1d">( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> (because the interceptors will decrease the cost to the agents if they pick edges from paths <math alttext="p_{i+1},...,p_{m}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.13.m13.3"><semantics id="S5.SS2.SSSx1.p2.13.m13.3a"><mrow id="S5.SS2.SSSx1.p2.13.m13.3.3.2" xref="S5.SS2.SSSx1.p2.13.m13.3.3.3.cmml"><msub id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.2" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.2.cmml">p</mi><mrow id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.cmml"><mi id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.2" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.2.cmml">i</mi><mo id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.1" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.1.cmml">+</mo><mn id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.3" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.SS2.SSSx1.p2.13.m13.3.3.2.3" xref="S5.SS2.SSSx1.p2.13.m13.3.3.3.cmml">,</mo><mi id="S5.SS2.SSSx1.p2.13.m13.1.1" mathvariant="normal" xref="S5.SS2.SSSx1.p2.13.m13.1.1.cmml">…</mi><mo id="S5.SS2.SSSx1.p2.13.m13.3.3.2.4" xref="S5.SS2.SSSx1.p2.13.m13.3.3.3.cmml">,</mo><msub id="S5.SS2.SSSx1.p2.13.m13.3.3.2.2" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.cmml"><mi id="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.2" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.2.cmml">p</mi><mi id="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.3" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.13.m13.3b"><list id="S5.SS2.SSSx1.p2.13.m13.3.3.3.cmml" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2"><apply id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.cmml" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.2">𝑝</ci><apply id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3"><plus id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.1.cmml" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.1"></plus><ci id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.2.cmml" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.2">𝑖</ci><cn id="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSSx1.p2.13.m13.2.2.1.1.3.3">1</cn></apply></apply><ci id="S5.SS2.SSSx1.p2.13.m13.1.1.cmml" xref="S5.SS2.SSSx1.p2.13.m13.1.1">…</ci><apply id="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.cmml" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.1.cmml" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2.2">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.2.cmml" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.2">𝑝</ci><ci id="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.3.cmml" xref="S5.SS2.SSSx1.p2.13.m13.3.3.2.2.3">𝑚</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.13.m13.3c">p_{i+1},...,p_{m}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.13.m13.3d">italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, in contrast with their goal of maximizing the agents’ cost). Let <math alttext="m_{1}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.14.m14.1"><semantics id="S5.SS2.SSSx1.p2.14.m14.1a"><msub id="S5.SS2.SSSx1.p2.14.m14.1.1" xref="S5.SS2.SSSx1.p2.14.m14.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.14.m14.1.1.2" xref="S5.SS2.SSSx1.p2.14.m14.1.1.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p2.14.m14.1.1.3" xref="S5.SS2.SSSx1.p2.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.14.m14.1b"><apply id="S5.SS2.SSSx1.p2.14.m14.1.1.cmml" xref="S5.SS2.SSSx1.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.14.m14.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.14.m14.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.14.m14.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.14.m14.1.1.2">𝑚</ci><cn id="S5.SS2.SSSx1.p2.14.m14.1.1.3.cmml" type="integer" xref="S5.SS2.SSSx1.p2.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.14.m14.1c">m_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.14.m14.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be the largest integer such that <math alttext="|p_{1}|=...=|p_{m_{1}}|" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p2.15.m15.2"><semantics id="S5.SS2.SSSx1.p2.15.m15.2a"><mrow id="S5.SS2.SSSx1.p2.15.m15.2.2" xref="S5.SS2.SSSx1.p2.15.m15.2.2.cmml"><mrow id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.2.cmml"><mo id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.2.1.cmml">|</mo><msub id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.2" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.2.cmml">p</mi><mn id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.3" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.SSSx1.p2.15.m15.2.2.4" xref="S5.SS2.SSSx1.p2.15.m15.2.2.4.cmml">=</mo><mi id="S5.SS2.SSSx1.p2.15.m15.2.2.5" mathvariant="normal" xref="S5.SS2.SSSx1.p2.15.m15.2.2.5.cmml">…</mi><mo id="S5.SS2.SSSx1.p2.15.m15.2.2.6" xref="S5.SS2.SSSx1.p2.15.m15.2.2.6.cmml">=</mo><mrow id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.2.cmml"><mo id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.2" stretchy="false" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.2.1.cmml">|</mo><msub id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.cmml"><mi id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.2" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.2.cmml">p</mi><msub id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.cmml"><mi id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.2" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.3" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.3.cmml">1</mn></msub></msub><mo id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.3" stretchy="false" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p2.15.m15.2b"><apply id="S5.SS2.SSSx1.p2.15.m15.2.2.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2"><and id="S5.SS2.SSSx1.p2.15.m15.2.2a.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2"></and><apply id="S5.SS2.SSSx1.p2.15.m15.2.2b.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2"><eq id="S5.SS2.SSSx1.p2.15.m15.2.2.4.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.4"></eq><apply id="S5.SS2.SSSx1.p2.15.m15.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1"><abs id="S5.SS2.SSSx1.p2.15.m15.1.1.1.2.1.cmml" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.2"></abs><apply id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.2">𝑝</ci><cn id="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSSx1.p2.15.m15.1.1.1.1.1.3">1</cn></apply></apply><ci id="S5.SS2.SSSx1.p2.15.m15.2.2.5.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.5">…</ci></apply><apply id="S5.SS2.SSSx1.p2.15.m15.2.2c.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2"><eq id="S5.SS2.SSSx1.p2.15.m15.2.2.6.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.6"></eq><share href="https://arxiv.org/html/2308.08047v3#S5.SS2.SSSx1.p2.15.m15.2.2.5.cmml" id="S5.SS2.SSSx1.p2.15.m15.2.2d.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2"></share><apply id="S5.SS2.SSSx1.p2.15.m15.2.2.2.2.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1"><abs id="S5.SS2.SSSx1.p2.15.m15.2.2.2.2.1.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.2"></abs><apply id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.1.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.2.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.2">𝑝</ci><apply id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.1.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.2.cmml" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.2">𝑚</ci><cn id="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSSx1.p2.15.m15.2.2.2.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p2.15.m15.2c">|p_{1}|=...=|p_{m_{1}}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p2.15.m15.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = … = | italic_p start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.SSSx1.p3"> <p class="ltx_p" id="S5.SS2.SSSx1.p3.8">Suppose the agents play only on <math alttext="G_{m_{1}}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.1.m1.1"><semantics id="S5.SS2.SSSx1.p3.1.m1.1a"><msub id="S5.SS2.SSSx1.p3.1.m1.1.1" xref="S5.SS2.SSSx1.p3.1.m1.1.1.cmml"><mi id="S5.SS2.SSSx1.p3.1.m1.1.1.2" xref="S5.SS2.SSSx1.p3.1.m1.1.1.2.cmml">G</mi><msub id="S5.SS2.SSSx1.p3.1.m1.1.1.3" xref="S5.SS2.SSSx1.p3.1.m1.1.1.3.cmml"><mi id="S5.SS2.SSSx1.p3.1.m1.1.1.3.2" xref="S5.SS2.SSSx1.p3.1.m1.1.1.3.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p3.1.m1.1.1.3.3" xref="S5.SS2.SSSx1.p3.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p3.1.m1.1b"><apply id="S5.SS2.SSSx1.p3.1.m1.1.1.cmml" xref="S5.SS2.SSSx1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.1.m1.1.1.1.cmml" xref="S5.SS2.SSSx1.p3.1.m1.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.1.m1.1.1.2.cmml" xref="S5.SS2.SSSx1.p3.1.m1.1.1.2">𝐺</ci><apply id="S5.SS2.SSSx1.p3.1.m1.1.1.3.cmml" xref="S5.SS2.SSSx1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.1.m1.1.1.3.1.cmml" xref="S5.SS2.SSSx1.p3.1.m1.1.1.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.1.m1.1.1.3.2.cmml" xref="S5.SS2.SSSx1.p3.1.m1.1.1.3.2">𝑚</ci><cn id="S5.SS2.SSSx1.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSSx1.p3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.1.m1.1c">G_{m_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.1.m1.1d">italic_G start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. All these paths have the same length, so intuitively the agents minimize their cost by minimizing their probability of picking a path with a large imposed delay. The best strategy for agents is then <math alttext="U_{P_{m_{1}}}^{n}\in\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.2.m2.1"><semantics id="S5.SS2.SSSx1.p3.2.m2.1a"><mrow id="S5.SS2.SSSx1.p3.2.m2.1.1" xref="S5.SS2.SSSx1.p3.2.m2.1.1.cmml"><msubsup id="S5.SS2.SSSx1.p3.2.m2.1.1.2" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.cmml"><mi id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.2" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.2.cmml">U</mi><msub id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.cmml"><mi id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.2" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.2.cmml">P</mi><msub id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.cmml"><mi id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.2" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.3" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.3.cmml">1</mn></msub></msub><mi id="S5.SS2.SSSx1.p3.2.m2.1.1.2.3" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.3.cmml">n</mi></msubsup><mo id="S5.SS2.SSSx1.p3.2.m2.1.1.1" xref="S5.SS2.SSSx1.p3.2.m2.1.1.1.cmml">∈</mo><msub id="S5.SS2.SSSx1.p3.2.m2.1.1.3" xref="S5.SS2.SSSx1.p3.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSSx1.p3.2.m2.1.1.3.2" xref="S5.SS2.SSSx1.p3.2.m2.1.1.3.2.cmml">𝒰</mi><mi id="S5.SS2.SSSx1.p3.2.m2.1.1.3.3" xref="S5.SS2.SSSx1.p3.2.m2.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p3.2.m2.1b"><apply id="S5.SS2.SSSx1.p3.2.m2.1.1.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1"><in id="S5.SS2.SSSx1.p3.2.m2.1.1.1.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.1"></in><apply id="S5.SS2.SSSx1.p3.2.m2.1.1.2.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.2.m2.1.1.2.1.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2">superscript</csymbol><apply id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.1.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.2.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.2">𝑈</ci><apply id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.1.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.2.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.2">𝑃</ci><apply id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.1.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.2.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.2">𝑚</ci><cn id="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.3.cmml" type="integer" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.2.3.3.3">1</cn></apply></apply></apply><ci id="S5.SS2.SSSx1.p3.2.m2.1.1.2.3.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.2.3">𝑛</ci></apply><apply id="S5.SS2.SSSx1.p3.2.m2.1.1.3.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.2.m2.1.1.3.1.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.2.m2.1.1.3.2.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.3.2">𝒰</ci><ci id="S5.SS2.SSSx1.p3.2.m2.1.1.3.3.cmml" xref="S5.SS2.SSSx1.p3.2.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.2.m2.1c">U_{P_{m_{1}}}^{n}\in\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Intuitively, the interceptors will respond by picking <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.3.m3.1"><semantics id="S5.SS2.SSSx1.p3.3.m3.1a"><mi id="S5.SS2.SSSx1.p3.3.m3.1.1" xref="S5.SS2.SSSx1.p3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p3.3.m3.1b"><ci id="S5.SS2.SSSx1.p3.3.m3.1.1.cmml" xref="S5.SS2.SSSx1.p3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.3.m3.1d">italic_k</annotation></semantics></math> edges from the same path (with some distribution over <math alttext="P_{m_{1}}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.4.m4.1"><semantics id="S5.SS2.SSSx1.p3.4.m4.1a"><msub id="S5.SS2.SSSx1.p3.4.m4.1.1" xref="S5.SS2.SSSx1.p3.4.m4.1.1.cmml"><mi id="S5.SS2.SSSx1.p3.4.m4.1.1.2" xref="S5.SS2.SSSx1.p3.4.m4.1.1.2.cmml">P</mi><msub id="S5.SS2.SSSx1.p3.4.m4.1.1.3" xref="S5.SS2.SSSx1.p3.4.m4.1.1.3.cmml"><mi id="S5.SS2.SSSx1.p3.4.m4.1.1.3.2" xref="S5.SS2.SSSx1.p3.4.m4.1.1.3.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p3.4.m4.1.1.3.3" xref="S5.SS2.SSSx1.p3.4.m4.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p3.4.m4.1b"><apply id="S5.SS2.SSSx1.p3.4.m4.1.1.cmml" xref="S5.SS2.SSSx1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.4.m4.1.1.1.cmml" xref="S5.SS2.SSSx1.p3.4.m4.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.4.m4.1.1.2.cmml" xref="S5.SS2.SSSx1.p3.4.m4.1.1.2">𝑃</ci><apply id="S5.SS2.SSSx1.p3.4.m4.1.1.3.cmml" xref="S5.SS2.SSSx1.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.4.m4.1.1.3.1.cmml" xref="S5.SS2.SSSx1.p3.4.m4.1.1.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.4.m4.1.1.3.2.cmml" xref="S5.SS2.SSSx1.p3.4.m4.1.1.3.2">𝑚</ci><cn id="S5.SS2.SSSx1.p3.4.m4.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSSx1.p3.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.4.m4.1c">P_{m_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>), because in each choice of paths and edges, the cost is determined only by a costliest path. The cost to the agents is <math alttext="|p_{1}|+\frac{\mathcal{E}_{m_{1},n}}{m_{1}}k\alpha" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.5.m5.3"><semantics id="S5.SS2.SSSx1.p3.5.m5.3a"><mrow id="S5.SS2.SSSx1.p3.5.m5.3.3" xref="S5.SS2.SSSx1.p3.5.m5.3.3.cmml"><mrow id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.2.cmml"><mo id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.2" stretchy="false" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.2.1.cmml">|</mo><msub id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.cmml"><mi id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.2" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.2.cmml">p</mi><mn id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.3" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.3" stretchy="false" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.SSSx1.p3.5.m5.3.3.2" xref="S5.SS2.SSSx1.p3.5.m5.3.3.2.cmml">+</mo><mrow id="S5.SS2.SSSx1.p3.5.m5.3.3.3" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.cmml"><mfrac id="S5.SS2.SSSx1.p3.5.m5.2.2" xref="S5.SS2.SSSx1.p3.5.m5.2.2.cmml"><msub id="S5.SS2.SSSx1.p3.5.m5.2.2.2" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSSx1.p3.5.m5.2.2.2.4" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.4.cmml">ℰ</mi><mrow id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.3.cmml"><msub id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.cmml"><mi id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.2" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.3" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.2" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.3.cmml">,</mo><mi id="S5.SS2.SSSx1.p3.5.m5.1.1.1.1.1.1" xref="S5.SS2.SSSx1.p3.5.m5.1.1.1.1.1.1.cmml">n</mi></mrow></msub><msub id="S5.SS2.SSSx1.p3.5.m5.2.2.4" xref="S5.SS2.SSSx1.p3.5.m5.2.2.4.cmml"><mi id="S5.SS2.SSSx1.p3.5.m5.2.2.4.2" xref="S5.SS2.SSSx1.p3.5.m5.2.2.4.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p3.5.m5.2.2.4.3" xref="S5.SS2.SSSx1.p3.5.m5.2.2.4.3.cmml">1</mn></msub></mfrac><mo id="S5.SS2.SSSx1.p3.5.m5.3.3.3.1" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.1.cmml">⁢</mo><mi id="S5.SS2.SSSx1.p3.5.m5.3.3.3.2" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.2.cmml">k</mi><mo id="S5.SS2.SSSx1.p3.5.m5.3.3.3.1a" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.1.cmml">⁢</mo><mi id="S5.SS2.SSSx1.p3.5.m5.3.3.3.3" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.3.cmml">α</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p3.5.m5.3b"><apply id="S5.SS2.SSSx1.p3.5.m5.3.3.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3"><plus id="S5.SS2.SSSx1.p3.5.m5.3.3.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.2"></plus><apply id="S5.SS2.SSSx1.p3.5.m5.3.3.1.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1"><abs id="S5.SS2.SSSx1.p3.5.m5.3.3.1.2.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.2"></abs><apply id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.2">𝑝</ci><cn id="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSSx1.p3.5.m5.3.3.1.1.1.3">1</cn></apply></apply><apply id="S5.SS2.SSSx1.p3.5.m5.3.3.3.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3"><times id="S5.SS2.SSSx1.p3.5.m5.3.3.3.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.1"></times><apply id="S5.SS2.SSSx1.p3.5.m5.2.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2"><divide id="S5.SS2.SSSx1.p3.5.m5.2.2.3.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2"></divide><apply id="S5.SS2.SSSx1.p3.5.m5.2.2.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.5.m5.2.2.2.3.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.5.m5.2.2.2.4.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.4">ℰ</ci><list id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.3.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2"><apply id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.2">𝑚</ci><cn id="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.3.cmml" type="integer" xref="S5.SS2.SSSx1.p3.5.m5.2.2.2.2.2.2.1.3">1</cn></apply><ci id="S5.SS2.SSSx1.p3.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.1.1.1.1.1.1">𝑛</ci></list></apply><apply id="S5.SS2.SSSx1.p3.5.m5.2.2.4.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.4"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.5.m5.2.2.4.1.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.4">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.5.m5.2.2.4.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.2.2.4.2">𝑚</ci><cn id="S5.SS2.SSSx1.p3.5.m5.2.2.4.3.cmml" type="integer" xref="S5.SS2.SSSx1.p3.5.m5.2.2.4.3">1</cn></apply></apply><ci id="S5.SS2.SSSx1.p3.5.m5.3.3.3.2.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.2">𝑘</ci><ci id="S5.SS2.SSSx1.p3.5.m5.3.3.3.3.cmml" xref="S5.SS2.SSSx1.p3.5.m5.3.3.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.5.m5.3c">|p_{1}|+\frac{\mathcal{E}_{m_{1},n}}{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.5.m5.3d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. Suppose that <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.6.m6.1"><semantics id="S5.SS2.SSSx1.p3.6.m6.1a"><mi id="S5.SS2.SSSx1.p3.6.m6.1.1" xref="S5.SS2.SSSx1.p3.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p3.6.m6.1b"><ci id="S5.SS2.SSSx1.p3.6.m6.1.1.cmml" xref="S5.SS2.SSSx1.p3.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.6.m6.1d">italic_α</annotation></semantics></math> is large so that <math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.7.m7.6"><semantics id="S5.SS2.SSSx1.p3.7.m7.6a"><mrow id="S5.SS2.SSSx1.p3.7.m7.6.6" xref="S5.SS2.SSSx1.p3.7.m7.6.6.cmml"><mrow id="S5.SS2.SSSx1.p3.7.m7.6.6.3" xref="S5.SS2.SSSx1.p3.7.m7.6.6.3.cmml"><mrow 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xref="S5.SS2.SSSx1.p3.7.m7.6.6.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.7.m7.6c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.7.m7.6d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. As we show next, this makes <span class="ltx_text ltx_font_italic" id="S5.SS2.SSSx1.p3.8.1">all</span> strategies in <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p3.8.m8.1"><semantics id="S5.SS2.SSSx1.p3.8.m8.1a"><msub id="S5.SS2.SSSx1.p3.8.m8.1.1" xref="S5.SS2.SSSx1.p3.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSSx1.p3.8.m8.1.1.2" xref="S5.SS2.SSSx1.p3.8.m8.1.1.2.cmml">𝒰</mi><mi id="S5.SS2.SSSx1.p3.8.m8.1.1.3" xref="S5.SS2.SSSx1.p3.8.m8.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p3.8.m8.1b"><apply id="S5.SS2.SSSx1.p3.8.m8.1.1.cmml" xref="S5.SS2.SSSx1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p3.8.m8.1.1.1.cmml" xref="S5.SS2.SSSx1.p3.8.m8.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p3.8.m8.1.1.2.cmml" xref="S5.SS2.SSSx1.p3.8.m8.1.1.2">𝒰</ci><ci id="S5.SS2.SSSx1.p3.8.m8.1.1.3.cmml" xref="S5.SS2.SSSx1.p3.8.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p3.8.m8.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p3.8.m8.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> no longer optimal for the agents. Below we make these observations concrete.</p> </div> <div class="ltx_para" id="S5.SS2.SSSx1.p4"> <p class="ltx_p" id="S5.SS2.SSSx1.p4.1">We begin with the key lemma, which holds for any graph of disjoint paths.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma14"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 14</span></h6> <div class="ltx_para" id="Thmlemma14.p1"> <p class="ltx_p" id="Thmlemma14.p1.5"><span class="ltx_text ltx_font_italic" id="Thmlemma14.p1.5.5">For every <math alttext="G" class="ltx_Math" display="inline" id="Thmlemma14.p1.1.1.m1.1"><semantics id="Thmlemma14.p1.1.1.m1.1a"><mi id="Thmlemma14.p1.1.1.m1.1.1" xref="Thmlemma14.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmlemma14.p1.1.1.m1.1b"><ci id="Thmlemma14.p1.1.1.m1.1.1.cmml" xref="Thmlemma14.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma14.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmlemma14.p1.1.1.m1.1d">italic_G</annotation></semantics></math> with <math alttext="m>1" class="ltx_Math" display="inline" id="Thmlemma14.p1.2.2.m2.1"><semantics id="Thmlemma14.p1.2.2.m2.1a"><mrow id="Thmlemma14.p1.2.2.m2.1.1" xref="Thmlemma14.p1.2.2.m2.1.1.cmml"><mi id="Thmlemma14.p1.2.2.m2.1.1.2" xref="Thmlemma14.p1.2.2.m2.1.1.2.cmml">m</mi><mo id="Thmlemma14.p1.2.2.m2.1.1.1" xref="Thmlemma14.p1.2.2.m2.1.1.1.cmml">></mo><mn id="Thmlemma14.p1.2.2.m2.1.1.3" xref="Thmlemma14.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma14.p1.2.2.m2.1b"><apply id="Thmlemma14.p1.2.2.m2.1.1.cmml" xref="Thmlemma14.p1.2.2.m2.1.1"><gt id="Thmlemma14.p1.2.2.m2.1.1.1.cmml" xref="Thmlemma14.p1.2.2.m2.1.1.1"></gt><ci id="Thmlemma14.p1.2.2.m2.1.1.2.cmml" xref="Thmlemma14.p1.2.2.m2.1.1.2">𝑚</ci><cn id="Thmlemma14.p1.2.2.m2.1.1.3.cmml" type="integer" xref="Thmlemma14.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma14.p1.2.2.m2.1c">m>1</annotation><annotation encoding="application/x-llamapun" id="Thmlemma14.p1.2.2.m2.1d">italic_m > 1</annotation></semantics></math> disjoint paths, for every <math alttext="n>1,k\geq 1" class="ltx_Math" display="inline" id="Thmlemma14.p1.3.3.m3.2"><semantics id="Thmlemma14.p1.3.3.m3.2a"><mrow id="Thmlemma14.p1.3.3.m3.2.2.2" xref="Thmlemma14.p1.3.3.m3.2.2.3.cmml"><mrow id="Thmlemma14.p1.3.3.m3.1.1.1.1" xref="Thmlemma14.p1.3.3.m3.1.1.1.1.cmml"><mi id="Thmlemma14.p1.3.3.m3.1.1.1.1.2" xref="Thmlemma14.p1.3.3.m3.1.1.1.1.2.cmml">n</mi><mo id="Thmlemma14.p1.3.3.m3.1.1.1.1.1" xref="Thmlemma14.p1.3.3.m3.1.1.1.1.1.cmml">></mo><mn id="Thmlemma14.p1.3.3.m3.1.1.1.1.3" xref="Thmlemma14.p1.3.3.m3.1.1.1.1.3.cmml">1</mn></mrow><mo id="Thmlemma14.p1.3.3.m3.2.2.2.3" xref="Thmlemma14.p1.3.3.m3.2.2.3a.cmml">,</mo><mrow id="Thmlemma14.p1.3.3.m3.2.2.2.2" 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id="Thmlemma14.p1.4.4.m4.2.2.2.2.2.1.cmml" xref="Thmlemma14.p1.4.4.m4.2.2.2.2.1.2"></abs><apply id="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1.cmml" xref="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1.1.cmml" xref="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1">subscript</csymbol><ci id="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1.2.cmml" xref="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1.2">𝑝</ci><cn id="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1.3.cmml" type="integer" xref="Thmlemma14.p1.4.4.m4.2.2.2.2.1.1.3">1</cn></apply></apply></apply><ci id="Thmlemma14.p1.4.4.m4.2.2.4.cmml" xref="Thmlemma14.p1.4.4.m4.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma14.p1.4.4.m4.2c">\alpha>\frac{|p_{m}|-|p_{1}|}{k}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma14.p1.4.4.m4.2d">italic_α > divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>: <math alttext="F_{n}(U_{P_{st}}^{n},\alpha)=f_{(m,...,m)}(U_{P_{st}}^{n},\alpha)>\max_{d\neq(% m,...,m)}f_{d}(U_{P_{st}}^{n},\alpha)" class="ltx_Math" display="inline" id="Thmlemma14.p1.5.5.m5.12"><semantics id="Thmlemma14.p1.5.5.m5.12a"><mrow id="Thmlemma14.p1.5.5.m5.12.12" xref="Thmlemma14.p1.5.5.m5.12.12.cmml"><mrow id="Thmlemma14.p1.5.5.m5.10.10.1" xref="Thmlemma14.p1.5.5.m5.10.10.1.cmml"><msub id="Thmlemma14.p1.5.5.m5.10.10.1.3" xref="Thmlemma14.p1.5.5.m5.10.10.1.3.cmml"><mi id="Thmlemma14.p1.5.5.m5.10.10.1.3.2" xref="Thmlemma14.p1.5.5.m5.10.10.1.3.2.cmml">F</mi><mi id="Thmlemma14.p1.5.5.m5.10.10.1.3.3" xref="Thmlemma14.p1.5.5.m5.10.10.1.3.3.cmml">n</mi></msub><mo id="Thmlemma14.p1.5.5.m5.10.10.1.2" xref="Thmlemma14.p1.5.5.m5.10.10.1.2.cmml">⁢</mo><mrow id="Thmlemma14.p1.5.5.m5.10.10.1.1.1" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.2.cmml"><mo id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.2" stretchy="false" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.2.cmml">(</mo><msubsup id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.cmml"><mi id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.2" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.2.cmml">U</mi><msub id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.cmml"><mi id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.2" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3.cmml"><mi id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3.2" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3.1" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3.3" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.3" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.3" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.2.cmml">,</mo><mi id="Thmlemma14.p1.5.5.m5.7.7" xref="Thmlemma14.p1.5.5.m5.7.7.cmml">α</mi><mo id="Thmlemma14.p1.5.5.m5.10.10.1.1.1.4" stretchy="false" xref="Thmlemma14.p1.5.5.m5.10.10.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma14.p1.5.5.m5.12.12.5" xref="Thmlemma14.p1.5.5.m5.12.12.5.cmml">=</mo><mrow id="Thmlemma14.p1.5.5.m5.11.11.2" xref="Thmlemma14.p1.5.5.m5.11.11.2.cmml"><msub id="Thmlemma14.p1.5.5.m5.11.11.2.3" xref="Thmlemma14.p1.5.5.m5.11.11.2.3.cmml"><mi id="Thmlemma14.p1.5.5.m5.11.11.2.3.2" xref="Thmlemma14.p1.5.5.m5.11.11.2.3.2.cmml">f</mi><mrow id="Thmlemma14.p1.5.5.m5.3.3.3.5" xref="Thmlemma14.p1.5.5.m5.3.3.3.4.cmml"><mo id="Thmlemma14.p1.5.5.m5.3.3.3.5.1" stretchy="false" xref="Thmlemma14.p1.5.5.m5.3.3.3.4.cmml">(</mo><mi id="Thmlemma14.p1.5.5.m5.1.1.1.1" xref="Thmlemma14.p1.5.5.m5.1.1.1.1.cmml">m</mi><mo id="Thmlemma14.p1.5.5.m5.3.3.3.5.2" xref="Thmlemma14.p1.5.5.m5.3.3.3.4.cmml">,</mo><mi id="Thmlemma14.p1.5.5.m5.2.2.2.2" mathvariant="normal" xref="Thmlemma14.p1.5.5.m5.2.2.2.2.cmml">…</mi><mo id="Thmlemma14.p1.5.5.m5.3.3.3.5.3" xref="Thmlemma14.p1.5.5.m5.3.3.3.4.cmml">,</mo><mi id="Thmlemma14.p1.5.5.m5.3.3.3.3" xref="Thmlemma14.p1.5.5.m5.3.3.3.3.cmml">m</mi><mo id="Thmlemma14.p1.5.5.m5.3.3.3.5.4" stretchy="false" xref="Thmlemma14.p1.5.5.m5.3.3.3.4.cmml">)</mo></mrow></msub><mo id="Thmlemma14.p1.5.5.m5.11.11.2.2" xref="Thmlemma14.p1.5.5.m5.11.11.2.2.cmml">⁢</mo><mrow id="Thmlemma14.p1.5.5.m5.11.11.2.1.1" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.2.cmml"><mo id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.2" stretchy="false" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.2.cmml">(</mo><msubsup id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.cmml"><mi id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.2" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.2.cmml">U</mi><msub id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.cmml"><mi id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.2" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3.cmml"><mi id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3.2" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3.1" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3.3" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.3" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.3" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.2.cmml">,</mo><mi id="Thmlemma14.p1.5.5.m5.8.8" xref="Thmlemma14.p1.5.5.m5.8.8.cmml">α</mi><mo id="Thmlemma14.p1.5.5.m5.11.11.2.1.1.4" stretchy="false" xref="Thmlemma14.p1.5.5.m5.11.11.2.1.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma14.p1.5.5.m5.12.12.6" xref="Thmlemma14.p1.5.5.m5.12.12.6.cmml">></mo><mrow id="Thmlemma14.p1.5.5.m5.12.12.3" xref="Thmlemma14.p1.5.5.m5.12.12.3.cmml"><mrow id="Thmlemma14.p1.5.5.m5.12.12.3.3" xref="Thmlemma14.p1.5.5.m5.12.12.3.3.cmml"><msub id="Thmlemma14.p1.5.5.m5.12.12.3.3.1" xref="Thmlemma14.p1.5.5.m5.12.12.3.3.1.cmml"><mi id="Thmlemma14.p1.5.5.m5.12.12.3.3.1.2" xref="Thmlemma14.p1.5.5.m5.12.12.3.3.1.2.cmml">max</mi><mrow id="Thmlemma14.p1.5.5.m5.6.6.3" xref="Thmlemma14.p1.5.5.m5.6.6.3.cmml"><mi id="Thmlemma14.p1.5.5.m5.6.6.3.5" xref="Thmlemma14.p1.5.5.m5.6.6.3.5.cmml">d</mi><mo id="Thmlemma14.p1.5.5.m5.6.6.3.4" xref="Thmlemma14.p1.5.5.m5.6.6.3.4.cmml">≠</mo><mrow id="Thmlemma14.p1.5.5.m5.6.6.3.6.2" xref="Thmlemma14.p1.5.5.m5.6.6.3.6.1.cmml"><mo id="Thmlemma14.p1.5.5.m5.6.6.3.6.2.1" stretchy="false" xref="Thmlemma14.p1.5.5.m5.6.6.3.6.1.cmml">(</mo><mi id="Thmlemma14.p1.5.5.m5.4.4.1.1" xref="Thmlemma14.p1.5.5.m5.4.4.1.1.cmml">m</mi><mo id="Thmlemma14.p1.5.5.m5.6.6.3.6.2.2" xref="Thmlemma14.p1.5.5.m5.6.6.3.6.1.cmml">,</mo><mi id="Thmlemma14.p1.5.5.m5.5.5.2.2" mathvariant="normal" xref="Thmlemma14.p1.5.5.m5.5.5.2.2.cmml">…</mi><mo id="Thmlemma14.p1.5.5.m5.6.6.3.6.2.3" xref="Thmlemma14.p1.5.5.m5.6.6.3.6.1.cmml">,</mo><mi id="Thmlemma14.p1.5.5.m5.6.6.3.3" xref="Thmlemma14.p1.5.5.m5.6.6.3.3.cmml">m</mi><mo id="Thmlemma14.p1.5.5.m5.6.6.3.6.2.4" stretchy="false" xref="Thmlemma14.p1.5.5.m5.6.6.3.6.1.cmml">)</mo></mrow></mrow></msub><mo id="Thmlemma14.p1.5.5.m5.12.12.3.3a" lspace="0.167em" xref="Thmlemma14.p1.5.5.m5.12.12.3.3.cmml">⁡</mo><msub id="Thmlemma14.p1.5.5.m5.12.12.3.3.2" xref="Thmlemma14.p1.5.5.m5.12.12.3.3.2.cmml"><mi id="Thmlemma14.p1.5.5.m5.12.12.3.3.2.2" xref="Thmlemma14.p1.5.5.m5.12.12.3.3.2.2.cmml">f</mi><mi id="Thmlemma14.p1.5.5.m5.12.12.3.3.2.3" xref="Thmlemma14.p1.5.5.m5.12.12.3.3.2.3.cmml">d</mi></msub></mrow><mo id="Thmlemma14.p1.5.5.m5.12.12.3.2" 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id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.3" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.3" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.3" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.2.cmml">,</mo><mi id="Thmlemma14.p1.5.5.m5.9.9" xref="Thmlemma14.p1.5.5.m5.9.9.cmml">α</mi><mo id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.4" stretchy="false" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma14.p1.5.5.m5.12b"><apply id="Thmlemma14.p1.5.5.m5.12.12.cmml" xref="Thmlemma14.p1.5.5.m5.12.12"><and id="Thmlemma14.p1.5.5.m5.12.12a.cmml" xref="Thmlemma14.p1.5.5.m5.12.12"></and><apply id="Thmlemma14.p1.5.5.m5.12.12b.cmml" xref="Thmlemma14.p1.5.5.m5.12.12"><eq id="Thmlemma14.p1.5.5.m5.12.12.5.cmml" xref="Thmlemma14.p1.5.5.m5.12.12.5"></eq><apply id="Thmlemma14.p1.5.5.m5.10.10.1.cmml" 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id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.1.cmml" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.1"></times><ci id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.2.cmml" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.2">𝑠</ci><ci id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.3.cmml" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.3.cmml" xref="Thmlemma14.p1.5.5.m5.12.12.3.1.1.1.3">𝑛</ci></apply><ci id="Thmlemma14.p1.5.5.m5.9.9.cmml" xref="Thmlemma14.p1.5.5.m5.9.9">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma14.p1.5.5.m5.12c">F_{n}(U_{P_{st}}^{n},\alpha)=f_{(m,...,m)}(U_{P_{st}}^{n},\alpha)>\max_{d\neq(% m,...,m)}f_{d}(U_{P_{st}}^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma14.p1.5.5.m5.12d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) > roman_max start_POSTSUBSCRIPT italic_d ≠ ( italic_m , … , italic_m ) end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS2.SSSx1.p5"> <p class="ltx_p" id="S5.SS2.SSSx1.p5.3">We apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma14" title="Lemma 14 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">14</span></a> to <math alttext="F_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p5.1.m1.1"><semantics id="S5.SS2.SSSx1.p5.1.m1.1a"><msub id="S5.SS2.SSSx1.p5.1.m1.1.1" xref="S5.SS2.SSSx1.p5.1.m1.1.1.cmml"><mi id="S5.SS2.SSSx1.p5.1.m1.1.1.2" xref="S5.SS2.SSSx1.p5.1.m1.1.1.2.cmml">F</mi><mi id="S5.SS2.SSSx1.p5.1.m1.1.1.3" xref="S5.SS2.SSSx1.p5.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p5.1.m1.1b"><apply id="S5.SS2.SSSx1.p5.1.m1.1.1.cmml" xref="S5.SS2.SSSx1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p5.1.m1.1.1.1.cmml" xref="S5.SS2.SSSx1.p5.1.m1.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p5.1.m1.1.1.2.cmml" xref="S5.SS2.SSSx1.p5.1.m1.1.1.2">𝐹</ci><ci id="S5.SS2.SSSx1.p5.1.m1.1.1.3.cmml" xref="S5.SS2.SSSx1.p5.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p5.1.m1.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p5.1.m1.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> on a subset of <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p5.2.m2.1"><semantics id="S5.SS2.SSSx1.p5.2.m2.1a"><msub id="S5.SS2.SSSx1.p5.2.m2.1.1" xref="S5.SS2.SSSx1.p5.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSSx1.p5.2.m2.1.1.2" xref="S5.SS2.SSSx1.p5.2.m2.1.1.2.cmml">𝒰</mi><mi id="S5.SS2.SSSx1.p5.2.m2.1.1.3" xref="S5.SS2.SSSx1.p5.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p5.2.m2.1b"><apply id="S5.SS2.SSSx1.p5.2.m2.1.1.cmml" xref="S5.SS2.SSSx1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p5.2.m2.1.1.1.cmml" xref="S5.SS2.SSSx1.p5.2.m2.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p5.2.m2.1.1.2.cmml" xref="S5.SS2.SSSx1.p5.2.m2.1.1.2">𝒰</ci><ci id="S5.SS2.SSSx1.p5.2.m2.1.1.3.cmml" xref="S5.SS2.SSSx1.p5.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p5.2.m2.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p5.2.m2.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, specifically those distributions which are supported on the first <math alttext="m_{1}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p5.3.m3.1"><semantics id="S5.SS2.SSSx1.p5.3.m3.1a"><msub id="S5.SS2.SSSx1.p5.3.m3.1.1" xref="S5.SS2.SSSx1.p5.3.m3.1.1.cmml"><mi id="S5.SS2.SSSx1.p5.3.m3.1.1.2" xref="S5.SS2.SSSx1.p5.3.m3.1.1.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p5.3.m3.1.1.3" xref="S5.SS2.SSSx1.p5.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p5.3.m3.1b"><apply id="S5.SS2.SSSx1.p5.3.m3.1.1.cmml" xref="S5.SS2.SSSx1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p5.3.m3.1.1.1.cmml" xref="S5.SS2.SSSx1.p5.3.m3.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p5.3.m3.1.1.2.cmml" xref="S5.SS2.SSSx1.p5.3.m3.1.1.2">𝑚</ci><cn id="S5.SS2.SSSx1.p5.3.m3.1.1.3.cmml" type="integer" xref="S5.SS2.SSSx1.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p5.3.m3.1c">m_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p5.3.m3.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> paths.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma15"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 15</span></h6> <div class="ltx_para" id="Thmlemma15.p1"> <p class="ltx_p" id="Thmlemma15.p1.3"><span class="ltx_text ltx_font_italic" id="Thmlemma15.p1.3.3">For every <math alttext="i\leq m_{1}" class="ltx_Math" display="inline" id="Thmlemma15.p1.1.1.m1.1"><semantics id="Thmlemma15.p1.1.1.m1.1a"><mrow id="Thmlemma15.p1.1.1.m1.1.1" xref="Thmlemma15.p1.1.1.m1.1.1.cmml"><mi id="Thmlemma15.p1.1.1.m1.1.1.2" xref="Thmlemma15.p1.1.1.m1.1.1.2.cmml">i</mi><mo id="Thmlemma15.p1.1.1.m1.1.1.1" xref="Thmlemma15.p1.1.1.m1.1.1.1.cmml">≤</mo><msub id="Thmlemma15.p1.1.1.m1.1.1.3" xref="Thmlemma15.p1.1.1.m1.1.1.3.cmml"><mi id="Thmlemma15.p1.1.1.m1.1.1.3.2" xref="Thmlemma15.p1.1.1.m1.1.1.3.2.cmml">m</mi><mn id="Thmlemma15.p1.1.1.m1.1.1.3.3" xref="Thmlemma15.p1.1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma15.p1.1.1.m1.1b"><apply id="Thmlemma15.p1.1.1.m1.1.1.cmml" xref="Thmlemma15.p1.1.1.m1.1.1"><leq id="Thmlemma15.p1.1.1.m1.1.1.1.cmml" xref="Thmlemma15.p1.1.1.m1.1.1.1"></leq><ci id="Thmlemma15.p1.1.1.m1.1.1.2.cmml" xref="Thmlemma15.p1.1.1.m1.1.1.2">𝑖</ci><apply id="Thmlemma15.p1.1.1.m1.1.1.3.cmml" xref="Thmlemma15.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma15.p1.1.1.m1.1.1.3.1.cmml" xref="Thmlemma15.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="Thmlemma15.p1.1.1.m1.1.1.3.2.cmml" xref="Thmlemma15.p1.1.1.m1.1.1.3.2">𝑚</ci><cn id="Thmlemma15.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="Thmlemma15.p1.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma15.p1.1.1.m1.1c">i\leq m_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma15.p1.1.1.m1.1d">italic_i ≤ italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="F_{n}(U_{P_{i}}^{n},\alpha)=|p_{1}|+\frac{\mathcal{E}_{i,n}}{i}k\alpha" class="ltx_Math" display="inline" id="Thmlemma15.p1.2.2.m2.5"><semantics id="Thmlemma15.p1.2.2.m2.5a"><mrow id="Thmlemma15.p1.2.2.m2.5.5" xref="Thmlemma15.p1.2.2.m2.5.5.cmml"><mrow id="Thmlemma15.p1.2.2.m2.4.4.1" xref="Thmlemma15.p1.2.2.m2.4.4.1.cmml"><msub id="Thmlemma15.p1.2.2.m2.4.4.1.3" xref="Thmlemma15.p1.2.2.m2.4.4.1.3.cmml"><mi id="Thmlemma15.p1.2.2.m2.4.4.1.3.2" xref="Thmlemma15.p1.2.2.m2.4.4.1.3.2.cmml">F</mi><mi id="Thmlemma15.p1.2.2.m2.4.4.1.3.3" xref="Thmlemma15.p1.2.2.m2.4.4.1.3.3.cmml">n</mi></msub><mo id="Thmlemma15.p1.2.2.m2.4.4.1.2" xref="Thmlemma15.p1.2.2.m2.4.4.1.2.cmml">⁢</mo><mrow id="Thmlemma15.p1.2.2.m2.4.4.1.1.1" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.2.cmml"><mo id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.2" stretchy="false" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.2.cmml">(</mo><msubsup id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.cmml"><mi id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.2" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.2.cmml">U</mi><msub id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.cmml"><mi id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.2" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.2.cmml">P</mi><mi id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.3" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.3.cmml">i</mi></msub><mi id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.3" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.3" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.2.cmml">,</mo><mi id="Thmlemma15.p1.2.2.m2.3.3" xref="Thmlemma15.p1.2.2.m2.3.3.cmml">α</mi><mo id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.4" stretchy="false" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma15.p1.2.2.m2.5.5.3" xref="Thmlemma15.p1.2.2.m2.5.5.3.cmml">=</mo><mrow id="Thmlemma15.p1.2.2.m2.5.5.2" xref="Thmlemma15.p1.2.2.m2.5.5.2.cmml"><mrow id="Thmlemma15.p1.2.2.m2.5.5.2.1.1" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.2.cmml"><mo id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.2" stretchy="false" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.2.1.cmml">|</mo><msub id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.cmml"><mi id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.2" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.2.cmml">p</mi><mn id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.3" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.3.cmml">1</mn></msub><mo id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.3" stretchy="false" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.2.1.cmml">|</mo></mrow><mo id="Thmlemma15.p1.2.2.m2.5.5.2.2" xref="Thmlemma15.p1.2.2.m2.5.5.2.2.cmml">+</mo><mrow id="Thmlemma15.p1.2.2.m2.5.5.2.3" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.cmml"><mfrac id="Thmlemma15.p1.2.2.m2.2.2" xref="Thmlemma15.p1.2.2.m2.2.2.cmml"><msub id="Thmlemma15.p1.2.2.m2.2.2.2" xref="Thmlemma15.p1.2.2.m2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma15.p1.2.2.m2.2.2.2.4" xref="Thmlemma15.p1.2.2.m2.2.2.2.4.cmml">ℰ</mi><mrow id="Thmlemma15.p1.2.2.m2.2.2.2.2.2.4" xref="Thmlemma15.p1.2.2.m2.2.2.2.2.2.3.cmml"><mi id="Thmlemma15.p1.2.2.m2.1.1.1.1.1.1" xref="Thmlemma15.p1.2.2.m2.1.1.1.1.1.1.cmml">i</mi><mo id="Thmlemma15.p1.2.2.m2.2.2.2.2.2.4.1" xref="Thmlemma15.p1.2.2.m2.2.2.2.2.2.3.cmml">,</mo><mi id="Thmlemma15.p1.2.2.m2.2.2.2.2.2.2" xref="Thmlemma15.p1.2.2.m2.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Thmlemma15.p1.2.2.m2.2.2.4" xref="Thmlemma15.p1.2.2.m2.2.2.4.cmml">i</mi></mfrac><mo id="Thmlemma15.p1.2.2.m2.5.5.2.3.1" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.1.cmml">⁢</mo><mi id="Thmlemma15.p1.2.2.m2.5.5.2.3.2" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.2.cmml">k</mi><mo id="Thmlemma15.p1.2.2.m2.5.5.2.3.1a" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.1.cmml">⁢</mo><mi id="Thmlemma15.p1.2.2.m2.5.5.2.3.3" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma15.p1.2.2.m2.5b"><apply id="Thmlemma15.p1.2.2.m2.5.5.cmml" xref="Thmlemma15.p1.2.2.m2.5.5"><eq id="Thmlemma15.p1.2.2.m2.5.5.3.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.3"></eq><apply id="Thmlemma15.p1.2.2.m2.4.4.1.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1"><times id="Thmlemma15.p1.2.2.m2.4.4.1.2.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.2"></times><apply id="Thmlemma15.p1.2.2.m2.4.4.1.3.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.3"><csymbol cd="ambiguous" id="Thmlemma15.p1.2.2.m2.4.4.1.3.1.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.3">subscript</csymbol><ci id="Thmlemma15.p1.2.2.m2.4.4.1.3.2.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.3.2">𝐹</ci><ci id="Thmlemma15.p1.2.2.m2.4.4.1.3.3.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.3.3">𝑛</ci></apply><interval closure="open" id="Thmlemma15.p1.2.2.m2.4.4.1.1.2.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1"><apply id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.1.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1">superscript</csymbol><apply id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.1.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1">subscript</csymbol><ci id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.2.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.2">𝑈</ci><apply id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.1.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3">subscript</csymbol><ci id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.2.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.2">𝑃</ci><ci id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.3.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.3.cmml" xref="Thmlemma15.p1.2.2.m2.4.4.1.1.1.1.3">𝑛</ci></apply><ci id="Thmlemma15.p1.2.2.m2.3.3.cmml" xref="Thmlemma15.p1.2.2.m2.3.3">𝛼</ci></interval></apply><apply id="Thmlemma15.p1.2.2.m2.5.5.2.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2"><plus id="Thmlemma15.p1.2.2.m2.5.5.2.2.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.2"></plus><apply id="Thmlemma15.p1.2.2.m2.5.5.2.1.2.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1"><abs id="Thmlemma15.p1.2.2.m2.5.5.2.1.2.1.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.2"></abs><apply id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.1.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1">subscript</csymbol><ci id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.2.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.2">𝑝</ci><cn id="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.3.cmml" type="integer" xref="Thmlemma15.p1.2.2.m2.5.5.2.1.1.1.3">1</cn></apply></apply><apply id="Thmlemma15.p1.2.2.m2.5.5.2.3.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.3"><times id="Thmlemma15.p1.2.2.m2.5.5.2.3.1.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.1"></times><apply id="Thmlemma15.p1.2.2.m2.2.2.cmml" xref="Thmlemma15.p1.2.2.m2.2.2"><divide id="Thmlemma15.p1.2.2.m2.2.2.3.cmml" xref="Thmlemma15.p1.2.2.m2.2.2"></divide><apply id="Thmlemma15.p1.2.2.m2.2.2.2.cmml" xref="Thmlemma15.p1.2.2.m2.2.2.2"><csymbol cd="ambiguous" id="Thmlemma15.p1.2.2.m2.2.2.2.3.cmml" xref="Thmlemma15.p1.2.2.m2.2.2.2">subscript</csymbol><ci id="Thmlemma15.p1.2.2.m2.2.2.2.4.cmml" xref="Thmlemma15.p1.2.2.m2.2.2.2.4">ℰ</ci><list id="Thmlemma15.p1.2.2.m2.2.2.2.2.2.3.cmml" xref="Thmlemma15.p1.2.2.m2.2.2.2.2.2.4"><ci id="Thmlemma15.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="Thmlemma15.p1.2.2.m2.1.1.1.1.1.1">𝑖</ci><ci id="Thmlemma15.p1.2.2.m2.2.2.2.2.2.2.cmml" xref="Thmlemma15.p1.2.2.m2.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Thmlemma15.p1.2.2.m2.2.2.4.cmml" xref="Thmlemma15.p1.2.2.m2.2.2.4">𝑖</ci></apply><ci id="Thmlemma15.p1.2.2.m2.5.5.2.3.2.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.2">𝑘</ci><ci id="Thmlemma15.p1.2.2.m2.5.5.2.3.3.cmml" xref="Thmlemma15.p1.2.2.m2.5.5.2.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma15.p1.2.2.m2.5c">F_{n}(U_{P_{i}}^{n},\alpha)=|p_{1}|+\frac{\mathcal{E}_{i,n}}{i}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmlemma15.p1.2.2.m2.5d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG italic_k italic_α</annotation></semantics></math>. We also have <math alttext="\min_{i}F_{n}(U_{P_{i}}^{n},\alpha)=F_{n}(U_{P_{m_{1}}}^{n},\alpha)" class="ltx_Math" display="inline" id="Thmlemma15.p1.3.3.m3.4"><semantics id="Thmlemma15.p1.3.3.m3.4a"><mrow id="Thmlemma15.p1.3.3.m3.4.4" xref="Thmlemma15.p1.3.3.m3.4.4.cmml"><mrow id="Thmlemma15.p1.3.3.m3.3.3.1" xref="Thmlemma15.p1.3.3.m3.3.3.1.cmml"><mrow id="Thmlemma15.p1.3.3.m3.3.3.1.3" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.cmml"><msub id="Thmlemma15.p1.3.3.m3.3.3.1.3.1" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.1.cmml"><mi id="Thmlemma15.p1.3.3.m3.3.3.1.3.1.2" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.1.2.cmml">min</mi><mi id="Thmlemma15.p1.3.3.m3.3.3.1.3.1.3" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.1.3.cmml">i</mi></msub><mo id="Thmlemma15.p1.3.3.m3.3.3.1.3a" lspace="0.167em" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.cmml">⁡</mo><msub id="Thmlemma15.p1.3.3.m3.3.3.1.3.2" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.2.cmml"><mi id="Thmlemma15.p1.3.3.m3.3.3.1.3.2.2" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.2.2.cmml">F</mi><mi id="Thmlemma15.p1.3.3.m3.3.3.1.3.2.3" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.2.3.cmml">n</mi></msub></mrow><mo id="Thmlemma15.p1.3.3.m3.3.3.1.2" xref="Thmlemma15.p1.3.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="Thmlemma15.p1.3.3.m3.3.3.1.1.1" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.2.cmml"><mo id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.2" stretchy="false" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.2.cmml">(</mo><msubsup id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.cmml"><mi id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.2" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.2.cmml">U</mi><msub id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.cmml"><mi id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.2" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.2.cmml">P</mi><mi id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.3" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.3.cmml">i</mi></msub><mi id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.3" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.3" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.2.cmml">,</mo><mi id="Thmlemma15.p1.3.3.m3.1.1" xref="Thmlemma15.p1.3.3.m3.1.1.cmml">α</mi><mo id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.4" stretchy="false" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmlemma15.p1.3.3.m3.4.4.3" xref="Thmlemma15.p1.3.3.m3.4.4.3.cmml">=</mo><mrow id="Thmlemma15.p1.3.3.m3.4.4.2" xref="Thmlemma15.p1.3.3.m3.4.4.2.cmml"><msub id="Thmlemma15.p1.3.3.m3.4.4.2.3" xref="Thmlemma15.p1.3.3.m3.4.4.2.3.cmml"><mi id="Thmlemma15.p1.3.3.m3.4.4.2.3.2" xref="Thmlemma15.p1.3.3.m3.4.4.2.3.2.cmml">F</mi><mi id="Thmlemma15.p1.3.3.m3.4.4.2.3.3" xref="Thmlemma15.p1.3.3.m3.4.4.2.3.3.cmml">n</mi></msub><mo id="Thmlemma15.p1.3.3.m3.4.4.2.2" xref="Thmlemma15.p1.3.3.m3.4.4.2.2.cmml">⁢</mo><mrow id="Thmlemma15.p1.3.3.m3.4.4.2.1.1" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.2.cmml"><mo id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.2" stretchy="false" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.2.cmml">(</mo><msubsup id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.cmml"><mi id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.2" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.2.cmml">U</mi><msub id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.cmml"><mi id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.2" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.2.cmml">P</mi><msub id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.cmml"><mi id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.2" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.2.cmml">m</mi><mn id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.3" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.3.cmml">1</mn></msub></msub><mi id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.3" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.3" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.2.cmml">,</mo><mi id="Thmlemma15.p1.3.3.m3.2.2" xref="Thmlemma15.p1.3.3.m3.2.2.cmml">α</mi><mo id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.4" stretchy="false" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma15.p1.3.3.m3.4b"><apply id="Thmlemma15.p1.3.3.m3.4.4.cmml" xref="Thmlemma15.p1.3.3.m3.4.4"><eq id="Thmlemma15.p1.3.3.m3.4.4.3.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.3"></eq><apply id="Thmlemma15.p1.3.3.m3.3.3.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1"><times id="Thmlemma15.p1.3.3.m3.3.3.1.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.2"></times><apply id="Thmlemma15.p1.3.3.m3.3.3.1.3.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3"><apply id="Thmlemma15.p1.3.3.m3.3.3.1.3.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.3.3.1.3.1.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.1">subscript</csymbol><min id="Thmlemma15.p1.3.3.m3.3.3.1.3.1.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.1.2"></min><ci id="Thmlemma15.p1.3.3.m3.3.3.1.3.1.3.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.1.3">𝑖</ci></apply><apply id="Thmlemma15.p1.3.3.m3.3.3.1.3.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.2"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.3.3.1.3.2.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.2">subscript</csymbol><ci id="Thmlemma15.p1.3.3.m3.3.3.1.3.2.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.2.2">𝐹</ci><ci id="Thmlemma15.p1.3.3.m3.3.3.1.3.2.3.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.3.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmlemma15.p1.3.3.m3.3.3.1.1.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1"><apply id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1">superscript</csymbol><apply id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1">subscript</csymbol><ci id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.2">𝑈</ci><apply id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.1.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3">subscript</csymbol><ci id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.2.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.2">𝑃</ci><ci id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.3.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.3.cmml" xref="Thmlemma15.p1.3.3.m3.3.3.1.1.1.1.3">𝑛</ci></apply><ci id="Thmlemma15.p1.3.3.m3.1.1.cmml" xref="Thmlemma15.p1.3.3.m3.1.1">𝛼</ci></interval></apply><apply id="Thmlemma15.p1.3.3.m3.4.4.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2"><times id="Thmlemma15.p1.3.3.m3.4.4.2.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.2"></times><apply id="Thmlemma15.p1.3.3.m3.4.4.2.3.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.3"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.4.4.2.3.1.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.3">subscript</csymbol><ci id="Thmlemma15.p1.3.3.m3.4.4.2.3.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.3.2">𝐹</ci><ci id="Thmlemma15.p1.3.3.m3.4.4.2.3.3.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.3.3">𝑛</ci></apply><interval closure="open" id="Thmlemma15.p1.3.3.m3.4.4.2.1.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1"><apply id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.1.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1">superscript</csymbol><apply id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.1.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1">subscript</csymbol><ci id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.2">𝑈</ci><apply id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.1.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3">subscript</csymbol><ci id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.2">𝑃</ci><apply id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3"><csymbol cd="ambiguous" id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.1.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3">subscript</csymbol><ci id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.2.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.2">𝑚</ci><cn id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.3.cmml" type="integer" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.2.3.3.3">1</cn></apply></apply></apply><ci id="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.3.cmml" xref="Thmlemma15.p1.3.3.m3.4.4.2.1.1.1.3">𝑛</ci></apply><ci id="Thmlemma15.p1.3.3.m3.2.2.cmml" xref="Thmlemma15.p1.3.3.m3.2.2">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma15.p1.3.3.m3.4c">\min_{i}F_{n}(U_{P_{i}}^{n},\alpha)=F_{n}(U_{P_{m_{1}}}^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma15.p1.3.3.m3.4d">roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS2.SSSx1.p6"> <p class="ltx_p" id="S5.SS2.SSSx1.p6.3">Next, an upper bound on the minimal <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p6.1.m1.1"><semantics id="S5.SS2.SSSx1.p6.1.m1.1a"><mi id="S5.SS2.SSSx1.p6.1.m1.1.1" xref="S5.SS2.SSSx1.p6.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p6.1.m1.1b"><ci id="S5.SS2.SSSx1.p6.1.m1.1.1.cmml" xref="S5.SS2.SSSx1.p6.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p6.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p6.1.m1.1d">italic_α</annotation></semantics></math> for which <math alttext="U_{P_{m_{1}}}^{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p6.2.m2.1"><semantics id="S5.SS2.SSSx1.p6.2.m2.1a"><msubsup id="S5.SS2.SSSx1.p6.2.m2.1.1" xref="S5.SS2.SSSx1.p6.2.m2.1.1.cmml"><mi id="S5.SS2.SSSx1.p6.2.m2.1.1.2.2" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.2.cmml">U</mi><msub id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.cmml"><mi id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.2" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.2.cmml">P</mi><msub id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.cmml"><mi id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.2" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.2.cmml">m</mi><mn id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.3" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.3.cmml">1</mn></msub></msub><mi id="S5.SS2.SSSx1.p6.2.m2.1.1.3" xref="S5.SS2.SSSx1.p6.2.m2.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p6.2.m2.1b"><apply id="S5.SS2.SSSx1.p6.2.m2.1.1.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p6.2.m2.1.1.1.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1">superscript</csymbol><apply id="S5.SS2.SSSx1.p6.2.m2.1.1.2.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p6.2.m2.1.1.2.1.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1">subscript</csymbol><ci id="S5.SS2.SSSx1.p6.2.m2.1.1.2.2.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.2">𝑈</ci><apply id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.1.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.2.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.2">𝑃</ci><apply id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.1.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3">subscript</csymbol><ci id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.2.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.2">𝑚</ci><cn id="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.3.cmml" type="integer" xref="S5.SS2.SSSx1.p6.2.m2.1.1.2.3.3.3">1</cn></apply></apply></apply><ci id="S5.SS2.SSSx1.p6.2.m2.1.1.3.cmml" xref="S5.SS2.SSSx1.p6.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p6.2.m2.1c">U_{P_{m_{1}}}^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p6.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is no longer optimal for the agents. Let <math alttext="E^{\prime}_{i}=\frac{\mathcal{E}_{i,n}}{i}" class="ltx_Math" display="inline" id="S5.SS2.SSSx1.p6.3.m3.2"><semantics id="S5.SS2.SSSx1.p6.3.m3.2a"><mrow id="S5.SS2.SSSx1.p6.3.m3.2.3" xref="S5.SS2.SSSx1.p6.3.m3.2.3.cmml"><msubsup id="S5.SS2.SSSx1.p6.3.m3.2.3.2" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2.cmml"><mi id="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.2" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.2.cmml">E</mi><mi id="S5.SS2.SSSx1.p6.3.m3.2.3.2.3" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2.3.cmml">i</mi><mo id="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.3" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.3.cmml">′</mo></msubsup><mo id="S5.SS2.SSSx1.p6.3.m3.2.3.1" xref="S5.SS2.SSSx1.p6.3.m3.2.3.1.cmml">=</mo><mfrac id="S5.SS2.SSSx1.p6.3.m3.2.2" xref="S5.SS2.SSSx1.p6.3.m3.2.2.cmml"><msub id="S5.SS2.SSSx1.p6.3.m3.2.2.2" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSSx1.p6.3.m3.2.2.2.4" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.4.cmml">ℰ</mi><mrow id="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.4" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.3.cmml"><mi id="S5.SS2.SSSx1.p6.3.m3.1.1.1.1.1.1" xref="S5.SS2.SSSx1.p6.3.m3.1.1.1.1.1.1.cmml">i</mi><mo id="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.4.1" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.3.cmml">,</mo><mi id="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.2" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="S5.SS2.SSSx1.p6.3.m3.2.2.4" xref="S5.SS2.SSSx1.p6.3.m3.2.2.4.cmml">i</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx1.p6.3.m3.2b"><apply id="S5.SS2.SSSx1.p6.3.m3.2.3.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3"><eq id="S5.SS2.SSSx1.p6.3.m3.2.3.1.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.1"></eq><apply id="S5.SS2.SSSx1.p6.3.m3.2.3.2.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p6.3.m3.2.3.2.1.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2">subscript</csymbol><apply id="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.1.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2">superscript</csymbol><ci id="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.2.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.2">𝐸</ci><ci id="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.3.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2.2.3">′</ci></apply><ci id="S5.SS2.SSSx1.p6.3.m3.2.3.2.3.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.3.2.3">𝑖</ci></apply><apply id="S5.SS2.SSSx1.p6.3.m3.2.2.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2"><divide id="S5.SS2.SSSx1.p6.3.m3.2.2.3.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2"></divide><apply id="S5.SS2.SSSx1.p6.3.m3.2.2.2.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSSx1.p6.3.m3.2.2.2.3.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2">subscript</csymbol><ci id="S5.SS2.SSSx1.p6.3.m3.2.2.2.4.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.4">ℰ</ci><list id="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.3.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.4"><ci id="S5.SS2.SSSx1.p6.3.m3.1.1.1.1.1.1.cmml" xref="S5.SS2.SSSx1.p6.3.m3.1.1.1.1.1.1">𝑖</ci><ci id="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.2.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="S5.SS2.SSSx1.p6.3.m3.2.2.4.cmml" xref="S5.SS2.SSSx1.p6.3.m3.2.2.4">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx1.p6.3.m3.2c">E^{\prime}_{i}=\frac{\mathcal{E}_{i,n}}{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx1.p6.3.m3.2d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma16"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 16</span></h6> <div class="ltx_para" id="Thmlemma16.p1"> <p class="ltx_p" id="Thmlemma16.p1.2"><span class="ltx_text ltx_font_italic" id="Thmlemma16.p1.2.2">Let <math alttext="\alpha>\frac{(|p_{m_{1}+1}|-|p_{1}|)E^{\prime}_{m_{1}+1}}{k(E^{\prime}_{m_{1}}% -E^{\prime}_{m_{1}+1})}" class="ltx_Math" display="inline" id="Thmlemma16.p1.1.1.m1.2"><semantics id="Thmlemma16.p1.1.1.m1.2a"><mrow id="Thmlemma16.p1.1.1.m1.2.3" xref="Thmlemma16.p1.1.1.m1.2.3.cmml"><mi id="Thmlemma16.p1.1.1.m1.2.3.2" xref="Thmlemma16.p1.1.1.m1.2.3.2.cmml">α</mi><mo id="Thmlemma16.p1.1.1.m1.2.3.1" xref="Thmlemma16.p1.1.1.m1.2.3.1.cmml">></mo><mfrac id="Thmlemma16.p1.1.1.m1.2.2" xref="Thmlemma16.p1.1.1.m1.2.2.cmml"><mrow id="Thmlemma16.p1.1.1.m1.1.1.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.cmml"><mrow id="Thmlemma16.p1.1.1.m1.1.1.1.1.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.cmml"><mrow id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.2.cmml"><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.2.1.cmml">|</mo><msub id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><msub id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.2.cmml">m</mi><mn id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml">1</mn></msub><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.3.cmml">−</mo><mrow id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.2.cmml"><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.2" stretchy="false" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.2.1.cmml">|</mo><msub id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.cmml"><mi id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.2.cmml">p</mi><mn id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.3.cmml">1</mn></msub><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.3" stretchy="false" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmlemma16.p1.1.1.m1.1.1.1.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.2.cmml">⁢</mo><msubsup id="Thmlemma16.p1.1.1.m1.1.1.1.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.cmml"><mi id="Thmlemma16.p1.1.1.m1.1.1.1.3.2.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.2.2.cmml">E</mi><mrow id="Thmlemma16.p1.1.1.m1.1.1.1.3.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.cmml"><msub id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.cmml"><mi id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.2" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.2.cmml">m</mi><mn id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.3.cmml">1</mn></msub><mo id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.1" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.1.cmml">+</mo><mn id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.3.cmml">1</mn></mrow><mo id="Thmlemma16.p1.1.1.m1.1.1.1.3.2.3" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.2.3.cmml">′</mo></msubsup></mrow><mrow id="Thmlemma16.p1.1.1.m1.2.2.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.cmml"><mi id="Thmlemma16.p1.1.1.m1.2.2.2.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.3.cmml">k</mi><mo id="Thmlemma16.p1.1.1.m1.2.2.2.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.2.cmml">⁢</mo><mrow id="Thmlemma16.p1.1.1.m1.2.2.2.1.1" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.cmml"><mo id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.2" stretchy="false" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.cmml"><msubsup id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.cmml"><mi id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.2.cmml">E</mi><msub id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.cmml"><mi id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.2.cmml">m</mi><mn id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.3.cmml">1</mn></msub><mo id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.1" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.1.cmml">−</mo><msubsup id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.cmml"><mi id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.2.cmml">E</mi><mrow id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.cmml"><msub id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.cmml"><mi id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.2" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.2.cmml">m</mi><mn id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.3.cmml">1</mn></msub><mo id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.1" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.1.cmml">+</mo><mn id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.3.cmml">1</mn></mrow><mo id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.3" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.3.cmml">′</mo></msubsup></mrow><mo id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.3" stretchy="false" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma16.p1.1.1.m1.2b"><apply id="Thmlemma16.p1.1.1.m1.2.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.3"><gt id="Thmlemma16.p1.1.1.m1.2.3.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.3.1"></gt><ci id="Thmlemma16.p1.1.1.m1.2.3.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.3.2">𝛼</ci><apply id="Thmlemma16.p1.1.1.m1.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2"><divide id="Thmlemma16.p1.1.1.m1.2.2.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.2"></divide><apply id="Thmlemma16.p1.1.1.m1.1.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1"><times id="Thmlemma16.p1.1.1.m1.1.1.1.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.2"></times><apply id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1"><minus id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.3"></minus><apply id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1"><abs id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.2"></abs><apply id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><apply id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3"><plus id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1"></plus><apply id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.2">𝑚</ci><cn id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.3">1</cn></apply><cn id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1"><abs id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.2"></abs><apply id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1">subscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.2">𝑝</ci><cn id="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.1.1.1.1.1.1.2.1.1.3">1</cn></apply></apply></apply><apply id="Thmlemma16.p1.1.1.m1.1.1.1.3.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.1.1.1.3.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3">subscript</csymbol><apply id="Thmlemma16.p1.1.1.m1.1.1.1.3.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.1.1.1.3.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3">superscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.1.1.1.3.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.2.2">𝐸</ci><ci id="Thmlemma16.p1.1.1.m1.1.1.1.3.2.3.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.2.3">′</ci></apply><apply id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3"><plus id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.1"></plus><apply id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2">subscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.2">𝑚</ci><cn id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.2.3">1</cn></apply><cn id="Thmlemma16.p1.1.1.m1.1.1.1.3.3.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.1.1.1.3.3.3">1</cn></apply></apply></apply><apply id="Thmlemma16.p1.1.1.m1.2.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2"><times id="Thmlemma16.p1.1.1.m1.2.2.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.2"></times><ci id="Thmlemma16.p1.1.1.m1.2.2.2.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.3">𝑘</ci><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1"><minus id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.1"></minus><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2">subscript</csymbol><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2">superscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.2">𝐸</ci><ci id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.2.3">′</ci></apply><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3">subscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.2">𝑚</ci><cn id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.2.3.3">1</cn></apply></apply><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3">subscript</csymbol><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3">superscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.2">𝐸</ci><ci id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.2.3">′</ci></apply><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3"><plus id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.1"></plus><apply id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.1.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2">subscript</csymbol><ci id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.2.cmml" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.2">𝑚</ci><cn id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.2.3">1</cn></apply><cn id="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.3.cmml" type="integer" xref="Thmlemma16.p1.1.1.m1.2.2.2.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma16.p1.1.1.m1.2c">\alpha>\frac{(|p_{m_{1}+1}|-|p_{1}|)E^{\prime}_{m_{1}+1}}{k(E^{\prime}_{m_{1}}% -E^{\prime}_{m_{1}+1})}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma16.p1.1.1.m1.2d">italic_α > divide start_ARG ( | italic_p start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ) italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_k ( italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT ) end_ARG</annotation></semantics></math>. Then <math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha" class="ltx_Math" display="inline" id="Thmlemma16.p1.2.2.m2.6"><semantics id="Thmlemma16.p1.2.2.m2.6a"><mrow id="Thmlemma16.p1.2.2.m2.6.6" xref="Thmlemma16.p1.2.2.m2.6.6.cmml"><mrow id="Thmlemma16.p1.2.2.m2.6.6.3" xref="Thmlemma16.p1.2.2.m2.6.6.3.cmml"><mrow id="Thmlemma16.p1.2.2.m2.6.6.3.2" xref="Thmlemma16.p1.2.2.m2.6.6.3.2.cmml"><msub id="Thmlemma16.p1.2.2.m2.6.6.3.2.1" xref="Thmlemma16.p1.2.2.m2.6.6.3.2.1.cmml"><mi id="Thmlemma16.p1.2.2.m2.6.6.3.2.1.2" xref="Thmlemma16.p1.2.2.m2.6.6.3.2.1.2.cmml">min</mi><mrow id="Thmlemma16.p1.2.2.m2.1.1.1" xref="Thmlemma16.p1.2.2.m2.1.1.1.cmml"><mi id="Thmlemma16.p1.2.2.m2.1.1.1.3" xref="Thmlemma16.p1.2.2.m2.1.1.1.3.cmml">σ</mi><mo id="Thmlemma16.p1.2.2.m2.1.1.1.2" xref="Thmlemma16.p1.2.2.m2.1.1.1.2.cmml">∈</mo><msup id="Thmlemma16.p1.2.2.m2.1.1.1.1" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.cmml"><mrow id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.cmml"><mo id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.cmml"><mi id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.3" mathvariant="normal" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.2" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.2" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmlemma16.p1.2.2.m2.1.1.1.1.3" xref="Thmlemma16.p1.2.2.m2.1.1.1.1.3.cmml">n</mi></msup></mrow></msub><mo id="Thmlemma16.p1.2.2.m2.6.6.3.2a" lspace="0.167em" xref="Thmlemma16.p1.2.2.m2.6.6.3.2.cmml">⁡</mo><msub id="Thmlemma16.p1.2.2.m2.6.6.3.2.2" xref="Thmlemma16.p1.2.2.m2.6.6.3.2.2.cmml"><mi id="Thmlemma16.p1.2.2.m2.6.6.3.2.2.2" xref="Thmlemma16.p1.2.2.m2.6.6.3.2.2.2.cmml">F</mi><mi id="Thmlemma16.p1.2.2.m2.6.6.3.2.2.3" xref="Thmlemma16.p1.2.2.m2.6.6.3.2.2.3.cmml">n</mi></msub></mrow><mo id="Thmlemma16.p1.2.2.m2.6.6.3.1" xref="Thmlemma16.p1.2.2.m2.6.6.3.1.cmml">⁢</mo><mrow id="Thmlemma16.p1.2.2.m2.6.6.3.3.2" xref="Thmlemma16.p1.2.2.m2.6.6.3.3.1.cmml"><mo id="Thmlemma16.p1.2.2.m2.6.6.3.3.2.1" stretchy="false" xref="Thmlemma16.p1.2.2.m2.6.6.3.3.1.cmml">(</mo><mi id="Thmlemma16.p1.2.2.m2.4.4" xref="Thmlemma16.p1.2.2.m2.4.4.cmml">σ</mi><mo id="Thmlemma16.p1.2.2.m2.6.6.3.3.2.2" xref="Thmlemma16.p1.2.2.m2.6.6.3.3.1.cmml">,</mo><mi id="Thmlemma16.p1.2.2.m2.5.5" xref="Thmlemma16.p1.2.2.m2.5.5.cmml">α</mi><mo id="Thmlemma16.p1.2.2.m2.6.6.3.3.2.3" stretchy="false" xref="Thmlemma16.p1.2.2.m2.6.6.3.3.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma16.p1.2.2.m2.6.6.2" xref="Thmlemma16.p1.2.2.m2.6.6.2.cmml"><</mo><mrow id="Thmlemma16.p1.2.2.m2.6.6.1" xref="Thmlemma16.p1.2.2.m2.6.6.1.cmml"><mrow id="Thmlemma16.p1.2.2.m2.6.6.1.1.1" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.2.cmml"><mo id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.2" stretchy="false" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.2.1.cmml">|</mo><msub id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.cmml"><mi id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.2" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.2.cmml">p</mi><mn id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.3" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.3" stretchy="false" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.2.1.cmml">|</mo></mrow><mo id="Thmlemma16.p1.2.2.m2.6.6.1.2" xref="Thmlemma16.p1.2.2.m2.6.6.1.2.cmml">+</mo><mrow id="Thmlemma16.p1.2.2.m2.6.6.1.3" xref="Thmlemma16.p1.2.2.m2.6.6.1.3.cmml"><mfrac id="Thmlemma16.p1.2.2.m2.3.3" xref="Thmlemma16.p1.2.2.m2.3.3.cmml"><msub id="Thmlemma16.p1.2.2.m2.3.3.2" xref="Thmlemma16.p1.2.2.m2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma16.p1.2.2.m2.3.3.2.4" xref="Thmlemma16.p1.2.2.m2.3.3.2.4.cmml">ℰ</mi><mrow id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.3.cmml"><msub id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.cmml"><mi id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.2" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.2.cmml">m</mi><mn id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.3" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.3.cmml">1</mn></msub><mo id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.2" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.3.cmml">,</mo><mi id="Thmlemma16.p1.2.2.m2.2.2.1.1.1.1" xref="Thmlemma16.p1.2.2.m2.2.2.1.1.1.1.cmml">n</mi></mrow></msub><msub id="Thmlemma16.p1.2.2.m2.3.3.4" xref="Thmlemma16.p1.2.2.m2.3.3.4.cmml"><mi id="Thmlemma16.p1.2.2.m2.3.3.4.2" xref="Thmlemma16.p1.2.2.m2.3.3.4.2.cmml">m</mi><mn id="Thmlemma16.p1.2.2.m2.3.3.4.3" xref="Thmlemma16.p1.2.2.m2.3.3.4.3.cmml">1</mn></msub></mfrac><mo 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xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.2"></abs><apply id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.cmml" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.1.cmml" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1">subscript</csymbol><ci id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.2.cmml" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.2">𝑝</ci><cn id="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.3.cmml" type="integer" xref="Thmlemma16.p1.2.2.m2.6.6.1.1.1.1.3">1</cn></apply></apply><apply id="Thmlemma16.p1.2.2.m2.6.6.1.3.cmml" xref="Thmlemma16.p1.2.2.m2.6.6.1.3"><times id="Thmlemma16.p1.2.2.m2.6.6.1.3.1.cmml" xref="Thmlemma16.p1.2.2.m2.6.6.1.3.1"></times><apply id="Thmlemma16.p1.2.2.m2.3.3.cmml" xref="Thmlemma16.p1.2.2.m2.3.3"><divide id="Thmlemma16.p1.2.2.m2.3.3.3.cmml" xref="Thmlemma16.p1.2.2.m2.3.3"></divide><apply id="Thmlemma16.p1.2.2.m2.3.3.2.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.2"><csymbol cd="ambiguous" id="Thmlemma16.p1.2.2.m2.3.3.2.3.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.2">subscript</csymbol><ci id="Thmlemma16.p1.2.2.m2.3.3.2.4.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.2.4">ℰ</ci><list id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.3.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2"><apply id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.1.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1">subscript</csymbol><ci id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.2.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.2">𝑚</ci><cn id="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.3.cmml" type="integer" xref="Thmlemma16.p1.2.2.m2.3.3.2.2.2.2.1.3">1</cn></apply><ci id="Thmlemma16.p1.2.2.m2.2.2.1.1.1.1.cmml" xref="Thmlemma16.p1.2.2.m2.2.2.1.1.1.1">𝑛</ci></list></apply><apply id="Thmlemma16.p1.2.2.m2.3.3.4.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.4"><csymbol cd="ambiguous" id="Thmlemma16.p1.2.2.m2.3.3.4.1.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.4">subscript</csymbol><ci id="Thmlemma16.p1.2.2.m2.3.3.4.2.cmml" xref="Thmlemma16.p1.2.2.m2.3.3.4.2">𝑚</ci><cn id="Thmlemma16.p1.2.2.m2.3.3.4.3.cmml" type="integer" xref="Thmlemma16.p1.2.2.m2.3.3.4.3">1</cn></apply></apply><ci id="Thmlemma16.p1.2.2.m2.6.6.1.3.2.cmml" xref="Thmlemma16.p1.2.2.m2.6.6.1.3.2">𝑘</ci><ci id="Thmlemma16.p1.2.2.m2.6.6.1.3.3.cmml" xref="Thmlemma16.p1.2.2.m2.6.6.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma16.p1.2.2.m2.6c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmlemma16.p1.2.2.m2.6d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS2.SSSx1.p7"> <p class="ltx_p" id="S5.SS2.SSSx1.p7.1">Now we can state the disjoint-paths version of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem1" title="Theorem 5.1 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.1</span></a>:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 5.2</span></h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.7"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem2.p1.7.7">Let <math alttext="G" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.1.m1.1"><semantics id="S5.Thmtheorem2.p1.1.1.m1.1a"><mi id="S5.Thmtheorem2.p1.1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.1.m1.1b"><ci id="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.1.m1.1d">italic_G</annotation></semantics></math> with <math alttext="m>1" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem2.p1.2.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">m</mi><mo id="S5.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S5.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.2.m2.1b"><apply id="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1"><gt id="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1"></gt><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2">𝑚</ci><cn id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.2.m2.1c">m>1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.2.m2.1d">italic_m > 1</annotation></semantics></math> disjoint paths where <math alttext="|p_{1}|<|p_{m}|" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.3.m3.2"><semantics id="S5.Thmtheorem2.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem2.p1.3.3.m3.2.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.cmml"><mrow id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.2.cmml"><mo id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.2.1.cmml">|</mo><msub id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2.cmml">p</mi><mn id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.Thmtheorem2.p1.3.3.m3.2.2.3" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.3.cmml"><</mo><mrow id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.2.cmml"><mo id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.2.1.cmml">|</mo><msub id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.cmml"><mi id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.2.cmml">p</mi><mi id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.3" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.3.cmml">m</mi></msub><mo id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.3.m3.2b"><apply id="S5.Thmtheorem2.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2"><lt id="S5.Thmtheorem2.p1.3.3.m3.2.2.3.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.3"></lt><apply id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1"><abs id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.2.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.2"></abs><apply id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2">𝑝</ci><cn id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1"><abs id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.2.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.2"></abs><apply id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.2">𝑝</ci><ci id="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.3.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.3.m3.2c">|p_{1}|<|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.3.m3.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math>. For every <math alttext="n,k>1" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.4.4.m4.2"><semantics id="S5.Thmtheorem2.p1.4.4.m4.2a"><mrow id="S5.Thmtheorem2.p1.4.4.m4.2.3" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.cmml"><mrow id="S5.Thmtheorem2.p1.4.4.m4.2.3.2.2" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.2.1.cmml"><mi id="S5.Thmtheorem2.p1.4.4.m4.1.1" xref="S5.Thmtheorem2.p1.4.4.m4.1.1.cmml">n</mi><mo id="S5.Thmtheorem2.p1.4.4.m4.2.3.2.2.1" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.2.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.4.4.m4.2.2" xref="S5.Thmtheorem2.p1.4.4.m4.2.2.cmml">k</mi></mrow><mo id="S5.Thmtheorem2.p1.4.4.m4.2.3.1" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.1.cmml">></mo><mn id="S5.Thmtheorem2.p1.4.4.m4.2.3.3" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.4.4.m4.2b"><apply id="S5.Thmtheorem2.p1.4.4.m4.2.3.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.2.3"><gt id="S5.Thmtheorem2.p1.4.4.m4.2.3.1.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.1"></gt><list id="S5.Thmtheorem2.p1.4.4.m4.2.3.2.1.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.2.2"><ci id="S5.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.1.1">𝑛</ci><ci id="S5.Thmtheorem2.p1.4.4.m4.2.2.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.2.2">𝑘</ci></list><cn id="S5.Thmtheorem2.p1.4.4.m4.2.3.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.4.4.m4.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.4.4.m4.2c">n,k>1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.4.4.m4.2d">italic_n , italic_k > 1</annotation></semantics></math> and for every <math alttext="\alpha>\frac{|p_{m}|-|p_{1}|}{k}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.5.5.m5.2"><semantics id="S5.Thmtheorem2.p1.5.5.m5.2a"><mrow id="S5.Thmtheorem2.p1.5.5.m5.2.3" xref="S5.Thmtheorem2.p1.5.5.m5.2.3.cmml"><mi id="S5.Thmtheorem2.p1.5.5.m5.2.3.2" xref="S5.Thmtheorem2.p1.5.5.m5.2.3.2.cmml">α</mi><mo id="S5.Thmtheorem2.p1.5.5.m5.2.3.1" xref="S5.Thmtheorem2.p1.5.5.m5.2.3.1.cmml">></mo><mfrac id="S5.Thmtheorem2.p1.5.5.m5.2.2" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.cmml"><mrow id="S5.Thmtheorem2.p1.5.5.m5.2.2.2" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.cmml"><mrow id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.2.cmml"><mo id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.2.1.cmml">|</mo><msub id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2.cmml">p</mi><mi id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.3" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.3.cmml">−</mo><mrow id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.2.cmml"><mo id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.2.1.cmml">|</mo><msub id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.cmml"><mi id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.2" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.2.cmml">p</mi><mn id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.3" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mi id="S5.Thmtheorem2.p1.5.5.m5.2.2.4" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.4.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.5.5.m5.2b"><apply id="S5.Thmtheorem2.p1.5.5.m5.2.3.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.3"><gt id="S5.Thmtheorem2.p1.5.5.m5.2.3.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.3.1"></gt><ci id="S5.Thmtheorem2.p1.5.5.m5.2.3.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.3.2">𝛼</ci><apply id="S5.Thmtheorem2.p1.5.5.m5.2.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2"><divide id="S5.Thmtheorem2.p1.5.5.m5.2.2.3.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2"></divide><apply id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2"><minus id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.3.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.3"></minus><apply id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1"><abs id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.2"></abs><apply id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2">𝑝</ci><ci id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3">𝑚</ci></apply></apply><apply id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1"><abs id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.2.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.2"></abs><apply id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.2">𝑝</ci><cn id="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.2.2.1.1.3">1</cn></apply></apply></apply><ci id="S5.Thmtheorem2.p1.5.5.m5.2.2.4.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.5.5.m5.2c">\alpha>\frac{|p_{m}|-|p_{1}|}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.5.5.m5.2d">italic_α > divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>, if <math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.6.6.m6.6"><semantics id="S5.Thmtheorem2.p1.6.6.m6.6a"><mrow id="S5.Thmtheorem2.p1.6.6.m6.6.6" xref="S5.Thmtheorem2.p1.6.6.m6.6.6.cmml"><mrow id="S5.Thmtheorem2.p1.6.6.m6.6.6.3" xref="S5.Thmtheorem2.p1.6.6.m6.6.6.3.cmml"><mrow id="S5.Thmtheorem2.p1.6.6.m6.6.6.3.2" xref="S5.Thmtheorem2.p1.6.6.m6.6.6.3.2.cmml"><msub 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id="S5.Thmtheorem2.p1.6.6.m6.3.3.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.3.3.2.2.2.2.1.2">𝑚</ci><cn id="S5.Thmtheorem2.p1.6.6.m6.3.3.2.2.2.2.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.6.6.m6.3.3.2.2.2.2.1.3">1</cn></apply><ci id="S5.Thmtheorem2.p1.6.6.m6.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.2.2.1.1.1.1">𝑛</ci></list></apply><apply id="S5.Thmtheorem2.p1.6.6.m6.3.3.4.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.3.3.4"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.6.6.m6.3.3.4.1.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.3.3.4">subscript</csymbol><ci id="S5.Thmtheorem2.p1.6.6.m6.3.3.4.2.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.3.3.4.2">𝑚</ci><cn id="S5.Thmtheorem2.p1.6.6.m6.3.3.4.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.6.6.m6.3.3.4.3">1</cn></apply></apply><ci id="S5.Thmtheorem2.p1.6.6.m6.6.6.1.3.2.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.6.6.1.3.2">𝑘</ci><ci id="S5.Thmtheorem2.p1.6.6.m6.6.6.1.3.3.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.6.6.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.6.6.m6.6c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.6.6.m6.6d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>, then <math 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xref="S5.Thmtheorem2.p1.7.7.m7.5.6.3.3.2"><ci id="S5.Thmtheorem2.p1.7.7.m7.4.4.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.4.4">𝜎</ci><ci id="S5.Thmtheorem2.p1.7.7.m7.5.5.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.5.5">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.7.7.m7.5c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<\min_{\sigma\in% \mathcal{U}_{n}}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.7.7.m7.5d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS2.SSSx1.p8"> <p class="ltx_p" id="S5.SS2.SSSx1.p8.1">We end with finishing the first part of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S1.Thmtheorem3" title="Theorem 1.3 (Informal version of Theorems 5.1 and 5.3) ‣ 1.1 Our results ‣ 1 Introduction ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1.3</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 5.3</span></h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.5.5">Let <math alttext="G" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.1.m1.1a"><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.1.m1.1b"><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.1.m1.1d">italic_G</annotation></semantics></math> with <math alttext="m" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.2.m2.1a"><mi id="S5.Thmtheorem3.p1.2.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.2.m2.1b"><ci id="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.2.m2.1d">italic_m</annotation></semantics></math> disjoint paths where <math alttext="|p_{1}|=|p_{m}|" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.3.m3.2"><semantics id="S5.Thmtheorem3.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem3.p1.3.3.m3.2.2" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.cmml"><mrow id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.2.cmml"><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.2.1.cmml">|</mo><msub id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.2.cmml">p</mi><mn id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.Thmtheorem3.p1.3.3.m3.2.2.3" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.3.cmml">=</mo><mrow id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.2.cmml"><mo id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.2.1.cmml">|</mo><msub id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.cmml"><mi id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2.cmml">p</mi><mi id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3.cmml">m</mi></msub><mo id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.3.m3.2b"><apply id="S5.Thmtheorem3.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2"><eq id="S5.Thmtheorem3.p1.3.3.m3.2.2.3.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.3"></eq><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1"><abs id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.2"></abs><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.2">𝑝</ci><cn id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1"><abs id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.2.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.2"></abs><apply id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2">𝑝</ci><ci id="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.3.m3.2c">|p_{1}|=|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.3.m3.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math>. For every <math alttext="n,k" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.4.4.m4.2"><semantics id="S5.Thmtheorem3.p1.4.4.m4.2a"><mrow id="S5.Thmtheorem3.p1.4.4.m4.2.3.2" xref="S5.Thmtheorem3.p1.4.4.m4.2.3.1.cmml"><mi id="S5.Thmtheorem3.p1.4.4.m4.1.1" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml">n</mi><mo id="S5.Thmtheorem3.p1.4.4.m4.2.3.2.1" xref="S5.Thmtheorem3.p1.4.4.m4.2.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.4.4.m4.2.2" xref="S5.Thmtheorem3.p1.4.4.m4.2.2.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.4.4.m4.2b"><list id="S5.Thmtheorem3.p1.4.4.m4.2.3.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.2.3.2"><ci id="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1">𝑛</ci><ci id="S5.Thmtheorem3.p1.4.4.m4.2.2.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.2.2">𝑘</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.4.4.m4.2c">n,k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.4.4.m4.2d">italic_n , italic_k</annotation></semantics></math> and for every <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.5.5.m5.1"><semantics id="S5.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml"><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml">α</mi><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.cmml">></mo><mn id="S5.Thmtheorem3.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.5.5.m5.1b"><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1"><gt id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1"></gt><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2">𝛼</ci><cn id="S5.Thmtheorem3.p1.5.5.m5.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.5.5.m5.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.5.5.m5.1d">italic_α > 0</annotation></semantics></math>:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx13"> <tbody id="S5.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)=F_{n}(U_% {P_{st}}^{n},\alpha)=|p_{1}|+\frac{\mathcal{E}_{m,n}}{m}k\alpha" class="ltx_Math" display="inline" id="S5.Ex12.m1.8"><semantics id="S5.Ex12.m1.8a"><mrow id="S5.Ex12.m1.8.8" xref="S5.Ex12.m1.8.8.cmml"><mrow id="S5.Ex12.m1.8.8.4" xref="S5.Ex12.m1.8.8.4.cmml"><mrow id="S5.Ex12.m1.8.8.4.2" xref="S5.Ex12.m1.8.8.4.2.cmml"><munder id="S5.Ex12.m1.8.8.4.2.1" xref="S5.Ex12.m1.8.8.4.2.1.cmml"><mi id="S5.Ex12.m1.8.8.4.2.1.2" xref="S5.Ex12.m1.8.8.4.2.1.2.cmml">min</mi><mrow id="S5.Ex12.m1.1.1.1" xref="S5.Ex12.m1.1.1.1.cmml"><mi id="S5.Ex12.m1.1.1.1.3" xref="S5.Ex12.m1.1.1.1.3.cmml">σ</mi><mo id="S5.Ex12.m1.1.1.1.2" xref="S5.Ex12.m1.1.1.1.2.cmml">∈</mo><msup id="S5.Ex12.m1.1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.cmml"><mrow id="S5.Ex12.m1.1.1.1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex12.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex12.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex12.m1.1.1.1.1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex12.m1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="S5.Ex12.m1.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="S5.Ex12.m1.1.1.1.1.1.1.1.2" xref="S5.Ex12.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3.1" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3.3" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex12.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex12.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex12.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.Ex12.m1.1.1.1.1.3" xref="S5.Ex12.m1.1.1.1.1.3.cmml">n</mi></msup></mrow></munder><mo id="S5.Ex12.m1.8.8.4.2a" lspace="0.167em" xref="S5.Ex12.m1.8.8.4.2.cmml">⁡</mo><msub id="S5.Ex12.m1.8.8.4.2.2" xref="S5.Ex12.m1.8.8.4.2.2.cmml"><mi id="S5.Ex12.m1.8.8.4.2.2.2" xref="S5.Ex12.m1.8.8.4.2.2.2.cmml">F</mi><mi id="S5.Ex12.m1.8.8.4.2.2.3" xref="S5.Ex12.m1.8.8.4.2.2.3.cmml">n</mi></msub></mrow><mo id="S5.Ex12.m1.8.8.4.1" xref="S5.Ex12.m1.8.8.4.1.cmml">⁢</mo><mrow id="S5.Ex12.m1.8.8.4.3.2" xref="S5.Ex12.m1.8.8.4.3.1.cmml"><mo id="S5.Ex12.m1.8.8.4.3.2.1" stretchy="false" xref="S5.Ex12.m1.8.8.4.3.1.cmml">(</mo><mi id="S5.Ex12.m1.4.4" xref="S5.Ex12.m1.4.4.cmml">σ</mi><mo id="S5.Ex12.m1.8.8.4.3.2.2" xref="S5.Ex12.m1.8.8.4.3.1.cmml">,</mo><mi id="S5.Ex12.m1.5.5" xref="S5.Ex12.m1.5.5.cmml">α</mi><mo id="S5.Ex12.m1.8.8.4.3.2.3" stretchy="false" xref="S5.Ex12.m1.8.8.4.3.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex12.m1.8.8.5" xref="S5.Ex12.m1.8.8.5.cmml">=</mo><mrow id="S5.Ex12.m1.7.7.1" xref="S5.Ex12.m1.7.7.1.cmml"><msub id="S5.Ex12.m1.7.7.1.3" xref="S5.Ex12.m1.7.7.1.3.cmml"><mi id="S5.Ex12.m1.7.7.1.3.2" xref="S5.Ex12.m1.7.7.1.3.2.cmml">F</mi><mi id="S5.Ex12.m1.7.7.1.3.3" xref="S5.Ex12.m1.7.7.1.3.3.cmml">n</mi></msub><mo id="S5.Ex12.m1.7.7.1.2" xref="S5.Ex12.m1.7.7.1.2.cmml">⁢</mo><mrow id="S5.Ex12.m1.7.7.1.1.1" xref="S5.Ex12.m1.7.7.1.1.2.cmml"><mo id="S5.Ex12.m1.7.7.1.1.1.2" stretchy="false" xref="S5.Ex12.m1.7.7.1.1.2.cmml">(</mo><msubsup id="S5.Ex12.m1.7.7.1.1.1.1" xref="S5.Ex12.m1.7.7.1.1.1.1.cmml"><mi id="S5.Ex12.m1.7.7.1.1.1.1.2.2" xref="S5.Ex12.m1.7.7.1.1.1.1.2.2.cmml">U</mi><msub id="S5.Ex12.m1.7.7.1.1.1.1.2.3" xref="S5.Ex12.m1.7.7.1.1.1.1.2.3.cmml"><mi id="S5.Ex12.m1.7.7.1.1.1.1.2.3.2" xref="S5.Ex12.m1.7.7.1.1.1.1.2.3.2.cmml">P</mi><mrow id="S5.Ex12.m1.7.7.1.1.1.1.2.3.3" xref="S5.Ex12.m1.7.7.1.1.1.1.2.3.3.cmml"><mi id="S5.Ex12.m1.7.7.1.1.1.1.2.3.3.2" xref="S5.Ex12.m1.7.7.1.1.1.1.2.3.3.2.cmml">s</mi><mo id="S5.Ex12.m1.7.7.1.1.1.1.2.3.3.1" xref="S5.Ex12.m1.7.7.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="S5.Ex12.m1.7.7.1.1.1.1.2.3.3.3" xref="S5.Ex12.m1.7.7.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="S5.Ex12.m1.7.7.1.1.1.1.3" xref="S5.Ex12.m1.7.7.1.1.1.1.3.cmml">n</mi></msubsup><mo id="S5.Ex12.m1.7.7.1.1.1.3" xref="S5.Ex12.m1.7.7.1.1.2.cmml">,</mo><mi id="S5.Ex12.m1.6.6" xref="S5.Ex12.m1.6.6.cmml">α</mi><mo id="S5.Ex12.m1.7.7.1.1.1.4" stretchy="false" xref="S5.Ex12.m1.7.7.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Ex12.m1.8.8.6" xref="S5.Ex12.m1.8.8.6.cmml">=</mo><mrow id="S5.Ex12.m1.8.8.2" xref="S5.Ex12.m1.8.8.2.cmml"><mrow id="S5.Ex12.m1.8.8.2.1.1" xref="S5.Ex12.m1.8.8.2.1.2.cmml"><mo id="S5.Ex12.m1.8.8.2.1.1.2" stretchy="false" xref="S5.Ex12.m1.8.8.2.1.2.1.cmml">|</mo><msub id="S5.Ex12.m1.8.8.2.1.1.1" xref="S5.Ex12.m1.8.8.2.1.1.1.cmml"><mi id="S5.Ex12.m1.8.8.2.1.1.1.2" xref="S5.Ex12.m1.8.8.2.1.1.1.2.cmml">p</mi><mn id="S5.Ex12.m1.8.8.2.1.1.1.3" xref="S5.Ex12.m1.8.8.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.Ex12.m1.8.8.2.1.1.3" stretchy="false" xref="S5.Ex12.m1.8.8.2.1.2.1.cmml">|</mo></mrow><mo id="S5.Ex12.m1.8.8.2.2" xref="S5.Ex12.m1.8.8.2.2.cmml">+</mo><mrow id="S5.Ex12.m1.8.8.2.3" 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italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> </section> <section class="ltx_subsubsection" id="S5.SS2.SSSx2"> <h4 class="ltx_title ltx_title_subsubsection">The <math alttext="k=1" class="ltx_Math" display="inline" id="S5.SS2.SSSx2.1.m1.1"><semantics id="S5.SS2.SSSx2.1.m1.1b"><mrow id="S5.SS2.SSSx2.1.m1.1.1" xref="S5.SS2.SSSx2.1.m1.1.1.cmml"><mi id="S5.SS2.SSSx2.1.m1.1.1.2" xref="S5.SS2.SSSx2.1.m1.1.1.2.cmml">k</mi><mo id="S5.SS2.SSSx2.1.m1.1.1.1" 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Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">9</span></a>:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma17"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 17</span></h6> <div class="ltx_para" id="Thmlemma17.p1"> <p class="ltx_p" id="Thmlemma17.p1.4"><span class="ltx_text ltx_font_italic" id="Thmlemma17.p1.4.4">When minimizing <math alttext="F_{n}(\cdot,\alpha)" class="ltx_Math" display="inline" id="Thmlemma17.p1.1.1.m1.2"><semantics id="Thmlemma17.p1.1.1.m1.2a"><mrow id="Thmlemma17.p1.1.1.m1.2.3" xref="Thmlemma17.p1.1.1.m1.2.3.cmml"><msub id="Thmlemma17.p1.1.1.m1.2.3.2" xref="Thmlemma17.p1.1.1.m1.2.3.2.cmml"><mi id="Thmlemma17.p1.1.1.m1.2.3.2.2" xref="Thmlemma17.p1.1.1.m1.2.3.2.2.cmml">F</mi><mi id="Thmlemma17.p1.1.1.m1.2.3.2.3" xref="Thmlemma17.p1.1.1.m1.2.3.2.3.cmml">n</mi></msub><mo id="Thmlemma17.p1.1.1.m1.2.3.1" xref="Thmlemma17.p1.1.1.m1.2.3.1.cmml">⁢</mo><mrow id="Thmlemma17.p1.1.1.m1.2.3.3.2" xref="Thmlemma17.p1.1.1.m1.2.3.3.1.cmml"><mo id="Thmlemma17.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="Thmlemma17.p1.1.1.m1.2.3.3.1.cmml">(</mo><mo id="Thmlemma17.p1.1.1.m1.1.1" lspace="0em" rspace="0em" xref="Thmlemma17.p1.1.1.m1.1.1.cmml">⋅</mo><mo id="Thmlemma17.p1.1.1.m1.2.3.3.2.2" xref="Thmlemma17.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="Thmlemma17.p1.1.1.m1.2.2" xref="Thmlemma17.p1.1.1.m1.2.2.cmml">α</mi><mo id="Thmlemma17.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="Thmlemma17.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma17.p1.1.1.m1.2b"><apply id="Thmlemma17.p1.1.1.m1.2.3.cmml" xref="Thmlemma17.p1.1.1.m1.2.3"><times id="Thmlemma17.p1.1.1.m1.2.3.1.cmml" xref="Thmlemma17.p1.1.1.m1.2.3.1"></times><apply id="Thmlemma17.p1.1.1.m1.2.3.2.cmml" xref="Thmlemma17.p1.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="Thmlemma17.p1.1.1.m1.2.3.2.1.cmml" xref="Thmlemma17.p1.1.1.m1.2.3.2">subscript</csymbol><ci id="Thmlemma17.p1.1.1.m1.2.3.2.2.cmml" xref="Thmlemma17.p1.1.1.m1.2.3.2.2">𝐹</ci><ci id="Thmlemma17.p1.1.1.m1.2.3.2.3.cmml" xref="Thmlemma17.p1.1.1.m1.2.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmlemma17.p1.1.1.m1.2.3.3.1.cmml" xref="Thmlemma17.p1.1.1.m1.2.3.3.2"><ci id="Thmlemma17.p1.1.1.m1.1.1.cmml" xref="Thmlemma17.p1.1.1.m1.1.1">⋅</ci><ci id="Thmlemma17.p1.1.1.m1.2.2.cmml" xref="Thmlemma17.p1.1.1.m1.2.2">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma17.p1.1.1.m1.2c">F_{n}(\cdot,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma17.p1.1.1.m1.2d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ⋅ , italic_α )</annotation></semantics></math> we can restrict ourselves to the set of decreasing distributions: all <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmlemma17.p1.2.2.m2.1"><semantics id="Thmlemma17.p1.2.2.m2.1a"><mi id="Thmlemma17.p1.2.2.m2.1.1" xref="Thmlemma17.p1.2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmlemma17.p1.2.2.m2.1b"><ci id="Thmlemma17.p1.2.2.m2.1.1.cmml" xref="Thmlemma17.p1.2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma17.p1.2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmlemma17.p1.2.2.m2.1d">italic_σ</annotation></semantics></math> such that for <math alttext="i<i^{\prime}" class="ltx_Math" display="inline" id="Thmlemma17.p1.3.3.m3.1"><semantics id="Thmlemma17.p1.3.3.m3.1a"><mrow id="Thmlemma17.p1.3.3.m3.1.1" xref="Thmlemma17.p1.3.3.m3.1.1.cmml"><mi id="Thmlemma17.p1.3.3.m3.1.1.2" xref="Thmlemma17.p1.3.3.m3.1.1.2.cmml">i</mi><mo id="Thmlemma17.p1.3.3.m3.1.1.1" xref="Thmlemma17.p1.3.3.m3.1.1.1.cmml"><</mo><msup id="Thmlemma17.p1.3.3.m3.1.1.3" xref="Thmlemma17.p1.3.3.m3.1.1.3.cmml"><mi id="Thmlemma17.p1.3.3.m3.1.1.3.2" xref="Thmlemma17.p1.3.3.m3.1.1.3.2.cmml">i</mi><mo id="Thmlemma17.p1.3.3.m3.1.1.3.3" xref="Thmlemma17.p1.3.3.m3.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma17.p1.3.3.m3.1b"><apply id="Thmlemma17.p1.3.3.m3.1.1.cmml" xref="Thmlemma17.p1.3.3.m3.1.1"><lt id="Thmlemma17.p1.3.3.m3.1.1.1.cmml" xref="Thmlemma17.p1.3.3.m3.1.1.1"></lt><ci id="Thmlemma17.p1.3.3.m3.1.1.2.cmml" xref="Thmlemma17.p1.3.3.m3.1.1.2">𝑖</ci><apply id="Thmlemma17.p1.3.3.m3.1.1.3.cmml" xref="Thmlemma17.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="Thmlemma17.p1.3.3.m3.1.1.3.1.cmml" xref="Thmlemma17.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="Thmlemma17.p1.3.3.m3.1.1.3.2.cmml" xref="Thmlemma17.p1.3.3.m3.1.1.3.2">𝑖</ci><ci id="Thmlemma17.p1.3.3.m3.1.1.3.3.cmml" xref="Thmlemma17.p1.3.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma17.p1.3.3.m3.1c">i<i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma17.p1.3.3.m3.1d">italic_i < italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\sigma(p_{i})\geq\sigma(p_{i^{\prime}})" class="ltx_Math" display="inline" id="Thmlemma17.p1.4.4.m4.2"><semantics id="Thmlemma17.p1.4.4.m4.2a"><mrow id="Thmlemma17.p1.4.4.m4.2.2" xref="Thmlemma17.p1.4.4.m4.2.2.cmml"><mrow id="Thmlemma17.p1.4.4.m4.1.1.1" xref="Thmlemma17.p1.4.4.m4.1.1.1.cmml"><mi id="Thmlemma17.p1.4.4.m4.1.1.1.3" xref="Thmlemma17.p1.4.4.m4.1.1.1.3.cmml">σ</mi><mo id="Thmlemma17.p1.4.4.m4.1.1.1.2" xref="Thmlemma17.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma17.p1.4.4.m4.1.1.1.1.1" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.2" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.3" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma17.p1.4.4.m4.2.2.3" xref="Thmlemma17.p1.4.4.m4.2.2.3.cmml">≥</mo><mrow id="Thmlemma17.p1.4.4.m4.2.2.2" xref="Thmlemma17.p1.4.4.m4.2.2.2.cmml"><mi id="Thmlemma17.p1.4.4.m4.2.2.2.3" xref="Thmlemma17.p1.4.4.m4.2.2.2.3.cmml">σ</mi><mo id="Thmlemma17.p1.4.4.m4.2.2.2.2" xref="Thmlemma17.p1.4.4.m4.2.2.2.2.cmml">⁢</mo><mrow id="Thmlemma17.p1.4.4.m4.2.2.2.1.1" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.cmml"><mo id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.2" stretchy="false" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.cmml">(</mo><msub id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.cmml"><mi id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.2" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.2.cmml">p</mi><msup id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.cmml"><mi id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.2" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.3" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.3" stretchy="false" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma17.p1.4.4.m4.2b"><apply id="Thmlemma17.p1.4.4.m4.2.2.cmml" xref="Thmlemma17.p1.4.4.m4.2.2"><geq id="Thmlemma17.p1.4.4.m4.2.2.3.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.3"></geq><apply id="Thmlemma17.p1.4.4.m4.1.1.1.cmml" xref="Thmlemma17.p1.4.4.m4.1.1.1"><times id="Thmlemma17.p1.4.4.m4.1.1.1.2.cmml" xref="Thmlemma17.p1.4.4.m4.1.1.1.2"></times><ci id="Thmlemma17.p1.4.4.m4.1.1.1.3.cmml" xref="Thmlemma17.p1.4.4.m4.1.1.1.3">𝜎</ci><apply id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmlemma17.p1.4.4.m4.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmlemma17.p1.4.4.m4.2.2.2.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2"><times id="Thmlemma17.p1.4.4.m4.2.2.2.2.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.2"></times><ci id="Thmlemma17.p1.4.4.m4.2.2.2.3.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.3">𝜎</ci><apply id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.1.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1">subscript</csymbol><ci id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.2.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.2">𝑝</ci><apply id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.1.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3">superscript</csymbol><ci id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.2.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.2">𝑖</ci><ci id="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.3.cmml" xref="Thmlemma17.p1.4.4.m4.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma17.p1.4.4.m4.2c">\sigma(p_{i})\geq\sigma(p_{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma17.p1.4.4.m4.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ italic_σ ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS2.SSSx2.p2"> <p class="ltx_p" id="S5.SS2.SSSx2.p2.1">Using Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma7" title="Lemma 7 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">7</span></a>, we get a corollary for MAX cost which is analogous to the SUM cost case:</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Corollary 3</span></h6> <div class="ltx_para" id="Thmcorollary3.p1"> <p class="ltx_p" id="Thmcorollary3.p1.1"><span class="ltx_text ltx_font_italic" id="Thmcorollary3.p1.1.1">When <math alttext="k=1" class="ltx_Math" display="inline" id="Thmcorollary3.p1.1.1.m1.1"><semantics id="Thmcorollary3.p1.1.1.m1.1a"><mrow id="Thmcorollary3.p1.1.1.m1.1.1" xref="Thmcorollary3.p1.1.1.m1.1.1.cmml"><mi id="Thmcorollary3.p1.1.1.m1.1.1.2" xref="Thmcorollary3.p1.1.1.m1.1.1.2.cmml">k</mi><mo id="Thmcorollary3.p1.1.1.m1.1.1.1" xref="Thmcorollary3.p1.1.1.m1.1.1.1.cmml">=</mo><mn id="Thmcorollary3.p1.1.1.m1.1.1.3" xref="Thmcorollary3.p1.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary3.p1.1.1.m1.1b"><apply id="Thmcorollary3.p1.1.1.m1.1.1.cmml" xref="Thmcorollary3.p1.1.1.m1.1.1"><eq id="Thmcorollary3.p1.1.1.m1.1.1.1.cmml" xref="Thmcorollary3.p1.1.1.m1.1.1.1"></eq><ci id="Thmcorollary3.p1.1.1.m1.1.1.2.cmml" xref="Thmcorollary3.p1.1.1.m1.1.1.2">𝑘</ci><cn id="Thmcorollary3.p1.1.1.m1.1.1.3.cmml" type="integer" xref="Thmcorollary3.p1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary3.p1.1.1.m1.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary3.p1.1.1.m1.1d">italic_k = 1</annotation></semantics></math>, there holds</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx14"> <tbody id="S5.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right 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start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = roman_min start_POSTSUBSCRIPT italic_σ ∈ roman_conv ( caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S5.SS2.SSSx2.p3"> <p class="ltx_p" id="S5.SS2.SSSx2.p3.2">But in contrast to the SUM cost case, Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem2" title="Theorem 5.2 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.2</span></a> implies that in many game instances <math alttext="F_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx2.p3.1.m1.1"><semantics id="S5.SS2.SSSx2.p3.1.m1.1a"><msub id="S5.SS2.SSSx2.p3.1.m1.1.1" xref="S5.SS2.SSSx2.p3.1.m1.1.1.cmml"><mi id="S5.SS2.SSSx2.p3.1.m1.1.1.2" xref="S5.SS2.SSSx2.p3.1.m1.1.1.2.cmml">F</mi><mi id="S5.SS2.SSSx2.p3.1.m1.1.1.3" xref="S5.SS2.SSSx2.p3.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx2.p3.1.m1.1b"><apply id="S5.SS2.SSSx2.p3.1.m1.1.1.cmml" xref="S5.SS2.SSSx2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx2.p3.1.m1.1.1.1.cmml" xref="S5.SS2.SSSx2.p3.1.m1.1.1">subscript</csymbol><ci id="S5.SS2.SSSx2.p3.1.m1.1.1.2.cmml" xref="S5.SS2.SSSx2.p3.1.m1.1.1.2">𝐹</ci><ci id="S5.SS2.SSSx2.p3.1.m1.1.1.3.cmml" xref="S5.SS2.SSSx2.p3.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx2.p3.1.m1.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx2.p3.1.m1.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is not concave on <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="S5.SS2.SSSx2.p3.2.m2.1"><semantics id="S5.SS2.SSSx2.p3.2.m2.1a"><msub id="S5.SS2.SSSx2.p3.2.m2.1.1" xref="S5.SS2.SSSx2.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSSx2.p3.2.m2.1.1.2" xref="S5.SS2.SSSx2.p3.2.m2.1.1.2.cmml">𝒰</mi><mi id="S5.SS2.SSSx2.p3.2.m2.1.1.3" xref="S5.SS2.SSSx2.p3.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSSx2.p3.2.m2.1b"><apply id="S5.SS2.SSSx2.p3.2.m2.1.1.cmml" xref="S5.SS2.SSSx2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSSx2.p3.2.m2.1.1.1.cmml" xref="S5.SS2.SSSx2.p3.2.m2.1.1">subscript</csymbol><ci id="S5.SS2.SSSx2.p3.2.m2.1.1.2.cmml" xref="S5.SS2.SSSx2.p3.2.m2.1.1.2">𝒰</ci><ci id="S5.SS2.SSSx2.p3.2.m2.1.1.3.cmml" xref="S5.SS2.SSSx2.p3.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSSx2.p3.2.m2.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSSx2.p3.2.m2.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>:</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary4"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Corollary 4</span></h6> <div class="ltx_para" id="Thmcorollary4.p1"> <p class="ltx_p" id="Thmcorollary4.p1.9"><span class="ltx_text ltx_font_italic" id="Thmcorollary4.p1.9.9">Let <math alttext="G" class="ltx_Math" display="inline" id="Thmcorollary4.p1.1.1.m1.1"><semantics id="Thmcorollary4.p1.1.1.m1.1a"><mi id="Thmcorollary4.p1.1.1.m1.1.1" xref="Thmcorollary4.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.1.1.m1.1b"><ci id="Thmcorollary4.p1.1.1.m1.1.1.cmml" xref="Thmcorollary4.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.1.1.m1.1d">italic_G</annotation></semantics></math> a graph with <math alttext="m" class="ltx_Math" display="inline" id="Thmcorollary4.p1.2.2.m2.1"><semantics id="Thmcorollary4.p1.2.2.m2.1a"><mi id="Thmcorollary4.p1.2.2.m2.1.1" xref="Thmcorollary4.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.2.2.m2.1b"><ci id="Thmcorollary4.p1.2.2.m2.1.1.cmml" xref="Thmcorollary4.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.2.2.m2.1d">italic_m</annotation></semantics></math> disjoint paths, and <math alttext="m_{1}" class="ltx_Math" display="inline" id="Thmcorollary4.p1.3.3.m3.1"><semantics id="Thmcorollary4.p1.3.3.m3.1a"><msub id="Thmcorollary4.p1.3.3.m3.1.1" xref="Thmcorollary4.p1.3.3.m3.1.1.cmml"><mi id="Thmcorollary4.p1.3.3.m3.1.1.2" xref="Thmcorollary4.p1.3.3.m3.1.1.2.cmml">m</mi><mn id="Thmcorollary4.p1.3.3.m3.1.1.3" xref="Thmcorollary4.p1.3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.3.3.m3.1b"><apply id="Thmcorollary4.p1.3.3.m3.1.1.cmml" xref="Thmcorollary4.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.3.3.m3.1.1.1.cmml" xref="Thmcorollary4.p1.3.3.m3.1.1">subscript</csymbol><ci id="Thmcorollary4.p1.3.3.m3.1.1.2.cmml" xref="Thmcorollary4.p1.3.3.m3.1.1.2">𝑚</ci><cn id="Thmcorollary4.p1.3.3.m3.1.1.3.cmml" type="integer" xref="Thmcorollary4.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.3.3.m3.1c">m_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.3.3.m3.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> shortest paths. Let <math alttext="k=1" class="ltx_Math" display="inline" id="Thmcorollary4.p1.4.4.m4.1"><semantics id="Thmcorollary4.p1.4.4.m4.1a"><mrow id="Thmcorollary4.p1.4.4.m4.1.1" xref="Thmcorollary4.p1.4.4.m4.1.1.cmml"><mi id="Thmcorollary4.p1.4.4.m4.1.1.2" xref="Thmcorollary4.p1.4.4.m4.1.1.2.cmml">k</mi><mo id="Thmcorollary4.p1.4.4.m4.1.1.1" xref="Thmcorollary4.p1.4.4.m4.1.1.1.cmml">=</mo><mn id="Thmcorollary4.p1.4.4.m4.1.1.3" xref="Thmcorollary4.p1.4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.4.4.m4.1b"><apply id="Thmcorollary4.p1.4.4.m4.1.1.cmml" xref="Thmcorollary4.p1.4.4.m4.1.1"><eq id="Thmcorollary4.p1.4.4.m4.1.1.1.cmml" xref="Thmcorollary4.p1.4.4.m4.1.1.1"></eq><ci id="Thmcorollary4.p1.4.4.m4.1.1.2.cmml" xref="Thmcorollary4.p1.4.4.m4.1.1.2">𝑘</ci><cn id="Thmcorollary4.p1.4.4.m4.1.1.3.cmml" type="integer" xref="Thmcorollary4.p1.4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.4.4.m4.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.4.4.m4.1d">italic_k = 1</annotation></semantics></math> and <math alttext="n>1" class="ltx_Math" display="inline" id="Thmcorollary4.p1.5.5.m5.1"><semantics id="Thmcorollary4.p1.5.5.m5.1a"><mrow id="Thmcorollary4.p1.5.5.m5.1.1" xref="Thmcorollary4.p1.5.5.m5.1.1.cmml"><mi id="Thmcorollary4.p1.5.5.m5.1.1.2" xref="Thmcorollary4.p1.5.5.m5.1.1.2.cmml">n</mi><mo id="Thmcorollary4.p1.5.5.m5.1.1.1" xref="Thmcorollary4.p1.5.5.m5.1.1.1.cmml">></mo><mn id="Thmcorollary4.p1.5.5.m5.1.1.3" xref="Thmcorollary4.p1.5.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.5.5.m5.1b"><apply id="Thmcorollary4.p1.5.5.m5.1.1.cmml" xref="Thmcorollary4.p1.5.5.m5.1.1"><gt id="Thmcorollary4.p1.5.5.m5.1.1.1.cmml" xref="Thmcorollary4.p1.5.5.m5.1.1.1"></gt><ci id="Thmcorollary4.p1.5.5.m5.1.1.2.cmml" xref="Thmcorollary4.p1.5.5.m5.1.1.2">𝑛</ci><cn id="Thmcorollary4.p1.5.5.m5.1.1.3.cmml" type="integer" xref="Thmcorollary4.p1.5.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.5.5.m5.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.5.5.m5.1d">italic_n > 1</annotation></semantics></math>. Let <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmcorollary4.p1.6.6.m6.1"><semantics id="Thmcorollary4.p1.6.6.m6.1a"><mi id="Thmcorollary4.p1.6.6.m6.1.1" xref="Thmcorollary4.p1.6.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.6.6.m6.1b"><ci id="Thmcorollary4.p1.6.6.m6.1.1.cmml" xref="Thmcorollary4.p1.6.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.6.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.6.6.m6.1d">italic_α</annotation></semantics></math> such that <math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha" class="ltx_Math" display="inline" id="Thmcorollary4.p1.7.7.m7.6"><semantics id="Thmcorollary4.p1.7.7.m7.6a"><mrow id="Thmcorollary4.p1.7.7.m7.6.6" xref="Thmcorollary4.p1.7.7.m7.6.6.cmml"><mrow id="Thmcorollary4.p1.7.7.m7.6.6.3" xref="Thmcorollary4.p1.7.7.m7.6.6.3.cmml"><mrow id="Thmcorollary4.p1.7.7.m7.6.6.3.2" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.cmml"><msub id="Thmcorollary4.p1.7.7.m7.6.6.3.2.1" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.1.cmml"><mi id="Thmcorollary4.p1.7.7.m7.6.6.3.2.1.2" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.1.2.cmml">min</mi><mrow id="Thmcorollary4.p1.7.7.m7.1.1.1" xref="Thmcorollary4.p1.7.7.m7.1.1.1.cmml"><mi id="Thmcorollary4.p1.7.7.m7.1.1.1.3" xref="Thmcorollary4.p1.7.7.m7.1.1.1.3.cmml">σ</mi><mo id="Thmcorollary4.p1.7.7.m7.1.1.1.2" xref="Thmcorollary4.p1.7.7.m7.1.1.1.2.cmml">∈</mo><msup id="Thmcorollary4.p1.7.7.m7.1.1.1.1" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.cmml"><mrow id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.cmml"><mo id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.2" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.cmml"><mi id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.3" mathvariant="normal" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.2" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.2" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.3" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmcorollary4.p1.7.7.m7.1.1.1.1.3" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.3.cmml">n</mi></msup></mrow></msub><mo id="Thmcorollary4.p1.7.7.m7.6.6.3.2a" lspace="0.167em" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.cmml">⁡</mo><msub id="Thmcorollary4.p1.7.7.m7.6.6.3.2.2" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.cmml"><mi id="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.2" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.2.cmml">F</mi><mi id="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.3" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.3.cmml">n</mi></msub></mrow><mo id="Thmcorollary4.p1.7.7.m7.6.6.3.1" xref="Thmcorollary4.p1.7.7.m7.6.6.3.1.cmml">⁢</mo><mrow id="Thmcorollary4.p1.7.7.m7.6.6.3.3.2" xref="Thmcorollary4.p1.7.7.m7.6.6.3.3.1.cmml"><mo id="Thmcorollary4.p1.7.7.m7.6.6.3.3.2.1" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.6.6.3.3.1.cmml">(</mo><mi id="Thmcorollary4.p1.7.7.m7.4.4" xref="Thmcorollary4.p1.7.7.m7.4.4.cmml">σ</mi><mo id="Thmcorollary4.p1.7.7.m7.6.6.3.3.2.2" xref="Thmcorollary4.p1.7.7.m7.6.6.3.3.1.cmml">,</mo><mi id="Thmcorollary4.p1.7.7.m7.5.5" xref="Thmcorollary4.p1.7.7.m7.5.5.cmml">α</mi><mo id="Thmcorollary4.p1.7.7.m7.6.6.3.3.2.3" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.6.6.3.3.1.cmml">)</mo></mrow></mrow><mo id="Thmcorollary4.p1.7.7.m7.6.6.2" xref="Thmcorollary4.p1.7.7.m7.6.6.2.cmml"><</mo><mrow id="Thmcorollary4.p1.7.7.m7.6.6.1" xref="Thmcorollary4.p1.7.7.m7.6.6.1.cmml"><mrow id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.2.cmml"><mo id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.2" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.2.1.cmml">|</mo><msub id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.cmml"><mi id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.2" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.2.cmml">p</mi><mn id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.3" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.3" stretchy="false" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.2.1.cmml">|</mo></mrow><mo id="Thmcorollary4.p1.7.7.m7.6.6.1.2" xref="Thmcorollary4.p1.7.7.m7.6.6.1.2.cmml">+</mo><mrow id="Thmcorollary4.p1.7.7.m7.6.6.1.3" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.cmml"><mfrac id="Thmcorollary4.p1.7.7.m7.3.3" xref="Thmcorollary4.p1.7.7.m7.3.3.cmml"><msub id="Thmcorollary4.p1.7.7.m7.3.3.2" xref="Thmcorollary4.p1.7.7.m7.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmcorollary4.p1.7.7.m7.3.3.2.4" xref="Thmcorollary4.p1.7.7.m7.3.3.2.4.cmml">ℰ</mi><mrow id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.3.cmml"><msub id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.cmml"><mi id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.2" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.2.cmml">m</mi><mn id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.3" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.3.cmml">1</mn></msub><mo id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.2" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.3.cmml">,</mo><mi id="Thmcorollary4.p1.7.7.m7.2.2.1.1.1.1" xref="Thmcorollary4.p1.7.7.m7.2.2.1.1.1.1.cmml">n</mi></mrow></msub><msub id="Thmcorollary4.p1.7.7.m7.3.3.4" xref="Thmcorollary4.p1.7.7.m7.3.3.4.cmml"><mi id="Thmcorollary4.p1.7.7.m7.3.3.4.2" xref="Thmcorollary4.p1.7.7.m7.3.3.4.2.cmml">m</mi><mn id="Thmcorollary4.p1.7.7.m7.3.3.4.3" xref="Thmcorollary4.p1.7.7.m7.3.3.4.3.cmml">1</mn></msub></mfrac><mo id="Thmcorollary4.p1.7.7.m7.6.6.1.3.1" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.1.cmml">⁢</mo><mi id="Thmcorollary4.p1.7.7.m7.6.6.1.3.2" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.2.cmml">k</mi><mo id="Thmcorollary4.p1.7.7.m7.6.6.1.3.1a" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.1.cmml">⁢</mo><mi id="Thmcorollary4.p1.7.7.m7.6.6.1.3.3" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.7.7.m7.6b"><apply id="Thmcorollary4.p1.7.7.m7.6.6.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6"><lt id="Thmcorollary4.p1.7.7.m7.6.6.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.2"></lt><apply id="Thmcorollary4.p1.7.7.m7.6.6.3.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3"><times id="Thmcorollary4.p1.7.7.m7.6.6.3.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.1"></times><apply id="Thmcorollary4.p1.7.7.m7.6.6.3.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2"><apply id="Thmcorollary4.p1.7.7.m7.6.6.3.2.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.6.6.3.2.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.1">subscript</csymbol><min id="Thmcorollary4.p1.7.7.m7.6.6.3.2.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.1.2"></min><apply id="Thmcorollary4.p1.7.7.m7.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1"><in id="Thmcorollary4.p1.7.7.m7.1.1.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.2"></in><ci id="Thmcorollary4.p1.7.7.m7.1.1.1.3.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.3">𝜎</ci><apply id="Thmcorollary4.p1.7.7.m7.1.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.1.1.1.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1">superscript</csymbol><apply id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1"><times id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.2"></times><ci id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.3.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.3">Δ</ci><apply id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3"><times id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="Thmcorollary4.p1.7.7.m7.1.1.1.1.3.cmml" xref="Thmcorollary4.p1.7.7.m7.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.2"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.2">subscript</csymbol><ci id="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.2">𝐹</ci><ci id="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.3.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmcorollary4.p1.7.7.m7.6.6.3.3.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.3.3.2"><ci id="Thmcorollary4.p1.7.7.m7.4.4.cmml" xref="Thmcorollary4.p1.7.7.m7.4.4">𝜎</ci><ci id="Thmcorollary4.p1.7.7.m7.5.5.cmml" xref="Thmcorollary4.p1.7.7.m7.5.5">𝛼</ci></interval></apply><apply id="Thmcorollary4.p1.7.7.m7.6.6.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1"><plus id="Thmcorollary4.p1.7.7.m7.6.6.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.2"></plus><apply id="Thmcorollary4.p1.7.7.m7.6.6.1.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1"><abs id="Thmcorollary4.p1.7.7.m7.6.6.1.1.2.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.2"></abs><apply id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1">subscript</csymbol><ci id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.2">𝑝</ci><cn id="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.3.cmml" type="integer" xref="Thmcorollary4.p1.7.7.m7.6.6.1.1.1.1.3">1</cn></apply></apply><apply id="Thmcorollary4.p1.7.7.m7.6.6.1.3.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3"><times id="Thmcorollary4.p1.7.7.m7.6.6.1.3.1.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.1"></times><apply id="Thmcorollary4.p1.7.7.m7.3.3.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3"><divide id="Thmcorollary4.p1.7.7.m7.3.3.3.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3"></divide><apply id="Thmcorollary4.p1.7.7.m7.3.3.2.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.2"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.3.3.2.3.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.2">subscript</csymbol><ci id="Thmcorollary4.p1.7.7.m7.3.3.2.4.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.2.4">ℰ</ci><list id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.3.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2"><apply id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1">subscript</csymbol><ci id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.2.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.2">𝑚</ci><cn id="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.3.cmml" type="integer" xref="Thmcorollary4.p1.7.7.m7.3.3.2.2.2.2.1.3">1</cn></apply><ci id="Thmcorollary4.p1.7.7.m7.2.2.1.1.1.1.cmml" xref="Thmcorollary4.p1.7.7.m7.2.2.1.1.1.1">𝑛</ci></list></apply><apply id="Thmcorollary4.p1.7.7.m7.3.3.4.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.4"><csymbol cd="ambiguous" id="Thmcorollary4.p1.7.7.m7.3.3.4.1.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.4">subscript</csymbol><ci id="Thmcorollary4.p1.7.7.m7.3.3.4.2.cmml" xref="Thmcorollary4.p1.7.7.m7.3.3.4.2">𝑚</ci><cn id="Thmcorollary4.p1.7.7.m7.3.3.4.3.cmml" type="integer" xref="Thmcorollary4.p1.7.7.m7.3.3.4.3">1</cn></apply></apply><ci id="Thmcorollary4.p1.7.7.m7.6.6.1.3.2.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.2">𝑘</ci><ci id="Thmcorollary4.p1.7.7.m7.6.6.1.3.3.cmml" xref="Thmcorollary4.p1.7.7.m7.6.6.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.7.7.m7.6c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{% \mathcal{E}_{m_{1},n}}{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.7.7.m7.6d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. Then <math alttext="F_{n}" class="ltx_Math" display="inline" id="Thmcorollary4.p1.8.8.m8.1"><semantics id="Thmcorollary4.p1.8.8.m8.1a"><msub id="Thmcorollary4.p1.8.8.m8.1.1" xref="Thmcorollary4.p1.8.8.m8.1.1.cmml"><mi id="Thmcorollary4.p1.8.8.m8.1.1.2" xref="Thmcorollary4.p1.8.8.m8.1.1.2.cmml">F</mi><mi id="Thmcorollary4.p1.8.8.m8.1.1.3" xref="Thmcorollary4.p1.8.8.m8.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.8.8.m8.1b"><apply id="Thmcorollary4.p1.8.8.m8.1.1.cmml" xref="Thmcorollary4.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.8.8.m8.1.1.1.cmml" xref="Thmcorollary4.p1.8.8.m8.1.1">subscript</csymbol><ci id="Thmcorollary4.p1.8.8.m8.1.1.2.cmml" xref="Thmcorollary4.p1.8.8.m8.1.1.2">𝐹</ci><ci id="Thmcorollary4.p1.8.8.m8.1.1.3.cmml" xref="Thmcorollary4.p1.8.8.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.8.8.m8.1c">F_{n}</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.8.8.m8.1d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is not concave on <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="Thmcorollary4.p1.9.9.m9.1"><semantics id="Thmcorollary4.p1.9.9.m9.1a"><msub id="Thmcorollary4.p1.9.9.m9.1.1" xref="Thmcorollary4.p1.9.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmcorollary4.p1.9.9.m9.1.1.2" xref="Thmcorollary4.p1.9.9.m9.1.1.2.cmml">𝒰</mi><mi id="Thmcorollary4.p1.9.9.m9.1.1.3" xref="Thmcorollary4.p1.9.9.m9.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.9.9.m9.1b"><apply id="Thmcorollary4.p1.9.9.m9.1.1.cmml" xref="Thmcorollary4.p1.9.9.m9.1.1"><csymbol cd="ambiguous" id="Thmcorollary4.p1.9.9.m9.1.1.1.cmml" xref="Thmcorollary4.p1.9.9.m9.1.1">subscript</csymbol><ci id="Thmcorollary4.p1.9.9.m9.1.1.2.cmml" xref="Thmcorollary4.p1.9.9.m9.1.1.2">𝒰</ci><ci id="Thmcorollary4.p1.9.9.m9.1.1.3.cmml" xref="Thmcorollary4.p1.9.9.m9.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.9.9.m9.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.9.9.m9.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS2.SSSx2.p4"> <span class="ltx_ERROR undefined" id="S5.SS2.SSSx2.p4.1">{credits}</span> </div> </section> <section class="ltx_subsubsection" id="S5.SS2.SSSx3"> <h4 class="ltx_title ltx_title_subsubsection">Acknowledgements.</h4> <div class="ltx_para" id="S5.SS2.SSSx3.p1"> <p class="ltx_p" id="S5.SS2.SSSx3.p1.1">We thank Farid Arthaud and Ziyu Zhang for helpful discussions and valuable feedback on earlier versions of the introduction. We also thank Yannai A. Gonczarowski for helpful discussions, and Kai Jia for valuable feedback on earlier versions of the introduction.</p> </div> <div class="ltx_para" id="S5.SS2.SSSx3.p2"> <span class="ltx_ERROR undefined" id="S5.SS2.SSSx3.p2.1">\printbibliography</span> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </section> </section> <section class="ltx_appendix" id="Pt0.A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix 0.A </span>Proofs of General Lemmas</h2> <div class="ltx_para" id="Pt0.A1.p1"> <p class="ltx_p" id="Pt0.A1.p1.2">Recall that <math alttext="\mathcal{E}_{\ell,n}=\ell\left(1-\left(1-\frac{1}{\ell}\right)^{n}\right)" class="ltx_Math" display="inline" id="Pt0.A1.p1.1.m1.3"><semantics id="Pt0.A1.p1.1.m1.3a"><mrow id="Pt0.A1.p1.1.m1.3.3" xref="Pt0.A1.p1.1.m1.3.3.cmml"><msub id="Pt0.A1.p1.1.m1.3.3.3" xref="Pt0.A1.p1.1.m1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.p1.1.m1.3.3.3.2" xref="Pt0.A1.p1.1.m1.3.3.3.2.cmml">ℰ</mi><mrow id="Pt0.A1.p1.1.m1.2.2.2.4" xref="Pt0.A1.p1.1.m1.2.2.2.3.cmml"><mi id="Pt0.A1.p1.1.m1.1.1.1.1" mathvariant="normal" xref="Pt0.A1.p1.1.m1.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.p1.1.m1.2.2.2.4.1" xref="Pt0.A1.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.p1.1.m1.2.2.2.2" xref="Pt0.A1.p1.1.m1.2.2.2.2.cmml">n</mi></mrow></msub><mo id="Pt0.A1.p1.1.m1.3.3.2" xref="Pt0.A1.p1.1.m1.3.3.2.cmml">=</mo><mrow id="Pt0.A1.p1.1.m1.3.3.1" xref="Pt0.A1.p1.1.m1.3.3.1.cmml"><mi id="Pt0.A1.p1.1.m1.3.3.1.3" mathvariant="normal" xref="Pt0.A1.p1.1.m1.3.3.1.3.cmml">ℓ</mi><mo id="Pt0.A1.p1.1.m1.3.3.1.2" xref="Pt0.A1.p1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="Pt0.A1.p1.1.m1.3.3.1.1.1" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.cmml"><mo id="Pt0.A1.p1.1.m1.3.3.1.1.1.2" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="Pt0.A1.p1.1.m1.3.3.1.1.1.1" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.cmml"><mn id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.3" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.3.cmml">1</mn><mo id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.2" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.2.cmml">−</mo><msup id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.cmml"><mrow id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.2" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mn id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.2" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.1" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.cmml"><mn id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.3" mathvariant="normal" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.3.cmml">ℓ</mi></mfrac></mrow><mo id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.3" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.3" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Pt0.A1.p1.1.m1.3.3.1.1.1.3" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.p1.1.m1.3b"><apply id="Pt0.A1.p1.1.m1.3.3.cmml" xref="Pt0.A1.p1.1.m1.3.3"><eq id="Pt0.A1.p1.1.m1.3.3.2.cmml" xref="Pt0.A1.p1.1.m1.3.3.2"></eq><apply id="Pt0.A1.p1.1.m1.3.3.3.cmml" xref="Pt0.A1.p1.1.m1.3.3.3"><csymbol cd="ambiguous" id="Pt0.A1.p1.1.m1.3.3.3.1.cmml" xref="Pt0.A1.p1.1.m1.3.3.3">subscript</csymbol><ci id="Pt0.A1.p1.1.m1.3.3.3.2.cmml" xref="Pt0.A1.p1.1.m1.3.3.3.2">ℰ</ci><list id="Pt0.A1.p1.1.m1.2.2.2.3.cmml" xref="Pt0.A1.p1.1.m1.2.2.2.4"><ci id="Pt0.A1.p1.1.m1.1.1.1.1.cmml" xref="Pt0.A1.p1.1.m1.1.1.1.1">ℓ</ci><ci id="Pt0.A1.p1.1.m1.2.2.2.2.cmml" xref="Pt0.A1.p1.1.m1.2.2.2.2">𝑛</ci></list></apply><apply id="Pt0.A1.p1.1.m1.3.3.1.cmml" xref="Pt0.A1.p1.1.m1.3.3.1"><times id="Pt0.A1.p1.1.m1.3.3.1.2.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.2"></times><ci id="Pt0.A1.p1.1.m1.3.3.1.3.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.3">ℓ</ci><apply id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1"><minus id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.2"></minus><cn id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.3">1</cn><apply id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.2.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1">superscript</csymbol><apply id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1"><minus id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.1"></minus><cn id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.2">1</cn><apply id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3"><divide id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3"></divide><cn id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.3.3">ℓ</ci></apply></apply><ci id="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.3.cmml" xref="Pt0.A1.p1.1.m1.3.3.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p1.1.m1.3c">\mathcal{E}_{\ell,n}=\ell\left(1-\left(1-\frac{1}{\ell}\right)^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p1.1.m1.3d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT = roman_ℓ ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG roman_ℓ end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>. We begin with basic properties of <math alttext="\mathcal{E}_{\ell,n}" class="ltx_Math" display="inline" id="Pt0.A1.p1.2.m2.2"><semantics id="Pt0.A1.p1.2.m2.2a"><msub id="Pt0.A1.p1.2.m2.2.3" xref="Pt0.A1.p1.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.p1.2.m2.2.3.2" xref="Pt0.A1.p1.2.m2.2.3.2.cmml">ℰ</mi><mrow id="Pt0.A1.p1.2.m2.2.2.2.4" xref="Pt0.A1.p1.2.m2.2.2.2.3.cmml"><mi id="Pt0.A1.p1.2.m2.1.1.1.1" mathvariant="normal" xref="Pt0.A1.p1.2.m2.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.p1.2.m2.2.2.2.4.1" xref="Pt0.A1.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.p1.2.m2.2.2.2.2" xref="Pt0.A1.p1.2.m2.2.2.2.2.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Pt0.A1.p1.2.m2.2b"><apply id="Pt0.A1.p1.2.m2.2.3.cmml" xref="Pt0.A1.p1.2.m2.2.3"><csymbol cd="ambiguous" id="Pt0.A1.p1.2.m2.2.3.1.cmml" xref="Pt0.A1.p1.2.m2.2.3">subscript</csymbol><ci id="Pt0.A1.p1.2.m2.2.3.2.cmml" xref="Pt0.A1.p1.2.m2.2.3.2">ℰ</ci><list id="Pt0.A1.p1.2.m2.2.2.2.3.cmml" xref="Pt0.A1.p1.2.m2.2.2.2.4"><ci id="Pt0.A1.p1.2.m2.1.1.1.1.cmml" xref="Pt0.A1.p1.2.m2.1.1.1.1">ℓ</ci><ci id="Pt0.A1.p1.2.m2.2.2.2.2.cmml" xref="Pt0.A1.p1.2.m2.2.2.2.2">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p1.2.m2.2c">\mathcal{E}_{\ell,n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p1.2.m2.2d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma18"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 18</span></h6> <div class="ltx_para" id="Thmlemma18.p1"> <ol class="ltx_enumerate" id="Pt0.A1.I1"> <li class="ltx_item" id="Pt0.A1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="Pt0.A1.I1.i1.p1"> <p class="ltx_p" id="Pt0.A1.I1.i1.p1.7"><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.1">When </span><math alttext="\ell,n\geq 1" class="ltx_Math" display="inline" id="Pt0.A1.I1.i1.p1.1.m1.2"><semantics id="Pt0.A1.I1.i1.p1.1.m1.2a"><mrow id="Pt0.A1.I1.i1.p1.1.m1.2.3" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.cmml"><mrow id="Pt0.A1.I1.i1.p1.1.m1.2.3.2.2" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.2.1.cmml"><mi id="Pt0.A1.I1.i1.p1.1.m1.1.1" mathvariant="normal" xref="Pt0.A1.I1.i1.p1.1.m1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.I1.i1.p1.1.m1.2.3.2.2.1" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="Pt0.A1.I1.i1.p1.1.m1.2.2" xref="Pt0.A1.I1.i1.p1.1.m1.2.2.cmml">n</mi></mrow><mo id="Pt0.A1.I1.i1.p1.1.m1.2.3.1" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.1.cmml">≥</mo><mn id="Pt0.A1.I1.i1.p1.1.m1.2.3.3" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i1.p1.1.m1.2b"><apply id="Pt0.A1.I1.i1.p1.1.m1.2.3.cmml" xref="Pt0.A1.I1.i1.p1.1.m1.2.3"><geq id="Pt0.A1.I1.i1.p1.1.m1.2.3.1.cmml" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.1"></geq><list id="Pt0.A1.I1.i1.p1.1.m1.2.3.2.1.cmml" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.2.2"><ci id="Pt0.A1.I1.i1.p1.1.m1.1.1.cmml" xref="Pt0.A1.I1.i1.p1.1.m1.1.1">ℓ</ci><ci id="Pt0.A1.I1.i1.p1.1.m1.2.2.cmml" xref="Pt0.A1.I1.i1.p1.1.m1.2.2">𝑛</ci></list><cn id="Pt0.A1.I1.i1.p1.1.m1.2.3.3.cmml" type="integer" xref="Pt0.A1.I1.i1.p1.1.m1.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i1.p1.1.m1.2c">\ell,n\geq 1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i1.p1.1.m1.2d">roman_ℓ , italic_n ≥ 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.2">, </span><math alttext="\mathcal{E}_{\ell,n}" class="ltx_Math" display="inline" id="Pt0.A1.I1.i1.p1.2.m2.2"><semantics id="Pt0.A1.I1.i1.p1.2.m2.2a"><msub id="Pt0.A1.I1.i1.p1.2.m2.2.3" xref="Pt0.A1.I1.i1.p1.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.I1.i1.p1.2.m2.2.3.2" xref="Pt0.A1.I1.i1.p1.2.m2.2.3.2.cmml">ℰ</mi><mrow id="Pt0.A1.I1.i1.p1.2.m2.2.2.2.4" xref="Pt0.A1.I1.i1.p1.2.m2.2.2.2.3.cmml"><mi id="Pt0.A1.I1.i1.p1.2.m2.1.1.1.1" mathvariant="normal" xref="Pt0.A1.I1.i1.p1.2.m2.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.I1.i1.p1.2.m2.2.2.2.4.1" xref="Pt0.A1.I1.i1.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.I1.i1.p1.2.m2.2.2.2.2" xref="Pt0.A1.I1.i1.p1.2.m2.2.2.2.2.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i1.p1.2.m2.2b"><apply id="Pt0.A1.I1.i1.p1.2.m2.2.3.cmml" xref="Pt0.A1.I1.i1.p1.2.m2.2.3"><csymbol cd="ambiguous" id="Pt0.A1.I1.i1.p1.2.m2.2.3.1.cmml" xref="Pt0.A1.I1.i1.p1.2.m2.2.3">subscript</csymbol><ci id="Pt0.A1.I1.i1.p1.2.m2.2.3.2.cmml" xref="Pt0.A1.I1.i1.p1.2.m2.2.3.2">ℰ</ci><list id="Pt0.A1.I1.i1.p1.2.m2.2.2.2.3.cmml" xref="Pt0.A1.I1.i1.p1.2.m2.2.2.2.4"><ci id="Pt0.A1.I1.i1.p1.2.m2.1.1.1.1.cmml" xref="Pt0.A1.I1.i1.p1.2.m2.1.1.1.1">ℓ</ci><ci id="Pt0.A1.I1.i1.p1.2.m2.2.2.2.2.cmml" xref="Pt0.A1.I1.i1.p1.2.m2.2.2.2.2">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i1.p1.2.m2.2c">\mathcal{E}_{\ell,n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i1.p1.2.m2.2d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.3"> is strictly increasing in </span><math alttext="\ell" class="ltx_Math" display="inline" id="Pt0.A1.I1.i1.p1.3.m3.1"><semantics id="Pt0.A1.I1.i1.p1.3.m3.1a"><mi id="Pt0.A1.I1.i1.p1.3.m3.1.1" mathvariant="normal" xref="Pt0.A1.I1.i1.p1.3.m3.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i1.p1.3.m3.1b"><ci id="Pt0.A1.I1.i1.p1.3.m3.1.1.cmml" xref="Pt0.A1.I1.i1.p1.3.m3.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i1.p1.3.m3.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i1.p1.3.m3.1d">roman_ℓ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.4"> and strictly increasing in </span><math alttext="n" class="ltx_Math" display="inline" id="Pt0.A1.I1.i1.p1.4.m4.1"><semantics id="Pt0.A1.I1.i1.p1.4.m4.1a"><mi id="Pt0.A1.I1.i1.p1.4.m4.1.1" xref="Pt0.A1.I1.i1.p1.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i1.p1.4.m4.1b"><ci id="Pt0.A1.I1.i1.p1.4.m4.1.1.cmml" xref="Pt0.A1.I1.i1.p1.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i1.p1.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i1.p1.4.m4.1d">italic_n</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.5">. On the other hand, </span><math alttext="\frac{\mathcal{E}_{\ell,n}}{\ell}" class="ltx_Math" display="inline" id="Pt0.A1.I1.i1.p1.5.m5.2"><semantics id="Pt0.A1.I1.i1.p1.5.m5.2a"><mfrac id="Pt0.A1.I1.i1.p1.5.m5.2.2" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.cmml"><msub id="Pt0.A1.I1.i1.p1.5.m5.2.2.2" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.4" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.4.cmml">ℰ</mi><mrow id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.4" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.3.cmml"><mi id="Pt0.A1.I1.i1.p1.5.m5.1.1.1.1.1.1" mathvariant="normal" xref="Pt0.A1.I1.i1.p1.5.m5.1.1.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.4.1" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.2" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Pt0.A1.I1.i1.p1.5.m5.2.2.4" mathvariant="normal" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.4.cmml">ℓ</mi></mfrac><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i1.p1.5.m5.2b"><apply id="Pt0.A1.I1.i1.p1.5.m5.2.2.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2"><divide id="Pt0.A1.I1.i1.p1.5.m5.2.2.3.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2"></divide><apply id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2"><csymbol cd="ambiguous" id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.3.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2">subscript</csymbol><ci id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.4.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.4">ℰ</ci><list id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.3.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.4"><ci id="Pt0.A1.I1.i1.p1.5.m5.1.1.1.1.1.1.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.1.1.1.1.1.1">ℓ</ci><ci id="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.2.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Pt0.A1.I1.i1.p1.5.m5.2.2.4.cmml" xref="Pt0.A1.I1.i1.p1.5.m5.2.2.4">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i1.p1.5.m5.2c">\frac{\mathcal{E}_{\ell,n}}{\ell}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i1.p1.5.m5.2d">divide start_ARG caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT end_ARG start_ARG roman_ℓ end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.6"> is strictly decreasing in </span><math alttext="\ell" class="ltx_Math" display="inline" id="Pt0.A1.I1.i1.p1.6.m6.1"><semantics id="Pt0.A1.I1.i1.p1.6.m6.1a"><mi id="Pt0.A1.I1.i1.p1.6.m6.1.1" mathvariant="normal" xref="Pt0.A1.I1.i1.p1.6.m6.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i1.p1.6.m6.1b"><ci id="Pt0.A1.I1.i1.p1.6.m6.1.1.cmml" xref="Pt0.A1.I1.i1.p1.6.m6.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i1.p1.6.m6.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i1.p1.6.m6.1d">roman_ℓ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.7"> and strictly increasing in </span><math alttext="n" class="ltx_Math" display="inline" id="Pt0.A1.I1.i1.p1.7.m7.1"><semantics id="Pt0.A1.I1.i1.p1.7.m7.1a"><mi id="Pt0.A1.I1.i1.p1.7.m7.1.1" xref="Pt0.A1.I1.i1.p1.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i1.p1.7.m7.1b"><ci id="Pt0.A1.I1.i1.p1.7.m7.1.1.cmml" xref="Pt0.A1.I1.i1.p1.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i1.p1.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i1.p1.7.m7.1d">italic_n</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i1.p1.7.8">.</span></p> </div> </li> <li class="ltx_item" id="Pt0.A1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="Pt0.A1.I1.i2.p1"> <p class="ltx_p" id="Pt0.A1.I1.i2.p1.2"><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i2.p1.2.1">Treat </span><math alttext="n=n(\ell)" class="ltx_Math" display="inline" id="Pt0.A1.I1.i2.p1.1.m1.1"><semantics id="Pt0.A1.I1.i2.p1.1.m1.1a"><mrow id="Pt0.A1.I1.i2.p1.1.m1.1.2" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.cmml"><mi id="Pt0.A1.I1.i2.p1.1.m1.1.2.2" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.2.cmml">n</mi><mo id="Pt0.A1.I1.i2.p1.1.m1.1.2.1" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.1.cmml">=</mo><mrow id="Pt0.A1.I1.i2.p1.1.m1.1.2.3" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.cmml"><mi id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.2" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.2.cmml">n</mi><mo id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.1" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.3.2" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.cmml"><mo id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.cmml">(</mo><mi id="Pt0.A1.I1.i2.p1.1.m1.1.1" mathvariant="normal" xref="Pt0.A1.I1.i2.p1.1.m1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i2.p1.1.m1.1b"><apply id="Pt0.A1.I1.i2.p1.1.m1.1.2.cmml" xref="Pt0.A1.I1.i2.p1.1.m1.1.2"><eq id="Pt0.A1.I1.i2.p1.1.m1.1.2.1.cmml" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.1"></eq><ci id="Pt0.A1.I1.i2.p1.1.m1.1.2.2.cmml" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.2">𝑛</ci><apply id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.cmml" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3"><times id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.1.cmml" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.1"></times><ci id="Pt0.A1.I1.i2.p1.1.m1.1.2.3.2.cmml" xref="Pt0.A1.I1.i2.p1.1.m1.1.2.3.2">𝑛</ci><ci id="Pt0.A1.I1.i2.p1.1.m1.1.1.cmml" xref="Pt0.A1.I1.i2.p1.1.m1.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i2.p1.1.m1.1c">n=n(\ell)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i2.p1.1.m1.1d">italic_n = italic_n ( roman_ℓ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i2.p1.2.2"> as a function of </span><math alttext="\ell" class="ltx_Math" display="inline" id="Pt0.A1.I1.i2.p1.2.m2.1"><semantics id="Pt0.A1.I1.i2.p1.2.m2.1a"><mi id="Pt0.A1.I1.i2.p1.2.m2.1.1" mathvariant="normal" xref="Pt0.A1.I1.i2.p1.2.m2.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i2.p1.2.m2.1b"><ci id="Pt0.A1.I1.i2.p1.2.m2.1.1.cmml" xref="Pt0.A1.I1.i2.p1.2.m2.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i2.p1.2.m2.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i2.p1.2.m2.1d">roman_ℓ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i2.p1.2.3">. Then</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx15"> <tbody id="Pt0.A1.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 1=\begin{cases}\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{n% }&n=o(\ell)\\ \lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{(1-e^{-x})\ell}&n=x\ell+o(\ell% ),~{}x\in(0,\infty)\\ \lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{\ell}&n=\omega(\ell)\end{cases}" class="ltx_Math" display="inline" id="Pt0.A1.Ex14.m1.6"><semantics id="Pt0.A1.Ex14.m1.6a"><mrow id="Pt0.A1.Ex14.m1.6.7" xref="Pt0.A1.Ex14.m1.6.7.cmml"><mn id="Pt0.A1.Ex14.m1.6.7.2" xref="Pt0.A1.Ex14.m1.6.7.2.cmml">1</mn><mo id="Pt0.A1.Ex14.m1.6.7.1" xref="Pt0.A1.Ex14.m1.6.7.1.cmml">=</mo><mrow id="Pt0.A1.Ex14.m1.6.6a" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mo id="Pt0.A1.Ex14.m1.6.6a.7" xref="Pt0.A1.Ex14.m1.6.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" id="Pt0.A1.Ex14.m1.6.6.6a" rowspacing="0pt" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mtr id="Pt0.A1.Ex14.m1.6.6.6aa" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="Pt0.A1.Ex14.m1.6.6.6ab" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mrow id="Pt0.A1.Ex14.m1.1.1.1.1.1.1" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.cmml"><msub id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.cmml"><mo id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.2" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.2.cmml">lim</mo><mrow id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3.cmml"><mi id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3.2" mathvariant="normal" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3.2.cmml">ℓ</mi><mo id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3.1" stretchy="false" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3.1.cmml">→</mo><mi id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3.3" mathvariant="normal" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.3.3.3.cmml">∞</mi></mrow></msub><mfrac id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.cmml"><msub id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.4" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.4.cmml">ℰ</mi><mrow id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.2.2.4" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.2.2.3.cmml"><mi id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.1.1.1.1.1" mathvariant="normal" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.2.2.4.1" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.2.2.2" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.4" xref="Pt0.A1.Ex14.m1.1.1.1.1.1.1.2.4.cmml">n</mi></mfrac></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="Pt0.A1.Ex14.m1.6.6.6ac" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mrow id="Pt0.A1.Ex14.m1.2.2.2.2.2.1" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.cmml"><mi id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.3" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.3.cmml">n</mi><mo id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.2" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.2.cmml">=</mo><mrow id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.cmml"><mi id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.2" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.2.cmml">o</mi><mo id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.1" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.1.cmml">⁢</mo><mrow id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.3.2" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.cmml"><mo id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.3.2.1" stretchy="false" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.cmml">(</mo><mi id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.1" mathvariant="normal" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.1.cmml">ℓ</mi><mo id="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.3.2.2" stretchy="false" xref="Pt0.A1.Ex14.m1.2.2.2.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="Pt0.A1.Ex14.m1.6.6.6ad" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="Pt0.A1.Ex14.m1.6.6.6ae" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mrow id="Pt0.A1.Ex14.m1.3.3.3.3.1.1" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.cmml"><msub id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.cmml"><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.2" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.2.cmml">lim</mo><mrow id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3.cmml"><mi id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3.2" mathvariant="normal" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3.2.cmml">ℓ</mi><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3.1" stretchy="false" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3.1.cmml">→</mo><mi id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3.3" mathvariant="normal" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.4.3.3.cmml">∞</mi></mrow></msub><mfrac id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.cmml"><msub id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.4" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.4.cmml">ℰ</mi><mrow id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.2.2.4" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.2.2.3.cmml"><mi id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.1.1.1.1.1" mathvariant="normal" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.1.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.2.2.4.1" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.2.2.2" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.2.2.2.2.2.cmml">n</mi></mrow></msub><mrow id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.cmml"><mrow id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.cmml"><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.2" stretchy="false" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.cmml">(</mo><mrow id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.cmml"><mn id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.2" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.2.cmml">1</mn><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.1" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.1.cmml">−</mo><msup id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.cmml"><mi id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.2" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.2.cmml">e</mi><mrow id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.3" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.3.cmml"><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.3a" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.3.cmml">−</mo><mi id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.3.2" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.3.3.2.cmml">x</mi></mrow></msup></mrow><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.3" stretchy="false" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.2" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.2.cmml">⁢</mo><mi id="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.3" mathvariant="normal" xref="Pt0.A1.Ex14.m1.3.3.3.3.1.1.3.3.3.cmml">ℓ</mi></mrow></mfrac></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="Pt0.A1.Ex14.m1.6.6.6af" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.6.cmml"><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.cmml"><mi id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.2.cmml">n</mi><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.1" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.1.cmml">=</mo><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.cmml"><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2.cmml"><mi id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2.2.cmml">x</mi><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2.1" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2.1.cmml">⁢</mo><mi id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2.3" mathvariant="normal" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.2.3.cmml">ℓ</mi></mrow><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.1" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.1.cmml">+</mo><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.cmml"><mi id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.2.cmml">o</mi><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.1" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.1.cmml">⁢</mo><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.3.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.cmml"><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.3.2.1" stretchy="false" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.cmml">(</mo><mi id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.1" mathvariant="normal" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.1.cmml">ℓ</mi><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.3.2.2" stretchy="false" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.4.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.3" rspace="0.497em" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.6a.cmml">,</mo><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.cmml"><mi id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.2.cmml">x</mi><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.1" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.1.cmml">∈</mo><mrow id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.1.cmml"><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.2.1" stretchy="false" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.1.cmml">(</mo><mn id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.2.cmml">0</mn><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.2.2" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.1.cmml">,</mo><mi id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.3" mathvariant="normal" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.3.cmml">∞</mi><mo id="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.2.3" stretchy="false" xref="Pt0.A1.Ex14.m1.4.4.4.4.2.1.5.2.3.1.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="Pt0.A1.Ex14.m1.6.6.6ag" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="Pt0.A1.Ex14.m1.6.6.6ah" xref="Pt0.A1.Ex14.m1.6.7.3.1.cmml"><mrow id="Pt0.A1.Ex14.m1.5.5.5.5.1.1" xref="Pt0.A1.Ex14.m1.5.5.5.5.1.1.cmml"><msub id="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3" xref="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.cmml"><mo id="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.2" xref="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.2.cmml">lim</mo><mrow id="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.3" xref="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.3.cmml"><mi id="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.3.2" mathvariant="normal" xref="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.3.2.cmml">ℓ</mi><mo id="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.3.1" stretchy="false" xref="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.3.1.cmml">→</mo><mi id="Pt0.A1.Ex14.m1.5.5.5.5.1.1.3.3.3" mathvariant="normal" 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xref="Pt0.A1.Ex14.m1.6.6.6.6.2.1.3">𝑛</ci><apply id="Pt0.A1.Ex14.m1.6.6.6.6.2.1.4.cmml" xref="Pt0.A1.Ex14.m1.6.6.6.6.2.1.4"><times id="Pt0.A1.Ex14.m1.6.6.6.6.2.1.4.1.cmml" xref="Pt0.A1.Ex14.m1.6.6.6.6.2.1.4.1"></times><ci id="Pt0.A1.Ex14.m1.6.6.6.6.2.1.4.2.cmml" xref="Pt0.A1.Ex14.m1.6.6.6.6.2.1.4.2">𝜔</ci><ci id="Pt0.A1.Ex14.m1.6.6.6.6.2.1.1.cmml" xref="Pt0.A1.Ex14.m1.6.6.6.6.2.1.1">ℓ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex14.m1.6c">\displaystyle 1=\begin{cases}\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{n% }&n=o(\ell)\\ \lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{(1-e^{-x})\ell}&n=x\ell+o(\ell% ),~{}x\in(0,\infty)\\ \lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{\ell}&n=\omega(\ell)\end{cases}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex14.m1.6d">1 = { start_ROW start_CELL roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_n end_ARG end_CELL start_CELL italic_n = italic_o ( roman_ℓ ) end_CELL end_ROW start_ROW start_CELL roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT end_ARG start_ARG ( 1 - italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT ) roman_ℓ end_ARG end_CELL start_CELL italic_n = italic_x roman_ℓ + italic_o ( roman_ℓ ) , italic_x ∈ ( 0 , ∞ ) end_CELL end_ROW start_ROW start_CELL roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT end_ARG start_ARG roman_ℓ end_ARG end_CELL start_CELL italic_n = italic_ω ( roman_ℓ ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A1.I1.i2.p1.4"><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i2.p1.4.1">i.e., </span><math alttext="\mathcal{E}_{\ell,n}" class="ltx_Math" display="inline" id="Pt0.A1.I1.i2.p1.3.m1.2"><semantics id="Pt0.A1.I1.i2.p1.3.m1.2a"><msub id="Pt0.A1.I1.i2.p1.3.m1.2.3" xref="Pt0.A1.I1.i2.p1.3.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.I1.i2.p1.3.m1.2.3.2" xref="Pt0.A1.I1.i2.p1.3.m1.2.3.2.cmml">ℰ</mi><mrow id="Pt0.A1.I1.i2.p1.3.m1.2.2.2.4" xref="Pt0.A1.I1.i2.p1.3.m1.2.2.2.3.cmml"><mi id="Pt0.A1.I1.i2.p1.3.m1.1.1.1.1" mathvariant="normal" xref="Pt0.A1.I1.i2.p1.3.m1.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.I1.i2.p1.3.m1.2.2.2.4.1" xref="Pt0.A1.I1.i2.p1.3.m1.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.I1.i2.p1.3.m1.2.2.2.2" xref="Pt0.A1.I1.i2.p1.3.m1.2.2.2.2.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i2.p1.3.m1.2b"><apply id="Pt0.A1.I1.i2.p1.3.m1.2.3.cmml" xref="Pt0.A1.I1.i2.p1.3.m1.2.3"><csymbol cd="ambiguous" id="Pt0.A1.I1.i2.p1.3.m1.2.3.1.cmml" xref="Pt0.A1.I1.i2.p1.3.m1.2.3">subscript</csymbol><ci id="Pt0.A1.I1.i2.p1.3.m1.2.3.2.cmml" xref="Pt0.A1.I1.i2.p1.3.m1.2.3.2">ℰ</ci><list id="Pt0.A1.I1.i2.p1.3.m1.2.2.2.3.cmml" xref="Pt0.A1.I1.i2.p1.3.m1.2.2.2.4"><ci id="Pt0.A1.I1.i2.p1.3.m1.1.1.1.1.cmml" xref="Pt0.A1.I1.i2.p1.3.m1.1.1.1.1">ℓ</ci><ci id="Pt0.A1.I1.i2.p1.3.m1.2.2.2.2.cmml" xref="Pt0.A1.I1.i2.p1.3.m1.2.2.2.2">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i2.p1.3.m1.2c">\mathcal{E}_{\ell,n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i2.p1.3.m1.2d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i2.p1.4.2"> converges to the functions subtracted from it in the limits above, depending on whether </span><math alttext="n=o(\ell)/\Theta(\ell)/\omega(\ell)" class="ltx_Math" display="inline" id="Pt0.A1.I1.i2.p1.4.m2.3"><semantics id="Pt0.A1.I1.i2.p1.4.m2.3a"><mrow id="Pt0.A1.I1.i2.p1.4.m2.3.4" 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xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.cmml">(</mo><mi id="Pt0.A1.I1.i2.p1.4.m2.1.1" mathvariant="normal" xref="Pt0.A1.I1.i2.p1.4.m2.1.1.cmml">ℓ</mi><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.3.2.2" stretchy="false" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.1" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.1.cmml">/</mo><mi id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.3" mathvariant="normal" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.3.cmml">Θ</mi></mrow><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.1" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.1.cmml">⁢</mo><mrow id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.3.2" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.cmml"><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.3.2.1" stretchy="false" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.cmml">(</mo><mi id="Pt0.A1.I1.i2.p1.4.m2.2.2" mathvariant="normal" xref="Pt0.A1.I1.i2.p1.4.m2.2.2.cmml">ℓ</mi><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.3.2.2" stretchy="false" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.1" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.1.cmml">/</mo><mi id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.3" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.3.cmml">ω</mi></mrow><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.1" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.1.cmml">⁢</mo><mrow id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.3.2" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.cmml"><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.3.2.1" stretchy="false" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.cmml">(</mo><mi id="Pt0.A1.I1.i2.p1.4.m2.3.3" mathvariant="normal" xref="Pt0.A1.I1.i2.p1.4.m2.3.3.cmml">ℓ</mi><mo id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.3.2.2" stretchy="false" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.I1.i2.p1.4.m2.3b"><apply id="Pt0.A1.I1.i2.p1.4.m2.3.4.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4"><eq id="Pt0.A1.I1.i2.p1.4.m2.3.4.1.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.1"></eq><ci id="Pt0.A1.I1.i2.p1.4.m2.3.4.2.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.2">𝑛</ci><apply id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3"><times id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.1.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.1"></times><apply id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2"><divide id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.1.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.1"></divide><apply id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2"><times id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.1.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.1"></times><apply id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2"><divide id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.1.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.1"></divide><apply id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2"><times id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.1.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.1"></times><ci id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.2.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.2.2">𝑜</ci><ci id="Pt0.A1.I1.i2.p1.4.m2.1.1.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.1.1">ℓ</ci></apply><ci id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.3.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.2.2.3">Θ</ci></apply><ci id="Pt0.A1.I1.i2.p1.4.m2.2.2.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.2.2">ℓ</ci></apply><ci id="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.3.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.4.3.2.3">𝜔</ci></apply><ci id="Pt0.A1.I1.i2.p1.4.m2.3.3.cmml" xref="Pt0.A1.I1.i2.p1.4.m2.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.I1.i2.p1.4.m2.3c">n=o(\ell)/\Theta(\ell)/\omega(\ell)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.I1.i2.p1.4.m2.3d">italic_n = italic_o ( roman_ℓ ) / roman_Θ ( roman_ℓ ) / italic_ω ( roman_ℓ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Pt0.A1.I1.i2.p1.4.3">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx1"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx1.p1"> <p class="ltx_p" id="Thmproofx1.p1.8">Item 1 is easy to verify, so we prove only Item 2. Suppose <math alttext="n=\omega(\ell)" class="ltx_Math" display="inline" id="Thmproofx1.p1.1.m1.1"><semantics id="Thmproofx1.p1.1.m1.1a"><mrow id="Thmproofx1.p1.1.m1.1.2" xref="Thmproofx1.p1.1.m1.1.2.cmml"><mi id="Thmproofx1.p1.1.m1.1.2.2" xref="Thmproofx1.p1.1.m1.1.2.2.cmml">n</mi><mo id="Thmproofx1.p1.1.m1.1.2.1" xref="Thmproofx1.p1.1.m1.1.2.1.cmml">=</mo><mrow id="Thmproofx1.p1.1.m1.1.2.3" xref="Thmproofx1.p1.1.m1.1.2.3.cmml"><mi id="Thmproofx1.p1.1.m1.1.2.3.2" xref="Thmproofx1.p1.1.m1.1.2.3.2.cmml">ω</mi><mo id="Thmproofx1.p1.1.m1.1.2.3.1" xref="Thmproofx1.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Thmproofx1.p1.1.m1.1.2.3.3.2" xref="Thmproofx1.p1.1.m1.1.2.3.cmml"><mo id="Thmproofx1.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="Thmproofx1.p1.1.m1.1.2.3.cmml">(</mo><mi id="Thmproofx1.p1.1.m1.1.1" mathvariant="normal" xref="Thmproofx1.p1.1.m1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="Thmproofx1.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.1.m1.1b"><apply id="Thmproofx1.p1.1.m1.1.2.cmml" xref="Thmproofx1.p1.1.m1.1.2"><eq id="Thmproofx1.p1.1.m1.1.2.1.cmml" xref="Thmproofx1.p1.1.m1.1.2.1"></eq><ci id="Thmproofx1.p1.1.m1.1.2.2.cmml" xref="Thmproofx1.p1.1.m1.1.2.2">𝑛</ci><apply id="Thmproofx1.p1.1.m1.1.2.3.cmml" xref="Thmproofx1.p1.1.m1.1.2.3"><times id="Thmproofx1.p1.1.m1.1.2.3.1.cmml" xref="Thmproofx1.p1.1.m1.1.2.3.1"></times><ci id="Thmproofx1.p1.1.m1.1.2.3.2.cmml" xref="Thmproofx1.p1.1.m1.1.2.3.2">𝜔</ci><ci id="Thmproofx1.p1.1.m1.1.1.cmml" xref="Thmproofx1.p1.1.m1.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.1.m1.1c">n=\omega(\ell)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.1.m1.1d">italic_n = italic_ω ( roman_ℓ )</annotation></semantics></math>. Then by elementary calculus, <math alttext="\lim_{\ell\to\infty}\left(1-\frac{1}{\ell}\right)^{n}=0" class="ltx_Math" display="inline" id="Thmproofx1.p1.2.m2.1"><semantics id="Thmproofx1.p1.2.m2.1a"><mrow id="Thmproofx1.p1.2.m2.1.1" xref="Thmproofx1.p1.2.m2.1.1.cmml"><mrow id="Thmproofx1.p1.2.m2.1.1.1" xref="Thmproofx1.p1.2.m2.1.1.1.cmml"><msub id="Thmproofx1.p1.2.m2.1.1.1.2" xref="Thmproofx1.p1.2.m2.1.1.1.2.cmml"><mo id="Thmproofx1.p1.2.m2.1.1.1.2.2" xref="Thmproofx1.p1.2.m2.1.1.1.2.2.cmml">lim</mo><mrow id="Thmproofx1.p1.2.m2.1.1.1.2.3" xref="Thmproofx1.p1.2.m2.1.1.1.2.3.cmml"><mi id="Thmproofx1.p1.2.m2.1.1.1.2.3.2" mathvariant="normal" xref="Thmproofx1.p1.2.m2.1.1.1.2.3.2.cmml">ℓ</mi><mo id="Thmproofx1.p1.2.m2.1.1.1.2.3.1" stretchy="false" xref="Thmproofx1.p1.2.m2.1.1.1.2.3.1.cmml">→</mo><mi id="Thmproofx1.p1.2.m2.1.1.1.2.3.3" mathvariant="normal" xref="Thmproofx1.p1.2.m2.1.1.1.2.3.3.cmml">∞</mi></mrow></msub><msup id="Thmproofx1.p1.2.m2.1.1.1.1" xref="Thmproofx1.p1.2.m2.1.1.1.1.cmml"><mrow id="Thmproofx1.p1.2.m2.1.1.1.1.1.1" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.2" lspace="0em" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.cmml"><mn id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.2" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.1" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.cmml"><mn id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.2" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.3.cmml">ℓ</mi></mfrac></mrow><mo id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.3" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx1.p1.2.m2.1.1.1.1.3" xref="Thmproofx1.p1.2.m2.1.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Thmproofx1.p1.2.m2.1.1.2" xref="Thmproofx1.p1.2.m2.1.1.2.cmml">=</mo><mn id="Thmproofx1.p1.2.m2.1.1.3" xref="Thmproofx1.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.2.m2.1b"><apply id="Thmproofx1.p1.2.m2.1.1.cmml" xref="Thmproofx1.p1.2.m2.1.1"><eq id="Thmproofx1.p1.2.m2.1.1.2.cmml" xref="Thmproofx1.p1.2.m2.1.1.2"></eq><apply id="Thmproofx1.p1.2.m2.1.1.1.cmml" xref="Thmproofx1.p1.2.m2.1.1.1"><apply id="Thmproofx1.p1.2.m2.1.1.1.2.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx1.p1.2.m2.1.1.1.2.1.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.2">subscript</csymbol><limit id="Thmproofx1.p1.2.m2.1.1.1.2.2.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.2.2"></limit><apply id="Thmproofx1.p1.2.m2.1.1.1.2.3.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.2.3"><ci id="Thmproofx1.p1.2.m2.1.1.1.2.3.1.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.2.3.1">→</ci><ci id="Thmproofx1.p1.2.m2.1.1.1.2.3.2.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.2.3.2">ℓ</ci><infinity id="Thmproofx1.p1.2.m2.1.1.1.2.3.3.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.2.3.3"></infinity></apply></apply><apply id="Thmproofx1.p1.2.m2.1.1.1.1.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx1.p1.2.m2.1.1.1.1.2.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1">superscript</csymbol><apply id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1"><minus id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.1"></minus><cn id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3"><divide id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3"></divide><cn id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.2">1</cn><ci id="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1.1.1.1.3.3">ℓ</ci></apply></apply><ci id="Thmproofx1.p1.2.m2.1.1.1.1.3.cmml" xref="Thmproofx1.p1.2.m2.1.1.1.1.3">𝑛</ci></apply></apply><cn id="Thmproofx1.p1.2.m2.1.1.3.cmml" type="integer" xref="Thmproofx1.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.2.m2.1c">\lim_{\ell\to\infty}\left(1-\frac{1}{\ell}\right)^{n}=0</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.2.m2.1d">roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT ( 1 - divide start_ARG 1 end_ARG start_ARG roman_ℓ end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = 0</annotation></semantics></math>. Therefore, <math alttext="\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{\ell}=1" class="ltx_Math" display="inline" id="Thmproofx1.p1.3.m3.2"><semantics id="Thmproofx1.p1.3.m3.2a"><mrow id="Thmproofx1.p1.3.m3.2.3" xref="Thmproofx1.p1.3.m3.2.3.cmml"><mrow id="Thmproofx1.p1.3.m3.2.3.2" xref="Thmproofx1.p1.3.m3.2.3.2.cmml"><msub id="Thmproofx1.p1.3.m3.2.3.2.1" xref="Thmproofx1.p1.3.m3.2.3.2.1.cmml"><mo id="Thmproofx1.p1.3.m3.2.3.2.1.2" xref="Thmproofx1.p1.3.m3.2.3.2.1.2.cmml">lim</mo><mrow id="Thmproofx1.p1.3.m3.2.3.2.1.3" xref="Thmproofx1.p1.3.m3.2.3.2.1.3.cmml"><mi id="Thmproofx1.p1.3.m3.2.3.2.1.3.2" mathvariant="normal" xref="Thmproofx1.p1.3.m3.2.3.2.1.3.2.cmml">ℓ</mi><mo id="Thmproofx1.p1.3.m3.2.3.2.1.3.1" stretchy="false" xref="Thmproofx1.p1.3.m3.2.3.2.1.3.1.cmml">→</mo><mi id="Thmproofx1.p1.3.m3.2.3.2.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.3.m3.2.3.2.1.3.3.cmml">∞</mi></mrow></msub><mfrac id="Thmproofx1.p1.3.m3.2.2" xref="Thmproofx1.p1.3.m3.2.2.cmml"><msub id="Thmproofx1.p1.3.m3.2.2.2" xref="Thmproofx1.p1.3.m3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx1.p1.3.m3.2.2.2.4" xref="Thmproofx1.p1.3.m3.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx1.p1.3.m3.2.2.2.2.2.4" xref="Thmproofx1.p1.3.m3.2.2.2.2.2.3.cmml"><mi id="Thmproofx1.p1.3.m3.1.1.1.1.1.1" mathvariant="normal" xref="Thmproofx1.p1.3.m3.1.1.1.1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.3.m3.2.2.2.2.2.4.1" xref="Thmproofx1.p1.3.m3.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx1.p1.3.m3.2.2.2.2.2.2" xref="Thmproofx1.p1.3.m3.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Thmproofx1.p1.3.m3.2.2.4" mathvariant="normal" xref="Thmproofx1.p1.3.m3.2.2.4.cmml">ℓ</mi></mfrac></mrow><mo id="Thmproofx1.p1.3.m3.2.3.1" xref="Thmproofx1.p1.3.m3.2.3.1.cmml">=</mo><mn id="Thmproofx1.p1.3.m3.2.3.3" xref="Thmproofx1.p1.3.m3.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.3.m3.2b"><apply id="Thmproofx1.p1.3.m3.2.3.cmml" xref="Thmproofx1.p1.3.m3.2.3"><eq id="Thmproofx1.p1.3.m3.2.3.1.cmml" xref="Thmproofx1.p1.3.m3.2.3.1"></eq><apply id="Thmproofx1.p1.3.m3.2.3.2.cmml" xref="Thmproofx1.p1.3.m3.2.3.2"><apply id="Thmproofx1.p1.3.m3.2.3.2.1.cmml" xref="Thmproofx1.p1.3.m3.2.3.2.1"><csymbol cd="ambiguous" id="Thmproofx1.p1.3.m3.2.3.2.1.1.cmml" xref="Thmproofx1.p1.3.m3.2.3.2.1">subscript</csymbol><limit id="Thmproofx1.p1.3.m3.2.3.2.1.2.cmml" xref="Thmproofx1.p1.3.m3.2.3.2.1.2"></limit><apply id="Thmproofx1.p1.3.m3.2.3.2.1.3.cmml" xref="Thmproofx1.p1.3.m3.2.3.2.1.3"><ci id="Thmproofx1.p1.3.m3.2.3.2.1.3.1.cmml" xref="Thmproofx1.p1.3.m3.2.3.2.1.3.1">→</ci><ci id="Thmproofx1.p1.3.m3.2.3.2.1.3.2.cmml" xref="Thmproofx1.p1.3.m3.2.3.2.1.3.2">ℓ</ci><infinity id="Thmproofx1.p1.3.m3.2.3.2.1.3.3.cmml" xref="Thmproofx1.p1.3.m3.2.3.2.1.3.3"></infinity></apply></apply><apply id="Thmproofx1.p1.3.m3.2.2.cmml" xref="Thmproofx1.p1.3.m3.2.2"><divide id="Thmproofx1.p1.3.m3.2.2.3.cmml" xref="Thmproofx1.p1.3.m3.2.2"></divide><apply id="Thmproofx1.p1.3.m3.2.2.2.cmml" xref="Thmproofx1.p1.3.m3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx1.p1.3.m3.2.2.2.3.cmml" xref="Thmproofx1.p1.3.m3.2.2.2">subscript</csymbol><ci id="Thmproofx1.p1.3.m3.2.2.2.4.cmml" xref="Thmproofx1.p1.3.m3.2.2.2.4">ℰ</ci><list id="Thmproofx1.p1.3.m3.2.2.2.2.2.3.cmml" xref="Thmproofx1.p1.3.m3.2.2.2.2.2.4"><ci id="Thmproofx1.p1.3.m3.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.3.m3.1.1.1.1.1.1">ℓ</ci><ci id="Thmproofx1.p1.3.m3.2.2.2.2.2.2.cmml" xref="Thmproofx1.p1.3.m3.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Thmproofx1.p1.3.m3.2.2.4.cmml" xref="Thmproofx1.p1.3.m3.2.2.4">ℓ</ci></apply></apply><cn id="Thmproofx1.p1.3.m3.2.3.3.cmml" type="integer" xref="Thmproofx1.p1.3.m3.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.3.m3.2c">\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{\ell}=1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.3.m3.2d">roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT end_ARG start_ARG roman_ℓ end_ARG = 1</annotation></semantics></math>. Now suppose that <math alttext="n=o(\ell)" class="ltx_Math" display="inline" id="Thmproofx1.p1.4.m4.1"><semantics id="Thmproofx1.p1.4.m4.1a"><mrow id="Thmproofx1.p1.4.m4.1.2" xref="Thmproofx1.p1.4.m4.1.2.cmml"><mi id="Thmproofx1.p1.4.m4.1.2.2" xref="Thmproofx1.p1.4.m4.1.2.2.cmml">n</mi><mo id="Thmproofx1.p1.4.m4.1.2.1" xref="Thmproofx1.p1.4.m4.1.2.1.cmml">=</mo><mrow id="Thmproofx1.p1.4.m4.1.2.3" xref="Thmproofx1.p1.4.m4.1.2.3.cmml"><mi id="Thmproofx1.p1.4.m4.1.2.3.2" xref="Thmproofx1.p1.4.m4.1.2.3.2.cmml">o</mi><mo id="Thmproofx1.p1.4.m4.1.2.3.1" xref="Thmproofx1.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="Thmproofx1.p1.4.m4.1.2.3.3.2" xref="Thmproofx1.p1.4.m4.1.2.3.cmml"><mo id="Thmproofx1.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="Thmproofx1.p1.4.m4.1.2.3.cmml">(</mo><mi id="Thmproofx1.p1.4.m4.1.1" mathvariant="normal" xref="Thmproofx1.p1.4.m4.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="Thmproofx1.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.4.m4.1b"><apply id="Thmproofx1.p1.4.m4.1.2.cmml" xref="Thmproofx1.p1.4.m4.1.2"><eq id="Thmproofx1.p1.4.m4.1.2.1.cmml" xref="Thmproofx1.p1.4.m4.1.2.1"></eq><ci id="Thmproofx1.p1.4.m4.1.2.2.cmml" xref="Thmproofx1.p1.4.m4.1.2.2">𝑛</ci><apply id="Thmproofx1.p1.4.m4.1.2.3.cmml" xref="Thmproofx1.p1.4.m4.1.2.3"><times id="Thmproofx1.p1.4.m4.1.2.3.1.cmml" xref="Thmproofx1.p1.4.m4.1.2.3.1"></times><ci id="Thmproofx1.p1.4.m4.1.2.3.2.cmml" xref="Thmproofx1.p1.4.m4.1.2.3.2">𝑜</ci><ci id="Thmproofx1.p1.4.m4.1.1.cmml" xref="Thmproofx1.p1.4.m4.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.4.m4.1c">n=o(\ell)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.4.m4.1d">italic_n = italic_o ( roman_ℓ )</annotation></semantics></math>. On the one hand, by Bernoulli’s inequality <math alttext="\mathcal{E}_{\ell,n}\leq\ell\left(1-\left(1-\frac{n}{\ell}\right)\right)=n" class="ltx_Math" display="inline" id="Thmproofx1.p1.5.m5.3"><semantics id="Thmproofx1.p1.5.m5.3a"><mrow id="Thmproofx1.p1.5.m5.3.3" xref="Thmproofx1.p1.5.m5.3.3.cmml"><msub id="Thmproofx1.p1.5.m5.3.3.3" xref="Thmproofx1.p1.5.m5.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx1.p1.5.m5.3.3.3.2" xref="Thmproofx1.p1.5.m5.3.3.3.2.cmml">ℰ</mi><mrow id="Thmproofx1.p1.5.m5.2.2.2.4" xref="Thmproofx1.p1.5.m5.2.2.2.3.cmml"><mi id="Thmproofx1.p1.5.m5.1.1.1.1" mathvariant="normal" xref="Thmproofx1.p1.5.m5.1.1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.5.m5.2.2.2.4.1" xref="Thmproofx1.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="Thmproofx1.p1.5.m5.2.2.2.2" xref="Thmproofx1.p1.5.m5.2.2.2.2.cmml">n</mi></mrow></msub><mo id="Thmproofx1.p1.5.m5.3.3.4" xref="Thmproofx1.p1.5.m5.3.3.4.cmml">≤</mo><mrow id="Thmproofx1.p1.5.m5.3.3.1" xref="Thmproofx1.p1.5.m5.3.3.1.cmml"><mi id="Thmproofx1.p1.5.m5.3.3.1.3" mathvariant="normal" xref="Thmproofx1.p1.5.m5.3.3.1.3.cmml">ℓ</mi><mo id="Thmproofx1.p1.5.m5.3.3.1.2" xref="Thmproofx1.p1.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="Thmproofx1.p1.5.m5.3.3.1.1.1" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.cmml"><mo id="Thmproofx1.p1.5.m5.3.3.1.1.1.2" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.5.m5.3.3.1.1.1.1" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.cmml"><mn id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.3" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.3.cmml">1</mn><mo id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.2" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.2.cmml">−</mo><mrow id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.2" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml"><mn id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.2" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.1" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.2" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.2.cmml">n</mi><mi id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.3.cmml">ℓ</mi></mfrac></mrow><mo id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.3" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx1.p1.5.m5.3.3.1.1.1.3" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx1.p1.5.m5.3.3.5" xref="Thmproofx1.p1.5.m5.3.3.5.cmml">=</mo><mi id="Thmproofx1.p1.5.m5.3.3.6" xref="Thmproofx1.p1.5.m5.3.3.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.5.m5.3b"><apply id="Thmproofx1.p1.5.m5.3.3.cmml" xref="Thmproofx1.p1.5.m5.3.3"><and id="Thmproofx1.p1.5.m5.3.3a.cmml" xref="Thmproofx1.p1.5.m5.3.3"></and><apply id="Thmproofx1.p1.5.m5.3.3b.cmml" xref="Thmproofx1.p1.5.m5.3.3"><leq id="Thmproofx1.p1.5.m5.3.3.4.cmml" xref="Thmproofx1.p1.5.m5.3.3.4"></leq><apply id="Thmproofx1.p1.5.m5.3.3.3.cmml" xref="Thmproofx1.p1.5.m5.3.3.3"><csymbol cd="ambiguous" id="Thmproofx1.p1.5.m5.3.3.3.1.cmml" xref="Thmproofx1.p1.5.m5.3.3.3">subscript</csymbol><ci id="Thmproofx1.p1.5.m5.3.3.3.2.cmml" xref="Thmproofx1.p1.5.m5.3.3.3.2">ℰ</ci><list id="Thmproofx1.p1.5.m5.2.2.2.3.cmml" xref="Thmproofx1.p1.5.m5.2.2.2.4"><ci id="Thmproofx1.p1.5.m5.1.1.1.1.cmml" xref="Thmproofx1.p1.5.m5.1.1.1.1">ℓ</ci><ci id="Thmproofx1.p1.5.m5.2.2.2.2.cmml" xref="Thmproofx1.p1.5.m5.2.2.2.2">𝑛</ci></list></apply><apply id="Thmproofx1.p1.5.m5.3.3.1.cmml" xref="Thmproofx1.p1.5.m5.3.3.1"><times id="Thmproofx1.p1.5.m5.3.3.1.2.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.2"></times><ci id="Thmproofx1.p1.5.m5.3.3.1.3.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.3">ℓ</ci><apply id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1"><minus id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.2.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.2"></minus><cn id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.3.cmml" type="integer" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.3">1</cn><apply id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1"><minus id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.1"></minus><cn id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3"><divide id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3"></divide><ci id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.2">𝑛</ci><ci id="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx1.p1.5.m5.3.3.1.1.1.1.1.1.1.3.3">ℓ</ci></apply></apply></apply></apply></apply><apply id="Thmproofx1.p1.5.m5.3.3c.cmml" xref="Thmproofx1.p1.5.m5.3.3"><eq id="Thmproofx1.p1.5.m5.3.3.5.cmml" xref="Thmproofx1.p1.5.m5.3.3.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx1.p1.5.m5.3.3.1.cmml" id="Thmproofx1.p1.5.m5.3.3d.cmml" xref="Thmproofx1.p1.5.m5.3.3"></share><ci id="Thmproofx1.p1.5.m5.3.3.6.cmml" xref="Thmproofx1.p1.5.m5.3.3.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.5.m5.3c">\mathcal{E}_{\ell,n}\leq\ell\left(1-\left(1-\frac{n}{\ell}\right)\right)=n</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.5.m5.3d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT ≤ roman_ℓ ( 1 - ( 1 - divide start_ARG italic_n end_ARG start_ARG roman_ℓ end_ARG ) ) = italic_n</annotation></semantics></math>. On the other hand, because <math alttext="\left(1-\frac{1}{\ell}\right)^{n}\leq n\left(1-\frac{1}{l}\right)" class="ltx_Math" display="inline" id="Thmproofx1.p1.6.m6.2"><semantics id="Thmproofx1.p1.6.m6.2a"><mrow id="Thmproofx1.p1.6.m6.2.2" xref="Thmproofx1.p1.6.m6.2.2.cmml"><msup id="Thmproofx1.p1.6.m6.1.1.1" xref="Thmproofx1.p1.6.m6.1.1.1.cmml"><mrow id="Thmproofx1.p1.6.m6.1.1.1.1.1" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmproofx1.p1.6.m6.1.1.1.1.1.2" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.6.m6.1.1.1.1.1.1" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.cmml"><mn id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.2" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.1" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.cmml"><mn id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.2" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.3.cmml">ℓ</mi></mfrac></mrow><mo id="Thmproofx1.p1.6.m6.1.1.1.1.1.3" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx1.p1.6.m6.1.1.1.3" xref="Thmproofx1.p1.6.m6.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx1.p1.6.m6.2.2.3" xref="Thmproofx1.p1.6.m6.2.2.3.cmml">≤</mo><mrow id="Thmproofx1.p1.6.m6.2.2.2" xref="Thmproofx1.p1.6.m6.2.2.2.cmml"><mi id="Thmproofx1.p1.6.m6.2.2.2.3" xref="Thmproofx1.p1.6.m6.2.2.2.3.cmml">n</mi><mo id="Thmproofx1.p1.6.m6.2.2.2.2" xref="Thmproofx1.p1.6.m6.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx1.p1.6.m6.2.2.2.1.1" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.cmml"><mo id="Thmproofx1.p1.6.m6.2.2.2.1.1.2" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.6.m6.2.2.2.1.1.1" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.cmml"><mn id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.2" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.1" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.cmml"><mn id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.2" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.2.cmml">1</mn><mi id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.3" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.3.cmml">l</mi></mfrac></mrow><mo id="Thmproofx1.p1.6.m6.2.2.2.1.1.3" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.6.m6.2b"><apply id="Thmproofx1.p1.6.m6.2.2.cmml" xref="Thmproofx1.p1.6.m6.2.2"><leq id="Thmproofx1.p1.6.m6.2.2.3.cmml" xref="Thmproofx1.p1.6.m6.2.2.3"></leq><apply id="Thmproofx1.p1.6.m6.1.1.1.cmml" xref="Thmproofx1.p1.6.m6.1.1.1"><csymbol cd="ambiguous" id="Thmproofx1.p1.6.m6.1.1.1.2.cmml" xref="Thmproofx1.p1.6.m6.1.1.1">superscript</csymbol><apply id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.6.m6.1.1.1.1.1"><minus id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.1"></minus><cn id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3"><divide id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3"></divide><cn id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.2">1</cn><ci id="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx1.p1.6.m6.1.1.1.1.1.1.3.3">ℓ</ci></apply></apply><ci id="Thmproofx1.p1.6.m6.1.1.1.3.cmml" xref="Thmproofx1.p1.6.m6.1.1.1.3">𝑛</ci></apply><apply id="Thmproofx1.p1.6.m6.2.2.2.cmml" xref="Thmproofx1.p1.6.m6.2.2.2"><times id="Thmproofx1.p1.6.m6.2.2.2.2.cmml" xref="Thmproofx1.p1.6.m6.2.2.2.2"></times><ci id="Thmproofx1.p1.6.m6.2.2.2.3.cmml" xref="Thmproofx1.p1.6.m6.2.2.2.3">𝑛</ci><apply id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.cmml" xref="Thmproofx1.p1.6.m6.2.2.2.1.1"><minus id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.1.cmml" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.1"></minus><cn id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.2">1</cn><apply id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.cmml" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3"><divide id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3"></divide><cn id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.2.cmml" type="integer" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.2">1</cn><ci id="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx1.p1.6.m6.2.2.2.1.1.1.3.3">𝑙</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.6.m6.2c">\left(1-\frac{1}{\ell}\right)^{n}\leq n\left(1-\frac{1}{l}\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.6.m6.2d">( 1 - divide start_ARG 1 end_ARG start_ARG roman_ℓ end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ≤ italic_n ( 1 - divide start_ARG 1 end_ARG start_ARG italic_l end_ARG )</annotation></semantics></math> for <math alttext="\ell,n\geq 1" class="ltx_Math" display="inline" id="Thmproofx1.p1.7.m7.2"><semantics id="Thmproofx1.p1.7.m7.2a"><mrow id="Thmproofx1.p1.7.m7.2.3" xref="Thmproofx1.p1.7.m7.2.3.cmml"><mrow id="Thmproofx1.p1.7.m7.2.3.2.2" xref="Thmproofx1.p1.7.m7.2.3.2.1.cmml"><mi id="Thmproofx1.p1.7.m7.1.1" mathvariant="normal" xref="Thmproofx1.p1.7.m7.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.7.m7.2.3.2.2.1" xref="Thmproofx1.p1.7.m7.2.3.2.1.cmml">,</mo><mi id="Thmproofx1.p1.7.m7.2.2" xref="Thmproofx1.p1.7.m7.2.2.cmml">n</mi></mrow><mo id="Thmproofx1.p1.7.m7.2.3.1" xref="Thmproofx1.p1.7.m7.2.3.1.cmml">≥</mo><mn id="Thmproofx1.p1.7.m7.2.3.3" xref="Thmproofx1.p1.7.m7.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.7.m7.2b"><apply id="Thmproofx1.p1.7.m7.2.3.cmml" xref="Thmproofx1.p1.7.m7.2.3"><geq id="Thmproofx1.p1.7.m7.2.3.1.cmml" xref="Thmproofx1.p1.7.m7.2.3.1"></geq><list id="Thmproofx1.p1.7.m7.2.3.2.1.cmml" xref="Thmproofx1.p1.7.m7.2.3.2.2"><ci id="Thmproofx1.p1.7.m7.1.1.cmml" xref="Thmproofx1.p1.7.m7.1.1">ℓ</ci><ci id="Thmproofx1.p1.7.m7.2.2.cmml" xref="Thmproofx1.p1.7.m7.2.2">𝑛</ci></list><cn id="Thmproofx1.p1.7.m7.2.3.3.cmml" type="integer" xref="Thmproofx1.p1.7.m7.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.7.m7.2c">\ell,n\geq 1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.7.m7.2d">roman_ℓ , italic_n ≥ 1</annotation></semantics></math>, we have <math alttext="\mathcal{E}_{\ell,n}\geq\ell\left(1-n\left(1-\frac{1}{\ell}\right)\right)=n-% \frac{n}{\ell}" class="ltx_Math" display="inline" id="Thmproofx1.p1.8.m8.3"><semantics id="Thmproofx1.p1.8.m8.3a"><mrow id="Thmproofx1.p1.8.m8.3.3" xref="Thmproofx1.p1.8.m8.3.3.cmml"><msub id="Thmproofx1.p1.8.m8.3.3.3" xref="Thmproofx1.p1.8.m8.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx1.p1.8.m8.3.3.3.2" xref="Thmproofx1.p1.8.m8.3.3.3.2.cmml">ℰ</mi><mrow id="Thmproofx1.p1.8.m8.2.2.2.4" xref="Thmproofx1.p1.8.m8.2.2.2.3.cmml"><mi id="Thmproofx1.p1.8.m8.1.1.1.1" mathvariant="normal" xref="Thmproofx1.p1.8.m8.1.1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.8.m8.2.2.2.4.1" xref="Thmproofx1.p1.8.m8.2.2.2.3.cmml">,</mo><mi id="Thmproofx1.p1.8.m8.2.2.2.2" xref="Thmproofx1.p1.8.m8.2.2.2.2.cmml">n</mi></mrow></msub><mo id="Thmproofx1.p1.8.m8.3.3.4" xref="Thmproofx1.p1.8.m8.3.3.4.cmml">≥</mo><mrow id="Thmproofx1.p1.8.m8.3.3.1" xref="Thmproofx1.p1.8.m8.3.3.1.cmml"><mi id="Thmproofx1.p1.8.m8.3.3.1.3" mathvariant="normal" xref="Thmproofx1.p1.8.m8.3.3.1.3.cmml">ℓ</mi><mo id="Thmproofx1.p1.8.m8.3.3.1.2" xref="Thmproofx1.p1.8.m8.3.3.1.2.cmml">⁢</mo><mrow id="Thmproofx1.p1.8.m8.3.3.1.1.1" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.cmml"><mo id="Thmproofx1.p1.8.m8.3.3.1.1.1.2" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.8.m8.3.3.1.1.1.1" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.cmml"><mn id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.3" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.3.cmml">1</mn><mo id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.2" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.2.cmml">−</mo><mrow id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.cmml"><mi id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.3" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.3.cmml">n</mi><mo id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.2" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.2" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.cmml"><mn id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.2" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.1" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.cmml"><mn id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.2" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.3.cmml">ℓ</mi></mfrac></mrow><mo id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.3" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx1.p1.8.m8.3.3.1.1.1.3" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx1.p1.8.m8.3.3.5" xref="Thmproofx1.p1.8.m8.3.3.5.cmml">=</mo><mrow id="Thmproofx1.p1.8.m8.3.3.6" xref="Thmproofx1.p1.8.m8.3.3.6.cmml"><mi id="Thmproofx1.p1.8.m8.3.3.6.2" xref="Thmproofx1.p1.8.m8.3.3.6.2.cmml">n</mi><mo id="Thmproofx1.p1.8.m8.3.3.6.1" xref="Thmproofx1.p1.8.m8.3.3.6.1.cmml">−</mo><mfrac id="Thmproofx1.p1.8.m8.3.3.6.3" xref="Thmproofx1.p1.8.m8.3.3.6.3.cmml"><mi id="Thmproofx1.p1.8.m8.3.3.6.3.2" xref="Thmproofx1.p1.8.m8.3.3.6.3.2.cmml">n</mi><mi id="Thmproofx1.p1.8.m8.3.3.6.3.3" mathvariant="normal" xref="Thmproofx1.p1.8.m8.3.3.6.3.3.cmml">ℓ</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.8.m8.3b"><apply id="Thmproofx1.p1.8.m8.3.3.cmml" xref="Thmproofx1.p1.8.m8.3.3"><and id="Thmproofx1.p1.8.m8.3.3a.cmml" xref="Thmproofx1.p1.8.m8.3.3"></and><apply id="Thmproofx1.p1.8.m8.3.3b.cmml" xref="Thmproofx1.p1.8.m8.3.3"><geq id="Thmproofx1.p1.8.m8.3.3.4.cmml" xref="Thmproofx1.p1.8.m8.3.3.4"></geq><apply id="Thmproofx1.p1.8.m8.3.3.3.cmml" xref="Thmproofx1.p1.8.m8.3.3.3"><csymbol cd="ambiguous" id="Thmproofx1.p1.8.m8.3.3.3.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.3">subscript</csymbol><ci id="Thmproofx1.p1.8.m8.3.3.3.2.cmml" xref="Thmproofx1.p1.8.m8.3.3.3.2">ℰ</ci><list id="Thmproofx1.p1.8.m8.2.2.2.3.cmml" xref="Thmproofx1.p1.8.m8.2.2.2.4"><ci id="Thmproofx1.p1.8.m8.1.1.1.1.cmml" xref="Thmproofx1.p1.8.m8.1.1.1.1">ℓ</ci><ci id="Thmproofx1.p1.8.m8.2.2.2.2.cmml" xref="Thmproofx1.p1.8.m8.2.2.2.2">𝑛</ci></list></apply><apply id="Thmproofx1.p1.8.m8.3.3.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.1"><times id="Thmproofx1.p1.8.m8.3.3.1.2.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.2"></times><ci id="Thmproofx1.p1.8.m8.3.3.1.3.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.3">ℓ</ci><apply id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1"><minus id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.2.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.2"></minus><cn id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.3.cmml" type="integer" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.3">1</cn><apply id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1"><times id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.2.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.2"></times><ci id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.3.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.3">𝑛</ci><apply id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1"><minus id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.1"></minus><cn id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3"><divide id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3"></divide><cn id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.3.3">ℓ</ci></apply></apply></apply></apply></apply></apply><apply id="Thmproofx1.p1.8.m8.3.3c.cmml" xref="Thmproofx1.p1.8.m8.3.3"><eq id="Thmproofx1.p1.8.m8.3.3.5.cmml" xref="Thmproofx1.p1.8.m8.3.3.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx1.p1.8.m8.3.3.1.cmml" id="Thmproofx1.p1.8.m8.3.3d.cmml" xref="Thmproofx1.p1.8.m8.3.3"></share><apply id="Thmproofx1.p1.8.m8.3.3.6.cmml" xref="Thmproofx1.p1.8.m8.3.3.6"><minus id="Thmproofx1.p1.8.m8.3.3.6.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.6.1"></minus><ci id="Thmproofx1.p1.8.m8.3.3.6.2.cmml" xref="Thmproofx1.p1.8.m8.3.3.6.2">𝑛</ci><apply id="Thmproofx1.p1.8.m8.3.3.6.3.cmml" xref="Thmproofx1.p1.8.m8.3.3.6.3"><divide id="Thmproofx1.p1.8.m8.3.3.6.3.1.cmml" xref="Thmproofx1.p1.8.m8.3.3.6.3"></divide><ci id="Thmproofx1.p1.8.m8.3.3.6.3.2.cmml" xref="Thmproofx1.p1.8.m8.3.3.6.3.2">𝑛</ci><ci id="Thmproofx1.p1.8.m8.3.3.6.3.3.cmml" xref="Thmproofx1.p1.8.m8.3.3.6.3.3">ℓ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.8.m8.3c">\mathcal{E}_{\ell,n}\geq\ell\left(1-n\left(1-\frac{1}{\ell}\right)\right)=n-% \frac{n}{\ell}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.8.m8.3d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT ≥ roman_ℓ ( 1 - italic_n ( 1 - divide start_ARG 1 end_ARG start_ARG roman_ℓ end_ARG ) ) = italic_n - divide start_ARG italic_n end_ARG start_ARG roman_ℓ end_ARG</annotation></semantics></math>. Therefore</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx16"> <tbody id="Pt0.A1.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 1\geq\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{n}\geq\lim_% {\ell\to\infty}\frac{n}{\frac{n}{\ell}-n}=1" class="ltx_Math" display="inline" id="Pt0.A1.Ex15.m1.2"><semantics id="Pt0.A1.Ex15.m1.2a"><mrow id="Pt0.A1.Ex15.m1.2.3" xref="Pt0.A1.Ex15.m1.2.3.cmml"><mn id="Pt0.A1.Ex15.m1.2.3.2" xref="Pt0.A1.Ex15.m1.2.3.2.cmml">1</mn><mo id="Pt0.A1.Ex15.m1.2.3.3" rspace="0.1389em" xref="Pt0.A1.Ex15.m1.2.3.3.cmml">≥</mo><mrow id="Pt0.A1.Ex15.m1.2.3.4" xref="Pt0.A1.Ex15.m1.2.3.4.cmml"><munder id="Pt0.A1.Ex15.m1.2.3.4.1" xref="Pt0.A1.Ex15.m1.2.3.4.1.cmml"><mo id="Pt0.A1.Ex15.m1.2.3.4.1.2" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="Pt0.A1.Ex15.m1.2.3.4.1.2.cmml">lim</mo><mrow id="Pt0.A1.Ex15.m1.2.3.4.1.3" xref="Pt0.A1.Ex15.m1.2.3.4.1.3.cmml"><mi id="Pt0.A1.Ex15.m1.2.3.4.1.3.2" mathvariant="normal" xref="Pt0.A1.Ex15.m1.2.3.4.1.3.2.cmml">ℓ</mi><mo id="Pt0.A1.Ex15.m1.2.3.4.1.3.1" stretchy="false" xref="Pt0.A1.Ex15.m1.2.3.4.1.3.1.cmml">→</mo><mi id="Pt0.A1.Ex15.m1.2.3.4.1.3.3" mathvariant="normal" xref="Pt0.A1.Ex15.m1.2.3.4.1.3.3.cmml">∞</mi></mrow></munder><mstyle displaystyle="true" id="Pt0.A1.Ex15.m1.2.2" xref="Pt0.A1.Ex15.m1.2.2.cmml"><mfrac id="Pt0.A1.Ex15.m1.2.2a" xref="Pt0.A1.Ex15.m1.2.2.cmml"><msub id="Pt0.A1.Ex15.m1.2.2.2" xref="Pt0.A1.Ex15.m1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.Ex15.m1.2.2.2.4" xref="Pt0.A1.Ex15.m1.2.2.2.4.cmml">ℰ</mi><mrow id="Pt0.A1.Ex15.m1.2.2.2.2.2.4" xref="Pt0.A1.Ex15.m1.2.2.2.2.2.3.cmml"><mi id="Pt0.A1.Ex15.m1.1.1.1.1.1.1" mathvariant="normal" xref="Pt0.A1.Ex15.m1.1.1.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.Ex15.m1.2.2.2.2.2.4.1" xref="Pt0.A1.Ex15.m1.2.2.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.Ex15.m1.2.2.2.2.2.2" xref="Pt0.A1.Ex15.m1.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Pt0.A1.Ex15.m1.2.2.4" xref="Pt0.A1.Ex15.m1.2.2.4.cmml">n</mi></mfrac></mstyle></mrow><mo id="Pt0.A1.Ex15.m1.2.3.5" rspace="0.1389em" xref="Pt0.A1.Ex15.m1.2.3.5.cmml">≥</mo><mrow id="Pt0.A1.Ex15.m1.2.3.6" xref="Pt0.A1.Ex15.m1.2.3.6.cmml"><munder id="Pt0.A1.Ex15.m1.2.3.6.1" xref="Pt0.A1.Ex15.m1.2.3.6.1.cmml"><mo id="Pt0.A1.Ex15.m1.2.3.6.1.2" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="Pt0.A1.Ex15.m1.2.3.6.1.2.cmml">lim</mo><mrow id="Pt0.A1.Ex15.m1.2.3.6.1.3" xref="Pt0.A1.Ex15.m1.2.3.6.1.3.cmml"><mi id="Pt0.A1.Ex15.m1.2.3.6.1.3.2" mathvariant="normal" xref="Pt0.A1.Ex15.m1.2.3.6.1.3.2.cmml">ℓ</mi><mo id="Pt0.A1.Ex15.m1.2.3.6.1.3.1" stretchy="false" xref="Pt0.A1.Ex15.m1.2.3.6.1.3.1.cmml">→</mo><mi id="Pt0.A1.Ex15.m1.2.3.6.1.3.3" mathvariant="normal" xref="Pt0.A1.Ex15.m1.2.3.6.1.3.3.cmml">∞</mi></mrow></munder><mstyle displaystyle="true" id="Pt0.A1.Ex15.m1.2.3.6.2" xref="Pt0.A1.Ex15.m1.2.3.6.2.cmml"><mfrac id="Pt0.A1.Ex15.m1.2.3.6.2a" xref="Pt0.A1.Ex15.m1.2.3.6.2.cmml"><mi id="Pt0.A1.Ex15.m1.2.3.6.2.2" xref="Pt0.A1.Ex15.m1.2.3.6.2.2.cmml">n</mi><mrow id="Pt0.A1.Ex15.m1.2.3.6.2.3" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.cmml"><mfrac id="Pt0.A1.Ex15.m1.2.3.6.2.3.2" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.2.cmml"><mi id="Pt0.A1.Ex15.m1.2.3.6.2.3.2.2" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.2.2.cmml">n</mi><mi id="Pt0.A1.Ex15.m1.2.3.6.2.3.2.3" mathvariant="normal" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.2.3.cmml">ℓ</mi></mfrac><mo id="Pt0.A1.Ex15.m1.2.3.6.2.3.1" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.1.cmml">−</mo><mi id="Pt0.A1.Ex15.m1.2.3.6.2.3.3" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.3.cmml">n</mi></mrow></mfrac></mstyle></mrow><mo id="Pt0.A1.Ex15.m1.2.3.7" xref="Pt0.A1.Ex15.m1.2.3.7.cmml">=</mo><mn id="Pt0.A1.Ex15.m1.2.3.8" xref="Pt0.A1.Ex15.m1.2.3.8.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.Ex15.m1.2b"><apply id="Pt0.A1.Ex15.m1.2.3.cmml" xref="Pt0.A1.Ex15.m1.2.3"><and id="Pt0.A1.Ex15.m1.2.3a.cmml" xref="Pt0.A1.Ex15.m1.2.3"></and><apply id="Pt0.A1.Ex15.m1.2.3b.cmml" xref="Pt0.A1.Ex15.m1.2.3"><geq id="Pt0.A1.Ex15.m1.2.3.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.3"></geq><cn id="Pt0.A1.Ex15.m1.2.3.2.cmml" type="integer" xref="Pt0.A1.Ex15.m1.2.3.2">1</cn><apply id="Pt0.A1.Ex15.m1.2.3.4.cmml" xref="Pt0.A1.Ex15.m1.2.3.4"><apply id="Pt0.A1.Ex15.m1.2.3.4.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.4.1"><csymbol cd="ambiguous" id="Pt0.A1.Ex15.m1.2.3.4.1.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.4.1">subscript</csymbol><limit id="Pt0.A1.Ex15.m1.2.3.4.1.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.4.1.2"></limit><apply id="Pt0.A1.Ex15.m1.2.3.4.1.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.4.1.3"><ci id="Pt0.A1.Ex15.m1.2.3.4.1.3.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.4.1.3.1">→</ci><ci id="Pt0.A1.Ex15.m1.2.3.4.1.3.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.4.1.3.2">ℓ</ci><infinity id="Pt0.A1.Ex15.m1.2.3.4.1.3.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.4.1.3.3"></infinity></apply></apply><apply id="Pt0.A1.Ex15.m1.2.2.cmml" xref="Pt0.A1.Ex15.m1.2.2"><divide id="Pt0.A1.Ex15.m1.2.2.3.cmml" xref="Pt0.A1.Ex15.m1.2.2"></divide><apply id="Pt0.A1.Ex15.m1.2.2.2.cmml" xref="Pt0.A1.Ex15.m1.2.2.2"><csymbol cd="ambiguous" id="Pt0.A1.Ex15.m1.2.2.2.3.cmml" xref="Pt0.A1.Ex15.m1.2.2.2">subscript</csymbol><ci id="Pt0.A1.Ex15.m1.2.2.2.4.cmml" xref="Pt0.A1.Ex15.m1.2.2.2.4">ℰ</ci><list id="Pt0.A1.Ex15.m1.2.2.2.2.2.3.cmml" xref="Pt0.A1.Ex15.m1.2.2.2.2.2.4"><ci id="Pt0.A1.Ex15.m1.1.1.1.1.1.1.cmml" xref="Pt0.A1.Ex15.m1.1.1.1.1.1.1">ℓ</ci><ci id="Pt0.A1.Ex15.m1.2.2.2.2.2.2.cmml" xref="Pt0.A1.Ex15.m1.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Pt0.A1.Ex15.m1.2.2.4.cmml" xref="Pt0.A1.Ex15.m1.2.2.4">𝑛</ci></apply></apply></apply><apply id="Pt0.A1.Ex15.m1.2.3c.cmml" xref="Pt0.A1.Ex15.m1.2.3"><geq id="Pt0.A1.Ex15.m1.2.3.5.cmml" xref="Pt0.A1.Ex15.m1.2.3.5"></geq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A1.Ex15.m1.2.3.4.cmml" id="Pt0.A1.Ex15.m1.2.3d.cmml" xref="Pt0.A1.Ex15.m1.2.3"></share><apply id="Pt0.A1.Ex15.m1.2.3.6.cmml" xref="Pt0.A1.Ex15.m1.2.3.6"><apply id="Pt0.A1.Ex15.m1.2.3.6.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.1"><csymbol cd="ambiguous" id="Pt0.A1.Ex15.m1.2.3.6.1.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.1">subscript</csymbol><limit id="Pt0.A1.Ex15.m1.2.3.6.1.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.1.2"></limit><apply id="Pt0.A1.Ex15.m1.2.3.6.1.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.1.3"><ci id="Pt0.A1.Ex15.m1.2.3.6.1.3.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.1.3.1">→</ci><ci id="Pt0.A1.Ex15.m1.2.3.6.1.3.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.1.3.2">ℓ</ci><infinity id="Pt0.A1.Ex15.m1.2.3.6.1.3.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.1.3.3"></infinity></apply></apply><apply id="Pt0.A1.Ex15.m1.2.3.6.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2"><divide id="Pt0.A1.Ex15.m1.2.3.6.2.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2"></divide><ci id="Pt0.A1.Ex15.m1.2.3.6.2.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.2">𝑛</ci><apply id="Pt0.A1.Ex15.m1.2.3.6.2.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.3"><minus id="Pt0.A1.Ex15.m1.2.3.6.2.3.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.1"></minus><apply id="Pt0.A1.Ex15.m1.2.3.6.2.3.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.2"><divide id="Pt0.A1.Ex15.m1.2.3.6.2.3.2.1.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.2"></divide><ci id="Pt0.A1.Ex15.m1.2.3.6.2.3.2.2.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.2.2">𝑛</ci><ci id="Pt0.A1.Ex15.m1.2.3.6.2.3.2.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.2.3">ℓ</ci></apply><ci id="Pt0.A1.Ex15.m1.2.3.6.2.3.3.cmml" xref="Pt0.A1.Ex15.m1.2.3.6.2.3.3">𝑛</ci></apply></apply></apply></apply><apply id="Pt0.A1.Ex15.m1.2.3e.cmml" xref="Pt0.A1.Ex15.m1.2.3"><eq id="Pt0.A1.Ex15.m1.2.3.7.cmml" xref="Pt0.A1.Ex15.m1.2.3.7"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A1.Ex15.m1.2.3.6.cmml" id="Pt0.A1.Ex15.m1.2.3f.cmml" xref="Pt0.A1.Ex15.m1.2.3"></share><cn id="Pt0.A1.Ex15.m1.2.3.8.cmml" type="integer" xref="Pt0.A1.Ex15.m1.2.3.8">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex15.m1.2c">\displaystyle 1\geq\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{n}\geq\lim_% {\ell\to\infty}\frac{n}{\frac{n}{\ell}-n}=1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex15.m1.2d">1 ≥ roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_n end_ARG ≥ roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG italic_n end_ARG start_ARG divide start_ARG italic_n end_ARG start_ARG roman_ℓ end_ARG - italic_n end_ARG = 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx1.p1.13">Now suppose that <math alttext="n=x\ell+o(\ell)" class="ltx_Math" display="inline" id="Thmproofx1.p1.9.m1.1"><semantics id="Thmproofx1.p1.9.m1.1a"><mrow id="Thmproofx1.p1.9.m1.1.2" xref="Thmproofx1.p1.9.m1.1.2.cmml"><mi id="Thmproofx1.p1.9.m1.1.2.2" xref="Thmproofx1.p1.9.m1.1.2.2.cmml">n</mi><mo id="Thmproofx1.p1.9.m1.1.2.1" xref="Thmproofx1.p1.9.m1.1.2.1.cmml">=</mo><mrow id="Thmproofx1.p1.9.m1.1.2.3" xref="Thmproofx1.p1.9.m1.1.2.3.cmml"><mrow id="Thmproofx1.p1.9.m1.1.2.3.2" xref="Thmproofx1.p1.9.m1.1.2.3.2.cmml"><mi id="Thmproofx1.p1.9.m1.1.2.3.2.2" xref="Thmproofx1.p1.9.m1.1.2.3.2.2.cmml">x</mi><mo id="Thmproofx1.p1.9.m1.1.2.3.2.1" xref="Thmproofx1.p1.9.m1.1.2.3.2.1.cmml">⁢</mo><mi id="Thmproofx1.p1.9.m1.1.2.3.2.3" mathvariant="normal" xref="Thmproofx1.p1.9.m1.1.2.3.2.3.cmml">ℓ</mi></mrow><mo id="Thmproofx1.p1.9.m1.1.2.3.1" xref="Thmproofx1.p1.9.m1.1.2.3.1.cmml">+</mo><mrow id="Thmproofx1.p1.9.m1.1.2.3.3" xref="Thmproofx1.p1.9.m1.1.2.3.3.cmml"><mi id="Thmproofx1.p1.9.m1.1.2.3.3.2" xref="Thmproofx1.p1.9.m1.1.2.3.3.2.cmml">o</mi><mo id="Thmproofx1.p1.9.m1.1.2.3.3.1" xref="Thmproofx1.p1.9.m1.1.2.3.3.1.cmml">⁢</mo><mrow id="Thmproofx1.p1.9.m1.1.2.3.3.3.2" xref="Thmproofx1.p1.9.m1.1.2.3.3.cmml"><mo id="Thmproofx1.p1.9.m1.1.2.3.3.3.2.1" stretchy="false" xref="Thmproofx1.p1.9.m1.1.2.3.3.cmml">(</mo><mi id="Thmproofx1.p1.9.m1.1.1" mathvariant="normal" xref="Thmproofx1.p1.9.m1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.9.m1.1.2.3.3.3.2.2" stretchy="false" xref="Thmproofx1.p1.9.m1.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.9.m1.1b"><apply id="Thmproofx1.p1.9.m1.1.2.cmml" xref="Thmproofx1.p1.9.m1.1.2"><eq id="Thmproofx1.p1.9.m1.1.2.1.cmml" xref="Thmproofx1.p1.9.m1.1.2.1"></eq><ci id="Thmproofx1.p1.9.m1.1.2.2.cmml" xref="Thmproofx1.p1.9.m1.1.2.2">𝑛</ci><apply id="Thmproofx1.p1.9.m1.1.2.3.cmml" xref="Thmproofx1.p1.9.m1.1.2.3"><plus id="Thmproofx1.p1.9.m1.1.2.3.1.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.1"></plus><apply id="Thmproofx1.p1.9.m1.1.2.3.2.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.2"><times id="Thmproofx1.p1.9.m1.1.2.3.2.1.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.2.1"></times><ci id="Thmproofx1.p1.9.m1.1.2.3.2.2.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.2.2">𝑥</ci><ci id="Thmproofx1.p1.9.m1.1.2.3.2.3.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.2.3">ℓ</ci></apply><apply id="Thmproofx1.p1.9.m1.1.2.3.3.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.3"><times id="Thmproofx1.p1.9.m1.1.2.3.3.1.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.3.1"></times><ci id="Thmproofx1.p1.9.m1.1.2.3.3.2.cmml" xref="Thmproofx1.p1.9.m1.1.2.3.3.2">𝑜</ci><ci id="Thmproofx1.p1.9.m1.1.1.cmml" xref="Thmproofx1.p1.9.m1.1.1">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.9.m1.1c">n=x\ell+o(\ell)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.9.m1.1d">italic_n = italic_x roman_ℓ + italic_o ( roman_ℓ )</annotation></semantics></math>. Then <math alttext="\mathcal{E}_{\ell,n}=\ell\left(1-\left(1-\frac{1}{\ell}\right)^{x\ell+o(\ell)}\right)" class="ltx_Math" display="inline" id="Thmproofx1.p1.10.m2.4"><semantics id="Thmproofx1.p1.10.m2.4a"><mrow id="Thmproofx1.p1.10.m2.4.4" xref="Thmproofx1.p1.10.m2.4.4.cmml"><msub id="Thmproofx1.p1.10.m2.4.4.3" xref="Thmproofx1.p1.10.m2.4.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx1.p1.10.m2.4.4.3.2" xref="Thmproofx1.p1.10.m2.4.4.3.2.cmml">ℰ</mi><mrow id="Thmproofx1.p1.10.m2.2.2.2.4" xref="Thmproofx1.p1.10.m2.2.2.2.3.cmml"><mi id="Thmproofx1.p1.10.m2.1.1.1.1" mathvariant="normal" xref="Thmproofx1.p1.10.m2.1.1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.10.m2.2.2.2.4.1" xref="Thmproofx1.p1.10.m2.2.2.2.3.cmml">,</mo><mi id="Thmproofx1.p1.10.m2.2.2.2.2" xref="Thmproofx1.p1.10.m2.2.2.2.2.cmml">n</mi></mrow></msub><mo id="Thmproofx1.p1.10.m2.4.4.2" xref="Thmproofx1.p1.10.m2.4.4.2.cmml">=</mo><mrow id="Thmproofx1.p1.10.m2.4.4.1" xref="Thmproofx1.p1.10.m2.4.4.1.cmml"><mi id="Thmproofx1.p1.10.m2.4.4.1.3" mathvariant="normal" xref="Thmproofx1.p1.10.m2.4.4.1.3.cmml">ℓ</mi><mo id="Thmproofx1.p1.10.m2.4.4.1.2" xref="Thmproofx1.p1.10.m2.4.4.1.2.cmml">⁢</mo><mrow id="Thmproofx1.p1.10.m2.4.4.1.1.1" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.cmml"><mo id="Thmproofx1.p1.10.m2.4.4.1.1.1.2" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.10.m2.4.4.1.1.1.1" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.cmml"><mn id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.3" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.3.cmml">1</mn><mo id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.2" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.2.cmml">−</mo><msup id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.cmml"><mrow id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.2" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.cmml"><mn id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.2" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.1" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.cmml"><mn id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.2" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.3.cmml">ℓ</mi></mfrac></mrow><mo id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.3" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="Thmproofx1.p1.10.m2.3.3.1" xref="Thmproofx1.p1.10.m2.3.3.1.cmml"><mrow id="Thmproofx1.p1.10.m2.3.3.1.3" xref="Thmproofx1.p1.10.m2.3.3.1.3.cmml"><mi id="Thmproofx1.p1.10.m2.3.3.1.3.2" xref="Thmproofx1.p1.10.m2.3.3.1.3.2.cmml">x</mi><mo id="Thmproofx1.p1.10.m2.3.3.1.3.1" xref="Thmproofx1.p1.10.m2.3.3.1.3.1.cmml">⁢</mo><mi id="Thmproofx1.p1.10.m2.3.3.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.10.m2.3.3.1.3.3.cmml">ℓ</mi></mrow><mo id="Thmproofx1.p1.10.m2.3.3.1.2" xref="Thmproofx1.p1.10.m2.3.3.1.2.cmml">+</mo><mrow id="Thmproofx1.p1.10.m2.3.3.1.4" xref="Thmproofx1.p1.10.m2.3.3.1.4.cmml"><mi id="Thmproofx1.p1.10.m2.3.3.1.4.2" xref="Thmproofx1.p1.10.m2.3.3.1.4.2.cmml">o</mi><mo id="Thmproofx1.p1.10.m2.3.3.1.4.1" xref="Thmproofx1.p1.10.m2.3.3.1.4.1.cmml">⁢</mo><mrow id="Thmproofx1.p1.10.m2.3.3.1.4.3.2" xref="Thmproofx1.p1.10.m2.3.3.1.4.cmml"><mo id="Thmproofx1.p1.10.m2.3.3.1.4.3.2.1" stretchy="false" xref="Thmproofx1.p1.10.m2.3.3.1.4.cmml">(</mo><mi id="Thmproofx1.p1.10.m2.3.3.1.1" mathvariant="normal" xref="Thmproofx1.p1.10.m2.3.3.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.10.m2.3.3.1.4.3.2.2" stretchy="false" xref="Thmproofx1.p1.10.m2.3.3.1.4.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="Thmproofx1.p1.10.m2.4.4.1.1.1.3" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.10.m2.4b"><apply id="Thmproofx1.p1.10.m2.4.4.cmml" xref="Thmproofx1.p1.10.m2.4.4"><eq id="Thmproofx1.p1.10.m2.4.4.2.cmml" xref="Thmproofx1.p1.10.m2.4.4.2"></eq><apply id="Thmproofx1.p1.10.m2.4.4.3.cmml" xref="Thmproofx1.p1.10.m2.4.4.3"><csymbol cd="ambiguous" id="Thmproofx1.p1.10.m2.4.4.3.1.cmml" xref="Thmproofx1.p1.10.m2.4.4.3">subscript</csymbol><ci id="Thmproofx1.p1.10.m2.4.4.3.2.cmml" xref="Thmproofx1.p1.10.m2.4.4.3.2">ℰ</ci><list id="Thmproofx1.p1.10.m2.2.2.2.3.cmml" xref="Thmproofx1.p1.10.m2.2.2.2.4"><ci id="Thmproofx1.p1.10.m2.1.1.1.1.cmml" xref="Thmproofx1.p1.10.m2.1.1.1.1">ℓ</ci><ci id="Thmproofx1.p1.10.m2.2.2.2.2.cmml" xref="Thmproofx1.p1.10.m2.2.2.2.2">𝑛</ci></list></apply><apply id="Thmproofx1.p1.10.m2.4.4.1.cmml" xref="Thmproofx1.p1.10.m2.4.4.1"><times id="Thmproofx1.p1.10.m2.4.4.1.2.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.2"></times><ci id="Thmproofx1.p1.10.m2.4.4.1.3.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.3">ℓ</ci><apply id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1"><minus id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.2.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.2"></minus><cn id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.3.cmml" type="integer" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.3">1</cn><apply id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.2.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1">superscript</csymbol><apply id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1"><minus id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.1"></minus><cn id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3"><divide id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3"></divide><cn id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx1.p1.10.m2.4.4.1.1.1.1.1.1.1.1.3.3">ℓ</ci></apply></apply><apply id="Thmproofx1.p1.10.m2.3.3.1.cmml" xref="Thmproofx1.p1.10.m2.3.3.1"><plus id="Thmproofx1.p1.10.m2.3.3.1.2.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.2"></plus><apply id="Thmproofx1.p1.10.m2.3.3.1.3.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.3"><times id="Thmproofx1.p1.10.m2.3.3.1.3.1.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.3.1"></times><ci id="Thmproofx1.p1.10.m2.3.3.1.3.2.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.3.2">𝑥</ci><ci id="Thmproofx1.p1.10.m2.3.3.1.3.3.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.3.3">ℓ</ci></apply><apply id="Thmproofx1.p1.10.m2.3.3.1.4.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.4"><times id="Thmproofx1.p1.10.m2.3.3.1.4.1.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.4.1"></times><ci id="Thmproofx1.p1.10.m2.3.3.1.4.2.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.4.2">𝑜</ci><ci id="Thmproofx1.p1.10.m2.3.3.1.1.cmml" xref="Thmproofx1.p1.10.m2.3.3.1.1">ℓ</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.10.m2.4c">\mathcal{E}_{\ell,n}=\ell\left(1-\left(1-\frac{1}{\ell}\right)^{x\ell+o(\ell)}\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.10.m2.4d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT = roman_ℓ ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG roman_ℓ end_ARG ) start_POSTSUPERSCRIPT italic_x roman_ℓ + italic_o ( roman_ℓ ) end_POSTSUPERSCRIPT )</annotation></semantics></math>. Again by calculus, <math alttext="\lim_{\ell\to\infty}\frac{\left(1-\frac{1}{\ell}\right)^{x\ell+o(\ell)}}{e^{-x% }}=1" class="ltx_Math" display="inline" id="Thmproofx1.p1.11.m3.2"><semantics id="Thmproofx1.p1.11.m3.2a"><mrow id="Thmproofx1.p1.11.m3.2.3" xref="Thmproofx1.p1.11.m3.2.3.cmml"><mrow id="Thmproofx1.p1.11.m3.2.3.2" xref="Thmproofx1.p1.11.m3.2.3.2.cmml"><msub id="Thmproofx1.p1.11.m3.2.3.2.1" xref="Thmproofx1.p1.11.m3.2.3.2.1.cmml"><mo id="Thmproofx1.p1.11.m3.2.3.2.1.2" xref="Thmproofx1.p1.11.m3.2.3.2.1.2.cmml">lim</mo><mrow id="Thmproofx1.p1.11.m3.2.3.2.1.3" xref="Thmproofx1.p1.11.m3.2.3.2.1.3.cmml"><mi id="Thmproofx1.p1.11.m3.2.3.2.1.3.2" mathvariant="normal" xref="Thmproofx1.p1.11.m3.2.3.2.1.3.2.cmml">ℓ</mi><mo id="Thmproofx1.p1.11.m3.2.3.2.1.3.1" stretchy="false" xref="Thmproofx1.p1.11.m3.2.3.2.1.3.1.cmml">→</mo><mi id="Thmproofx1.p1.11.m3.2.3.2.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.11.m3.2.3.2.1.3.3.cmml">∞</mi></mrow></msub><mfrac id="Thmproofx1.p1.11.m3.2.2" xref="Thmproofx1.p1.11.m3.2.2.cmml"><msup id="Thmproofx1.p1.11.m3.2.2.2" xref="Thmproofx1.p1.11.m3.2.2.2.cmml"><mrow id="Thmproofx1.p1.11.m3.2.2.2.2.1" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.cmml"><mo id="Thmproofx1.p1.11.m3.2.2.2.2.1.2" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.11.m3.2.2.2.2.1.1" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.cmml"><mn id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.2" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.1" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.1.cmml">−</mo><mfrac id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.cmml"><mn id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.2" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.2.cmml">1</mn><mi id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.3.cmml">ℓ</mi></mfrac></mrow><mo id="Thmproofx1.p1.11.m3.2.2.2.2.1.3" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.cmml">)</mo></mrow><mrow id="Thmproofx1.p1.11.m3.1.1.1.1.1" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.cmml"><mrow id="Thmproofx1.p1.11.m3.1.1.1.1.1.3" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3.cmml"><mi id="Thmproofx1.p1.11.m3.1.1.1.1.1.3.2" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3.2.cmml">x</mi><mo id="Thmproofx1.p1.11.m3.1.1.1.1.1.3.1" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx1.p1.11.m3.1.1.1.1.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3.3.cmml">ℓ</mi></mrow><mo id="Thmproofx1.p1.11.m3.1.1.1.1.1.2" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.2.cmml">+</mo><mrow id="Thmproofx1.p1.11.m3.1.1.1.1.1.4" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.cmml"><mi id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.2" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.2.cmml">o</mi><mo id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.1" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.3.2" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.cmml"><mo id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.3.2.1" stretchy="false" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.cmml">(</mo><mi id="Thmproofx1.p1.11.m3.1.1.1.1.1.1" mathvariant="normal" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.3.2.2" stretchy="false" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msup><msup id="Thmproofx1.p1.11.m3.2.2.4" xref="Thmproofx1.p1.11.m3.2.2.4.cmml"><mi id="Thmproofx1.p1.11.m3.2.2.4.2" xref="Thmproofx1.p1.11.m3.2.2.4.2.cmml">e</mi><mrow id="Thmproofx1.p1.11.m3.2.2.4.3" xref="Thmproofx1.p1.11.m3.2.2.4.3.cmml"><mo id="Thmproofx1.p1.11.m3.2.2.4.3a" xref="Thmproofx1.p1.11.m3.2.2.4.3.cmml">−</mo><mi id="Thmproofx1.p1.11.m3.2.2.4.3.2" xref="Thmproofx1.p1.11.m3.2.2.4.3.2.cmml">x</mi></mrow></msup></mfrac></mrow><mo id="Thmproofx1.p1.11.m3.2.3.1" xref="Thmproofx1.p1.11.m3.2.3.1.cmml">=</mo><mn id="Thmproofx1.p1.11.m3.2.3.3" xref="Thmproofx1.p1.11.m3.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.11.m3.2b"><apply id="Thmproofx1.p1.11.m3.2.3.cmml" xref="Thmproofx1.p1.11.m3.2.3"><eq id="Thmproofx1.p1.11.m3.2.3.1.cmml" xref="Thmproofx1.p1.11.m3.2.3.1"></eq><apply id="Thmproofx1.p1.11.m3.2.3.2.cmml" xref="Thmproofx1.p1.11.m3.2.3.2"><apply id="Thmproofx1.p1.11.m3.2.3.2.1.cmml" xref="Thmproofx1.p1.11.m3.2.3.2.1"><csymbol cd="ambiguous" id="Thmproofx1.p1.11.m3.2.3.2.1.1.cmml" xref="Thmproofx1.p1.11.m3.2.3.2.1">subscript</csymbol><limit id="Thmproofx1.p1.11.m3.2.3.2.1.2.cmml" xref="Thmproofx1.p1.11.m3.2.3.2.1.2"></limit><apply id="Thmproofx1.p1.11.m3.2.3.2.1.3.cmml" xref="Thmproofx1.p1.11.m3.2.3.2.1.3"><ci id="Thmproofx1.p1.11.m3.2.3.2.1.3.1.cmml" xref="Thmproofx1.p1.11.m3.2.3.2.1.3.1">→</ci><ci id="Thmproofx1.p1.11.m3.2.3.2.1.3.2.cmml" xref="Thmproofx1.p1.11.m3.2.3.2.1.3.2">ℓ</ci><infinity id="Thmproofx1.p1.11.m3.2.3.2.1.3.3.cmml" xref="Thmproofx1.p1.11.m3.2.3.2.1.3.3"></infinity></apply></apply><apply id="Thmproofx1.p1.11.m3.2.2.cmml" xref="Thmproofx1.p1.11.m3.2.2"><divide id="Thmproofx1.p1.11.m3.2.2.3.cmml" xref="Thmproofx1.p1.11.m3.2.2"></divide><apply id="Thmproofx1.p1.11.m3.2.2.2.cmml" xref="Thmproofx1.p1.11.m3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx1.p1.11.m3.2.2.2.3.cmml" xref="Thmproofx1.p1.11.m3.2.2.2">superscript</csymbol><apply id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.cmml" xref="Thmproofx1.p1.11.m3.2.2.2.2.1"><minus id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.1.cmml" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.1"></minus><cn id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.2">1</cn><apply id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.cmml" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3"><divide id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.1.cmml" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3"></divide><cn id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.2.cmml" type="integer" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.2">1</cn><ci id="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.3.cmml" xref="Thmproofx1.p1.11.m3.2.2.2.2.1.1.3.3">ℓ</ci></apply></apply><apply id="Thmproofx1.p1.11.m3.1.1.1.1.1.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1"><plus id="Thmproofx1.p1.11.m3.1.1.1.1.1.2.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.2"></plus><apply id="Thmproofx1.p1.11.m3.1.1.1.1.1.3.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3"><times id="Thmproofx1.p1.11.m3.1.1.1.1.1.3.1.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3.1"></times><ci id="Thmproofx1.p1.11.m3.1.1.1.1.1.3.2.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3.2">𝑥</ci><ci id="Thmproofx1.p1.11.m3.1.1.1.1.1.3.3.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.3.3">ℓ</ci></apply><apply id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4"><times id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.1.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.1"></times><ci id="Thmproofx1.p1.11.m3.1.1.1.1.1.4.2.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.4.2">𝑜</ci><ci id="Thmproofx1.p1.11.m3.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.11.m3.1.1.1.1.1.1">ℓ</ci></apply></apply></apply><apply id="Thmproofx1.p1.11.m3.2.2.4.cmml" xref="Thmproofx1.p1.11.m3.2.2.4"><csymbol cd="ambiguous" id="Thmproofx1.p1.11.m3.2.2.4.1.cmml" xref="Thmproofx1.p1.11.m3.2.2.4">superscript</csymbol><ci id="Thmproofx1.p1.11.m3.2.2.4.2.cmml" xref="Thmproofx1.p1.11.m3.2.2.4.2">𝑒</ci><apply id="Thmproofx1.p1.11.m3.2.2.4.3.cmml" xref="Thmproofx1.p1.11.m3.2.2.4.3"><minus id="Thmproofx1.p1.11.m3.2.2.4.3.1.cmml" xref="Thmproofx1.p1.11.m3.2.2.4.3"></minus><ci id="Thmproofx1.p1.11.m3.2.2.4.3.2.cmml" xref="Thmproofx1.p1.11.m3.2.2.4.3.2">𝑥</ci></apply></apply></apply></apply><cn id="Thmproofx1.p1.11.m3.2.3.3.cmml" type="integer" xref="Thmproofx1.p1.11.m3.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.11.m3.2c">\lim_{\ell\to\infty}\frac{\left(1-\frac{1}{\ell}\right)^{x\ell+o(\ell)}}{e^{-x% }}=1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.11.m3.2d">roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG roman_ℓ end_ARG ) start_POSTSUPERSCRIPT italic_x roman_ℓ + italic_o ( roman_ℓ ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT end_ARG = 1</annotation></semantics></math>, as <math alttext="x" class="ltx_Math" display="inline" id="Thmproofx1.p1.12.m4.1"><semantics id="Thmproofx1.p1.12.m4.1a"><mi id="Thmproofx1.p1.12.m4.1.1" xref="Thmproofx1.p1.12.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.12.m4.1b"><ci id="Thmproofx1.p1.12.m4.1.1.cmml" xref="Thmproofx1.p1.12.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.12.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.12.m4.1d">italic_x</annotation></semantics></math> is constant. So <math alttext="\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{(1-e^{-x})\ell}=1" class="ltx_Math" display="inline" id="Thmproofx1.p1.13.m5.3"><semantics id="Thmproofx1.p1.13.m5.3a"><mrow id="Thmproofx1.p1.13.m5.3.4" xref="Thmproofx1.p1.13.m5.3.4.cmml"><mrow id="Thmproofx1.p1.13.m5.3.4.2" xref="Thmproofx1.p1.13.m5.3.4.2.cmml"><msub id="Thmproofx1.p1.13.m5.3.4.2.1" xref="Thmproofx1.p1.13.m5.3.4.2.1.cmml"><mo id="Thmproofx1.p1.13.m5.3.4.2.1.2" xref="Thmproofx1.p1.13.m5.3.4.2.1.2.cmml">lim</mo><mrow id="Thmproofx1.p1.13.m5.3.4.2.1.3" xref="Thmproofx1.p1.13.m5.3.4.2.1.3.cmml"><mi id="Thmproofx1.p1.13.m5.3.4.2.1.3.2" mathvariant="normal" xref="Thmproofx1.p1.13.m5.3.4.2.1.3.2.cmml">ℓ</mi><mo id="Thmproofx1.p1.13.m5.3.4.2.1.3.1" stretchy="false" xref="Thmproofx1.p1.13.m5.3.4.2.1.3.1.cmml">→</mo><mi id="Thmproofx1.p1.13.m5.3.4.2.1.3.3" mathvariant="normal" xref="Thmproofx1.p1.13.m5.3.4.2.1.3.3.cmml">∞</mi></mrow></msub><mfrac id="Thmproofx1.p1.13.m5.3.3" xref="Thmproofx1.p1.13.m5.3.3.cmml"><msub id="Thmproofx1.p1.13.m5.2.2.2" xref="Thmproofx1.p1.13.m5.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx1.p1.13.m5.2.2.2.4" xref="Thmproofx1.p1.13.m5.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx1.p1.13.m5.2.2.2.2.2.4" xref="Thmproofx1.p1.13.m5.2.2.2.2.2.3.cmml"><mi id="Thmproofx1.p1.13.m5.1.1.1.1.1.1" mathvariant="normal" xref="Thmproofx1.p1.13.m5.1.1.1.1.1.1.cmml">ℓ</mi><mo id="Thmproofx1.p1.13.m5.2.2.2.2.2.4.1" xref="Thmproofx1.p1.13.m5.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx1.p1.13.m5.2.2.2.2.2.2" xref="Thmproofx1.p1.13.m5.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mrow id="Thmproofx1.p1.13.m5.3.3.3" xref="Thmproofx1.p1.13.m5.3.3.3.cmml"><mrow id="Thmproofx1.p1.13.m5.3.3.3.1.1" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.cmml"><mo id="Thmproofx1.p1.13.m5.3.3.3.1.1.2" stretchy="false" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.cmml">(</mo><mrow id="Thmproofx1.p1.13.m5.3.3.3.1.1.1" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.cmml"><mn id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.2" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.2.cmml">1</mn><mo id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.1" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.1.cmml">−</mo><msup id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.cmml"><mi id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.2" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.2.cmml">e</mi><mrow id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.cmml"><mo id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3a" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.cmml">−</mo><mi id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.2" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.2.cmml">x</mi></mrow></msup></mrow><mo id="Thmproofx1.p1.13.m5.3.3.3.1.1.3" stretchy="false" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.cmml">)</mo></mrow><mo id="Thmproofx1.p1.13.m5.3.3.3.2" xref="Thmproofx1.p1.13.m5.3.3.3.2.cmml">⁢</mo><mi id="Thmproofx1.p1.13.m5.3.3.3.3" mathvariant="normal" xref="Thmproofx1.p1.13.m5.3.3.3.3.cmml">ℓ</mi></mrow></mfrac></mrow><mo id="Thmproofx1.p1.13.m5.3.4.1" xref="Thmproofx1.p1.13.m5.3.4.1.cmml">=</mo><mn id="Thmproofx1.p1.13.m5.3.4.3" xref="Thmproofx1.p1.13.m5.3.4.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.13.m5.3b"><apply id="Thmproofx1.p1.13.m5.3.4.cmml" xref="Thmproofx1.p1.13.m5.3.4"><eq id="Thmproofx1.p1.13.m5.3.4.1.cmml" xref="Thmproofx1.p1.13.m5.3.4.1"></eq><apply id="Thmproofx1.p1.13.m5.3.4.2.cmml" xref="Thmproofx1.p1.13.m5.3.4.2"><apply id="Thmproofx1.p1.13.m5.3.4.2.1.cmml" xref="Thmproofx1.p1.13.m5.3.4.2.1"><csymbol cd="ambiguous" id="Thmproofx1.p1.13.m5.3.4.2.1.1.cmml" xref="Thmproofx1.p1.13.m5.3.4.2.1">subscript</csymbol><limit id="Thmproofx1.p1.13.m5.3.4.2.1.2.cmml" xref="Thmproofx1.p1.13.m5.3.4.2.1.2"></limit><apply id="Thmproofx1.p1.13.m5.3.4.2.1.3.cmml" xref="Thmproofx1.p1.13.m5.3.4.2.1.3"><ci id="Thmproofx1.p1.13.m5.3.4.2.1.3.1.cmml" xref="Thmproofx1.p1.13.m5.3.4.2.1.3.1">→</ci><ci id="Thmproofx1.p1.13.m5.3.4.2.1.3.2.cmml" xref="Thmproofx1.p1.13.m5.3.4.2.1.3.2">ℓ</ci><infinity id="Thmproofx1.p1.13.m5.3.4.2.1.3.3.cmml" xref="Thmproofx1.p1.13.m5.3.4.2.1.3.3"></infinity></apply></apply><apply id="Thmproofx1.p1.13.m5.3.3.cmml" xref="Thmproofx1.p1.13.m5.3.3"><divide id="Thmproofx1.p1.13.m5.3.3.4.cmml" xref="Thmproofx1.p1.13.m5.3.3"></divide><apply id="Thmproofx1.p1.13.m5.2.2.2.cmml" xref="Thmproofx1.p1.13.m5.2.2.2"><csymbol cd="ambiguous" id="Thmproofx1.p1.13.m5.2.2.2.3.cmml" xref="Thmproofx1.p1.13.m5.2.2.2">subscript</csymbol><ci id="Thmproofx1.p1.13.m5.2.2.2.4.cmml" xref="Thmproofx1.p1.13.m5.2.2.2.4">ℰ</ci><list id="Thmproofx1.p1.13.m5.2.2.2.2.2.3.cmml" xref="Thmproofx1.p1.13.m5.2.2.2.2.2.4"><ci id="Thmproofx1.p1.13.m5.1.1.1.1.1.1.cmml" xref="Thmproofx1.p1.13.m5.1.1.1.1.1.1">ℓ</ci><ci id="Thmproofx1.p1.13.m5.2.2.2.2.2.2.cmml" xref="Thmproofx1.p1.13.m5.2.2.2.2.2.2">𝑛</ci></list></apply><apply id="Thmproofx1.p1.13.m5.3.3.3.cmml" xref="Thmproofx1.p1.13.m5.3.3.3"><times id="Thmproofx1.p1.13.m5.3.3.3.2.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.2"></times><apply id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1"><minus id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.1.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.1"></minus><cn id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.2.cmml" type="integer" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.2">1</cn><apply id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.1.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3">superscript</csymbol><ci id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.2.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.2">𝑒</ci><apply id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3"><minus id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.1.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3"></minus><ci id="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.2.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.1.1.1.3.3.2">𝑥</ci></apply></apply></apply><ci id="Thmproofx1.p1.13.m5.3.3.3.3.cmml" xref="Thmproofx1.p1.13.m5.3.3.3.3">ℓ</ci></apply></apply></apply><cn id="Thmproofx1.p1.13.m5.3.4.3.cmml" type="integer" xref="Thmproofx1.p1.13.m5.3.4.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.13.m5.3c">\lim_{\ell\to\infty}\frac{\mathcal{E}_{\ell,n}}{(1-e^{-x})\ell}=1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.13.m5.3d">roman_lim start_POSTSUBSCRIPT roman_ℓ → ∞ end_POSTSUBSCRIPT divide start_ARG caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT end_ARG start_ARG ( 1 - italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT ) roman_ℓ end_ARG = 1</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="Pt0.A1.p2"> <p class="ltx_p" id="Pt0.A1.p2.5">This means that unless <math alttext="n=x\ell+o(\ell)" class="ltx_Math" display="inline" id="Pt0.A1.p2.1.m1.1"><semantics id="Pt0.A1.p2.1.m1.1a"><mrow id="Pt0.A1.p2.1.m1.1.2" xref="Pt0.A1.p2.1.m1.1.2.cmml"><mi id="Pt0.A1.p2.1.m1.1.2.2" xref="Pt0.A1.p2.1.m1.1.2.2.cmml">n</mi><mo id="Pt0.A1.p2.1.m1.1.2.1" xref="Pt0.A1.p2.1.m1.1.2.1.cmml">=</mo><mrow id="Pt0.A1.p2.1.m1.1.2.3" xref="Pt0.A1.p2.1.m1.1.2.3.cmml"><mrow id="Pt0.A1.p2.1.m1.1.2.3.2" xref="Pt0.A1.p2.1.m1.1.2.3.2.cmml"><mi id="Pt0.A1.p2.1.m1.1.2.3.2.2" xref="Pt0.A1.p2.1.m1.1.2.3.2.2.cmml">x</mi><mo id="Pt0.A1.p2.1.m1.1.2.3.2.1" xref="Pt0.A1.p2.1.m1.1.2.3.2.1.cmml">⁢</mo><mi id="Pt0.A1.p2.1.m1.1.2.3.2.3" mathvariant="normal" xref="Pt0.A1.p2.1.m1.1.2.3.2.3.cmml">ℓ</mi></mrow><mo id="Pt0.A1.p2.1.m1.1.2.3.1" xref="Pt0.A1.p2.1.m1.1.2.3.1.cmml">+</mo><mrow id="Pt0.A1.p2.1.m1.1.2.3.3" xref="Pt0.A1.p2.1.m1.1.2.3.3.cmml"><mi id="Pt0.A1.p2.1.m1.1.2.3.3.2" xref="Pt0.A1.p2.1.m1.1.2.3.3.2.cmml">o</mi><mo id="Pt0.A1.p2.1.m1.1.2.3.3.1" xref="Pt0.A1.p2.1.m1.1.2.3.3.1.cmml">⁢</mo><mrow id="Pt0.A1.p2.1.m1.1.2.3.3.3.2" xref="Pt0.A1.p2.1.m1.1.2.3.3.cmml"><mo id="Pt0.A1.p2.1.m1.1.2.3.3.3.2.1" stretchy="false" xref="Pt0.A1.p2.1.m1.1.2.3.3.cmml">(</mo><mi id="Pt0.A1.p2.1.m1.1.1" mathvariant="normal" xref="Pt0.A1.p2.1.m1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.p2.1.m1.1.2.3.3.3.2.2" stretchy="false" xref="Pt0.A1.p2.1.m1.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.p2.1.m1.1b"><apply id="Pt0.A1.p2.1.m1.1.2.cmml" xref="Pt0.A1.p2.1.m1.1.2"><eq id="Pt0.A1.p2.1.m1.1.2.1.cmml" xref="Pt0.A1.p2.1.m1.1.2.1"></eq><ci id="Pt0.A1.p2.1.m1.1.2.2.cmml" xref="Pt0.A1.p2.1.m1.1.2.2">𝑛</ci><apply id="Pt0.A1.p2.1.m1.1.2.3.cmml" xref="Pt0.A1.p2.1.m1.1.2.3"><plus id="Pt0.A1.p2.1.m1.1.2.3.1.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.1"></plus><apply id="Pt0.A1.p2.1.m1.1.2.3.2.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.2"><times id="Pt0.A1.p2.1.m1.1.2.3.2.1.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.2.1"></times><ci id="Pt0.A1.p2.1.m1.1.2.3.2.2.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.2.2">𝑥</ci><ci id="Pt0.A1.p2.1.m1.1.2.3.2.3.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.2.3">ℓ</ci></apply><apply id="Pt0.A1.p2.1.m1.1.2.3.3.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.3"><times id="Pt0.A1.p2.1.m1.1.2.3.3.1.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.3.1"></times><ci id="Pt0.A1.p2.1.m1.1.2.3.3.2.cmml" xref="Pt0.A1.p2.1.m1.1.2.3.3.2">𝑜</ci><ci id="Pt0.A1.p2.1.m1.1.1.cmml" xref="Pt0.A1.p2.1.m1.1.1">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p2.1.m1.1c">n=x\ell+o(\ell)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p2.1.m1.1d">italic_n = italic_x roman_ℓ + italic_o ( roman_ℓ )</annotation></semantics></math>, then <math alttext="\mathcal{E}_{\ell,n}=(1+o(1))\min(\ell,n)" class="ltx_Math" display="inline" id="Pt0.A1.p2.2.m2.7"><semantics id="Pt0.A1.p2.2.m2.7a"><mrow id="Pt0.A1.p2.2.m2.7.7" xref="Pt0.A1.p2.2.m2.7.7.cmml"><msub id="Pt0.A1.p2.2.m2.7.7.3" xref="Pt0.A1.p2.2.m2.7.7.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.p2.2.m2.7.7.3.2" xref="Pt0.A1.p2.2.m2.7.7.3.2.cmml">ℰ</mi><mrow id="Pt0.A1.p2.2.m2.2.2.2.4" xref="Pt0.A1.p2.2.m2.2.2.2.3.cmml"><mi id="Pt0.A1.p2.2.m2.1.1.1.1" mathvariant="normal" xref="Pt0.A1.p2.2.m2.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.p2.2.m2.2.2.2.4.1" xref="Pt0.A1.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.p2.2.m2.2.2.2.2" xref="Pt0.A1.p2.2.m2.2.2.2.2.cmml">n</mi></mrow></msub><mo id="Pt0.A1.p2.2.m2.7.7.2" xref="Pt0.A1.p2.2.m2.7.7.2.cmml">=</mo><mrow id="Pt0.A1.p2.2.m2.7.7.1" xref="Pt0.A1.p2.2.m2.7.7.1.cmml"><mrow id="Pt0.A1.p2.2.m2.7.7.1.1.1" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.cmml"><mo id="Pt0.A1.p2.2.m2.7.7.1.1.1.2" stretchy="false" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.cmml">(</mo><mrow id="Pt0.A1.p2.2.m2.7.7.1.1.1.1" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.cmml"><mn id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.2" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.1" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.1.cmml">+</mo><mrow id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.cmml"><mi id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.2" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.2.cmml">o</mi><mo id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.1" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.1.cmml">⁢</mo><mrow id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.3.2" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.cmml"><mo id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.3.2.1" stretchy="false" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.cmml">(</mo><mn id="Pt0.A1.p2.2.m2.3.3" xref="Pt0.A1.p2.2.m2.3.3.cmml">1</mn><mo id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.3.2.2" stretchy="false" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A1.p2.2.m2.7.7.1.1.1.3" stretchy="false" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A1.p2.2.m2.7.7.1.2" lspace="0.167em" xref="Pt0.A1.p2.2.m2.7.7.1.2.cmml">⁢</mo><mrow id="Pt0.A1.p2.2.m2.7.7.1.3.2" xref="Pt0.A1.p2.2.m2.7.7.1.3.1.cmml"><mi id="Pt0.A1.p2.2.m2.4.4" xref="Pt0.A1.p2.2.m2.4.4.cmml">min</mi><mo id="Pt0.A1.p2.2.m2.7.7.1.3.2a" xref="Pt0.A1.p2.2.m2.7.7.1.3.1.cmml">⁡</mo><mrow id="Pt0.A1.p2.2.m2.7.7.1.3.2.1" xref="Pt0.A1.p2.2.m2.7.7.1.3.1.cmml"><mo id="Pt0.A1.p2.2.m2.7.7.1.3.2.1.1" stretchy="false" xref="Pt0.A1.p2.2.m2.7.7.1.3.1.cmml">(</mo><mi id="Pt0.A1.p2.2.m2.5.5" mathvariant="normal" xref="Pt0.A1.p2.2.m2.5.5.cmml">ℓ</mi><mo id="Pt0.A1.p2.2.m2.7.7.1.3.2.1.2" xref="Pt0.A1.p2.2.m2.7.7.1.3.1.cmml">,</mo><mi id="Pt0.A1.p2.2.m2.6.6" xref="Pt0.A1.p2.2.m2.6.6.cmml">n</mi><mo id="Pt0.A1.p2.2.m2.7.7.1.3.2.1.3" stretchy="false" xref="Pt0.A1.p2.2.m2.7.7.1.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.p2.2.m2.7b"><apply id="Pt0.A1.p2.2.m2.7.7.cmml" xref="Pt0.A1.p2.2.m2.7.7"><eq id="Pt0.A1.p2.2.m2.7.7.2.cmml" xref="Pt0.A1.p2.2.m2.7.7.2"></eq><apply id="Pt0.A1.p2.2.m2.7.7.3.cmml" xref="Pt0.A1.p2.2.m2.7.7.3"><csymbol cd="ambiguous" id="Pt0.A1.p2.2.m2.7.7.3.1.cmml" xref="Pt0.A1.p2.2.m2.7.7.3">subscript</csymbol><ci id="Pt0.A1.p2.2.m2.7.7.3.2.cmml" xref="Pt0.A1.p2.2.m2.7.7.3.2">ℰ</ci><list id="Pt0.A1.p2.2.m2.2.2.2.3.cmml" xref="Pt0.A1.p2.2.m2.2.2.2.4"><ci id="Pt0.A1.p2.2.m2.1.1.1.1.cmml" xref="Pt0.A1.p2.2.m2.1.1.1.1">ℓ</ci><ci id="Pt0.A1.p2.2.m2.2.2.2.2.cmml" xref="Pt0.A1.p2.2.m2.2.2.2.2">𝑛</ci></list></apply><apply id="Pt0.A1.p2.2.m2.7.7.1.cmml" xref="Pt0.A1.p2.2.m2.7.7.1"><times id="Pt0.A1.p2.2.m2.7.7.1.2.cmml" xref="Pt0.A1.p2.2.m2.7.7.1.2"></times><apply id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.cmml" xref="Pt0.A1.p2.2.m2.7.7.1.1.1"><plus id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.1.cmml" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.1"></plus><cn id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.2.cmml" type="integer" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.2">1</cn><apply id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.cmml" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3"><times id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.1.cmml" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.1"></times><ci id="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.2.cmml" xref="Pt0.A1.p2.2.m2.7.7.1.1.1.1.3.2">𝑜</ci><cn id="Pt0.A1.p2.2.m2.3.3.cmml" type="integer" xref="Pt0.A1.p2.2.m2.3.3">1</cn></apply></apply><apply id="Pt0.A1.p2.2.m2.7.7.1.3.1.cmml" xref="Pt0.A1.p2.2.m2.7.7.1.3.2"><min id="Pt0.A1.p2.2.m2.4.4.cmml" xref="Pt0.A1.p2.2.m2.4.4"></min><ci id="Pt0.A1.p2.2.m2.5.5.cmml" xref="Pt0.A1.p2.2.m2.5.5">ℓ</ci><ci id="Pt0.A1.p2.2.m2.6.6.cmml" xref="Pt0.A1.p2.2.m2.6.6">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p2.2.m2.7c">\mathcal{E}_{\ell,n}=(1+o(1))\min(\ell,n)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p2.2.m2.7d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT = ( 1 + italic_o ( 1 ) ) roman_min ( roman_ℓ , italic_n )</annotation></semantics></math>, where <math alttext="o(1)=o_{\ell,n}(1)" class="ltx_Math" display="inline" id="Pt0.A1.p2.3.m3.4"><semantics id="Pt0.A1.p2.3.m3.4a"><mrow id="Pt0.A1.p2.3.m3.4.5" xref="Pt0.A1.p2.3.m3.4.5.cmml"><mrow id="Pt0.A1.p2.3.m3.4.5.2" xref="Pt0.A1.p2.3.m3.4.5.2.cmml"><mi id="Pt0.A1.p2.3.m3.4.5.2.2" xref="Pt0.A1.p2.3.m3.4.5.2.2.cmml">o</mi><mo id="Pt0.A1.p2.3.m3.4.5.2.1" xref="Pt0.A1.p2.3.m3.4.5.2.1.cmml">⁢</mo><mrow id="Pt0.A1.p2.3.m3.4.5.2.3.2" xref="Pt0.A1.p2.3.m3.4.5.2.cmml"><mo id="Pt0.A1.p2.3.m3.4.5.2.3.2.1" stretchy="false" xref="Pt0.A1.p2.3.m3.4.5.2.cmml">(</mo><mn id="Pt0.A1.p2.3.m3.3.3" xref="Pt0.A1.p2.3.m3.3.3.cmml">1</mn><mo id="Pt0.A1.p2.3.m3.4.5.2.3.2.2" stretchy="false" xref="Pt0.A1.p2.3.m3.4.5.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.p2.3.m3.4.5.1" xref="Pt0.A1.p2.3.m3.4.5.1.cmml">=</mo><mrow id="Pt0.A1.p2.3.m3.4.5.3" xref="Pt0.A1.p2.3.m3.4.5.3.cmml"><msub id="Pt0.A1.p2.3.m3.4.5.3.2" xref="Pt0.A1.p2.3.m3.4.5.3.2.cmml"><mi id="Pt0.A1.p2.3.m3.4.5.3.2.2" xref="Pt0.A1.p2.3.m3.4.5.3.2.2.cmml">o</mi><mrow id="Pt0.A1.p2.3.m3.2.2.2.4" xref="Pt0.A1.p2.3.m3.2.2.2.3.cmml"><mi id="Pt0.A1.p2.3.m3.1.1.1.1" mathvariant="normal" xref="Pt0.A1.p2.3.m3.1.1.1.1.cmml">ℓ</mi><mo id="Pt0.A1.p2.3.m3.2.2.2.4.1" xref="Pt0.A1.p2.3.m3.2.2.2.3.cmml">,</mo><mi id="Pt0.A1.p2.3.m3.2.2.2.2" xref="Pt0.A1.p2.3.m3.2.2.2.2.cmml">n</mi></mrow></msub><mo id="Pt0.A1.p2.3.m3.4.5.3.1" xref="Pt0.A1.p2.3.m3.4.5.3.1.cmml">⁢</mo><mrow id="Pt0.A1.p2.3.m3.4.5.3.3.2" xref="Pt0.A1.p2.3.m3.4.5.3.cmml"><mo id="Pt0.A1.p2.3.m3.4.5.3.3.2.1" stretchy="false" xref="Pt0.A1.p2.3.m3.4.5.3.cmml">(</mo><mn id="Pt0.A1.p2.3.m3.4.4" xref="Pt0.A1.p2.3.m3.4.4.cmml">1</mn><mo id="Pt0.A1.p2.3.m3.4.5.3.3.2.2" stretchy="false" xref="Pt0.A1.p2.3.m3.4.5.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.p2.3.m3.4b"><apply id="Pt0.A1.p2.3.m3.4.5.cmml" xref="Pt0.A1.p2.3.m3.4.5"><eq id="Pt0.A1.p2.3.m3.4.5.1.cmml" xref="Pt0.A1.p2.3.m3.4.5.1"></eq><apply id="Pt0.A1.p2.3.m3.4.5.2.cmml" xref="Pt0.A1.p2.3.m3.4.5.2"><times id="Pt0.A1.p2.3.m3.4.5.2.1.cmml" xref="Pt0.A1.p2.3.m3.4.5.2.1"></times><ci id="Pt0.A1.p2.3.m3.4.5.2.2.cmml" xref="Pt0.A1.p2.3.m3.4.5.2.2">𝑜</ci><cn id="Pt0.A1.p2.3.m3.3.3.cmml" type="integer" xref="Pt0.A1.p2.3.m3.3.3">1</cn></apply><apply id="Pt0.A1.p2.3.m3.4.5.3.cmml" xref="Pt0.A1.p2.3.m3.4.5.3"><times id="Pt0.A1.p2.3.m3.4.5.3.1.cmml" xref="Pt0.A1.p2.3.m3.4.5.3.1"></times><apply id="Pt0.A1.p2.3.m3.4.5.3.2.cmml" xref="Pt0.A1.p2.3.m3.4.5.3.2"><csymbol cd="ambiguous" id="Pt0.A1.p2.3.m3.4.5.3.2.1.cmml" xref="Pt0.A1.p2.3.m3.4.5.3.2">subscript</csymbol><ci id="Pt0.A1.p2.3.m3.4.5.3.2.2.cmml" xref="Pt0.A1.p2.3.m3.4.5.3.2.2">𝑜</ci><list id="Pt0.A1.p2.3.m3.2.2.2.3.cmml" xref="Pt0.A1.p2.3.m3.2.2.2.4"><ci id="Pt0.A1.p2.3.m3.1.1.1.1.cmml" xref="Pt0.A1.p2.3.m3.1.1.1.1">ℓ</ci><ci id="Pt0.A1.p2.3.m3.2.2.2.2.cmml" xref="Pt0.A1.p2.3.m3.2.2.2.2">𝑛</ci></list></apply><cn id="Pt0.A1.p2.3.m3.4.4.cmml" type="integer" xref="Pt0.A1.p2.3.m3.4.4">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p2.3.m3.4c">o(1)=o_{\ell,n}(1)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p2.3.m3.4d">italic_o ( 1 ) = italic_o start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT ( 1 )</annotation></semantics></math>. If <math alttext="n=x\ell+o(\ell)" class="ltx_Math" display="inline" id="Pt0.A1.p2.4.m4.1"><semantics id="Pt0.A1.p2.4.m4.1a"><mrow id="Pt0.A1.p2.4.m4.1.2" xref="Pt0.A1.p2.4.m4.1.2.cmml"><mi id="Pt0.A1.p2.4.m4.1.2.2" xref="Pt0.A1.p2.4.m4.1.2.2.cmml">n</mi><mo id="Pt0.A1.p2.4.m4.1.2.1" xref="Pt0.A1.p2.4.m4.1.2.1.cmml">=</mo><mrow id="Pt0.A1.p2.4.m4.1.2.3" xref="Pt0.A1.p2.4.m4.1.2.3.cmml"><mrow id="Pt0.A1.p2.4.m4.1.2.3.2" xref="Pt0.A1.p2.4.m4.1.2.3.2.cmml"><mi id="Pt0.A1.p2.4.m4.1.2.3.2.2" xref="Pt0.A1.p2.4.m4.1.2.3.2.2.cmml">x</mi><mo id="Pt0.A1.p2.4.m4.1.2.3.2.1" xref="Pt0.A1.p2.4.m4.1.2.3.2.1.cmml">⁢</mo><mi id="Pt0.A1.p2.4.m4.1.2.3.2.3" mathvariant="normal" xref="Pt0.A1.p2.4.m4.1.2.3.2.3.cmml">ℓ</mi></mrow><mo id="Pt0.A1.p2.4.m4.1.2.3.1" xref="Pt0.A1.p2.4.m4.1.2.3.1.cmml">+</mo><mrow id="Pt0.A1.p2.4.m4.1.2.3.3" xref="Pt0.A1.p2.4.m4.1.2.3.3.cmml"><mi id="Pt0.A1.p2.4.m4.1.2.3.3.2" xref="Pt0.A1.p2.4.m4.1.2.3.3.2.cmml">o</mi><mo id="Pt0.A1.p2.4.m4.1.2.3.3.1" xref="Pt0.A1.p2.4.m4.1.2.3.3.1.cmml">⁢</mo><mrow id="Pt0.A1.p2.4.m4.1.2.3.3.3.2" xref="Pt0.A1.p2.4.m4.1.2.3.3.cmml"><mo id="Pt0.A1.p2.4.m4.1.2.3.3.3.2.1" stretchy="false" xref="Pt0.A1.p2.4.m4.1.2.3.3.cmml">(</mo><mi id="Pt0.A1.p2.4.m4.1.1" mathvariant="normal" xref="Pt0.A1.p2.4.m4.1.1.cmml">ℓ</mi><mo id="Pt0.A1.p2.4.m4.1.2.3.3.3.2.2" stretchy="false" xref="Pt0.A1.p2.4.m4.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.p2.4.m4.1b"><apply id="Pt0.A1.p2.4.m4.1.2.cmml" xref="Pt0.A1.p2.4.m4.1.2"><eq id="Pt0.A1.p2.4.m4.1.2.1.cmml" xref="Pt0.A1.p2.4.m4.1.2.1"></eq><ci id="Pt0.A1.p2.4.m4.1.2.2.cmml" xref="Pt0.A1.p2.4.m4.1.2.2">𝑛</ci><apply id="Pt0.A1.p2.4.m4.1.2.3.cmml" xref="Pt0.A1.p2.4.m4.1.2.3"><plus id="Pt0.A1.p2.4.m4.1.2.3.1.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.1"></plus><apply id="Pt0.A1.p2.4.m4.1.2.3.2.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.2"><times id="Pt0.A1.p2.4.m4.1.2.3.2.1.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.2.1"></times><ci id="Pt0.A1.p2.4.m4.1.2.3.2.2.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.2.2">𝑥</ci><ci id="Pt0.A1.p2.4.m4.1.2.3.2.3.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.2.3">ℓ</ci></apply><apply id="Pt0.A1.p2.4.m4.1.2.3.3.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.3"><times id="Pt0.A1.p2.4.m4.1.2.3.3.1.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.3.1"></times><ci id="Pt0.A1.p2.4.m4.1.2.3.3.2.cmml" xref="Pt0.A1.p2.4.m4.1.2.3.3.2">𝑜</ci><ci id="Pt0.A1.p2.4.m4.1.1.cmml" xref="Pt0.A1.p2.4.m4.1.1">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p2.4.m4.1c">n=x\ell+o(\ell)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p2.4.m4.1d">italic_n = italic_x roman_ℓ + italic_o ( roman_ℓ )</annotation></semantics></math>, then <math alttext="\mathcal{E}_{\ell,n}=(1+o(1))(1-e^{-x})\ell\in\left[(1-e^{-1})\ell,\ell\right]" class="ltx_Math" display="inline" id="Pt0.A1.p2.5.m5.7"><semantics id="Pt0.A1.p2.5.m5.7a"><mrow id="Pt0.A1.p2.5.m5.7.7" xref="Pt0.A1.p2.5.m5.7.7.cmml"><msub 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xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.1.cmml">+</mo><mrow id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.cmml"><mi id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.2" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.2.cmml">o</mi><mo id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.1" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.1.cmml">⁢</mo><mrow id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.3.2" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.cmml"><mo id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.3.2.1" stretchy="false" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.cmml">(</mo><mn id="Pt0.A1.p2.5.m5.3.3" xref="Pt0.A1.p2.5.m5.3.3.cmml">1</mn><mo id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.3.2.2" stretchy="false" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A1.p2.5.m5.5.5.1.1.1.3" stretchy="false" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A1.p2.5.m5.6.6.2.3" xref="Pt0.A1.p2.5.m5.6.6.2.3.cmml">⁢</mo><mrow id="Pt0.A1.p2.5.m5.6.6.2.2.1" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.cmml"><mo id="Pt0.A1.p2.5.m5.6.6.2.2.1.2" stretchy="false" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.cmml">(</mo><mrow id="Pt0.A1.p2.5.m5.6.6.2.2.1.1" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.cmml"><mn id="Pt0.A1.p2.5.m5.6.6.2.2.1.1.2" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.2.cmml">1</mn><mo id="Pt0.A1.p2.5.m5.6.6.2.2.1.1.1" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.1.cmml">−</mo><msup id="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.cmml"><mi id="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.2" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.2.cmml">e</mi><mrow id="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.3" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.3.cmml"><mo id="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.3a" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.3.cmml">−</mo><mi id="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.3.2" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.3.3.2.cmml">x</mi></mrow></msup></mrow><mo id="Pt0.A1.p2.5.m5.6.6.2.2.1.3" stretchy="false" xref="Pt0.A1.p2.5.m5.6.6.2.2.1.1.cmml">)</mo></mrow><mo id="Pt0.A1.p2.5.m5.6.6.2.3a" xref="Pt0.A1.p2.5.m5.6.6.2.3.cmml">⁢</mo><mi id="Pt0.A1.p2.5.m5.6.6.2.4" mathvariant="normal" xref="Pt0.A1.p2.5.m5.6.6.2.4.cmml">ℓ</mi></mrow><mo id="Pt0.A1.p2.5.m5.7.7.7" xref="Pt0.A1.p2.5.m5.7.7.7.cmml">∈</mo><mrow id="Pt0.A1.p2.5.m5.7.7.3.1" xref="Pt0.A1.p2.5.m5.7.7.3.2.cmml"><mo id="Pt0.A1.p2.5.m5.7.7.3.1.2" xref="Pt0.A1.p2.5.m5.7.7.3.2.cmml">[</mo><mrow id="Pt0.A1.p2.5.m5.7.7.3.1.1" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.cmml"><mrow id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.cmml"><mo id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.2" stretchy="false" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.cmml"><mn id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.2" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.1" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.1.cmml">−</mo><msup id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.cmml"><mi id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.2" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.2.cmml">e</mi><mrow 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id="Pt0.A1.p2.5.m5.7.7a.cmml" xref="Pt0.A1.p2.5.m5.7.7"></and><apply id="Pt0.A1.p2.5.m5.7.7b.cmml" xref="Pt0.A1.p2.5.m5.7.7"><eq id="Pt0.A1.p2.5.m5.7.7.6.cmml" xref="Pt0.A1.p2.5.m5.7.7.6"></eq><apply id="Pt0.A1.p2.5.m5.7.7.5.cmml" xref="Pt0.A1.p2.5.m5.7.7.5"><csymbol cd="ambiguous" id="Pt0.A1.p2.5.m5.7.7.5.1.cmml" xref="Pt0.A1.p2.5.m5.7.7.5">subscript</csymbol><ci id="Pt0.A1.p2.5.m5.7.7.5.2.cmml" xref="Pt0.A1.p2.5.m5.7.7.5.2">ℰ</ci><list id="Pt0.A1.p2.5.m5.2.2.2.3.cmml" xref="Pt0.A1.p2.5.m5.2.2.2.4"><ci id="Pt0.A1.p2.5.m5.1.1.1.1.cmml" xref="Pt0.A1.p2.5.m5.1.1.1.1">ℓ</ci><ci id="Pt0.A1.p2.5.m5.2.2.2.2.cmml" xref="Pt0.A1.p2.5.m5.2.2.2.2">𝑛</ci></list></apply><apply id="Pt0.A1.p2.5.m5.6.6.2.cmml" xref="Pt0.A1.p2.5.m5.6.6.2"><times id="Pt0.A1.p2.5.m5.6.6.2.3.cmml" xref="Pt0.A1.p2.5.m5.6.6.2.3"></times><apply id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.cmml" xref="Pt0.A1.p2.5.m5.5.5.1.1.1"><plus id="Pt0.A1.p2.5.m5.5.5.1.1.1.1.1.cmml" xref="Pt0.A1.p2.5.m5.5.5.1.1.1.1.1"></plus><cn 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xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1"><minus id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.1.cmml" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.1"></minus><cn id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.2.cmml" type="integer" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.2">1</cn><apply id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.cmml" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.1.cmml" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3">superscript</csymbol><ci id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.2.cmml" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.2">𝑒</ci><apply id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.3.cmml" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.3"><minus id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.3.1.cmml" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.3"></minus><cn id="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.1.1.1.3.3.2">1</cn></apply></apply></apply><ci id="Pt0.A1.p2.5.m5.7.7.3.1.1.3.cmml" xref="Pt0.A1.p2.5.m5.7.7.3.1.1.3">ℓ</ci></apply><ci id="Pt0.A1.p2.5.m5.4.4.cmml" xref="Pt0.A1.p2.5.m5.4.4">ℓ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p2.5.m5.7c">\mathcal{E}_{\ell,n}=(1+o(1))(1-e^{-x})\ell\in\left[(1-e^{-1})\ell,\ell\right]</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p2.5.m5.7d">caligraphic_E start_POSTSUBSCRIPT roman_ℓ , italic_n end_POSTSUBSCRIPT = ( 1 + italic_o ( 1 ) ) ( 1 - italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT ) roman_ℓ ∈ [ ( 1 - italic_e start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) roman_ℓ , roman_ℓ ]</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Pt0.A1.p3"> <p class="ltx_p" id="Pt0.A1.p3.1">We now prove the lower bound <math alttext="\beta" class="ltx_Math" display="inline" id="Pt0.A1.p3.1.m1.1"><semantics id="Pt0.A1.p3.1.m1.1a"><mi id="Pt0.A1.p3.1.m1.1.1" xref="Pt0.A1.p3.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="Pt0.A1.p3.1.m1.1b"><ci id="Pt0.A1.p3.1.m1.1.1.cmml" xref="Pt0.A1.p3.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.p3.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.p3.1.m1.1d">italic_β</annotation></semantics></math> of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma4" title="Lemma 4 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a>. For that we have the following lemma:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma19"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 19</span></h6> <div class="ltx_para" id="Thmlemma19.p1"> <p class="ltx_p" id="Thmlemma19.p1.2"><span class="ltx_text ltx_font_italic" id="Thmlemma19.p1.2.2">Let <math alttext="p\in P\subseteq P_{st}" class="ltx_Math" display="inline" id="Thmlemma19.p1.1.1.m1.1"><semantics id="Thmlemma19.p1.1.1.m1.1a"><mrow id="Thmlemma19.p1.1.1.m1.1.1" xref="Thmlemma19.p1.1.1.m1.1.1.cmml"><mi id="Thmlemma19.p1.1.1.m1.1.1.2" xref="Thmlemma19.p1.1.1.m1.1.1.2.cmml">p</mi><mo id="Thmlemma19.p1.1.1.m1.1.1.3" xref="Thmlemma19.p1.1.1.m1.1.1.3.cmml">∈</mo><mi id="Thmlemma19.p1.1.1.m1.1.1.4" xref="Thmlemma19.p1.1.1.m1.1.1.4.cmml">P</mi><mo id="Thmlemma19.p1.1.1.m1.1.1.5" xref="Thmlemma19.p1.1.1.m1.1.1.5.cmml">⊆</mo><msub id="Thmlemma19.p1.1.1.m1.1.1.6" xref="Thmlemma19.p1.1.1.m1.1.1.6.cmml"><mi id="Thmlemma19.p1.1.1.m1.1.1.6.2" xref="Thmlemma19.p1.1.1.m1.1.1.6.2.cmml">P</mi><mrow id="Thmlemma19.p1.1.1.m1.1.1.6.3" xref="Thmlemma19.p1.1.1.m1.1.1.6.3.cmml"><mi id="Thmlemma19.p1.1.1.m1.1.1.6.3.2" xref="Thmlemma19.p1.1.1.m1.1.1.6.3.2.cmml">s</mi><mo id="Thmlemma19.p1.1.1.m1.1.1.6.3.1" xref="Thmlemma19.p1.1.1.m1.1.1.6.3.1.cmml">⁢</mo><mi id="Thmlemma19.p1.1.1.m1.1.1.6.3.3" xref="Thmlemma19.p1.1.1.m1.1.1.6.3.3.cmml">t</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma19.p1.1.1.m1.1b"><apply id="Thmlemma19.p1.1.1.m1.1.1.cmml" xref="Thmlemma19.p1.1.1.m1.1.1"><and id="Thmlemma19.p1.1.1.m1.1.1a.cmml" xref="Thmlemma19.p1.1.1.m1.1.1"></and><apply id="Thmlemma19.p1.1.1.m1.1.1b.cmml" xref="Thmlemma19.p1.1.1.m1.1.1"><in id="Thmlemma19.p1.1.1.m1.1.1.3.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.3"></in><ci id="Thmlemma19.p1.1.1.m1.1.1.2.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.2">𝑝</ci><ci id="Thmlemma19.p1.1.1.m1.1.1.4.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.4">𝑃</ci></apply><apply id="Thmlemma19.p1.1.1.m1.1.1c.cmml" xref="Thmlemma19.p1.1.1.m1.1.1"><subset id="Thmlemma19.p1.1.1.m1.1.1.5.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.5"></subset><share href="https://arxiv.org/html/2308.08047v3#Thmlemma19.p1.1.1.m1.1.1.4.cmml" id="Thmlemma19.p1.1.1.m1.1.1d.cmml" xref="Thmlemma19.p1.1.1.m1.1.1"></share><apply id="Thmlemma19.p1.1.1.m1.1.1.6.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.6"><csymbol cd="ambiguous" id="Thmlemma19.p1.1.1.m1.1.1.6.1.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.6">subscript</csymbol><ci id="Thmlemma19.p1.1.1.m1.1.1.6.2.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.6.2">𝑃</ci><apply id="Thmlemma19.p1.1.1.m1.1.1.6.3.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.6.3"><times id="Thmlemma19.p1.1.1.m1.1.1.6.3.1.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.6.3.1"></times><ci id="Thmlemma19.p1.1.1.m1.1.1.6.3.2.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.6.3.2">𝑠</ci><ci id="Thmlemma19.p1.1.1.m1.1.1.6.3.3.cmml" xref="Thmlemma19.p1.1.1.m1.1.1.6.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma19.p1.1.1.m1.1c">p\in P\subseteq P_{st}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma19.p1.1.1.m1.1d">italic_p ∈ italic_P ⊆ italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. The distribution <math alttext="U_{P}^{n}" class="ltx_Math" display="inline" id="Thmlemma19.p1.2.2.m2.1"><semantics id="Thmlemma19.p1.2.2.m2.1a"><msubsup id="Thmlemma19.p1.2.2.m2.1.1" xref="Thmlemma19.p1.2.2.m2.1.1.cmml"><mi id="Thmlemma19.p1.2.2.m2.1.1.2.2" xref="Thmlemma19.p1.2.2.m2.1.1.2.2.cmml">U</mi><mi id="Thmlemma19.p1.2.2.m2.1.1.2.3" xref="Thmlemma19.p1.2.2.m2.1.1.2.3.cmml">P</mi><mi id="Thmlemma19.p1.2.2.m2.1.1.3" xref="Thmlemma19.p1.2.2.m2.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmlemma19.p1.2.2.m2.1b"><apply id="Thmlemma19.p1.2.2.m2.1.1.cmml" xref="Thmlemma19.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmlemma19.p1.2.2.m2.1.1.1.cmml" xref="Thmlemma19.p1.2.2.m2.1.1">superscript</csymbol><apply id="Thmlemma19.p1.2.2.m2.1.1.2.cmml" xref="Thmlemma19.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmlemma19.p1.2.2.m2.1.1.2.1.cmml" xref="Thmlemma19.p1.2.2.m2.1.1">subscript</csymbol><ci id="Thmlemma19.p1.2.2.m2.1.1.2.2.cmml" xref="Thmlemma19.p1.2.2.m2.1.1.2.2">𝑈</ci><ci id="Thmlemma19.p1.2.2.m2.1.1.2.3.cmml" xref="Thmlemma19.p1.2.2.m2.1.1.2.3">𝑃</ci></apply><ci id="Thmlemma19.p1.2.2.m2.1.1.3.cmml" xref="Thmlemma19.p1.2.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma19.p1.2.2.m2.1c">U_{P}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma19.p1.2.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a solution of the maximization problem</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx17"> <tbody id="Pt0.A1.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\max_{\sigma\in(\Delta(P_{st}))^{n}}~{}~{}~{}" class="ltx_Math" display="inline" 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xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.2"></times><ci id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.3">Δ</ci><apply id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3"><times id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="Pt0.A1.Ex16.m1.1.1.1.1.3.cmml" xref="Pt0.A1.Ex16.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex16.m1.1c">\displaystyle\max_{\sigma\in(\Delta(P_{st}))^{n}}~{}~{}~{}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex16.m1.1d">roman_max start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\prod_{j}(1-\sigma_{j}(p))" class="ltx_Math" display="inline" id="Pt0.A1.Ex16.m2.2"><semantics id="Pt0.A1.Ex16.m2.2a"><mrow id="Pt0.A1.Ex16.m2.2.2" xref="Pt0.A1.Ex16.m2.2.2.cmml"><mstyle displaystyle="true" id="Pt0.A1.Ex16.m2.2.2.2" xref="Pt0.A1.Ex16.m2.2.2.2.cmml"><munder id="Pt0.A1.Ex16.m2.2.2.2a" xref="Pt0.A1.Ex16.m2.2.2.2.cmml"><mo id="Pt0.A1.Ex16.m2.2.2.2.2" movablelimits="false" 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href="https://arxiv.org/html/2308.08047v3#Pt0.A1.Ex17.m2.2.2.1.cmml" id="Pt0.A1.Ex17.m2.2.2d.cmml" xref="Pt0.A1.Ex17.m2.2.2"></share><cn id="Pt0.A1.Ex17.m2.2.2.6.cmml" type="integer" xref="Pt0.A1.Ex17.m2.2.2.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex17.m2.2c">\displaystyle\forall j\in[n]~{}~{}~{}\sum_{p^{\prime}\in P}\sigma_{j}(p^{% \prime})=1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex17.m2.2d">∀ italic_j ∈ [ italic_n ] ∑ start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_P end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A1.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td 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{j}(1-\sigma_{j}(p^{\prime}))</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex18.m1.3d">∀ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_P ∏ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p ) ) = ∏ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx2"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx2.p1"> <p class="ltx_p" id="Thmproofx2.p1.5">By the AMGM inequality</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx18"> <tbody id="Pt0.A1.Ex19"><tr 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start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p ) end_ARG start_ARG italic_n end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx2.p1.6">By the second constraint we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx19"> <tbody id="Pt0.A1.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall p^{\prime}\in P~{}~{}~{}\sqrt[n]{\prod_{j}(1-\sigma_{j}(p% ))}\leq\frac{n-\sum_{j}\sigma_{j}(p^{\prime})}{n}" class="ltx_Math" display="inline" id="Pt0.A1.Ex20.m1.3"><semantics id="Pt0.A1.Ex20.m1.3a"><mrow id="Pt0.A1.Ex20.m1.3.4" xref="Pt0.A1.Ex20.m1.3.4.cmml"><mrow id="Pt0.A1.Ex20.m1.3.4.2" xref="Pt0.A1.Ex20.m1.3.4.2.cmml"><mo id="Pt0.A1.Ex20.m1.3.4.2.1" rspace="0.167em" 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xref="Pt0.A1.Ex21.m1.1.1.3">𝑛</ci><apply id="Pt0.A1.Ex21.m1.1.1.1.2.cmml" xref="Pt0.A1.Ex21.m1.1.1.1.3"><abs id="Pt0.A1.Ex21.m1.1.1.1.2.1.cmml" xref="Pt0.A1.Ex21.m1.1.1.1.3.1"></abs><ci id="Pt0.A1.Ex21.m1.1.1.1.1.cmml" xref="Pt0.A1.Ex21.m1.1.1.1.1">𝑃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex21.m1.3c">\displaystyle\sum_{p^{\prime}\in P}\sum_{j}\sigma_{j}(p^{\prime})=n~{}% \Rightarrow~{}\exists p^{\prime}~{}~{}~{}\sum_{j}\sigma_{j}(p^{\prime})\geq% \frac{n}{|P|}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex21.m1.3d">∑ start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_P end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_n ⇒ ∃ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≥ divide start_ARG italic_n end_ARG start_ARG | italic_P | end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx2.p1.1">Therefore for that <math alttext="p^{\prime}" class="ltx_Math" display="inline" id="Thmproofx2.p1.1.m1.1"><semantics id="Thmproofx2.p1.1.m1.1a"><msup id="Thmproofx2.p1.1.m1.1.1" xref="Thmproofx2.p1.1.m1.1.1.cmml"><mi id="Thmproofx2.p1.1.m1.1.1.2" xref="Thmproofx2.p1.1.m1.1.1.2.cmml">p</mi><mo id="Thmproofx2.p1.1.m1.1.1.3" xref="Thmproofx2.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.1.m1.1b"><apply id="Thmproofx2.p1.1.m1.1.1.cmml" xref="Thmproofx2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx2.p1.1.m1.1.1.1.cmml" xref="Thmproofx2.p1.1.m1.1.1">superscript</csymbol><ci id="Thmproofx2.p1.1.m1.1.1.2.cmml" xref="Thmproofx2.p1.1.m1.1.1.2">𝑝</ci><ci id="Thmproofx2.p1.1.m1.1.1.3.cmml" xref="Thmproofx2.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.1.m1.1c">p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.1.m1.1d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx21"> <tbody id="Pt0.A1.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sqrt[n]{\prod_{j}(1-\sigma_{j}(p))}\leq 1-\frac{1}{|P|}% \Rightarrow\prod_{j}(1-\sigma_{j}(p))\leq\left(1-\frac{1}{|P|}\right)^{n}" class="ltx_Math" display="inline" id="Pt0.A1.Ex22.m1.7"><semantics id="Pt0.A1.Ex22.m1.7a"><mrow id="Pt0.A1.Ex22.m1.7.7" xref="Pt0.A1.Ex22.m1.7.7.cmml"><mroot id="Pt0.A1.Ex22.m1.2.2" 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xref="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.1"></times><apply id="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2.cmml" xref="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2"><csymbol cd="ambiguous" id="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2.1.cmml" xref="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2">subscript</csymbol><ci id="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2.2.cmml" xref="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2.2">𝜎</ci><ci id="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2.3.cmml" xref="Pt0.A1.Ex22.m1.6.6.1.1.1.1.3.2.3">𝑗</ci></apply><ci id="Pt0.A1.Ex22.m1.5.5.cmml" xref="Pt0.A1.Ex22.m1.5.5">𝑝</ci></apply></apply></apply></apply><apply id="Pt0.A1.Ex22.m1.7.7e.cmml" xref="Pt0.A1.Ex22.m1.7.7"><leq id="Pt0.A1.Ex22.m1.7.7.7.cmml" xref="Pt0.A1.Ex22.m1.7.7.7"></leq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A1.Ex22.m1.6.6.1.cmml" id="Pt0.A1.Ex22.m1.7.7f.cmml" xref="Pt0.A1.Ex22.m1.7.7"></share><apply id="Pt0.A1.Ex22.m1.7.7.2.cmml" xref="Pt0.A1.Ex22.m1.7.7.2"><csymbol cd="ambiguous" id="Pt0.A1.Ex22.m1.7.7.2.2.cmml" xref="Pt0.A1.Ex22.m1.7.7.2">superscript</csymbol><apply id="Pt0.A1.Ex22.m1.7.7.2.1.1.1.cmml" xref="Pt0.A1.Ex22.m1.7.7.2.1.1"><minus id="Pt0.A1.Ex22.m1.7.7.2.1.1.1.1.cmml" xref="Pt0.A1.Ex22.m1.7.7.2.1.1.1.1"></minus><cn id="Pt0.A1.Ex22.m1.7.7.2.1.1.1.2.cmml" type="integer" xref="Pt0.A1.Ex22.m1.7.7.2.1.1.1.2">1</cn><apply id="Pt0.A1.Ex22.m1.4.4.cmml" xref="Pt0.A1.Ex22.m1.4.4"><divide id="Pt0.A1.Ex22.m1.4.4.2.cmml" xref="Pt0.A1.Ex22.m1.4.4"></divide><cn id="Pt0.A1.Ex22.m1.4.4.3.cmml" type="integer" xref="Pt0.A1.Ex22.m1.4.4.3">1</cn><apply id="Pt0.A1.Ex22.m1.4.4.1.2.cmml" xref="Pt0.A1.Ex22.m1.4.4.1.3"><abs id="Pt0.A1.Ex22.m1.4.4.1.2.1.cmml" xref="Pt0.A1.Ex22.m1.4.4.1.3.1"></abs><ci id="Pt0.A1.Ex22.m1.4.4.1.1.cmml" xref="Pt0.A1.Ex22.m1.4.4.1.1">𝑃</ci></apply></apply></apply><ci id="Pt0.A1.Ex22.m1.7.7.2.3.cmml" xref="Pt0.A1.Ex22.m1.7.7.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex22.m1.7c">\displaystyle\sqrt[n]{\prod_{j}(1-\sigma_{j}(p))}\leq 1-\frac{1}{|P|}% \Rightarrow\prod_{j}(1-\sigma_{j}(p))\leq\left(1-\frac{1}{|P|}\right)^{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex22.m1.7d">nth-root start_ARG italic_n end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p ) ) end_ARG ≤ 1 - divide start_ARG 1 end_ARG start_ARG | italic_P | end_ARG ⇒ ∏ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p ) ) ≤ ( 1 - divide start_ARG 1 end_ARG start_ARG | italic_P | end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx2.p1.4">so <math alttext="\left(1-\frac{1}{|P|}\right)^{n}" class="ltx_Math" display="inline" id="Thmproofx2.p1.2.m1.2"><semantics id="Thmproofx2.p1.2.m1.2a"><msup id="Thmproofx2.p1.2.m1.2.2" xref="Thmproofx2.p1.2.m1.2.2.cmml"><mrow id="Thmproofx2.p1.2.m1.2.2.1.1" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.cmml"><mo id="Thmproofx2.p1.2.m1.2.2.1.1.2" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx2.p1.2.m1.2.2.1.1.1" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.cmml"><mn id="Thmproofx2.p1.2.m1.2.2.1.1.1.2" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.2.cmml">1</mn><mo id="Thmproofx2.p1.2.m1.2.2.1.1.1.1" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx2.p1.2.m1.1.1" xref="Thmproofx2.p1.2.m1.1.1.cmml"><mn id="Thmproofx2.p1.2.m1.1.1.3" xref="Thmproofx2.p1.2.m1.1.1.3.cmml">1</mn><mrow id="Thmproofx2.p1.2.m1.1.1.1.3" xref="Thmproofx2.p1.2.m1.1.1.1.2.cmml"><mo id="Thmproofx2.p1.2.m1.1.1.1.3.1" stretchy="false" xref="Thmproofx2.p1.2.m1.1.1.1.2.1.cmml">|</mo><mi id="Thmproofx2.p1.2.m1.1.1.1.1" xref="Thmproofx2.p1.2.m1.1.1.1.1.cmml">P</mi><mo id="Thmproofx2.p1.2.m1.1.1.1.3.2" stretchy="false" xref="Thmproofx2.p1.2.m1.1.1.1.2.1.cmml">|</mo></mrow></mfrac></mrow><mo id="Thmproofx2.p1.2.m1.2.2.1.1.3" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx2.p1.2.m1.2.2.3" xref="Thmproofx2.p1.2.m1.2.2.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.2.m1.2b"><apply id="Thmproofx2.p1.2.m1.2.2.cmml" xref="Thmproofx2.p1.2.m1.2.2"><csymbol cd="ambiguous" id="Thmproofx2.p1.2.m1.2.2.2.cmml" xref="Thmproofx2.p1.2.m1.2.2">superscript</csymbol><apply id="Thmproofx2.p1.2.m1.2.2.1.1.1.cmml" xref="Thmproofx2.p1.2.m1.2.2.1.1"><minus id="Thmproofx2.p1.2.m1.2.2.1.1.1.1.cmml" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.1"></minus><cn id="Thmproofx2.p1.2.m1.2.2.1.1.1.2.cmml" type="integer" xref="Thmproofx2.p1.2.m1.2.2.1.1.1.2">1</cn><apply id="Thmproofx2.p1.2.m1.1.1.cmml" xref="Thmproofx2.p1.2.m1.1.1"><divide id="Thmproofx2.p1.2.m1.1.1.2.cmml" xref="Thmproofx2.p1.2.m1.1.1"></divide><cn id="Thmproofx2.p1.2.m1.1.1.3.cmml" type="integer" xref="Thmproofx2.p1.2.m1.1.1.3">1</cn><apply id="Thmproofx2.p1.2.m1.1.1.1.2.cmml" xref="Thmproofx2.p1.2.m1.1.1.1.3"><abs id="Thmproofx2.p1.2.m1.1.1.1.2.1.cmml" xref="Thmproofx2.p1.2.m1.1.1.1.3.1"></abs><ci id="Thmproofx2.p1.2.m1.1.1.1.1.cmml" xref="Thmproofx2.p1.2.m1.1.1.1.1">𝑃</ci></apply></apply></apply><ci id="Thmproofx2.p1.2.m1.2.2.3.cmml" xref="Thmproofx2.p1.2.m1.2.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.2.m1.2c">\left(1-\frac{1}{|P|}\right)^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.2.m1.2d">( 1 - divide start_ARG 1 end_ARG start_ARG | italic_P | end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is an upper bound on the maximal value of <math alttext="\prod_{j}(1-\sigma_{j}(p))" class="ltx_Math" display="inline" id="Thmproofx2.p1.3.m2.2"><semantics id="Thmproofx2.p1.3.m2.2a"><mrow id="Thmproofx2.p1.3.m2.2.2" xref="Thmproofx2.p1.3.m2.2.2.cmml"><msub id="Thmproofx2.p1.3.m2.2.2.2" xref="Thmproofx2.p1.3.m2.2.2.2.cmml"><mo id="Thmproofx2.p1.3.m2.2.2.2.2" xref="Thmproofx2.p1.3.m2.2.2.2.2.cmml">∏</mo><mi id="Thmproofx2.p1.3.m2.2.2.2.3" xref="Thmproofx2.p1.3.m2.2.2.2.3.cmml">j</mi></msub><mrow id="Thmproofx2.p1.3.m2.2.2.1.1" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.cmml"><mo id="Thmproofx2.p1.3.m2.2.2.1.1.2" lspace="0em" stretchy="false" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx2.p1.3.m2.2.2.1.1.1" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.cmml"><mn id="Thmproofx2.p1.3.m2.2.2.1.1.1.2" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.2.cmml">1</mn><mo id="Thmproofx2.p1.3.m2.2.2.1.1.1.1" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.1.cmml">−</mo><mrow id="Thmproofx2.p1.3.m2.2.2.1.1.1.3" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.cmml"><msub id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.cmml"><mi id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.2" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.2.cmml">σ</mi><mi id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.3" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.3.cmml">j</mi></msub><mo id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.1" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.1.cmml">⁢</mo><mrow id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.3.2" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.cmml"><mo id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.3.2.1" stretchy="false" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.cmml">(</mo><mi id="Thmproofx2.p1.3.m2.1.1" xref="Thmproofx2.p1.3.m2.1.1.cmml">p</mi><mo id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.3.2.2" stretchy="false" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx2.p1.3.m2.2.2.1.1.3" stretchy="false" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.3.m2.2b"><apply id="Thmproofx2.p1.3.m2.2.2.cmml" xref="Thmproofx2.p1.3.m2.2.2"><apply id="Thmproofx2.p1.3.m2.2.2.2.cmml" xref="Thmproofx2.p1.3.m2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx2.p1.3.m2.2.2.2.1.cmml" xref="Thmproofx2.p1.3.m2.2.2.2">subscript</csymbol><csymbol cd="latexml" id="Thmproofx2.p1.3.m2.2.2.2.2.cmml" xref="Thmproofx2.p1.3.m2.2.2.2.2">product</csymbol><ci id="Thmproofx2.p1.3.m2.2.2.2.3.cmml" xref="Thmproofx2.p1.3.m2.2.2.2.3">𝑗</ci></apply><apply id="Thmproofx2.p1.3.m2.2.2.1.1.1.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1"><minus id="Thmproofx2.p1.3.m2.2.2.1.1.1.1.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.1"></minus><cn id="Thmproofx2.p1.3.m2.2.2.1.1.1.2.cmml" type="integer" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.2">1</cn><apply id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3"><times id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.1.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.1"></times><apply id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2"><csymbol cd="ambiguous" id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.1.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2">subscript</csymbol><ci id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.2.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.2">𝜎</ci><ci id="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.3.cmml" xref="Thmproofx2.p1.3.m2.2.2.1.1.1.3.2.3">𝑗</ci></apply><ci id="Thmproofx2.p1.3.m2.1.1.cmml" xref="Thmproofx2.p1.3.m2.1.1">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.3.m2.2c">\prod_{j}(1-\sigma_{j}(p))</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.3.m2.2d">∏ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p ) )</annotation></semantics></math>, and <math alttext="\sigma=U_{P}^{n}" class="ltx_Math" display="inline" id="Thmproofx2.p1.4.m3.1"><semantics id="Thmproofx2.p1.4.m3.1a"><mrow id="Thmproofx2.p1.4.m3.1.1" xref="Thmproofx2.p1.4.m3.1.1.cmml"><mi id="Thmproofx2.p1.4.m3.1.1.2" xref="Thmproofx2.p1.4.m3.1.1.2.cmml">σ</mi><mo id="Thmproofx2.p1.4.m3.1.1.1" xref="Thmproofx2.p1.4.m3.1.1.1.cmml">=</mo><msubsup id="Thmproofx2.p1.4.m3.1.1.3" xref="Thmproofx2.p1.4.m3.1.1.3.cmml"><mi id="Thmproofx2.p1.4.m3.1.1.3.2.2" xref="Thmproofx2.p1.4.m3.1.1.3.2.2.cmml">U</mi><mi id="Thmproofx2.p1.4.m3.1.1.3.2.3" xref="Thmproofx2.p1.4.m3.1.1.3.2.3.cmml">P</mi><mi id="Thmproofx2.p1.4.m3.1.1.3.3" xref="Thmproofx2.p1.4.m3.1.1.3.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.4.m3.1b"><apply id="Thmproofx2.p1.4.m3.1.1.cmml" xref="Thmproofx2.p1.4.m3.1.1"><eq id="Thmproofx2.p1.4.m3.1.1.1.cmml" xref="Thmproofx2.p1.4.m3.1.1.1"></eq><ci id="Thmproofx2.p1.4.m3.1.1.2.cmml" xref="Thmproofx2.p1.4.m3.1.1.2">𝜎</ci><apply id="Thmproofx2.p1.4.m3.1.1.3.cmml" xref="Thmproofx2.p1.4.m3.1.1.3"><csymbol cd="ambiguous" id="Thmproofx2.p1.4.m3.1.1.3.1.cmml" xref="Thmproofx2.p1.4.m3.1.1.3">superscript</csymbol><apply id="Thmproofx2.p1.4.m3.1.1.3.2.cmml" xref="Thmproofx2.p1.4.m3.1.1.3"><csymbol cd="ambiguous" id="Thmproofx2.p1.4.m3.1.1.3.2.1.cmml" xref="Thmproofx2.p1.4.m3.1.1.3">subscript</csymbol><ci id="Thmproofx2.p1.4.m3.1.1.3.2.2.cmml" xref="Thmproofx2.p1.4.m3.1.1.3.2.2">𝑈</ci><ci id="Thmproofx2.p1.4.m3.1.1.3.2.3.cmml" xref="Thmproofx2.p1.4.m3.1.1.3.2.3">𝑃</ci></apply><ci id="Thmproofx2.p1.4.m3.1.1.3.3.cmml" xref="Thmproofx2.p1.4.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.4.m3.1c">\sigma=U_{P}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.4.m3.1d">italic_σ = italic_U start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> attains this bound. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx3"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma4" title="Lemma 4 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a>)</h6> <div class="ltx_para" id="Thmproofx3.p1"> <p class="ltx_p" id="Thmproofx3.p1.32">Let <math alttext="E^{\prime},P^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p1.1.m1.2"><semantics id="Thmproofx3.p1.1.m1.2a"><mrow id="Thmproofx3.p1.1.m1.2.2.2" xref="Thmproofx3.p1.1.m1.2.2.3.cmml"><msup id="Thmproofx3.p1.1.m1.1.1.1.1" xref="Thmproofx3.p1.1.m1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.1.m1.1.1.1.1.2" xref="Thmproofx3.p1.1.m1.1.1.1.1.2.cmml">E</mi><mo id="Thmproofx3.p1.1.m1.1.1.1.1.3" xref="Thmproofx3.p1.1.m1.1.1.1.1.3.cmml">′</mo></msup><mo id="Thmproofx3.p1.1.m1.2.2.2.3" xref="Thmproofx3.p1.1.m1.2.2.3.cmml">,</mo><msup id="Thmproofx3.p1.1.m1.2.2.2.2" xref="Thmproofx3.p1.1.m1.2.2.2.2.cmml"><mi id="Thmproofx3.p1.1.m1.2.2.2.2.2" xref="Thmproofx3.p1.1.m1.2.2.2.2.2.cmml">P</mi><mo id="Thmproofx3.p1.1.m1.2.2.2.2.3" xref="Thmproofx3.p1.1.m1.2.2.2.2.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.1.m1.2b"><list id="Thmproofx3.p1.1.m1.2.2.3.cmml" xref="Thmproofx3.p1.1.m1.2.2.2"><apply id="Thmproofx3.p1.1.m1.1.1.1.1.cmml" xref="Thmproofx3.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.1.m1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.1.m1.1.1.1.1">superscript</csymbol><ci id="Thmproofx3.p1.1.m1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.1.m1.1.1.1.1.2">𝐸</ci><ci id="Thmproofx3.p1.1.m1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.1.m1.1.1.1.1.3">′</ci></apply><apply id="Thmproofx3.p1.1.m1.2.2.2.2.cmml" xref="Thmproofx3.p1.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.1.m1.2.2.2.2.1.cmml" xref="Thmproofx3.p1.1.m1.2.2.2.2">superscript</csymbol><ci id="Thmproofx3.p1.1.m1.2.2.2.2.2.cmml" xref="Thmproofx3.p1.1.m1.2.2.2.2.2">𝑃</ci><ci id="Thmproofx3.p1.1.m1.2.2.2.2.3.cmml" xref="Thmproofx3.p1.1.m1.2.2.2.2.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.1.m1.2c">E^{\prime},P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.1.m1.2d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the sets of edges, <math alttext="s-t" class="ltx_Math" display="inline" id="Thmproofx3.p1.2.m2.1"><semantics id="Thmproofx3.p1.2.m2.1a"><mrow id="Thmproofx3.p1.2.m2.1.1" xref="Thmproofx3.p1.2.m2.1.1.cmml"><mi id="Thmproofx3.p1.2.m2.1.1.2" xref="Thmproofx3.p1.2.m2.1.1.2.cmml">s</mi><mo id="Thmproofx3.p1.2.m2.1.1.1" xref="Thmproofx3.p1.2.m2.1.1.1.cmml">−</mo><mi id="Thmproofx3.p1.2.m2.1.1.3" xref="Thmproofx3.p1.2.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.2.m2.1b"><apply id="Thmproofx3.p1.2.m2.1.1.cmml" xref="Thmproofx3.p1.2.m2.1.1"><minus id="Thmproofx3.p1.2.m2.1.1.1.cmml" xref="Thmproofx3.p1.2.m2.1.1.1"></minus><ci id="Thmproofx3.p1.2.m2.1.1.2.cmml" xref="Thmproofx3.p1.2.m2.1.1.2">𝑠</ci><ci id="Thmproofx3.p1.2.m2.1.1.3.cmml" xref="Thmproofx3.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.2.m2.1c">s-t</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.2.m2.1d">italic_s - italic_t</annotation></semantics></math> paths of <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p1.3.m3.1"><semantics id="Thmproofx3.p1.3.m3.1a"><msup id="Thmproofx3.p1.3.m3.1.1" xref="Thmproofx3.p1.3.m3.1.1.cmml"><mi id="Thmproofx3.p1.3.m3.1.1.2" xref="Thmproofx3.p1.3.m3.1.1.2.cmml">G</mi><mo id="Thmproofx3.p1.3.m3.1.1.3" xref="Thmproofx3.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.3.m3.1b"><apply id="Thmproofx3.p1.3.m3.1.1.cmml" xref="Thmproofx3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.3.m3.1.1.1.cmml" xref="Thmproofx3.p1.3.m3.1.1">superscript</csymbol><ci id="Thmproofx3.p1.3.m3.1.1.2.cmml" xref="Thmproofx3.p1.3.m3.1.1.2">𝐺</ci><ci id="Thmproofx3.p1.3.m3.1.1.3.cmml" xref="Thmproofx3.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.3.m3.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.3.m3.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Take a maximum flow <math alttext="f" class="ltx_Math" display="inline" id="Thmproofx3.p1.4.m4.1"><semantics id="Thmproofx3.p1.4.m4.1a"><mi id="Thmproofx3.p1.4.m4.1.1" xref="Thmproofx3.p1.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.4.m4.1b"><ci id="Thmproofx3.p1.4.m4.1.1.cmml" xref="Thmproofx3.p1.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.4.m4.1d">italic_f</annotation></semantics></math> of <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p1.5.m5.1"><semantics id="Thmproofx3.p1.5.m5.1a"><msup id="Thmproofx3.p1.5.m5.1.1" xref="Thmproofx3.p1.5.m5.1.1.cmml"><mi id="Thmproofx3.p1.5.m5.1.1.2" xref="Thmproofx3.p1.5.m5.1.1.2.cmml">G</mi><mo id="Thmproofx3.p1.5.m5.1.1.3" xref="Thmproofx3.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.5.m5.1b"><apply id="Thmproofx3.p1.5.m5.1.1.cmml" xref="Thmproofx3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.5.m5.1.1.1.cmml" xref="Thmproofx3.p1.5.m5.1.1">superscript</csymbol><ci id="Thmproofx3.p1.5.m5.1.1.2.cmml" xref="Thmproofx3.p1.5.m5.1.1.2">𝐺</ci><ci id="Thmproofx3.p1.5.m5.1.1.3.cmml" xref="Thmproofx3.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.5.m5.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.5.m5.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="|f|=|P_{f}|=m^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p1.6.m6.2"><semantics id="Thmproofx3.p1.6.m6.2a"><mrow id="Thmproofx3.p1.6.m6.2.2" xref="Thmproofx3.p1.6.m6.2.2.cmml"><mrow id="Thmproofx3.p1.6.m6.2.2.3.2" xref="Thmproofx3.p1.6.m6.2.2.3.1.cmml"><mo id="Thmproofx3.p1.6.m6.2.2.3.2.1" stretchy="false" xref="Thmproofx3.p1.6.m6.2.2.3.1.1.cmml">|</mo><mi id="Thmproofx3.p1.6.m6.1.1" xref="Thmproofx3.p1.6.m6.1.1.cmml">f</mi><mo id="Thmproofx3.p1.6.m6.2.2.3.2.2" stretchy="false" xref="Thmproofx3.p1.6.m6.2.2.3.1.1.cmml">|</mo></mrow><mo id="Thmproofx3.p1.6.m6.2.2.4" xref="Thmproofx3.p1.6.m6.2.2.4.cmml">=</mo><mrow id="Thmproofx3.p1.6.m6.2.2.1.1" xref="Thmproofx3.p1.6.m6.2.2.1.2.cmml"><mo id="Thmproofx3.p1.6.m6.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p1.6.m6.2.2.1.2.1.cmml">|</mo><msub id="Thmproofx3.p1.6.m6.2.2.1.1.1" xref="Thmproofx3.p1.6.m6.2.2.1.1.1.cmml"><mi id="Thmproofx3.p1.6.m6.2.2.1.1.1.2" xref="Thmproofx3.p1.6.m6.2.2.1.1.1.2.cmml">P</mi><mi id="Thmproofx3.p1.6.m6.2.2.1.1.1.3" xref="Thmproofx3.p1.6.m6.2.2.1.1.1.3.cmml">f</mi></msub><mo id="Thmproofx3.p1.6.m6.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p1.6.m6.2.2.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx3.p1.6.m6.2.2.5" xref="Thmproofx3.p1.6.m6.2.2.5.cmml">=</mo><msup id="Thmproofx3.p1.6.m6.2.2.6" xref="Thmproofx3.p1.6.m6.2.2.6.cmml"><mi id="Thmproofx3.p1.6.m6.2.2.6.2" xref="Thmproofx3.p1.6.m6.2.2.6.2.cmml">m</mi><mo id="Thmproofx3.p1.6.m6.2.2.6.3" xref="Thmproofx3.p1.6.m6.2.2.6.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.6.m6.2b"><apply id="Thmproofx3.p1.6.m6.2.2.cmml" xref="Thmproofx3.p1.6.m6.2.2"><and id="Thmproofx3.p1.6.m6.2.2a.cmml" xref="Thmproofx3.p1.6.m6.2.2"></and><apply id="Thmproofx3.p1.6.m6.2.2b.cmml" xref="Thmproofx3.p1.6.m6.2.2"><eq id="Thmproofx3.p1.6.m6.2.2.4.cmml" xref="Thmproofx3.p1.6.m6.2.2.4"></eq><apply id="Thmproofx3.p1.6.m6.2.2.3.1.cmml" xref="Thmproofx3.p1.6.m6.2.2.3.2"><abs id="Thmproofx3.p1.6.m6.2.2.3.1.1.cmml" xref="Thmproofx3.p1.6.m6.2.2.3.2.1"></abs><ci id="Thmproofx3.p1.6.m6.1.1.cmml" xref="Thmproofx3.p1.6.m6.1.1">𝑓</ci></apply><apply id="Thmproofx3.p1.6.m6.2.2.1.2.cmml" xref="Thmproofx3.p1.6.m6.2.2.1.1"><abs id="Thmproofx3.p1.6.m6.2.2.1.2.1.cmml" xref="Thmproofx3.p1.6.m6.2.2.1.1.2"></abs><apply id="Thmproofx3.p1.6.m6.2.2.1.1.1.cmml" xref="Thmproofx3.p1.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.6.m6.2.2.1.1.1.1.cmml" xref="Thmproofx3.p1.6.m6.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.6.m6.2.2.1.1.1.2.cmml" xref="Thmproofx3.p1.6.m6.2.2.1.1.1.2">𝑃</ci><ci id="Thmproofx3.p1.6.m6.2.2.1.1.1.3.cmml" xref="Thmproofx3.p1.6.m6.2.2.1.1.1.3">𝑓</ci></apply></apply></apply><apply id="Thmproofx3.p1.6.m6.2.2c.cmml" xref="Thmproofx3.p1.6.m6.2.2"><eq id="Thmproofx3.p1.6.m6.2.2.5.cmml" xref="Thmproofx3.p1.6.m6.2.2.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx3.p1.6.m6.2.2.1.cmml" id="Thmproofx3.p1.6.m6.2.2d.cmml" xref="Thmproofx3.p1.6.m6.2.2"></share><apply id="Thmproofx3.p1.6.m6.2.2.6.cmml" xref="Thmproofx3.p1.6.m6.2.2.6"><csymbol cd="ambiguous" id="Thmproofx3.p1.6.m6.2.2.6.1.cmml" xref="Thmproofx3.p1.6.m6.2.2.6">superscript</csymbol><ci id="Thmproofx3.p1.6.m6.2.2.6.2.cmml" xref="Thmproofx3.p1.6.m6.2.2.6.2">𝑚</ci><ci id="Thmproofx3.p1.6.m6.2.2.6.3.cmml" xref="Thmproofx3.p1.6.m6.2.2.6.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.6.m6.2c">|f|=|P_{f}|=m^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.6.m6.2d">| italic_f | = | italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | = italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and denote <math alttext="P_{f}=\left\{p_{1},...,p_{m^{\prime}}\right\}" class="ltx_Math" display="inline" id="Thmproofx3.p1.7.m7.3"><semantics id="Thmproofx3.p1.7.m7.3a"><mrow id="Thmproofx3.p1.7.m7.3.3" xref="Thmproofx3.p1.7.m7.3.3.cmml"><msub id="Thmproofx3.p1.7.m7.3.3.4" xref="Thmproofx3.p1.7.m7.3.3.4.cmml"><mi id="Thmproofx3.p1.7.m7.3.3.4.2" xref="Thmproofx3.p1.7.m7.3.3.4.2.cmml">P</mi><mi id="Thmproofx3.p1.7.m7.3.3.4.3" xref="Thmproofx3.p1.7.m7.3.3.4.3.cmml">f</mi></msub><mo id="Thmproofx3.p1.7.m7.3.3.3" xref="Thmproofx3.p1.7.m7.3.3.3.cmml">=</mo><mrow id="Thmproofx3.p1.7.m7.3.3.2.2" xref="Thmproofx3.p1.7.m7.3.3.2.3.cmml"><mo id="Thmproofx3.p1.7.m7.3.3.2.2.3" xref="Thmproofx3.p1.7.m7.3.3.2.3.cmml">{</mo><msub id="Thmproofx3.p1.7.m7.2.2.1.1.1" xref="Thmproofx3.p1.7.m7.2.2.1.1.1.cmml"><mi id="Thmproofx3.p1.7.m7.2.2.1.1.1.2" xref="Thmproofx3.p1.7.m7.2.2.1.1.1.2.cmml">p</mi><mn id="Thmproofx3.p1.7.m7.2.2.1.1.1.3" xref="Thmproofx3.p1.7.m7.2.2.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx3.p1.7.m7.3.3.2.2.4" xref="Thmproofx3.p1.7.m7.3.3.2.3.cmml">,</mo><mi id="Thmproofx3.p1.7.m7.1.1" mathvariant="normal" xref="Thmproofx3.p1.7.m7.1.1.cmml">…</mi><mo id="Thmproofx3.p1.7.m7.3.3.2.2.5" xref="Thmproofx3.p1.7.m7.3.3.2.3.cmml">,</mo><msub id="Thmproofx3.p1.7.m7.3.3.2.2.2" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.cmml"><mi id="Thmproofx3.p1.7.m7.3.3.2.2.2.2" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.2.cmml">p</mi><msup id="Thmproofx3.p1.7.m7.3.3.2.2.2.3" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.3.cmml"><mi id="Thmproofx3.p1.7.m7.3.3.2.2.2.3.2" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.3.2.cmml">m</mi><mo id="Thmproofx3.p1.7.m7.3.3.2.2.2.3.3" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.7.m7.3.3.2.2.6" xref="Thmproofx3.p1.7.m7.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.7.m7.3b"><apply id="Thmproofx3.p1.7.m7.3.3.cmml" xref="Thmproofx3.p1.7.m7.3.3"><eq id="Thmproofx3.p1.7.m7.3.3.3.cmml" xref="Thmproofx3.p1.7.m7.3.3.3"></eq><apply id="Thmproofx3.p1.7.m7.3.3.4.cmml" xref="Thmproofx3.p1.7.m7.3.3.4"><csymbol cd="ambiguous" id="Thmproofx3.p1.7.m7.3.3.4.1.cmml" xref="Thmproofx3.p1.7.m7.3.3.4">subscript</csymbol><ci id="Thmproofx3.p1.7.m7.3.3.4.2.cmml" xref="Thmproofx3.p1.7.m7.3.3.4.2">𝑃</ci><ci id="Thmproofx3.p1.7.m7.3.3.4.3.cmml" xref="Thmproofx3.p1.7.m7.3.3.4.3">𝑓</ci></apply><set id="Thmproofx3.p1.7.m7.3.3.2.3.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2"><apply id="Thmproofx3.p1.7.m7.2.2.1.1.1.cmml" xref="Thmproofx3.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.7.m7.2.2.1.1.1.1.cmml" xref="Thmproofx3.p1.7.m7.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.7.m7.2.2.1.1.1.2.cmml" xref="Thmproofx3.p1.7.m7.2.2.1.1.1.2">𝑝</ci><cn id="Thmproofx3.p1.7.m7.2.2.1.1.1.3.cmml" type="integer" xref="Thmproofx3.p1.7.m7.2.2.1.1.1.3">1</cn></apply><ci id="Thmproofx3.p1.7.m7.1.1.cmml" xref="Thmproofx3.p1.7.m7.1.1">…</ci><apply id="Thmproofx3.p1.7.m7.3.3.2.2.2.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.7.m7.3.3.2.2.2.1.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2.2">subscript</csymbol><ci id="Thmproofx3.p1.7.m7.3.3.2.2.2.2.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.2">𝑝</ci><apply id="Thmproofx3.p1.7.m7.3.3.2.2.2.3.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.7.m7.3.3.2.2.2.3.1.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.3">superscript</csymbol><ci id="Thmproofx3.p1.7.m7.3.3.2.2.2.3.2.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.3.2">𝑚</ci><ci id="Thmproofx3.p1.7.m7.3.3.2.2.2.3.3.cmml" xref="Thmproofx3.p1.7.m7.3.3.2.2.2.3.3">′</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.7.m7.3c">P_{f}=\left\{p_{1},...,p_{m^{\prime}}\right\}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.7.m7.3d">italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math>. For every <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx3.p1.8.m8.1"><semantics id="Thmproofx3.p1.8.m8.1a"><mi id="Thmproofx3.p1.8.m8.1.1" xref="Thmproofx3.p1.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.8.m8.1b"><ci id="Thmproofx3.p1.8.m8.1.1.cmml" xref="Thmproofx3.p1.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.8.m8.1d">italic_i</annotation></semantics></math> pick an edge <math alttext="e_{i}\in p_{i}" class="ltx_Math" display="inline" id="Thmproofx3.p1.9.m9.1"><semantics id="Thmproofx3.p1.9.m9.1a"><mrow id="Thmproofx3.p1.9.m9.1.1" xref="Thmproofx3.p1.9.m9.1.1.cmml"><msub id="Thmproofx3.p1.9.m9.1.1.2" xref="Thmproofx3.p1.9.m9.1.1.2.cmml"><mi id="Thmproofx3.p1.9.m9.1.1.2.2" xref="Thmproofx3.p1.9.m9.1.1.2.2.cmml">e</mi><mi id="Thmproofx3.p1.9.m9.1.1.2.3" xref="Thmproofx3.p1.9.m9.1.1.2.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.9.m9.1.1.1" xref="Thmproofx3.p1.9.m9.1.1.1.cmml">∈</mo><msub id="Thmproofx3.p1.9.m9.1.1.3" xref="Thmproofx3.p1.9.m9.1.1.3.cmml"><mi id="Thmproofx3.p1.9.m9.1.1.3.2" xref="Thmproofx3.p1.9.m9.1.1.3.2.cmml">p</mi><mi id="Thmproofx3.p1.9.m9.1.1.3.3" xref="Thmproofx3.p1.9.m9.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.9.m9.1b"><apply id="Thmproofx3.p1.9.m9.1.1.cmml" xref="Thmproofx3.p1.9.m9.1.1"><in id="Thmproofx3.p1.9.m9.1.1.1.cmml" xref="Thmproofx3.p1.9.m9.1.1.1"></in><apply id="Thmproofx3.p1.9.m9.1.1.2.cmml" xref="Thmproofx3.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.9.m9.1.1.2.1.cmml" xref="Thmproofx3.p1.9.m9.1.1.2">subscript</csymbol><ci id="Thmproofx3.p1.9.m9.1.1.2.2.cmml" xref="Thmproofx3.p1.9.m9.1.1.2.2">𝑒</ci><ci id="Thmproofx3.p1.9.m9.1.1.2.3.cmml" xref="Thmproofx3.p1.9.m9.1.1.2.3">𝑖</ci></apply><apply id="Thmproofx3.p1.9.m9.1.1.3.cmml" xref="Thmproofx3.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.9.m9.1.1.3.1.cmml" xref="Thmproofx3.p1.9.m9.1.1.3">subscript</csymbol><ci id="Thmproofx3.p1.9.m9.1.1.3.2.cmml" xref="Thmproofx3.p1.9.m9.1.1.3.2">𝑝</ci><ci id="Thmproofx3.p1.9.m9.1.1.3.3.cmml" xref="Thmproofx3.p1.9.m9.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.9.m9.1c">e_{i}\in p_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.9.m9.1d">italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\left\{e_{1},...,e_{m^{\prime}}\right\}" class="ltx_Math" display="inline" id="Thmproofx3.p1.10.m10.3"><semantics id="Thmproofx3.p1.10.m10.3a"><mrow id="Thmproofx3.p1.10.m10.3.3.2" xref="Thmproofx3.p1.10.m10.3.3.3.cmml"><mo id="Thmproofx3.p1.10.m10.3.3.2.3" xref="Thmproofx3.p1.10.m10.3.3.3.cmml">{</mo><msub id="Thmproofx3.p1.10.m10.2.2.1.1" xref="Thmproofx3.p1.10.m10.2.2.1.1.cmml"><mi id="Thmproofx3.p1.10.m10.2.2.1.1.2" xref="Thmproofx3.p1.10.m10.2.2.1.1.2.cmml">e</mi><mn id="Thmproofx3.p1.10.m10.2.2.1.1.3" xref="Thmproofx3.p1.10.m10.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproofx3.p1.10.m10.3.3.2.4" xref="Thmproofx3.p1.10.m10.3.3.3.cmml">,</mo><mi id="Thmproofx3.p1.10.m10.1.1" mathvariant="normal" xref="Thmproofx3.p1.10.m10.1.1.cmml">…</mi><mo id="Thmproofx3.p1.10.m10.3.3.2.5" xref="Thmproofx3.p1.10.m10.3.3.3.cmml">,</mo><msub id="Thmproofx3.p1.10.m10.3.3.2.2" xref="Thmproofx3.p1.10.m10.3.3.2.2.cmml"><mi id="Thmproofx3.p1.10.m10.3.3.2.2.2" xref="Thmproofx3.p1.10.m10.3.3.2.2.2.cmml">e</mi><msup id="Thmproofx3.p1.10.m10.3.3.2.2.3" xref="Thmproofx3.p1.10.m10.3.3.2.2.3.cmml"><mi id="Thmproofx3.p1.10.m10.3.3.2.2.3.2" xref="Thmproofx3.p1.10.m10.3.3.2.2.3.2.cmml">m</mi><mo id="Thmproofx3.p1.10.m10.3.3.2.2.3.3" xref="Thmproofx3.p1.10.m10.3.3.2.2.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.10.m10.3.3.2.6" xref="Thmproofx3.p1.10.m10.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.10.m10.3b"><set id="Thmproofx3.p1.10.m10.3.3.3.cmml" xref="Thmproofx3.p1.10.m10.3.3.2"><apply id="Thmproofx3.p1.10.m10.2.2.1.1.cmml" xref="Thmproofx3.p1.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.10.m10.2.2.1.1.1.cmml" xref="Thmproofx3.p1.10.m10.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p1.10.m10.2.2.1.1.2.cmml" xref="Thmproofx3.p1.10.m10.2.2.1.1.2">𝑒</ci><cn id="Thmproofx3.p1.10.m10.2.2.1.1.3.cmml" type="integer" xref="Thmproofx3.p1.10.m10.2.2.1.1.3">1</cn></apply><ci id="Thmproofx3.p1.10.m10.1.1.cmml" xref="Thmproofx3.p1.10.m10.1.1">…</ci><apply id="Thmproofx3.p1.10.m10.3.3.2.2.cmml" xref="Thmproofx3.p1.10.m10.3.3.2.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.10.m10.3.3.2.2.1.cmml" xref="Thmproofx3.p1.10.m10.3.3.2.2">subscript</csymbol><ci id="Thmproofx3.p1.10.m10.3.3.2.2.2.cmml" xref="Thmproofx3.p1.10.m10.3.3.2.2.2">𝑒</ci><apply id="Thmproofx3.p1.10.m10.3.3.2.2.3.cmml" xref="Thmproofx3.p1.10.m10.3.3.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.10.m10.3.3.2.2.3.1.cmml" xref="Thmproofx3.p1.10.m10.3.3.2.2.3">superscript</csymbol><ci id="Thmproofx3.p1.10.m10.3.3.2.2.3.2.cmml" xref="Thmproofx3.p1.10.m10.3.3.2.2.3.2">𝑚</ci><ci id="Thmproofx3.p1.10.m10.3.3.2.2.3.3.cmml" xref="Thmproofx3.p1.10.m10.3.3.2.2.3.3">′</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.10.m10.3c">\left\{e_{1},...,e_{m^{\prime}}\right\}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.10.m10.3d">{ italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math> forms a mincut of <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p1.11.m11.1"><semantics id="Thmproofx3.p1.11.m11.1a"><msup id="Thmproofx3.p1.11.m11.1.1" xref="Thmproofx3.p1.11.m11.1.1.cmml"><mi id="Thmproofx3.p1.11.m11.1.1.2" xref="Thmproofx3.p1.11.m11.1.1.2.cmml">G</mi><mo id="Thmproofx3.p1.11.m11.1.1.3" xref="Thmproofx3.p1.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.11.m11.1b"><apply id="Thmproofx3.p1.11.m11.1.1.cmml" xref="Thmproofx3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.11.m11.1.1.1.cmml" xref="Thmproofx3.p1.11.m11.1.1">superscript</csymbol><ci id="Thmproofx3.p1.11.m11.1.1.2.cmml" xref="Thmproofx3.p1.11.m11.1.1.2">𝐺</ci><ci id="Thmproofx3.p1.11.m11.1.1.3.cmml" xref="Thmproofx3.p1.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.11.m11.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.11.m11.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> (which is possible because <math alttext="f" class="ltx_Math" display="inline" id="Thmproofx3.p1.12.m12.1"><semantics id="Thmproofx3.p1.12.m12.1a"><mi id="Thmproofx3.p1.12.m12.1.1" xref="Thmproofx3.p1.12.m12.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.12.m12.1b"><ci id="Thmproofx3.p1.12.m12.1.1.cmml" xref="Thmproofx3.p1.12.m12.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.12.m12.1c">f</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.12.m12.1d">italic_f</annotation></semantics></math> saturates some mincut). Then <math alttext="\beta(\sigma)\geq\max_{i}\sigma^{E^{\prime}}(e_{i})" class="ltx_Math" display="inline" id="Thmproofx3.p1.13.m13.2"><semantics id="Thmproofx3.p1.13.m13.2a"><mrow id="Thmproofx3.p1.13.m13.2.2" xref="Thmproofx3.p1.13.m13.2.2.cmml"><mrow id="Thmproofx3.p1.13.m13.2.2.3" xref="Thmproofx3.p1.13.m13.2.2.3.cmml"><mi id="Thmproofx3.p1.13.m13.2.2.3.2" xref="Thmproofx3.p1.13.m13.2.2.3.2.cmml">β</mi><mo id="Thmproofx3.p1.13.m13.2.2.3.1" xref="Thmproofx3.p1.13.m13.2.2.3.1.cmml">⁢</mo><mrow id="Thmproofx3.p1.13.m13.2.2.3.3.2" xref="Thmproofx3.p1.13.m13.2.2.3.cmml"><mo id="Thmproofx3.p1.13.m13.2.2.3.3.2.1" stretchy="false" xref="Thmproofx3.p1.13.m13.2.2.3.cmml">(</mo><mi id="Thmproofx3.p1.13.m13.1.1" xref="Thmproofx3.p1.13.m13.1.1.cmml">σ</mi><mo id="Thmproofx3.p1.13.m13.2.2.3.3.2.2" stretchy="false" xref="Thmproofx3.p1.13.m13.2.2.3.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p1.13.m13.2.2.2" xref="Thmproofx3.p1.13.m13.2.2.2.cmml">≥</mo><mrow id="Thmproofx3.p1.13.m13.2.2.1" xref="Thmproofx3.p1.13.m13.2.2.1.cmml"><mrow id="Thmproofx3.p1.13.m13.2.2.1.3" xref="Thmproofx3.p1.13.m13.2.2.1.3.cmml"><msub id="Thmproofx3.p1.13.m13.2.2.1.3.1" xref="Thmproofx3.p1.13.m13.2.2.1.3.1.cmml"><mi id="Thmproofx3.p1.13.m13.2.2.1.3.1.2" xref="Thmproofx3.p1.13.m13.2.2.1.3.1.2.cmml">max</mi><mi id="Thmproofx3.p1.13.m13.2.2.1.3.1.3" xref="Thmproofx3.p1.13.m13.2.2.1.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.13.m13.2.2.1.3a" lspace="0.167em" xref="Thmproofx3.p1.13.m13.2.2.1.3.cmml">⁡</mo><msup id="Thmproofx3.p1.13.m13.2.2.1.3.2" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.cmml"><mi id="Thmproofx3.p1.13.m13.2.2.1.3.2.2" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.2.cmml">σ</mi><msup id="Thmproofx3.p1.13.m13.2.2.1.3.2.3" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.3.cmml"><mi id="Thmproofx3.p1.13.m13.2.2.1.3.2.3.2" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.3.2.cmml">E</mi><mo id="Thmproofx3.p1.13.m13.2.2.1.3.2.3.3" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.3.3.cmml">′</mo></msup></msup></mrow><mo id="Thmproofx3.p1.13.m13.2.2.1.2" xref="Thmproofx3.p1.13.m13.2.2.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.13.m13.2.2.1.1.1" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.cmml"><mo id="Thmproofx3.p1.13.m13.2.2.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.13.m13.2.2.1.1.1.1" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.cmml"><mi id="Thmproofx3.p1.13.m13.2.2.1.1.1.1.2" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.2.cmml">e</mi><mi id="Thmproofx3.p1.13.m13.2.2.1.1.1.1.3" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.13.m13.2.2.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.13.m13.2b"><apply id="Thmproofx3.p1.13.m13.2.2.cmml" xref="Thmproofx3.p1.13.m13.2.2"><geq id="Thmproofx3.p1.13.m13.2.2.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.2"></geq><apply id="Thmproofx3.p1.13.m13.2.2.3.cmml" xref="Thmproofx3.p1.13.m13.2.2.3"><times id="Thmproofx3.p1.13.m13.2.2.3.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.3.1"></times><ci id="Thmproofx3.p1.13.m13.2.2.3.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.3.2">𝛽</ci><ci id="Thmproofx3.p1.13.m13.1.1.cmml" xref="Thmproofx3.p1.13.m13.1.1">𝜎</ci></apply><apply id="Thmproofx3.p1.13.m13.2.2.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.1"><times id="Thmproofx3.p1.13.m13.2.2.1.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.2"></times><apply id="Thmproofx3.p1.13.m13.2.2.1.3.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3"><apply id="Thmproofx3.p1.13.m13.2.2.1.3.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.13.m13.2.2.1.3.1.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.1">subscript</csymbol><max id="Thmproofx3.p1.13.m13.2.2.1.3.1.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.1.2"></max><ci id="Thmproofx3.p1.13.m13.2.2.1.3.1.3.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.1.3">𝑖</ci></apply><apply id="Thmproofx3.p1.13.m13.2.2.1.3.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.13.m13.2.2.1.3.2.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.2">superscript</csymbol><ci id="Thmproofx3.p1.13.m13.2.2.1.3.2.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.2">𝜎</ci><apply id="Thmproofx3.p1.13.m13.2.2.1.3.2.3.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.13.m13.2.2.1.3.2.3.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.3">superscript</csymbol><ci id="Thmproofx3.p1.13.m13.2.2.1.3.2.3.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.3.2">𝐸</ci><ci id="Thmproofx3.p1.13.m13.2.2.1.3.2.3.3.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.3.2.3.3">′</ci></apply></apply></apply><apply id="Thmproofx3.p1.13.m13.2.2.1.1.1.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.13.m13.2.2.1.1.1.1.1.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.13.m13.2.2.1.1.1.1.2.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.2">𝑒</ci><ci id="Thmproofx3.p1.13.m13.2.2.1.1.1.1.3.cmml" xref="Thmproofx3.p1.13.m13.2.2.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.13.m13.2c">\beta(\sigma)\geq\max_{i}\sigma^{E^{\prime}}(e_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.13.m13.2d">italic_β ( italic_σ ) ≥ roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. <math alttext="f" class="ltx_Math" display="inline" id="Thmproofx3.p1.14.m14.1"><semantics id="Thmproofx3.p1.14.m14.1a"><mi id="Thmproofx3.p1.14.m14.1.1" xref="Thmproofx3.p1.14.m14.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.14.m14.1b"><ci id="Thmproofx3.p1.14.m14.1.1.cmml" xref="Thmproofx3.p1.14.m14.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.14.m14.1c">f</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.14.m14.1d">italic_f</annotation></semantics></math> is maximal and <math alttext="\left\{e_{1},...,e_{m^{\prime}}\right\}" class="ltx_Math" display="inline" id="Thmproofx3.p1.15.m15.3"><semantics id="Thmproofx3.p1.15.m15.3a"><mrow id="Thmproofx3.p1.15.m15.3.3.2" xref="Thmproofx3.p1.15.m15.3.3.3.cmml"><mo id="Thmproofx3.p1.15.m15.3.3.2.3" xref="Thmproofx3.p1.15.m15.3.3.3.cmml">{</mo><msub id="Thmproofx3.p1.15.m15.2.2.1.1" xref="Thmproofx3.p1.15.m15.2.2.1.1.cmml"><mi id="Thmproofx3.p1.15.m15.2.2.1.1.2" xref="Thmproofx3.p1.15.m15.2.2.1.1.2.cmml">e</mi><mn id="Thmproofx3.p1.15.m15.2.2.1.1.3" xref="Thmproofx3.p1.15.m15.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproofx3.p1.15.m15.3.3.2.4" xref="Thmproofx3.p1.15.m15.3.3.3.cmml">,</mo><mi id="Thmproofx3.p1.15.m15.1.1" mathvariant="normal" xref="Thmproofx3.p1.15.m15.1.1.cmml">…</mi><mo id="Thmproofx3.p1.15.m15.3.3.2.5" xref="Thmproofx3.p1.15.m15.3.3.3.cmml">,</mo><msub id="Thmproofx3.p1.15.m15.3.3.2.2" xref="Thmproofx3.p1.15.m15.3.3.2.2.cmml"><mi id="Thmproofx3.p1.15.m15.3.3.2.2.2" xref="Thmproofx3.p1.15.m15.3.3.2.2.2.cmml">e</mi><msup id="Thmproofx3.p1.15.m15.3.3.2.2.3" xref="Thmproofx3.p1.15.m15.3.3.2.2.3.cmml"><mi id="Thmproofx3.p1.15.m15.3.3.2.2.3.2" xref="Thmproofx3.p1.15.m15.3.3.2.2.3.2.cmml">m</mi><mo id="Thmproofx3.p1.15.m15.3.3.2.2.3.3" xref="Thmproofx3.p1.15.m15.3.3.2.2.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.15.m15.3.3.2.6" xref="Thmproofx3.p1.15.m15.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.15.m15.3b"><set id="Thmproofx3.p1.15.m15.3.3.3.cmml" xref="Thmproofx3.p1.15.m15.3.3.2"><apply id="Thmproofx3.p1.15.m15.2.2.1.1.cmml" xref="Thmproofx3.p1.15.m15.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.15.m15.2.2.1.1.1.cmml" xref="Thmproofx3.p1.15.m15.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p1.15.m15.2.2.1.1.2.cmml" xref="Thmproofx3.p1.15.m15.2.2.1.1.2">𝑒</ci><cn id="Thmproofx3.p1.15.m15.2.2.1.1.3.cmml" type="integer" xref="Thmproofx3.p1.15.m15.2.2.1.1.3">1</cn></apply><ci id="Thmproofx3.p1.15.m15.1.1.cmml" xref="Thmproofx3.p1.15.m15.1.1">…</ci><apply id="Thmproofx3.p1.15.m15.3.3.2.2.cmml" xref="Thmproofx3.p1.15.m15.3.3.2.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.15.m15.3.3.2.2.1.cmml" xref="Thmproofx3.p1.15.m15.3.3.2.2">subscript</csymbol><ci id="Thmproofx3.p1.15.m15.3.3.2.2.2.cmml" xref="Thmproofx3.p1.15.m15.3.3.2.2.2">𝑒</ci><apply id="Thmproofx3.p1.15.m15.3.3.2.2.3.cmml" xref="Thmproofx3.p1.15.m15.3.3.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.15.m15.3.3.2.2.3.1.cmml" xref="Thmproofx3.p1.15.m15.3.3.2.2.3">superscript</csymbol><ci id="Thmproofx3.p1.15.m15.3.3.2.2.3.2.cmml" xref="Thmproofx3.p1.15.m15.3.3.2.2.3.2">𝑚</ci><ci id="Thmproofx3.p1.15.m15.3.3.2.2.3.3.cmml" xref="Thmproofx3.p1.15.m15.3.3.2.2.3.3">′</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.15.m15.3c">\left\{e_{1},...,e_{m^{\prime}}\right\}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.15.m15.3d">{ italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math> intersects with every path in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p1.16.m16.1"><semantics id="Thmproofx3.p1.16.m16.1a"><msup id="Thmproofx3.p1.16.m16.1.1" xref="Thmproofx3.p1.16.m16.1.1.cmml"><mi id="Thmproofx3.p1.16.m16.1.1.2" xref="Thmproofx3.p1.16.m16.1.1.2.cmml">P</mi><mo id="Thmproofx3.p1.16.m16.1.1.3" xref="Thmproofx3.p1.16.m16.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.16.m16.1b"><apply id="Thmproofx3.p1.16.m16.1.1.cmml" xref="Thmproofx3.p1.16.m16.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.16.m16.1.1.1.cmml" xref="Thmproofx3.p1.16.m16.1.1">superscript</csymbol><ci id="Thmproofx3.p1.16.m16.1.1.2.cmml" xref="Thmproofx3.p1.16.m16.1.1.2">𝑃</ci><ci id="Thmproofx3.p1.16.m16.1.1.3.cmml" xref="Thmproofx3.p1.16.m16.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.16.m16.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.16.m16.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. So we argue that <math alttext="\max_{i}\sigma^{E^{\prime}}(e_{i})" class="ltx_Math" display="inline" id="Thmproofx3.p1.17.m17.1"><semantics id="Thmproofx3.p1.17.m17.1a"><mrow id="Thmproofx3.p1.17.m17.1.1" xref="Thmproofx3.p1.17.m17.1.1.cmml"><mrow id="Thmproofx3.p1.17.m17.1.1.3" xref="Thmproofx3.p1.17.m17.1.1.3.cmml"><msub id="Thmproofx3.p1.17.m17.1.1.3.1" xref="Thmproofx3.p1.17.m17.1.1.3.1.cmml"><mi id="Thmproofx3.p1.17.m17.1.1.3.1.2" xref="Thmproofx3.p1.17.m17.1.1.3.1.2.cmml">max</mi><mi id="Thmproofx3.p1.17.m17.1.1.3.1.3" xref="Thmproofx3.p1.17.m17.1.1.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.17.m17.1.1.3a" lspace="0.167em" xref="Thmproofx3.p1.17.m17.1.1.3.cmml">⁡</mo><msup id="Thmproofx3.p1.17.m17.1.1.3.2" xref="Thmproofx3.p1.17.m17.1.1.3.2.cmml"><mi id="Thmproofx3.p1.17.m17.1.1.3.2.2" xref="Thmproofx3.p1.17.m17.1.1.3.2.2.cmml">σ</mi><msup id="Thmproofx3.p1.17.m17.1.1.3.2.3" xref="Thmproofx3.p1.17.m17.1.1.3.2.3.cmml"><mi id="Thmproofx3.p1.17.m17.1.1.3.2.3.2" xref="Thmproofx3.p1.17.m17.1.1.3.2.3.2.cmml">E</mi><mo id="Thmproofx3.p1.17.m17.1.1.3.2.3.3" xref="Thmproofx3.p1.17.m17.1.1.3.2.3.3.cmml">′</mo></msup></msup></mrow><mo id="Thmproofx3.p1.17.m17.1.1.2" xref="Thmproofx3.p1.17.m17.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.17.m17.1.1.1.1" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.cmml"><mo id="Thmproofx3.p1.17.m17.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.17.m17.1.1.1.1.1" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.17.m17.1.1.1.1.1.2" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.2.cmml">e</mi><mi id="Thmproofx3.p1.17.m17.1.1.1.1.1.3" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.17.m17.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.17.m17.1b"><apply id="Thmproofx3.p1.17.m17.1.1.cmml" xref="Thmproofx3.p1.17.m17.1.1"><times id="Thmproofx3.p1.17.m17.1.1.2.cmml" xref="Thmproofx3.p1.17.m17.1.1.2"></times><apply id="Thmproofx3.p1.17.m17.1.1.3.cmml" xref="Thmproofx3.p1.17.m17.1.1.3"><apply id="Thmproofx3.p1.17.m17.1.1.3.1.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.17.m17.1.1.3.1.1.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.1">subscript</csymbol><max id="Thmproofx3.p1.17.m17.1.1.3.1.2.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.1.2"></max><ci id="Thmproofx3.p1.17.m17.1.1.3.1.3.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.1.3">𝑖</ci></apply><apply id="Thmproofx3.p1.17.m17.1.1.3.2.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.17.m17.1.1.3.2.1.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.2">superscript</csymbol><ci id="Thmproofx3.p1.17.m17.1.1.3.2.2.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.2.2">𝜎</ci><apply id="Thmproofx3.p1.17.m17.1.1.3.2.3.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.17.m17.1.1.3.2.3.1.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.2.3">superscript</csymbol><ci id="Thmproofx3.p1.17.m17.1.1.3.2.3.2.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.2.3.2">𝐸</ci><ci id="Thmproofx3.p1.17.m17.1.1.3.2.3.3.cmml" xref="Thmproofx3.p1.17.m17.1.1.3.2.3.3">′</ci></apply></apply></apply><apply id="Thmproofx3.p1.17.m17.1.1.1.1.1.cmml" xref="Thmproofx3.p1.17.m17.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.17.m17.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.17.m17.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.17.m17.1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.2">𝑒</ci><ci id="Thmproofx3.p1.17.m17.1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.17.m17.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.17.m17.1c">\max_{i}\sigma^{E^{\prime}}(e_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.17.m17.1d">roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is minimal when <math alttext="\bigcup_{j}\operatorname{supp}\sigma_{j}\subseteq P_{f}" class="ltx_Math" display="inline" id="Thmproofx3.p1.18.m18.1"><semantics id="Thmproofx3.p1.18.m18.1a"><mrow id="Thmproofx3.p1.18.m18.1.1" xref="Thmproofx3.p1.18.m18.1.1.cmml"><mrow id="Thmproofx3.p1.18.m18.1.1.2" xref="Thmproofx3.p1.18.m18.1.1.2.cmml"><msub id="Thmproofx3.p1.18.m18.1.1.2.1" xref="Thmproofx3.p1.18.m18.1.1.2.1.cmml"><mo id="Thmproofx3.p1.18.m18.1.1.2.1.2" xref="Thmproofx3.p1.18.m18.1.1.2.1.2.cmml">⋃</mo><mi id="Thmproofx3.p1.18.m18.1.1.2.1.3" xref="Thmproofx3.p1.18.m18.1.1.2.1.3.cmml">j</mi></msub><mrow id="Thmproofx3.p1.18.m18.1.1.2.2" xref="Thmproofx3.p1.18.m18.1.1.2.2.cmml"><mi id="Thmproofx3.p1.18.m18.1.1.2.2.1" xref="Thmproofx3.p1.18.m18.1.1.2.2.1.cmml">supp</mi><mo id="Thmproofx3.p1.18.m18.1.1.2.2a" lspace="0.167em" xref="Thmproofx3.p1.18.m18.1.1.2.2.cmml">⁡</mo><msub id="Thmproofx3.p1.18.m18.1.1.2.2.2" xref="Thmproofx3.p1.18.m18.1.1.2.2.2.cmml"><mi id="Thmproofx3.p1.18.m18.1.1.2.2.2.2" xref="Thmproofx3.p1.18.m18.1.1.2.2.2.2.cmml">σ</mi><mi id="Thmproofx3.p1.18.m18.1.1.2.2.2.3" xref="Thmproofx3.p1.18.m18.1.1.2.2.2.3.cmml">j</mi></msub></mrow></mrow><mo id="Thmproofx3.p1.18.m18.1.1.1" xref="Thmproofx3.p1.18.m18.1.1.1.cmml">⊆</mo><msub id="Thmproofx3.p1.18.m18.1.1.3" xref="Thmproofx3.p1.18.m18.1.1.3.cmml"><mi id="Thmproofx3.p1.18.m18.1.1.3.2" xref="Thmproofx3.p1.18.m18.1.1.3.2.cmml">P</mi><mi id="Thmproofx3.p1.18.m18.1.1.3.3" xref="Thmproofx3.p1.18.m18.1.1.3.3.cmml">f</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.18.m18.1b"><apply id="Thmproofx3.p1.18.m18.1.1.cmml" xref="Thmproofx3.p1.18.m18.1.1"><subset id="Thmproofx3.p1.18.m18.1.1.1.cmml" xref="Thmproofx3.p1.18.m18.1.1.1"></subset><apply id="Thmproofx3.p1.18.m18.1.1.2.cmml" xref="Thmproofx3.p1.18.m18.1.1.2"><apply id="Thmproofx3.p1.18.m18.1.1.2.1.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.18.m18.1.1.2.1.1.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.1">subscript</csymbol><union id="Thmproofx3.p1.18.m18.1.1.2.1.2.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.1.2"></union><ci id="Thmproofx3.p1.18.m18.1.1.2.1.3.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.1.3">𝑗</ci></apply><apply id="Thmproofx3.p1.18.m18.1.1.2.2.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.2"><ci id="Thmproofx3.p1.18.m18.1.1.2.2.1.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.2.1">supp</ci><apply id="Thmproofx3.p1.18.m18.1.1.2.2.2.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.18.m18.1.1.2.2.2.1.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.2.2">subscript</csymbol><ci id="Thmproofx3.p1.18.m18.1.1.2.2.2.2.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.2.2.2">𝜎</ci><ci id="Thmproofx3.p1.18.m18.1.1.2.2.2.3.cmml" xref="Thmproofx3.p1.18.m18.1.1.2.2.2.3">𝑗</ci></apply></apply></apply><apply id="Thmproofx3.p1.18.m18.1.1.3.cmml" xref="Thmproofx3.p1.18.m18.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.18.m18.1.1.3.1.cmml" xref="Thmproofx3.p1.18.m18.1.1.3">subscript</csymbol><ci id="Thmproofx3.p1.18.m18.1.1.3.2.cmml" xref="Thmproofx3.p1.18.m18.1.1.3.2">𝑃</ci><ci id="Thmproofx3.p1.18.m18.1.1.3.3.cmml" xref="Thmproofx3.p1.18.m18.1.1.3.3">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.18.m18.1c">\bigcup_{j}\operatorname{supp}\sigma_{j}\subseteq P_{f}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.18.m18.1d">⋃ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_supp italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊆ italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math>. This is because if <math alttext="\sigma(p)>0" class="ltx_Math" display="inline" id="Thmproofx3.p1.19.m19.1"><semantics id="Thmproofx3.p1.19.m19.1a"><mrow id="Thmproofx3.p1.19.m19.1.2" xref="Thmproofx3.p1.19.m19.1.2.cmml"><mrow id="Thmproofx3.p1.19.m19.1.2.2" xref="Thmproofx3.p1.19.m19.1.2.2.cmml"><mi id="Thmproofx3.p1.19.m19.1.2.2.2" xref="Thmproofx3.p1.19.m19.1.2.2.2.cmml">σ</mi><mo id="Thmproofx3.p1.19.m19.1.2.2.1" xref="Thmproofx3.p1.19.m19.1.2.2.1.cmml">⁢</mo><mrow id="Thmproofx3.p1.19.m19.1.2.2.3.2" xref="Thmproofx3.p1.19.m19.1.2.2.cmml"><mo id="Thmproofx3.p1.19.m19.1.2.2.3.2.1" stretchy="false" xref="Thmproofx3.p1.19.m19.1.2.2.cmml">(</mo><mi id="Thmproofx3.p1.19.m19.1.1" xref="Thmproofx3.p1.19.m19.1.1.cmml">p</mi><mo id="Thmproofx3.p1.19.m19.1.2.2.3.2.2" stretchy="false" xref="Thmproofx3.p1.19.m19.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p1.19.m19.1.2.1" xref="Thmproofx3.p1.19.m19.1.2.1.cmml">></mo><mn id="Thmproofx3.p1.19.m19.1.2.3" xref="Thmproofx3.p1.19.m19.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.19.m19.1b"><apply id="Thmproofx3.p1.19.m19.1.2.cmml" xref="Thmproofx3.p1.19.m19.1.2"><gt id="Thmproofx3.p1.19.m19.1.2.1.cmml" xref="Thmproofx3.p1.19.m19.1.2.1"></gt><apply id="Thmproofx3.p1.19.m19.1.2.2.cmml" xref="Thmproofx3.p1.19.m19.1.2.2"><times id="Thmproofx3.p1.19.m19.1.2.2.1.cmml" xref="Thmproofx3.p1.19.m19.1.2.2.1"></times><ci id="Thmproofx3.p1.19.m19.1.2.2.2.cmml" xref="Thmproofx3.p1.19.m19.1.2.2.2">𝜎</ci><ci id="Thmproofx3.p1.19.m19.1.1.cmml" xref="Thmproofx3.p1.19.m19.1.1">𝑝</ci></apply><cn id="Thmproofx3.p1.19.m19.1.2.3.cmml" type="integer" xref="Thmproofx3.p1.19.m19.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.19.m19.1c">\sigma(p)>0</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.19.m19.1d">italic_σ ( italic_p ) > 0</annotation></semantics></math> for <math alttext="p\notin P_{f}" class="ltx_Math" display="inline" id="Thmproofx3.p1.20.m20.1"><semantics id="Thmproofx3.p1.20.m20.1a"><mrow id="Thmproofx3.p1.20.m20.1.1" xref="Thmproofx3.p1.20.m20.1.1.cmml"><mi id="Thmproofx3.p1.20.m20.1.1.2" xref="Thmproofx3.p1.20.m20.1.1.2.cmml">p</mi><mo id="Thmproofx3.p1.20.m20.1.1.1" xref="Thmproofx3.p1.20.m20.1.1.1.cmml">∉</mo><msub id="Thmproofx3.p1.20.m20.1.1.3" xref="Thmproofx3.p1.20.m20.1.1.3.cmml"><mi id="Thmproofx3.p1.20.m20.1.1.3.2" xref="Thmproofx3.p1.20.m20.1.1.3.2.cmml">P</mi><mi id="Thmproofx3.p1.20.m20.1.1.3.3" xref="Thmproofx3.p1.20.m20.1.1.3.3.cmml">f</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.20.m20.1b"><apply id="Thmproofx3.p1.20.m20.1.1.cmml" xref="Thmproofx3.p1.20.m20.1.1"><notin id="Thmproofx3.p1.20.m20.1.1.1.cmml" xref="Thmproofx3.p1.20.m20.1.1.1"></notin><ci id="Thmproofx3.p1.20.m20.1.1.2.cmml" xref="Thmproofx3.p1.20.m20.1.1.2">𝑝</ci><apply id="Thmproofx3.p1.20.m20.1.1.3.cmml" xref="Thmproofx3.p1.20.m20.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.20.m20.1.1.3.1.cmml" xref="Thmproofx3.p1.20.m20.1.1.3">subscript</csymbol><ci id="Thmproofx3.p1.20.m20.1.1.3.2.cmml" xref="Thmproofx3.p1.20.m20.1.1.3.2">𝑃</ci><ci id="Thmproofx3.p1.20.m20.1.1.3.3.cmml" xref="Thmproofx3.p1.20.m20.1.1.3.3">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.20.m20.1c">p\notin P_{f}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.20.m20.1d">italic_p ∉ italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math>, then for <math alttext="e_{i}" class="ltx_Math" display="inline" id="Thmproofx3.p1.21.m21.1"><semantics id="Thmproofx3.p1.21.m21.1a"><msub id="Thmproofx3.p1.21.m21.1.1" xref="Thmproofx3.p1.21.m21.1.1.cmml"><mi id="Thmproofx3.p1.21.m21.1.1.2" xref="Thmproofx3.p1.21.m21.1.1.2.cmml">e</mi><mi id="Thmproofx3.p1.21.m21.1.1.3" xref="Thmproofx3.p1.21.m21.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.21.m21.1b"><apply id="Thmproofx3.p1.21.m21.1.1.cmml" xref="Thmproofx3.p1.21.m21.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.21.m21.1.1.1.cmml" xref="Thmproofx3.p1.21.m21.1.1">subscript</csymbol><ci id="Thmproofx3.p1.21.m21.1.1.2.cmml" xref="Thmproofx3.p1.21.m21.1.1.2">𝑒</ci><ci id="Thmproofx3.p1.21.m21.1.1.3.cmml" xref="Thmproofx3.p1.21.m21.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.21.m21.1c">e_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.21.m21.1d">italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="e_{i}\in p" class="ltx_Math" display="inline" id="Thmproofx3.p1.22.m22.1"><semantics id="Thmproofx3.p1.22.m22.1a"><mrow id="Thmproofx3.p1.22.m22.1.1" xref="Thmproofx3.p1.22.m22.1.1.cmml"><msub id="Thmproofx3.p1.22.m22.1.1.2" xref="Thmproofx3.p1.22.m22.1.1.2.cmml"><mi id="Thmproofx3.p1.22.m22.1.1.2.2" xref="Thmproofx3.p1.22.m22.1.1.2.2.cmml">e</mi><mi id="Thmproofx3.p1.22.m22.1.1.2.3" xref="Thmproofx3.p1.22.m22.1.1.2.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.22.m22.1.1.1" xref="Thmproofx3.p1.22.m22.1.1.1.cmml">∈</mo><mi id="Thmproofx3.p1.22.m22.1.1.3" xref="Thmproofx3.p1.22.m22.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.22.m22.1b"><apply id="Thmproofx3.p1.22.m22.1.1.cmml" xref="Thmproofx3.p1.22.m22.1.1"><in id="Thmproofx3.p1.22.m22.1.1.1.cmml" xref="Thmproofx3.p1.22.m22.1.1.1"></in><apply id="Thmproofx3.p1.22.m22.1.1.2.cmml" xref="Thmproofx3.p1.22.m22.1.1.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.22.m22.1.1.2.1.cmml" xref="Thmproofx3.p1.22.m22.1.1.2">subscript</csymbol><ci id="Thmproofx3.p1.22.m22.1.1.2.2.cmml" xref="Thmproofx3.p1.22.m22.1.1.2.2">𝑒</ci><ci id="Thmproofx3.p1.22.m22.1.1.2.3.cmml" xref="Thmproofx3.p1.22.m22.1.1.2.3">𝑖</ci></apply><ci id="Thmproofx3.p1.22.m22.1.1.3.cmml" xref="Thmproofx3.p1.22.m22.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.22.m22.1c">e_{i}\in p</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.22.m22.1d">italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_p</annotation></semantics></math> we can move the weight of <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmproofx3.p1.23.m23.1"><semantics id="Thmproofx3.p1.23.m23.1a"><mi id="Thmproofx3.p1.23.m23.1.1" xref="Thmproofx3.p1.23.m23.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.23.m23.1b"><ci id="Thmproofx3.p1.23.m23.1.1.cmml" xref="Thmproofx3.p1.23.m23.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.23.m23.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.23.m23.1d">italic_σ</annotation></semantics></math> on <math alttext="p" class="ltx_Math" display="inline" id="Thmproofx3.p1.24.m24.1"><semantics id="Thmproofx3.p1.24.m24.1a"><mi id="Thmproofx3.p1.24.m24.1.1" xref="Thmproofx3.p1.24.m24.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.24.m24.1b"><ci id="Thmproofx3.p1.24.m24.1.1.cmml" xref="Thmproofx3.p1.24.m24.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.24.m24.1c">p</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.24.m24.1d">italic_p</annotation></semantics></math> to <math alttext="p_{i}" class="ltx_Math" display="inline" id="Thmproofx3.p1.25.m25.1"><semantics id="Thmproofx3.p1.25.m25.1a"><msub id="Thmproofx3.p1.25.m25.1.1" xref="Thmproofx3.p1.25.m25.1.1.cmml"><mi id="Thmproofx3.p1.25.m25.1.1.2" xref="Thmproofx3.p1.25.m25.1.1.2.cmml">p</mi><mi id="Thmproofx3.p1.25.m25.1.1.3" xref="Thmproofx3.p1.25.m25.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.25.m25.1b"><apply id="Thmproofx3.p1.25.m25.1.1.cmml" xref="Thmproofx3.p1.25.m25.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.25.m25.1.1.1.cmml" xref="Thmproofx3.p1.25.m25.1.1">subscript</csymbol><ci id="Thmproofx3.p1.25.m25.1.1.2.cmml" xref="Thmproofx3.p1.25.m25.1.1.2">𝑝</ci><ci id="Thmproofx3.p1.25.m25.1.1.3.cmml" xref="Thmproofx3.p1.25.m25.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.25.m25.1c">p_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.25.m25.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and get the corresponding <math alttext="\mu" class="ltx_Math" display="inline" id="Thmproofx3.p1.26.m26.1"><semantics id="Thmproofx3.p1.26.m26.1a"><mi id="Thmproofx3.p1.26.m26.1.1" xref="Thmproofx3.p1.26.m26.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.26.m26.1b"><ci id="Thmproofx3.p1.26.m26.1.1.cmml" xref="Thmproofx3.p1.26.m26.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.26.m26.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.26.m26.1d">italic_μ</annotation></semantics></math>. It satisfies <math alttext="\mu^{E^{\prime}}(e_{i})\stackrel{{\scriptstyle\text{def}}}{{=}}\sigma^{E^{% \prime}}(e_{i})" class="ltx_Math" display="inline" id="Thmproofx3.p1.27.m27.2"><semantics id="Thmproofx3.p1.27.m27.2a"><mrow id="Thmproofx3.p1.27.m27.2.2" xref="Thmproofx3.p1.27.m27.2.2.cmml"><mrow id="Thmproofx3.p1.27.m27.1.1.1" xref="Thmproofx3.p1.27.m27.1.1.1.cmml"><msup id="Thmproofx3.p1.27.m27.1.1.1.3" xref="Thmproofx3.p1.27.m27.1.1.1.3.cmml"><mi id="Thmproofx3.p1.27.m27.1.1.1.3.2" xref="Thmproofx3.p1.27.m27.1.1.1.3.2.cmml">μ</mi><msup id="Thmproofx3.p1.27.m27.1.1.1.3.3" xref="Thmproofx3.p1.27.m27.1.1.1.3.3.cmml"><mi id="Thmproofx3.p1.27.m27.1.1.1.3.3.2" xref="Thmproofx3.p1.27.m27.1.1.1.3.3.2.cmml">E</mi><mo id="Thmproofx3.p1.27.m27.1.1.1.3.3.3" xref="Thmproofx3.p1.27.m27.1.1.1.3.3.3.cmml">′</mo></msup></msup><mo id="Thmproofx3.p1.27.m27.1.1.1.2" xref="Thmproofx3.p1.27.m27.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.27.m27.1.1.1.1.1" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p1.27.m27.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.27.m27.1.1.1.1.1.1" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.27.m27.1.1.1.1.1.1.2" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.2.cmml">e</mi><mi id="Thmproofx3.p1.27.m27.1.1.1.1.1.1.3" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.27.m27.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mover id="Thmproofx3.p1.27.m27.2.2.3" xref="Thmproofx3.p1.27.m27.2.2.3.cmml"><mo id="Thmproofx3.p1.27.m27.2.2.3.2" xref="Thmproofx3.p1.27.m27.2.2.3.2.cmml">=</mo><mtext id="Thmproofx3.p1.27.m27.2.2.3.3" xref="Thmproofx3.p1.27.m27.2.2.3.3a.cmml">def</mtext></mover><mrow id="Thmproofx3.p1.27.m27.2.2.2" xref="Thmproofx3.p1.27.m27.2.2.2.cmml"><msup id="Thmproofx3.p1.27.m27.2.2.2.3" xref="Thmproofx3.p1.27.m27.2.2.2.3.cmml"><mi id="Thmproofx3.p1.27.m27.2.2.2.3.2" xref="Thmproofx3.p1.27.m27.2.2.2.3.2.cmml">σ</mi><msup id="Thmproofx3.p1.27.m27.2.2.2.3.3" xref="Thmproofx3.p1.27.m27.2.2.2.3.3.cmml"><mi id="Thmproofx3.p1.27.m27.2.2.2.3.3.2" xref="Thmproofx3.p1.27.m27.2.2.2.3.3.2.cmml">E</mi><mo id="Thmproofx3.p1.27.m27.2.2.2.3.3.3" xref="Thmproofx3.p1.27.m27.2.2.2.3.3.3.cmml">′</mo></msup></msup><mo id="Thmproofx3.p1.27.m27.2.2.2.2" xref="Thmproofx3.p1.27.m27.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.27.m27.2.2.2.1.1" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p1.27.m27.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.27.m27.2.2.2.1.1.1" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p1.27.m27.2.2.2.1.1.1.2" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.2.cmml">e</mi><mi id="Thmproofx3.p1.27.m27.2.2.2.1.1.1.3" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.27.m27.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.27.m27.2b"><apply id="Thmproofx3.p1.27.m27.2.2.cmml" xref="Thmproofx3.p1.27.m27.2.2"><apply id="Thmproofx3.p1.27.m27.2.2.3.cmml" xref="Thmproofx3.p1.27.m27.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.27.m27.2.2.3.1.cmml" xref="Thmproofx3.p1.27.m27.2.2.3">superscript</csymbol><eq id="Thmproofx3.p1.27.m27.2.2.3.2.cmml" xref="Thmproofx3.p1.27.m27.2.2.3.2"></eq><ci id="Thmproofx3.p1.27.m27.2.2.3.3a.cmml" xref="Thmproofx3.p1.27.m27.2.2.3.3"><mtext id="Thmproofx3.p1.27.m27.2.2.3.3.cmml" mathsize="70%" xref="Thmproofx3.p1.27.m27.2.2.3.3">def</mtext></ci></apply><apply id="Thmproofx3.p1.27.m27.1.1.1.cmml" xref="Thmproofx3.p1.27.m27.1.1.1"><times id="Thmproofx3.p1.27.m27.1.1.1.2.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.2"></times><apply id="Thmproofx3.p1.27.m27.1.1.1.3.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.27.m27.1.1.1.3.1.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.27.m27.1.1.1.3.2.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.3.2">𝜇</ci><apply id="Thmproofx3.p1.27.m27.1.1.1.3.3.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.27.m27.1.1.1.3.3.1.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.3.3">superscript</csymbol><ci id="Thmproofx3.p1.27.m27.1.1.1.3.3.2.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.3.3.2">𝐸</ci><ci id="Thmproofx3.p1.27.m27.1.1.1.3.3.3.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.3.3.3">′</ci></apply></apply><apply id="Thmproofx3.p1.27.m27.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.27.m27.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.27.m27.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.2">𝑒</ci><ci id="Thmproofx3.p1.27.m27.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.27.m27.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx3.p1.27.m27.2.2.2.cmml" xref="Thmproofx3.p1.27.m27.2.2.2"><times id="Thmproofx3.p1.27.m27.2.2.2.2.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.2"></times><apply id="Thmproofx3.p1.27.m27.2.2.2.3.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.27.m27.2.2.2.3.1.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.3">superscript</csymbol><ci id="Thmproofx3.p1.27.m27.2.2.2.3.2.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.3.2">𝜎</ci><apply id="Thmproofx3.p1.27.m27.2.2.2.3.3.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.3.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.27.m27.2.2.2.3.3.1.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.3.3">superscript</csymbol><ci id="Thmproofx3.p1.27.m27.2.2.2.3.3.2.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.3.3.2">𝐸</ci><ci id="Thmproofx3.p1.27.m27.2.2.2.3.3.3.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.3.3.3">′</ci></apply></apply><apply id="Thmproofx3.p1.27.m27.2.2.2.1.1.1.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.27.m27.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p1.27.m27.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.2">𝑒</ci><ci id="Thmproofx3.p1.27.m27.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p1.27.m27.2.2.2.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.27.m27.2c">\mu^{E^{\prime}}(e_{i})\stackrel{{\scriptstyle\text{def}}}{{=}}\sigma^{E^{% \prime}}(e_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.27.m27.2d">italic_μ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP italic_σ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="\mu^{E^{\prime}}(e_{i^{\prime}})=\sigma^{E^{\prime}}(e_{i^{\prime}})" class="ltx_Math" display="inline" id="Thmproofx3.p1.28.m28.2"><semantics id="Thmproofx3.p1.28.m28.2a"><mrow id="Thmproofx3.p1.28.m28.2.2" xref="Thmproofx3.p1.28.m28.2.2.cmml"><mrow id="Thmproofx3.p1.28.m28.1.1.1" xref="Thmproofx3.p1.28.m28.1.1.1.cmml"><msup id="Thmproofx3.p1.28.m28.1.1.1.3" xref="Thmproofx3.p1.28.m28.1.1.1.3.cmml"><mi id="Thmproofx3.p1.28.m28.1.1.1.3.2" xref="Thmproofx3.p1.28.m28.1.1.1.3.2.cmml">μ</mi><msup id="Thmproofx3.p1.28.m28.1.1.1.3.3" xref="Thmproofx3.p1.28.m28.1.1.1.3.3.cmml"><mi id="Thmproofx3.p1.28.m28.1.1.1.3.3.2" xref="Thmproofx3.p1.28.m28.1.1.1.3.3.2.cmml">E</mi><mo id="Thmproofx3.p1.28.m28.1.1.1.3.3.3" xref="Thmproofx3.p1.28.m28.1.1.1.3.3.3.cmml">′</mo></msup></msup><mo id="Thmproofx3.p1.28.m28.1.1.1.2" xref="Thmproofx3.p1.28.m28.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.28.m28.1.1.1.1.1" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p1.28.m28.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.28.m28.1.1.1.1.1.1" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.2" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.2.cmml">e</mi><msup id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.2" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.3" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.28.m28.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p1.28.m28.2.2.3" xref="Thmproofx3.p1.28.m28.2.2.3.cmml">=</mo><mrow id="Thmproofx3.p1.28.m28.2.2.2" xref="Thmproofx3.p1.28.m28.2.2.2.cmml"><msup id="Thmproofx3.p1.28.m28.2.2.2.3" xref="Thmproofx3.p1.28.m28.2.2.2.3.cmml"><mi id="Thmproofx3.p1.28.m28.2.2.2.3.2" xref="Thmproofx3.p1.28.m28.2.2.2.3.2.cmml">σ</mi><msup id="Thmproofx3.p1.28.m28.2.2.2.3.3" xref="Thmproofx3.p1.28.m28.2.2.2.3.3.cmml"><mi id="Thmproofx3.p1.28.m28.2.2.2.3.3.2" xref="Thmproofx3.p1.28.m28.2.2.2.3.3.2.cmml">E</mi><mo id="Thmproofx3.p1.28.m28.2.2.2.3.3.3" xref="Thmproofx3.p1.28.m28.2.2.2.3.3.3.cmml">′</mo></msup></msup><mo id="Thmproofx3.p1.28.m28.2.2.2.2" xref="Thmproofx3.p1.28.m28.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.28.m28.2.2.2.1.1" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p1.28.m28.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.28.m28.2.2.2.1.1.1" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.2" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.2.cmml">e</mi><msup id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.cmml"><mi id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.2" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.3" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.28.m28.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.28.m28.2b"><apply id="Thmproofx3.p1.28.m28.2.2.cmml" xref="Thmproofx3.p1.28.m28.2.2"><eq id="Thmproofx3.p1.28.m28.2.2.3.cmml" xref="Thmproofx3.p1.28.m28.2.2.3"></eq><apply id="Thmproofx3.p1.28.m28.1.1.1.cmml" xref="Thmproofx3.p1.28.m28.1.1.1"><times id="Thmproofx3.p1.28.m28.1.1.1.2.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.2"></times><apply id="Thmproofx3.p1.28.m28.1.1.1.3.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.1.1.1.3.1.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.28.m28.1.1.1.3.2.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.3.2">𝜇</ci><apply id="Thmproofx3.p1.28.m28.1.1.1.3.3.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.1.1.1.3.3.1.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.3.3">superscript</csymbol><ci id="Thmproofx3.p1.28.m28.1.1.1.3.3.2.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.3.3.2">𝐸</ci><ci id="Thmproofx3.p1.28.m28.1.1.1.3.3.3.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.3.3.3">′</ci></apply></apply><apply id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.2">𝑒</ci><apply id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx3.p1.28.m28.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply><apply id="Thmproofx3.p1.28.m28.2.2.2.cmml" xref="Thmproofx3.p1.28.m28.2.2.2"><times id="Thmproofx3.p1.28.m28.2.2.2.2.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.2"></times><apply id="Thmproofx3.p1.28.m28.2.2.2.3.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.2.2.2.3.1.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.3">superscript</csymbol><ci id="Thmproofx3.p1.28.m28.2.2.2.3.2.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.3.2">𝜎</ci><apply id="Thmproofx3.p1.28.m28.2.2.2.3.3.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.3.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.2.2.2.3.3.1.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.3.3">superscript</csymbol><ci id="Thmproofx3.p1.28.m28.2.2.2.3.3.2.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.3.3.2">𝐸</ci><ci id="Thmproofx3.p1.28.m28.2.2.2.3.3.3.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.3.3.3">′</ci></apply></apply><apply id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.2">𝑒</ci><apply id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.2.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.2">𝑖</ci><ci id="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx3.p1.28.m28.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.28.m28.2c">\mu^{E^{\prime}}(e_{i^{\prime}})=\sigma^{E^{\prime}}(e_{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.28.m28.2d">italic_μ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_σ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="e_{i^{\prime}}\notin p" class="ltx_Math" display="inline" id="Thmproofx3.p1.29.m29.1"><semantics id="Thmproofx3.p1.29.m29.1a"><mrow id="Thmproofx3.p1.29.m29.1.1" xref="Thmproofx3.p1.29.m29.1.1.cmml"><msub id="Thmproofx3.p1.29.m29.1.1.2" xref="Thmproofx3.p1.29.m29.1.1.2.cmml"><mi id="Thmproofx3.p1.29.m29.1.1.2.2" xref="Thmproofx3.p1.29.m29.1.1.2.2.cmml">e</mi><msup id="Thmproofx3.p1.29.m29.1.1.2.3" xref="Thmproofx3.p1.29.m29.1.1.2.3.cmml"><mi id="Thmproofx3.p1.29.m29.1.1.2.3.2" xref="Thmproofx3.p1.29.m29.1.1.2.3.2.cmml">i</mi><mo id="Thmproofx3.p1.29.m29.1.1.2.3.3" xref="Thmproofx3.p1.29.m29.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.29.m29.1.1.1" xref="Thmproofx3.p1.29.m29.1.1.1.cmml">∉</mo><mi id="Thmproofx3.p1.29.m29.1.1.3" xref="Thmproofx3.p1.29.m29.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.29.m29.1b"><apply id="Thmproofx3.p1.29.m29.1.1.cmml" xref="Thmproofx3.p1.29.m29.1.1"><notin id="Thmproofx3.p1.29.m29.1.1.1.cmml" xref="Thmproofx3.p1.29.m29.1.1.1"></notin><apply id="Thmproofx3.p1.29.m29.1.1.2.cmml" xref="Thmproofx3.p1.29.m29.1.1.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.29.m29.1.1.2.1.cmml" xref="Thmproofx3.p1.29.m29.1.1.2">subscript</csymbol><ci id="Thmproofx3.p1.29.m29.1.1.2.2.cmml" xref="Thmproofx3.p1.29.m29.1.1.2.2">𝑒</ci><apply id="Thmproofx3.p1.29.m29.1.1.2.3.cmml" xref="Thmproofx3.p1.29.m29.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.29.m29.1.1.2.3.1.cmml" xref="Thmproofx3.p1.29.m29.1.1.2.3">superscript</csymbol><ci id="Thmproofx3.p1.29.m29.1.1.2.3.2.cmml" xref="Thmproofx3.p1.29.m29.1.1.2.3.2">𝑖</ci><ci id="Thmproofx3.p1.29.m29.1.1.2.3.3.cmml" xref="Thmproofx3.p1.29.m29.1.1.2.3.3">′</ci></apply></apply><ci id="Thmproofx3.p1.29.m29.1.1.3.cmml" xref="Thmproofx3.p1.29.m29.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.29.m29.1c">e_{i^{\prime}}\notin p</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.29.m29.1d">italic_e start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∉ italic_p</annotation></semantics></math>, and <math alttext="\mu^{E^{\prime}}(e_{i^{\prime}})<\sigma^{E^{\prime}}(e_{i^{\prime}})" class="ltx_Math" display="inline" id="Thmproofx3.p1.30.m30.2"><semantics id="Thmproofx3.p1.30.m30.2a"><mrow id="Thmproofx3.p1.30.m30.2.2" xref="Thmproofx3.p1.30.m30.2.2.cmml"><mrow id="Thmproofx3.p1.30.m30.1.1.1" xref="Thmproofx3.p1.30.m30.1.1.1.cmml"><msup id="Thmproofx3.p1.30.m30.1.1.1.3" xref="Thmproofx3.p1.30.m30.1.1.1.3.cmml"><mi id="Thmproofx3.p1.30.m30.1.1.1.3.2" xref="Thmproofx3.p1.30.m30.1.1.1.3.2.cmml">μ</mi><msup id="Thmproofx3.p1.30.m30.1.1.1.3.3" xref="Thmproofx3.p1.30.m30.1.1.1.3.3.cmml"><mi id="Thmproofx3.p1.30.m30.1.1.1.3.3.2" xref="Thmproofx3.p1.30.m30.1.1.1.3.3.2.cmml">E</mi><mo id="Thmproofx3.p1.30.m30.1.1.1.3.3.3" xref="Thmproofx3.p1.30.m30.1.1.1.3.3.3.cmml">′</mo></msup></msup><mo id="Thmproofx3.p1.30.m30.1.1.1.2" xref="Thmproofx3.p1.30.m30.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.30.m30.1.1.1.1.1" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p1.30.m30.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.30.m30.1.1.1.1.1.1" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.2" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.2.cmml">e</mi><msup id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.2" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.3" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.30.m30.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p1.30.m30.2.2.3" xref="Thmproofx3.p1.30.m30.2.2.3.cmml"><</mo><mrow id="Thmproofx3.p1.30.m30.2.2.2" xref="Thmproofx3.p1.30.m30.2.2.2.cmml"><msup id="Thmproofx3.p1.30.m30.2.2.2.3" xref="Thmproofx3.p1.30.m30.2.2.2.3.cmml"><mi id="Thmproofx3.p1.30.m30.2.2.2.3.2" xref="Thmproofx3.p1.30.m30.2.2.2.3.2.cmml">σ</mi><msup id="Thmproofx3.p1.30.m30.2.2.2.3.3" xref="Thmproofx3.p1.30.m30.2.2.2.3.3.cmml"><mi id="Thmproofx3.p1.30.m30.2.2.2.3.3.2" xref="Thmproofx3.p1.30.m30.2.2.2.3.3.2.cmml">E</mi><mo id="Thmproofx3.p1.30.m30.2.2.2.3.3.3" xref="Thmproofx3.p1.30.m30.2.2.2.3.3.3.cmml">′</mo></msup></msup><mo id="Thmproofx3.p1.30.m30.2.2.2.2" xref="Thmproofx3.p1.30.m30.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.30.m30.2.2.2.1.1" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p1.30.m30.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.30.m30.2.2.2.1.1.1" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.2" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.2.cmml">e</mi><msup id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.cmml"><mi id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.2" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.3" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.30.m30.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.30.m30.2b"><apply id="Thmproofx3.p1.30.m30.2.2.cmml" xref="Thmproofx3.p1.30.m30.2.2"><lt id="Thmproofx3.p1.30.m30.2.2.3.cmml" xref="Thmproofx3.p1.30.m30.2.2.3"></lt><apply id="Thmproofx3.p1.30.m30.1.1.1.cmml" xref="Thmproofx3.p1.30.m30.1.1.1"><times id="Thmproofx3.p1.30.m30.1.1.1.2.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.2"></times><apply id="Thmproofx3.p1.30.m30.1.1.1.3.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.1.1.1.3.1.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.30.m30.1.1.1.3.2.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.3.2">𝜇</ci><apply id="Thmproofx3.p1.30.m30.1.1.1.3.3.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.1.1.1.3.3.1.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.3.3">superscript</csymbol><ci id="Thmproofx3.p1.30.m30.1.1.1.3.3.2.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.3.3.2">𝐸</ci><ci id="Thmproofx3.p1.30.m30.1.1.1.3.3.3.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.3.3.3">′</ci></apply></apply><apply id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.2">𝑒</ci><apply id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx3.p1.30.m30.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply><apply id="Thmproofx3.p1.30.m30.2.2.2.cmml" xref="Thmproofx3.p1.30.m30.2.2.2"><times id="Thmproofx3.p1.30.m30.2.2.2.2.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.2"></times><apply id="Thmproofx3.p1.30.m30.2.2.2.3.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.2.2.2.3.1.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.3">superscript</csymbol><ci id="Thmproofx3.p1.30.m30.2.2.2.3.2.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.3.2">𝜎</ci><apply id="Thmproofx3.p1.30.m30.2.2.2.3.3.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.3.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.2.2.2.3.3.1.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.3.3">superscript</csymbol><ci id="Thmproofx3.p1.30.m30.2.2.2.3.3.2.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.3.3.2">𝐸</ci><ci id="Thmproofx3.p1.30.m30.2.2.2.3.3.3.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.3.3.3">′</ci></apply></apply><apply id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.2">𝑒</ci><apply id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.2.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.2">𝑖</ci><ci id="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx3.p1.30.m30.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.30.m30.2c">\mu^{E^{\prime}}(e_{i^{\prime}})<\sigma^{E^{\prime}}(e_{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.30.m30.2d">italic_μ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) < italic_σ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="e_{i^{\prime}}\in p" class="ltx_Math" display="inline" id="Thmproofx3.p1.31.m31.1"><semantics id="Thmproofx3.p1.31.m31.1a"><mrow id="Thmproofx3.p1.31.m31.1.1" xref="Thmproofx3.p1.31.m31.1.1.cmml"><msub id="Thmproofx3.p1.31.m31.1.1.2" xref="Thmproofx3.p1.31.m31.1.1.2.cmml"><mi id="Thmproofx3.p1.31.m31.1.1.2.2" xref="Thmproofx3.p1.31.m31.1.1.2.2.cmml">e</mi><msup id="Thmproofx3.p1.31.m31.1.1.2.3" xref="Thmproofx3.p1.31.m31.1.1.2.3.cmml"><mi id="Thmproofx3.p1.31.m31.1.1.2.3.2" xref="Thmproofx3.p1.31.m31.1.1.2.3.2.cmml">i</mi><mo id="Thmproofx3.p1.31.m31.1.1.2.3.3" xref="Thmproofx3.p1.31.m31.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p1.31.m31.1.1.1" xref="Thmproofx3.p1.31.m31.1.1.1.cmml">∈</mo><mi id="Thmproofx3.p1.31.m31.1.1.3" xref="Thmproofx3.p1.31.m31.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.31.m31.1b"><apply id="Thmproofx3.p1.31.m31.1.1.cmml" xref="Thmproofx3.p1.31.m31.1.1"><in id="Thmproofx3.p1.31.m31.1.1.1.cmml" xref="Thmproofx3.p1.31.m31.1.1.1"></in><apply id="Thmproofx3.p1.31.m31.1.1.2.cmml" xref="Thmproofx3.p1.31.m31.1.1.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.31.m31.1.1.2.1.cmml" xref="Thmproofx3.p1.31.m31.1.1.2">subscript</csymbol><ci id="Thmproofx3.p1.31.m31.1.1.2.2.cmml" xref="Thmproofx3.p1.31.m31.1.1.2.2">𝑒</ci><apply id="Thmproofx3.p1.31.m31.1.1.2.3.cmml" xref="Thmproofx3.p1.31.m31.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.31.m31.1.1.2.3.1.cmml" xref="Thmproofx3.p1.31.m31.1.1.2.3">superscript</csymbol><ci id="Thmproofx3.p1.31.m31.1.1.2.3.2.cmml" xref="Thmproofx3.p1.31.m31.1.1.2.3.2">𝑖</ci><ci id="Thmproofx3.p1.31.m31.1.1.2.3.3.cmml" xref="Thmproofx3.p1.31.m31.1.1.2.3.3">′</ci></apply></apply><ci id="Thmproofx3.p1.31.m31.1.1.3.cmml" xref="Thmproofx3.p1.31.m31.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.31.m31.1c">e_{i^{\prime}}\in p</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.31.m31.1d">italic_e start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∈ italic_p</annotation></semantics></math>. Therefore at the minimum, <math alttext="\sigma^{E^{\prime}}(e_{i})=\sigma(p_{i})=1-\prod_{j}(1-\sigma_{j}(p_{i}))" class="ltx_Math" display="inline" id="Thmproofx3.p1.32.m32.3"><semantics id="Thmproofx3.p1.32.m32.3a"><mrow id="Thmproofx3.p1.32.m32.3.3" xref="Thmproofx3.p1.32.m32.3.3.cmml"><mrow id="Thmproofx3.p1.32.m32.1.1.1" xref="Thmproofx3.p1.32.m32.1.1.1.cmml"><msup id="Thmproofx3.p1.32.m32.1.1.1.3" xref="Thmproofx3.p1.32.m32.1.1.1.3.cmml"><mi id="Thmproofx3.p1.32.m32.1.1.1.3.2" xref="Thmproofx3.p1.32.m32.1.1.1.3.2.cmml">σ</mi><msup id="Thmproofx3.p1.32.m32.1.1.1.3.3" xref="Thmproofx3.p1.32.m32.1.1.1.3.3.cmml"><mi id="Thmproofx3.p1.32.m32.1.1.1.3.3.2" xref="Thmproofx3.p1.32.m32.1.1.1.3.3.2.cmml">E</mi><mo id="Thmproofx3.p1.32.m32.1.1.1.3.3.3" xref="Thmproofx3.p1.32.m32.1.1.1.3.3.3.cmml">′</mo></msup></msup><mo id="Thmproofx3.p1.32.m32.1.1.1.2" xref="Thmproofx3.p1.32.m32.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.32.m32.1.1.1.1.1" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p1.32.m32.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.32.m32.1.1.1.1.1.1" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.32.m32.1.1.1.1.1.1.2" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.2.cmml">e</mi><mi id="Thmproofx3.p1.32.m32.1.1.1.1.1.1.3" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.32.m32.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p1.32.m32.3.3.5" xref="Thmproofx3.p1.32.m32.3.3.5.cmml">=</mo><mrow id="Thmproofx3.p1.32.m32.2.2.2" xref="Thmproofx3.p1.32.m32.2.2.2.cmml"><mi id="Thmproofx3.p1.32.m32.2.2.2.3" xref="Thmproofx3.p1.32.m32.2.2.2.3.cmml">σ</mi><mo id="Thmproofx3.p1.32.m32.2.2.2.2" xref="Thmproofx3.p1.32.m32.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.32.m32.2.2.2.1.1" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p1.32.m32.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.32.m32.2.2.2.1.1.1" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p1.32.m32.2.2.2.1.1.1.2" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p1.32.m32.2.2.2.1.1.1.3" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.32.m32.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p1.32.m32.3.3.6" xref="Thmproofx3.p1.32.m32.3.3.6.cmml">=</mo><mrow id="Thmproofx3.p1.32.m32.3.3.3" xref="Thmproofx3.p1.32.m32.3.3.3.cmml"><mn id="Thmproofx3.p1.32.m32.3.3.3.3" xref="Thmproofx3.p1.32.m32.3.3.3.3.cmml">1</mn><mo id="Thmproofx3.p1.32.m32.3.3.3.2" rspace="0.055em" xref="Thmproofx3.p1.32.m32.3.3.3.2.cmml">−</mo><mrow id="Thmproofx3.p1.32.m32.3.3.3.1" xref="Thmproofx3.p1.32.m32.3.3.3.1.cmml"><msub id="Thmproofx3.p1.32.m32.3.3.3.1.2" xref="Thmproofx3.p1.32.m32.3.3.3.1.2.cmml"><mo id="Thmproofx3.p1.32.m32.3.3.3.1.2.2" rspace="0em" xref="Thmproofx3.p1.32.m32.3.3.3.1.2.2.cmml">∏</mo><mi id="Thmproofx3.p1.32.m32.3.3.3.1.2.3" xref="Thmproofx3.p1.32.m32.3.3.3.1.2.3.cmml">j</mi></msub><mrow id="Thmproofx3.p1.32.m32.3.3.3.1.1.1" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.cmml"><mo id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.cmml">(</mo><mrow id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.cmml"><mn id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.3" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.3.cmml">1</mn><mo id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.2" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.2.cmml">−</mo><mrow id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.cmml"><msub id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.cmml"><mi id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.2" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.2.cmml">σ</mi><mi id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.3" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.3.cmml">j</mi></msub><mo id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.2" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.2" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.3" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.3" stretchy="false" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.32.m32.3b"><apply id="Thmproofx3.p1.32.m32.3.3.cmml" xref="Thmproofx3.p1.32.m32.3.3"><and id="Thmproofx3.p1.32.m32.3.3a.cmml" xref="Thmproofx3.p1.32.m32.3.3"></and><apply id="Thmproofx3.p1.32.m32.3.3b.cmml" xref="Thmproofx3.p1.32.m32.3.3"><eq id="Thmproofx3.p1.32.m32.3.3.5.cmml" xref="Thmproofx3.p1.32.m32.3.3.5"></eq><apply id="Thmproofx3.p1.32.m32.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.1.1.1"><times id="Thmproofx3.p1.32.m32.1.1.1.2.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.2"></times><apply id="Thmproofx3.p1.32.m32.1.1.1.3.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.32.m32.1.1.1.3.1.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p1.32.m32.1.1.1.3.2.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.3.2">𝜎</ci><apply id="Thmproofx3.p1.32.m32.1.1.1.3.3.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.32.m32.1.1.1.3.3.1.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.3.3">superscript</csymbol><ci id="Thmproofx3.p1.32.m32.1.1.1.3.3.2.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.3.3.2">𝐸</ci><ci id="Thmproofx3.p1.32.m32.1.1.1.3.3.3.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.3.3.3">′</ci></apply></apply><apply id="Thmproofx3.p1.32.m32.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.32.m32.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.32.m32.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.2">𝑒</ci><ci id="Thmproofx3.p1.32.m32.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.32.m32.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx3.p1.32.m32.2.2.2.cmml" xref="Thmproofx3.p1.32.m32.2.2.2"><times id="Thmproofx3.p1.32.m32.2.2.2.2.cmml" xref="Thmproofx3.p1.32.m32.2.2.2.2"></times><ci id="Thmproofx3.p1.32.m32.2.2.2.3.cmml" xref="Thmproofx3.p1.32.m32.2.2.2.3">𝜎</ci><apply id="Thmproofx3.p1.32.m32.2.2.2.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.32.m32.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p1.32.m32.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p1.32.m32.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p1.32.m32.2.2.2.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="Thmproofx3.p1.32.m32.3.3c.cmml" xref="Thmproofx3.p1.32.m32.3.3"><eq id="Thmproofx3.p1.32.m32.3.3.6.cmml" xref="Thmproofx3.p1.32.m32.3.3.6"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx3.p1.32.m32.2.2.2.cmml" id="Thmproofx3.p1.32.m32.3.3d.cmml" xref="Thmproofx3.p1.32.m32.3.3"></share><apply id="Thmproofx3.p1.32.m32.3.3.3.cmml" xref="Thmproofx3.p1.32.m32.3.3.3"><minus id="Thmproofx3.p1.32.m32.3.3.3.2.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.2"></minus><cn id="Thmproofx3.p1.32.m32.3.3.3.3.cmml" type="integer" xref="Thmproofx3.p1.32.m32.3.3.3.3">1</cn><apply id="Thmproofx3.p1.32.m32.3.3.3.1.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1"><apply id="Thmproofx3.p1.32.m32.3.3.3.1.2.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.32.m32.3.3.3.1.2.1.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.2">subscript</csymbol><csymbol cd="latexml" id="Thmproofx3.p1.32.m32.3.3.3.1.2.2.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.2.2">product</csymbol><ci id="Thmproofx3.p1.32.m32.3.3.3.1.2.3.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.2.3">𝑗</ci></apply><apply id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1"><minus id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.2.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.2"></minus><cn id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.3.cmml" type="integer" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.3">1</cn><apply id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1"><times id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.2"></times><apply id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.1.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3">subscript</csymbol><ci id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.2.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.2">𝜎</ci><ci id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.3.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.3.3">𝑗</ci></apply><apply id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p1.32.m32.3.3.3.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.32.m32.3c">\sigma^{E^{\prime}}(e_{i})=\sigma(p_{i})=1-\prod_{j}(1-\sigma_{j}(p_{i}))</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.32.m32.3d">italic_σ start_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 1 - ∏ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx3.p2"> <p class="ltx_p" id="Thmproofx3.p2.9">Now, suppose that <math alttext="\min_{i}\sigma(p_{i})<\max_{i}\sigma(p_{i})" class="ltx_Math" display="inline" id="Thmproofx3.p2.1.m1.2"><semantics id="Thmproofx3.p2.1.m1.2a"><mrow id="Thmproofx3.p2.1.m1.2.2" xref="Thmproofx3.p2.1.m1.2.2.cmml"><mrow id="Thmproofx3.p2.1.m1.1.1.1" xref="Thmproofx3.p2.1.m1.1.1.1.cmml"><mrow id="Thmproofx3.p2.1.m1.1.1.1.3" xref="Thmproofx3.p2.1.m1.1.1.1.3.cmml"><msub id="Thmproofx3.p2.1.m1.1.1.1.3.1" xref="Thmproofx3.p2.1.m1.1.1.1.3.1.cmml"><mi id="Thmproofx3.p2.1.m1.1.1.1.3.1.2" xref="Thmproofx3.p2.1.m1.1.1.1.3.1.2.cmml">min</mi><mi id="Thmproofx3.p2.1.m1.1.1.1.3.1.3" xref="Thmproofx3.p2.1.m1.1.1.1.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.1.m1.1.1.1.3a" lspace="0.167em" xref="Thmproofx3.p2.1.m1.1.1.1.3.cmml">⁡</mo><mi id="Thmproofx3.p2.1.m1.1.1.1.3.2" xref="Thmproofx3.p2.1.m1.1.1.1.3.2.cmml">σ</mi></mrow><mo id="Thmproofx3.p2.1.m1.1.1.1.2" xref="Thmproofx3.p2.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.1.m1.1.1.1.1.1" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.1.m1.1.1.1.1.1.1" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p2.1.m1.1.1.1.1.1.1.2" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p2.1.m1.1.1.1.1.1.1.3" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p2.1.m1.2.2.3" xref="Thmproofx3.p2.1.m1.2.2.3.cmml"><</mo><mrow id="Thmproofx3.p2.1.m1.2.2.2" xref="Thmproofx3.p2.1.m1.2.2.2.cmml"><mrow id="Thmproofx3.p2.1.m1.2.2.2.3" xref="Thmproofx3.p2.1.m1.2.2.2.3.cmml"><msub id="Thmproofx3.p2.1.m1.2.2.2.3.1" xref="Thmproofx3.p2.1.m1.2.2.2.3.1.cmml"><mi id="Thmproofx3.p2.1.m1.2.2.2.3.1.2" xref="Thmproofx3.p2.1.m1.2.2.2.3.1.2.cmml">max</mi><mi id="Thmproofx3.p2.1.m1.2.2.2.3.1.3" xref="Thmproofx3.p2.1.m1.2.2.2.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.1.m1.2.2.2.3a" lspace="0.167em" xref="Thmproofx3.p2.1.m1.2.2.2.3.cmml">⁡</mo><mi id="Thmproofx3.p2.1.m1.2.2.2.3.2" xref="Thmproofx3.p2.1.m1.2.2.2.3.2.cmml">σ</mi></mrow><mo id="Thmproofx3.p2.1.m1.2.2.2.2" xref="Thmproofx3.p2.1.m1.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.1.m1.2.2.2.1.1" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p2.1.m1.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.1.m1.2.2.2.1.1.1" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p2.1.m1.2.2.2.1.1.1.2" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p2.1.m1.2.2.2.1.1.1.3" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.1.m1.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.1.m1.2b"><apply id="Thmproofx3.p2.1.m1.2.2.cmml" xref="Thmproofx3.p2.1.m1.2.2"><lt id="Thmproofx3.p2.1.m1.2.2.3.cmml" xref="Thmproofx3.p2.1.m1.2.2.3"></lt><apply id="Thmproofx3.p2.1.m1.1.1.1.cmml" xref="Thmproofx3.p2.1.m1.1.1.1"><times id="Thmproofx3.p2.1.m1.1.1.1.2.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.2"></times><apply id="Thmproofx3.p2.1.m1.1.1.1.3.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.3"><apply id="Thmproofx3.p2.1.m1.1.1.1.3.1.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.1.m1.1.1.1.3.1.1.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.3.1">subscript</csymbol><min id="Thmproofx3.p2.1.m1.1.1.1.3.1.2.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.3.1.2"></min><ci id="Thmproofx3.p2.1.m1.1.1.1.3.1.3.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.3.1.3">𝑖</ci></apply><ci id="Thmproofx3.p2.1.m1.1.1.1.3.2.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.3.2">𝜎</ci></apply><apply id="Thmproofx3.p2.1.m1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p2.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx3.p2.1.m1.2.2.2.cmml" xref="Thmproofx3.p2.1.m1.2.2.2"><times id="Thmproofx3.p2.1.m1.2.2.2.2.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.2"></times><apply id="Thmproofx3.p2.1.m1.2.2.2.3.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.3"><apply id="Thmproofx3.p2.1.m1.2.2.2.3.1.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.1.m1.2.2.2.3.1.1.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.3.1">subscript</csymbol><max id="Thmproofx3.p2.1.m1.2.2.2.3.1.2.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.3.1.2"></max><ci id="Thmproofx3.p2.1.m1.2.2.2.3.1.3.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.3.1.3">𝑖</ci></apply><ci id="Thmproofx3.p2.1.m1.2.2.2.3.2.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.3.2">𝜎</ci></apply><apply id="Thmproofx3.p2.1.m1.2.2.2.1.1.1.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.1.m1.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p2.1.m1.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p2.1.m1.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p2.1.m1.2.2.2.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.1.m1.2c">\min_{i}\sigma(p_{i})<\max_{i}\sigma(p_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.1.m1.2d">roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) < roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. Then by continuity we can slightly increase <math alttext="\min_{i}\sigma(p_{i})" class="ltx_Math" display="inline" id="Thmproofx3.p2.2.m2.1"><semantics id="Thmproofx3.p2.2.m2.1a"><mrow id="Thmproofx3.p2.2.m2.1.1" xref="Thmproofx3.p2.2.m2.1.1.cmml"><mrow id="Thmproofx3.p2.2.m2.1.1.3" xref="Thmproofx3.p2.2.m2.1.1.3.cmml"><msub id="Thmproofx3.p2.2.m2.1.1.3.1" xref="Thmproofx3.p2.2.m2.1.1.3.1.cmml"><mi id="Thmproofx3.p2.2.m2.1.1.3.1.2" xref="Thmproofx3.p2.2.m2.1.1.3.1.2.cmml">min</mi><mi id="Thmproofx3.p2.2.m2.1.1.3.1.3" xref="Thmproofx3.p2.2.m2.1.1.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.2.m2.1.1.3a" lspace="0.167em" xref="Thmproofx3.p2.2.m2.1.1.3.cmml">⁡</mo><mi id="Thmproofx3.p2.2.m2.1.1.3.2" xref="Thmproofx3.p2.2.m2.1.1.3.2.cmml">σ</mi></mrow><mo id="Thmproofx3.p2.2.m2.1.1.2" xref="Thmproofx3.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.2.m2.1.1.1.1" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.cmml"><mo id="Thmproofx3.p2.2.m2.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.2.m2.1.1.1.1.1" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.cmml"><mi id="Thmproofx3.p2.2.m2.1.1.1.1.1.2" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p2.2.m2.1.1.1.1.1.3" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.2.m2.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.2.m2.1b"><apply id="Thmproofx3.p2.2.m2.1.1.cmml" xref="Thmproofx3.p2.2.m2.1.1"><times id="Thmproofx3.p2.2.m2.1.1.2.cmml" xref="Thmproofx3.p2.2.m2.1.1.2"></times><apply id="Thmproofx3.p2.2.m2.1.1.3.cmml" xref="Thmproofx3.p2.2.m2.1.1.3"><apply id="Thmproofx3.p2.2.m2.1.1.3.1.cmml" xref="Thmproofx3.p2.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.2.m2.1.1.3.1.1.cmml" xref="Thmproofx3.p2.2.m2.1.1.3.1">subscript</csymbol><min id="Thmproofx3.p2.2.m2.1.1.3.1.2.cmml" xref="Thmproofx3.p2.2.m2.1.1.3.1.2"></min><ci id="Thmproofx3.p2.2.m2.1.1.3.1.3.cmml" xref="Thmproofx3.p2.2.m2.1.1.3.1.3">𝑖</ci></apply><ci id="Thmproofx3.p2.2.m2.1.1.3.2.cmml" xref="Thmproofx3.p2.2.m2.1.1.3.2">𝜎</ci></apply><apply id="Thmproofx3.p2.2.m2.1.1.1.1.1.cmml" xref="Thmproofx3.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.2.m2.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.2.m2.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p2.2.m2.1.1.1.1.1.2.cmml" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p2.2.m2.1.1.1.1.1.3.cmml" xref="Thmproofx3.p2.2.m2.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.2.m2.1c">\min_{i}\sigma(p_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.2.m2.1d">roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and slightly decrease each <math alttext="\sigma(p_{i^{\prime}})=\max_{i}\sigma(p_{i})" class="ltx_Math" display="inline" id="Thmproofx3.p2.3.m3.2"><semantics id="Thmproofx3.p2.3.m3.2a"><mrow id="Thmproofx3.p2.3.m3.2.2" xref="Thmproofx3.p2.3.m3.2.2.cmml"><mrow id="Thmproofx3.p2.3.m3.1.1.1" xref="Thmproofx3.p2.3.m3.1.1.1.cmml"><mi id="Thmproofx3.p2.3.m3.1.1.1.3" xref="Thmproofx3.p2.3.m3.1.1.1.3.cmml">σ</mi><mo id="Thmproofx3.p2.3.m3.1.1.1.2" xref="Thmproofx3.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.3.m3.1.1.1.1.1" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p2.3.m3.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.3.m3.1.1.1.1.1.1" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.2" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.2.cmml">p</mi><msup id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.2" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.3" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p2.3.m3.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p2.3.m3.2.2.3" xref="Thmproofx3.p2.3.m3.2.2.3.cmml">=</mo><mrow id="Thmproofx3.p2.3.m3.2.2.2" xref="Thmproofx3.p2.3.m3.2.2.2.cmml"><mrow id="Thmproofx3.p2.3.m3.2.2.2.3" xref="Thmproofx3.p2.3.m3.2.2.2.3.cmml"><msub id="Thmproofx3.p2.3.m3.2.2.2.3.1" xref="Thmproofx3.p2.3.m3.2.2.2.3.1.cmml"><mi id="Thmproofx3.p2.3.m3.2.2.2.3.1.2" xref="Thmproofx3.p2.3.m3.2.2.2.3.1.2.cmml">max</mi><mi id="Thmproofx3.p2.3.m3.2.2.2.3.1.3" xref="Thmproofx3.p2.3.m3.2.2.2.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.3.m3.2.2.2.3a" lspace="0.167em" xref="Thmproofx3.p2.3.m3.2.2.2.3.cmml">⁡</mo><mi id="Thmproofx3.p2.3.m3.2.2.2.3.2" xref="Thmproofx3.p2.3.m3.2.2.2.3.2.cmml">σ</mi></mrow><mo id="Thmproofx3.p2.3.m3.2.2.2.2" xref="Thmproofx3.p2.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.3.m3.2.2.2.1.1" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p2.3.m3.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.3.m3.2.2.2.1.1.1" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p2.3.m3.2.2.2.1.1.1.2" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p2.3.m3.2.2.2.1.1.1.3" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.3.m3.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.3.m3.2b"><apply id="Thmproofx3.p2.3.m3.2.2.cmml" xref="Thmproofx3.p2.3.m3.2.2"><eq id="Thmproofx3.p2.3.m3.2.2.3.cmml" xref="Thmproofx3.p2.3.m3.2.2.3"></eq><apply id="Thmproofx3.p2.3.m3.1.1.1.cmml" xref="Thmproofx3.p2.3.m3.1.1.1"><times id="Thmproofx3.p2.3.m3.1.1.1.2.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.2"></times><ci id="Thmproofx3.p2.3.m3.1.1.1.3.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.3">𝜎</ci><apply id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.2">𝑝</ci><apply id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx3.p2.3.m3.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply><apply id="Thmproofx3.p2.3.m3.2.2.2.cmml" xref="Thmproofx3.p2.3.m3.2.2.2"><times id="Thmproofx3.p2.3.m3.2.2.2.2.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.2"></times><apply id="Thmproofx3.p2.3.m3.2.2.2.3.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.3"><apply id="Thmproofx3.p2.3.m3.2.2.2.3.1.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.3.m3.2.2.2.3.1.1.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.3.1">subscript</csymbol><max id="Thmproofx3.p2.3.m3.2.2.2.3.1.2.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.3.1.2"></max><ci id="Thmproofx3.p2.3.m3.2.2.2.3.1.3.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.3.1.3">𝑖</ci></apply><ci id="Thmproofx3.p2.3.m3.2.2.2.3.2.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.3.2">𝜎</ci></apply><apply id="Thmproofx3.p2.3.m3.2.2.2.1.1.1.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.3.m3.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p2.3.m3.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p2.3.m3.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p2.3.m3.2.2.2.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.3.m3.2c">\sigma(p_{i^{\prime}})=\max_{i}\sigma(p_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.3.m3.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>, to get a product distribution <math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p2.4.m4.1"><semantics id="Thmproofx3.p2.4.m4.1a"><msup id="Thmproofx3.p2.4.m4.1.1" xref="Thmproofx3.p2.4.m4.1.1.cmml"><mi id="Thmproofx3.p2.4.m4.1.1.2" xref="Thmproofx3.p2.4.m4.1.1.2.cmml">σ</mi><mo id="Thmproofx3.p2.4.m4.1.1.3" xref="Thmproofx3.p2.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.4.m4.1b"><apply id="Thmproofx3.p2.4.m4.1.1.cmml" xref="Thmproofx3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.4.m4.1.1.1.cmml" xref="Thmproofx3.p2.4.m4.1.1">superscript</csymbol><ci id="Thmproofx3.p2.4.m4.1.1.2.cmml" xref="Thmproofx3.p2.4.m4.1.1.2">𝜎</ci><ci id="Thmproofx3.p2.4.m4.1.1.3.cmml" xref="Thmproofx3.p2.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.4.m4.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.4.m4.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\max_{i}\sigma^{\prime}(p_{i})<\max_{i}\sigma(p_{i})" class="ltx_Math" display="inline" id="Thmproofx3.p2.5.m5.2"><semantics id="Thmproofx3.p2.5.m5.2a"><mrow id="Thmproofx3.p2.5.m5.2.2" xref="Thmproofx3.p2.5.m5.2.2.cmml"><mrow id="Thmproofx3.p2.5.m5.1.1.1" xref="Thmproofx3.p2.5.m5.1.1.1.cmml"><mrow id="Thmproofx3.p2.5.m5.1.1.1.3" xref="Thmproofx3.p2.5.m5.1.1.1.3.cmml"><msub id="Thmproofx3.p2.5.m5.1.1.1.3.1" xref="Thmproofx3.p2.5.m5.1.1.1.3.1.cmml"><mi id="Thmproofx3.p2.5.m5.1.1.1.3.1.2" xref="Thmproofx3.p2.5.m5.1.1.1.3.1.2.cmml">max</mi><mi id="Thmproofx3.p2.5.m5.1.1.1.3.1.3" xref="Thmproofx3.p2.5.m5.1.1.1.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.5.m5.1.1.1.3a" lspace="0.167em" xref="Thmproofx3.p2.5.m5.1.1.1.3.cmml">⁡</mo><msup id="Thmproofx3.p2.5.m5.1.1.1.3.2" xref="Thmproofx3.p2.5.m5.1.1.1.3.2.cmml"><mi id="Thmproofx3.p2.5.m5.1.1.1.3.2.2" xref="Thmproofx3.p2.5.m5.1.1.1.3.2.2.cmml">σ</mi><mo id="Thmproofx3.p2.5.m5.1.1.1.3.2.3" xref="Thmproofx3.p2.5.m5.1.1.1.3.2.3.cmml">′</mo></msup></mrow><mo id="Thmproofx3.p2.5.m5.1.1.1.2" xref="Thmproofx3.p2.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.5.m5.1.1.1.1.1" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p2.5.m5.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.5.m5.1.1.1.1.1.1" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p2.5.m5.1.1.1.1.1.1.2" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p2.5.m5.1.1.1.1.1.1.3" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.5.m5.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p2.5.m5.2.2.3" xref="Thmproofx3.p2.5.m5.2.2.3.cmml"><</mo><mrow id="Thmproofx3.p2.5.m5.2.2.2" xref="Thmproofx3.p2.5.m5.2.2.2.cmml"><mrow id="Thmproofx3.p2.5.m5.2.2.2.3" xref="Thmproofx3.p2.5.m5.2.2.2.3.cmml"><msub id="Thmproofx3.p2.5.m5.2.2.2.3.1" xref="Thmproofx3.p2.5.m5.2.2.2.3.1.cmml"><mi id="Thmproofx3.p2.5.m5.2.2.2.3.1.2" xref="Thmproofx3.p2.5.m5.2.2.2.3.1.2.cmml">max</mi><mi id="Thmproofx3.p2.5.m5.2.2.2.3.1.3" xref="Thmproofx3.p2.5.m5.2.2.2.3.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.5.m5.2.2.2.3a" lspace="0.167em" xref="Thmproofx3.p2.5.m5.2.2.2.3.cmml">⁡</mo><mi id="Thmproofx3.p2.5.m5.2.2.2.3.2" xref="Thmproofx3.p2.5.m5.2.2.2.3.2.cmml">σ</mi></mrow><mo id="Thmproofx3.p2.5.m5.2.2.2.2" xref="Thmproofx3.p2.5.m5.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.5.m5.2.2.2.1.1" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p2.5.m5.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.5.m5.2.2.2.1.1.1" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p2.5.m5.2.2.2.1.1.1.2" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p2.5.m5.2.2.2.1.1.1.3" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.5.m5.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.5.m5.2b"><apply id="Thmproofx3.p2.5.m5.2.2.cmml" xref="Thmproofx3.p2.5.m5.2.2"><lt id="Thmproofx3.p2.5.m5.2.2.3.cmml" xref="Thmproofx3.p2.5.m5.2.2.3"></lt><apply id="Thmproofx3.p2.5.m5.1.1.1.cmml" xref="Thmproofx3.p2.5.m5.1.1.1"><times id="Thmproofx3.p2.5.m5.1.1.1.2.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.2"></times><apply id="Thmproofx3.p2.5.m5.1.1.1.3.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3"><apply id="Thmproofx3.p2.5.m5.1.1.1.3.1.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.5.m5.1.1.1.3.1.1.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.1">subscript</csymbol><max id="Thmproofx3.p2.5.m5.1.1.1.3.1.2.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.1.2"></max><ci id="Thmproofx3.p2.5.m5.1.1.1.3.1.3.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.1.3">𝑖</ci></apply><apply id="Thmproofx3.p2.5.m5.1.1.1.3.2.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.2"><csymbol cd="ambiguous" id="Thmproofx3.p2.5.m5.1.1.1.3.2.1.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.2">superscript</csymbol><ci id="Thmproofx3.p2.5.m5.1.1.1.3.2.2.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.2.2">𝜎</ci><ci id="Thmproofx3.p2.5.m5.1.1.1.3.2.3.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.3.2.3">′</ci></apply></apply><apply id="Thmproofx3.p2.5.m5.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p2.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx3.p2.5.m5.2.2.2.cmml" xref="Thmproofx3.p2.5.m5.2.2.2"><times id="Thmproofx3.p2.5.m5.2.2.2.2.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.2"></times><apply id="Thmproofx3.p2.5.m5.2.2.2.3.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.3"><apply id="Thmproofx3.p2.5.m5.2.2.2.3.1.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.3.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.5.m5.2.2.2.3.1.1.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.3.1">subscript</csymbol><max id="Thmproofx3.p2.5.m5.2.2.2.3.1.2.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.3.1.2"></max><ci id="Thmproofx3.p2.5.m5.2.2.2.3.1.3.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.3.1.3">𝑖</ci></apply><ci id="Thmproofx3.p2.5.m5.2.2.2.3.2.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.3.2">𝜎</ci></apply><apply id="Thmproofx3.p2.5.m5.2.2.2.1.1.1.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.5.m5.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p2.5.m5.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p2.5.m5.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p2.5.m5.2.2.2.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.5.m5.2c">\max_{i}\sigma^{\prime}(p_{i})<\max_{i}\sigma(p_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.5.m5.2d">roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) < roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. Therefore, at the minimum we have <math alttext="\sigma(p_{i})=\sigma(p_{i^{\prime}})" class="ltx_Math" display="inline" id="Thmproofx3.p2.6.m6.2"><semantics id="Thmproofx3.p2.6.m6.2a"><mrow id="Thmproofx3.p2.6.m6.2.2" xref="Thmproofx3.p2.6.m6.2.2.cmml"><mrow id="Thmproofx3.p2.6.m6.1.1.1" xref="Thmproofx3.p2.6.m6.1.1.1.cmml"><mi id="Thmproofx3.p2.6.m6.1.1.1.3" xref="Thmproofx3.p2.6.m6.1.1.1.3.cmml">σ</mi><mo id="Thmproofx3.p2.6.m6.1.1.1.2" xref="Thmproofx3.p2.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.6.m6.1.1.1.1.1" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmproofx3.p2.6.m6.1.1.1.1.1.2" stretchy="false" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.6.m6.1.1.1.1.1.1" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.cmml"><mi id="Thmproofx3.p2.6.m6.1.1.1.1.1.1.2" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx3.p2.6.m6.1.1.1.1.1.1.3" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx3.p2.6.m6.1.1.1.1.1.3" stretchy="false" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx3.p2.6.m6.2.2.3" xref="Thmproofx3.p2.6.m6.2.2.3.cmml">=</mo><mrow id="Thmproofx3.p2.6.m6.2.2.2" xref="Thmproofx3.p2.6.m6.2.2.2.cmml"><mi id="Thmproofx3.p2.6.m6.2.2.2.3" xref="Thmproofx3.p2.6.m6.2.2.2.3.cmml">σ</mi><mo id="Thmproofx3.p2.6.m6.2.2.2.2" xref="Thmproofx3.p2.6.m6.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx3.p2.6.m6.2.2.2.1.1" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.cmml"><mo id="Thmproofx3.p2.6.m6.2.2.2.1.1.2" stretchy="false" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx3.p2.6.m6.2.2.2.1.1.1" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.cmml"><mi id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.2" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.2.cmml">p</mi><msup id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.cmml"><mi id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.2" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.3" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx3.p2.6.m6.2.2.2.1.1.3" stretchy="false" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.6.m6.2b"><apply id="Thmproofx3.p2.6.m6.2.2.cmml" xref="Thmproofx3.p2.6.m6.2.2"><eq id="Thmproofx3.p2.6.m6.2.2.3.cmml" xref="Thmproofx3.p2.6.m6.2.2.3"></eq><apply id="Thmproofx3.p2.6.m6.1.1.1.cmml" xref="Thmproofx3.p2.6.m6.1.1.1"><times id="Thmproofx3.p2.6.m6.1.1.1.2.cmml" xref="Thmproofx3.p2.6.m6.1.1.1.2"></times><ci id="Thmproofx3.p2.6.m6.1.1.1.3.cmml" xref="Thmproofx3.p2.6.m6.1.1.1.3">𝜎</ci><apply id="Thmproofx3.p2.6.m6.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmproofx3.p2.6.m6.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx3.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx3.p2.6.m6.1.1.1.1.1.1.3.cmml" xref="Thmproofx3.p2.6.m6.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx3.p2.6.m6.2.2.2.cmml" xref="Thmproofx3.p2.6.m6.2.2.2"><times id="Thmproofx3.p2.6.m6.2.2.2.2.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.2"></times><ci id="Thmproofx3.p2.6.m6.2.2.2.3.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.3">𝜎</ci><apply id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.1.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.2.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.2">𝑝</ci><apply id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3">superscript</csymbol><ci id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.2.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.2">𝑖</ci><ci id="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx3.p2.6.m6.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.6.m6.2c">\sigma(p_{i})=\sigma(p_{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.6.m6.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_σ ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math> for every <math alttext="i,i^{\prime}\leq m^{\prime}" class="ltx_Math" display="inline" id="Thmproofx3.p2.7.m7.2"><semantics id="Thmproofx3.p2.7.m7.2a"><mrow id="Thmproofx3.p2.7.m7.2.2" xref="Thmproofx3.p2.7.m7.2.2.cmml"><mrow id="Thmproofx3.p2.7.m7.2.2.1.1" xref="Thmproofx3.p2.7.m7.2.2.1.2.cmml"><mi id="Thmproofx3.p2.7.m7.1.1" xref="Thmproofx3.p2.7.m7.1.1.cmml">i</mi><mo id="Thmproofx3.p2.7.m7.2.2.1.1.2" xref="Thmproofx3.p2.7.m7.2.2.1.2.cmml">,</mo><msup id="Thmproofx3.p2.7.m7.2.2.1.1.1" xref="Thmproofx3.p2.7.m7.2.2.1.1.1.cmml"><mi id="Thmproofx3.p2.7.m7.2.2.1.1.1.2" xref="Thmproofx3.p2.7.m7.2.2.1.1.1.2.cmml">i</mi><mo id="Thmproofx3.p2.7.m7.2.2.1.1.1.3" xref="Thmproofx3.p2.7.m7.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="Thmproofx3.p2.7.m7.2.2.2" xref="Thmproofx3.p2.7.m7.2.2.2.cmml">≤</mo><msup id="Thmproofx3.p2.7.m7.2.2.3" xref="Thmproofx3.p2.7.m7.2.2.3.cmml"><mi id="Thmproofx3.p2.7.m7.2.2.3.2" xref="Thmproofx3.p2.7.m7.2.2.3.2.cmml">m</mi><mo id="Thmproofx3.p2.7.m7.2.2.3.3" xref="Thmproofx3.p2.7.m7.2.2.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.7.m7.2b"><apply id="Thmproofx3.p2.7.m7.2.2.cmml" xref="Thmproofx3.p2.7.m7.2.2"><leq id="Thmproofx3.p2.7.m7.2.2.2.cmml" xref="Thmproofx3.p2.7.m7.2.2.2"></leq><list id="Thmproofx3.p2.7.m7.2.2.1.2.cmml" xref="Thmproofx3.p2.7.m7.2.2.1.1"><ci id="Thmproofx3.p2.7.m7.1.1.cmml" xref="Thmproofx3.p2.7.m7.1.1">𝑖</ci><apply id="Thmproofx3.p2.7.m7.2.2.1.1.1.cmml" xref="Thmproofx3.p2.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p2.7.m7.2.2.1.1.1.1.cmml" xref="Thmproofx3.p2.7.m7.2.2.1.1.1">superscript</csymbol><ci id="Thmproofx3.p2.7.m7.2.2.1.1.1.2.cmml" xref="Thmproofx3.p2.7.m7.2.2.1.1.1.2">𝑖</ci><ci id="Thmproofx3.p2.7.m7.2.2.1.1.1.3.cmml" xref="Thmproofx3.p2.7.m7.2.2.1.1.1.3">′</ci></apply></list><apply id="Thmproofx3.p2.7.m7.2.2.3.cmml" xref="Thmproofx3.p2.7.m7.2.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p2.7.m7.2.2.3.1.cmml" xref="Thmproofx3.p2.7.m7.2.2.3">superscript</csymbol><ci id="Thmproofx3.p2.7.m7.2.2.3.2.cmml" xref="Thmproofx3.p2.7.m7.2.2.3.2">𝑚</ci><ci id="Thmproofx3.p2.7.m7.2.2.3.3.cmml" xref="Thmproofx3.p2.7.m7.2.2.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.7.m7.2c">i,i^{\prime}\leq m^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.7.m7.2d">italic_i , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We finish the proof by applying Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma19" title="Lemma 19 ‣ Appendix 0.A Proofs of General Lemmas ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">19</span></a> and with it evaluating <math alttext="\beta" class="ltx_Math" display="inline" id="Thmproofx3.p2.8.m8.1"><semantics id="Thmproofx3.p2.8.m8.1a"><mi id="Thmproofx3.p2.8.m8.1.1" xref="Thmproofx3.p2.8.m8.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.8.m8.1b"><ci id="Thmproofx3.p2.8.m8.1.1.cmml" xref="Thmproofx3.p2.8.m8.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.8.m8.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.8.m8.1d">italic_β</annotation></semantics></math> at <math alttext="\sigma=U_{P_{f}}^{n}" class="ltx_Math" display="inline" id="Thmproofx3.p2.9.m9.1"><semantics id="Thmproofx3.p2.9.m9.1a"><mrow id="Thmproofx3.p2.9.m9.1.1" xref="Thmproofx3.p2.9.m9.1.1.cmml"><mi id="Thmproofx3.p2.9.m9.1.1.2" xref="Thmproofx3.p2.9.m9.1.1.2.cmml">σ</mi><mo id="Thmproofx3.p2.9.m9.1.1.1" xref="Thmproofx3.p2.9.m9.1.1.1.cmml">=</mo><msubsup id="Thmproofx3.p2.9.m9.1.1.3" xref="Thmproofx3.p2.9.m9.1.1.3.cmml"><mi id="Thmproofx3.p2.9.m9.1.1.3.2.2" xref="Thmproofx3.p2.9.m9.1.1.3.2.2.cmml">U</mi><msub id="Thmproofx3.p2.9.m9.1.1.3.2.3" xref="Thmproofx3.p2.9.m9.1.1.3.2.3.cmml"><mi id="Thmproofx3.p2.9.m9.1.1.3.2.3.2" xref="Thmproofx3.p2.9.m9.1.1.3.2.3.2.cmml">P</mi><mi id="Thmproofx3.p2.9.m9.1.1.3.2.3.3" xref="Thmproofx3.p2.9.m9.1.1.3.2.3.3.cmml">f</mi></msub><mi id="Thmproofx3.p2.9.m9.1.1.3.3" xref="Thmproofx3.p2.9.m9.1.1.3.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p2.9.m9.1b"><apply id="Thmproofx3.p2.9.m9.1.1.cmml" xref="Thmproofx3.p2.9.m9.1.1"><eq id="Thmproofx3.p2.9.m9.1.1.1.cmml" xref="Thmproofx3.p2.9.m9.1.1.1"></eq><ci id="Thmproofx3.p2.9.m9.1.1.2.cmml" xref="Thmproofx3.p2.9.m9.1.1.2">𝜎</ci><apply id="Thmproofx3.p2.9.m9.1.1.3.cmml" xref="Thmproofx3.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p2.9.m9.1.1.3.1.cmml" xref="Thmproofx3.p2.9.m9.1.1.3">superscript</csymbol><apply id="Thmproofx3.p2.9.m9.1.1.3.2.cmml" xref="Thmproofx3.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="Thmproofx3.p2.9.m9.1.1.3.2.1.cmml" xref="Thmproofx3.p2.9.m9.1.1.3">subscript</csymbol><ci id="Thmproofx3.p2.9.m9.1.1.3.2.2.cmml" xref="Thmproofx3.p2.9.m9.1.1.3.2.2">𝑈</ci><apply id="Thmproofx3.p2.9.m9.1.1.3.2.3.cmml" xref="Thmproofx3.p2.9.m9.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmproofx3.p2.9.m9.1.1.3.2.3.1.cmml" xref="Thmproofx3.p2.9.m9.1.1.3.2.3">subscript</csymbol><ci id="Thmproofx3.p2.9.m9.1.1.3.2.3.2.cmml" xref="Thmproofx3.p2.9.m9.1.1.3.2.3.2">𝑃</ci><ci id="Thmproofx3.p2.9.m9.1.1.3.2.3.3.cmml" xref="Thmproofx3.p2.9.m9.1.1.3.2.3.3">𝑓</ci></apply></apply><ci id="Thmproofx3.p2.9.m9.1.1.3.3.cmml" xref="Thmproofx3.p2.9.m9.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p2.9.m9.1c">\sigma=U_{P_{f}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p2.9.m9.1d">italic_σ = italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx4"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma5" title="Lemma 5 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a>)</h6> <div class="ltx_para" id="Thmproofx4.p1"> <p class="ltx_p" id="Thmproofx4.p1.4">Let <math alttext="\sigma\in\Delta(P_{st})" class="ltx_Math" display="inline" id="Thmproofx4.p1.1.m1.1"><semantics id="Thmproofx4.p1.1.m1.1a"><mrow id="Thmproofx4.p1.1.m1.1.1" xref="Thmproofx4.p1.1.m1.1.1.cmml"><mi id="Thmproofx4.p1.1.m1.1.1.3" xref="Thmproofx4.p1.1.m1.1.1.3.cmml">σ</mi><mo id="Thmproofx4.p1.1.m1.1.1.2" xref="Thmproofx4.p1.1.m1.1.1.2.cmml">∈</mo><mrow id="Thmproofx4.p1.1.m1.1.1.1" xref="Thmproofx4.p1.1.m1.1.1.1.cmml"><mi id="Thmproofx4.p1.1.m1.1.1.1.3" mathvariant="normal" xref="Thmproofx4.p1.1.m1.1.1.1.3.cmml">Δ</mi><mo id="Thmproofx4.p1.1.m1.1.1.1.2" xref="Thmproofx4.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx4.p1.1.m1.1.1.1.1.1" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmproofx4.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx4.p1.1.m1.1.1.1.1.1.1" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.2" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.2" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.1" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.3" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmproofx4.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.1.m1.1b"><apply id="Thmproofx4.p1.1.m1.1.1.cmml" xref="Thmproofx4.p1.1.m1.1.1"><in id="Thmproofx4.p1.1.m1.1.1.2.cmml" xref="Thmproofx4.p1.1.m1.1.1.2"></in><ci id="Thmproofx4.p1.1.m1.1.1.3.cmml" xref="Thmproofx4.p1.1.m1.1.1.3">𝜎</ci><apply id="Thmproofx4.p1.1.m1.1.1.1.cmml" xref="Thmproofx4.p1.1.m1.1.1.1"><times id="Thmproofx4.p1.1.m1.1.1.1.2.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.2"></times><ci id="Thmproofx4.p1.1.m1.1.1.1.3.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.3">Δ</ci><apply id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3"><times id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.1"></times><ci id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx4.p1.1.m1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.1.m1.1c">\sigma\in\Delta(P_{st})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.1.m1.1d">italic_σ ∈ roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT )</annotation></semantics></math>. The inequality <math alttext="\beta(\sigma^{n})\leq m_{c}\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx4.p1.2.m2.2"><semantics id="Thmproofx4.p1.2.m2.2a"><mrow id="Thmproofx4.p1.2.m2.2.2" xref="Thmproofx4.p1.2.m2.2.2.cmml"><mrow id="Thmproofx4.p1.2.m2.2.2.1" xref="Thmproofx4.p1.2.m2.2.2.1.cmml"><mi id="Thmproofx4.p1.2.m2.2.2.1.3" xref="Thmproofx4.p1.2.m2.2.2.1.3.cmml">β</mi><mo id="Thmproofx4.p1.2.m2.2.2.1.2" xref="Thmproofx4.p1.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="Thmproofx4.p1.2.m2.2.2.1.1.1" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.cmml"><mo id="Thmproofx4.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.cmml">(</mo><msup id="Thmproofx4.p1.2.m2.2.2.1.1.1.1" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.cmml"><mi id="Thmproofx4.p1.2.m2.2.2.1.1.1.1.2" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.2.cmml">σ</mi><mi id="Thmproofx4.p1.2.m2.2.2.1.1.1.1.3" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx4.p1.2.m2.2.2.1.1.1.3" stretchy="false" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.2.m2.2.2.2" xref="Thmproofx4.p1.2.m2.2.2.2.cmml">≤</mo><mrow id="Thmproofx4.p1.2.m2.2.2.3" xref="Thmproofx4.p1.2.m2.2.2.3.cmml"><msub id="Thmproofx4.p1.2.m2.2.2.3.2" xref="Thmproofx4.p1.2.m2.2.2.3.2.cmml"><mi id="Thmproofx4.p1.2.m2.2.2.3.2.2" xref="Thmproofx4.p1.2.m2.2.2.3.2.2.cmml">m</mi><mi id="Thmproofx4.p1.2.m2.2.2.3.2.3" xref="Thmproofx4.p1.2.m2.2.2.3.2.3.cmml">c</mi></msub><mo id="Thmproofx4.p1.2.m2.2.2.3.1" xref="Thmproofx4.p1.2.m2.2.2.3.1.cmml">⁢</mo><mi id="Thmproofx4.p1.2.m2.2.2.3.3" xref="Thmproofx4.p1.2.m2.2.2.3.3.cmml">β</mi><mo id="Thmproofx4.p1.2.m2.2.2.3.1a" xref="Thmproofx4.p1.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.2.m2.2.2.3.4.2" xref="Thmproofx4.p1.2.m2.2.2.3.cmml"><mo id="Thmproofx4.p1.2.m2.2.2.3.4.2.1" stretchy="false" xref="Thmproofx4.p1.2.m2.2.2.3.cmml">(</mo><mi id="Thmproofx4.p1.2.m2.1.1" xref="Thmproofx4.p1.2.m2.1.1.cmml">σ</mi><mo id="Thmproofx4.p1.2.m2.2.2.3.4.2.2" stretchy="false" xref="Thmproofx4.p1.2.m2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.2.m2.2b"><apply id="Thmproofx4.p1.2.m2.2.2.cmml" xref="Thmproofx4.p1.2.m2.2.2"><leq id="Thmproofx4.p1.2.m2.2.2.2.cmml" xref="Thmproofx4.p1.2.m2.2.2.2"></leq><apply id="Thmproofx4.p1.2.m2.2.2.1.cmml" xref="Thmproofx4.p1.2.m2.2.2.1"><times id="Thmproofx4.p1.2.m2.2.2.1.2.cmml" xref="Thmproofx4.p1.2.m2.2.2.1.2"></times><ci id="Thmproofx4.p1.2.m2.2.2.1.3.cmml" xref="Thmproofx4.p1.2.m2.2.2.1.3">𝛽</ci><apply id="Thmproofx4.p1.2.m2.2.2.1.1.1.1.cmml" xref="Thmproofx4.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="Thmproofx4.p1.2.m2.2.2.1.1.1">superscript</csymbol><ci id="Thmproofx4.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.2">𝜎</ci><ci id="Thmproofx4.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="Thmproofx4.p1.2.m2.2.2.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmproofx4.p1.2.m2.2.2.3.cmml" xref="Thmproofx4.p1.2.m2.2.2.3"><times id="Thmproofx4.p1.2.m2.2.2.3.1.cmml" xref="Thmproofx4.p1.2.m2.2.2.3.1"></times><apply id="Thmproofx4.p1.2.m2.2.2.3.2.cmml" xref="Thmproofx4.p1.2.m2.2.2.3.2"><csymbol cd="ambiguous" id="Thmproofx4.p1.2.m2.2.2.3.2.1.cmml" xref="Thmproofx4.p1.2.m2.2.2.3.2">subscript</csymbol><ci id="Thmproofx4.p1.2.m2.2.2.3.2.2.cmml" xref="Thmproofx4.p1.2.m2.2.2.3.2.2">𝑚</ci><ci id="Thmproofx4.p1.2.m2.2.2.3.2.3.cmml" xref="Thmproofx4.p1.2.m2.2.2.3.2.3">𝑐</ci></apply><ci id="Thmproofx4.p1.2.m2.2.2.3.3.cmml" xref="Thmproofx4.p1.2.m2.2.2.3.3">𝛽</ci><ci id="Thmproofx4.p1.2.m2.1.1.cmml" xref="Thmproofx4.p1.2.m2.1.1">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.2.m2.2c">\beta(\sigma^{n})\leq m_{c}\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.2.m2.2d">italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ≤ italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_β ( italic_σ )</annotation></semantics></math> follows from <math alttext="\beta(\sigma)\geq\frac{1}{m_{c}}" class="ltx_Math" display="inline" id="Thmproofx4.p1.3.m3.1"><semantics id="Thmproofx4.p1.3.m3.1a"><mrow id="Thmproofx4.p1.3.m3.1.2" xref="Thmproofx4.p1.3.m3.1.2.cmml"><mrow id="Thmproofx4.p1.3.m3.1.2.2" xref="Thmproofx4.p1.3.m3.1.2.2.cmml"><mi id="Thmproofx4.p1.3.m3.1.2.2.2" xref="Thmproofx4.p1.3.m3.1.2.2.2.cmml">β</mi><mo id="Thmproofx4.p1.3.m3.1.2.2.1" xref="Thmproofx4.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.3.m3.1.2.2.3.2" xref="Thmproofx4.p1.3.m3.1.2.2.cmml"><mo id="Thmproofx4.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="Thmproofx4.p1.3.m3.1.2.2.cmml">(</mo><mi id="Thmproofx4.p1.3.m3.1.1" xref="Thmproofx4.p1.3.m3.1.1.cmml">σ</mi><mo id="Thmproofx4.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="Thmproofx4.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.3.m3.1.2.1" xref="Thmproofx4.p1.3.m3.1.2.1.cmml">≥</mo><mfrac id="Thmproofx4.p1.3.m3.1.2.3" xref="Thmproofx4.p1.3.m3.1.2.3.cmml"><mn id="Thmproofx4.p1.3.m3.1.2.3.2" xref="Thmproofx4.p1.3.m3.1.2.3.2.cmml">1</mn><msub id="Thmproofx4.p1.3.m3.1.2.3.3" xref="Thmproofx4.p1.3.m3.1.2.3.3.cmml"><mi id="Thmproofx4.p1.3.m3.1.2.3.3.2" xref="Thmproofx4.p1.3.m3.1.2.3.3.2.cmml">m</mi><mi id="Thmproofx4.p1.3.m3.1.2.3.3.3" xref="Thmproofx4.p1.3.m3.1.2.3.3.3.cmml">c</mi></msub></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.3.m3.1b"><apply id="Thmproofx4.p1.3.m3.1.2.cmml" xref="Thmproofx4.p1.3.m3.1.2"><geq id="Thmproofx4.p1.3.m3.1.2.1.cmml" xref="Thmproofx4.p1.3.m3.1.2.1"></geq><apply id="Thmproofx4.p1.3.m3.1.2.2.cmml" xref="Thmproofx4.p1.3.m3.1.2.2"><times id="Thmproofx4.p1.3.m3.1.2.2.1.cmml" xref="Thmproofx4.p1.3.m3.1.2.2.1"></times><ci id="Thmproofx4.p1.3.m3.1.2.2.2.cmml" xref="Thmproofx4.p1.3.m3.1.2.2.2">𝛽</ci><ci id="Thmproofx4.p1.3.m3.1.1.cmml" xref="Thmproofx4.p1.3.m3.1.1">𝜎</ci></apply><apply id="Thmproofx4.p1.3.m3.1.2.3.cmml" xref="Thmproofx4.p1.3.m3.1.2.3"><divide id="Thmproofx4.p1.3.m3.1.2.3.1.cmml" xref="Thmproofx4.p1.3.m3.1.2.3"></divide><cn id="Thmproofx4.p1.3.m3.1.2.3.2.cmml" type="integer" xref="Thmproofx4.p1.3.m3.1.2.3.2">1</cn><apply id="Thmproofx4.p1.3.m3.1.2.3.3.cmml" xref="Thmproofx4.p1.3.m3.1.2.3.3"><csymbol cd="ambiguous" id="Thmproofx4.p1.3.m3.1.2.3.3.1.cmml" xref="Thmproofx4.p1.3.m3.1.2.3.3">subscript</csymbol><ci id="Thmproofx4.p1.3.m3.1.2.3.3.2.cmml" xref="Thmproofx4.p1.3.m3.1.2.3.3.2">𝑚</ci><ci id="Thmproofx4.p1.3.m3.1.2.3.3.3.cmml" xref="Thmproofx4.p1.3.m3.1.2.3.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.3.m3.1c">\beta(\sigma)\geq\frac{1}{m_{c}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.3.m3.1d">italic_β ( italic_σ ) ≥ divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> (Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma4" title="Lemma 4 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a>) and <math alttext="\beta(\sigma^{n})\leq 1" class="ltx_Math" display="inline" id="Thmproofx4.p1.4.m4.1"><semantics id="Thmproofx4.p1.4.m4.1a"><mrow id="Thmproofx4.p1.4.m4.1.1" xref="Thmproofx4.p1.4.m4.1.1.cmml"><mrow id="Thmproofx4.p1.4.m4.1.1.1" xref="Thmproofx4.p1.4.m4.1.1.1.cmml"><mi id="Thmproofx4.p1.4.m4.1.1.1.3" xref="Thmproofx4.p1.4.m4.1.1.1.3.cmml">β</mi><mo id="Thmproofx4.p1.4.m4.1.1.1.2" xref="Thmproofx4.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx4.p1.4.m4.1.1.1.1.1" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="Thmproofx4.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="Thmproofx4.p1.4.m4.1.1.1.1.1.1" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmproofx4.p1.4.m4.1.1.1.1.1.1.2" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.2.cmml">σ</mi><mi id="Thmproofx4.p1.4.m4.1.1.1.1.1.1.3" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx4.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.4.m4.1.1.2" xref="Thmproofx4.p1.4.m4.1.1.2.cmml">≤</mo><mn id="Thmproofx4.p1.4.m4.1.1.3" xref="Thmproofx4.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.4.m4.1b"><apply id="Thmproofx4.p1.4.m4.1.1.cmml" xref="Thmproofx4.p1.4.m4.1.1"><leq id="Thmproofx4.p1.4.m4.1.1.2.cmml" xref="Thmproofx4.p1.4.m4.1.1.2"></leq><apply id="Thmproofx4.p1.4.m4.1.1.1.cmml" xref="Thmproofx4.p1.4.m4.1.1.1"><times id="Thmproofx4.p1.4.m4.1.1.1.2.cmml" xref="Thmproofx4.p1.4.m4.1.1.1.2"></times><ci id="Thmproofx4.p1.4.m4.1.1.1.3.cmml" xref="Thmproofx4.p1.4.m4.1.1.1.3">𝛽</ci><apply id="Thmproofx4.p1.4.m4.1.1.1.1.1.1.cmml" xref="Thmproofx4.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmproofx4.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx4.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.2">𝜎</ci><ci id="Thmproofx4.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmproofx4.p1.4.m4.1.1.1.1.1.1.3">𝑛</ci></apply></apply><cn id="Thmproofx4.p1.4.m4.1.1.3.cmml" type="integer" xref="Thmproofx4.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.4.m4.1c">\beta(\sigma^{n})\leq 1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.4.m4.1d">italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ≤ 1</annotation></semantics></math>. By Bernoulli’s inequality,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx22"> <tbody id="Pt0.A1.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\beta(\sigma^{n})=\max_{e}\left(1-(1-\sigma^{E}(e))^{n}\right)% \leq n\max_{e}\sigma^{E}(e)=n\beta(\sigma)" class="ltx_Math" display="inline" id="Pt0.A1.Ex23.m1.6"><semantics id="Pt0.A1.Ex23.m1.6a"><mrow id="Pt0.A1.Ex23.m1.6.6" xref="Pt0.A1.Ex23.m1.6.6.cmml"><mrow id="Pt0.A1.Ex23.m1.4.4.1" xref="Pt0.A1.Ex23.m1.4.4.1.cmml"><mi id="Pt0.A1.Ex23.m1.4.4.1.3" xref="Pt0.A1.Ex23.m1.4.4.1.3.cmml">β</mi><mo id="Pt0.A1.Ex23.m1.4.4.1.2" xref="Pt0.A1.Ex23.m1.4.4.1.2.cmml">⁢</mo><mrow id="Pt0.A1.Ex23.m1.4.4.1.1.1" xref="Pt0.A1.Ex23.m1.4.4.1.1.1.1.cmml"><mo id="Pt0.A1.Ex23.m1.4.4.1.1.1.2" stretchy="false" xref="Pt0.A1.Ex23.m1.4.4.1.1.1.1.cmml">(</mo><msup id="Pt0.A1.Ex23.m1.4.4.1.1.1.1" xref="Pt0.A1.Ex23.m1.4.4.1.1.1.1.cmml"><mi id="Pt0.A1.Ex23.m1.4.4.1.1.1.1.2" xref="Pt0.A1.Ex23.m1.4.4.1.1.1.1.2.cmml">σ</mi><mi id="Pt0.A1.Ex23.m1.4.4.1.1.1.1.3" xref="Pt0.A1.Ex23.m1.4.4.1.1.1.1.3.cmml">n</mi></msup><mo id="Pt0.A1.Ex23.m1.4.4.1.1.1.3" stretchy="false" xref="Pt0.A1.Ex23.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.Ex23.m1.6.6.5" xref="Pt0.A1.Ex23.m1.6.6.5.cmml">=</mo><mrow id="Pt0.A1.Ex23.m1.6.6.3.2" xref="Pt0.A1.Ex23.m1.6.6.3.3.cmml"><munder id="Pt0.A1.Ex23.m1.5.5.2.1.1" xref="Pt0.A1.Ex23.m1.5.5.2.1.1.cmml"><mi id="Pt0.A1.Ex23.m1.5.5.2.1.1.2" xref="Pt0.A1.Ex23.m1.5.5.2.1.1.2.cmml">max</mi><mi id="Pt0.A1.Ex23.m1.5.5.2.1.1.3" xref="Pt0.A1.Ex23.m1.5.5.2.1.1.3.cmml">e</mi></munder><mo id="Pt0.A1.Ex23.m1.6.6.3.2a" xref="Pt0.A1.Ex23.m1.6.6.3.3.cmml">⁡</mo><mrow id="Pt0.A1.Ex23.m1.6.6.3.2.2" xref="Pt0.A1.Ex23.m1.6.6.3.3.cmml"><mo id="Pt0.A1.Ex23.m1.6.6.3.2.2.2" xref="Pt0.A1.Ex23.m1.6.6.3.3.cmml">(</mo><mrow id="Pt0.A1.Ex23.m1.6.6.3.2.2.1" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.cmml"><mn id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.3" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.3.cmml">1</mn><mo id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.2" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.2.cmml">−</mo><msup id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.cmml"><mrow id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.cmml"><mo id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.2" stretchy="false" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.cmml"><mn id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.2" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.1" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.1.cmml">−</mo><mrow id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.3" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.3.cmml"><msup id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.3.2" 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xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.3.cmml">n</mi></msup></mrow><mo id="Pt0.A1.Ex23.m1.6.6.3.2.2.3" xref="Pt0.A1.Ex23.m1.6.6.3.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.Ex23.m1.6.6.6" xref="Pt0.A1.Ex23.m1.6.6.6.cmml">≤</mo><mrow id="Pt0.A1.Ex23.m1.6.6.7" xref="Pt0.A1.Ex23.m1.6.6.7.cmml"><mi id="Pt0.A1.Ex23.m1.6.6.7.2" xref="Pt0.A1.Ex23.m1.6.6.7.2.cmml">n</mi><mo id="Pt0.A1.Ex23.m1.6.6.7.1" lspace="0.167em" xref="Pt0.A1.Ex23.m1.6.6.7.1.cmml">⁢</mo><mrow id="Pt0.A1.Ex23.m1.6.6.7.3" xref="Pt0.A1.Ex23.m1.6.6.7.3.cmml"><munder id="Pt0.A1.Ex23.m1.6.6.7.3.1" xref="Pt0.A1.Ex23.m1.6.6.7.3.1.cmml"><mi id="Pt0.A1.Ex23.m1.6.6.7.3.1.2" xref="Pt0.A1.Ex23.m1.6.6.7.3.1.2.cmml">max</mi><mi id="Pt0.A1.Ex23.m1.6.6.7.3.1.3" xref="Pt0.A1.Ex23.m1.6.6.7.3.1.3.cmml">e</mi></munder><mo id="Pt0.A1.Ex23.m1.6.6.7.3a" lspace="0.167em" xref="Pt0.A1.Ex23.m1.6.6.7.3.cmml">⁡</mo><msup id="Pt0.A1.Ex23.m1.6.6.7.3.2" xref="Pt0.A1.Ex23.m1.6.6.7.3.2.cmml"><mi id="Pt0.A1.Ex23.m1.6.6.7.3.2.2" xref="Pt0.A1.Ex23.m1.6.6.7.3.2.2.cmml">σ</mi><mi id="Pt0.A1.Ex23.m1.6.6.7.3.2.3" xref="Pt0.A1.Ex23.m1.6.6.7.3.2.3.cmml">E</mi></msup></mrow><mo id="Pt0.A1.Ex23.m1.6.6.7.1a" xref="Pt0.A1.Ex23.m1.6.6.7.1.cmml">⁢</mo><mrow id="Pt0.A1.Ex23.m1.6.6.7.4.2" xref="Pt0.A1.Ex23.m1.6.6.7.cmml"><mo id="Pt0.A1.Ex23.m1.6.6.7.4.2.1" stretchy="false" xref="Pt0.A1.Ex23.m1.6.6.7.cmml">(</mo><mi id="Pt0.A1.Ex23.m1.2.2" xref="Pt0.A1.Ex23.m1.2.2.cmml">e</mi><mo id="Pt0.A1.Ex23.m1.6.6.7.4.2.2" stretchy="false" xref="Pt0.A1.Ex23.m1.6.6.7.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.Ex23.m1.6.6.8" xref="Pt0.A1.Ex23.m1.6.6.8.cmml">=</mo><mrow id="Pt0.A1.Ex23.m1.6.6.9" xref="Pt0.A1.Ex23.m1.6.6.9.cmml"><mi id="Pt0.A1.Ex23.m1.6.6.9.2" xref="Pt0.A1.Ex23.m1.6.6.9.2.cmml">n</mi><mo id="Pt0.A1.Ex23.m1.6.6.9.1" xref="Pt0.A1.Ex23.m1.6.6.9.1.cmml">⁢</mo><mi id="Pt0.A1.Ex23.m1.6.6.9.3" xref="Pt0.A1.Ex23.m1.6.6.9.3.cmml">β</mi><mo id="Pt0.A1.Ex23.m1.6.6.9.1a" xref="Pt0.A1.Ex23.m1.6.6.9.1.cmml">⁢</mo><mrow 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id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.3.2.2.cmml" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.3.2.2">𝜎</ci><ci id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.3.2.3.cmml" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.1.1.1.3.2.3">𝐸</ci></apply><ci id="Pt0.A1.Ex23.m1.1.1.cmml" xref="Pt0.A1.Ex23.m1.1.1">𝑒</ci></apply></apply><ci id="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.3.cmml" xref="Pt0.A1.Ex23.m1.6.6.3.2.2.1.1.3">𝑛</ci></apply></apply></apply></apply><apply id="Pt0.A1.Ex23.m1.6.6c.cmml" xref="Pt0.A1.Ex23.m1.6.6"><leq id="Pt0.A1.Ex23.m1.6.6.6.cmml" xref="Pt0.A1.Ex23.m1.6.6.6"></leq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A1.Ex23.m1.6.6.3.cmml" id="Pt0.A1.Ex23.m1.6.6d.cmml" xref="Pt0.A1.Ex23.m1.6.6"></share><apply id="Pt0.A1.Ex23.m1.6.6.7.cmml" xref="Pt0.A1.Ex23.m1.6.6.7"><times id="Pt0.A1.Ex23.m1.6.6.7.1.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.1"></times><ci id="Pt0.A1.Ex23.m1.6.6.7.2.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.2">𝑛</ci><apply id="Pt0.A1.Ex23.m1.6.6.7.3.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3"><apply id="Pt0.A1.Ex23.m1.6.6.7.3.1.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.1"><csymbol cd="ambiguous" id="Pt0.A1.Ex23.m1.6.6.7.3.1.1.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.1">subscript</csymbol><max id="Pt0.A1.Ex23.m1.6.6.7.3.1.2.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.1.2"></max><ci id="Pt0.A1.Ex23.m1.6.6.7.3.1.3.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.1.3">𝑒</ci></apply><apply id="Pt0.A1.Ex23.m1.6.6.7.3.2.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.2"><csymbol cd="ambiguous" id="Pt0.A1.Ex23.m1.6.6.7.3.2.1.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.2">superscript</csymbol><ci id="Pt0.A1.Ex23.m1.6.6.7.3.2.2.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.2.2">𝜎</ci><ci id="Pt0.A1.Ex23.m1.6.6.7.3.2.3.cmml" xref="Pt0.A1.Ex23.m1.6.6.7.3.2.3">𝐸</ci></apply></apply><ci id="Pt0.A1.Ex23.m1.2.2.cmml" xref="Pt0.A1.Ex23.m1.2.2">𝑒</ci></apply></apply><apply id="Pt0.A1.Ex23.m1.6.6e.cmml" xref="Pt0.A1.Ex23.m1.6.6"><eq id="Pt0.A1.Ex23.m1.6.6.8.cmml" xref="Pt0.A1.Ex23.m1.6.6.8"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A1.Ex23.m1.6.6.7.cmml" id="Pt0.A1.Ex23.m1.6.6f.cmml" xref="Pt0.A1.Ex23.m1.6.6"></share><apply id="Pt0.A1.Ex23.m1.6.6.9.cmml" xref="Pt0.A1.Ex23.m1.6.6.9"><times id="Pt0.A1.Ex23.m1.6.6.9.1.cmml" xref="Pt0.A1.Ex23.m1.6.6.9.1"></times><ci id="Pt0.A1.Ex23.m1.6.6.9.2.cmml" xref="Pt0.A1.Ex23.m1.6.6.9.2">𝑛</ci><ci id="Pt0.A1.Ex23.m1.6.6.9.3.cmml" xref="Pt0.A1.Ex23.m1.6.6.9.3">𝛽</ci><ci id="Pt0.A1.Ex23.m1.3.3.cmml" xref="Pt0.A1.Ex23.m1.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex23.m1.6c">\displaystyle\beta(\sigma^{n})=\max_{e}\left(1-(1-\sigma^{E}(e))^{n}\right)% \leq n\max_{e}\sigma^{E}(e)=n\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex23.m1.6d">italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) = roman_max start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( 1 - ( 1 - italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ≤ italic_n roman_max start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e ) = italic_n italic_β ( italic_σ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx4.p1.9">so <math alttext="\beta(\sigma^{n})\leq\min(m_{c},n)\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx4.p1.5.m1.5"><semantics id="Thmproofx4.p1.5.m1.5a"><mrow id="Thmproofx4.p1.5.m1.5.5" xref="Thmproofx4.p1.5.m1.5.5.cmml"><mrow id="Thmproofx4.p1.5.m1.4.4.1" xref="Thmproofx4.p1.5.m1.4.4.1.cmml"><mi id="Thmproofx4.p1.5.m1.4.4.1.3" xref="Thmproofx4.p1.5.m1.4.4.1.3.cmml">β</mi><mo id="Thmproofx4.p1.5.m1.4.4.1.2" xref="Thmproofx4.p1.5.m1.4.4.1.2.cmml">⁢</mo><mrow id="Thmproofx4.p1.5.m1.4.4.1.1.1" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.cmml"><mo id="Thmproofx4.p1.5.m1.4.4.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.cmml">(</mo><msup id="Thmproofx4.p1.5.m1.4.4.1.1.1.1" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.cmml"><mi id="Thmproofx4.p1.5.m1.4.4.1.1.1.1.2" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.2.cmml">σ</mi><mi id="Thmproofx4.p1.5.m1.4.4.1.1.1.1.3" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx4.p1.5.m1.4.4.1.1.1.3" stretchy="false" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.5.m1.5.5.3" xref="Thmproofx4.p1.5.m1.5.5.3.cmml">≤</mo><mrow id="Thmproofx4.p1.5.m1.5.5.2" xref="Thmproofx4.p1.5.m1.5.5.2.cmml"><mrow id="Thmproofx4.p1.5.m1.5.5.2.1.1" xref="Thmproofx4.p1.5.m1.5.5.2.1.2.cmml"><mi id="Thmproofx4.p1.5.m1.1.1" xref="Thmproofx4.p1.5.m1.1.1.cmml">min</mi><mo id="Thmproofx4.p1.5.m1.5.5.2.1.1a" xref="Thmproofx4.p1.5.m1.5.5.2.1.2.cmml">⁡</mo><mrow id="Thmproofx4.p1.5.m1.5.5.2.1.1.1" xref="Thmproofx4.p1.5.m1.5.5.2.1.2.cmml"><mo id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.5.m1.5.5.2.1.2.cmml">(</mo><msub id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1" xref="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.cmml"><mi id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.2" xref="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.2.cmml">m</mi><mi id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.3" xref="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.3.cmml">c</mi></msub><mo id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.3" xref="Thmproofx4.p1.5.m1.5.5.2.1.2.cmml">,</mo><mi id="Thmproofx4.p1.5.m1.2.2" xref="Thmproofx4.p1.5.m1.2.2.cmml">n</mi><mo id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.4" stretchy="false" xref="Thmproofx4.p1.5.m1.5.5.2.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.5.m1.5.5.2.2" xref="Thmproofx4.p1.5.m1.5.5.2.2.cmml">⁢</mo><mi id="Thmproofx4.p1.5.m1.5.5.2.3" xref="Thmproofx4.p1.5.m1.5.5.2.3.cmml">β</mi><mo id="Thmproofx4.p1.5.m1.5.5.2.2a" xref="Thmproofx4.p1.5.m1.5.5.2.2.cmml">⁢</mo><mrow id="Thmproofx4.p1.5.m1.5.5.2.4.2" xref="Thmproofx4.p1.5.m1.5.5.2.cmml"><mo id="Thmproofx4.p1.5.m1.5.5.2.4.2.1" stretchy="false" xref="Thmproofx4.p1.5.m1.5.5.2.cmml">(</mo><mi id="Thmproofx4.p1.5.m1.3.3" xref="Thmproofx4.p1.5.m1.3.3.cmml">σ</mi><mo id="Thmproofx4.p1.5.m1.5.5.2.4.2.2" stretchy="false" xref="Thmproofx4.p1.5.m1.5.5.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.5.m1.5b"><apply id="Thmproofx4.p1.5.m1.5.5.cmml" xref="Thmproofx4.p1.5.m1.5.5"><leq id="Thmproofx4.p1.5.m1.5.5.3.cmml" xref="Thmproofx4.p1.5.m1.5.5.3"></leq><apply id="Thmproofx4.p1.5.m1.4.4.1.cmml" xref="Thmproofx4.p1.5.m1.4.4.1"><times id="Thmproofx4.p1.5.m1.4.4.1.2.cmml" xref="Thmproofx4.p1.5.m1.4.4.1.2"></times><ci id="Thmproofx4.p1.5.m1.4.4.1.3.cmml" xref="Thmproofx4.p1.5.m1.4.4.1.3">𝛽</ci><apply id="Thmproofx4.p1.5.m1.4.4.1.1.1.1.cmml" xref="Thmproofx4.p1.5.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.5.m1.4.4.1.1.1.1.1.cmml" xref="Thmproofx4.p1.5.m1.4.4.1.1.1">superscript</csymbol><ci id="Thmproofx4.p1.5.m1.4.4.1.1.1.1.2.cmml" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.2">𝜎</ci><ci id="Thmproofx4.p1.5.m1.4.4.1.1.1.1.3.cmml" xref="Thmproofx4.p1.5.m1.4.4.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmproofx4.p1.5.m1.5.5.2.cmml" xref="Thmproofx4.p1.5.m1.5.5.2"><times id="Thmproofx4.p1.5.m1.5.5.2.2.cmml" xref="Thmproofx4.p1.5.m1.5.5.2.2"></times><apply id="Thmproofx4.p1.5.m1.5.5.2.1.2.cmml" xref="Thmproofx4.p1.5.m1.5.5.2.1.1"><min id="Thmproofx4.p1.5.m1.1.1.cmml" xref="Thmproofx4.p1.5.m1.1.1"></min><apply id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.cmml" xref="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.1.cmml" xref="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1">subscript</csymbol><ci id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.2.cmml" xref="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.2">𝑚</ci><ci id="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.3.cmml" xref="Thmproofx4.p1.5.m1.5.5.2.1.1.1.1.3">𝑐</ci></apply><ci id="Thmproofx4.p1.5.m1.2.2.cmml" xref="Thmproofx4.p1.5.m1.2.2">𝑛</ci></apply><ci id="Thmproofx4.p1.5.m1.5.5.2.3.cmml" xref="Thmproofx4.p1.5.m1.5.5.2.3">𝛽</ci><ci id="Thmproofx4.p1.5.m1.3.3.cmml" xref="Thmproofx4.p1.5.m1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.5.m1.5c">\beta(\sigma^{n})\leq\min(m_{c},n)\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.5.m1.5d">italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ≤ roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n ) italic_β ( italic_σ )</annotation></semantics></math>. Now we prove that <math alttext="\beta(\sigma)\leq\beta(\sigma^{n})" class="ltx_Math" display="inline" id="Thmproofx4.p1.6.m2.2"><semantics id="Thmproofx4.p1.6.m2.2a"><mrow id="Thmproofx4.p1.6.m2.2.2" xref="Thmproofx4.p1.6.m2.2.2.cmml"><mrow id="Thmproofx4.p1.6.m2.2.2.3" xref="Thmproofx4.p1.6.m2.2.2.3.cmml"><mi id="Thmproofx4.p1.6.m2.2.2.3.2" xref="Thmproofx4.p1.6.m2.2.2.3.2.cmml">β</mi><mo id="Thmproofx4.p1.6.m2.2.2.3.1" xref="Thmproofx4.p1.6.m2.2.2.3.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.6.m2.2.2.3.3.2" xref="Thmproofx4.p1.6.m2.2.2.3.cmml"><mo id="Thmproofx4.p1.6.m2.2.2.3.3.2.1" stretchy="false" xref="Thmproofx4.p1.6.m2.2.2.3.cmml">(</mo><mi id="Thmproofx4.p1.6.m2.1.1" xref="Thmproofx4.p1.6.m2.1.1.cmml">σ</mi><mo id="Thmproofx4.p1.6.m2.2.2.3.3.2.2" stretchy="false" xref="Thmproofx4.p1.6.m2.2.2.3.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.6.m2.2.2.2" xref="Thmproofx4.p1.6.m2.2.2.2.cmml">≤</mo><mrow id="Thmproofx4.p1.6.m2.2.2.1" xref="Thmproofx4.p1.6.m2.2.2.1.cmml"><mi id="Thmproofx4.p1.6.m2.2.2.1.3" xref="Thmproofx4.p1.6.m2.2.2.1.3.cmml">β</mi><mo id="Thmproofx4.p1.6.m2.2.2.1.2" xref="Thmproofx4.p1.6.m2.2.2.1.2.cmml">⁢</mo><mrow id="Thmproofx4.p1.6.m2.2.2.1.1.1" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.cmml"><mo id="Thmproofx4.p1.6.m2.2.2.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.cmml">(</mo><msup id="Thmproofx4.p1.6.m2.2.2.1.1.1.1" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.cmml"><mi id="Thmproofx4.p1.6.m2.2.2.1.1.1.1.2" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.2.cmml">σ</mi><mi id="Thmproofx4.p1.6.m2.2.2.1.1.1.1.3" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx4.p1.6.m2.2.2.1.1.1.3" stretchy="false" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.6.m2.2b"><apply id="Thmproofx4.p1.6.m2.2.2.cmml" xref="Thmproofx4.p1.6.m2.2.2"><leq id="Thmproofx4.p1.6.m2.2.2.2.cmml" xref="Thmproofx4.p1.6.m2.2.2.2"></leq><apply id="Thmproofx4.p1.6.m2.2.2.3.cmml" xref="Thmproofx4.p1.6.m2.2.2.3"><times id="Thmproofx4.p1.6.m2.2.2.3.1.cmml" xref="Thmproofx4.p1.6.m2.2.2.3.1"></times><ci id="Thmproofx4.p1.6.m2.2.2.3.2.cmml" xref="Thmproofx4.p1.6.m2.2.2.3.2">𝛽</ci><ci id="Thmproofx4.p1.6.m2.1.1.cmml" xref="Thmproofx4.p1.6.m2.1.1">𝜎</ci></apply><apply id="Thmproofx4.p1.6.m2.2.2.1.cmml" xref="Thmproofx4.p1.6.m2.2.2.1"><times id="Thmproofx4.p1.6.m2.2.2.1.2.cmml" xref="Thmproofx4.p1.6.m2.2.2.1.2"></times><ci id="Thmproofx4.p1.6.m2.2.2.1.3.cmml" xref="Thmproofx4.p1.6.m2.2.2.1.3">𝛽</ci><apply id="Thmproofx4.p1.6.m2.2.2.1.1.1.1.cmml" xref="Thmproofx4.p1.6.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.6.m2.2.2.1.1.1.1.1.cmml" xref="Thmproofx4.p1.6.m2.2.2.1.1.1">superscript</csymbol><ci id="Thmproofx4.p1.6.m2.2.2.1.1.1.1.2.cmml" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.2">𝜎</ci><ci id="Thmproofx4.p1.6.m2.2.2.1.1.1.1.3.cmml" xref="Thmproofx4.p1.6.m2.2.2.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.6.m2.2c">\beta(\sigma)\leq\beta(\sigma^{n})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.6.m2.2d">italic_β ( italic_σ ) ≤ italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>. Suppose that <math alttext="\beta(\sigma)=\sigma^{E}(e)" class="ltx_Math" display="inline" id="Thmproofx4.p1.7.m3.2"><semantics id="Thmproofx4.p1.7.m3.2a"><mrow id="Thmproofx4.p1.7.m3.2.3" xref="Thmproofx4.p1.7.m3.2.3.cmml"><mrow id="Thmproofx4.p1.7.m3.2.3.2" xref="Thmproofx4.p1.7.m3.2.3.2.cmml"><mi id="Thmproofx4.p1.7.m3.2.3.2.2" xref="Thmproofx4.p1.7.m3.2.3.2.2.cmml">β</mi><mo id="Thmproofx4.p1.7.m3.2.3.2.1" xref="Thmproofx4.p1.7.m3.2.3.2.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.7.m3.2.3.2.3.2" xref="Thmproofx4.p1.7.m3.2.3.2.cmml"><mo id="Thmproofx4.p1.7.m3.2.3.2.3.2.1" stretchy="false" xref="Thmproofx4.p1.7.m3.2.3.2.cmml">(</mo><mi id="Thmproofx4.p1.7.m3.1.1" xref="Thmproofx4.p1.7.m3.1.1.cmml">σ</mi><mo id="Thmproofx4.p1.7.m3.2.3.2.3.2.2" stretchy="false" xref="Thmproofx4.p1.7.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.7.m3.2.3.1" xref="Thmproofx4.p1.7.m3.2.3.1.cmml">=</mo><mrow id="Thmproofx4.p1.7.m3.2.3.3" xref="Thmproofx4.p1.7.m3.2.3.3.cmml"><msup id="Thmproofx4.p1.7.m3.2.3.3.2" xref="Thmproofx4.p1.7.m3.2.3.3.2.cmml"><mi id="Thmproofx4.p1.7.m3.2.3.3.2.2" xref="Thmproofx4.p1.7.m3.2.3.3.2.2.cmml">σ</mi><mi id="Thmproofx4.p1.7.m3.2.3.3.2.3" xref="Thmproofx4.p1.7.m3.2.3.3.2.3.cmml">E</mi></msup><mo id="Thmproofx4.p1.7.m3.2.3.3.1" xref="Thmproofx4.p1.7.m3.2.3.3.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.7.m3.2.3.3.3.2" xref="Thmproofx4.p1.7.m3.2.3.3.cmml"><mo id="Thmproofx4.p1.7.m3.2.3.3.3.2.1" stretchy="false" xref="Thmproofx4.p1.7.m3.2.3.3.cmml">(</mo><mi id="Thmproofx4.p1.7.m3.2.2" xref="Thmproofx4.p1.7.m3.2.2.cmml">e</mi><mo id="Thmproofx4.p1.7.m3.2.3.3.3.2.2" stretchy="false" xref="Thmproofx4.p1.7.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.7.m3.2b"><apply id="Thmproofx4.p1.7.m3.2.3.cmml" xref="Thmproofx4.p1.7.m3.2.3"><eq id="Thmproofx4.p1.7.m3.2.3.1.cmml" xref="Thmproofx4.p1.7.m3.2.3.1"></eq><apply id="Thmproofx4.p1.7.m3.2.3.2.cmml" xref="Thmproofx4.p1.7.m3.2.3.2"><times id="Thmproofx4.p1.7.m3.2.3.2.1.cmml" xref="Thmproofx4.p1.7.m3.2.3.2.1"></times><ci id="Thmproofx4.p1.7.m3.2.3.2.2.cmml" xref="Thmproofx4.p1.7.m3.2.3.2.2">𝛽</ci><ci id="Thmproofx4.p1.7.m3.1.1.cmml" xref="Thmproofx4.p1.7.m3.1.1">𝜎</ci></apply><apply id="Thmproofx4.p1.7.m3.2.3.3.cmml" xref="Thmproofx4.p1.7.m3.2.3.3"><times id="Thmproofx4.p1.7.m3.2.3.3.1.cmml" xref="Thmproofx4.p1.7.m3.2.3.3.1"></times><apply id="Thmproofx4.p1.7.m3.2.3.3.2.cmml" xref="Thmproofx4.p1.7.m3.2.3.3.2"><csymbol cd="ambiguous" id="Thmproofx4.p1.7.m3.2.3.3.2.1.cmml" xref="Thmproofx4.p1.7.m3.2.3.3.2">superscript</csymbol><ci id="Thmproofx4.p1.7.m3.2.3.3.2.2.cmml" xref="Thmproofx4.p1.7.m3.2.3.3.2.2">𝜎</ci><ci id="Thmproofx4.p1.7.m3.2.3.3.2.3.cmml" xref="Thmproofx4.p1.7.m3.2.3.3.2.3">𝐸</ci></apply><ci id="Thmproofx4.p1.7.m3.2.2.cmml" xref="Thmproofx4.p1.7.m3.2.2">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.7.m3.2c">\beta(\sigma)=\sigma^{E}(e)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.7.m3.2d">italic_β ( italic_σ ) = italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e )</annotation></semantics></math>. As <math alttext="\beta(\sigma^{n})\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{e^{\prime}}% \left(1-(1-\sigma^{E}(e^{\prime})^{n}\right)" class="ltx_math_unparsed" display="inline" id="Thmproofx4.p1.8.m4.1"><semantics id="Thmproofx4.p1.8.m4.1a"><mrow id="Thmproofx4.p1.8.m4.1b"><mi id="Thmproofx4.p1.8.m4.1.1">β</mi><mrow id="Thmproofx4.p1.8.m4.1.2"><mo id="Thmproofx4.p1.8.m4.1.2.1" stretchy="false">(</mo><msup id="Thmproofx4.p1.8.m4.1.2.2"><mi id="Thmproofx4.p1.8.m4.1.2.2.2">σ</mi><mi id="Thmproofx4.p1.8.m4.1.2.2.3">n</mi></msup><mo id="Thmproofx4.p1.8.m4.1.2.3" stretchy="false">)</mo></mrow><mover id="Thmproofx4.p1.8.m4.1.3"><mo id="Thmproofx4.p1.8.m4.1.3.2">=</mo><mtext id="Thmproofx4.p1.8.m4.1.3.3">def</mtext></mover><msub id="Thmproofx4.p1.8.m4.1.4"><mi id="Thmproofx4.p1.8.m4.1.4.2">max</mi><msup id="Thmproofx4.p1.8.m4.1.4.3"><mi id="Thmproofx4.p1.8.m4.1.4.3.2">e</mi><mo id="Thmproofx4.p1.8.m4.1.4.3.3">′</mo></msup></msub><mrow id="Thmproofx4.p1.8.m4.1.5"><mo id="Thmproofx4.p1.8.m4.1.5.1">(</mo><mn id="Thmproofx4.p1.8.m4.1.5.2">1</mn><mo id="Thmproofx4.p1.8.m4.1.5.3">−</mo><mrow id="Thmproofx4.p1.8.m4.1.5.4"><mo id="Thmproofx4.p1.8.m4.1.5.4.1" stretchy="false">(</mo><mn id="Thmproofx4.p1.8.m4.1.5.4.2">1</mn><mo id="Thmproofx4.p1.8.m4.1.5.4.3">−</mo><msup id="Thmproofx4.p1.8.m4.1.5.4.4"><mi id="Thmproofx4.p1.8.m4.1.5.4.4.2">σ</mi><mi id="Thmproofx4.p1.8.m4.1.5.4.4.3">E</mi></msup><msup id="Thmproofx4.p1.8.m4.1.5.4.5"><mrow id="Thmproofx4.p1.8.m4.1.5.4.5.2"><mo id="Thmproofx4.p1.8.m4.1.5.4.5.2.1" stretchy="false">(</mo><msup id="Thmproofx4.p1.8.m4.1.5.4.5.2.2"><mi id="Thmproofx4.p1.8.m4.1.5.4.5.2.2.2">e</mi><mo id="Thmproofx4.p1.8.m4.1.5.4.5.2.2.3">′</mo></msup><mo id="Thmproofx4.p1.8.m4.1.5.4.5.2.3" stretchy="false">)</mo></mrow><mi id="Thmproofx4.p1.8.m4.1.5.4.5.3">n</mi></msup><mo id="Thmproofx4.p1.8.m4.1.5.4.6">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="Thmproofx4.p1.8.m4.1c">\beta(\sigma^{n})\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{e^{\prime}}% \left(1-(1-\sigma^{E}(e^{\prime})^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.8.m4.1d">italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP roman_max start_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( 1 - ( 1 - italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>, by monotonicity <math alttext="\beta(\sigma^{n})=1-(1-\sigma^{E}(e))^{n}\geq\sigma^{E}(e)=\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx4.p1.9.m5.5"><semantics id="Thmproofx4.p1.9.m5.5a"><mrow id="Thmproofx4.p1.9.m5.5.5" xref="Thmproofx4.p1.9.m5.5.5.cmml"><mrow id="Thmproofx4.p1.9.m5.4.4.1" xref="Thmproofx4.p1.9.m5.4.4.1.cmml"><mi id="Thmproofx4.p1.9.m5.4.4.1.3" xref="Thmproofx4.p1.9.m5.4.4.1.3.cmml">β</mi><mo id="Thmproofx4.p1.9.m5.4.4.1.2" xref="Thmproofx4.p1.9.m5.4.4.1.2.cmml">⁢</mo><mrow id="Thmproofx4.p1.9.m5.4.4.1.1.1" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.cmml"><mo id="Thmproofx4.p1.9.m5.4.4.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.cmml">(</mo><msup id="Thmproofx4.p1.9.m5.4.4.1.1.1.1" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.cmml"><mi id="Thmproofx4.p1.9.m5.4.4.1.1.1.1.2" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.2.cmml">σ</mi><mi id="Thmproofx4.p1.9.m5.4.4.1.1.1.1.3" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx4.p1.9.m5.4.4.1.1.1.3" stretchy="false" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.9.m5.5.5.4" xref="Thmproofx4.p1.9.m5.5.5.4.cmml">=</mo><mrow id="Thmproofx4.p1.9.m5.5.5.2" xref="Thmproofx4.p1.9.m5.5.5.2.cmml"><mn id="Thmproofx4.p1.9.m5.5.5.2.3" xref="Thmproofx4.p1.9.m5.5.5.2.3.cmml">1</mn><mo id="Thmproofx4.p1.9.m5.5.5.2.2" xref="Thmproofx4.p1.9.m5.5.5.2.2.cmml">−</mo><msup id="Thmproofx4.p1.9.m5.5.5.2.1" xref="Thmproofx4.p1.9.m5.5.5.2.1.cmml"><mrow id="Thmproofx4.p1.9.m5.5.5.2.1.1.1" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.cmml"><mo id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.2" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.cmml">(</mo><mrow id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.cmml"><mn id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.2" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.1" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.1.cmml">−</mo><mrow id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.cmml"><msup id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.2" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.2.cmml"><mi id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.2.2" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.2.2.cmml">σ</mi><mi id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.2.3" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.2.3.cmml">E</mi></msup><mo id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.1" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.3.2" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.cmml"><mo id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.3.2.1" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.cmml">(</mo><mi id="Thmproofx4.p1.9.m5.1.1" xref="Thmproofx4.p1.9.m5.1.1.cmml">e</mi><mo id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.3.2.2" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.3" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx4.p1.9.m5.5.5.2.1.3" xref="Thmproofx4.p1.9.m5.5.5.2.1.3.cmml">n</mi></msup></mrow><mo id="Thmproofx4.p1.9.m5.5.5.5" xref="Thmproofx4.p1.9.m5.5.5.5.cmml">≥</mo><mrow id="Thmproofx4.p1.9.m5.5.5.6" xref="Thmproofx4.p1.9.m5.5.5.6.cmml"><msup id="Thmproofx4.p1.9.m5.5.5.6.2" xref="Thmproofx4.p1.9.m5.5.5.6.2.cmml"><mi id="Thmproofx4.p1.9.m5.5.5.6.2.2" xref="Thmproofx4.p1.9.m5.5.5.6.2.2.cmml">σ</mi><mi id="Thmproofx4.p1.9.m5.5.5.6.2.3" xref="Thmproofx4.p1.9.m5.5.5.6.2.3.cmml">E</mi></msup><mo id="Thmproofx4.p1.9.m5.5.5.6.1" xref="Thmproofx4.p1.9.m5.5.5.6.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.9.m5.5.5.6.3.2" xref="Thmproofx4.p1.9.m5.5.5.6.cmml"><mo id="Thmproofx4.p1.9.m5.5.5.6.3.2.1" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.6.cmml">(</mo><mi id="Thmproofx4.p1.9.m5.2.2" xref="Thmproofx4.p1.9.m5.2.2.cmml">e</mi><mo id="Thmproofx4.p1.9.m5.5.5.6.3.2.2" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.6.cmml">)</mo></mrow></mrow><mo id="Thmproofx4.p1.9.m5.5.5.7" xref="Thmproofx4.p1.9.m5.5.5.7.cmml">=</mo><mrow id="Thmproofx4.p1.9.m5.5.5.8" xref="Thmproofx4.p1.9.m5.5.5.8.cmml"><mi id="Thmproofx4.p1.9.m5.5.5.8.2" xref="Thmproofx4.p1.9.m5.5.5.8.2.cmml">β</mi><mo id="Thmproofx4.p1.9.m5.5.5.8.1" xref="Thmproofx4.p1.9.m5.5.5.8.1.cmml">⁢</mo><mrow id="Thmproofx4.p1.9.m5.5.5.8.3.2" xref="Thmproofx4.p1.9.m5.5.5.8.cmml"><mo id="Thmproofx4.p1.9.m5.5.5.8.3.2.1" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.8.cmml">(</mo><mi id="Thmproofx4.p1.9.m5.3.3" xref="Thmproofx4.p1.9.m5.3.3.cmml">σ</mi><mo id="Thmproofx4.p1.9.m5.5.5.8.3.2.2" stretchy="false" xref="Thmproofx4.p1.9.m5.5.5.8.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.9.m5.5b"><apply id="Thmproofx4.p1.9.m5.5.5.cmml" xref="Thmproofx4.p1.9.m5.5.5"><and id="Thmproofx4.p1.9.m5.5.5a.cmml" xref="Thmproofx4.p1.9.m5.5.5"></and><apply id="Thmproofx4.p1.9.m5.5.5b.cmml" xref="Thmproofx4.p1.9.m5.5.5"><eq id="Thmproofx4.p1.9.m5.5.5.4.cmml" xref="Thmproofx4.p1.9.m5.5.5.4"></eq><apply id="Thmproofx4.p1.9.m5.4.4.1.cmml" xref="Thmproofx4.p1.9.m5.4.4.1"><times id="Thmproofx4.p1.9.m5.4.4.1.2.cmml" xref="Thmproofx4.p1.9.m5.4.4.1.2"></times><ci id="Thmproofx4.p1.9.m5.4.4.1.3.cmml" xref="Thmproofx4.p1.9.m5.4.4.1.3">𝛽</ci><apply id="Thmproofx4.p1.9.m5.4.4.1.1.1.1.cmml" xref="Thmproofx4.p1.9.m5.4.4.1.1.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.9.m5.4.4.1.1.1.1.1.cmml" xref="Thmproofx4.p1.9.m5.4.4.1.1.1">superscript</csymbol><ci id="Thmproofx4.p1.9.m5.4.4.1.1.1.1.2.cmml" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.2">𝜎</ci><ci id="Thmproofx4.p1.9.m5.4.4.1.1.1.1.3.cmml" xref="Thmproofx4.p1.9.m5.4.4.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmproofx4.p1.9.m5.5.5.2.cmml" xref="Thmproofx4.p1.9.m5.5.5.2"><minus id="Thmproofx4.p1.9.m5.5.5.2.2.cmml" xref="Thmproofx4.p1.9.m5.5.5.2.2"></minus><cn id="Thmproofx4.p1.9.m5.5.5.2.3.cmml" type="integer" xref="Thmproofx4.p1.9.m5.5.5.2.3">1</cn><apply id="Thmproofx4.p1.9.m5.5.5.2.1.cmml" xref="Thmproofx4.p1.9.m5.5.5.2.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.9.m5.5.5.2.1.2.cmml" xref="Thmproofx4.p1.9.m5.5.5.2.1">superscript</csymbol><apply id="Thmproofx4.p1.9.m5.5.5.2.1.1.1.1.cmml" 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id="Thmproofx4.p1.9.m5.5.5.2.1.3.cmml" xref="Thmproofx4.p1.9.m5.5.5.2.1.3">𝑛</ci></apply></apply></apply><apply id="Thmproofx4.p1.9.m5.5.5c.cmml" xref="Thmproofx4.p1.9.m5.5.5"><geq id="Thmproofx4.p1.9.m5.5.5.5.cmml" xref="Thmproofx4.p1.9.m5.5.5.5"></geq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx4.p1.9.m5.5.5.2.cmml" id="Thmproofx4.p1.9.m5.5.5d.cmml" xref="Thmproofx4.p1.9.m5.5.5"></share><apply id="Thmproofx4.p1.9.m5.5.5.6.cmml" xref="Thmproofx4.p1.9.m5.5.5.6"><times id="Thmproofx4.p1.9.m5.5.5.6.1.cmml" xref="Thmproofx4.p1.9.m5.5.5.6.1"></times><apply id="Thmproofx4.p1.9.m5.5.5.6.2.cmml" xref="Thmproofx4.p1.9.m5.5.5.6.2"><csymbol cd="ambiguous" id="Thmproofx4.p1.9.m5.5.5.6.2.1.cmml" xref="Thmproofx4.p1.9.m5.5.5.6.2">superscript</csymbol><ci id="Thmproofx4.p1.9.m5.5.5.6.2.2.cmml" xref="Thmproofx4.p1.9.m5.5.5.6.2.2">𝜎</ci><ci id="Thmproofx4.p1.9.m5.5.5.6.2.3.cmml" xref="Thmproofx4.p1.9.m5.5.5.6.2.3">𝐸</ci></apply><ci id="Thmproofx4.p1.9.m5.2.2.cmml" xref="Thmproofx4.p1.9.m5.2.2">𝑒</ci></apply></apply><apply id="Thmproofx4.p1.9.m5.5.5e.cmml" xref="Thmproofx4.p1.9.m5.5.5"><eq id="Thmproofx4.p1.9.m5.5.5.7.cmml" xref="Thmproofx4.p1.9.m5.5.5.7"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx4.p1.9.m5.5.5.6.cmml" id="Thmproofx4.p1.9.m5.5.5f.cmml" xref="Thmproofx4.p1.9.m5.5.5"></share><apply id="Thmproofx4.p1.9.m5.5.5.8.cmml" xref="Thmproofx4.p1.9.m5.5.5.8"><times id="Thmproofx4.p1.9.m5.5.5.8.1.cmml" xref="Thmproofx4.p1.9.m5.5.5.8.1"></times><ci id="Thmproofx4.p1.9.m5.5.5.8.2.cmml" xref="Thmproofx4.p1.9.m5.5.5.8.2">𝛽</ci><ci id="Thmproofx4.p1.9.m5.3.3.cmml" xref="Thmproofx4.p1.9.m5.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.9.m5.5c">\beta(\sigma^{n})=1-(1-\sigma^{E}(e))^{n}\geq\sigma^{E}(e)=\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.9.m5.5d">italic_β ( italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) = 1 - ( 1 - italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ≥ italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e ) = italic_β ( italic_σ )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx5"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma6" title="Lemma 6 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">6</span></a>)</h6> <div class="ltx_para" id="Thmproofx5.p1"> <p class="ltx_p" id="Thmproofx5.p1.1">Take <math alttext="\nu\in\operatorname*{arg\,min}_{\sigma\in\Delta(P_{st})}F_{1}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Thmproofx5.p1.1.m1.3"><semantics id="Thmproofx5.p1.1.m1.3a"><mrow id="Thmproofx5.p1.1.m1.3.4" xref="Thmproofx5.p1.1.m1.3.4.cmml"><mi id="Thmproofx5.p1.1.m1.3.4.2" xref="Thmproofx5.p1.1.m1.3.4.2.cmml">ν</mi><mo id="Thmproofx5.p1.1.m1.3.4.1" xref="Thmproofx5.p1.1.m1.3.4.1.cmml">∈</mo><mrow id="Thmproofx5.p1.1.m1.3.4.3" xref="Thmproofx5.p1.1.m1.3.4.3.cmml"><mrow id="Thmproofx5.p1.1.m1.3.4.3.2" xref="Thmproofx5.p1.1.m1.3.4.3.2.cmml"><msub id="Thmproofx5.p1.1.m1.3.4.3.2.1" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.cmml"><mrow id="Thmproofx5.p1.1.m1.3.4.3.2.1.2" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2.cmml"><mi id="Thmproofx5.p1.1.m1.3.4.3.2.1.2.2" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2.2.cmml">arg</mi><mo id="Thmproofx5.p1.1.m1.3.4.3.2.1.2.1" lspace="0.170em" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2.1.cmml">⁢</mo><mi id="Thmproofx5.p1.1.m1.3.4.3.2.1.2.3" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2.3.cmml">min</mi></mrow><mrow id="Thmproofx5.p1.1.m1.1.1.1" xref="Thmproofx5.p1.1.m1.1.1.1.cmml"><mi id="Thmproofx5.p1.1.m1.1.1.1.3" xref="Thmproofx5.p1.1.m1.1.1.1.3.cmml">σ</mi><mo id="Thmproofx5.p1.1.m1.1.1.1.2" xref="Thmproofx5.p1.1.m1.1.1.1.2.cmml">∈</mo><mrow id="Thmproofx5.p1.1.m1.1.1.1.1" xref="Thmproofx5.p1.1.m1.1.1.1.1.cmml"><mi id="Thmproofx5.p1.1.m1.1.1.1.1.3" mathvariant="normal" xref="Thmproofx5.p1.1.m1.1.1.1.1.3.cmml">Δ</mi><mo id="Thmproofx5.p1.1.m1.1.1.1.1.2" xref="Thmproofx5.p1.1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx5.p1.1.m1.1.1.1.1.1.1" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.2" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.2" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.1" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.3" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mo id="Thmproofx5.p1.1.m1.3.4.3.2a" xref="Thmproofx5.p1.1.m1.3.4.3.2.cmml">⁡</mo><msub id="Thmproofx5.p1.1.m1.3.4.3.2.2" xref="Thmproofx5.p1.1.m1.3.4.3.2.2.cmml"><mi id="Thmproofx5.p1.1.m1.3.4.3.2.2.2" xref="Thmproofx5.p1.1.m1.3.4.3.2.2.2.cmml">F</mi><mn id="Thmproofx5.p1.1.m1.3.4.3.2.2.3" xref="Thmproofx5.p1.1.m1.3.4.3.2.2.3.cmml">1</mn></msub></mrow><mo id="Thmproofx5.p1.1.m1.3.4.3.1" xref="Thmproofx5.p1.1.m1.3.4.3.1.cmml">⁢</mo><mrow id="Thmproofx5.p1.1.m1.3.4.3.3.2" xref="Thmproofx5.p1.1.m1.3.4.3.3.1.cmml"><mo id="Thmproofx5.p1.1.m1.3.4.3.3.2.1" stretchy="false" xref="Thmproofx5.p1.1.m1.3.4.3.3.1.cmml">(</mo><mi id="Thmproofx5.p1.1.m1.2.2" xref="Thmproofx5.p1.1.m1.2.2.cmml">σ</mi><mo id="Thmproofx5.p1.1.m1.3.4.3.3.2.2" xref="Thmproofx5.p1.1.m1.3.4.3.3.1.cmml">,</mo><mi id="Thmproofx5.p1.1.m1.3.3" xref="Thmproofx5.p1.1.m1.3.3.cmml">α</mi><mo id="Thmproofx5.p1.1.m1.3.4.3.3.2.3" stretchy="false" xref="Thmproofx5.p1.1.m1.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx5.p1.1.m1.3b"><apply id="Thmproofx5.p1.1.m1.3.4.cmml" xref="Thmproofx5.p1.1.m1.3.4"><in id="Thmproofx5.p1.1.m1.3.4.1.cmml" xref="Thmproofx5.p1.1.m1.3.4.1"></in><ci id="Thmproofx5.p1.1.m1.3.4.2.cmml" xref="Thmproofx5.p1.1.m1.3.4.2">𝜈</ci><apply id="Thmproofx5.p1.1.m1.3.4.3.cmml" xref="Thmproofx5.p1.1.m1.3.4.3"><times id="Thmproofx5.p1.1.m1.3.4.3.1.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.1"></times><apply id="Thmproofx5.p1.1.m1.3.4.3.2.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2"><apply id="Thmproofx5.p1.1.m1.3.4.3.2.1.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.1"><csymbol cd="ambiguous" id="Thmproofx5.p1.1.m1.3.4.3.2.1.1.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.1">subscript</csymbol><apply id="Thmproofx5.p1.1.m1.3.4.3.2.1.2.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2"><times id="Thmproofx5.p1.1.m1.3.4.3.2.1.2.1.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2.1"></times><ci id="Thmproofx5.p1.1.m1.3.4.3.2.1.2.2.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2.2">arg</ci><ci id="Thmproofx5.p1.1.m1.3.4.3.2.1.2.3.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.1.2.3">min</ci></apply><apply id="Thmproofx5.p1.1.m1.1.1.1.cmml" xref="Thmproofx5.p1.1.m1.1.1.1"><in id="Thmproofx5.p1.1.m1.1.1.1.2.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.2"></in><ci id="Thmproofx5.p1.1.m1.1.1.1.3.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.3">𝜎</ci><apply id="Thmproofx5.p1.1.m1.1.1.1.1.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1"><times id="Thmproofx5.p1.1.m1.1.1.1.1.2.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.2"></times><ci id="Thmproofx5.p1.1.m1.1.1.1.1.3.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.3">Δ</ci><apply id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3"><times id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.1"></times><ci id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx5.p1.1.m1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></apply><apply id="Thmproofx5.p1.1.m1.3.4.3.2.2.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="Thmproofx5.p1.1.m1.3.4.3.2.2.1.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.2">subscript</csymbol><ci id="Thmproofx5.p1.1.m1.3.4.3.2.2.2.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.2.2.2">𝐹</ci><cn id="Thmproofx5.p1.1.m1.3.4.3.2.2.3.cmml" type="integer" xref="Thmproofx5.p1.1.m1.3.4.3.2.2.3">1</cn></apply></apply><interval closure="open" id="Thmproofx5.p1.1.m1.3.4.3.3.1.cmml" xref="Thmproofx5.p1.1.m1.3.4.3.3.2"><ci id="Thmproofx5.p1.1.m1.2.2.cmml" xref="Thmproofx5.p1.1.m1.2.2">𝜎</ci><ci id="Thmproofx5.p1.1.m1.3.3.cmml" xref="Thmproofx5.p1.1.m1.3.3">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx5.p1.1.m1.3c">\nu\in\operatorname*{arg\,min}_{\sigma\in\Delta(P_{st})}F_{1}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx5.p1.1.m1.3d">italic_ν ∈ start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_σ ∈ roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="Pt0.A1.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="r(G,n,k,\alpha)=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)% }{F_{1}(\nu,\alpha)}\leq\frac{F_{n}(\nu^{n},\alpha)}{F_{1}(\nu,\alpha)}\leq x" class="ltx_Math" display="block" id="Pt0.A1.Ex24.m1.13"><semantics id="Pt0.A1.Ex24.m1.13a"><mrow id="Pt0.A1.Ex24.m1.13.14" xref="Pt0.A1.Ex24.m1.13.14.cmml"><mrow id="Pt0.A1.Ex24.m1.13.14.2" xref="Pt0.A1.Ex24.m1.13.14.2.cmml"><mi id="Pt0.A1.Ex24.m1.13.14.2.2" xref="Pt0.A1.Ex24.m1.13.14.2.2.cmml">r</mi><mo id="Pt0.A1.Ex24.m1.13.14.2.1" xref="Pt0.A1.Ex24.m1.13.14.2.1.cmml">⁢</mo><mrow id="Pt0.A1.Ex24.m1.13.14.2.3.2" xref="Pt0.A1.Ex24.m1.13.14.2.3.1.cmml"><mo id="Pt0.A1.Ex24.m1.13.14.2.3.2.1" stretchy="false" xref="Pt0.A1.Ex24.m1.13.14.2.3.1.cmml">(</mo><mi id="Pt0.A1.Ex24.m1.10.10" xref="Pt0.A1.Ex24.m1.10.10.cmml">G</mi><mo id="Pt0.A1.Ex24.m1.13.14.2.3.2.2" xref="Pt0.A1.Ex24.m1.13.14.2.3.1.cmml">,</mo><mi id="Pt0.A1.Ex24.m1.11.11" xref="Pt0.A1.Ex24.m1.11.11.cmml">n</mi><mo id="Pt0.A1.Ex24.m1.13.14.2.3.2.3" xref="Pt0.A1.Ex24.m1.13.14.2.3.1.cmml">,</mo><mi id="Pt0.A1.Ex24.m1.12.12" xref="Pt0.A1.Ex24.m1.12.12.cmml">k</mi><mo id="Pt0.A1.Ex24.m1.13.14.2.3.2.4" xref="Pt0.A1.Ex24.m1.13.14.2.3.1.cmml">,</mo><mi id="Pt0.A1.Ex24.m1.13.13" xref="Pt0.A1.Ex24.m1.13.13.cmml">α</mi><mo id="Pt0.A1.Ex24.m1.13.14.2.3.2.5" stretchy="false" xref="Pt0.A1.Ex24.m1.13.14.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.Ex24.m1.13.14.3" xref="Pt0.A1.Ex24.m1.13.14.3.cmml">=</mo><mfrac id="Pt0.A1.Ex24.m1.5.5" xref="Pt0.A1.Ex24.m1.5.5.cmml"><mrow id="Pt0.A1.Ex24.m1.3.3.3" xref="Pt0.A1.Ex24.m1.3.3.3.cmml"><mrow id="Pt0.A1.Ex24.m1.3.3.3.5" xref="Pt0.A1.Ex24.m1.3.3.3.5.cmml"><msub id="Pt0.A1.Ex24.m1.3.3.3.5.1" xref="Pt0.A1.Ex24.m1.3.3.3.5.1.cmml"><mi id="Pt0.A1.Ex24.m1.3.3.3.5.1.2" xref="Pt0.A1.Ex24.m1.3.3.3.5.1.2.cmml">min</mi><mrow id="Pt0.A1.Ex24.m1.1.1.1.1.1" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.cmml"><mi id="Pt0.A1.Ex24.m1.1.1.1.1.1.3" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.3.cmml">σ</mi><mo id="Pt0.A1.Ex24.m1.1.1.1.1.1.2" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.2.cmml">∈</mo><msup id="Pt0.A1.Ex24.m1.1.1.1.1.1.1" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.cmml"><mrow id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A1.Ex24.m1.1.1.1.1.1.1.1.1.3" stretchy="false" 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xref="Pt0.A1.Ex24.m1.7.7.2.2.1.1"><csymbol cd="ambiguous" id="Pt0.A1.Ex24.m1.7.7.2.2.1.1.1.cmml" xref="Pt0.A1.Ex24.m1.7.7.2.2.1.1">superscript</csymbol><ci id="Pt0.A1.Ex24.m1.7.7.2.2.1.1.2.cmml" xref="Pt0.A1.Ex24.m1.7.7.2.2.1.1.2">𝜈</ci><ci id="Pt0.A1.Ex24.m1.7.7.2.2.1.1.3.cmml" xref="Pt0.A1.Ex24.m1.7.7.2.2.1.1.3">𝑛</ci></apply><ci id="Pt0.A1.Ex24.m1.6.6.1.1.cmml" xref="Pt0.A1.Ex24.m1.6.6.1.1">𝛼</ci></interval></apply><apply id="Pt0.A1.Ex24.m1.9.9.4.cmml" xref="Pt0.A1.Ex24.m1.9.9.4"><times id="Pt0.A1.Ex24.m1.9.9.4.3.cmml" xref="Pt0.A1.Ex24.m1.9.9.4.3"></times><apply id="Pt0.A1.Ex24.m1.9.9.4.4.cmml" xref="Pt0.A1.Ex24.m1.9.9.4.4"><csymbol cd="ambiguous" id="Pt0.A1.Ex24.m1.9.9.4.4.1.cmml" xref="Pt0.A1.Ex24.m1.9.9.4.4">subscript</csymbol><ci id="Pt0.A1.Ex24.m1.9.9.4.4.2.cmml" xref="Pt0.A1.Ex24.m1.9.9.4.4.2">𝐹</ci><cn id="Pt0.A1.Ex24.m1.9.9.4.4.3.cmml" type="integer" xref="Pt0.A1.Ex24.m1.9.9.4.4.3">1</cn></apply><interval closure="open" id="Pt0.A1.Ex24.m1.9.9.4.5.1.cmml" xref="Pt0.A1.Ex24.m1.9.9.4.5.2"><ci id="Pt0.A1.Ex24.m1.8.8.3.1.cmml" xref="Pt0.A1.Ex24.m1.8.8.3.1">𝜈</ci><ci id="Pt0.A1.Ex24.m1.9.9.4.2.cmml" xref="Pt0.A1.Ex24.m1.9.9.4.2">𝛼</ci></interval></apply></apply></apply><apply id="Pt0.A1.Ex24.m1.13.14e.cmml" xref="Pt0.A1.Ex24.m1.13.14"><leq id="Pt0.A1.Ex24.m1.13.14.5.cmml" xref="Pt0.A1.Ex24.m1.13.14.5"></leq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A1.Ex24.m1.9.9.cmml" id="Pt0.A1.Ex24.m1.13.14f.cmml" xref="Pt0.A1.Ex24.m1.13.14"></share><ci id="Pt0.A1.Ex24.m1.13.14.6.cmml" xref="Pt0.A1.Ex24.m1.13.14.6">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex24.m1.13c">r(G,n,k,\alpha)=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)% }{F_{1}(\nu,\alpha)}\leq\frac{F_{n}(\nu^{n},\alpha)}{F_{1}(\nu,\alpha)}\leq x</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex24.m1.13d">italic_r ( italic_G , italic_n , italic_k , italic_α ) = divide start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) end_ARG start_ARG italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ν , italic_α ) end_ARG ≤ divide start_ARG italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_ν start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) end_ARG start_ARG italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ν , italic_α ) end_ARG ≤ italic_x</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx5.p1.2">∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx6"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma7" title="Lemma 7 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">7</span></a>)</h6> <div class="ltx_para" id="Thmproofx6.p1"> <p class="ltx_p" id="Thmproofx6.p1.11">Let <math alttext="P_{i}=\left\{p_{1},...,p_{i}\right\}" class="ltx_Math" display="inline" id="Thmproofx6.p1.1.m1.3"><semantics id="Thmproofx6.p1.1.m1.3a"><mrow id="Thmproofx6.p1.1.m1.3.3" xref="Thmproofx6.p1.1.m1.3.3.cmml"><msub id="Thmproofx6.p1.1.m1.3.3.4" xref="Thmproofx6.p1.1.m1.3.3.4.cmml"><mi id="Thmproofx6.p1.1.m1.3.3.4.2" xref="Thmproofx6.p1.1.m1.3.3.4.2.cmml">P</mi><mi id="Thmproofx6.p1.1.m1.3.3.4.3" xref="Thmproofx6.p1.1.m1.3.3.4.3.cmml">i</mi></msub><mo id="Thmproofx6.p1.1.m1.3.3.3" xref="Thmproofx6.p1.1.m1.3.3.3.cmml">=</mo><mrow id="Thmproofx6.p1.1.m1.3.3.2.2" xref="Thmproofx6.p1.1.m1.3.3.2.3.cmml"><mo id="Thmproofx6.p1.1.m1.3.3.2.2.3" xref="Thmproofx6.p1.1.m1.3.3.2.3.cmml">{</mo><msub id="Thmproofx6.p1.1.m1.2.2.1.1.1" xref="Thmproofx6.p1.1.m1.2.2.1.1.1.cmml"><mi id="Thmproofx6.p1.1.m1.2.2.1.1.1.2" xref="Thmproofx6.p1.1.m1.2.2.1.1.1.2.cmml">p</mi><mn id="Thmproofx6.p1.1.m1.2.2.1.1.1.3" xref="Thmproofx6.p1.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx6.p1.1.m1.3.3.2.2.4" xref="Thmproofx6.p1.1.m1.3.3.2.3.cmml">,</mo><mi id="Thmproofx6.p1.1.m1.1.1" mathvariant="normal" xref="Thmproofx6.p1.1.m1.1.1.cmml">…</mi><mo id="Thmproofx6.p1.1.m1.3.3.2.2.5" xref="Thmproofx6.p1.1.m1.3.3.2.3.cmml">,</mo><msub id="Thmproofx6.p1.1.m1.3.3.2.2.2" xref="Thmproofx6.p1.1.m1.3.3.2.2.2.cmml"><mi id="Thmproofx6.p1.1.m1.3.3.2.2.2.2" xref="Thmproofx6.p1.1.m1.3.3.2.2.2.2.cmml">p</mi><mi id="Thmproofx6.p1.1.m1.3.3.2.2.2.3" xref="Thmproofx6.p1.1.m1.3.3.2.2.2.3.cmml">i</mi></msub><mo id="Thmproofx6.p1.1.m1.3.3.2.2.6" xref="Thmproofx6.p1.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.1.m1.3b"><apply id="Thmproofx6.p1.1.m1.3.3.cmml" xref="Thmproofx6.p1.1.m1.3.3"><eq id="Thmproofx6.p1.1.m1.3.3.3.cmml" xref="Thmproofx6.p1.1.m1.3.3.3"></eq><apply id="Thmproofx6.p1.1.m1.3.3.4.cmml" xref="Thmproofx6.p1.1.m1.3.3.4"><csymbol cd="ambiguous" id="Thmproofx6.p1.1.m1.3.3.4.1.cmml" xref="Thmproofx6.p1.1.m1.3.3.4">subscript</csymbol><ci id="Thmproofx6.p1.1.m1.3.3.4.2.cmml" xref="Thmproofx6.p1.1.m1.3.3.4.2">𝑃</ci><ci id="Thmproofx6.p1.1.m1.3.3.4.3.cmml" xref="Thmproofx6.p1.1.m1.3.3.4.3">𝑖</ci></apply><set id="Thmproofx6.p1.1.m1.3.3.2.3.cmml" xref="Thmproofx6.p1.1.m1.3.3.2.2"><apply id="Thmproofx6.p1.1.m1.2.2.1.1.1.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.1.m1.2.2.1.1.1.1.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx6.p1.1.m1.2.2.1.1.1.2.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.1.2">𝑝</ci><cn id="Thmproofx6.p1.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="Thmproofx6.p1.1.m1.2.2.1.1.1.3">1</cn></apply><ci id="Thmproofx6.p1.1.m1.1.1.cmml" xref="Thmproofx6.p1.1.m1.1.1">…</ci><apply id="Thmproofx6.p1.1.m1.3.3.2.2.2.cmml" xref="Thmproofx6.p1.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.1.m1.3.3.2.2.2.1.cmml" xref="Thmproofx6.p1.1.m1.3.3.2.2.2">subscript</csymbol><ci id="Thmproofx6.p1.1.m1.3.3.2.2.2.2.cmml" xref="Thmproofx6.p1.1.m1.3.3.2.2.2.2">𝑝</ci><ci id="Thmproofx6.p1.1.m1.3.3.2.2.2.3.cmml" xref="Thmproofx6.p1.1.m1.3.3.2.2.2.3">𝑖</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.1.m1.3c">P_{i}=\left\{p_{1},...,p_{i}\right\}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.1.m1.3d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math>. Let <math alttext="X=(x_{1},...,x_{m})\in S" class="ltx_Math" display="inline" id="Thmproofx6.p1.2.m2.3"><semantics id="Thmproofx6.p1.2.m2.3a"><mrow id="Thmproofx6.p1.2.m2.3.3" xref="Thmproofx6.p1.2.m2.3.3.cmml"><mi id="Thmproofx6.p1.2.m2.3.3.4" xref="Thmproofx6.p1.2.m2.3.3.4.cmml">X</mi><mo id="Thmproofx6.p1.2.m2.3.3.5" xref="Thmproofx6.p1.2.m2.3.3.5.cmml">=</mo><mrow id="Thmproofx6.p1.2.m2.3.3.2.2" xref="Thmproofx6.p1.2.m2.3.3.2.3.cmml"><mo id="Thmproofx6.p1.2.m2.3.3.2.2.3" stretchy="false" xref="Thmproofx6.p1.2.m2.3.3.2.3.cmml">(</mo><msub id="Thmproofx6.p1.2.m2.2.2.1.1.1" xref="Thmproofx6.p1.2.m2.2.2.1.1.1.cmml"><mi id="Thmproofx6.p1.2.m2.2.2.1.1.1.2" xref="Thmproofx6.p1.2.m2.2.2.1.1.1.2.cmml">x</mi><mn id="Thmproofx6.p1.2.m2.2.2.1.1.1.3" xref="Thmproofx6.p1.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx6.p1.2.m2.3.3.2.2.4" xref="Thmproofx6.p1.2.m2.3.3.2.3.cmml">,</mo><mi id="Thmproofx6.p1.2.m2.1.1" mathvariant="normal" xref="Thmproofx6.p1.2.m2.1.1.cmml">…</mi><mo id="Thmproofx6.p1.2.m2.3.3.2.2.5" xref="Thmproofx6.p1.2.m2.3.3.2.3.cmml">,</mo><msub id="Thmproofx6.p1.2.m2.3.3.2.2.2" xref="Thmproofx6.p1.2.m2.3.3.2.2.2.cmml"><mi id="Thmproofx6.p1.2.m2.3.3.2.2.2.2" xref="Thmproofx6.p1.2.m2.3.3.2.2.2.2.cmml">x</mi><mi id="Thmproofx6.p1.2.m2.3.3.2.2.2.3" xref="Thmproofx6.p1.2.m2.3.3.2.2.2.3.cmml">m</mi></msub><mo id="Thmproofx6.p1.2.m2.3.3.2.2.6" stretchy="false" xref="Thmproofx6.p1.2.m2.3.3.2.3.cmml">)</mo></mrow><mo id="Thmproofx6.p1.2.m2.3.3.6" xref="Thmproofx6.p1.2.m2.3.3.6.cmml">∈</mo><mi id="Thmproofx6.p1.2.m2.3.3.7" xref="Thmproofx6.p1.2.m2.3.3.7.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.2.m2.3b"><apply id="Thmproofx6.p1.2.m2.3.3.cmml" xref="Thmproofx6.p1.2.m2.3.3"><and id="Thmproofx6.p1.2.m2.3.3a.cmml" xref="Thmproofx6.p1.2.m2.3.3"></and><apply id="Thmproofx6.p1.2.m2.3.3b.cmml" xref="Thmproofx6.p1.2.m2.3.3"><eq id="Thmproofx6.p1.2.m2.3.3.5.cmml" xref="Thmproofx6.p1.2.m2.3.3.5"></eq><ci id="Thmproofx6.p1.2.m2.3.3.4.cmml" xref="Thmproofx6.p1.2.m2.3.3.4">𝑋</ci><vector id="Thmproofx6.p1.2.m2.3.3.2.3.cmml" xref="Thmproofx6.p1.2.m2.3.3.2.2"><apply id="Thmproofx6.p1.2.m2.2.2.1.1.1.cmml" xref="Thmproofx6.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.2.m2.2.2.1.1.1.1.cmml" xref="Thmproofx6.p1.2.m2.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx6.p1.2.m2.2.2.1.1.1.2.cmml" xref="Thmproofx6.p1.2.m2.2.2.1.1.1.2">𝑥</ci><cn id="Thmproofx6.p1.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="Thmproofx6.p1.2.m2.2.2.1.1.1.3">1</cn></apply><ci id="Thmproofx6.p1.2.m2.1.1.cmml" xref="Thmproofx6.p1.2.m2.1.1">…</ci><apply id="Thmproofx6.p1.2.m2.3.3.2.2.2.cmml" xref="Thmproofx6.p1.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.2.m2.3.3.2.2.2.1.cmml" xref="Thmproofx6.p1.2.m2.3.3.2.2.2">subscript</csymbol><ci id="Thmproofx6.p1.2.m2.3.3.2.2.2.2.cmml" xref="Thmproofx6.p1.2.m2.3.3.2.2.2.2">𝑥</ci><ci id="Thmproofx6.p1.2.m2.3.3.2.2.2.3.cmml" xref="Thmproofx6.p1.2.m2.3.3.2.2.2.3">𝑚</ci></apply></vector></apply><apply id="Thmproofx6.p1.2.m2.3.3c.cmml" xref="Thmproofx6.p1.2.m2.3.3"><in id="Thmproofx6.p1.2.m2.3.3.6.cmml" xref="Thmproofx6.p1.2.m2.3.3.6"></in><share href="https://arxiv.org/html/2308.08047v3#Thmproofx6.p1.2.m2.3.3.2.cmml" id="Thmproofx6.p1.2.m2.3.3d.cmml" xref="Thmproofx6.p1.2.m2.3.3"></share><ci id="Thmproofx6.p1.2.m2.3.3.7.cmml" xref="Thmproofx6.p1.2.m2.3.3.7">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.2.m2.3c">X=(x_{1},...,x_{m})\in S</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.2.m2.3d">italic_X = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ∈ italic_S</annotation></semantics></math>. Define <math alttext="y_{m}=mx_{m}" class="ltx_Math" display="inline" id="Thmproofx6.p1.3.m3.1"><semantics id="Thmproofx6.p1.3.m3.1a"><mrow id="Thmproofx6.p1.3.m3.1.1" xref="Thmproofx6.p1.3.m3.1.1.cmml"><msub id="Thmproofx6.p1.3.m3.1.1.2" xref="Thmproofx6.p1.3.m3.1.1.2.cmml"><mi id="Thmproofx6.p1.3.m3.1.1.2.2" xref="Thmproofx6.p1.3.m3.1.1.2.2.cmml">y</mi><mi id="Thmproofx6.p1.3.m3.1.1.2.3" xref="Thmproofx6.p1.3.m3.1.1.2.3.cmml">m</mi></msub><mo id="Thmproofx6.p1.3.m3.1.1.1" xref="Thmproofx6.p1.3.m3.1.1.1.cmml">=</mo><mrow id="Thmproofx6.p1.3.m3.1.1.3" xref="Thmproofx6.p1.3.m3.1.1.3.cmml"><mi id="Thmproofx6.p1.3.m3.1.1.3.2" xref="Thmproofx6.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="Thmproofx6.p1.3.m3.1.1.3.1" xref="Thmproofx6.p1.3.m3.1.1.3.1.cmml">⁢</mo><msub id="Thmproofx6.p1.3.m3.1.1.3.3" xref="Thmproofx6.p1.3.m3.1.1.3.3.cmml"><mi id="Thmproofx6.p1.3.m3.1.1.3.3.2" xref="Thmproofx6.p1.3.m3.1.1.3.3.2.cmml">x</mi><mi id="Thmproofx6.p1.3.m3.1.1.3.3.3" xref="Thmproofx6.p1.3.m3.1.1.3.3.3.cmml">m</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.3.m3.1b"><apply id="Thmproofx6.p1.3.m3.1.1.cmml" xref="Thmproofx6.p1.3.m3.1.1"><eq id="Thmproofx6.p1.3.m3.1.1.1.cmml" xref="Thmproofx6.p1.3.m3.1.1.1"></eq><apply id="Thmproofx6.p1.3.m3.1.1.2.cmml" xref="Thmproofx6.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.3.m3.1.1.2.1.cmml" xref="Thmproofx6.p1.3.m3.1.1.2">subscript</csymbol><ci id="Thmproofx6.p1.3.m3.1.1.2.2.cmml" xref="Thmproofx6.p1.3.m3.1.1.2.2">𝑦</ci><ci id="Thmproofx6.p1.3.m3.1.1.2.3.cmml" xref="Thmproofx6.p1.3.m3.1.1.2.3">𝑚</ci></apply><apply id="Thmproofx6.p1.3.m3.1.1.3.cmml" xref="Thmproofx6.p1.3.m3.1.1.3"><times id="Thmproofx6.p1.3.m3.1.1.3.1.cmml" xref="Thmproofx6.p1.3.m3.1.1.3.1"></times><ci id="Thmproofx6.p1.3.m3.1.1.3.2.cmml" xref="Thmproofx6.p1.3.m3.1.1.3.2">𝑚</ci><apply id="Thmproofx6.p1.3.m3.1.1.3.3.cmml" xref="Thmproofx6.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx6.p1.3.m3.1.1.3.3.1.cmml" xref="Thmproofx6.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="Thmproofx6.p1.3.m3.1.1.3.3.2.cmml" xref="Thmproofx6.p1.3.m3.1.1.3.3.2">𝑥</ci><ci id="Thmproofx6.p1.3.m3.1.1.3.3.3.cmml" xref="Thmproofx6.p1.3.m3.1.1.3.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.3.m3.1c">y_{m}=mx_{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.3.m3.1d">italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_m italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="y_{i}=i(x_{i}-x_{i+1})" class="ltx_Math" display="inline" id="Thmproofx6.p1.4.m4.1"><semantics id="Thmproofx6.p1.4.m4.1a"><mrow id="Thmproofx6.p1.4.m4.1.1" xref="Thmproofx6.p1.4.m4.1.1.cmml"><msub id="Thmproofx6.p1.4.m4.1.1.3" xref="Thmproofx6.p1.4.m4.1.1.3.cmml"><mi id="Thmproofx6.p1.4.m4.1.1.3.2" xref="Thmproofx6.p1.4.m4.1.1.3.2.cmml">y</mi><mi id="Thmproofx6.p1.4.m4.1.1.3.3" xref="Thmproofx6.p1.4.m4.1.1.3.3.cmml">i</mi></msub><mo id="Thmproofx6.p1.4.m4.1.1.2" xref="Thmproofx6.p1.4.m4.1.1.2.cmml">=</mo><mrow id="Thmproofx6.p1.4.m4.1.1.1" xref="Thmproofx6.p1.4.m4.1.1.1.cmml"><mi id="Thmproofx6.p1.4.m4.1.1.1.3" xref="Thmproofx6.p1.4.m4.1.1.1.3.cmml">i</mi><mo id="Thmproofx6.p1.4.m4.1.1.1.2" xref="Thmproofx6.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx6.p1.4.m4.1.1.1.1.1" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="Thmproofx6.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx6.p1.4.m4.1.1.1.1.1.1" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.cmml"><msub id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.cmml"><mi id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.2" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.2.cmml">x</mi><mi id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.3" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.1" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.1.cmml">−</mo><msub id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.2" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.2.cmml">x</mi><mrow id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.cmml"><mi id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.2" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.2.cmml">i</mi><mo id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.1" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.3" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="Thmproofx6.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.4.m4.1b"><apply id="Thmproofx6.p1.4.m4.1.1.cmml" xref="Thmproofx6.p1.4.m4.1.1"><eq id="Thmproofx6.p1.4.m4.1.1.2.cmml" xref="Thmproofx6.p1.4.m4.1.1.2"></eq><apply id="Thmproofx6.p1.4.m4.1.1.3.cmml" xref="Thmproofx6.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="Thmproofx6.p1.4.m4.1.1.3.1.cmml" xref="Thmproofx6.p1.4.m4.1.1.3">subscript</csymbol><ci id="Thmproofx6.p1.4.m4.1.1.3.2.cmml" xref="Thmproofx6.p1.4.m4.1.1.3.2">𝑦</ci><ci id="Thmproofx6.p1.4.m4.1.1.3.3.cmml" xref="Thmproofx6.p1.4.m4.1.1.3.3">𝑖</ci></apply><apply id="Thmproofx6.p1.4.m4.1.1.1.cmml" xref="Thmproofx6.p1.4.m4.1.1.1"><times id="Thmproofx6.p1.4.m4.1.1.1.2.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.2"></times><ci id="Thmproofx6.p1.4.m4.1.1.1.3.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.3">𝑖</ci><apply id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1"><minus id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.1"></minus><apply id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.1.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2">subscript</csymbol><ci id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.2.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.2">𝑥</ci><ci id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.3.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.2">𝑥</ci><apply id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3"><plus id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.1.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.1"></plus><ci id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.2.cmml" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.2">𝑖</ci><cn id="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="Thmproofx6.p1.4.m4.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.4.m4.1c">y_{i}=i(x_{i}-x_{i+1})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.4.m4.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_i ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math> for every <math alttext="1\leq i<m" class="ltx_Math" display="inline" id="Thmproofx6.p1.5.m5.1"><semantics id="Thmproofx6.p1.5.m5.1a"><mrow id="Thmproofx6.p1.5.m5.1.1" xref="Thmproofx6.p1.5.m5.1.1.cmml"><mn id="Thmproofx6.p1.5.m5.1.1.2" xref="Thmproofx6.p1.5.m5.1.1.2.cmml">1</mn><mo id="Thmproofx6.p1.5.m5.1.1.3" xref="Thmproofx6.p1.5.m5.1.1.3.cmml">≤</mo><mi id="Thmproofx6.p1.5.m5.1.1.4" xref="Thmproofx6.p1.5.m5.1.1.4.cmml">i</mi><mo id="Thmproofx6.p1.5.m5.1.1.5" xref="Thmproofx6.p1.5.m5.1.1.5.cmml"><</mo><mi id="Thmproofx6.p1.5.m5.1.1.6" xref="Thmproofx6.p1.5.m5.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.5.m5.1b"><apply id="Thmproofx6.p1.5.m5.1.1.cmml" xref="Thmproofx6.p1.5.m5.1.1"><and id="Thmproofx6.p1.5.m5.1.1a.cmml" xref="Thmproofx6.p1.5.m5.1.1"></and><apply id="Thmproofx6.p1.5.m5.1.1b.cmml" xref="Thmproofx6.p1.5.m5.1.1"><leq id="Thmproofx6.p1.5.m5.1.1.3.cmml" xref="Thmproofx6.p1.5.m5.1.1.3"></leq><cn id="Thmproofx6.p1.5.m5.1.1.2.cmml" type="integer" xref="Thmproofx6.p1.5.m5.1.1.2">1</cn><ci id="Thmproofx6.p1.5.m5.1.1.4.cmml" xref="Thmproofx6.p1.5.m5.1.1.4">𝑖</ci></apply><apply id="Thmproofx6.p1.5.m5.1.1c.cmml" xref="Thmproofx6.p1.5.m5.1.1"><lt id="Thmproofx6.p1.5.m5.1.1.5.cmml" xref="Thmproofx6.p1.5.m5.1.1.5"></lt><share href="https://arxiv.org/html/2308.08047v3#Thmproofx6.p1.5.m5.1.1.4.cmml" id="Thmproofx6.p1.5.m5.1.1d.cmml" xref="Thmproofx6.p1.5.m5.1.1"></share><ci id="Thmproofx6.p1.5.m5.1.1.6.cmml" xref="Thmproofx6.p1.5.m5.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.5.m5.1c">1\leq i<m</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.5.m5.1d">1 ≤ italic_i < italic_m</annotation></semantics></math>. Then <math alttext="y_{i}\in[0,1]" class="ltx_Math" display="inline" id="Thmproofx6.p1.6.m6.2"><semantics id="Thmproofx6.p1.6.m6.2a"><mrow id="Thmproofx6.p1.6.m6.2.3" xref="Thmproofx6.p1.6.m6.2.3.cmml"><msub id="Thmproofx6.p1.6.m6.2.3.2" xref="Thmproofx6.p1.6.m6.2.3.2.cmml"><mi id="Thmproofx6.p1.6.m6.2.3.2.2" xref="Thmproofx6.p1.6.m6.2.3.2.2.cmml">y</mi><mi id="Thmproofx6.p1.6.m6.2.3.2.3" xref="Thmproofx6.p1.6.m6.2.3.2.3.cmml">i</mi></msub><mo id="Thmproofx6.p1.6.m6.2.3.1" xref="Thmproofx6.p1.6.m6.2.3.1.cmml">∈</mo><mrow id="Thmproofx6.p1.6.m6.2.3.3.2" xref="Thmproofx6.p1.6.m6.2.3.3.1.cmml"><mo id="Thmproofx6.p1.6.m6.2.3.3.2.1" stretchy="false" xref="Thmproofx6.p1.6.m6.2.3.3.1.cmml">[</mo><mn id="Thmproofx6.p1.6.m6.1.1" xref="Thmproofx6.p1.6.m6.1.1.cmml">0</mn><mo id="Thmproofx6.p1.6.m6.2.3.3.2.2" xref="Thmproofx6.p1.6.m6.2.3.3.1.cmml">,</mo><mn id="Thmproofx6.p1.6.m6.2.2" xref="Thmproofx6.p1.6.m6.2.2.cmml">1</mn><mo id="Thmproofx6.p1.6.m6.2.3.3.2.3" stretchy="false" xref="Thmproofx6.p1.6.m6.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.6.m6.2b"><apply id="Thmproofx6.p1.6.m6.2.3.cmml" xref="Thmproofx6.p1.6.m6.2.3"><in id="Thmproofx6.p1.6.m6.2.3.1.cmml" xref="Thmproofx6.p1.6.m6.2.3.1"></in><apply id="Thmproofx6.p1.6.m6.2.3.2.cmml" xref="Thmproofx6.p1.6.m6.2.3.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.6.m6.2.3.2.1.cmml" xref="Thmproofx6.p1.6.m6.2.3.2">subscript</csymbol><ci id="Thmproofx6.p1.6.m6.2.3.2.2.cmml" xref="Thmproofx6.p1.6.m6.2.3.2.2">𝑦</ci><ci id="Thmproofx6.p1.6.m6.2.3.2.3.cmml" xref="Thmproofx6.p1.6.m6.2.3.2.3">𝑖</ci></apply><interval closure="closed" id="Thmproofx6.p1.6.m6.2.3.3.1.cmml" xref="Thmproofx6.p1.6.m6.2.3.3.2"><cn id="Thmproofx6.p1.6.m6.1.1.cmml" type="integer" xref="Thmproofx6.p1.6.m6.1.1">0</cn><cn id="Thmproofx6.p1.6.m6.2.2.cmml" type="integer" xref="Thmproofx6.p1.6.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.6.m6.2c">y_{i}\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.6.m6.2d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ 0 , 1 ]</annotation></semantics></math>, <math alttext="\sum y_{i}=\sum x_{i}=1" class="ltx_Math" display="inline" id="Thmproofx6.p1.7.m7.1"><semantics id="Thmproofx6.p1.7.m7.1a"><mrow id="Thmproofx6.p1.7.m7.1.1" xref="Thmproofx6.p1.7.m7.1.1.cmml"><mrow id="Thmproofx6.p1.7.m7.1.1.2" xref="Thmproofx6.p1.7.m7.1.1.2.cmml"><mo id="Thmproofx6.p1.7.m7.1.1.2.1" xref="Thmproofx6.p1.7.m7.1.1.2.1.cmml">∑</mo><msub id="Thmproofx6.p1.7.m7.1.1.2.2" xref="Thmproofx6.p1.7.m7.1.1.2.2.cmml"><mi id="Thmproofx6.p1.7.m7.1.1.2.2.2" xref="Thmproofx6.p1.7.m7.1.1.2.2.2.cmml">y</mi><mi id="Thmproofx6.p1.7.m7.1.1.2.2.3" xref="Thmproofx6.p1.7.m7.1.1.2.2.3.cmml">i</mi></msub></mrow><mo id="Thmproofx6.p1.7.m7.1.1.3" rspace="0.111em" xref="Thmproofx6.p1.7.m7.1.1.3.cmml">=</mo><mrow id="Thmproofx6.p1.7.m7.1.1.4" xref="Thmproofx6.p1.7.m7.1.1.4.cmml"><mo id="Thmproofx6.p1.7.m7.1.1.4.1" xref="Thmproofx6.p1.7.m7.1.1.4.1.cmml">∑</mo><msub id="Thmproofx6.p1.7.m7.1.1.4.2" xref="Thmproofx6.p1.7.m7.1.1.4.2.cmml"><mi id="Thmproofx6.p1.7.m7.1.1.4.2.2" xref="Thmproofx6.p1.7.m7.1.1.4.2.2.cmml">x</mi><mi id="Thmproofx6.p1.7.m7.1.1.4.2.3" xref="Thmproofx6.p1.7.m7.1.1.4.2.3.cmml">i</mi></msub></mrow><mo id="Thmproofx6.p1.7.m7.1.1.5" xref="Thmproofx6.p1.7.m7.1.1.5.cmml">=</mo><mn id="Thmproofx6.p1.7.m7.1.1.6" xref="Thmproofx6.p1.7.m7.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.7.m7.1b"><apply id="Thmproofx6.p1.7.m7.1.1.cmml" xref="Thmproofx6.p1.7.m7.1.1"><and id="Thmproofx6.p1.7.m7.1.1a.cmml" xref="Thmproofx6.p1.7.m7.1.1"></and><apply id="Thmproofx6.p1.7.m7.1.1b.cmml" xref="Thmproofx6.p1.7.m7.1.1"><eq id="Thmproofx6.p1.7.m7.1.1.3.cmml" xref="Thmproofx6.p1.7.m7.1.1.3"></eq><apply id="Thmproofx6.p1.7.m7.1.1.2.cmml" xref="Thmproofx6.p1.7.m7.1.1.2"><sum id="Thmproofx6.p1.7.m7.1.1.2.1.cmml" xref="Thmproofx6.p1.7.m7.1.1.2.1"></sum><apply id="Thmproofx6.p1.7.m7.1.1.2.2.cmml" xref="Thmproofx6.p1.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.7.m7.1.1.2.2.1.cmml" xref="Thmproofx6.p1.7.m7.1.1.2.2">subscript</csymbol><ci id="Thmproofx6.p1.7.m7.1.1.2.2.2.cmml" xref="Thmproofx6.p1.7.m7.1.1.2.2.2">𝑦</ci><ci id="Thmproofx6.p1.7.m7.1.1.2.2.3.cmml" xref="Thmproofx6.p1.7.m7.1.1.2.2.3">𝑖</ci></apply></apply><apply id="Thmproofx6.p1.7.m7.1.1.4.cmml" xref="Thmproofx6.p1.7.m7.1.1.4"><sum id="Thmproofx6.p1.7.m7.1.1.4.1.cmml" xref="Thmproofx6.p1.7.m7.1.1.4.1"></sum><apply id="Thmproofx6.p1.7.m7.1.1.4.2.cmml" xref="Thmproofx6.p1.7.m7.1.1.4.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.7.m7.1.1.4.2.1.cmml" xref="Thmproofx6.p1.7.m7.1.1.4.2">subscript</csymbol><ci id="Thmproofx6.p1.7.m7.1.1.4.2.2.cmml" xref="Thmproofx6.p1.7.m7.1.1.4.2.2">𝑥</ci><ci id="Thmproofx6.p1.7.m7.1.1.4.2.3.cmml" xref="Thmproofx6.p1.7.m7.1.1.4.2.3">𝑖</ci></apply></apply></apply><apply id="Thmproofx6.p1.7.m7.1.1c.cmml" xref="Thmproofx6.p1.7.m7.1.1"><eq id="Thmproofx6.p1.7.m7.1.1.5.cmml" xref="Thmproofx6.p1.7.m7.1.1.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx6.p1.7.m7.1.1.4.cmml" id="Thmproofx6.p1.7.m7.1.1d.cmml" xref="Thmproofx6.p1.7.m7.1.1"></share><cn id="Thmproofx6.p1.7.m7.1.1.6.cmml" type="integer" xref="Thmproofx6.p1.7.m7.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.7.m7.1c">\sum y_{i}=\sum x_{i}=1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.7.m7.1d">∑ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math>, and treating <math alttext="X,U_{P_{i}}" class="ltx_Math" display="inline" id="Thmproofx6.p1.8.m8.2"><semantics id="Thmproofx6.p1.8.m8.2a"><mrow id="Thmproofx6.p1.8.m8.2.2.1" xref="Thmproofx6.p1.8.m8.2.2.2.cmml"><mi id="Thmproofx6.p1.8.m8.1.1" xref="Thmproofx6.p1.8.m8.1.1.cmml">X</mi><mo id="Thmproofx6.p1.8.m8.2.2.1.2" xref="Thmproofx6.p1.8.m8.2.2.2.cmml">,</mo><msub id="Thmproofx6.p1.8.m8.2.2.1.1" xref="Thmproofx6.p1.8.m8.2.2.1.1.cmml"><mi id="Thmproofx6.p1.8.m8.2.2.1.1.2" xref="Thmproofx6.p1.8.m8.2.2.1.1.2.cmml">U</mi><msub id="Thmproofx6.p1.8.m8.2.2.1.1.3" xref="Thmproofx6.p1.8.m8.2.2.1.1.3.cmml"><mi id="Thmproofx6.p1.8.m8.2.2.1.1.3.2" xref="Thmproofx6.p1.8.m8.2.2.1.1.3.2.cmml">P</mi><mi id="Thmproofx6.p1.8.m8.2.2.1.1.3.3" xref="Thmproofx6.p1.8.m8.2.2.1.1.3.3.cmml">i</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.8.m8.2b"><list id="Thmproofx6.p1.8.m8.2.2.2.cmml" xref="Thmproofx6.p1.8.m8.2.2.1"><ci id="Thmproofx6.p1.8.m8.1.1.cmml" xref="Thmproofx6.p1.8.m8.1.1">𝑋</ci><apply id="Thmproofx6.p1.8.m8.2.2.1.1.cmml" xref="Thmproofx6.p1.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.8.m8.2.2.1.1.1.cmml" xref="Thmproofx6.p1.8.m8.2.2.1.1">subscript</csymbol><ci id="Thmproofx6.p1.8.m8.2.2.1.1.2.cmml" xref="Thmproofx6.p1.8.m8.2.2.1.1.2">𝑈</ci><apply id="Thmproofx6.p1.8.m8.2.2.1.1.3.cmml" xref="Thmproofx6.p1.8.m8.2.2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx6.p1.8.m8.2.2.1.1.3.1.cmml" xref="Thmproofx6.p1.8.m8.2.2.1.1.3">subscript</csymbol><ci id="Thmproofx6.p1.8.m8.2.2.1.1.3.2.cmml" xref="Thmproofx6.p1.8.m8.2.2.1.1.3.2">𝑃</ci><ci id="Thmproofx6.p1.8.m8.2.2.1.1.3.3.cmml" xref="Thmproofx6.p1.8.m8.2.2.1.1.3.3">𝑖</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.8.m8.2c">X,U_{P_{i}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.8.m8.2d">italic_X , italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> as vectors in <math alttext="\mathbb{R}^{m}" class="ltx_Math" display="inline" id="Thmproofx6.p1.9.m9.1"><semantics id="Thmproofx6.p1.9.m9.1a"><msup id="Thmproofx6.p1.9.m9.1.1" xref="Thmproofx6.p1.9.m9.1.1.cmml"><mi id="Thmproofx6.p1.9.m9.1.1.2" xref="Thmproofx6.p1.9.m9.1.1.2.cmml">ℝ</mi><mi id="Thmproofx6.p1.9.m9.1.1.3" xref="Thmproofx6.p1.9.m9.1.1.3.cmml">m</mi></msup><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.9.m9.1b"><apply id="Thmproofx6.p1.9.m9.1.1.cmml" xref="Thmproofx6.p1.9.m9.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.9.m9.1.1.1.cmml" xref="Thmproofx6.p1.9.m9.1.1">superscript</csymbol><ci id="Thmproofx6.p1.9.m9.1.1.2.cmml" xref="Thmproofx6.p1.9.m9.1.1.2">ℝ</ci><ci id="Thmproofx6.p1.9.m9.1.1.3.cmml" xref="Thmproofx6.p1.9.m9.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.9.m9.1c">\mathbb{R}^{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.9.m9.1d">blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> we get <math alttext="\sum y_{i}U_{P_{i}}=X" class="ltx_Math" display="inline" id="Thmproofx6.p1.10.m10.1"><semantics id="Thmproofx6.p1.10.m10.1a"><mrow id="Thmproofx6.p1.10.m10.1.1" xref="Thmproofx6.p1.10.m10.1.1.cmml"><mrow id="Thmproofx6.p1.10.m10.1.1.2" xref="Thmproofx6.p1.10.m10.1.1.2.cmml"><mo id="Thmproofx6.p1.10.m10.1.1.2.1" xref="Thmproofx6.p1.10.m10.1.1.2.1.cmml">∑</mo><mrow id="Thmproofx6.p1.10.m10.1.1.2.2" xref="Thmproofx6.p1.10.m10.1.1.2.2.cmml"><msub id="Thmproofx6.p1.10.m10.1.1.2.2.2" xref="Thmproofx6.p1.10.m10.1.1.2.2.2.cmml"><mi id="Thmproofx6.p1.10.m10.1.1.2.2.2.2" xref="Thmproofx6.p1.10.m10.1.1.2.2.2.2.cmml">y</mi><mi id="Thmproofx6.p1.10.m10.1.1.2.2.2.3" xref="Thmproofx6.p1.10.m10.1.1.2.2.2.3.cmml">i</mi></msub><mo id="Thmproofx6.p1.10.m10.1.1.2.2.1" xref="Thmproofx6.p1.10.m10.1.1.2.2.1.cmml">⁢</mo><msub id="Thmproofx6.p1.10.m10.1.1.2.2.3" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.cmml"><mi id="Thmproofx6.p1.10.m10.1.1.2.2.3.2" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.2.cmml">U</mi><msub id="Thmproofx6.p1.10.m10.1.1.2.2.3.3" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.3.cmml"><mi id="Thmproofx6.p1.10.m10.1.1.2.2.3.3.2" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.3.2.cmml">P</mi><mi id="Thmproofx6.p1.10.m10.1.1.2.2.3.3.3" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.3.3.cmml">i</mi></msub></msub></mrow></mrow><mo id="Thmproofx6.p1.10.m10.1.1.1" xref="Thmproofx6.p1.10.m10.1.1.1.cmml">=</mo><mi id="Thmproofx6.p1.10.m10.1.1.3" xref="Thmproofx6.p1.10.m10.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.10.m10.1b"><apply id="Thmproofx6.p1.10.m10.1.1.cmml" xref="Thmproofx6.p1.10.m10.1.1"><eq id="Thmproofx6.p1.10.m10.1.1.1.cmml" xref="Thmproofx6.p1.10.m10.1.1.1"></eq><apply id="Thmproofx6.p1.10.m10.1.1.2.cmml" xref="Thmproofx6.p1.10.m10.1.1.2"><sum id="Thmproofx6.p1.10.m10.1.1.2.1.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.1"></sum><apply id="Thmproofx6.p1.10.m10.1.1.2.2.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2"><times id="Thmproofx6.p1.10.m10.1.1.2.2.1.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.1"></times><apply id="Thmproofx6.p1.10.m10.1.1.2.2.2.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx6.p1.10.m10.1.1.2.2.2.1.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.2">subscript</csymbol><ci id="Thmproofx6.p1.10.m10.1.1.2.2.2.2.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.2.2">𝑦</ci><ci id="Thmproofx6.p1.10.m10.1.1.2.2.2.3.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.2.3">𝑖</ci></apply><apply id="Thmproofx6.p1.10.m10.1.1.2.2.3.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.3"><csymbol cd="ambiguous" id="Thmproofx6.p1.10.m10.1.1.2.2.3.1.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.3">subscript</csymbol><ci id="Thmproofx6.p1.10.m10.1.1.2.2.3.2.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.2">𝑈</ci><apply id="Thmproofx6.p1.10.m10.1.1.2.2.3.3.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.3"><csymbol cd="ambiguous" id="Thmproofx6.p1.10.m10.1.1.2.2.3.3.1.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.3">subscript</csymbol><ci id="Thmproofx6.p1.10.m10.1.1.2.2.3.3.2.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.3.2">𝑃</ci><ci id="Thmproofx6.p1.10.m10.1.1.2.2.3.3.3.cmml" xref="Thmproofx6.p1.10.m10.1.1.2.2.3.3.3">𝑖</ci></apply></apply></apply></apply><ci id="Thmproofx6.p1.10.m10.1.1.3.cmml" xref="Thmproofx6.p1.10.m10.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.10.m10.1c">\sum y_{i}U_{P_{i}}=X</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.10.m10.1d">∑ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_X</annotation></semantics></math>. Hence <math alttext="X\in\operatorname{conv}\left(\left\{U_{P_{i}}\right\}_{i\in[m]}\right)% \stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname{conv}\left(\mathcal{U}_{% 1}\right)" class="ltx_Math" display="inline" id="Thmproofx6.p1.11.m11.5"><semantics id="Thmproofx6.p1.11.m11.5a"><mrow id="Thmproofx6.p1.11.m11.5.5" xref="Thmproofx6.p1.11.m11.5.5.cmml"><mi id="Thmproofx6.p1.11.m11.5.5.4" xref="Thmproofx6.p1.11.m11.5.5.4.cmml">X</mi><mo id="Thmproofx6.p1.11.m11.5.5.5" xref="Thmproofx6.p1.11.m11.5.5.5.cmml">∈</mo><mrow id="Thmproofx6.p1.11.m11.4.4.1.1" xref="Thmproofx6.p1.11.m11.4.4.1.2.cmml"><mi id="Thmproofx6.p1.11.m11.2.2" xref="Thmproofx6.p1.11.m11.2.2.cmml">conv</mi><mo id="Thmproofx6.p1.11.m11.4.4.1.1a" xref="Thmproofx6.p1.11.m11.4.4.1.2.cmml">⁡</mo><mrow id="Thmproofx6.p1.11.m11.4.4.1.1.1" xref="Thmproofx6.p1.11.m11.4.4.1.2.cmml"><mo id="Thmproofx6.p1.11.m11.4.4.1.1.1.2" xref="Thmproofx6.p1.11.m11.4.4.1.2.cmml">(</mo><msub id="Thmproofx6.p1.11.m11.4.4.1.1.1.1" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.cmml"><mrow id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.2.cmml"><mo id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.2" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.2.cmml">{</mo><msub id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.2" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.2.cmml">U</mi><msub id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.2" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.2.cmml">P</mi><mi id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.3" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.3.cmml">i</mi></msub></msub><mo id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.3" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="Thmproofx6.p1.11.m11.1.1.1" xref="Thmproofx6.p1.11.m11.1.1.1.cmml"><mi id="Thmproofx6.p1.11.m11.1.1.1.3" xref="Thmproofx6.p1.11.m11.1.1.1.3.cmml">i</mi><mo id="Thmproofx6.p1.11.m11.1.1.1.2" xref="Thmproofx6.p1.11.m11.1.1.1.2.cmml">∈</mo><mrow id="Thmproofx6.p1.11.m11.1.1.1.4.2" xref="Thmproofx6.p1.11.m11.1.1.1.4.1.cmml"><mo id="Thmproofx6.p1.11.m11.1.1.1.4.2.1" stretchy="false" xref="Thmproofx6.p1.11.m11.1.1.1.4.1.1.cmml">[</mo><mi id="Thmproofx6.p1.11.m11.1.1.1.1" xref="Thmproofx6.p1.11.m11.1.1.1.1.cmml">m</mi><mo id="Thmproofx6.p1.11.m11.1.1.1.4.2.2" stretchy="false" xref="Thmproofx6.p1.11.m11.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mo id="Thmproofx6.p1.11.m11.4.4.1.1.1.3" xref="Thmproofx6.p1.11.m11.4.4.1.2.cmml">)</mo></mrow></mrow><mover id="Thmproofx6.p1.11.m11.5.5.6" xref="Thmproofx6.p1.11.m11.5.5.6.cmml"><mo id="Thmproofx6.p1.11.m11.5.5.6.2" xref="Thmproofx6.p1.11.m11.5.5.6.2.cmml">=</mo><mtext id="Thmproofx6.p1.11.m11.5.5.6.3" xref="Thmproofx6.p1.11.m11.5.5.6.3a.cmml">def</mtext></mover><mrow id="Thmproofx6.p1.11.m11.5.5.2.1" xref="Thmproofx6.p1.11.m11.5.5.2.2.cmml"><mi id="Thmproofx6.p1.11.m11.3.3" xref="Thmproofx6.p1.11.m11.3.3.cmml">conv</mi><mo id="Thmproofx6.p1.11.m11.5.5.2.1a" xref="Thmproofx6.p1.11.m11.5.5.2.2.cmml">⁡</mo><mrow id="Thmproofx6.p1.11.m11.5.5.2.1.1" xref="Thmproofx6.p1.11.m11.5.5.2.2.cmml"><mo id="Thmproofx6.p1.11.m11.5.5.2.1.1.2" xref="Thmproofx6.p1.11.m11.5.5.2.2.cmml">(</mo><msub id="Thmproofx6.p1.11.m11.5.5.2.1.1.1" xref="Thmproofx6.p1.11.m11.5.5.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx6.p1.11.m11.5.5.2.1.1.1.2" xref="Thmproofx6.p1.11.m11.5.5.2.1.1.1.2.cmml">𝒰</mi><mn id="Thmproofx6.p1.11.m11.5.5.2.1.1.1.3" xref="Thmproofx6.p1.11.m11.5.5.2.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx6.p1.11.m11.5.5.2.1.1.3" xref="Thmproofx6.p1.11.m11.5.5.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.11.m11.5b"><apply id="Thmproofx6.p1.11.m11.5.5.cmml" xref="Thmproofx6.p1.11.m11.5.5"><and id="Thmproofx6.p1.11.m11.5.5a.cmml" xref="Thmproofx6.p1.11.m11.5.5"></and><apply id="Thmproofx6.p1.11.m11.5.5b.cmml" xref="Thmproofx6.p1.11.m11.5.5"><in id="Thmproofx6.p1.11.m11.5.5.5.cmml" xref="Thmproofx6.p1.11.m11.5.5.5"></in><ci id="Thmproofx6.p1.11.m11.5.5.4.cmml" xref="Thmproofx6.p1.11.m11.5.5.4">𝑋</ci><apply id="Thmproofx6.p1.11.m11.4.4.1.2.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1"><ci id="Thmproofx6.p1.11.m11.2.2.cmml" xref="Thmproofx6.p1.11.m11.2.2">conv</ci><apply id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.2.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1">subscript</csymbol><set id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.2.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1"><apply id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.2">𝑈</ci><apply id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.2">𝑃</ci><ci id="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx6.p1.11.m11.4.4.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></set><apply id="Thmproofx6.p1.11.m11.1.1.1.cmml" xref="Thmproofx6.p1.11.m11.1.1.1"><in id="Thmproofx6.p1.11.m11.1.1.1.2.cmml" xref="Thmproofx6.p1.11.m11.1.1.1.2"></in><ci id="Thmproofx6.p1.11.m11.1.1.1.3.cmml" xref="Thmproofx6.p1.11.m11.1.1.1.3">𝑖</ci><apply id="Thmproofx6.p1.11.m11.1.1.1.4.1.cmml" xref="Thmproofx6.p1.11.m11.1.1.1.4.2"><csymbol cd="latexml" id="Thmproofx6.p1.11.m11.1.1.1.4.1.1.cmml" xref="Thmproofx6.p1.11.m11.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Thmproofx6.p1.11.m11.1.1.1.1.cmml" xref="Thmproofx6.p1.11.m11.1.1.1.1">𝑚</ci></apply></apply></apply></apply></apply><apply id="Thmproofx6.p1.11.m11.5.5c.cmml" xref="Thmproofx6.p1.11.m11.5.5"><apply id="Thmproofx6.p1.11.m11.5.5.6.cmml" xref="Thmproofx6.p1.11.m11.5.5.6"><csymbol cd="ambiguous" id="Thmproofx6.p1.11.m11.5.5.6.1.cmml" xref="Thmproofx6.p1.11.m11.5.5.6">superscript</csymbol><eq id="Thmproofx6.p1.11.m11.5.5.6.2.cmml" xref="Thmproofx6.p1.11.m11.5.5.6.2"></eq><ci id="Thmproofx6.p1.11.m11.5.5.6.3a.cmml" xref="Thmproofx6.p1.11.m11.5.5.6.3"><mtext id="Thmproofx6.p1.11.m11.5.5.6.3.cmml" mathsize="70%" xref="Thmproofx6.p1.11.m11.5.5.6.3">def</mtext></ci></apply><share href="https://arxiv.org/html/2308.08047v3#Thmproofx6.p1.11.m11.4.4.1.cmml" id="Thmproofx6.p1.11.m11.5.5d.cmml" xref="Thmproofx6.p1.11.m11.5.5"></share><apply id="Thmproofx6.p1.11.m11.5.5.2.2.cmml" xref="Thmproofx6.p1.11.m11.5.5.2.1"><ci id="Thmproofx6.p1.11.m11.3.3.cmml" xref="Thmproofx6.p1.11.m11.3.3">conv</ci><apply id="Thmproofx6.p1.11.m11.5.5.2.1.1.1.cmml" xref="Thmproofx6.p1.11.m11.5.5.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.11.m11.5.5.2.1.1.1.1.cmml" xref="Thmproofx6.p1.11.m11.5.5.2.1.1.1">subscript</csymbol><ci id="Thmproofx6.p1.11.m11.5.5.2.1.1.1.2.cmml" xref="Thmproofx6.p1.11.m11.5.5.2.1.1.1.2">𝒰</ci><cn id="Thmproofx6.p1.11.m11.5.5.2.1.1.1.3.cmml" type="integer" xref="Thmproofx6.p1.11.m11.5.5.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.11.m11.5c">X\in\operatorname{conv}\left(\left\{U_{P_{i}}\right\}_{i\in[m]}\right)% \stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname{conv}\left(\mathcal{U}_{% 1}\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.11.m11.5d">italic_X ∈ roman_conv ( { italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. The other inclusion follows by definition.</p> </div> <div class="ltx_para" id="Thmproofx6.p2"> <p class="ltx_p" id="Thmproofx6.p2.4">Now we prove the second equality in the statement. Because all distributions in its right-hand side (rhs) are iid,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx23"> <tbody id="Pt0.A1.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\{X^{n}\mid X\in\operatorname{conv}(\mathcal{U}_{1})\right\}" class="ltx_Math" display="inline" id="Pt0.A1.Ex25.m1.3"><semantics id="Pt0.A1.Ex25.m1.3a"><mrow id="Pt0.A1.Ex25.m1.3.3.2" xref="Pt0.A1.Ex25.m1.3.3.3.cmml"><mo id="Pt0.A1.Ex25.m1.3.3.2.3" xref="Pt0.A1.Ex25.m1.3.3.3.1.cmml">{</mo><msup id="Pt0.A1.Ex25.m1.2.2.1.1" xref="Pt0.A1.Ex25.m1.2.2.1.1.cmml"><mi id="Pt0.A1.Ex25.m1.2.2.1.1.2" xref="Pt0.A1.Ex25.m1.2.2.1.1.2.cmml">X</mi><mi id="Pt0.A1.Ex25.m1.2.2.1.1.3" xref="Pt0.A1.Ex25.m1.2.2.1.1.3.cmml">n</mi></msup><mo fence="true" id="Pt0.A1.Ex25.m1.3.3.2.4" lspace="0em" rspace="0em" xref="Pt0.A1.Ex25.m1.3.3.3.1.cmml">∣</mo><mrow id="Pt0.A1.Ex25.m1.3.3.2.2" xref="Pt0.A1.Ex25.m1.3.3.2.2.cmml"><mi id="Pt0.A1.Ex25.m1.3.3.2.2.3" xref="Pt0.A1.Ex25.m1.3.3.2.2.3.cmml">X</mi><mo id="Pt0.A1.Ex25.m1.3.3.2.2.2" xref="Pt0.A1.Ex25.m1.3.3.2.2.2.cmml">∈</mo><mrow id="Pt0.A1.Ex25.m1.3.3.2.2.1.1" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.2.cmml"><mi id="Pt0.A1.Ex25.m1.1.1" xref="Pt0.A1.Ex25.m1.1.1.cmml">conv</mi><mo id="Pt0.A1.Ex25.m1.3.3.2.2.1.1a" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.2.cmml">⁡</mo><mrow id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.2.cmml"><mo id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.2" stretchy="false" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.2.cmml">(</mo><msub id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.2" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.2.cmml">𝒰</mi><mn id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.3" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.3" stretchy="false" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A1.Ex25.m1.3.3.2.5" xref="Pt0.A1.Ex25.m1.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A1.Ex25.m1.3b"><apply id="Pt0.A1.Ex25.m1.3.3.3.cmml" xref="Pt0.A1.Ex25.m1.3.3.2"><csymbol cd="latexml" id="Pt0.A1.Ex25.m1.3.3.3.1.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.3">conditional-set</csymbol><apply id="Pt0.A1.Ex25.m1.2.2.1.1.cmml" xref="Pt0.A1.Ex25.m1.2.2.1.1"><csymbol cd="ambiguous" id="Pt0.A1.Ex25.m1.2.2.1.1.1.cmml" xref="Pt0.A1.Ex25.m1.2.2.1.1">superscript</csymbol><ci id="Pt0.A1.Ex25.m1.2.2.1.1.2.cmml" xref="Pt0.A1.Ex25.m1.2.2.1.1.2">𝑋</ci><ci id="Pt0.A1.Ex25.m1.2.2.1.1.3.cmml" xref="Pt0.A1.Ex25.m1.2.2.1.1.3">𝑛</ci></apply><apply id="Pt0.A1.Ex25.m1.3.3.2.2.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.2"><in id="Pt0.A1.Ex25.m1.3.3.2.2.2.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.2.2"></in><ci id="Pt0.A1.Ex25.m1.3.3.2.2.3.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.2.3">𝑋</ci><apply id="Pt0.A1.Ex25.m1.3.3.2.2.1.2.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1"><ci id="Pt0.A1.Ex25.m1.1.1.cmml" xref="Pt0.A1.Ex25.m1.1.1">conv</ci><apply id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.1.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1">subscript</csymbol><ci id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.2.cmml" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.2">𝒰</ci><cn id="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.3.cmml" type="integer" xref="Pt0.A1.Ex25.m1.3.3.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex25.m1.3c">\displaystyle\left\{X^{n}\mid X\in\operatorname{conv}(\mathcal{U}_{1})\right\}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex25.m1.3d">{ italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∣ italic_X ∈ roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 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xref="Pt0.A1.Ex26.m1.4.4.2.2.2.2.2.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A1.Ex26.m1.4c">\displaystyle=\left\{\sum_{i}x_{i}U_{P_{i}}^{n}\mid x_{i}\in[0,1],~{}~{}\sum_{% i}x_{i}=1\right\}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A1.Ex26.m1.4d">= { ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∣ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ 0 , 1 ] , ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A1.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname{conv}(% \mathcal{U}_{n})" class="ltx_Math" display="inline" id="Pt0.A1.Ex27.m1.2"><semantics id="Pt0.A1.Ex27.m1.2a"><mrow id="Pt0.A1.Ex27.m1.2.2" xref="Pt0.A1.Ex27.m1.2.2.cmml"><mi id="Pt0.A1.Ex27.m1.2.2.3" xref="Pt0.A1.Ex27.m1.2.2.3.cmml"></mi><mover id="Pt0.A1.Ex27.m1.2.2.2" xref="Pt0.A1.Ex27.m1.2.2.2.cmml"><mo id="Pt0.A1.Ex27.m1.2.2.2.2" xref="Pt0.A1.Ex27.m1.2.2.2.2.cmml">=</mo><mtext id="Pt0.A1.Ex27.m1.2.2.2.3" xref="Pt0.A1.Ex27.m1.2.2.2.3a.cmml">def</mtext></mover><mrow id="Pt0.A1.Ex27.m1.2.2.1.1" xref="Pt0.A1.Ex27.m1.2.2.1.2.cmml"><mi id="Pt0.A1.Ex27.m1.1.1" xref="Pt0.A1.Ex27.m1.1.1.cmml">conv</mi><mo id="Pt0.A1.Ex27.m1.2.2.1.1a" xref="Pt0.A1.Ex27.m1.2.2.1.2.cmml">⁡</mo><mrow id="Pt0.A1.Ex27.m1.2.2.1.1.1" xref="Pt0.A1.Ex27.m1.2.2.1.2.cmml"><mo id="Pt0.A1.Ex27.m1.2.2.1.1.1.2" 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end_RELOP roman_conv ( caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx6.p2.3">where we treated <math alttext="U_{P_{i}}^{n}" class="ltx_Math" display="inline" id="Thmproofx6.p2.1.m1.1"><semantics id="Thmproofx6.p2.1.m1.1a"><msubsup id="Thmproofx6.p2.1.m1.1.1" xref="Thmproofx6.p2.1.m1.1.1.cmml"><mi id="Thmproofx6.p2.1.m1.1.1.2.2" xref="Thmproofx6.p2.1.m1.1.1.2.2.cmml">U</mi><msub id="Thmproofx6.p2.1.m1.1.1.2.3" xref="Thmproofx6.p2.1.m1.1.1.2.3.cmml"><mi id="Thmproofx6.p2.1.m1.1.1.2.3.2" xref="Thmproofx6.p2.1.m1.1.1.2.3.2.cmml">P</mi><mi id="Thmproofx6.p2.1.m1.1.1.2.3.3" xref="Thmproofx6.p2.1.m1.1.1.2.3.3.cmml">i</mi></msub><mi id="Thmproofx6.p2.1.m1.1.1.3" xref="Thmproofx6.p2.1.m1.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx6.p2.1.m1.1b"><apply id="Thmproofx6.p2.1.m1.1.1.cmml" 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encoding="application/x-llamapun" id="Thmproofx6.p2.1.m1.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> (and <math alttext="X" class="ltx_Math" display="inline" id="Thmproofx6.p2.2.m2.1"><semantics id="Thmproofx6.p2.2.m2.1a"><mi id="Thmproofx6.p2.2.m2.1.1" xref="Thmproofx6.p2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="Thmproofx6.p2.2.m2.1b"><ci id="Thmproofx6.p2.2.m2.1.1.cmml" xref="Thmproofx6.p2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p2.2.m2.1d">italic_X</annotation></semantics></math>) as a vector in <math alttext="\mathbb{R}^{m\times n}" class="ltx_Math" display="inline" id="Thmproofx6.p2.3.m3.1"><semantics id="Thmproofx6.p2.3.m3.1a"><msup id="Thmproofx6.p2.3.m3.1.1" xref="Thmproofx6.p2.3.m3.1.1.cmml"><mi id="Thmproofx6.p2.3.m3.1.1.2" xref="Thmproofx6.p2.3.m3.1.1.2.cmml">ℝ</mi><mrow id="Thmproofx6.p2.3.m3.1.1.3" xref="Thmproofx6.p2.3.m3.1.1.3.cmml"><mi id="Thmproofx6.p2.3.m3.1.1.3.2" xref="Thmproofx6.p2.3.m3.1.1.3.2.cmml">m</mi><mo id="Thmproofx6.p2.3.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="Thmproofx6.p2.3.m3.1.1.3.1.cmml">×</mo><mi id="Thmproofx6.p2.3.m3.1.1.3.3" xref="Thmproofx6.p2.3.m3.1.1.3.3.cmml">n</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="Thmproofx6.p2.3.m3.1b"><apply id="Thmproofx6.p2.3.m3.1.1.cmml" xref="Thmproofx6.p2.3.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx6.p2.3.m3.1.1.1.cmml" xref="Thmproofx6.p2.3.m3.1.1">superscript</csymbol><ci id="Thmproofx6.p2.3.m3.1.1.2.cmml" xref="Thmproofx6.p2.3.m3.1.1.2">ℝ</ci><apply id="Thmproofx6.p2.3.m3.1.1.3.cmml" xref="Thmproofx6.p2.3.m3.1.1.3"><times id="Thmproofx6.p2.3.m3.1.1.3.1.cmml" xref="Thmproofx6.p2.3.m3.1.1.3.1"></times><ci id="Thmproofx6.p2.3.m3.1.1.3.2.cmml" xref="Thmproofx6.p2.3.m3.1.1.3.2">𝑚</ci><ci id="Thmproofx6.p2.3.m3.1.1.3.3.cmml" xref="Thmproofx6.p2.3.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p2.3.m3.1c">\mathbb{R}^{m\times n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p2.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_m × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_appendix" id="Pt0.A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix 0.B </span>Connections of Previous Results to Our Model</h2> <section class="ltx_paragraph" id="Pt0.A2.SS0.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Converting payoffs to costs</h5> <div class="ltx_para" id="Pt0.A2.SS0.SSS0.Px1.p1"> <span class="ltx_ERROR undefined" id="Pt0.A2.SS0.SSS0.Px1.p1.11">\citet</span> <p class="ltx_p" id="Pt0.A2.SS0.SSS0.Px1.p1.10">basilico2017team studied games of team agents vs. an adversary, where the team agents aim to <span class="ltx_text ltx_font_italic" id="Pt0.A2.SS0.SSS0.Px1.p1.10.1">maximize their payoff</span>. For a payoff function <math alttext="u" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1a"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1b"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.1.m1.1d">italic_u</annotation></semantics></math>, let <math alttext="\tilde{r}(G,n,k,u)=\frac{\max_{\sigma\in\Delta((P_{st})^{n})}\min_{\tau}u(% \sigma,\tau)}{\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{\tau}u(\sigma,\tau)}" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.10"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.10a"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.10.11" 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xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mo id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5a" lspace="0.167em" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.cmml">⁡</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.cmml"><msub id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.1.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.1.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.1.2.cmml">min</mi><mi id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.1.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.1.3.cmml">τ</mi></msub><mo id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2a" lspace="0.167em" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.cmml">⁡</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.5.2.2.cmml">u</mi></mrow></mrow><mo id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.4" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.4.cmml">⁢</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.6.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.6.1.cmml"><mo id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.3.3.3.6.2.1" 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id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.1">subscript</csymbol><max id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.1.2"></max><apply id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1"><in id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.2"></in><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.3">𝜎</ci><apply id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1"><csymbol cd="ambiguous" id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1">superscript</csymbol><apply id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1"><times id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.2"></times><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.3">Δ</ci><apply id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3"><times id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.4.4.4.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2"><apply id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1"><csymbol cd="ambiguous" id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1">subscript</csymbol><min id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1.2"></min><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.1.3">𝜏</ci></apply><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.5.2.2">𝑢</ci></apply></apply><interval closure="open" id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.6.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.6.2"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.5.5.5.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.5.5.5.2">𝜎</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.6.6.6.3">𝜏</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.10c">\tilde{r}(G,n,k,u)=\frac{\max_{\sigma\in\Delta((P_{st})^{n})}\min_{\tau}u(% \sigma,\tau)}{\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{\tau}u(\sigma,\tau)}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.2.m2.10d">over~ start_ARG italic_r end_ARG ( italic_G , italic_n , italic_k , italic_u ) = divide start_ARG roman_max start_POSTSUBSCRIPT italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT italic_u ( italic_σ , italic_τ ) end_ARG start_ARG roman_max start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT italic_u ( italic_σ , italic_τ ) end_ARG</annotation></semantics></math>. From <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">von1997team</span>]</cite> it follows that <math alttext="\tilde{r}" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1a"><mover accent="true" id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.2.cmml">r</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1b"><apply id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.1">~</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1.1.2">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1c">\tilde{r}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.3.m3.1d">over~ start_ARG italic_r end_ARG</annotation></semantics></math> is indeed the ratio of the cost at TME to the cost at TMECor. <span class="ltx_ERROR undefined" id="Pt0.A2.SS0.SSS0.Px1.p1.10.2">\citet</span>basilico2017team showed that for a <math alttext="G" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1a"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1b"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.4.m4.1d">italic_G</annotation></semantics></math> that contains only <math alttext="s,t" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2a"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.3.1.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml">s</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.3.2.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.3.1.cmml">,</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.2.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2b"><list id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.3.2"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.1.1">𝑠</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2.2">𝑡</ci></list></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2c">s,t</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.5.m5.2d">italic_s , italic_t</annotation></semantics></math> and <math alttext="m" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1a"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1b"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1c">m</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.6.m6.1d">italic_m</annotation></semantics></math> disjoint paths of length 1 between them, the payoff function <math alttext="u(a,d)=1\left\{\bigcup a\cap\bigcup d=\emptyset\right\}" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3a"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.cmml"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.2.cmml">u</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.1.cmml">⁢</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.1.cmml"><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.2.1" stretchy="false" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.1.cmml">(</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.1.1.cmml">a</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.1.cmml">,</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.2.2.cmml">d</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.2.3" stretchy="false" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.2.cmml">=</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.cmml"><mn id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.3.cmml">1</mn><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.2.cmml">⁢</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.2.cmml"><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.2.cmml">{</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.cmml"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.cmml"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.cmml"><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.1" lspace="0em" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.1.cmml">⋃</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.2.cmml">a</mi></mrow><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.1" rspace="0.055em" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.1.cmml">∩</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.cmml"><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.1.cmml">⋃</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.2.cmml">d</mi></mrow></mrow><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.1.cmml">=</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.3" mathvariant="normal" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.3.cmml">∅</mi></mrow><mo id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3b"><apply id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3"><eq id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.2"></eq><apply id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3"><times id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.1"></times><ci id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.2">𝑢</ci><interval closure="open" id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.3.3.2"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.1.1">𝑎</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.2.2">𝑑</ci></interval></apply><apply id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1"><times id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.2"></times><cn id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.3.cmml" type="integer" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.3">1</cn><set id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1"><apply id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1"><eq id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.1"></eq><apply id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2"><intersect id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.1"></intersect><apply id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2"><union id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.1"></union><ci id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.2.2">𝑎</ci></apply><apply id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3"><union id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.1"></union><ci id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.2.3.2">𝑑</ci></apply></apply><emptyset id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3.3.1.1.1.1.3"></emptyset></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3c">u(a,d)=1\left\{\bigcup a\cap\bigcup d=\emptyset\right\}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.7.m7.3d">italic_u ( italic_a , italic_d ) = 1 { ⋃ italic_a ∩ ⋃ italic_d = ∅ }</annotation></semantics></math> and <math alttext="k=m-1" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1a"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.2.cmml">k</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.1.cmml">=</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.2.cmml">m</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.1.cmml">−</mo><mn id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1b"><apply id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1"><eq id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.1"></eq><ci id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.2">𝑘</ci><apply id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3"><minus id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.1"></minus><ci id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.2">𝑚</ci><cn id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.3.cmml" type="integer" xref="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1c">k=m-1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.8.m8.1d">italic_k = italic_m - 1</annotation></semantics></math>, the PoU is <math alttext="\tilde{r}(G,n,k,u)=m^{n-1}" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4a"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.cmml"><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.cmml"><mover accent="true" id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.2.cmml">r</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.1.cmml">~</mo></mover><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.1.cmml">⁢</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.1.cmml"><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.2.1" stretchy="false" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.1.cmml">(</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.1.1.cmml">G</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.1.cmml">,</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.2.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.2.2.cmml">n</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.2.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.1.cmml">,</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.3.3.cmml">k</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.2.4" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.1.cmml">,</mo><mi id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.4" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.4.cmml">u</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.2.5" stretchy="false" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.1.cmml">=</mo><msup id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.2.cmml">m</mi><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.2.cmml">n</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.1.cmml">−</mo><mn id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4b"><apply id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5"><eq id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.1"></eq><apply id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2"><times id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.1"></times><apply id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.1">~</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.2.2">𝑟</ci></apply><vector id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.2.3.2"><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.1.1">𝐺</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.2.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.2.2">𝑛</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.3.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.3.3">𝑘</ci><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.4.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.4">𝑢</ci></vector></apply><apply id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3"><csymbol cd="ambiguous" id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3">superscript</csymbol><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.2">𝑚</ci><apply id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3"><minus id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.1"></minus><ci id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.2">𝑛</ci><cn id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.3.cmml" type="integer" xref="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4.5.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4c">\tilde{r}(G,n,k,u)=m^{n-1}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.9.m9.4d">over~ start_ARG italic_r end_ARG ( italic_G , italic_n , italic_k , italic_u ) = italic_m start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. On the other hand, they showed that for every such zero-sum game, this PoU is at most <math alttext="m^{n-1}" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1"><semantics id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1a"><msup id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.2.cmml">m</mi><mrow id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.2.cmml">n</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.1.cmml">−</mo><mn id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1b"><apply id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1">superscript</csymbol><ci id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.2">𝑚</ci><apply id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3"><minus id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.1.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.1"></minus><ci id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.2">𝑛</ci><cn id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.3.cmml" type="integer" xref="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1c">m^{n-1}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p1.10.m10.1d">italic_m start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. By allowing arbitrary payoffs, this formulation captures every normal-form game that is considered in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">basilico2017team</span>]</cite>. The same results hold when we assume that the team agents minimize a cost function. This is because one can transform a payoff function to a cost function while preserving the PoU (up to an additional assumption):</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma20"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Lemma 20</span></h6> <div class="ltx_para" id="Thmlemma20.p1"> <p class="ltx_p" id="Thmlemma20.p1.2"><span class="ltx_text ltx_font_italic" id="Thmlemma20.p1.2.2">Let <math alttext="G,n,k" class="ltx_Math" display="inline" id="Thmlemma20.p1.1.1.m1.3"><semantics id="Thmlemma20.p1.1.1.m1.3a"><mrow id="Thmlemma20.p1.1.1.m1.3.4.2" xref="Thmlemma20.p1.1.1.m1.3.4.1.cmml"><mi id="Thmlemma20.p1.1.1.m1.1.1" xref="Thmlemma20.p1.1.1.m1.1.1.cmml">G</mi><mo id="Thmlemma20.p1.1.1.m1.3.4.2.1" xref="Thmlemma20.p1.1.1.m1.3.4.1.cmml">,</mo><mi id="Thmlemma20.p1.1.1.m1.2.2" xref="Thmlemma20.p1.1.1.m1.2.2.cmml">n</mi><mo id="Thmlemma20.p1.1.1.m1.3.4.2.2" xref="Thmlemma20.p1.1.1.m1.3.4.1.cmml">,</mo><mi id="Thmlemma20.p1.1.1.m1.3.3" xref="Thmlemma20.p1.1.1.m1.3.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma20.p1.1.1.m1.3b"><list id="Thmlemma20.p1.1.1.m1.3.4.1.cmml" xref="Thmlemma20.p1.1.1.m1.3.4.2"><ci id="Thmlemma20.p1.1.1.m1.1.1.cmml" xref="Thmlemma20.p1.1.1.m1.1.1">𝐺</ci><ci id="Thmlemma20.p1.1.1.m1.2.2.cmml" xref="Thmlemma20.p1.1.1.m1.2.2">𝑛</ci><ci id="Thmlemma20.p1.1.1.m1.3.3.cmml" xref="Thmlemma20.p1.1.1.m1.3.3">𝑘</ci></list></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma20.p1.1.1.m1.3c">G,n,k</annotation><annotation encoding="application/x-llamapun" id="Thmlemma20.p1.1.1.m1.3d">italic_G , italic_n , italic_k</annotation></semantics></math> and a payoff function <math alttext="u" class="ltx_Math" display="inline" id="Thmlemma20.p1.2.2.m2.1"><semantics id="Thmlemma20.p1.2.2.m2.1a"><mi id="Thmlemma20.p1.2.2.m2.1.1" xref="Thmlemma20.p1.2.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="Thmlemma20.p1.2.2.m2.1b"><ci id="Thmlemma20.p1.2.2.m2.1.1.cmml" xref="Thmlemma20.p1.2.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma20.p1.2.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="Thmlemma20.p1.2.2.m2.1d">italic_u</annotation></semantics></math> such that</span></p> <table class="ltx_equation ltx_eqn_table" id="Pt0.A2.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{d\in E^{k}}u(\sigma,d)>0" class="ltx_Math" display="block" id="Pt0.A2.Ex28.m1.3"><semantics id="Pt0.A2.Ex28.m1.3a"><mrow id="Pt0.A2.Ex28.m1.3.4" xref="Pt0.A2.Ex28.m1.3.4.cmml"><mrow id="Pt0.A2.Ex28.m1.3.4.2" xref="Pt0.A2.Ex28.m1.3.4.2.cmml"><mrow id="Pt0.A2.Ex28.m1.3.4.2.2" xref="Pt0.A2.Ex28.m1.3.4.2.2.cmml"><munder id="Pt0.A2.Ex28.m1.3.4.2.2.1" xref="Pt0.A2.Ex28.m1.3.4.2.2.1.cmml"><mi id="Pt0.A2.Ex28.m1.3.4.2.2.1.2" xref="Pt0.A2.Ex28.m1.3.4.2.2.1.2.cmml">max</mi><mrow id="Pt0.A2.Ex28.m1.1.1.1" xref="Pt0.A2.Ex28.m1.1.1.1.cmml"><mi id="Pt0.A2.Ex28.m1.1.1.1.3" xref="Pt0.A2.Ex28.m1.1.1.1.3.cmml">σ</mi><mo id="Pt0.A2.Ex28.m1.1.1.1.2" xref="Pt0.A2.Ex28.m1.1.1.1.2.cmml">∈</mo><msup id="Pt0.A2.Ex28.m1.1.1.1.1" xref="Pt0.A2.Ex28.m1.1.1.1.1.cmml"><mrow id="Pt0.A2.Ex28.m1.1.1.1.1.1.1" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.2" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A2.Ex28.m1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.Ex28.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A2.Ex28.m1.1.1.1.1.3" xref="Pt0.A2.Ex28.m1.1.1.1.1.3.cmml">n</mi></msup></mrow></munder><mo id="Pt0.A2.Ex28.m1.3.4.2.2a" lspace="0.167em" xref="Pt0.A2.Ex28.m1.3.4.2.2.cmml">⁡</mo><mrow id="Pt0.A2.Ex28.m1.3.4.2.2.2" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.cmml"><munder id="Pt0.A2.Ex28.m1.3.4.2.2.2.1" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.cmml"><mi id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.2" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.2.cmml">min</mi><mrow id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.cmml"><mi id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.2" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.2.cmml">d</mi><mo id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.1" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.1.cmml">∈</mo><msup id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.cmml"><mi id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.2" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.2.cmml">E</mi><mi id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.3" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.3.cmml">k</mi></msup></mrow></munder><mo id="Pt0.A2.Ex28.m1.3.4.2.2.2a" lspace="0.167em" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.cmml">⁡</mo><mi id="Pt0.A2.Ex28.m1.3.4.2.2.2.2" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.2.cmml">u</mi></mrow></mrow><mo id="Pt0.A2.Ex28.m1.3.4.2.1" xref="Pt0.A2.Ex28.m1.3.4.2.1.cmml">⁢</mo><mrow id="Pt0.A2.Ex28.m1.3.4.2.3.2" xref="Pt0.A2.Ex28.m1.3.4.2.3.1.cmml"><mo id="Pt0.A2.Ex28.m1.3.4.2.3.2.1" stretchy="false" xref="Pt0.A2.Ex28.m1.3.4.2.3.1.cmml">(</mo><mi id="Pt0.A2.Ex28.m1.2.2" xref="Pt0.A2.Ex28.m1.2.2.cmml">σ</mi><mo id="Pt0.A2.Ex28.m1.3.4.2.3.2.2" xref="Pt0.A2.Ex28.m1.3.4.2.3.1.cmml">,</mo><mi id="Pt0.A2.Ex28.m1.3.3" xref="Pt0.A2.Ex28.m1.3.3.cmml">d</mi><mo id="Pt0.A2.Ex28.m1.3.4.2.3.2.3" stretchy="false" xref="Pt0.A2.Ex28.m1.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A2.Ex28.m1.3.4.1" xref="Pt0.A2.Ex28.m1.3.4.1.cmml">></mo><mn id="Pt0.A2.Ex28.m1.3.4.3" xref="Pt0.A2.Ex28.m1.3.4.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A2.Ex28.m1.3b"><apply id="Pt0.A2.Ex28.m1.3.4.cmml" xref="Pt0.A2.Ex28.m1.3.4"><gt 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id="Pt0.A2.Ex28.m1.1.1.1.1.3.cmml" xref="Pt0.A2.Ex28.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Pt0.A2.Ex28.m1.3.4.2.2.2.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2"><apply id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1"><csymbol cd="ambiguous" id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.1.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1">subscript</csymbol><min id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.2.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.2"></min><apply id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3"><in id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.1.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.1"></in><ci id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.2.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.2">𝑑</ci><apply id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3"><csymbol cd="ambiguous" id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.1.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3">superscript</csymbol><ci id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.2.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.2">𝐸</ci><ci id="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.3.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.1.3.3.3">𝑘</ci></apply></apply></apply><ci id="Pt0.A2.Ex28.m1.3.4.2.2.2.2.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.2.2.2">𝑢</ci></apply></apply><interval closure="open" id="Pt0.A2.Ex28.m1.3.4.2.3.1.cmml" xref="Pt0.A2.Ex28.m1.3.4.2.3.2"><ci id="Pt0.A2.Ex28.m1.2.2.cmml" xref="Pt0.A2.Ex28.m1.2.2">𝜎</ci><ci id="Pt0.A2.Ex28.m1.3.3.cmml" xref="Pt0.A2.Ex28.m1.3.3">𝑑</ci></interval></apply><cn id="Pt0.A2.Ex28.m1.3.4.3.cmml" type="integer" xref="Pt0.A2.Ex28.m1.3.4.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.Ex28.m1.3c">\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{d\in E^{k}}u(\sigma,d)>0</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.Ex28.m1.3d">roman_max start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_u ( italic_σ , italic_d ) > 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmlemma20.p1.4"><span class="ltx_text ltx_font_italic" id="Thmlemma20.p1.4.2">Then there is a cost function <math alttext="C" class="ltx_Math" display="inline" id="Thmlemma20.p1.3.1.m1.1"><semantics id="Thmlemma20.p1.3.1.m1.1a"><mi id="Thmlemma20.p1.3.1.m1.1.1" xref="Thmlemma20.p1.3.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="Thmlemma20.p1.3.1.m1.1b"><ci id="Thmlemma20.p1.3.1.m1.1.1.cmml" xref="Thmlemma20.p1.3.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma20.p1.3.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="Thmlemma20.p1.3.1.m1.1d">italic_C</annotation></semantics></math> such that <math 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xref="Thmlemma20.p1.4.2.m2.8.9.1"></eq><apply id="Thmlemma20.p1.4.2.m2.8.9.2.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.2"><times id="Thmlemma20.p1.4.2.m2.8.9.2.1.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.2.1"></times><ci id="Thmlemma20.p1.4.2.m2.8.9.2.2.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.2.2">𝑟</ci><vector id="Thmlemma20.p1.4.2.m2.8.9.2.3.1.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.2.3.2"><ci id="Thmlemma20.p1.4.2.m2.1.1.cmml" xref="Thmlemma20.p1.4.2.m2.1.1">𝐺</ci><ci id="Thmlemma20.p1.4.2.m2.2.2.cmml" xref="Thmlemma20.p1.4.2.m2.2.2">𝑛</ci><ci id="Thmlemma20.p1.4.2.m2.3.3.cmml" xref="Thmlemma20.p1.4.2.m2.3.3">𝑘</ci><ci id="Thmlemma20.p1.4.2.m2.4.4.cmml" xref="Thmlemma20.p1.4.2.m2.4.4">𝐶</ci></vector></apply><apply id="Thmlemma20.p1.4.2.m2.8.9.3.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.3"><times id="Thmlemma20.p1.4.2.m2.8.9.3.1.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.3.1"></times><apply id="Thmlemma20.p1.4.2.m2.8.9.3.2.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.3.2"><ci id="Thmlemma20.p1.4.2.m2.8.9.3.2.1.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.3.2.1">~</ci><ci id="Thmlemma20.p1.4.2.m2.8.9.3.2.2.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.3.2.2">𝑟</ci></apply><vector id="Thmlemma20.p1.4.2.m2.8.9.3.3.1.cmml" xref="Thmlemma20.p1.4.2.m2.8.9.3.3.2"><ci id="Thmlemma20.p1.4.2.m2.5.5.cmml" xref="Thmlemma20.p1.4.2.m2.5.5">𝐺</ci><ci id="Thmlemma20.p1.4.2.m2.6.6.cmml" xref="Thmlemma20.p1.4.2.m2.6.6">𝑛</ci><ci id="Thmlemma20.p1.4.2.m2.7.7.cmml" xref="Thmlemma20.p1.4.2.m2.7.7">𝑘</ci><ci id="Thmlemma20.p1.4.2.m2.8.8.cmml" xref="Thmlemma20.p1.4.2.m2.8.8">𝑢</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma20.p1.4.2.m2.8c">r(G,n,k,C)=\tilde{r}(G,n,k,u)</annotation><annotation encoding="application/x-llamapun" id="Thmlemma20.p1.4.2.m2.8d">italic_r ( italic_G , italic_n , italic_k , italic_C ) = over~ start_ARG italic_r end_ARG ( italic_G , italic_n , italic_k , italic_u )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx7"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx7.p1"> <p class="ltx_p" id="Thmproofx7.p1.4">Define</p> <table class="ltx_equation ltx_eqn_table" id="Pt0.A2.Ex29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="T^{\prime}=\max_{\sigma\in\Delta((P_{st})^{n})}\min_{d\in E^{k}}u(\sigma,d),~{% }T^{\prime\prime}=\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{d\in E^{k}}u(% \sigma,d)" class="ltx_Math" display="block" id="Pt0.A2.Ex29.m1.8"><semantics id="Pt0.A2.Ex29.m1.8a"><mrow id="Pt0.A2.Ex29.m1.8.8.2" xref="Pt0.A2.Ex29.m1.8.8.3.cmml"><mrow id="Pt0.A2.Ex29.m1.7.7.1.1" xref="Pt0.A2.Ex29.m1.7.7.1.1.cmml"><msup id="Pt0.A2.Ex29.m1.7.7.1.1.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.2.cmml"><mi id="Pt0.A2.Ex29.m1.7.7.1.1.2.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.2.2.cmml">T</mi><mo id="Pt0.A2.Ex29.m1.7.7.1.1.2.3" xref="Pt0.A2.Ex29.m1.7.7.1.1.2.3.cmml">′</mo></msup><mo id="Pt0.A2.Ex29.m1.7.7.1.1.1" xref="Pt0.A2.Ex29.m1.7.7.1.1.1.cmml">=</mo><mrow id="Pt0.A2.Ex29.m1.7.7.1.1.3" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.cmml"><mrow id="Pt0.A2.Ex29.m1.7.7.1.1.3.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.cmml"><munder id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.1" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.1.cmml"><mi id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.1.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.1.2.cmml">max</mi><mrow id="Pt0.A2.Ex29.m1.1.1.1" xref="Pt0.A2.Ex29.m1.1.1.1.cmml"><mi id="Pt0.A2.Ex29.m1.1.1.1.3" xref="Pt0.A2.Ex29.m1.1.1.1.3.cmml">σ</mi><mo id="Pt0.A2.Ex29.m1.1.1.1.2" xref="Pt0.A2.Ex29.m1.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A2.Ex29.m1.1.1.1.1" xref="Pt0.A2.Ex29.m1.1.1.1.1.cmml"><mi id="Pt0.A2.Ex29.m1.1.1.1.1.3" mathvariant="normal" xref="Pt0.A2.Ex29.m1.1.1.1.1.3.cmml">Δ</mi><mo id="Pt0.A2.Ex29.m1.1.1.1.1.2" xref="Pt0.A2.Ex29.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A2.Ex29.m1.1.1.1.1.1.1" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.cmml">(</mo><msup id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.cmml"><mrow id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.3" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="Pt0.A2.Ex29.m1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.Ex29.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mo id="Pt0.A2.Ex29.m1.7.7.1.1.3.2a" lspace="0.167em" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.cmml">⁡</mo><mrow id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.cmml"><munder id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.cmml"><mi id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.2.cmml">min</mi><mrow id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.cmml"><mi id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.2.cmml">d</mi><mo id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.1" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.1.cmml">∈</mo><msup id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.3" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.3.cmml"><mi id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.3.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.3.2.cmml">E</mi><mi id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.3.3" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.1.3.3.3.cmml">k</mi></msup></mrow></munder><mo id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2a" lspace="0.167em" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.cmml">⁡</mo><mi id="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.2.2.2.cmml">u</mi></mrow></mrow><mo id="Pt0.A2.Ex29.m1.7.7.1.1.3.1" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.1.cmml">⁢</mo><mrow id="Pt0.A2.Ex29.m1.7.7.1.1.3.3.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.3.1.cmml"><mo id="Pt0.A2.Ex29.m1.7.7.1.1.3.3.2.1" stretchy="false" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.3.1.cmml">(</mo><mi id="Pt0.A2.Ex29.m1.3.3" xref="Pt0.A2.Ex29.m1.3.3.cmml">σ</mi><mo id="Pt0.A2.Ex29.m1.7.7.1.1.3.3.2.2" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.3.1.cmml">,</mo><mi id="Pt0.A2.Ex29.m1.4.4" xref="Pt0.A2.Ex29.m1.4.4.cmml">d</mi><mo id="Pt0.A2.Ex29.m1.7.7.1.1.3.3.2.3" stretchy="false" xref="Pt0.A2.Ex29.m1.7.7.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A2.Ex29.m1.8.8.2.3" rspace="0.497em" xref="Pt0.A2.Ex29.m1.8.8.3a.cmml">,</mo><mrow id="Pt0.A2.Ex29.m1.8.8.2.2" xref="Pt0.A2.Ex29.m1.8.8.2.2.cmml"><msup id="Pt0.A2.Ex29.m1.8.8.2.2.2" xref="Pt0.A2.Ex29.m1.8.8.2.2.2.cmml"><mi id="Pt0.A2.Ex29.m1.8.8.2.2.2.2" xref="Pt0.A2.Ex29.m1.8.8.2.2.2.2.cmml">T</mi><mo id="Pt0.A2.Ex29.m1.8.8.2.2.2.3" xref="Pt0.A2.Ex29.m1.8.8.2.2.2.3.cmml">′′</mo></msup><mo id="Pt0.A2.Ex29.m1.8.8.2.2.1" xref="Pt0.A2.Ex29.m1.8.8.2.2.1.cmml">=</mo><mrow id="Pt0.A2.Ex29.m1.8.8.2.2.3" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.cmml"><mrow id="Pt0.A2.Ex29.m1.8.8.2.2.3.2" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.cmml"><munder id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.1" 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id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1"><csymbol cd="ambiguous" id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.1.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1">subscript</csymbol><min id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.2.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.2"></min><apply id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3"><in id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.1.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.1"></in><ci id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.2.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.2">𝑑</ci><apply id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3"><csymbol cd="ambiguous" id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3.1.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3">superscript</csymbol><ci id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3.2.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3.2">𝐸</ci><ci id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3.3.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.1.3.3.3">𝑘</ci></apply></apply></apply><ci id="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.2.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.2.2.2">𝑢</ci></apply></apply><interval closure="open" id="Pt0.A2.Ex29.m1.8.8.2.2.3.3.1.cmml" xref="Pt0.A2.Ex29.m1.8.8.2.2.3.3.2"><ci id="Pt0.A2.Ex29.m1.5.5.cmml" xref="Pt0.A2.Ex29.m1.5.5">𝜎</ci><ci id="Pt0.A2.Ex29.m1.6.6.cmml" xref="Pt0.A2.Ex29.m1.6.6">𝑑</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.Ex29.m1.8c">T^{\prime}=\max_{\sigma\in\Delta((P_{st})^{n})}\min_{d\in E^{k}}u(\sigma,d),~{% }T^{\prime\prime}=\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{d\in E^{k}}u(% \sigma,d)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.Ex29.m1.8d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_max start_POSTSUBSCRIPT italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_u ( italic_σ , italic_d ) , italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = roman_max start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_u ( italic_σ , italic_d )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx7.p1.3">Let <math alttext="T=T^{\prime}+T^{\prime\prime}" class="ltx_Math" display="inline" id="Thmproofx7.p1.1.m1.1"><semantics id="Thmproofx7.p1.1.m1.1a"><mrow id="Thmproofx7.p1.1.m1.1.1" xref="Thmproofx7.p1.1.m1.1.1.cmml"><mi id="Thmproofx7.p1.1.m1.1.1.2" xref="Thmproofx7.p1.1.m1.1.1.2.cmml">T</mi><mo id="Thmproofx7.p1.1.m1.1.1.1" xref="Thmproofx7.p1.1.m1.1.1.1.cmml">=</mo><mrow id="Thmproofx7.p1.1.m1.1.1.3" xref="Thmproofx7.p1.1.m1.1.1.3.cmml"><msup id="Thmproofx7.p1.1.m1.1.1.3.2" xref="Thmproofx7.p1.1.m1.1.1.3.2.cmml"><mi id="Thmproofx7.p1.1.m1.1.1.3.2.2" xref="Thmproofx7.p1.1.m1.1.1.3.2.2.cmml">T</mi><mo id="Thmproofx7.p1.1.m1.1.1.3.2.3" xref="Thmproofx7.p1.1.m1.1.1.3.2.3.cmml">′</mo></msup><mo id="Thmproofx7.p1.1.m1.1.1.3.1" xref="Thmproofx7.p1.1.m1.1.1.3.1.cmml">+</mo><msup id="Thmproofx7.p1.1.m1.1.1.3.3" xref="Thmproofx7.p1.1.m1.1.1.3.3.cmml"><mi id="Thmproofx7.p1.1.m1.1.1.3.3.2" xref="Thmproofx7.p1.1.m1.1.1.3.3.2.cmml">T</mi><mo id="Thmproofx7.p1.1.m1.1.1.3.3.3" xref="Thmproofx7.p1.1.m1.1.1.3.3.3.cmml">′′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx7.p1.1.m1.1b"><apply id="Thmproofx7.p1.1.m1.1.1.cmml" xref="Thmproofx7.p1.1.m1.1.1"><eq id="Thmproofx7.p1.1.m1.1.1.1.cmml" xref="Thmproofx7.p1.1.m1.1.1.1"></eq><ci id="Thmproofx7.p1.1.m1.1.1.2.cmml" xref="Thmproofx7.p1.1.m1.1.1.2">𝑇</ci><apply id="Thmproofx7.p1.1.m1.1.1.3.cmml" xref="Thmproofx7.p1.1.m1.1.1.3"><plus id="Thmproofx7.p1.1.m1.1.1.3.1.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.1"></plus><apply id="Thmproofx7.p1.1.m1.1.1.3.2.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="Thmproofx7.p1.1.m1.1.1.3.2.1.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="Thmproofx7.p1.1.m1.1.1.3.2.2.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.2.2">𝑇</ci><ci id="Thmproofx7.p1.1.m1.1.1.3.2.3.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.2.3">′</ci></apply><apply id="Thmproofx7.p1.1.m1.1.1.3.3.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx7.p1.1.m1.1.1.3.3.1.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="Thmproofx7.p1.1.m1.1.1.3.3.2.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.3.2">𝑇</ci><ci id="Thmproofx7.p1.1.m1.1.1.3.3.3.cmml" xref="Thmproofx7.p1.1.m1.1.1.3.3.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx7.p1.1.m1.1c">T=T^{\prime}+T^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx7.p1.1.m1.1d">italic_T = italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and define the cost function <math alttext="C(a,d)=T-u(a,d)" class="ltx_Math" display="inline" id="Thmproofx7.p1.2.m2.4"><semantics id="Thmproofx7.p1.2.m2.4a"><mrow id="Thmproofx7.p1.2.m2.4.5" xref="Thmproofx7.p1.2.m2.4.5.cmml"><mrow id="Thmproofx7.p1.2.m2.4.5.2" xref="Thmproofx7.p1.2.m2.4.5.2.cmml"><mi id="Thmproofx7.p1.2.m2.4.5.2.2" xref="Thmproofx7.p1.2.m2.4.5.2.2.cmml">C</mi><mo id="Thmproofx7.p1.2.m2.4.5.2.1" xref="Thmproofx7.p1.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx7.p1.2.m2.4.5.2.3.2" xref="Thmproofx7.p1.2.m2.4.5.2.3.1.cmml"><mo id="Thmproofx7.p1.2.m2.4.5.2.3.2.1" stretchy="false" xref="Thmproofx7.p1.2.m2.4.5.2.3.1.cmml">(</mo><mi id="Thmproofx7.p1.2.m2.1.1" xref="Thmproofx7.p1.2.m2.1.1.cmml">a</mi><mo id="Thmproofx7.p1.2.m2.4.5.2.3.2.2" xref="Thmproofx7.p1.2.m2.4.5.2.3.1.cmml">,</mo><mi id="Thmproofx7.p1.2.m2.2.2" xref="Thmproofx7.p1.2.m2.2.2.cmml">d</mi><mo id="Thmproofx7.p1.2.m2.4.5.2.3.2.3" stretchy="false" xref="Thmproofx7.p1.2.m2.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx7.p1.2.m2.4.5.1" xref="Thmproofx7.p1.2.m2.4.5.1.cmml">=</mo><mrow id="Thmproofx7.p1.2.m2.4.5.3" xref="Thmproofx7.p1.2.m2.4.5.3.cmml"><mi id="Thmproofx7.p1.2.m2.4.5.3.2" xref="Thmproofx7.p1.2.m2.4.5.3.2.cmml">T</mi><mo id="Thmproofx7.p1.2.m2.4.5.3.1" xref="Thmproofx7.p1.2.m2.4.5.3.1.cmml">−</mo><mrow id="Thmproofx7.p1.2.m2.4.5.3.3" xref="Thmproofx7.p1.2.m2.4.5.3.3.cmml"><mi id="Thmproofx7.p1.2.m2.4.5.3.3.2" xref="Thmproofx7.p1.2.m2.4.5.3.3.2.cmml">u</mi><mo id="Thmproofx7.p1.2.m2.4.5.3.3.1" xref="Thmproofx7.p1.2.m2.4.5.3.3.1.cmml">⁢</mo><mrow id="Thmproofx7.p1.2.m2.4.5.3.3.3.2" xref="Thmproofx7.p1.2.m2.4.5.3.3.3.1.cmml"><mo id="Thmproofx7.p1.2.m2.4.5.3.3.3.2.1" stretchy="false" xref="Thmproofx7.p1.2.m2.4.5.3.3.3.1.cmml">(</mo><mi id="Thmproofx7.p1.2.m2.3.3" xref="Thmproofx7.p1.2.m2.3.3.cmml">a</mi><mo id="Thmproofx7.p1.2.m2.4.5.3.3.3.2.2" xref="Thmproofx7.p1.2.m2.4.5.3.3.3.1.cmml">,</mo><mi id="Thmproofx7.p1.2.m2.4.4" xref="Thmproofx7.p1.2.m2.4.4.cmml">d</mi><mo id="Thmproofx7.p1.2.m2.4.5.3.3.3.2.3" stretchy="false" xref="Thmproofx7.p1.2.m2.4.5.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx7.p1.2.m2.4b"><apply id="Thmproofx7.p1.2.m2.4.5.cmml" xref="Thmproofx7.p1.2.m2.4.5"><eq id="Thmproofx7.p1.2.m2.4.5.1.cmml" xref="Thmproofx7.p1.2.m2.4.5.1"></eq><apply id="Thmproofx7.p1.2.m2.4.5.2.cmml" xref="Thmproofx7.p1.2.m2.4.5.2"><times id="Thmproofx7.p1.2.m2.4.5.2.1.cmml" xref="Thmproofx7.p1.2.m2.4.5.2.1"></times><ci id="Thmproofx7.p1.2.m2.4.5.2.2.cmml" xref="Thmproofx7.p1.2.m2.4.5.2.2">𝐶</ci><interval closure="open" id="Thmproofx7.p1.2.m2.4.5.2.3.1.cmml" xref="Thmproofx7.p1.2.m2.4.5.2.3.2"><ci id="Thmproofx7.p1.2.m2.1.1.cmml" xref="Thmproofx7.p1.2.m2.1.1">𝑎</ci><ci id="Thmproofx7.p1.2.m2.2.2.cmml" xref="Thmproofx7.p1.2.m2.2.2">𝑑</ci></interval></apply><apply id="Thmproofx7.p1.2.m2.4.5.3.cmml" xref="Thmproofx7.p1.2.m2.4.5.3"><minus id="Thmproofx7.p1.2.m2.4.5.3.1.cmml" xref="Thmproofx7.p1.2.m2.4.5.3.1"></minus><ci id="Thmproofx7.p1.2.m2.4.5.3.2.cmml" xref="Thmproofx7.p1.2.m2.4.5.3.2">𝑇</ci><apply id="Thmproofx7.p1.2.m2.4.5.3.3.cmml" xref="Thmproofx7.p1.2.m2.4.5.3.3"><times id="Thmproofx7.p1.2.m2.4.5.3.3.1.cmml" xref="Thmproofx7.p1.2.m2.4.5.3.3.1"></times><ci id="Thmproofx7.p1.2.m2.4.5.3.3.2.cmml" xref="Thmproofx7.p1.2.m2.4.5.3.3.2">𝑢</ci><interval closure="open" id="Thmproofx7.p1.2.m2.4.5.3.3.3.1.cmml" xref="Thmproofx7.p1.2.m2.4.5.3.3.3.2"><ci id="Thmproofx7.p1.2.m2.3.3.cmml" xref="Thmproofx7.p1.2.m2.3.3">𝑎</ci><ci id="Thmproofx7.p1.2.m2.4.4.cmml" xref="Thmproofx7.p1.2.m2.4.4">𝑑</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx7.p1.2.m2.4c">C(a,d)=T-u(a,d)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx7.p1.2.m2.4d">italic_C ( italic_a , italic_d ) = italic_T - italic_u ( italic_a , italic_d )</annotation></semantics></math>. By linearity, <math alttext="C(\sigma,\tau)=\operatorname*{\mathbb{E}}_{(a,d)\sim(\sigma,\tau)}(T-u(a,d))=T% -u(\sigma,\tau)" class="ltx_Math" display="inline" id="Thmproofx7.p1.3.m3.12"><semantics id="Thmproofx7.p1.3.m3.12a"><mrow id="Thmproofx7.p1.3.m3.12.12" xref="Thmproofx7.p1.3.m3.12.12.cmml"><mrow id="Thmproofx7.p1.3.m3.12.12.4" xref="Thmproofx7.p1.3.m3.12.12.4.cmml"><mi id="Thmproofx7.p1.3.m3.12.12.4.2" xref="Thmproofx7.p1.3.m3.12.12.4.2.cmml">C</mi><mo id="Thmproofx7.p1.3.m3.12.12.4.1" xref="Thmproofx7.p1.3.m3.12.12.4.1.cmml">⁢</mo><mrow id="Thmproofx7.p1.3.m3.12.12.4.3.2" xref="Thmproofx7.p1.3.m3.12.12.4.3.1.cmml"><mo id="Thmproofx7.p1.3.m3.12.12.4.3.2.1" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.4.3.1.cmml">(</mo><mi id="Thmproofx7.p1.3.m3.5.5" xref="Thmproofx7.p1.3.m3.5.5.cmml">σ</mi><mo id="Thmproofx7.p1.3.m3.12.12.4.3.2.2" xref="Thmproofx7.p1.3.m3.12.12.4.3.1.cmml">,</mo><mi id="Thmproofx7.p1.3.m3.6.6" xref="Thmproofx7.p1.3.m3.6.6.cmml">τ</mi><mo id="Thmproofx7.p1.3.m3.12.12.4.3.2.3" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.4.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx7.p1.3.m3.12.12.5" rspace="0.1389em" xref="Thmproofx7.p1.3.m3.12.12.5.cmml">=</mo><mrow id="Thmproofx7.p1.3.m3.12.12.2.2" xref="Thmproofx7.p1.3.m3.12.12.2.3.cmml"><msub id="Thmproofx7.p1.3.m3.11.11.1.1.1" xref="Thmproofx7.p1.3.m3.11.11.1.1.1.cmml"><mo id="Thmproofx7.p1.3.m3.11.11.1.1.1.2" lspace="0.1389em" rspace="0em" xref="Thmproofx7.p1.3.m3.11.11.1.1.1.2.cmml">𝔼</mo><mrow id="Thmproofx7.p1.3.m3.4.4.4" xref="Thmproofx7.p1.3.m3.4.4.4.cmml"><mrow id="Thmproofx7.p1.3.m3.4.4.4.6.2" xref="Thmproofx7.p1.3.m3.4.4.4.6.1.cmml"><mo id="Thmproofx7.p1.3.m3.4.4.4.6.2.1" stretchy="false" xref="Thmproofx7.p1.3.m3.4.4.4.6.1.cmml">(</mo><mi id="Thmproofx7.p1.3.m3.1.1.1.1" xref="Thmproofx7.p1.3.m3.1.1.1.1.cmml">a</mi><mo id="Thmproofx7.p1.3.m3.4.4.4.6.2.2" xref="Thmproofx7.p1.3.m3.4.4.4.6.1.cmml">,</mo><mi id="Thmproofx7.p1.3.m3.2.2.2.2" xref="Thmproofx7.p1.3.m3.2.2.2.2.cmml">d</mi><mo id="Thmproofx7.p1.3.m3.4.4.4.6.2.3" stretchy="false" xref="Thmproofx7.p1.3.m3.4.4.4.6.1.cmml">)</mo></mrow><mo id="Thmproofx7.p1.3.m3.4.4.4.5" xref="Thmproofx7.p1.3.m3.4.4.4.5.cmml">∼</mo><mrow id="Thmproofx7.p1.3.m3.4.4.4.7.2" xref="Thmproofx7.p1.3.m3.4.4.4.7.1.cmml"><mo id="Thmproofx7.p1.3.m3.4.4.4.7.2.1" stretchy="false" xref="Thmproofx7.p1.3.m3.4.4.4.7.1.cmml">(</mo><mi id="Thmproofx7.p1.3.m3.3.3.3.3" xref="Thmproofx7.p1.3.m3.3.3.3.3.cmml">σ</mi><mo id="Thmproofx7.p1.3.m3.4.4.4.7.2.2" xref="Thmproofx7.p1.3.m3.4.4.4.7.1.cmml">,</mo><mi id="Thmproofx7.p1.3.m3.4.4.4.4" xref="Thmproofx7.p1.3.m3.4.4.4.4.cmml">τ</mi><mo id="Thmproofx7.p1.3.m3.4.4.4.7.2.3" stretchy="false" xref="Thmproofx7.p1.3.m3.4.4.4.7.1.cmml">)</mo></mrow></mrow></msub><mrow id="Thmproofx7.p1.3.m3.12.12.2.2.2" xref="Thmproofx7.p1.3.m3.12.12.2.3.cmml"><mo id="Thmproofx7.p1.3.m3.12.12.2.2.2.2" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.2.3.cmml">(</mo><mrow id="Thmproofx7.p1.3.m3.12.12.2.2.2.1" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.cmml"><mi id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.2" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.2.cmml">T</mi><mo id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.1" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.1.cmml">−</mo><mrow id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.cmml"><mi id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.2" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.2.cmml">u</mi><mo id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.1" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.1.cmml">⁢</mo><mrow id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.2" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.1.cmml"><mo id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.2.1" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.1.cmml">(</mo><mi id="Thmproofx7.p1.3.m3.7.7" xref="Thmproofx7.p1.3.m3.7.7.cmml">a</mi><mo id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.2.2" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.1.cmml">,</mo><mi id="Thmproofx7.p1.3.m3.8.8" xref="Thmproofx7.p1.3.m3.8.8.cmml">d</mi><mo id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.2.3" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx7.p1.3.m3.12.12.2.2.2.3" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.2.3.cmml">)</mo></mrow></mrow><mo id="Thmproofx7.p1.3.m3.12.12.6" xref="Thmproofx7.p1.3.m3.12.12.6.cmml">=</mo><mrow id="Thmproofx7.p1.3.m3.12.12.7" xref="Thmproofx7.p1.3.m3.12.12.7.cmml"><mi id="Thmproofx7.p1.3.m3.12.12.7.2" xref="Thmproofx7.p1.3.m3.12.12.7.2.cmml">T</mi><mo id="Thmproofx7.p1.3.m3.12.12.7.1" xref="Thmproofx7.p1.3.m3.12.12.7.1.cmml">−</mo><mrow id="Thmproofx7.p1.3.m3.12.12.7.3" xref="Thmproofx7.p1.3.m3.12.12.7.3.cmml"><mi id="Thmproofx7.p1.3.m3.12.12.7.3.2" xref="Thmproofx7.p1.3.m3.12.12.7.3.2.cmml">u</mi><mo id="Thmproofx7.p1.3.m3.12.12.7.3.1" xref="Thmproofx7.p1.3.m3.12.12.7.3.1.cmml">⁢</mo><mrow id="Thmproofx7.p1.3.m3.12.12.7.3.3.2" xref="Thmproofx7.p1.3.m3.12.12.7.3.3.1.cmml"><mo id="Thmproofx7.p1.3.m3.12.12.7.3.3.2.1" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.7.3.3.1.cmml">(</mo><mi id="Thmproofx7.p1.3.m3.9.9" xref="Thmproofx7.p1.3.m3.9.9.cmml">σ</mi><mo id="Thmproofx7.p1.3.m3.12.12.7.3.3.2.2" xref="Thmproofx7.p1.3.m3.12.12.7.3.3.1.cmml">,</mo><mi id="Thmproofx7.p1.3.m3.10.10" xref="Thmproofx7.p1.3.m3.10.10.cmml">τ</mi><mo id="Thmproofx7.p1.3.m3.12.12.7.3.3.2.3" stretchy="false" xref="Thmproofx7.p1.3.m3.12.12.7.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx7.p1.3.m3.12b"><apply id="Thmproofx7.p1.3.m3.12.12.cmml" xref="Thmproofx7.p1.3.m3.12.12"><and id="Thmproofx7.p1.3.m3.12.12a.cmml" xref="Thmproofx7.p1.3.m3.12.12"></and><apply id="Thmproofx7.p1.3.m3.12.12b.cmml" xref="Thmproofx7.p1.3.m3.12.12"><eq id="Thmproofx7.p1.3.m3.12.12.5.cmml" xref="Thmproofx7.p1.3.m3.12.12.5"></eq><apply id="Thmproofx7.p1.3.m3.12.12.4.cmml" xref="Thmproofx7.p1.3.m3.12.12.4"><times id="Thmproofx7.p1.3.m3.12.12.4.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.4.1"></times><ci id="Thmproofx7.p1.3.m3.12.12.4.2.cmml" xref="Thmproofx7.p1.3.m3.12.12.4.2">𝐶</ci><interval closure="open" id="Thmproofx7.p1.3.m3.12.12.4.3.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.4.3.2"><ci id="Thmproofx7.p1.3.m3.5.5.cmml" xref="Thmproofx7.p1.3.m3.5.5">𝜎</ci><ci id="Thmproofx7.p1.3.m3.6.6.cmml" xref="Thmproofx7.p1.3.m3.6.6">𝜏</ci></interval></apply><apply id="Thmproofx7.p1.3.m3.12.12.2.3.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2"><apply id="Thmproofx7.p1.3.m3.11.11.1.1.1.cmml" xref="Thmproofx7.p1.3.m3.11.11.1.1.1"><csymbol cd="ambiguous" id="Thmproofx7.p1.3.m3.11.11.1.1.1.1.cmml" xref="Thmproofx7.p1.3.m3.11.11.1.1.1">subscript</csymbol><ci id="Thmproofx7.p1.3.m3.11.11.1.1.1.2.cmml" xref="Thmproofx7.p1.3.m3.11.11.1.1.1.2">𝔼</ci><apply id="Thmproofx7.p1.3.m3.4.4.4.cmml" xref="Thmproofx7.p1.3.m3.4.4.4"><csymbol cd="latexml" id="Thmproofx7.p1.3.m3.4.4.4.5.cmml" xref="Thmproofx7.p1.3.m3.4.4.4.5">similar-to</csymbol><interval closure="open" id="Thmproofx7.p1.3.m3.4.4.4.6.1.cmml" xref="Thmproofx7.p1.3.m3.4.4.4.6.2"><ci id="Thmproofx7.p1.3.m3.1.1.1.1.cmml" xref="Thmproofx7.p1.3.m3.1.1.1.1">𝑎</ci><ci id="Thmproofx7.p1.3.m3.2.2.2.2.cmml" xref="Thmproofx7.p1.3.m3.2.2.2.2">𝑑</ci></interval><interval closure="open" id="Thmproofx7.p1.3.m3.4.4.4.7.1.cmml" xref="Thmproofx7.p1.3.m3.4.4.4.7.2"><ci id="Thmproofx7.p1.3.m3.3.3.3.3.cmml" xref="Thmproofx7.p1.3.m3.3.3.3.3">𝜎</ci><ci id="Thmproofx7.p1.3.m3.4.4.4.4.cmml" xref="Thmproofx7.p1.3.m3.4.4.4.4">𝜏</ci></interval></apply></apply><apply id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1"><minus id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.1"></minus><ci id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.2.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.2">𝑇</ci><apply id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3"><times id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.1"></times><ci id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.2.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.2">𝑢</ci><interval closure="open" id="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.2.2.2.1.3.3.2"><ci id="Thmproofx7.p1.3.m3.7.7.cmml" xref="Thmproofx7.p1.3.m3.7.7">𝑎</ci><ci id="Thmproofx7.p1.3.m3.8.8.cmml" xref="Thmproofx7.p1.3.m3.8.8">𝑑</ci></interval></apply></apply></apply></apply><apply id="Thmproofx7.p1.3.m3.12.12c.cmml" xref="Thmproofx7.p1.3.m3.12.12"><eq id="Thmproofx7.p1.3.m3.12.12.6.cmml" xref="Thmproofx7.p1.3.m3.12.12.6"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx7.p1.3.m3.12.12.2.cmml" id="Thmproofx7.p1.3.m3.12.12d.cmml" xref="Thmproofx7.p1.3.m3.12.12"></share><apply id="Thmproofx7.p1.3.m3.12.12.7.cmml" xref="Thmproofx7.p1.3.m3.12.12.7"><minus id="Thmproofx7.p1.3.m3.12.12.7.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.7.1"></minus><ci id="Thmproofx7.p1.3.m3.12.12.7.2.cmml" xref="Thmproofx7.p1.3.m3.12.12.7.2">𝑇</ci><apply id="Thmproofx7.p1.3.m3.12.12.7.3.cmml" xref="Thmproofx7.p1.3.m3.12.12.7.3"><times id="Thmproofx7.p1.3.m3.12.12.7.3.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.7.3.1"></times><ci id="Thmproofx7.p1.3.m3.12.12.7.3.2.cmml" xref="Thmproofx7.p1.3.m3.12.12.7.3.2">𝑢</ci><interval closure="open" id="Thmproofx7.p1.3.m3.12.12.7.3.3.1.cmml" xref="Thmproofx7.p1.3.m3.12.12.7.3.3.2"><ci id="Thmproofx7.p1.3.m3.9.9.cmml" xref="Thmproofx7.p1.3.m3.9.9">𝜎</ci><ci id="Thmproofx7.p1.3.m3.10.10.cmml" xref="Thmproofx7.p1.3.m3.10.10">𝜏</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx7.p1.3.m3.12c">C(\sigma,\tau)=\operatorname*{\mathbb{E}}_{(a,d)\sim(\sigma,\tau)}(T-u(a,d))=T% -u(\sigma,\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx7.p1.3.m3.12d">italic_C ( italic_σ , italic_τ ) = blackboard_E start_POSTSUBSCRIPT ( italic_a , italic_d ) ∼ ( italic_σ , italic_τ ) end_POSTSUBSCRIPT ( italic_T - italic_u ( italic_a , italic_d ) ) = italic_T - italic_u ( italic_σ , italic_τ )</annotation></semantics></math>, so</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx24"> <tbody id="Pt0.A2.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(G,n,k,C)" class="ltx_Math" display="inline" id="Pt0.A2.Ex30.m1.4"><semantics id="Pt0.A2.Ex30.m1.4a"><mrow id="Pt0.A2.Ex30.m1.4.5" xref="Pt0.A2.Ex30.m1.4.5.cmml"><mi id="Pt0.A2.Ex30.m1.4.5.2" xref="Pt0.A2.Ex30.m1.4.5.2.cmml">r</mi><mo id="Pt0.A2.Ex30.m1.4.5.1" xref="Pt0.A2.Ex30.m1.4.5.1.cmml">⁢</mo><mrow id="Pt0.A2.Ex30.m1.4.5.3.2" xref="Pt0.A2.Ex30.m1.4.5.3.1.cmml"><mo id="Pt0.A2.Ex30.m1.4.5.3.2.1" stretchy="false" xref="Pt0.A2.Ex30.m1.4.5.3.1.cmml">(</mo><mi id="Pt0.A2.Ex30.m1.1.1" xref="Pt0.A2.Ex30.m1.1.1.cmml">G</mi><mo id="Pt0.A2.Ex30.m1.4.5.3.2.2" xref="Pt0.A2.Ex30.m1.4.5.3.1.cmml">,</mo><mi id="Pt0.A2.Ex30.m1.2.2" xref="Pt0.A2.Ex30.m1.2.2.cmml">n</mi><mo id="Pt0.A2.Ex30.m1.4.5.3.2.3" xref="Pt0.A2.Ex30.m1.4.5.3.1.cmml">,</mo><mi id="Pt0.A2.Ex30.m1.3.3" xref="Pt0.A2.Ex30.m1.3.3.cmml">k</mi><mo id="Pt0.A2.Ex30.m1.4.5.3.2.4" xref="Pt0.A2.Ex30.m1.4.5.3.1.cmml">,</mo><mi id="Pt0.A2.Ex30.m1.4.4" xref="Pt0.A2.Ex30.m1.4.4.cmml">C</mi><mo id="Pt0.A2.Ex30.m1.4.5.3.2.5" stretchy="false" xref="Pt0.A2.Ex30.m1.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A2.Ex30.m1.4b"><apply id="Pt0.A2.Ex30.m1.4.5.cmml" xref="Pt0.A2.Ex30.m1.4.5"><times id="Pt0.A2.Ex30.m1.4.5.1.cmml" xref="Pt0.A2.Ex30.m1.4.5.1"></times><ci id="Pt0.A2.Ex30.m1.4.5.2.cmml" xref="Pt0.A2.Ex30.m1.4.5.2">𝑟</ci><vector id="Pt0.A2.Ex30.m1.4.5.3.1.cmml" xref="Pt0.A2.Ex30.m1.4.5.3.2"><ci id="Pt0.A2.Ex30.m1.1.1.cmml" xref="Pt0.A2.Ex30.m1.1.1">𝐺</ci><ci id="Pt0.A2.Ex30.m1.2.2.cmml" xref="Pt0.A2.Ex30.m1.2.2">𝑛</ci><ci id="Pt0.A2.Ex30.m1.3.3.cmml" xref="Pt0.A2.Ex30.m1.3.3">𝑘</ci><ci id="Pt0.A2.Ex30.m1.4.4.cmml" xref="Pt0.A2.Ex30.m1.4.4">𝐶</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.Ex30.m1.4c">\displaystyle r(G,n,k,C)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.Ex30.m1.4d">italic_r ( italic_G , italic_n , italic_k , italic_C )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}\max_{d\in E^{k}}(T-u(% \sigma,d))}{\min_{\sigma\in\Delta((P_{st})^{n})}\max_{d\in E^{k}}(T-u(\sigma,d% ))}" class="ltx_Math" display="inline" id="Pt0.A2.Ex30.m2.10"><semantics id="Pt0.A2.Ex30.m2.10a"><mrow id="Pt0.A2.Ex30.m2.10.11" xref="Pt0.A2.Ex30.m2.10.11.cmml"><mi id="Pt0.A2.Ex30.m2.10.11.2" xref="Pt0.A2.Ex30.m2.10.11.2.cmml"></mi><mo id="Pt0.A2.Ex30.m2.10.11.1" xref="Pt0.A2.Ex30.m2.10.11.1.cmml">=</mo><mstyle displaystyle="true" id="Pt0.A2.Ex30.m2.10.10" xref="Pt0.A2.Ex30.m2.10.10.cmml"><mfrac id="Pt0.A2.Ex30.m2.10.10a" xref="Pt0.A2.Ex30.m2.10.10.cmml"><mrow id="Pt0.A2.Ex30.m2.5.5.5" xref="Pt0.A2.Ex30.m2.5.5.5.cmml"><msub id="Pt0.A2.Ex30.m2.5.5.5.6" xref="Pt0.A2.Ex30.m2.5.5.5.6.cmml"><mi id="Pt0.A2.Ex30.m2.5.5.5.6.2" xref="Pt0.A2.Ex30.m2.5.5.5.6.2.cmml">min</mi><mrow id="Pt0.A2.Ex30.m2.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.cmml"><mi id="Pt0.A2.Ex30.m2.1.1.1.1.1.3" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.3.cmml">σ</mi><mo id="Pt0.A2.Ex30.m2.1.1.1.1.1.2" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.2.cmml">∈</mo><msup id="Pt0.A2.Ex30.m2.1.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.cmml"><mrow id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A2.Ex30.m2.1.1.1.1.1.1.3" xref="Pt0.A2.Ex30.m2.1.1.1.1.1.1.3.cmml">n</mi></msup></mrow></msub><mo id="Pt0.A2.Ex30.m2.5.5.5a" lspace="0.167em" xref="Pt0.A2.Ex30.m2.5.5.5.cmml">⁡</mo><mrow id="Pt0.A2.Ex30.m2.5.5.5.5.2" xref="Pt0.A2.Ex30.m2.5.5.5.5.3.cmml"><msub id="Pt0.A2.Ex30.m2.4.4.4.4.1.1" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.cmml"><mi id="Pt0.A2.Ex30.m2.4.4.4.4.1.1.2" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.2.cmml">max</mi><mrow id="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.cmml"><mi id="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.2" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.2.cmml">d</mi><mo id="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.1" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.1.cmml">∈</mo><msup id="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.3" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.3.cmml"><mi id="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.3.2" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.3.2.cmml">E</mi><mi id="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.3.3" xref="Pt0.A2.Ex30.m2.4.4.4.4.1.1.3.3.3.cmml">k</mi></msup></mrow></msub><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2a" xref="Pt0.A2.Ex30.m2.5.5.5.5.3.cmml">⁡</mo><mrow id="Pt0.A2.Ex30.m2.5.5.5.5.2.2" xref="Pt0.A2.Ex30.m2.5.5.5.5.3.cmml"><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.2" stretchy="false" xref="Pt0.A2.Ex30.m2.5.5.5.5.3.cmml">(</mo><mrow id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.cmml"><mi id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.2" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.2.cmml">T</mi><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.1" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.1.cmml">−</mo><mrow id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.cmml"><mi id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.2" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.2.cmml">u</mi><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.1" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.1.cmml">⁢</mo><mrow id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.2" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.1.cmml"><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.2.1" stretchy="false" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.1.cmml">(</mo><mi id="Pt0.A2.Ex30.m2.2.2.2.2" xref="Pt0.A2.Ex30.m2.2.2.2.2.cmml">σ</mi><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.2.2" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.1.cmml">,</mo><mi id="Pt0.A2.Ex30.m2.3.3.3.3" xref="Pt0.A2.Ex30.m2.3.3.3.3.cmml">d</mi><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.2.3" stretchy="false" xref="Pt0.A2.Ex30.m2.5.5.5.5.2.2.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A2.Ex30.m2.5.5.5.5.2.2.3" stretchy="false" xref="Pt0.A2.Ex30.m2.5.5.5.5.3.cmml">)</mo></mrow></mrow></mrow><mrow id="Pt0.A2.Ex30.m2.10.10.10" xref="Pt0.A2.Ex30.m2.10.10.10.cmml"><msub id="Pt0.A2.Ex30.m2.10.10.10.6" xref="Pt0.A2.Ex30.m2.10.10.10.6.cmml"><mi id="Pt0.A2.Ex30.m2.10.10.10.6.2" xref="Pt0.A2.Ex30.m2.10.10.10.6.2.cmml">min</mi><mrow id="Pt0.A2.Ex30.m2.6.6.6.1.1" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.cmml"><mi id="Pt0.A2.Ex30.m2.6.6.6.1.1.3" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.3.cmml">σ</mi><mo id="Pt0.A2.Ex30.m2.6.6.6.1.1.2" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.2.cmml">∈</mo><mrow id="Pt0.A2.Ex30.m2.6.6.6.1.1.1" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.1.cmml"><mi id="Pt0.A2.Ex30.m2.6.6.6.1.1.1.3" mathvariant="normal" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.1.3.cmml">Δ</mi><mo id="Pt0.A2.Ex30.m2.6.6.6.1.1.1.2" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A2.Ex30.m2.6.6.6.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.1.1.1.1.cmml"><mo id="Pt0.A2.Ex30.m2.6.6.6.1.1.1.1.1.2" stretchy="false" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.1.1.1.1.cmml">(</mo><msup id="Pt0.A2.Ex30.m2.6.6.6.1.1.1.1.1.1" xref="Pt0.A2.Ex30.m2.6.6.6.1.1.1.1.1.1.cmml"><mrow 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xref="Pt0.A2.Ex30.m2.10.10.10.5.2.2.1.3.3.2"><ci id="Pt0.A2.Ex30.m2.7.7.7.2.cmml" xref="Pt0.A2.Ex30.m2.7.7.7.2">𝜎</ci><ci id="Pt0.A2.Ex30.m2.8.8.8.3.cmml" xref="Pt0.A2.Ex30.m2.8.8.8.3">𝑑</ci></interval></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.Ex30.m2.10c">\displaystyle=\frac{\min_{\sigma\in(\Delta(P_{st}))^{n}}\max_{d\in E^{k}}(T-u(% \sigma,d))}{\min_{\sigma\in\Delta((P_{st})^{n})}\max_{d\in E^{k}}(T-u(\sigma,d% ))}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.Ex30.m2.10d">= divide start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_T - italic_u ( italic_σ , italic_d ) ) end_ARG start_ARG roman_min start_POSTSUBSCRIPT italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_T - italic_u ( italic_σ , italic_d ) ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A2.Ex31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{T-\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{d\in E^{k}}u(% \sigma,d)}{T-\max_{\sigma\in\Delta((P_{st})^{n})}\min_{d\in E^{k}}u(\sigma,d)}% =\frac{T-T^{\prime\prime}}{T-T^{\prime}}=\frac{T^{\prime}}{T^{\prime\prime}}=% \tilde{r}(G,n,k,u)" class="ltx_Math" display="inline" 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xref="Pt0.A2.Ex31.m1.10.11.7.3.3">′′</ci></apply></apply></apply><apply id="Pt0.A2.Ex31.m1.10.11g.cmml" xref="Pt0.A2.Ex31.m1.10.11"><eq id="Pt0.A2.Ex31.m1.10.11.8.cmml" xref="Pt0.A2.Ex31.m1.10.11.8"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A2.Ex31.m1.10.11.7.cmml" id="Pt0.A2.Ex31.m1.10.11h.cmml" xref="Pt0.A2.Ex31.m1.10.11"></share><apply id="Pt0.A2.Ex31.m1.10.11.9.cmml" xref="Pt0.A2.Ex31.m1.10.11.9"><times id="Pt0.A2.Ex31.m1.10.11.9.1.cmml" xref="Pt0.A2.Ex31.m1.10.11.9.1"></times><apply id="Pt0.A2.Ex31.m1.10.11.9.2.cmml" xref="Pt0.A2.Ex31.m1.10.11.9.2"><ci id="Pt0.A2.Ex31.m1.10.11.9.2.1.cmml" xref="Pt0.A2.Ex31.m1.10.11.9.2.1">~</ci><ci id="Pt0.A2.Ex31.m1.10.11.9.2.2.cmml" xref="Pt0.A2.Ex31.m1.10.11.9.2.2">𝑟</ci></apply><vector id="Pt0.A2.Ex31.m1.10.11.9.3.1.cmml" xref="Pt0.A2.Ex31.m1.10.11.9.3.2"><ci id="Pt0.A2.Ex31.m1.7.7.cmml" xref="Pt0.A2.Ex31.m1.7.7">𝐺</ci><ci id="Pt0.A2.Ex31.m1.8.8.cmml" xref="Pt0.A2.Ex31.m1.8.8">𝑛</ci><ci id="Pt0.A2.Ex31.m1.9.9.cmml" xref="Pt0.A2.Ex31.m1.9.9">𝑘</ci><ci id="Pt0.A2.Ex31.m1.10.10.cmml" xref="Pt0.A2.Ex31.m1.10.10">𝑢</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.Ex31.m1.10c">\displaystyle=\frac{T-\max_{\sigma\in(\Delta(P_{st}))^{n}}\min_{d\in E^{k}}u(% \sigma,d)}{T-\max_{\sigma\in\Delta((P_{st})^{n})}\min_{d\in E^{k}}u(\sigma,d)}% =\frac{T-T^{\prime\prime}}{T-T^{\prime}}=\frac{T^{\prime}}{T^{\prime\prime}}=% \tilde{r}(G,n,k,u)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.Ex31.m1.10d">= divide start_ARG italic_T - roman_max start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_u ( italic_σ , italic_d ) end_ARG start_ARG italic_T - roman_max start_POSTSUBSCRIPT italic_σ ∈ roman_Δ ( ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_u ( italic_σ , italic_d ) end_ARG = divide start_ARG italic_T - italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_T - italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG = divide start_ARG italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG = over~ start_ARG italic_r end_ARG ( italic_G , italic_n , italic_k , italic_u )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx7.p1.5">∎</p> </div> </div> <div class="ltx_para" id="Pt0.A2.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="Pt0.A2.SS0.SSS0.Px1.p2.1">When converting the payoff function of <span class="ltx_ERROR undefined" id="Pt0.A2.SS0.SSS0.Px1.p2.1.1">\citet</span>basilico2017team, we get the cost function <math alttext="C(a,d)=\frac{1}{m}+\frac{1}{m^{n}}-1\left\{\bigcup a\cap\bigcup d=\emptyset\right\}" class="ltx_Math" display="inline" id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3"><semantics id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3a"><mrow id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.cmml"><mrow id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.cmml"><mi id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.2.cmml">C</mi><mo id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.1" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.3.2" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.3.1.cmml"><mo id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.3.2.1" stretchy="false" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.3.3.1.cmml">(</mo><mi 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xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.1.1.1.1.1.2.3.1"></union><ci id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.1.1.1.1.1.2.3.2.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.1.1.1.1.1.2.3.2">𝑑</ci></apply></apply><emptyset id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.1.1.1.1.1.3.cmml" xref="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3.3.1.1.1.1.1.3"></emptyset></apply></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3c">C(a,d)=\frac{1}{m}+\frac{1}{m^{n}}-1\left\{\bigcup a\cap\bigcup d=\emptyset\right\}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A2.SS0.SSS0.Px1.p2.1.m1.3d">italic_C ( italic_a , italic_d ) = divide start_ARG 1 end_ARG start_ARG italic_m end_ARG + divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG - 1 { ⋃ italic_a ∩ ⋃ italic_d = ∅ }</annotation></semantics></math>. A similar proof shows how to convert a cost function to a payoff function while preserving the PoU, under an analogous assumption on the cost function.</p> </div> </section> </section> <section class="ltx_appendix" id="Pt0.A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix 0.C </span>Omitted Proofs in Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4" title="4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a> </h2> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx8"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma8" title="Lemma 8 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">8</span></a>)</h6> <div class="ltx_para" id="Thmproofx8.p1"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx25"> <tbody id="Pt0.A3.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C(\sigma,d,\alpha)" class="ltx_Math" display="inline" id="Pt0.A3.Ex32.m1.3"><semantics id="Pt0.A3.Ex32.m1.3a"><mrow id="Pt0.A3.Ex32.m1.3.4" xref="Pt0.A3.Ex32.m1.3.4.cmml"><mi id="Pt0.A3.Ex32.m1.3.4.2" xref="Pt0.A3.Ex32.m1.3.4.2.cmml">C</mi><mo id="Pt0.A3.Ex32.m1.3.4.1" xref="Pt0.A3.Ex32.m1.3.4.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex32.m1.3.4.3.2" xref="Pt0.A3.Ex32.m1.3.4.3.1.cmml"><mo id="Pt0.A3.Ex32.m1.3.4.3.2.1" stretchy="false" xref="Pt0.A3.Ex32.m1.3.4.3.1.cmml">(</mo><mi id="Pt0.A3.Ex32.m1.1.1" xref="Pt0.A3.Ex32.m1.1.1.cmml">σ</mi><mo id="Pt0.A3.Ex32.m1.3.4.3.2.2" xref="Pt0.A3.Ex32.m1.3.4.3.1.cmml">,</mo><mi 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xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.2.3.3">𝑘</ci></apply></apply><apply id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1"><times id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.2.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.2"></times><cn id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.3.cmml" type="integer" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.3">1</cn><set id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.2.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1"><apply id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1"><in id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.1"></in><apply id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2.1.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2.2.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2.2">𝑑</ci><ci id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2.3.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.2.3">𝑗</ci></apply><apply id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.3"><union id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.3.1"></union><ci id="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A3.Ex32.m2.5.5.3.3.1.1.1.1.1.1.1.3.2">𝑎</ci></apply></apply></set></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex32.m2.5c">\displaystyle=\operatorname*{\mathbb{E}}_{a\sim\sigma}\left(\left|\bigcup a% \right|\right)+\alpha\sum_{a\in(P_{st})^{n}}\sigma(a)\sum_{j\leq k}1\left\{d_{% j}\in\bigcup a\right\}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex32.m2.5d">= blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( | ⋃ italic_a | ) + italic_α ∑ start_POSTSUBSCRIPT italic_a ∈ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_σ ( italic_a ) ∑ start_POSTSUBSCRIPT italic_j ≤ italic_k end_POSTSUBSCRIPT 1 { italic_d start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ ⋃ italic_a }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname*{\mathbb{E}}_{a\sim\sigma}\left(\left|\bigcup a% \right|\right)+\alpha\sum_{j\leq k}\sum_{a\in(P_{st})^{n}}\sigma(a)1\left\{d_{% j}\in\bigcup a\right\}" class="ltx_Math" display="inline" id="Pt0.A3.Ex33.m1.5"><semantics id="Pt0.A3.Ex33.m1.5a"><mrow id="Pt0.A3.Ex33.m1.5.5" 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id="Pt0.A3.Ex34.m1.3.3.3.3.1.2.3.1.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.2.3.1"></leq><ci id="Pt0.A3.Ex34.m1.3.3.3.3.1.2.3.2.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.2.3.2">𝑗</ci><ci id="Pt0.A3.Ex34.m1.3.3.3.3.1.2.3.3.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.2.3.3">𝑘</ci></apply></apply><apply id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1"><times id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.2.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.2"></times><apply id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3.1.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3">superscript</csymbol><ci id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3.2.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3.2">𝜎</ci><ci id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3.3.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.3.3">𝐸</ci></apply><apply id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1.1.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1">subscript</csymbol><ci id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1.1.2.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1.1.2">𝑑</ci><ci id="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1.1.3.cmml" xref="Pt0.A3.Ex34.m1.3.3.3.3.1.1.1.1.1.3">𝑗</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex34.m1.3c">\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname*{\mathbb{E}% }_{a\sim\sigma}\left(\left|\bigcup a\right|\right)+\alpha\sum_{j\leq k}\sigma^% {E}(d_{j})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex34.m1.3d">start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( | ⋃ italic_a | ) + italic_α ∑ start_POSTSUBSCRIPT italic_j ≤ italic_k end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_d start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx8.p1.5">Now we see that <math alttext="\max_{d}C(\sigma,d,\alpha)" class="ltx_Math" display="inline" id="Thmproofx8.p1.1.m1.3"><semantics id="Thmproofx8.p1.1.m1.3a"><mrow id="Thmproofx8.p1.1.m1.3.4" xref="Thmproofx8.p1.1.m1.3.4.cmml"><mrow id="Thmproofx8.p1.1.m1.3.4.2" xref="Thmproofx8.p1.1.m1.3.4.2.cmml"><msub id="Thmproofx8.p1.1.m1.3.4.2.1" xref="Thmproofx8.p1.1.m1.3.4.2.1.cmml"><mi id="Thmproofx8.p1.1.m1.3.4.2.1.2" xref="Thmproofx8.p1.1.m1.3.4.2.1.2.cmml">max</mi><mi id="Thmproofx8.p1.1.m1.3.4.2.1.3" xref="Thmproofx8.p1.1.m1.3.4.2.1.3.cmml">d</mi></msub><mo id="Thmproofx8.p1.1.m1.3.4.2a" lspace="0.167em" xref="Thmproofx8.p1.1.m1.3.4.2.cmml">⁡</mo><mi id="Thmproofx8.p1.1.m1.3.4.2.2" xref="Thmproofx8.p1.1.m1.3.4.2.2.cmml">C</mi></mrow><mo id="Thmproofx8.p1.1.m1.3.4.1" xref="Thmproofx8.p1.1.m1.3.4.1.cmml">⁢</mo><mrow id="Thmproofx8.p1.1.m1.3.4.3.2" xref="Thmproofx8.p1.1.m1.3.4.3.1.cmml"><mo id="Thmproofx8.p1.1.m1.3.4.3.2.1" stretchy="false" xref="Thmproofx8.p1.1.m1.3.4.3.1.cmml">(</mo><mi id="Thmproofx8.p1.1.m1.1.1" xref="Thmproofx8.p1.1.m1.1.1.cmml">σ</mi><mo id="Thmproofx8.p1.1.m1.3.4.3.2.2" xref="Thmproofx8.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="Thmproofx8.p1.1.m1.2.2" xref="Thmproofx8.p1.1.m1.2.2.cmml">d</mi><mo id="Thmproofx8.p1.1.m1.3.4.3.2.3" xref="Thmproofx8.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="Thmproofx8.p1.1.m1.3.3" xref="Thmproofx8.p1.1.m1.3.3.cmml">α</mi><mo id="Thmproofx8.p1.1.m1.3.4.3.2.4" stretchy="false" xref="Thmproofx8.p1.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.1.m1.3b"><apply id="Thmproofx8.p1.1.m1.3.4.cmml" xref="Thmproofx8.p1.1.m1.3.4"><times id="Thmproofx8.p1.1.m1.3.4.1.cmml" xref="Thmproofx8.p1.1.m1.3.4.1"></times><apply id="Thmproofx8.p1.1.m1.3.4.2.cmml" xref="Thmproofx8.p1.1.m1.3.4.2"><apply id="Thmproofx8.p1.1.m1.3.4.2.1.cmml" xref="Thmproofx8.p1.1.m1.3.4.2.1"><csymbol cd="ambiguous" id="Thmproofx8.p1.1.m1.3.4.2.1.1.cmml" xref="Thmproofx8.p1.1.m1.3.4.2.1">subscript</csymbol><max id="Thmproofx8.p1.1.m1.3.4.2.1.2.cmml" xref="Thmproofx8.p1.1.m1.3.4.2.1.2"></max><ci id="Thmproofx8.p1.1.m1.3.4.2.1.3.cmml" xref="Thmproofx8.p1.1.m1.3.4.2.1.3">𝑑</ci></apply><ci id="Thmproofx8.p1.1.m1.3.4.2.2.cmml" xref="Thmproofx8.p1.1.m1.3.4.2.2">𝐶</ci></apply><vector id="Thmproofx8.p1.1.m1.3.4.3.1.cmml" xref="Thmproofx8.p1.1.m1.3.4.3.2"><ci id="Thmproofx8.p1.1.m1.1.1.cmml" xref="Thmproofx8.p1.1.m1.1.1">𝜎</ci><ci id="Thmproofx8.p1.1.m1.2.2.cmml" xref="Thmproofx8.p1.1.m1.2.2">𝑑</ci><ci id="Thmproofx8.p1.1.m1.3.3.cmml" xref="Thmproofx8.p1.1.m1.3.3">𝛼</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.1.m1.3c">\max_{d}C(\sigma,d,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.1.m1.3d">roman_max start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_C ( italic_σ , italic_d , italic_α )</annotation></semantics></math> is obtained for <math alttext="d" class="ltx_Math" display="inline" id="Thmproofx8.p1.2.m2.1"><semantics id="Thmproofx8.p1.2.m2.1a"><mi id="Thmproofx8.p1.2.m2.1.1" xref="Thmproofx8.p1.2.m2.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.2.m2.1b"><ci id="Thmproofx8.p1.2.m2.1.1.cmml" xref="Thmproofx8.p1.2.m2.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.2.m2.1c">d</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.2.m2.1d">italic_d</annotation></semantics></math> such that <math alttext="d_{j}\in\operatorname*{arg\,max}_{e}\sigma^{E}(e)" class="ltx_Math" display="inline" id="Thmproofx8.p1.3.m3.1"><semantics id="Thmproofx8.p1.3.m3.1a"><mrow id="Thmproofx8.p1.3.m3.1.2" xref="Thmproofx8.p1.3.m3.1.2.cmml"><msub id="Thmproofx8.p1.3.m3.1.2.2" xref="Thmproofx8.p1.3.m3.1.2.2.cmml"><mi id="Thmproofx8.p1.3.m3.1.2.2.2" xref="Thmproofx8.p1.3.m3.1.2.2.2.cmml">d</mi><mi id="Thmproofx8.p1.3.m3.1.2.2.3" xref="Thmproofx8.p1.3.m3.1.2.2.3.cmml">j</mi></msub><mo id="Thmproofx8.p1.3.m3.1.2.1" xref="Thmproofx8.p1.3.m3.1.2.1.cmml">∈</mo><mrow id="Thmproofx8.p1.3.m3.1.2.3" xref="Thmproofx8.p1.3.m3.1.2.3.cmml"><mrow id="Thmproofx8.p1.3.m3.1.2.3.2" xref="Thmproofx8.p1.3.m3.1.2.3.2.cmml"><msub id="Thmproofx8.p1.3.m3.1.2.3.2.1" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.cmml"><mrow id="Thmproofx8.p1.3.m3.1.2.3.2.1.2" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2.cmml"><mi id="Thmproofx8.p1.3.m3.1.2.3.2.1.2.2" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2.2.cmml">arg</mi><mo id="Thmproofx8.p1.3.m3.1.2.3.2.1.2.1" lspace="0.170em" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2.1.cmml">⁢</mo><mi id="Thmproofx8.p1.3.m3.1.2.3.2.1.2.3" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2.3.cmml">max</mi></mrow><mi id="Thmproofx8.p1.3.m3.1.2.3.2.1.3" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.3.cmml">e</mi></msub><mo id="Thmproofx8.p1.3.m3.1.2.3.2a" xref="Thmproofx8.p1.3.m3.1.2.3.2.cmml">⁡</mo><msup id="Thmproofx8.p1.3.m3.1.2.3.2.2" xref="Thmproofx8.p1.3.m3.1.2.3.2.2.cmml"><mi id="Thmproofx8.p1.3.m3.1.2.3.2.2.2" xref="Thmproofx8.p1.3.m3.1.2.3.2.2.2.cmml">σ</mi><mi id="Thmproofx8.p1.3.m3.1.2.3.2.2.3" xref="Thmproofx8.p1.3.m3.1.2.3.2.2.3.cmml">E</mi></msup></mrow><mo id="Thmproofx8.p1.3.m3.1.2.3.1" xref="Thmproofx8.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="Thmproofx8.p1.3.m3.1.2.3.3.2" xref="Thmproofx8.p1.3.m3.1.2.3.cmml"><mo id="Thmproofx8.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="Thmproofx8.p1.3.m3.1.2.3.cmml">(</mo><mi id="Thmproofx8.p1.3.m3.1.1" xref="Thmproofx8.p1.3.m3.1.1.cmml">e</mi><mo id="Thmproofx8.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="Thmproofx8.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.3.m3.1b"><apply id="Thmproofx8.p1.3.m3.1.2.cmml" xref="Thmproofx8.p1.3.m3.1.2"><in id="Thmproofx8.p1.3.m3.1.2.1.cmml" xref="Thmproofx8.p1.3.m3.1.2.1"></in><apply id="Thmproofx8.p1.3.m3.1.2.2.cmml" xref="Thmproofx8.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="Thmproofx8.p1.3.m3.1.2.2.1.cmml" xref="Thmproofx8.p1.3.m3.1.2.2">subscript</csymbol><ci id="Thmproofx8.p1.3.m3.1.2.2.2.cmml" xref="Thmproofx8.p1.3.m3.1.2.2.2">𝑑</ci><ci id="Thmproofx8.p1.3.m3.1.2.2.3.cmml" xref="Thmproofx8.p1.3.m3.1.2.2.3">𝑗</ci></apply><apply id="Thmproofx8.p1.3.m3.1.2.3.cmml" xref="Thmproofx8.p1.3.m3.1.2.3"><times id="Thmproofx8.p1.3.m3.1.2.3.1.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.1"></times><apply id="Thmproofx8.p1.3.m3.1.2.3.2.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2"><apply id="Thmproofx8.p1.3.m3.1.2.3.2.1.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.1"><csymbol cd="ambiguous" id="Thmproofx8.p1.3.m3.1.2.3.2.1.1.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.1">subscript</csymbol><apply id="Thmproofx8.p1.3.m3.1.2.3.2.1.2.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2"><times id="Thmproofx8.p1.3.m3.1.2.3.2.1.2.1.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2.1"></times><ci id="Thmproofx8.p1.3.m3.1.2.3.2.1.2.2.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2.2">arg</ci><ci id="Thmproofx8.p1.3.m3.1.2.3.2.1.2.3.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.2.3">max</ci></apply><ci id="Thmproofx8.p1.3.m3.1.2.3.2.1.3.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.1.3">𝑒</ci></apply><apply id="Thmproofx8.p1.3.m3.1.2.3.2.2.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.2"><csymbol cd="ambiguous" id="Thmproofx8.p1.3.m3.1.2.3.2.2.1.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.2">superscript</csymbol><ci id="Thmproofx8.p1.3.m3.1.2.3.2.2.2.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.2.2">𝜎</ci><ci id="Thmproofx8.p1.3.m3.1.2.3.2.2.3.cmml" xref="Thmproofx8.p1.3.m3.1.2.3.2.2.3">𝐸</ci></apply></apply><ci id="Thmproofx8.p1.3.m3.1.1.cmml" xref="Thmproofx8.p1.3.m3.1.1">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.3.m3.1c">d_{j}\in\operatorname*{arg\,max}_{e}\sigma^{E}(e)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.3.m3.1d">italic_d start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e )</annotation></semantics></math> for every <math alttext="j" class="ltx_Math" display="inline" id="Thmproofx8.p1.4.m4.1"><semantics id="Thmproofx8.p1.4.m4.1a"><mi id="Thmproofx8.p1.4.m4.1.1" xref="Thmproofx8.p1.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.4.m4.1b"><ci id="Thmproofx8.p1.4.m4.1.1.cmml" xref="Thmproofx8.p1.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.4.m4.1d">italic_j</annotation></semantics></math>. For such a <math alttext="d" class="ltx_Math" display="inline" id="Thmproofx8.p1.5.m5.1"><semantics id="Thmproofx8.p1.5.m5.1a"><mi id="Thmproofx8.p1.5.m5.1.1" xref="Thmproofx8.p1.5.m5.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.5.m5.1b"><ci id="Thmproofx8.p1.5.m5.1.1.cmml" xref="Thmproofx8.p1.5.m5.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.5.m5.1c">d</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.5.m5.1d">italic_d</annotation></semantics></math>, the above becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx26"> <tbody id="Pt0.A3.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{\mathbb{E}}_{a\sim\sigma}\left(\left|\bigcup a% \right|\right)+\alpha\sum_{j\leq k}\beta(\sigma)=\operatorname*{\mathbb{E}}_{a% \sim\sigma}\left(\left|\bigcup a\right|\right)+k\alpha\beta(\sigma)" class="ltx_Math" display="inline" id="Pt0.A3.Ex35.m1.6"><semantics id="Pt0.A3.Ex35.m1.6a"><mrow id="Pt0.A3.Ex35.m1.6.6" xref="Pt0.A3.Ex35.m1.6.6.cmml"><mrow id="Pt0.A3.Ex35.m1.4.4.2" xref="Pt0.A3.Ex35.m1.4.4.2.cmml"><mrow id="Pt0.A3.Ex35.m1.4.4.2.2.2" xref="Pt0.A3.Ex35.m1.4.4.2.2.3.cmml"><munder id="Pt0.A3.Ex35.m1.3.3.1.1.1.1" xref="Pt0.A3.Ex35.m1.3.3.1.1.1.1.cmml"><mo id="Pt0.A3.Ex35.m1.3.3.1.1.1.1.2" xref="Pt0.A3.Ex35.m1.3.3.1.1.1.1.2.cmml">𝔼</mo><mrow id="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3" xref="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3.2" xref="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3.2.cmml">a</mi><mo id="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3.1" xref="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3.1.cmml">∼</mo><mi id="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3.3" xref="Pt0.A3.Ex35.m1.3.3.1.1.1.1.3.3.cmml">σ</mi></mrow></munder><mrow id="Pt0.A3.Ex35.m1.4.4.2.2.2.2" xref="Pt0.A3.Ex35.m1.4.4.2.2.3.cmml"><mo id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.2" lspace="0em" xref="Pt0.A3.Ex35.m1.4.4.2.2.3.cmml">(</mo><mrow id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1" xref="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.2.cmml"><mo id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.2" xref="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.2.1.cmml">|</mo><mstyle displaystyle="true" id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1" xref="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1.cmml"><mrow id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1a" xref="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1.cmml"><mo id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1.1" movablelimits="false" xref="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1.1.cmml">⋃</mo><mi id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1.2" xref="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.1.2.cmml">a</mi></mrow></mstyle><mo id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.1.3" xref="Pt0.A3.Ex35.m1.4.4.2.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A3.Ex35.m1.4.4.2.2.2.2.3" xref="Pt0.A3.Ex35.m1.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A3.Ex35.m1.4.4.2.3" xref="Pt0.A3.Ex35.m1.4.4.2.3.cmml">+</mo><mrow id="Pt0.A3.Ex35.m1.4.4.2.4" xref="Pt0.A3.Ex35.m1.4.4.2.4.cmml"><mi id="Pt0.A3.Ex35.m1.4.4.2.4.2" xref="Pt0.A3.Ex35.m1.4.4.2.4.2.cmml">α</mi><mo id="Pt0.A3.Ex35.m1.4.4.2.4.1" xref="Pt0.A3.Ex35.m1.4.4.2.4.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex35.m1.4.4.2.4.3" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex35.m1.4.4.2.4.3.1" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.1.cmml"><munder id="Pt0.A3.Ex35.m1.4.4.2.4.3.1a" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.1.cmml"><mo id="Pt0.A3.Ex35.m1.4.4.2.4.3.1.2" movablelimits="false" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.1.2.cmml">∑</mo><mrow id="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3.cmml"><mi id="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3.2" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3.2.cmml">j</mi><mo id="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3.1" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3.1.cmml">≤</mo><mi id="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3.3" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.1.3.3.cmml">k</mi></mrow></munder></mstyle><mrow id="Pt0.A3.Ex35.m1.4.4.2.4.3.2" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.2.cmml"><mi id="Pt0.A3.Ex35.m1.4.4.2.4.3.2.2" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.2.2.cmml">β</mi><mo id="Pt0.A3.Ex35.m1.4.4.2.4.3.2.1" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.2.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex35.m1.4.4.2.4.3.2.3.2" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.2.cmml"><mo id="Pt0.A3.Ex35.m1.4.4.2.4.3.2.3.2.1" stretchy="false" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.2.cmml">(</mo><mi id="Pt0.A3.Ex35.m1.1.1" xref="Pt0.A3.Ex35.m1.1.1.cmml">σ</mi><mo id="Pt0.A3.Ex35.m1.4.4.2.4.3.2.3.2.2" stretchy="false" xref="Pt0.A3.Ex35.m1.4.4.2.4.3.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><mo id="Pt0.A3.Ex35.m1.6.6.5" rspace="0.1389em" xref="Pt0.A3.Ex35.m1.6.6.5.cmml">=</mo><mrow id="Pt0.A3.Ex35.m1.6.6.4" xref="Pt0.A3.Ex35.m1.6.6.4.cmml"><mrow id="Pt0.A3.Ex35.m1.6.6.4.2.2" xref="Pt0.A3.Ex35.m1.6.6.4.2.3.cmml"><munder id="Pt0.A3.Ex35.m1.5.5.3.1.1.1" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.cmml"><mo id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.2" lspace="0.1389em" rspace="0em" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.2.cmml">𝔼</mo><mrow id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.cmml"><mi id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.2" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.2.cmml">a</mi><mo id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.1" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.1.cmml">∼</mo><mi id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.3" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.3.cmml">σ</mi></mrow></munder><mrow id="Pt0.A3.Ex35.m1.6.6.4.2.2.2" xref="Pt0.A3.Ex35.m1.6.6.4.2.3.cmml"><mo id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.2" xref="Pt0.A3.Ex35.m1.6.6.4.2.3.cmml">(</mo><mrow id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.2.cmml"><mo id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.2" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.2.1.cmml">|</mo><mstyle displaystyle="true" id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.cmml"><mrow id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1a" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.cmml"><mo id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.1" movablelimits="false" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.1.cmml">⋃</mo><mi id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.2" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.2.cmml">a</mi></mrow></mstyle><mo id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.3" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.3" xref="Pt0.A3.Ex35.m1.6.6.4.2.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A3.Ex35.m1.6.6.4.3" xref="Pt0.A3.Ex35.m1.6.6.4.3.cmml">+</mo><mrow id="Pt0.A3.Ex35.m1.6.6.4.4" xref="Pt0.A3.Ex35.m1.6.6.4.4.cmml"><mi id="Pt0.A3.Ex35.m1.6.6.4.4.2" xref="Pt0.A3.Ex35.m1.6.6.4.4.2.cmml">k</mi><mo id="Pt0.A3.Ex35.m1.6.6.4.4.1" xref="Pt0.A3.Ex35.m1.6.6.4.4.1.cmml">⁢</mo><mi id="Pt0.A3.Ex35.m1.6.6.4.4.3" xref="Pt0.A3.Ex35.m1.6.6.4.4.3.cmml">α</mi><mo id="Pt0.A3.Ex35.m1.6.6.4.4.1a" xref="Pt0.A3.Ex35.m1.6.6.4.4.1.cmml">⁢</mo><mi id="Pt0.A3.Ex35.m1.6.6.4.4.4" 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cd="latexml" id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.1.cmml" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.1">similar-to</csymbol><ci id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.2.cmml" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.2">𝑎</ci><ci id="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.3.cmml" xref="Pt0.A3.Ex35.m1.5.5.3.1.1.1.3.3">𝜎</ci></apply></apply><apply id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.2.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1"><abs id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.2.1.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.2"></abs><apply id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1"><union id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.1.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.1"></union><ci id="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.2.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.2.2.2.1.1.1.2">𝑎</ci></apply></apply></apply><apply id="Pt0.A3.Ex35.m1.6.6.4.4.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.4"><times id="Pt0.A3.Ex35.m1.6.6.4.4.1.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.4.1"></times><ci id="Pt0.A3.Ex35.m1.6.6.4.4.2.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.4.2">𝑘</ci><ci id="Pt0.A3.Ex35.m1.6.6.4.4.3.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.4.3">𝛼</ci><ci id="Pt0.A3.Ex35.m1.6.6.4.4.4.cmml" xref="Pt0.A3.Ex35.m1.6.6.4.4.4">𝛽</ci><ci id="Pt0.A3.Ex35.m1.2.2.cmml" xref="Pt0.A3.Ex35.m1.2.2">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex35.m1.6c">\displaystyle\operatorname*{\mathbb{E}}_{a\sim\sigma}\left(\left|\bigcup a% \right|\right)+\alpha\sum_{j\leq k}\beta(\sigma)=\operatorname*{\mathbb{E}}_{a% \sim\sigma}\left(\left|\bigcup a\right|\right)+k\alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex35.m1.6d">blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( | ⋃ italic_a | ) + italic_α ∑ start_POSTSUBSCRIPT italic_j ≤ italic_k end_POSTSUBSCRIPT italic_β ( italic_σ ) = blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( | ⋃ italic_a | ) + italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx8.p1.8">Now suppose <math alttext="P:=\bigcup_{j}\operatorname{supp}\sigma_{j}" class="ltx_Math" display="inline" id="Thmproofx8.p1.6.m1.1"><semantics id="Thmproofx8.p1.6.m1.1a"><mrow id="Thmproofx8.p1.6.m1.1.1" xref="Thmproofx8.p1.6.m1.1.1.cmml"><mi id="Thmproofx8.p1.6.m1.1.1.2" xref="Thmproofx8.p1.6.m1.1.1.2.cmml">P</mi><mo id="Thmproofx8.p1.6.m1.1.1.1" lspace="0.278em" rspace="0.111em" xref="Thmproofx8.p1.6.m1.1.1.1.cmml">:=</mo><mrow id="Thmproofx8.p1.6.m1.1.1.3" xref="Thmproofx8.p1.6.m1.1.1.3.cmml"><msub id="Thmproofx8.p1.6.m1.1.1.3.1" xref="Thmproofx8.p1.6.m1.1.1.3.1.cmml"><mo id="Thmproofx8.p1.6.m1.1.1.3.1.2" xref="Thmproofx8.p1.6.m1.1.1.3.1.2.cmml">⋃</mo><mi id="Thmproofx8.p1.6.m1.1.1.3.1.3" xref="Thmproofx8.p1.6.m1.1.1.3.1.3.cmml">j</mi></msub><mrow id="Thmproofx8.p1.6.m1.1.1.3.2" xref="Thmproofx8.p1.6.m1.1.1.3.2.cmml"><mi id="Thmproofx8.p1.6.m1.1.1.3.2.1" xref="Thmproofx8.p1.6.m1.1.1.3.2.1.cmml">supp</mi><mo id="Thmproofx8.p1.6.m1.1.1.3.2a" lspace="0.167em" xref="Thmproofx8.p1.6.m1.1.1.3.2.cmml">⁡</mo><msub id="Thmproofx8.p1.6.m1.1.1.3.2.2" xref="Thmproofx8.p1.6.m1.1.1.3.2.2.cmml"><mi id="Thmproofx8.p1.6.m1.1.1.3.2.2.2" xref="Thmproofx8.p1.6.m1.1.1.3.2.2.2.cmml">σ</mi><mi id="Thmproofx8.p1.6.m1.1.1.3.2.2.3" xref="Thmproofx8.p1.6.m1.1.1.3.2.2.3.cmml">j</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.6.m1.1b"><apply id="Thmproofx8.p1.6.m1.1.1.cmml" xref="Thmproofx8.p1.6.m1.1.1"><csymbol cd="latexml" id="Thmproofx8.p1.6.m1.1.1.1.cmml" xref="Thmproofx8.p1.6.m1.1.1.1">assign</csymbol><ci id="Thmproofx8.p1.6.m1.1.1.2.cmml" xref="Thmproofx8.p1.6.m1.1.1.2">𝑃</ci><apply id="Thmproofx8.p1.6.m1.1.1.3.cmml" xref="Thmproofx8.p1.6.m1.1.1.3"><apply id="Thmproofx8.p1.6.m1.1.1.3.1.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.1"><csymbol cd="ambiguous" id="Thmproofx8.p1.6.m1.1.1.3.1.1.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.1">subscript</csymbol><union id="Thmproofx8.p1.6.m1.1.1.3.1.2.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.1.2"></union><ci id="Thmproofx8.p1.6.m1.1.1.3.1.3.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.1.3">𝑗</ci></apply><apply id="Thmproofx8.p1.6.m1.1.1.3.2.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.2"><ci id="Thmproofx8.p1.6.m1.1.1.3.2.1.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.2.1">supp</ci><apply id="Thmproofx8.p1.6.m1.1.1.3.2.2.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="Thmproofx8.p1.6.m1.1.1.3.2.2.1.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.2.2">subscript</csymbol><ci id="Thmproofx8.p1.6.m1.1.1.3.2.2.2.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.2.2.2">𝜎</ci><ci id="Thmproofx8.p1.6.m1.1.1.3.2.2.3.cmml" xref="Thmproofx8.p1.6.m1.1.1.3.2.2.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.6.m1.1c">P:=\bigcup_{j}\operatorname{supp}\sigma_{j}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.6.m1.1d">italic_P := ⋃ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_supp italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is a set of disjoint paths. Then for every <math alttext="e" class="ltx_Math" display="inline" id="Thmproofx8.p1.7.m2.1"><semantics id="Thmproofx8.p1.7.m2.1a"><mi id="Thmproofx8.p1.7.m2.1.1" xref="Thmproofx8.p1.7.m2.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.7.m2.1b"><ci id="Thmproofx8.p1.7.m2.1.1.cmml" xref="Thmproofx8.p1.7.m2.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.7.m2.1c">e</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.7.m2.1d">italic_e</annotation></semantics></math> edge, <math alttext="\sigma^{E}(e)=\sum_{p\in P_{st,e}}\sigma(p)" class="ltx_Math" display="inline" id="Thmproofx8.p1.8.m3.4"><semantics id="Thmproofx8.p1.8.m3.4a"><mrow id="Thmproofx8.p1.8.m3.4.5" xref="Thmproofx8.p1.8.m3.4.5.cmml"><mrow id="Thmproofx8.p1.8.m3.4.5.2" xref="Thmproofx8.p1.8.m3.4.5.2.cmml"><msup id="Thmproofx8.p1.8.m3.4.5.2.2" xref="Thmproofx8.p1.8.m3.4.5.2.2.cmml"><mi id="Thmproofx8.p1.8.m3.4.5.2.2.2" xref="Thmproofx8.p1.8.m3.4.5.2.2.2.cmml">σ</mi><mi id="Thmproofx8.p1.8.m3.4.5.2.2.3" xref="Thmproofx8.p1.8.m3.4.5.2.2.3.cmml">E</mi></msup><mo id="Thmproofx8.p1.8.m3.4.5.2.1" xref="Thmproofx8.p1.8.m3.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx8.p1.8.m3.4.5.2.3.2" xref="Thmproofx8.p1.8.m3.4.5.2.cmml"><mo id="Thmproofx8.p1.8.m3.4.5.2.3.2.1" stretchy="false" xref="Thmproofx8.p1.8.m3.4.5.2.cmml">(</mo><mi id="Thmproofx8.p1.8.m3.3.3" xref="Thmproofx8.p1.8.m3.3.3.cmml">e</mi><mo id="Thmproofx8.p1.8.m3.4.5.2.3.2.2" stretchy="false" xref="Thmproofx8.p1.8.m3.4.5.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx8.p1.8.m3.4.5.1" rspace="0.111em" xref="Thmproofx8.p1.8.m3.4.5.1.cmml">=</mo><mrow id="Thmproofx8.p1.8.m3.4.5.3" xref="Thmproofx8.p1.8.m3.4.5.3.cmml"><msub id="Thmproofx8.p1.8.m3.4.5.3.1" xref="Thmproofx8.p1.8.m3.4.5.3.1.cmml"><mo id="Thmproofx8.p1.8.m3.4.5.3.1.2" xref="Thmproofx8.p1.8.m3.4.5.3.1.2.cmml">∑</mo><mrow id="Thmproofx8.p1.8.m3.2.2.2" xref="Thmproofx8.p1.8.m3.2.2.2.cmml"><mi id="Thmproofx8.p1.8.m3.2.2.2.4" xref="Thmproofx8.p1.8.m3.2.2.2.4.cmml">p</mi><mo id="Thmproofx8.p1.8.m3.2.2.2.3" xref="Thmproofx8.p1.8.m3.2.2.2.3.cmml">∈</mo><msub id="Thmproofx8.p1.8.m3.2.2.2.5" xref="Thmproofx8.p1.8.m3.2.2.2.5.cmml"><mi id="Thmproofx8.p1.8.m3.2.2.2.5.2" xref="Thmproofx8.p1.8.m3.2.2.2.5.2.cmml">P</mi><mrow id="Thmproofx8.p1.8.m3.2.2.2.2.2.2" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.3.cmml"><mrow id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.cmml"><mi id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.2" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.2.cmml">s</mi><mo id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.1" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.3" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.3.cmml">t</mi></mrow><mo id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.2" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx8.p1.8.m3.1.1.1.1.1.1" xref="Thmproofx8.p1.8.m3.1.1.1.1.1.1.cmml">e</mi></mrow></msub></mrow></msub><mrow id="Thmproofx8.p1.8.m3.4.5.3.2" xref="Thmproofx8.p1.8.m3.4.5.3.2.cmml"><mi id="Thmproofx8.p1.8.m3.4.5.3.2.2" xref="Thmproofx8.p1.8.m3.4.5.3.2.2.cmml">σ</mi><mo id="Thmproofx8.p1.8.m3.4.5.3.2.1" xref="Thmproofx8.p1.8.m3.4.5.3.2.1.cmml">⁢</mo><mrow id="Thmproofx8.p1.8.m3.4.5.3.2.3.2" xref="Thmproofx8.p1.8.m3.4.5.3.2.cmml"><mo id="Thmproofx8.p1.8.m3.4.5.3.2.3.2.1" stretchy="false" xref="Thmproofx8.p1.8.m3.4.5.3.2.cmml">(</mo><mi id="Thmproofx8.p1.8.m3.4.4" xref="Thmproofx8.p1.8.m3.4.4.cmml">p</mi><mo id="Thmproofx8.p1.8.m3.4.5.3.2.3.2.2" stretchy="false" xref="Thmproofx8.p1.8.m3.4.5.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx8.p1.8.m3.4b"><apply id="Thmproofx8.p1.8.m3.4.5.cmml" xref="Thmproofx8.p1.8.m3.4.5"><eq id="Thmproofx8.p1.8.m3.4.5.1.cmml" xref="Thmproofx8.p1.8.m3.4.5.1"></eq><apply id="Thmproofx8.p1.8.m3.4.5.2.cmml" xref="Thmproofx8.p1.8.m3.4.5.2"><times id="Thmproofx8.p1.8.m3.4.5.2.1.cmml" xref="Thmproofx8.p1.8.m3.4.5.2.1"></times><apply id="Thmproofx8.p1.8.m3.4.5.2.2.cmml" xref="Thmproofx8.p1.8.m3.4.5.2.2"><csymbol cd="ambiguous" id="Thmproofx8.p1.8.m3.4.5.2.2.1.cmml" xref="Thmproofx8.p1.8.m3.4.5.2.2">superscript</csymbol><ci id="Thmproofx8.p1.8.m3.4.5.2.2.2.cmml" xref="Thmproofx8.p1.8.m3.4.5.2.2.2">𝜎</ci><ci id="Thmproofx8.p1.8.m3.4.5.2.2.3.cmml" xref="Thmproofx8.p1.8.m3.4.5.2.2.3">𝐸</ci></apply><ci id="Thmproofx8.p1.8.m3.3.3.cmml" xref="Thmproofx8.p1.8.m3.3.3">𝑒</ci></apply><apply id="Thmproofx8.p1.8.m3.4.5.3.cmml" xref="Thmproofx8.p1.8.m3.4.5.3"><apply id="Thmproofx8.p1.8.m3.4.5.3.1.cmml" xref="Thmproofx8.p1.8.m3.4.5.3.1"><csymbol cd="ambiguous" id="Thmproofx8.p1.8.m3.4.5.3.1.1.cmml" xref="Thmproofx8.p1.8.m3.4.5.3.1">subscript</csymbol><sum id="Thmproofx8.p1.8.m3.4.5.3.1.2.cmml" xref="Thmproofx8.p1.8.m3.4.5.3.1.2"></sum><apply id="Thmproofx8.p1.8.m3.2.2.2.cmml" xref="Thmproofx8.p1.8.m3.2.2.2"><in id="Thmproofx8.p1.8.m3.2.2.2.3.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.3"></in><ci id="Thmproofx8.p1.8.m3.2.2.2.4.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.4">𝑝</ci><apply id="Thmproofx8.p1.8.m3.2.2.2.5.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.5"><csymbol cd="ambiguous" id="Thmproofx8.p1.8.m3.2.2.2.5.1.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.5">subscript</csymbol><ci id="Thmproofx8.p1.8.m3.2.2.2.5.2.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.5.2">𝑃</ci><list id="Thmproofx8.p1.8.m3.2.2.2.2.2.3.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2"><apply id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1"><times id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.1.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.1"></times><ci id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.2.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.2">𝑠</ci><ci id="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.3.cmml" xref="Thmproofx8.p1.8.m3.2.2.2.2.2.2.1.3">𝑡</ci></apply><ci id="Thmproofx8.p1.8.m3.1.1.1.1.1.1.cmml" xref="Thmproofx8.p1.8.m3.1.1.1.1.1.1">𝑒</ci></list></apply></apply></apply><apply id="Thmproofx8.p1.8.m3.4.5.3.2.cmml" xref="Thmproofx8.p1.8.m3.4.5.3.2"><times id="Thmproofx8.p1.8.m3.4.5.3.2.1.cmml" xref="Thmproofx8.p1.8.m3.4.5.3.2.1"></times><ci id="Thmproofx8.p1.8.m3.4.5.3.2.2.cmml" xref="Thmproofx8.p1.8.m3.4.5.3.2.2">𝜎</ci><ci id="Thmproofx8.p1.8.m3.4.4.cmml" xref="Thmproofx8.p1.8.m3.4.4">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx8.p1.8.m3.4c">\sigma^{E}(e)=\sum_{p\in P_{st,e}}\sigma(p)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx8.p1.8.m3.4d">italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e ) = ∑ start_POSTSUBSCRIPT italic_p ∈ italic_P start_POSTSUBSCRIPT italic_s italic_t , italic_e end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( italic_p )</annotation></semantics></math> as a sum of disjoint events. So</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx27"> <tbody id="Pt0.A3.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{\mathbb{E}}_{a\sim\sigma}\left(\left|\bigcup a% \right|\right)" class="ltx_Math" display="inline" id="Pt0.A3.Ex36.m1.2"><semantics id="Pt0.A3.Ex36.m1.2a"><mrow id="Pt0.A3.Ex36.m1.2.2.2" xref="Pt0.A3.Ex36.m1.2.2.3.cmml"><munder id="Pt0.A3.Ex36.m1.1.1.1.1" xref="Pt0.A3.Ex36.m1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex36.m1.1.1.1.1.2" xref="Pt0.A3.Ex36.m1.1.1.1.1.2.cmml">𝔼</mo><mrow id="Pt0.A3.Ex36.m1.1.1.1.1.3" xref="Pt0.A3.Ex36.m1.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex36.m1.1.1.1.1.3.2" xref="Pt0.A3.Ex36.m1.1.1.1.1.3.2.cmml">a</mi><mo id="Pt0.A3.Ex36.m1.1.1.1.1.3.1" xref="Pt0.A3.Ex36.m1.1.1.1.1.3.1.cmml">∼</mo><mi id="Pt0.A3.Ex36.m1.1.1.1.1.3.3" xref="Pt0.A3.Ex36.m1.1.1.1.1.3.3.cmml">σ</mi></mrow></munder><mrow id="Pt0.A3.Ex36.m1.2.2.2.2" xref="Pt0.A3.Ex36.m1.2.2.3.cmml"><mo id="Pt0.A3.Ex36.m1.2.2.2.2.2" lspace="0em" xref="Pt0.A3.Ex36.m1.2.2.3.cmml">(</mo><mrow id="Pt0.A3.Ex36.m1.2.2.2.2.1.1" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.2.cmml"><mo id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.2" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.2.1.cmml">|</mo><mstyle displaystyle="true" id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.cmml"><mrow id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1a" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.cmml"><mo id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.1" movablelimits="false" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.1.cmml">⋃</mo><mi id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.2" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.2.cmml">a</mi></mrow></mstyle><mo id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.3" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A3.Ex36.m1.2.2.2.2.3" xref="Pt0.A3.Ex36.m1.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex36.m1.2b"><apply id="Pt0.A3.Ex36.m1.2.2.3.cmml" xref="Pt0.A3.Ex36.m1.2.2.2"><apply id="Pt0.A3.Ex36.m1.1.1.1.1.cmml" xref="Pt0.A3.Ex36.m1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex36.m1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex36.m1.1.1.1.1">subscript</csymbol><ci id="Pt0.A3.Ex36.m1.1.1.1.1.2.cmml" xref="Pt0.A3.Ex36.m1.1.1.1.1.2">𝔼</ci><apply id="Pt0.A3.Ex36.m1.1.1.1.1.3.cmml" xref="Pt0.A3.Ex36.m1.1.1.1.1.3"><csymbol cd="latexml" id="Pt0.A3.Ex36.m1.1.1.1.1.3.1.cmml" xref="Pt0.A3.Ex36.m1.1.1.1.1.3.1">similar-to</csymbol><ci id="Pt0.A3.Ex36.m1.1.1.1.1.3.2.cmml" xref="Pt0.A3.Ex36.m1.1.1.1.1.3.2">𝑎</ci><ci id="Pt0.A3.Ex36.m1.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex36.m1.1.1.1.1.3.3">𝜎</ci></apply></apply><apply id="Pt0.A3.Ex36.m1.2.2.2.2.1.2.cmml" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1"><abs id="Pt0.A3.Ex36.m1.2.2.2.2.1.2.1.cmml" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.2"></abs><apply id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.cmml" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1"><union id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.1.cmml" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.1"></union><ci id="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.2.cmml" xref="Pt0.A3.Ex36.m1.2.2.2.2.1.1.1.2">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex36.m1.2c">\displaystyle\operatorname*{\mathbb{E}}_{a\sim\sigma}\left(\left|\bigcup a% \right|\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex36.m1.2d">blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( | ⋃ italic_a | )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{e\in E}\sigma^{E}(e)=\sum_{e\in E}\sum_{p\in P_{st,e}}% \sigma(p)=\sum_{e\in E}\sum_{p\in P_{st}}1_{e\in p}\sigma(p)" class="ltx_Math" display="inline" id="Pt0.A3.Ex36.m2.5"><semantics id="Pt0.A3.Ex36.m2.5a"><mrow id="Pt0.A3.Ex36.m2.5.6" xref="Pt0.A3.Ex36.m2.5.6.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.2" xref="Pt0.A3.Ex36.m2.5.6.2.cmml"></mi><mo id="Pt0.A3.Ex36.m2.5.6.3" xref="Pt0.A3.Ex36.m2.5.6.3.cmml">=</mo><mrow id="Pt0.A3.Ex36.m2.5.6.4" xref="Pt0.A3.Ex36.m2.5.6.4.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex36.m2.5.6.4.1" xref="Pt0.A3.Ex36.m2.5.6.4.1.cmml"><munder id="Pt0.A3.Ex36.m2.5.6.4.1a" xref="Pt0.A3.Ex36.m2.5.6.4.1.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.4.1.2" movablelimits="false" xref="Pt0.A3.Ex36.m2.5.6.4.1.2.cmml">∑</mo><mrow id="Pt0.A3.Ex36.m2.5.6.4.1.3" xref="Pt0.A3.Ex36.m2.5.6.4.1.3.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.4.1.3.2" xref="Pt0.A3.Ex36.m2.5.6.4.1.3.2.cmml">e</mi><mo id="Pt0.A3.Ex36.m2.5.6.4.1.3.1" xref="Pt0.A3.Ex36.m2.5.6.4.1.3.1.cmml">∈</mo><mi id="Pt0.A3.Ex36.m2.5.6.4.1.3.3" xref="Pt0.A3.Ex36.m2.5.6.4.1.3.3.cmml">E</mi></mrow></munder></mstyle><mrow id="Pt0.A3.Ex36.m2.5.6.4.2" xref="Pt0.A3.Ex36.m2.5.6.4.2.cmml"><msup id="Pt0.A3.Ex36.m2.5.6.4.2.2" xref="Pt0.A3.Ex36.m2.5.6.4.2.2.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.4.2.2.2" xref="Pt0.A3.Ex36.m2.5.6.4.2.2.2.cmml">σ</mi><mi id="Pt0.A3.Ex36.m2.5.6.4.2.2.3" xref="Pt0.A3.Ex36.m2.5.6.4.2.2.3.cmml">E</mi></msup><mo id="Pt0.A3.Ex36.m2.5.6.4.2.1" xref="Pt0.A3.Ex36.m2.5.6.4.2.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex36.m2.5.6.4.2.3.2" xref="Pt0.A3.Ex36.m2.5.6.4.2.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.4.2.3.2.1" stretchy="false" xref="Pt0.A3.Ex36.m2.5.6.4.2.cmml">(</mo><mi id="Pt0.A3.Ex36.m2.3.3" xref="Pt0.A3.Ex36.m2.3.3.cmml">e</mi><mo id="Pt0.A3.Ex36.m2.5.6.4.2.3.2.2" stretchy="false" xref="Pt0.A3.Ex36.m2.5.6.4.2.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A3.Ex36.m2.5.6.5" xref="Pt0.A3.Ex36.m2.5.6.5.cmml">=</mo><mrow id="Pt0.A3.Ex36.m2.5.6.6" xref="Pt0.A3.Ex36.m2.5.6.6.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex36.m2.5.6.6.1" xref="Pt0.A3.Ex36.m2.5.6.6.1.cmml"><munder id="Pt0.A3.Ex36.m2.5.6.6.1a" xref="Pt0.A3.Ex36.m2.5.6.6.1.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.6.1.2" movablelimits="false" xref="Pt0.A3.Ex36.m2.5.6.6.1.2.cmml">∑</mo><mrow id="Pt0.A3.Ex36.m2.5.6.6.1.3" xref="Pt0.A3.Ex36.m2.5.6.6.1.3.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.6.1.3.2" xref="Pt0.A3.Ex36.m2.5.6.6.1.3.2.cmml">e</mi><mo id="Pt0.A3.Ex36.m2.5.6.6.1.3.1" xref="Pt0.A3.Ex36.m2.5.6.6.1.3.1.cmml">∈</mo><mi id="Pt0.A3.Ex36.m2.5.6.6.1.3.3" xref="Pt0.A3.Ex36.m2.5.6.6.1.3.3.cmml">E</mi></mrow></munder></mstyle><mrow id="Pt0.A3.Ex36.m2.5.6.6.2" xref="Pt0.A3.Ex36.m2.5.6.6.2.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex36.m2.5.6.6.2.1" xref="Pt0.A3.Ex36.m2.5.6.6.2.1.cmml"><munder id="Pt0.A3.Ex36.m2.5.6.6.2.1a" xref="Pt0.A3.Ex36.m2.5.6.6.2.1.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.6.2.1.2" movablelimits="false" xref="Pt0.A3.Ex36.m2.5.6.6.2.1.2.cmml">∑</mo><mrow id="Pt0.A3.Ex36.m2.2.2.2" xref="Pt0.A3.Ex36.m2.2.2.2.cmml"><mi id="Pt0.A3.Ex36.m2.2.2.2.4" xref="Pt0.A3.Ex36.m2.2.2.2.4.cmml">p</mi><mo id="Pt0.A3.Ex36.m2.2.2.2.3" xref="Pt0.A3.Ex36.m2.2.2.2.3.cmml">∈</mo><msub id="Pt0.A3.Ex36.m2.2.2.2.5" xref="Pt0.A3.Ex36.m2.2.2.2.5.cmml"><mi id="Pt0.A3.Ex36.m2.2.2.2.5.2" xref="Pt0.A3.Ex36.m2.2.2.2.5.2.cmml">P</mi><mrow id="Pt0.A3.Ex36.m2.2.2.2.2.2.2" xref="Pt0.A3.Ex36.m2.2.2.2.2.2.3.cmml"><mrow id="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1" xref="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1.cmml"><mi id="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1.2" xref="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1.2.cmml">s</mi><mo id="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1.1" xref="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1.3" xref="Pt0.A3.Ex36.m2.2.2.2.2.2.2.1.3.cmml">t</mi></mrow><mo id="Pt0.A3.Ex36.m2.2.2.2.2.2.2.2" xref="Pt0.A3.Ex36.m2.2.2.2.2.2.3.cmml">,</mo><mi id="Pt0.A3.Ex36.m2.1.1.1.1.1.1" xref="Pt0.A3.Ex36.m2.1.1.1.1.1.1.cmml">e</mi></mrow></msub></mrow></munder></mstyle><mrow id="Pt0.A3.Ex36.m2.5.6.6.2.2" xref="Pt0.A3.Ex36.m2.5.6.6.2.2.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.6.2.2.2" xref="Pt0.A3.Ex36.m2.5.6.6.2.2.2.cmml">σ</mi><mo id="Pt0.A3.Ex36.m2.5.6.6.2.2.1" xref="Pt0.A3.Ex36.m2.5.6.6.2.2.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex36.m2.5.6.6.2.2.3.2" xref="Pt0.A3.Ex36.m2.5.6.6.2.2.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.6.2.2.3.2.1" stretchy="false" xref="Pt0.A3.Ex36.m2.5.6.6.2.2.cmml">(</mo><mi id="Pt0.A3.Ex36.m2.4.4" xref="Pt0.A3.Ex36.m2.4.4.cmml">p</mi><mo id="Pt0.A3.Ex36.m2.5.6.6.2.2.3.2.2" stretchy="false" xref="Pt0.A3.Ex36.m2.5.6.6.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="Pt0.A3.Ex36.m2.5.6.7" xref="Pt0.A3.Ex36.m2.5.6.7.cmml">=</mo><mrow id="Pt0.A3.Ex36.m2.5.6.8" xref="Pt0.A3.Ex36.m2.5.6.8.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex36.m2.5.6.8.1" xref="Pt0.A3.Ex36.m2.5.6.8.1.cmml"><munder id="Pt0.A3.Ex36.m2.5.6.8.1a" xref="Pt0.A3.Ex36.m2.5.6.8.1.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.8.1.2" movablelimits="false" xref="Pt0.A3.Ex36.m2.5.6.8.1.2.cmml">∑</mo><mrow id="Pt0.A3.Ex36.m2.5.6.8.1.3" xref="Pt0.A3.Ex36.m2.5.6.8.1.3.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.8.1.3.2" xref="Pt0.A3.Ex36.m2.5.6.8.1.3.2.cmml">e</mi><mo id="Pt0.A3.Ex36.m2.5.6.8.1.3.1" xref="Pt0.A3.Ex36.m2.5.6.8.1.3.1.cmml">∈</mo><mi id="Pt0.A3.Ex36.m2.5.6.8.1.3.3" xref="Pt0.A3.Ex36.m2.5.6.8.1.3.3.cmml">E</mi></mrow></munder></mstyle><mrow id="Pt0.A3.Ex36.m2.5.6.8.2" xref="Pt0.A3.Ex36.m2.5.6.8.2.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex36.m2.5.6.8.2.1" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.cmml"><munder id="Pt0.A3.Ex36.m2.5.6.8.2.1a" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.8.2.1.2" movablelimits="false" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.2.cmml">∑</mo><mrow id="Pt0.A3.Ex36.m2.5.6.8.2.1.3" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.3.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.8.2.1.3.2" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.3.2.cmml">p</mi><mo id="Pt0.A3.Ex36.m2.5.6.8.2.1.3.1" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.3.1.cmml">∈</mo><msub id="Pt0.A3.Ex36.m2.5.6.8.2.1.3.3" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.3.3.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.8.2.1.3.3.2" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.3.3.2.cmml">P</mi><mrow id="Pt0.A3.Ex36.m2.5.6.8.2.1.3.3.3" xref="Pt0.A3.Ex36.m2.5.6.8.2.1.3.3.3.cmml"><mi id="Pt0.A3.Ex36.m2.5.6.8.2.1.3.3.3.2" 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xref="Pt0.A3.Ex36.m2.5.6.8.2.2.3.cmml">σ</mi><mo id="Pt0.A3.Ex36.m2.5.6.8.2.2.1a" xref="Pt0.A3.Ex36.m2.5.6.8.2.2.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex36.m2.5.6.8.2.2.4.2" xref="Pt0.A3.Ex36.m2.5.6.8.2.2.cmml"><mo id="Pt0.A3.Ex36.m2.5.6.8.2.2.4.2.1" stretchy="false" xref="Pt0.A3.Ex36.m2.5.6.8.2.2.cmml">(</mo><mi id="Pt0.A3.Ex36.m2.5.5" xref="Pt0.A3.Ex36.m2.5.5.cmml">p</mi><mo id="Pt0.A3.Ex36.m2.5.6.8.2.2.4.2.2" stretchy="false" xref="Pt0.A3.Ex36.m2.5.6.8.2.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex36.m2.5b"><apply id="Pt0.A3.Ex36.m2.5.6.cmml" xref="Pt0.A3.Ex36.m2.5.6"><and id="Pt0.A3.Ex36.m2.5.6a.cmml" xref="Pt0.A3.Ex36.m2.5.6"></and><apply id="Pt0.A3.Ex36.m2.5.6b.cmml" xref="Pt0.A3.Ex36.m2.5.6"><eq id="Pt0.A3.Ex36.m2.5.6.3.cmml" xref="Pt0.A3.Ex36.m2.5.6.3"></eq><csymbol cd="latexml" id="Pt0.A3.Ex36.m2.5.6.2.cmml" xref="Pt0.A3.Ex36.m2.5.6.2">absent</csymbol><apply id="Pt0.A3.Ex36.m2.5.6.4.cmml" xref="Pt0.A3.Ex36.m2.5.6.4"><apply 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id="Pt0.A3.Ex37.m1.3.4.4.2.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex37.m1.3.4.4.2.1.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.1">subscript</csymbol><sum id="Pt0.A3.Ex37.m1.3.4.4.2.1.2.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.1.2"></sum><apply id="Pt0.A3.Ex37.m1.3.4.4.2.1.3.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.1.3"><in id="Pt0.A3.Ex37.m1.3.4.4.2.1.3.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.1.3.1"></in><ci id="Pt0.A3.Ex37.m1.3.4.4.2.1.3.2.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.1.3.2">𝑒</ci><ci id="Pt0.A3.Ex37.m1.3.4.4.2.1.3.3.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.1.3.3">𝑝</ci></apply></apply><apply id="Pt0.A3.Ex37.m1.3.4.4.2.2.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.2"><times id="Pt0.A3.Ex37.m1.3.4.4.2.2.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.2.1"></times><ci id="Pt0.A3.Ex37.m1.3.4.4.2.2.2.cmml" xref="Pt0.A3.Ex37.m1.3.4.4.2.2.2">𝜎</ci><ci id="Pt0.A3.Ex37.m1.1.1.cmml" xref="Pt0.A3.Ex37.m1.1.1">𝑝</ci></apply></apply></apply></apply><apply id="Pt0.A3.Ex37.m1.3.4c.cmml" xref="Pt0.A3.Ex37.m1.3.4"><eq id="Pt0.A3.Ex37.m1.3.4.5.cmml" xref="Pt0.A3.Ex37.m1.3.4.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A3.Ex37.m1.3.4.4.cmml" id="Pt0.A3.Ex37.m1.3.4d.cmml" xref="Pt0.A3.Ex37.m1.3.4"></share><apply id="Pt0.A3.Ex37.m1.3.4.6.cmml" xref="Pt0.A3.Ex37.m1.3.4.6"><apply id="Pt0.A3.Ex37.m1.3.4.6.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex37.m1.3.4.6.1.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.1">subscript</csymbol><sum id="Pt0.A3.Ex37.m1.3.4.6.1.2.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.1.2"></sum><ci id="Pt0.A3.Ex37.m1.3.4.6.1.3.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.1.3">𝑝</ci></apply><apply id="Pt0.A3.Ex37.m1.3.4.6.2.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.2"><times id="Pt0.A3.Ex37.m1.3.4.6.2.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.2.1"></times><ci id="Pt0.A3.Ex37.m1.3.4.6.2.2.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.2.2">𝜎</ci><ci id="Pt0.A3.Ex37.m1.2.2.cmml" xref="Pt0.A3.Ex37.m1.2.2">𝑝</ci><apply id="Pt0.A3.Ex37.m1.3.4.6.2.4.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.2.4.2"><abs id="Pt0.A3.Ex37.m1.3.4.6.2.4.1.1.cmml" xref="Pt0.A3.Ex37.m1.3.4.6.2.4.2.1"></abs><ci id="Pt0.A3.Ex37.m1.3.3.cmml" xref="Pt0.A3.Ex37.m1.3.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex37.m1.3c">\displaystyle=\sum_{p}\sum_{e\in p}\sigma(p)=\sum_{p}\sigma(p)|p|</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex37.m1.3d">= ∑ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_e ∈ italic_p end_POSTSUBSCRIPT italic_σ ( italic_p ) = ∑ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_σ ( italic_p ) | italic_p |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx8.p1.9">∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx9"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.1</span></a>)</h6> <div class="ltx_para" id="Thmproofx9.p1"> <p class="ltx_p" id="Thmproofx9.p1.5">Let <math alttext="f\in MF" class="ltx_Math" display="inline" id="Thmproofx9.p1.1.m1.1"><semantics id="Thmproofx9.p1.1.m1.1a"><mrow id="Thmproofx9.p1.1.m1.1.1" xref="Thmproofx9.p1.1.m1.1.1.cmml"><mi id="Thmproofx9.p1.1.m1.1.1.2" xref="Thmproofx9.p1.1.m1.1.1.2.cmml">f</mi><mo id="Thmproofx9.p1.1.m1.1.1.1" xref="Thmproofx9.p1.1.m1.1.1.1.cmml">∈</mo><mrow id="Thmproofx9.p1.1.m1.1.1.3" xref="Thmproofx9.p1.1.m1.1.1.3.cmml"><mi id="Thmproofx9.p1.1.m1.1.1.3.2" xref="Thmproofx9.p1.1.m1.1.1.3.2.cmml">M</mi><mo id="Thmproofx9.p1.1.m1.1.1.3.1" xref="Thmproofx9.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.1.m1.1.1.3.3" xref="Thmproofx9.p1.1.m1.1.1.3.3.cmml">F</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx9.p1.1.m1.1b"><apply id="Thmproofx9.p1.1.m1.1.1.cmml" xref="Thmproofx9.p1.1.m1.1.1"><in id="Thmproofx9.p1.1.m1.1.1.1.cmml" xref="Thmproofx9.p1.1.m1.1.1.1"></in><ci id="Thmproofx9.p1.1.m1.1.1.2.cmml" xref="Thmproofx9.p1.1.m1.1.1.2">𝑓</ci><apply id="Thmproofx9.p1.1.m1.1.1.3.cmml" xref="Thmproofx9.p1.1.m1.1.1.3"><times id="Thmproofx9.p1.1.m1.1.1.3.1.cmml" xref="Thmproofx9.p1.1.m1.1.1.3.1"></times><ci id="Thmproofx9.p1.1.m1.1.1.3.2.cmml" xref="Thmproofx9.p1.1.m1.1.1.3.2">𝑀</ci><ci id="Thmproofx9.p1.1.m1.1.1.3.3.cmml" xref="Thmproofx9.p1.1.m1.1.1.3.3">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx9.p1.1.m1.1c">f\in MF</annotation><annotation encoding="application/x-llamapun" id="Thmproofx9.p1.1.m1.1d">italic_f ∈ italic_M italic_F</annotation></semantics></math> with <math alttext="|f|=m_{c}" class="ltx_Math" display="inline" id="Thmproofx9.p1.2.m2.1"><semantics id="Thmproofx9.p1.2.m2.1a"><mrow id="Thmproofx9.p1.2.m2.1.2" xref="Thmproofx9.p1.2.m2.1.2.cmml"><mrow id="Thmproofx9.p1.2.m2.1.2.2.2" xref="Thmproofx9.p1.2.m2.1.2.2.1.cmml"><mo id="Thmproofx9.p1.2.m2.1.2.2.2.1" stretchy="false" xref="Thmproofx9.p1.2.m2.1.2.2.1.1.cmml">|</mo><mi id="Thmproofx9.p1.2.m2.1.1" xref="Thmproofx9.p1.2.m2.1.1.cmml">f</mi><mo id="Thmproofx9.p1.2.m2.1.2.2.2.2" stretchy="false" xref="Thmproofx9.p1.2.m2.1.2.2.1.1.cmml">|</mo></mrow><mo id="Thmproofx9.p1.2.m2.1.2.1" xref="Thmproofx9.p1.2.m2.1.2.1.cmml">=</mo><msub id="Thmproofx9.p1.2.m2.1.2.3" xref="Thmproofx9.p1.2.m2.1.2.3.cmml"><mi id="Thmproofx9.p1.2.m2.1.2.3.2" xref="Thmproofx9.p1.2.m2.1.2.3.2.cmml">m</mi><mi id="Thmproofx9.p1.2.m2.1.2.3.3" xref="Thmproofx9.p1.2.m2.1.2.3.3.cmml">c</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx9.p1.2.m2.1b"><apply id="Thmproofx9.p1.2.m2.1.2.cmml" xref="Thmproofx9.p1.2.m2.1.2"><eq id="Thmproofx9.p1.2.m2.1.2.1.cmml" xref="Thmproofx9.p1.2.m2.1.2.1"></eq><apply id="Thmproofx9.p1.2.m2.1.2.2.1.cmml" xref="Thmproofx9.p1.2.m2.1.2.2.2"><abs id="Thmproofx9.p1.2.m2.1.2.2.1.1.cmml" xref="Thmproofx9.p1.2.m2.1.2.2.2.1"></abs><ci id="Thmproofx9.p1.2.m2.1.1.cmml" xref="Thmproofx9.p1.2.m2.1.1">𝑓</ci></apply><apply id="Thmproofx9.p1.2.m2.1.2.3.cmml" xref="Thmproofx9.p1.2.m2.1.2.3"><csymbol cd="ambiguous" id="Thmproofx9.p1.2.m2.1.2.3.1.cmml" xref="Thmproofx9.p1.2.m2.1.2.3">subscript</csymbol><ci id="Thmproofx9.p1.2.m2.1.2.3.2.cmml" xref="Thmproofx9.p1.2.m2.1.2.3.2">𝑚</ci><ci id="Thmproofx9.p1.2.m2.1.2.3.3.cmml" xref="Thmproofx9.p1.2.m2.1.2.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx9.p1.2.m2.1c">|f|=m_{c}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx9.p1.2.m2.1d">| italic_f | = italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>. Then by definition, <math alttext="F_{n}(U_{P_{f}}^{n},\alpha)\leq n\left|p_{max}\right|+\frac{\mathcal{E}_{m_{c}% ,n}}{m_{c}}k\alpha" class="ltx_Math" display="inline" id="Thmproofx9.p1.3.m3.5"><semantics id="Thmproofx9.p1.3.m3.5a"><mrow id="Thmproofx9.p1.3.m3.5.5" xref="Thmproofx9.p1.3.m3.5.5.cmml"><mrow id="Thmproofx9.p1.3.m3.4.4.1" xref="Thmproofx9.p1.3.m3.4.4.1.cmml"><msub id="Thmproofx9.p1.3.m3.4.4.1.3" xref="Thmproofx9.p1.3.m3.4.4.1.3.cmml"><mi id="Thmproofx9.p1.3.m3.4.4.1.3.2" xref="Thmproofx9.p1.3.m3.4.4.1.3.2.cmml">F</mi><mi id="Thmproofx9.p1.3.m3.4.4.1.3.3" xref="Thmproofx9.p1.3.m3.4.4.1.3.3.cmml">n</mi></msub><mo id="Thmproofx9.p1.3.m3.4.4.1.2" xref="Thmproofx9.p1.3.m3.4.4.1.2.cmml">⁢</mo><mrow id="Thmproofx9.p1.3.m3.4.4.1.1.1" xref="Thmproofx9.p1.3.m3.4.4.1.1.2.cmml"><mo id="Thmproofx9.p1.3.m3.4.4.1.1.1.2" stretchy="false" xref="Thmproofx9.p1.3.m3.4.4.1.1.2.cmml">(</mo><msubsup id="Thmproofx9.p1.3.m3.4.4.1.1.1.1" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.cmml"><mi id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.2" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.cmml"><mi id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.2" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.2.cmml">P</mi><mi id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.3" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.3.cmml">f</mi></msub><mi id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.3" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx9.p1.3.m3.4.4.1.1.1.3" xref="Thmproofx9.p1.3.m3.4.4.1.1.2.cmml">,</mo><mi id="Thmproofx9.p1.3.m3.3.3" xref="Thmproofx9.p1.3.m3.3.3.cmml">α</mi><mo id="Thmproofx9.p1.3.m3.4.4.1.1.1.4" stretchy="false" xref="Thmproofx9.p1.3.m3.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx9.p1.3.m3.5.5.3" xref="Thmproofx9.p1.3.m3.5.5.3.cmml">≤</mo><mrow id="Thmproofx9.p1.3.m3.5.5.2" xref="Thmproofx9.p1.3.m3.5.5.2.cmml"><mrow id="Thmproofx9.p1.3.m3.5.5.2.1" xref="Thmproofx9.p1.3.m3.5.5.2.1.cmml"><mi id="Thmproofx9.p1.3.m3.5.5.2.1.3" xref="Thmproofx9.p1.3.m3.5.5.2.1.3.cmml">n</mi><mo id="Thmproofx9.p1.3.m3.5.5.2.1.2" xref="Thmproofx9.p1.3.m3.5.5.2.1.2.cmml">⁢</mo><mrow id="Thmproofx9.p1.3.m3.5.5.2.1.1.1" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.2.cmml"><mo id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.2" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.2.1.cmml">|</mo><msub id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.cmml"><mi id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.2" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.cmml"><mi id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.2" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.1" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.3" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.3.cmml">a</mi><mo id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.1a" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.4" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.3" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="Thmproofx9.p1.3.m3.5.5.2.2" xref="Thmproofx9.p1.3.m3.5.5.2.2.cmml">+</mo><mrow id="Thmproofx9.p1.3.m3.5.5.2.3" xref="Thmproofx9.p1.3.m3.5.5.2.3.cmml"><mfrac id="Thmproofx9.p1.3.m3.2.2" xref="Thmproofx9.p1.3.m3.2.2.cmml"><msub id="Thmproofx9.p1.3.m3.2.2.2" xref="Thmproofx9.p1.3.m3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx9.p1.3.m3.2.2.2.4" xref="Thmproofx9.p1.3.m3.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx9.p1.3.m3.2.2.2.2.2.2" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.3.cmml"><msub id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.cmml"><mi id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.2" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.2.cmml">m</mi><mi id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.3" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.3.cmml">c</mi></msub><mo id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.2" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx9.p1.3.m3.1.1.1.1.1.1" xref="Thmproofx9.p1.3.m3.1.1.1.1.1.1.cmml">n</mi></mrow></msub><msub id="Thmproofx9.p1.3.m3.2.2.4" xref="Thmproofx9.p1.3.m3.2.2.4.cmml"><mi id="Thmproofx9.p1.3.m3.2.2.4.2" xref="Thmproofx9.p1.3.m3.2.2.4.2.cmml">m</mi><mi id="Thmproofx9.p1.3.m3.2.2.4.3" xref="Thmproofx9.p1.3.m3.2.2.4.3.cmml">c</mi></msub></mfrac><mo id="Thmproofx9.p1.3.m3.5.5.2.3.1" xref="Thmproofx9.p1.3.m3.5.5.2.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.3.m3.5.5.2.3.2" xref="Thmproofx9.p1.3.m3.5.5.2.3.2.cmml">k</mi><mo id="Thmproofx9.p1.3.m3.5.5.2.3.1a" xref="Thmproofx9.p1.3.m3.5.5.2.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.3.m3.5.5.2.3.3" xref="Thmproofx9.p1.3.m3.5.5.2.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml 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xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1">superscript</csymbol><apply id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.1.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1">subscript</csymbol><ci id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.2.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.2">𝑈</ci><apply id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.1.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.2.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.2">𝑃</ci><ci id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.3.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.2.3.3">𝑓</ci></apply></apply><ci id="Thmproofx9.p1.3.m3.4.4.1.1.1.1.3.cmml" xref="Thmproofx9.p1.3.m3.4.4.1.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx9.p1.3.m3.3.3.cmml" xref="Thmproofx9.p1.3.m3.3.3">𝛼</ci></interval></apply><apply id="Thmproofx9.p1.3.m3.5.5.2.cmml" xref="Thmproofx9.p1.3.m3.5.5.2"><plus id="Thmproofx9.p1.3.m3.5.5.2.2.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.2"></plus><apply id="Thmproofx9.p1.3.m3.5.5.2.1.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1"><times id="Thmproofx9.p1.3.m3.5.5.2.1.2.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.2"></times><ci id="Thmproofx9.p1.3.m3.5.5.2.1.3.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.3">𝑛</ci><apply id="Thmproofx9.p1.3.m3.5.5.2.1.1.2.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1"><abs id="Thmproofx9.p1.3.m3.5.5.2.1.1.2.1.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.2"></abs><apply id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.1.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1">subscript</csymbol><ci id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.2.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.2">𝑝</ci><apply id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3"><times id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.1.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.1"></times><ci id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.2.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.3.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.3">𝑎</ci><ci id="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.4.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.1.1.1.1.3.4">𝑥</ci></apply></apply></apply></apply><apply id="Thmproofx9.p1.3.m3.5.5.2.3.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.3"><times id="Thmproofx9.p1.3.m3.5.5.2.3.1.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.3.1"></times><apply id="Thmproofx9.p1.3.m3.2.2.cmml" xref="Thmproofx9.p1.3.m3.2.2"><divide id="Thmproofx9.p1.3.m3.2.2.3.cmml" xref="Thmproofx9.p1.3.m3.2.2"></divide><apply id="Thmproofx9.p1.3.m3.2.2.2.cmml" xref="Thmproofx9.p1.3.m3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx9.p1.3.m3.2.2.2.3.cmml" xref="Thmproofx9.p1.3.m3.2.2.2">subscript</csymbol><ci id="Thmproofx9.p1.3.m3.2.2.2.4.cmml" xref="Thmproofx9.p1.3.m3.2.2.2.4">ℰ</ci><list id="Thmproofx9.p1.3.m3.2.2.2.2.2.3.cmml" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2"><apply id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.cmml" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.1.cmml" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1">subscript</csymbol><ci id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.2.cmml" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.2">𝑚</ci><ci id="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.3.cmml" xref="Thmproofx9.p1.3.m3.2.2.2.2.2.2.1.3">𝑐</ci></apply><ci id="Thmproofx9.p1.3.m3.1.1.1.1.1.1.cmml" xref="Thmproofx9.p1.3.m3.1.1.1.1.1.1">𝑛</ci></list></apply><apply id="Thmproofx9.p1.3.m3.2.2.4.cmml" xref="Thmproofx9.p1.3.m3.2.2.4"><csymbol cd="ambiguous" id="Thmproofx9.p1.3.m3.2.2.4.1.cmml" xref="Thmproofx9.p1.3.m3.2.2.4">subscript</csymbol><ci id="Thmproofx9.p1.3.m3.2.2.4.2.cmml" xref="Thmproofx9.p1.3.m3.2.2.4.2">𝑚</ci><ci id="Thmproofx9.p1.3.m3.2.2.4.3.cmml" xref="Thmproofx9.p1.3.m3.2.2.4.3">𝑐</ci></apply></apply><ci id="Thmproofx9.p1.3.m3.5.5.2.3.2.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.3.2">𝑘</ci><ci id="Thmproofx9.p1.3.m3.5.5.2.3.3.cmml" xref="Thmproofx9.p1.3.m3.5.5.2.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx9.p1.3.m3.5c">F_{n}(U_{P_{f}}^{n},\alpha)\leq n\left|p_{max}\right|+\frac{\mathcal{E}_{m_{c}% ,n}}{m_{c}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx9.p1.3.m3.5d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) ≤ italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. On the other hand, <math alttext="F_{n}(\sigma,\alpha)\geq\left|p_{min}\right|+k\alpha\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx9.p1.4.m4.4"><semantics id="Thmproofx9.p1.4.m4.4a"><mrow id="Thmproofx9.p1.4.m4.4.4" xref="Thmproofx9.p1.4.m4.4.4.cmml"><mrow id="Thmproofx9.p1.4.m4.4.4.3" xref="Thmproofx9.p1.4.m4.4.4.3.cmml"><msub id="Thmproofx9.p1.4.m4.4.4.3.2" xref="Thmproofx9.p1.4.m4.4.4.3.2.cmml"><mi id="Thmproofx9.p1.4.m4.4.4.3.2.2" xref="Thmproofx9.p1.4.m4.4.4.3.2.2.cmml">F</mi><mi id="Thmproofx9.p1.4.m4.4.4.3.2.3" xref="Thmproofx9.p1.4.m4.4.4.3.2.3.cmml">n</mi></msub><mo id="Thmproofx9.p1.4.m4.4.4.3.1" xref="Thmproofx9.p1.4.m4.4.4.3.1.cmml">⁢</mo><mrow id="Thmproofx9.p1.4.m4.4.4.3.3.2" xref="Thmproofx9.p1.4.m4.4.4.3.3.1.cmml"><mo id="Thmproofx9.p1.4.m4.4.4.3.3.2.1" stretchy="false" xref="Thmproofx9.p1.4.m4.4.4.3.3.1.cmml">(</mo><mi id="Thmproofx9.p1.4.m4.1.1" xref="Thmproofx9.p1.4.m4.1.1.cmml">σ</mi><mo id="Thmproofx9.p1.4.m4.4.4.3.3.2.2" xref="Thmproofx9.p1.4.m4.4.4.3.3.1.cmml">,</mo><mi id="Thmproofx9.p1.4.m4.2.2" xref="Thmproofx9.p1.4.m4.2.2.cmml">α</mi><mo id="Thmproofx9.p1.4.m4.4.4.3.3.2.3" stretchy="false" xref="Thmproofx9.p1.4.m4.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx9.p1.4.m4.4.4.2" xref="Thmproofx9.p1.4.m4.4.4.2.cmml">≥</mo><mrow id="Thmproofx9.p1.4.m4.4.4.1" xref="Thmproofx9.p1.4.m4.4.4.1.cmml"><mrow id="Thmproofx9.p1.4.m4.4.4.1.1.1" xref="Thmproofx9.p1.4.m4.4.4.1.1.2.cmml"><mo id="Thmproofx9.p1.4.m4.4.4.1.1.1.2" xref="Thmproofx9.p1.4.m4.4.4.1.1.2.1.cmml">|</mo><msub id="Thmproofx9.p1.4.m4.4.4.1.1.1.1" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.cmml"><mi id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.2" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.cmml"><mi id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.2" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.1" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.3" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.3.cmml">i</mi><mo id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.1a" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.4" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx9.p1.4.m4.4.4.1.1.1.3" xref="Thmproofx9.p1.4.m4.4.4.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx9.p1.4.m4.4.4.1.2" xref="Thmproofx9.p1.4.m4.4.4.1.2.cmml">+</mo><mrow id="Thmproofx9.p1.4.m4.4.4.1.3" xref="Thmproofx9.p1.4.m4.4.4.1.3.cmml"><mi id="Thmproofx9.p1.4.m4.4.4.1.3.2" xref="Thmproofx9.p1.4.m4.4.4.1.3.2.cmml">k</mi><mo id="Thmproofx9.p1.4.m4.4.4.1.3.1" xref="Thmproofx9.p1.4.m4.4.4.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.4.m4.4.4.1.3.3" xref="Thmproofx9.p1.4.m4.4.4.1.3.3.cmml">α</mi><mo id="Thmproofx9.p1.4.m4.4.4.1.3.1a" xref="Thmproofx9.p1.4.m4.4.4.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.4.m4.4.4.1.3.4" xref="Thmproofx9.p1.4.m4.4.4.1.3.4.cmml">β</mi><mo id="Thmproofx9.p1.4.m4.4.4.1.3.1b" xref="Thmproofx9.p1.4.m4.4.4.1.3.1.cmml">⁢</mo><mrow id="Thmproofx9.p1.4.m4.4.4.1.3.5.2" xref="Thmproofx9.p1.4.m4.4.4.1.3.cmml"><mo id="Thmproofx9.p1.4.m4.4.4.1.3.5.2.1" stretchy="false" xref="Thmproofx9.p1.4.m4.4.4.1.3.cmml">(</mo><mi id="Thmproofx9.p1.4.m4.3.3" xref="Thmproofx9.p1.4.m4.3.3.cmml">σ</mi><mo id="Thmproofx9.p1.4.m4.4.4.1.3.5.2.2" stretchy="false" xref="Thmproofx9.p1.4.m4.4.4.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx9.p1.4.m4.4b"><apply id="Thmproofx9.p1.4.m4.4.4.cmml" xref="Thmproofx9.p1.4.m4.4.4"><geq id="Thmproofx9.p1.4.m4.4.4.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.2"></geq><apply id="Thmproofx9.p1.4.m4.4.4.3.cmml" xref="Thmproofx9.p1.4.m4.4.4.3"><times id="Thmproofx9.p1.4.m4.4.4.3.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.3.1"></times><apply id="Thmproofx9.p1.4.m4.4.4.3.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.3.2"><csymbol cd="ambiguous" id="Thmproofx9.p1.4.m4.4.4.3.2.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.3.2">subscript</csymbol><ci id="Thmproofx9.p1.4.m4.4.4.3.2.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.3.2.2">𝐹</ci><ci id="Thmproofx9.p1.4.m4.4.4.3.2.3.cmml" xref="Thmproofx9.p1.4.m4.4.4.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx9.p1.4.m4.4.4.3.3.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.3.3.2"><ci id="Thmproofx9.p1.4.m4.1.1.cmml" xref="Thmproofx9.p1.4.m4.1.1">𝜎</ci><ci id="Thmproofx9.p1.4.m4.2.2.cmml" xref="Thmproofx9.p1.4.m4.2.2">𝛼</ci></interval></apply><apply id="Thmproofx9.p1.4.m4.4.4.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.1"><plus id="Thmproofx9.p1.4.m4.4.4.1.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.2"></plus><apply id="Thmproofx9.p1.4.m4.4.4.1.1.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1"><abs id="Thmproofx9.p1.4.m4.4.4.1.1.2.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.2"></abs><apply id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1">subscript</csymbol><ci id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.2">𝑝</ci><apply id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3"><times id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.1"></times><ci id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.3.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.3">𝑖</ci><ci id="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.4.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Thmproofx9.p1.4.m4.4.4.1.3.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.3"><times id="Thmproofx9.p1.4.m4.4.4.1.3.1.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.3.1"></times><ci id="Thmproofx9.p1.4.m4.4.4.1.3.2.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.3.2">𝑘</ci><ci id="Thmproofx9.p1.4.m4.4.4.1.3.3.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.3.3">𝛼</ci><ci id="Thmproofx9.p1.4.m4.4.4.1.3.4.cmml" xref="Thmproofx9.p1.4.m4.4.4.1.3.4">𝛽</ci><ci id="Thmproofx9.p1.4.m4.3.3.cmml" xref="Thmproofx9.p1.4.m4.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx9.p1.4.m4.4c">F_{n}(\sigma,\alpha)\geq\left|p_{min}\right|+k\alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx9.p1.4.m4.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≥ | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma4" title="Lemma 4 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4</span></a> we get the lower bound <math alttext="\left|p_{min}\right|+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha" class="ltx_Math" display="inline" id="Thmproofx9.p1.5.m5.3"><semantics id="Thmproofx9.p1.5.m5.3a"><mrow id="Thmproofx9.p1.5.m5.3.3" xref="Thmproofx9.p1.5.m5.3.3.cmml"><mrow id="Thmproofx9.p1.5.m5.3.3.1.1" xref="Thmproofx9.p1.5.m5.3.3.1.2.cmml"><mo id="Thmproofx9.p1.5.m5.3.3.1.1.2" xref="Thmproofx9.p1.5.m5.3.3.1.2.1.cmml">|</mo><msub id="Thmproofx9.p1.5.m5.3.3.1.1.1" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.cmml"><mi id="Thmproofx9.p1.5.m5.3.3.1.1.1.2" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.2.cmml">p</mi><mrow id="Thmproofx9.p1.5.m5.3.3.1.1.1.3" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.cmml"><mi id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.2" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.1" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.3" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.3.cmml">i</mi><mo id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.1a" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.4" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx9.p1.5.m5.3.3.1.1.3" xref="Thmproofx9.p1.5.m5.3.3.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx9.p1.5.m5.3.3.2" xref="Thmproofx9.p1.5.m5.3.3.2.cmml">+</mo><mrow id="Thmproofx9.p1.5.m5.3.3.3" xref="Thmproofx9.p1.5.m5.3.3.3.cmml"><mfrac id="Thmproofx9.p1.5.m5.2.2" xref="Thmproofx9.p1.5.m5.2.2.cmml"><msub id="Thmproofx9.p1.5.m5.2.2.2" xref="Thmproofx9.p1.5.m5.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx9.p1.5.m5.2.2.2.4" xref="Thmproofx9.p1.5.m5.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx9.p1.5.m5.2.2.2.2.2.2" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.3.cmml"><msub id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.cmml"><mi id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.2" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.2.cmml">m</mi><mi id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.3" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.3.cmml">c</mi></msub><mo id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.2" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx9.p1.5.m5.1.1.1.1.1.1" xref="Thmproofx9.p1.5.m5.1.1.1.1.1.1.cmml">n</mi></mrow></msub><msub id="Thmproofx9.p1.5.m5.2.2.4" xref="Thmproofx9.p1.5.m5.2.2.4.cmml"><mi id="Thmproofx9.p1.5.m5.2.2.4.2" xref="Thmproofx9.p1.5.m5.2.2.4.2.cmml">m</mi><mi id="Thmproofx9.p1.5.m5.2.2.4.3" xref="Thmproofx9.p1.5.m5.2.2.4.3.cmml">c</mi></msub></mfrac><mo id="Thmproofx9.p1.5.m5.3.3.3.1" xref="Thmproofx9.p1.5.m5.3.3.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.5.m5.3.3.3.2" xref="Thmproofx9.p1.5.m5.3.3.3.2.cmml">k</mi><mo id="Thmproofx9.p1.5.m5.3.3.3.1a" xref="Thmproofx9.p1.5.m5.3.3.3.1.cmml">⁢</mo><mi id="Thmproofx9.p1.5.m5.3.3.3.3" xref="Thmproofx9.p1.5.m5.3.3.3.3.cmml">α</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx9.p1.5.m5.3b"><apply id="Thmproofx9.p1.5.m5.3.3.cmml" xref="Thmproofx9.p1.5.m5.3.3"><plus id="Thmproofx9.p1.5.m5.3.3.2.cmml" xref="Thmproofx9.p1.5.m5.3.3.2"></plus><apply id="Thmproofx9.p1.5.m5.3.3.1.2.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1"><abs id="Thmproofx9.p1.5.m5.3.3.1.2.1.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.2"></abs><apply id="Thmproofx9.p1.5.m5.3.3.1.1.1.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="Thmproofx9.p1.5.m5.3.3.1.1.1.1.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1">subscript</csymbol><ci id="Thmproofx9.p1.5.m5.3.3.1.1.1.2.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.2">𝑝</ci><apply id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3"><times id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.1.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.1"></times><ci id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.2.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.2">𝑚</ci><ci id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.3.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.3">𝑖</ci><ci id="Thmproofx9.p1.5.m5.3.3.1.1.1.3.4.cmml" xref="Thmproofx9.p1.5.m5.3.3.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Thmproofx9.p1.5.m5.3.3.3.cmml" xref="Thmproofx9.p1.5.m5.3.3.3"><times id="Thmproofx9.p1.5.m5.3.3.3.1.cmml" xref="Thmproofx9.p1.5.m5.3.3.3.1"></times><apply id="Thmproofx9.p1.5.m5.2.2.cmml" xref="Thmproofx9.p1.5.m5.2.2"><divide id="Thmproofx9.p1.5.m5.2.2.3.cmml" xref="Thmproofx9.p1.5.m5.2.2"></divide><apply id="Thmproofx9.p1.5.m5.2.2.2.cmml" xref="Thmproofx9.p1.5.m5.2.2.2"><csymbol cd="ambiguous" id="Thmproofx9.p1.5.m5.2.2.2.3.cmml" xref="Thmproofx9.p1.5.m5.2.2.2">subscript</csymbol><ci id="Thmproofx9.p1.5.m5.2.2.2.4.cmml" xref="Thmproofx9.p1.5.m5.2.2.2.4">ℰ</ci><list id="Thmproofx9.p1.5.m5.2.2.2.2.2.3.cmml" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2"><apply id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.cmml" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.1.cmml" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1">subscript</csymbol><ci id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.2.cmml" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.2">𝑚</ci><ci id="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.3.cmml" xref="Thmproofx9.p1.5.m5.2.2.2.2.2.2.1.3">𝑐</ci></apply><ci id="Thmproofx9.p1.5.m5.1.1.1.1.1.1.cmml" xref="Thmproofx9.p1.5.m5.1.1.1.1.1.1">𝑛</ci></list></apply><apply id="Thmproofx9.p1.5.m5.2.2.4.cmml" xref="Thmproofx9.p1.5.m5.2.2.4"><csymbol cd="ambiguous" id="Thmproofx9.p1.5.m5.2.2.4.1.cmml" xref="Thmproofx9.p1.5.m5.2.2.4">subscript</csymbol><ci id="Thmproofx9.p1.5.m5.2.2.4.2.cmml" xref="Thmproofx9.p1.5.m5.2.2.4.2">𝑚</ci><ci id="Thmproofx9.p1.5.m5.2.2.4.3.cmml" xref="Thmproofx9.p1.5.m5.2.2.4.3">𝑐</ci></apply></apply><ci id="Thmproofx9.p1.5.m5.3.3.3.2.cmml" xref="Thmproofx9.p1.5.m5.3.3.3.2">𝑘</ci><ci id="Thmproofx9.p1.5.m5.3.3.3.3.cmml" xref="Thmproofx9.p1.5.m5.3.3.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx9.p1.5.m5.3c">\left|p_{min}\right|+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx9.p1.5.m5.3d">| italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. Therefore, for the inequality to hold it is sufficient to have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx28"> <tbody id="Pt0.A3.Ex38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle n\left|p_{max}\right|+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha% \leq(1+\varepsilon)\left(b+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha\right)" class="ltx_Math" display="inline" id="Pt0.A3.Ex38.m1.7"><semantics id="Pt0.A3.Ex38.m1.7a"><mrow id="Pt0.A3.Ex38.m1.7.7" xref="Pt0.A3.Ex38.m1.7.7.cmml"><mrow id="Pt0.A3.Ex38.m1.5.5.1" xref="Pt0.A3.Ex38.m1.5.5.1.cmml"><mrow id="Pt0.A3.Ex38.m1.5.5.1.1" xref="Pt0.A3.Ex38.m1.5.5.1.1.cmml"><mi id="Pt0.A3.Ex38.m1.5.5.1.1.3" xref="Pt0.A3.Ex38.m1.5.5.1.1.3.cmml">n</mi><mo id="Pt0.A3.Ex38.m1.5.5.1.1.2" xref="Pt0.A3.Ex38.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex38.m1.5.5.1.1.1.1" 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xref="Pt0.A3.Ex38.m1.6.6.2.1.1.1.2">1</cn><ci id="Pt0.A3.Ex38.m1.6.6.2.1.1.1.3.cmml" xref="Pt0.A3.Ex38.m1.6.6.2.1.1.1.3">𝜀</ci></apply><apply id="Pt0.A3.Ex38.m1.7.7.3.2.1.1.cmml" xref="Pt0.A3.Ex38.m1.7.7.3.2.1"><plus id="Pt0.A3.Ex38.m1.7.7.3.2.1.1.1.cmml" xref="Pt0.A3.Ex38.m1.7.7.3.2.1.1.1"></plus><ci id="Pt0.A3.Ex38.m1.7.7.3.2.1.1.2.cmml" xref="Pt0.A3.Ex38.m1.7.7.3.2.1.1.2">𝑏</ci><apply id="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3.cmml" xref="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3"><times id="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3.1.cmml" xref="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3.1"></times><apply id="Pt0.A3.Ex38.m1.4.4.cmml" xref="Pt0.A3.Ex38.m1.4.4"><divide id="Pt0.A3.Ex38.m1.4.4.3.cmml" xref="Pt0.A3.Ex38.m1.4.4"></divide><apply id="Pt0.A3.Ex38.m1.4.4.2.cmml" xref="Pt0.A3.Ex38.m1.4.4.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex38.m1.4.4.2.3.cmml" xref="Pt0.A3.Ex38.m1.4.4.2">subscript</csymbol><ci id="Pt0.A3.Ex38.m1.4.4.2.4.cmml" xref="Pt0.A3.Ex38.m1.4.4.2.4">ℰ</ci><list id="Pt0.A3.Ex38.m1.4.4.2.2.2.3.cmml" xref="Pt0.A3.Ex38.m1.4.4.2.2.2.2"><apply id="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1.cmml" xref="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1.1.cmml" xref="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1">subscript</csymbol><ci id="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1.2.cmml" xref="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1.2">𝑚</ci><ci id="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1.3.cmml" xref="Pt0.A3.Ex38.m1.4.4.2.2.2.2.1.3">𝑐</ci></apply><ci id="Pt0.A3.Ex38.m1.3.3.1.1.1.1.cmml" xref="Pt0.A3.Ex38.m1.3.3.1.1.1.1">𝑛</ci></list></apply><apply id="Pt0.A3.Ex38.m1.4.4.4.cmml" xref="Pt0.A3.Ex38.m1.4.4.4"><csymbol cd="ambiguous" id="Pt0.A3.Ex38.m1.4.4.4.1.cmml" xref="Pt0.A3.Ex38.m1.4.4.4">subscript</csymbol><ci id="Pt0.A3.Ex38.m1.4.4.4.2.cmml" xref="Pt0.A3.Ex38.m1.4.4.4.2">𝑚</ci><ci id="Pt0.A3.Ex38.m1.4.4.4.3.cmml" xref="Pt0.A3.Ex38.m1.4.4.4.3">𝑐</ci></apply></apply><ci id="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3.2.cmml" xref="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3.2">𝑘</ci><ci id="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3.3.cmml" xref="Pt0.A3.Ex38.m1.7.7.3.2.1.1.3.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex38.m1.7c">\displaystyle n\left|p_{max}\right|+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha% \leq(1+\varepsilon)\left(b+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex38.m1.7d">italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α ≤ ( 1 + italic_ε ) ( italic_b + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx9.p1.6">and this holds for our choice of <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx9.p1.6.m1.1"><semantics id="Thmproofx9.p1.6.m1.1a"><mi id="Thmproofx9.p1.6.m1.1.1" xref="Thmproofx9.p1.6.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx9.p1.6.m1.1b"><ci id="Thmproofx9.p1.6.m1.1.1.cmml" xref="Thmproofx9.p1.6.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx9.p1.6.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx9.p1.6.m1.1d">italic_α</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx10"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem3" title="Theorem 4.3 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.3</span></a>)</h6> <div class="ltx_para" id="Thmproofx10.p1"> <p class="ltx_p" id="Thmproofx10.p1.2">We begin with proving Item 1. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma6" title="Lemma 6 ‣ 3.3 General lemmas ‣ 3 Preliminaries ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">6</span></a>, <math alttext="r(G,n,k,\alpha)\leq n" class="ltx_Math" display="inline" id="Thmproofx10.p1.1.m1.4"><semantics id="Thmproofx10.p1.1.m1.4a"><mrow id="Thmproofx10.p1.1.m1.4.5" xref="Thmproofx10.p1.1.m1.4.5.cmml"><mrow id="Thmproofx10.p1.1.m1.4.5.2" xref="Thmproofx10.p1.1.m1.4.5.2.cmml"><mi id="Thmproofx10.p1.1.m1.4.5.2.2" xref="Thmproofx10.p1.1.m1.4.5.2.2.cmml">r</mi><mo id="Thmproofx10.p1.1.m1.4.5.2.1" xref="Thmproofx10.p1.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx10.p1.1.m1.4.5.2.3.2" xref="Thmproofx10.p1.1.m1.4.5.2.3.1.cmml"><mo id="Thmproofx10.p1.1.m1.4.5.2.3.2.1" stretchy="false" xref="Thmproofx10.p1.1.m1.4.5.2.3.1.cmml">(</mo><mi id="Thmproofx10.p1.1.m1.1.1" xref="Thmproofx10.p1.1.m1.1.1.cmml">G</mi><mo id="Thmproofx10.p1.1.m1.4.5.2.3.2.2" xref="Thmproofx10.p1.1.m1.4.5.2.3.1.cmml">,</mo><mi id="Thmproofx10.p1.1.m1.2.2" xref="Thmproofx10.p1.1.m1.2.2.cmml">n</mi><mo id="Thmproofx10.p1.1.m1.4.5.2.3.2.3" xref="Thmproofx10.p1.1.m1.4.5.2.3.1.cmml">,</mo><mi id="Thmproofx10.p1.1.m1.3.3" xref="Thmproofx10.p1.1.m1.3.3.cmml">k</mi><mo id="Thmproofx10.p1.1.m1.4.5.2.3.2.4" xref="Thmproofx10.p1.1.m1.4.5.2.3.1.cmml">,</mo><mi id="Thmproofx10.p1.1.m1.4.4" xref="Thmproofx10.p1.1.m1.4.4.cmml">α</mi><mo id="Thmproofx10.p1.1.m1.4.5.2.3.2.5" stretchy="false" xref="Thmproofx10.p1.1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx10.p1.1.m1.4.5.1" xref="Thmproofx10.p1.1.m1.4.5.1.cmml">≤</mo><mi id="Thmproofx10.p1.1.m1.4.5.3" xref="Thmproofx10.p1.1.m1.4.5.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx10.p1.1.m1.4b"><apply id="Thmproofx10.p1.1.m1.4.5.cmml" xref="Thmproofx10.p1.1.m1.4.5"><leq id="Thmproofx10.p1.1.m1.4.5.1.cmml" xref="Thmproofx10.p1.1.m1.4.5.1"></leq><apply id="Thmproofx10.p1.1.m1.4.5.2.cmml" xref="Thmproofx10.p1.1.m1.4.5.2"><times id="Thmproofx10.p1.1.m1.4.5.2.1.cmml" xref="Thmproofx10.p1.1.m1.4.5.2.1"></times><ci id="Thmproofx10.p1.1.m1.4.5.2.2.cmml" xref="Thmproofx10.p1.1.m1.4.5.2.2">𝑟</ci><vector id="Thmproofx10.p1.1.m1.4.5.2.3.1.cmml" xref="Thmproofx10.p1.1.m1.4.5.2.3.2"><ci id="Thmproofx10.p1.1.m1.1.1.cmml" xref="Thmproofx10.p1.1.m1.1.1">𝐺</ci><ci id="Thmproofx10.p1.1.m1.2.2.cmml" xref="Thmproofx10.p1.1.m1.2.2">𝑛</ci><ci id="Thmproofx10.p1.1.m1.3.3.cmml" xref="Thmproofx10.p1.1.m1.3.3">𝑘</ci><ci id="Thmproofx10.p1.1.m1.4.4.cmml" xref="Thmproofx10.p1.1.m1.4.4">𝛼</ci></vector></apply><ci id="Thmproofx10.p1.1.m1.4.5.3.cmml" xref="Thmproofx10.p1.1.m1.4.5.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p1.1.m1.4c">r(G,n,k,\alpha)\leq n</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p1.1.m1.4d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≤ italic_n</annotation></semantics></math>. We have <math alttext="F_{1}(\sigma,\alpha)=L(\sigma)+k\alpha\beta(\sigma)\geq b+\frac{k\alpha}{m_{c}}" class="ltx_Math" display="inline" id="Thmproofx10.p1.2.m2.4"><semantics id="Thmproofx10.p1.2.m2.4a"><mrow id="Thmproofx10.p1.2.m2.4.5" xref="Thmproofx10.p1.2.m2.4.5.cmml"><mrow id="Thmproofx10.p1.2.m2.4.5.2" xref="Thmproofx10.p1.2.m2.4.5.2.cmml"><msub id="Thmproofx10.p1.2.m2.4.5.2.2" xref="Thmproofx10.p1.2.m2.4.5.2.2.cmml"><mi id="Thmproofx10.p1.2.m2.4.5.2.2.2" xref="Thmproofx10.p1.2.m2.4.5.2.2.2.cmml">F</mi><mn id="Thmproofx10.p1.2.m2.4.5.2.2.3" xref="Thmproofx10.p1.2.m2.4.5.2.2.3.cmml">1</mn></msub><mo id="Thmproofx10.p1.2.m2.4.5.2.1" xref="Thmproofx10.p1.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx10.p1.2.m2.4.5.2.3.2" xref="Thmproofx10.p1.2.m2.4.5.2.3.1.cmml"><mo id="Thmproofx10.p1.2.m2.4.5.2.3.2.1" stretchy="false" xref="Thmproofx10.p1.2.m2.4.5.2.3.1.cmml">(</mo><mi id="Thmproofx10.p1.2.m2.1.1" xref="Thmproofx10.p1.2.m2.1.1.cmml">σ</mi><mo id="Thmproofx10.p1.2.m2.4.5.2.3.2.2" xref="Thmproofx10.p1.2.m2.4.5.2.3.1.cmml">,</mo><mi id="Thmproofx10.p1.2.m2.2.2" xref="Thmproofx10.p1.2.m2.2.2.cmml">α</mi><mo id="Thmproofx10.p1.2.m2.4.5.2.3.2.3" stretchy="false" xref="Thmproofx10.p1.2.m2.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx10.p1.2.m2.4.5.3" xref="Thmproofx10.p1.2.m2.4.5.3.cmml">=</mo><mrow id="Thmproofx10.p1.2.m2.4.5.4" xref="Thmproofx10.p1.2.m2.4.5.4.cmml"><mrow id="Thmproofx10.p1.2.m2.4.5.4.2" xref="Thmproofx10.p1.2.m2.4.5.4.2.cmml"><mi id="Thmproofx10.p1.2.m2.4.5.4.2.2" xref="Thmproofx10.p1.2.m2.4.5.4.2.2.cmml">L</mi><mo id="Thmproofx10.p1.2.m2.4.5.4.2.1" xref="Thmproofx10.p1.2.m2.4.5.4.2.1.cmml">⁢</mo><mrow id="Thmproofx10.p1.2.m2.4.5.4.2.3.2" xref="Thmproofx10.p1.2.m2.4.5.4.2.cmml"><mo id="Thmproofx10.p1.2.m2.4.5.4.2.3.2.1" stretchy="false" xref="Thmproofx10.p1.2.m2.4.5.4.2.cmml">(</mo><mi id="Thmproofx10.p1.2.m2.3.3" xref="Thmproofx10.p1.2.m2.3.3.cmml">σ</mi><mo id="Thmproofx10.p1.2.m2.4.5.4.2.3.2.2" stretchy="false" xref="Thmproofx10.p1.2.m2.4.5.4.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx10.p1.2.m2.4.5.4.1" xref="Thmproofx10.p1.2.m2.4.5.4.1.cmml">+</mo><mrow id="Thmproofx10.p1.2.m2.4.5.4.3" xref="Thmproofx10.p1.2.m2.4.5.4.3.cmml"><mi id="Thmproofx10.p1.2.m2.4.5.4.3.2" xref="Thmproofx10.p1.2.m2.4.5.4.3.2.cmml">k</mi><mo id="Thmproofx10.p1.2.m2.4.5.4.3.1" xref="Thmproofx10.p1.2.m2.4.5.4.3.1.cmml">⁢</mo><mi id="Thmproofx10.p1.2.m2.4.5.4.3.3" xref="Thmproofx10.p1.2.m2.4.5.4.3.3.cmml">α</mi><mo id="Thmproofx10.p1.2.m2.4.5.4.3.1a" xref="Thmproofx10.p1.2.m2.4.5.4.3.1.cmml">⁢</mo><mi id="Thmproofx10.p1.2.m2.4.5.4.3.4" xref="Thmproofx10.p1.2.m2.4.5.4.3.4.cmml">β</mi><mo id="Thmproofx10.p1.2.m2.4.5.4.3.1b" xref="Thmproofx10.p1.2.m2.4.5.4.3.1.cmml">⁢</mo><mrow id="Thmproofx10.p1.2.m2.4.5.4.3.5.2" xref="Thmproofx10.p1.2.m2.4.5.4.3.cmml"><mo id="Thmproofx10.p1.2.m2.4.5.4.3.5.2.1" stretchy="false" xref="Thmproofx10.p1.2.m2.4.5.4.3.cmml">(</mo><mi id="Thmproofx10.p1.2.m2.4.4" xref="Thmproofx10.p1.2.m2.4.4.cmml">σ</mi><mo id="Thmproofx10.p1.2.m2.4.5.4.3.5.2.2" stretchy="false" xref="Thmproofx10.p1.2.m2.4.5.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx10.p1.2.m2.4.5.5" xref="Thmproofx10.p1.2.m2.4.5.5.cmml">≥</mo><mrow id="Thmproofx10.p1.2.m2.4.5.6" xref="Thmproofx10.p1.2.m2.4.5.6.cmml"><mi id="Thmproofx10.p1.2.m2.4.5.6.2" xref="Thmproofx10.p1.2.m2.4.5.6.2.cmml">b</mi><mo id="Thmproofx10.p1.2.m2.4.5.6.1" xref="Thmproofx10.p1.2.m2.4.5.6.1.cmml">+</mo><mfrac id="Thmproofx10.p1.2.m2.4.5.6.3" xref="Thmproofx10.p1.2.m2.4.5.6.3.cmml"><mrow id="Thmproofx10.p1.2.m2.4.5.6.3.2" xref="Thmproofx10.p1.2.m2.4.5.6.3.2.cmml"><mi id="Thmproofx10.p1.2.m2.4.5.6.3.2.2" xref="Thmproofx10.p1.2.m2.4.5.6.3.2.2.cmml">k</mi><mo id="Thmproofx10.p1.2.m2.4.5.6.3.2.1" xref="Thmproofx10.p1.2.m2.4.5.6.3.2.1.cmml">⁢</mo><mi id="Thmproofx10.p1.2.m2.4.5.6.3.2.3" xref="Thmproofx10.p1.2.m2.4.5.6.3.2.3.cmml">α</mi></mrow><msub id="Thmproofx10.p1.2.m2.4.5.6.3.3" 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id="Thmproofx10.p1.2.m2.4.5.4.3.cmml" xref="Thmproofx10.p1.2.m2.4.5.4.3"><times id="Thmproofx10.p1.2.m2.4.5.4.3.1.cmml" xref="Thmproofx10.p1.2.m2.4.5.4.3.1"></times><ci id="Thmproofx10.p1.2.m2.4.5.4.3.2.cmml" xref="Thmproofx10.p1.2.m2.4.5.4.3.2">𝑘</ci><ci id="Thmproofx10.p1.2.m2.4.5.4.3.3.cmml" xref="Thmproofx10.p1.2.m2.4.5.4.3.3">𝛼</ci><ci id="Thmproofx10.p1.2.m2.4.5.4.3.4.cmml" xref="Thmproofx10.p1.2.m2.4.5.4.3.4">𝛽</ci><ci id="Thmproofx10.p1.2.m2.4.4.cmml" xref="Thmproofx10.p1.2.m2.4.4">𝜎</ci></apply></apply></apply><apply id="Thmproofx10.p1.2.m2.4.5c.cmml" xref="Thmproofx10.p1.2.m2.4.5"><geq id="Thmproofx10.p1.2.m2.4.5.5.cmml" xref="Thmproofx10.p1.2.m2.4.5.5"></geq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx10.p1.2.m2.4.5.4.cmml" id="Thmproofx10.p1.2.m2.4.5d.cmml" xref="Thmproofx10.p1.2.m2.4.5"></share><apply id="Thmproofx10.p1.2.m2.4.5.6.cmml" xref="Thmproofx10.p1.2.m2.4.5.6"><plus id="Thmproofx10.p1.2.m2.4.5.6.1.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.1"></plus><ci id="Thmproofx10.p1.2.m2.4.5.6.2.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.2">𝑏</ci><apply id="Thmproofx10.p1.2.m2.4.5.6.3.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3"><divide id="Thmproofx10.p1.2.m2.4.5.6.3.1.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3"></divide><apply id="Thmproofx10.p1.2.m2.4.5.6.3.2.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.2"><times id="Thmproofx10.p1.2.m2.4.5.6.3.2.1.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.2.1"></times><ci id="Thmproofx10.p1.2.m2.4.5.6.3.2.2.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.2.2">𝑘</ci><ci id="Thmproofx10.p1.2.m2.4.5.6.3.2.3.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.2.3">𝛼</ci></apply><apply id="Thmproofx10.p1.2.m2.4.5.6.3.3.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.3"><csymbol cd="ambiguous" id="Thmproofx10.p1.2.m2.4.5.6.3.3.1.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.3">subscript</csymbol><ci id="Thmproofx10.p1.2.m2.4.5.6.3.3.2.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.3.2">𝑚</ci><ci id="Thmproofx10.p1.2.m2.4.5.6.3.3.3.cmml" xref="Thmproofx10.p1.2.m2.4.5.6.3.3.3">𝑐</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p1.2.m2.4c">F_{1}(\sigma,\alpha)=L(\sigma)+k\alpha\beta(\sigma)\geq b+\frac{k\alpha}{m_{c}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p1.2.m2.4d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_σ , italic_α ) = italic_L ( italic_σ ) + italic_k italic_α italic_β ( italic_σ ) ≥ italic_b + divide start_ARG italic_k italic_α end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>, and</p> <table class="ltx_equation ltx_eqn_table" id="Pt0.A3.Ex39"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\min_{\sigma}F_{n}(\sigma,\alpha)\leq F_{n}(1_{p_{min}}^{n},\alpha)=F_{1}(1_{p% _{min}},\alpha)=\left|p_{min}\right|+k\alpha" class="ltx_Math" display="block" id="Pt0.A3.Ex39.m1.7"><semantics id="Pt0.A3.Ex39.m1.7a"><mrow id="Pt0.A3.Ex39.m1.7.7" xref="Pt0.A3.Ex39.m1.7.7.cmml"><mrow id="Pt0.A3.Ex39.m1.7.7.5" xref="Pt0.A3.Ex39.m1.7.7.5.cmml"><mrow id="Pt0.A3.Ex39.m1.7.7.5.2" xref="Pt0.A3.Ex39.m1.7.7.5.2.cmml"><munder id="Pt0.A3.Ex39.m1.7.7.5.2.1" xref="Pt0.A3.Ex39.m1.7.7.5.2.1.cmml"><mi id="Pt0.A3.Ex39.m1.7.7.5.2.1.2" xref="Pt0.A3.Ex39.m1.7.7.5.2.1.2.cmml">min</mi><mi id="Pt0.A3.Ex39.m1.7.7.5.2.1.3" xref="Pt0.A3.Ex39.m1.7.7.5.2.1.3.cmml">σ</mi></munder><mo id="Pt0.A3.Ex39.m1.7.7.5.2a" lspace="0.167em" xref="Pt0.A3.Ex39.m1.7.7.5.2.cmml">⁡</mo><msub id="Pt0.A3.Ex39.m1.7.7.5.2.2" xref="Pt0.A3.Ex39.m1.7.7.5.2.2.cmml"><mi id="Pt0.A3.Ex39.m1.7.7.5.2.2.2" xref="Pt0.A3.Ex39.m1.7.7.5.2.2.2.cmml">F</mi><mi id="Pt0.A3.Ex39.m1.7.7.5.2.2.3" xref="Pt0.A3.Ex39.m1.7.7.5.2.2.3.cmml">n</mi></msub></mrow><mo id="Pt0.A3.Ex39.m1.7.7.5.1" xref="Pt0.A3.Ex39.m1.7.7.5.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex39.m1.7.7.5.3.2" xref="Pt0.A3.Ex39.m1.7.7.5.3.1.cmml"><mo id="Pt0.A3.Ex39.m1.7.7.5.3.2.1" stretchy="false" xref="Pt0.A3.Ex39.m1.7.7.5.3.1.cmml">(</mo><mi id="Pt0.A3.Ex39.m1.1.1" xref="Pt0.A3.Ex39.m1.1.1.cmml">σ</mi><mo id="Pt0.A3.Ex39.m1.7.7.5.3.2.2" xref="Pt0.A3.Ex39.m1.7.7.5.3.1.cmml">,</mo><mi id="Pt0.A3.Ex39.m1.2.2" xref="Pt0.A3.Ex39.m1.2.2.cmml">α</mi><mo id="Pt0.A3.Ex39.m1.7.7.5.3.2.3" stretchy="false" xref="Pt0.A3.Ex39.m1.7.7.5.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A3.Ex39.m1.7.7.6" xref="Pt0.A3.Ex39.m1.7.7.6.cmml">≤</mo><mrow id="Pt0.A3.Ex39.m1.5.5.1" xref="Pt0.A3.Ex39.m1.5.5.1.cmml"><msub id="Pt0.A3.Ex39.m1.5.5.1.3" xref="Pt0.A3.Ex39.m1.5.5.1.3.cmml"><mi id="Pt0.A3.Ex39.m1.5.5.1.3.2" xref="Pt0.A3.Ex39.m1.5.5.1.3.2.cmml">F</mi><mi id="Pt0.A3.Ex39.m1.5.5.1.3.3" xref="Pt0.A3.Ex39.m1.5.5.1.3.3.cmml">n</mi></msub><mo id="Pt0.A3.Ex39.m1.5.5.1.2" xref="Pt0.A3.Ex39.m1.5.5.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex39.m1.5.5.1.1.1" xref="Pt0.A3.Ex39.m1.5.5.1.1.2.cmml"><mo id="Pt0.A3.Ex39.m1.5.5.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex39.m1.5.5.1.1.2.cmml">(</mo><msubsup id="Pt0.A3.Ex39.m1.5.5.1.1.1.1" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.cmml"><mn id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.2" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.2.cmml">1</mn><msub id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.cmml"><mi id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.2" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.2.cmml">p</mi><mrow id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.2" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.2.cmml">m</mi><mo id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.1" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.3" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.3.cmml">i</mi><mo id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.1a" xref="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Pt0.A3.Ex39.m1.5.5.1.1.1.1.2.3.3.4" 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xref="Pt0.A3.Ex39.m1.7.7.3.1.1.2"></abs><apply id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.1.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1">subscript</csymbol><ci id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.2.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1.2">𝑝</ci><apply id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3"><times id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.1.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.1"></times><ci id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.2.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.2">𝑚</ci><ci id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.3.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.3">𝑖</ci><ci id="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.4.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Pt0.A3.Ex39.m1.7.7.3.3.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.3"><times id="Pt0.A3.Ex39.m1.7.7.3.3.1.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.3.1"></times><ci id="Pt0.A3.Ex39.m1.7.7.3.3.2.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.3.2">𝑘</ci><ci id="Pt0.A3.Ex39.m1.7.7.3.3.3.cmml" xref="Pt0.A3.Ex39.m1.7.7.3.3.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex39.m1.7c">\min_{\sigma}F_{n}(\sigma,\alpha)\leq F_{n}(1_{p_{min}}^{n},\alpha)=F_{1}(1_{p% _{min}},\alpha)=\left|p_{min}\right|+k\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex39.m1.7d">roman_min start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≤ italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 1 start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 1 start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_α ) = | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx10.p1.3">So in total</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx29"> <tbody id="Pt0.A3.Ex40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(G,n,k,\alpha)\leq\min\left(\frac{\left|p_{min}\right|+k\alpha}% {\left|p_{min}\right|+\frac{k\alpha}{m_{c}}},n\right)\leq\min(m_{c},n)" class="ltx_Math" display="inline" id="Pt0.A3.Ex40.m1.11"><semantics id="Pt0.A3.Ex40.m1.11a"><mrow id="Pt0.A3.Ex40.m1.11.11" xref="Pt0.A3.Ex40.m1.11.11.cmml"><mrow id="Pt0.A3.Ex40.m1.11.11.3" xref="Pt0.A3.Ex40.m1.11.11.3.cmml"><mi id="Pt0.A3.Ex40.m1.11.11.3.2" 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id="Pt0.A3.Ex40.m1.11.11.1.2.cmml" xref="Pt0.A3.Ex40.m1.11.11.1.1"><min id="Pt0.A3.Ex40.m1.9.9.cmml" xref="Pt0.A3.Ex40.m1.9.9"></min><apply id="Pt0.A3.Ex40.m1.11.11.1.1.1.1.cmml" xref="Pt0.A3.Ex40.m1.11.11.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex40.m1.11.11.1.1.1.1.1.cmml" xref="Pt0.A3.Ex40.m1.11.11.1.1.1.1">subscript</csymbol><ci id="Pt0.A3.Ex40.m1.11.11.1.1.1.1.2.cmml" xref="Pt0.A3.Ex40.m1.11.11.1.1.1.1.2">𝑚</ci><ci id="Pt0.A3.Ex40.m1.11.11.1.1.1.1.3.cmml" xref="Pt0.A3.Ex40.m1.11.11.1.1.1.1.3">𝑐</ci></apply><ci id="Pt0.A3.Ex40.m1.10.10.cmml" xref="Pt0.A3.Ex40.m1.10.10">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex40.m1.11c">\displaystyle r(G,n,k,\alpha)\leq\min\left(\frac{\left|p_{min}\right|+k\alpha}% {\left|p_{min}\right|+\frac{k\alpha}{m_{c}}},n\right)\leq\min(m_{c},n)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex40.m1.11d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≤ roman_min ( divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + italic_k italic_α end_ARG start_ARG | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + divide start_ARG italic_k italic_α end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG end_ARG , italic_n ) ≤ roman_min ( italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="Thmproofx10.p2"> <p class="ltx_p" id="Thmproofx10.p2.2">Now we prove Item 2. <math alttext="F_{n}(\sigma,\alpha)\leq F_{n}(U_{P_{f}}^{n},\alpha)\leq n\left|p_{max}\right|% +\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha" class="ltx_Math" display="inline" id="Thmproofx10.p2.1.m1.7"><semantics id="Thmproofx10.p2.1.m1.7a"><mrow id="Thmproofx10.p2.1.m1.7.7" xref="Thmproofx10.p2.1.m1.7.7.cmml"><mrow 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xref="Thmproofx10.p2.1.m1.7.7"></and><apply id="Thmproofx10.p2.1.m1.7.7b.cmml" xref="Thmproofx10.p2.1.m1.7.7"><leq id="Thmproofx10.p2.1.m1.7.7.5.cmml" xref="Thmproofx10.p2.1.m1.7.7.5"></leq><apply id="Thmproofx10.p2.1.m1.7.7.4.cmml" xref="Thmproofx10.p2.1.m1.7.7.4"><times id="Thmproofx10.p2.1.m1.7.7.4.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.4.1"></times><apply id="Thmproofx10.p2.1.m1.7.7.4.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.4.2"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.7.7.4.2.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.4.2">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.7.7.4.2.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.4.2.2">𝐹</ci><ci id="Thmproofx10.p2.1.m1.7.7.4.2.3.cmml" xref="Thmproofx10.p2.1.m1.7.7.4.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx10.p2.1.m1.7.7.4.3.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.4.3.2"><ci id="Thmproofx10.p2.1.m1.3.3.cmml" xref="Thmproofx10.p2.1.m1.3.3">𝜎</ci><ci id="Thmproofx10.p2.1.m1.4.4.cmml" xref="Thmproofx10.p2.1.m1.4.4">𝛼</ci></interval></apply><apply id="Thmproofx10.p2.1.m1.6.6.1.cmml" xref="Thmproofx10.p2.1.m1.6.6.1"><times id="Thmproofx10.p2.1.m1.6.6.1.2.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.2"></times><apply id="Thmproofx10.p2.1.m1.6.6.1.3.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.3"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.6.6.1.3.1.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.3">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.6.6.1.3.2.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.3.2">𝐹</ci><ci id="Thmproofx10.p2.1.m1.6.6.1.3.3.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.3.3">𝑛</ci></apply><interval closure="open" id="Thmproofx10.p2.1.m1.6.6.1.1.2.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1"><apply id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.1.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1">superscript</csymbol><apply id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.1.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.2.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.2">𝑈</ci><apply id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3.1.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3.2.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3.2">𝑃</ci><ci id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3.3.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1.2.3.3">𝑓</ci></apply></apply><ci id="Thmproofx10.p2.1.m1.6.6.1.1.1.1.3.cmml" xref="Thmproofx10.p2.1.m1.6.6.1.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx10.p2.1.m1.5.5.cmml" xref="Thmproofx10.p2.1.m1.5.5">𝛼</ci></interval></apply></apply><apply id="Thmproofx10.p2.1.m1.7.7c.cmml" xref="Thmproofx10.p2.1.m1.7.7"><leq id="Thmproofx10.p2.1.m1.7.7.6.cmml" xref="Thmproofx10.p2.1.m1.7.7.6"></leq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx10.p2.1.m1.6.6.1.cmml" id="Thmproofx10.p2.1.m1.7.7d.cmml" xref="Thmproofx10.p2.1.m1.7.7"></share><apply id="Thmproofx10.p2.1.m1.7.7.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.2"><plus id="Thmproofx10.p2.1.m1.7.7.2.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.2"></plus><apply id="Thmproofx10.p2.1.m1.7.7.2.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1"><times id="Thmproofx10.p2.1.m1.7.7.2.1.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.2"></times><ci id="Thmproofx10.p2.1.m1.7.7.2.1.3.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.3">𝑛</ci><apply id="Thmproofx10.p2.1.m1.7.7.2.1.1.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1"><abs id="Thmproofx10.p2.1.m1.7.7.2.1.1.2.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.2"></abs><apply id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.2">𝑝</ci><apply id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3"><times id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.1"></times><ci id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.3.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.3">𝑎</ci><ci id="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.4.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.1.1.1.1.3.4">𝑥</ci></apply></apply></apply></apply><apply id="Thmproofx10.p2.1.m1.7.7.2.3.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.3"><times id="Thmproofx10.p2.1.m1.7.7.2.3.1.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.3.1"></times><apply id="Thmproofx10.p2.1.m1.2.2.cmml" xref="Thmproofx10.p2.1.m1.2.2"><divide id="Thmproofx10.p2.1.m1.2.2.3.cmml" xref="Thmproofx10.p2.1.m1.2.2"></divide><apply id="Thmproofx10.p2.1.m1.2.2.2.cmml" xref="Thmproofx10.p2.1.m1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.2.2.2.3.cmml" xref="Thmproofx10.p2.1.m1.2.2.2">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.2.2.2.4.cmml" xref="Thmproofx10.p2.1.m1.2.2.2.4">ℰ</ci><list id="Thmproofx10.p2.1.m1.2.2.2.2.2.3.cmml" xref="Thmproofx10.p2.1.m1.2.2.2.2.2.2"><apply id="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1.cmml" xref="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1.1.cmml" xref="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1.2.cmml" xref="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1.2">𝑚</ci><ci id="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1.3.cmml" xref="Thmproofx10.p2.1.m1.2.2.2.2.2.2.1.3">𝑐</ci></apply><ci id="Thmproofx10.p2.1.m1.1.1.1.1.1.1.cmml" xref="Thmproofx10.p2.1.m1.1.1.1.1.1.1">𝑛</ci></list></apply><apply id="Thmproofx10.p2.1.m1.2.2.4.cmml" xref="Thmproofx10.p2.1.m1.2.2.4"><csymbol cd="ambiguous" id="Thmproofx10.p2.1.m1.2.2.4.1.cmml" xref="Thmproofx10.p2.1.m1.2.2.4">subscript</csymbol><ci id="Thmproofx10.p2.1.m1.2.2.4.2.cmml" xref="Thmproofx10.p2.1.m1.2.2.4.2">𝑚</ci><ci id="Thmproofx10.p2.1.m1.2.2.4.3.cmml" xref="Thmproofx10.p2.1.m1.2.2.4.3">𝑐</ci></apply></apply><ci id="Thmproofx10.p2.1.m1.7.7.2.3.2.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.3.2">𝑘</ci><ci id="Thmproofx10.p2.1.m1.7.7.2.3.3.cmml" xref="Thmproofx10.p2.1.m1.7.7.2.3.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p2.1.m1.7c">F_{n}(\sigma,\alpha)\leq F_{n}(U_{P_{f}}^{n},\alpha)\leq n\left|p_{max}\right|% +\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p2.1.m1.7d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≤ italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) ≤ italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. In total we have for the chosen <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx10.p2.2.m2.1"><semantics id="Thmproofx10.p2.2.m2.1a"><mi id="Thmproofx10.p2.2.m2.1.1" xref="Thmproofx10.p2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx10.p2.2.m2.1b"><ci id="Thmproofx10.p2.2.m2.1.1.cmml" xref="Thmproofx10.p2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p2.2.m2.1d">italic_α</annotation></semantics></math> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx30"> <tbody id="Pt0.A3.Ex41"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(G,n,k,\alpha)\leq\frac{B+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k% 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xref="Pt0.A3.Ex41.m1.9.9"></share><apply id="Pt0.A3.Ex41.m1.9.9.1.cmml" xref="Pt0.A3.Ex41.m1.9.9.1"><times id="Pt0.A3.Ex41.m1.9.9.1.2.cmml" xref="Pt0.A3.Ex41.m1.9.9.1.2"></times><apply id="Pt0.A3.Ex41.m1.9.9.1.1.1.1.cmml" xref="Pt0.A3.Ex41.m1.9.9.1.1.1"><plus id="Pt0.A3.Ex41.m1.9.9.1.1.1.1.1.cmml" xref="Pt0.A3.Ex41.m1.9.9.1.1.1.1.1"></plus><cn id="Pt0.A3.Ex41.m1.9.9.1.1.1.1.2.cmml" type="integer" xref="Pt0.A3.Ex41.m1.9.9.1.1.1.1.2">1</cn><ci id="Pt0.A3.Ex41.m1.9.9.1.1.1.1.3.cmml" xref="Pt0.A3.Ex41.m1.9.9.1.1.1.1.3">𝜀</ci></apply><apply id="Pt0.A3.Ex41.m1.9.9.1.3.cmml" xref="Pt0.A3.Ex41.m1.9.9.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex41.m1.9.9.1.3.1.cmml" xref="Pt0.A3.Ex41.m1.9.9.1.3">subscript</csymbol><ci id="Pt0.A3.Ex41.m1.9.9.1.3.2.cmml" xref="Pt0.A3.Ex41.m1.9.9.1.3.2">ℰ</ci><list id="Pt0.A3.Ex41.m1.4.4.2.3.cmml" xref="Pt0.A3.Ex41.m1.4.4.2.2"><apply id="Pt0.A3.Ex41.m1.4.4.2.2.1.cmml" xref="Pt0.A3.Ex41.m1.4.4.2.2.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex41.m1.4.4.2.2.1.1.cmml" xref="Pt0.A3.Ex41.m1.4.4.2.2.1">subscript</csymbol><ci id="Pt0.A3.Ex41.m1.4.4.2.2.1.2.cmml" xref="Pt0.A3.Ex41.m1.4.4.2.2.1.2">𝑚</ci><ci id="Pt0.A3.Ex41.m1.4.4.2.2.1.3.cmml" xref="Pt0.A3.Ex41.m1.4.4.2.2.1.3">𝑐</ci></apply><ci id="Pt0.A3.Ex41.m1.3.3.1.1.cmml" xref="Pt0.A3.Ex41.m1.3.3.1.1">𝑛</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex41.m1.9c">\displaystyle r(G,n,k,\alpha)\leq\frac{B+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k% \alpha}{b+\frac{k\alpha}{m_{c}}}\leq(1+\varepsilon)\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex41.m1.9d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≤ divide start_ARG italic_B + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α end_ARG start_ARG italic_b + divide start_ARG italic_k italic_α end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG end_ARG ≤ ( 1 + italic_ε ) caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="Thmproofx10.p3"> <p class="ltx_p" id="Thmproofx10.p3.5">Now Item 3. By definition <math alttext="F_{1}(\sigma,\alpha)\leq F_{1}(U_{P_{f}},\alpha)\leq n\left|p_{max}\right|+% \frac{k\alpha}{m_{c}}" class="ltx_Math" display="inline" id="Thmproofx10.p3.1.m1.5"><semantics id="Thmproofx10.p3.1.m1.5a"><mrow id="Thmproofx10.p3.1.m1.5.5" xref="Thmproofx10.p3.1.m1.5.5.cmml"><mrow id="Thmproofx10.p3.1.m1.5.5.4" xref="Thmproofx10.p3.1.m1.5.5.4.cmml"><msub id="Thmproofx10.p3.1.m1.5.5.4.2" xref="Thmproofx10.p3.1.m1.5.5.4.2.cmml"><mi id="Thmproofx10.p3.1.m1.5.5.4.2.2" xref="Thmproofx10.p3.1.m1.5.5.4.2.2.cmml">F</mi><mn id="Thmproofx10.p3.1.m1.5.5.4.2.3" xref="Thmproofx10.p3.1.m1.5.5.4.2.3.cmml">1</mn></msub><mo id="Thmproofx10.p3.1.m1.5.5.4.1" xref="Thmproofx10.p3.1.m1.5.5.4.1.cmml">⁢</mo><mrow id="Thmproofx10.p3.1.m1.5.5.4.3.2" xref="Thmproofx10.p3.1.m1.5.5.4.3.1.cmml"><mo id="Thmproofx10.p3.1.m1.5.5.4.3.2.1" stretchy="false" xref="Thmproofx10.p3.1.m1.5.5.4.3.1.cmml">(</mo><mi id="Thmproofx10.p3.1.m1.1.1" xref="Thmproofx10.p3.1.m1.1.1.cmml">σ</mi><mo 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href="https://arxiv.org/html/2308.08047v3#Thmproofx10.p3.1.m1.4.4.1.cmml" id="Thmproofx10.p3.1.m1.5.5d.cmml" xref="Thmproofx10.p3.1.m1.5.5"></share><apply id="Thmproofx10.p3.1.m1.5.5.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2"><plus id="Thmproofx10.p3.1.m1.5.5.2.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.2"></plus><apply id="Thmproofx10.p3.1.m1.5.5.2.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1"><times id="Thmproofx10.p3.1.m1.5.5.2.1.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.2"></times><ci id="Thmproofx10.p3.1.m1.5.5.2.1.3.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.3">𝑛</ci><apply id="Thmproofx10.p3.1.m1.5.5.2.1.1.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1"><abs id="Thmproofx10.p3.1.m1.5.5.2.1.1.2.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.2"></abs><apply id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1">subscript</csymbol><ci id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.2">𝑝</ci><apply id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3"><times id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.1"></times><ci id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.3.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.3">𝑎</ci><ci id="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.4.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.1.1.1.1.3.4">𝑥</ci></apply></apply></apply></apply><apply id="Thmproofx10.p3.1.m1.5.5.2.3.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3"><divide id="Thmproofx10.p3.1.m1.5.5.2.3.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3"></divide><apply id="Thmproofx10.p3.1.m1.5.5.2.3.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.2"><times id="Thmproofx10.p3.1.m1.5.5.2.3.2.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.2.1"></times><ci id="Thmproofx10.p3.1.m1.5.5.2.3.2.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.2.2">𝑘</ci><ci id="Thmproofx10.p3.1.m1.5.5.2.3.2.3.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.2.3">𝛼</ci></apply><apply id="Thmproofx10.p3.1.m1.5.5.2.3.3.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.3"><csymbol cd="ambiguous" id="Thmproofx10.p3.1.m1.5.5.2.3.3.1.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.3">subscript</csymbol><ci id="Thmproofx10.p3.1.m1.5.5.2.3.3.2.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.3.2">𝑚</ci><ci id="Thmproofx10.p3.1.m1.5.5.2.3.3.3.cmml" xref="Thmproofx10.p3.1.m1.5.5.2.3.3.3">𝑐</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p3.1.m1.5c">F_{1}(\sigma,\alpha)\leq F_{1}(U_{P_{f}},\alpha)\leq n\left|p_{max}\right|+% \frac{k\alpha}{m_{c}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p3.1.m1.5d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≤ italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_α ) ≤ italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + divide start_ARG italic_k italic_α end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> for <math alttext="f\in MF" class="ltx_Math" display="inline" id="Thmproofx10.p3.2.m2.1"><semantics id="Thmproofx10.p3.2.m2.1a"><mrow id="Thmproofx10.p3.2.m2.1.1" xref="Thmproofx10.p3.2.m2.1.1.cmml"><mi id="Thmproofx10.p3.2.m2.1.1.2" xref="Thmproofx10.p3.2.m2.1.1.2.cmml">f</mi><mo id="Thmproofx10.p3.2.m2.1.1.1" xref="Thmproofx10.p3.2.m2.1.1.1.cmml">∈</mo><mrow id="Thmproofx10.p3.2.m2.1.1.3" xref="Thmproofx10.p3.2.m2.1.1.3.cmml"><mi id="Thmproofx10.p3.2.m2.1.1.3.2" xref="Thmproofx10.p3.2.m2.1.1.3.2.cmml">M</mi><mo id="Thmproofx10.p3.2.m2.1.1.3.1" xref="Thmproofx10.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.2.m2.1.1.3.3" xref="Thmproofx10.p3.2.m2.1.1.3.3.cmml">F</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx10.p3.2.m2.1b"><apply id="Thmproofx10.p3.2.m2.1.1.cmml" xref="Thmproofx10.p3.2.m2.1.1"><in id="Thmproofx10.p3.2.m2.1.1.1.cmml" xref="Thmproofx10.p3.2.m2.1.1.1"></in><ci id="Thmproofx10.p3.2.m2.1.1.2.cmml" xref="Thmproofx10.p3.2.m2.1.1.2">𝑓</ci><apply id="Thmproofx10.p3.2.m2.1.1.3.cmml" xref="Thmproofx10.p3.2.m2.1.1.3"><times id="Thmproofx10.p3.2.m2.1.1.3.1.cmml" xref="Thmproofx10.p3.2.m2.1.1.3.1"></times><ci id="Thmproofx10.p3.2.m2.1.1.3.2.cmml" xref="Thmproofx10.p3.2.m2.1.1.3.2">𝑀</ci><ci id="Thmproofx10.p3.2.m2.1.1.3.3.cmml" xref="Thmproofx10.p3.2.m2.1.1.3.3">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p3.2.m2.1c">f\in MF</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p3.2.m2.1d">italic_f ∈ italic_M italic_F</annotation></semantics></math> with <math alttext="|f|=m_{c}" class="ltx_Math" display="inline" id="Thmproofx10.p3.3.m3.1"><semantics id="Thmproofx10.p3.3.m3.1a"><mrow id="Thmproofx10.p3.3.m3.1.2" xref="Thmproofx10.p3.3.m3.1.2.cmml"><mrow id="Thmproofx10.p3.3.m3.1.2.2.2" xref="Thmproofx10.p3.3.m3.1.2.2.1.cmml"><mo id="Thmproofx10.p3.3.m3.1.2.2.2.1" stretchy="false" xref="Thmproofx10.p3.3.m3.1.2.2.1.1.cmml">|</mo><mi id="Thmproofx10.p3.3.m3.1.1" xref="Thmproofx10.p3.3.m3.1.1.cmml">f</mi><mo id="Thmproofx10.p3.3.m3.1.2.2.2.2" stretchy="false" xref="Thmproofx10.p3.3.m3.1.2.2.1.1.cmml">|</mo></mrow><mo id="Thmproofx10.p3.3.m3.1.2.1" xref="Thmproofx10.p3.3.m3.1.2.1.cmml">=</mo><msub id="Thmproofx10.p3.3.m3.1.2.3" xref="Thmproofx10.p3.3.m3.1.2.3.cmml"><mi id="Thmproofx10.p3.3.m3.1.2.3.2" xref="Thmproofx10.p3.3.m3.1.2.3.2.cmml">m</mi><mi id="Thmproofx10.p3.3.m3.1.2.3.3" xref="Thmproofx10.p3.3.m3.1.2.3.3.cmml">c</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx10.p3.3.m3.1b"><apply id="Thmproofx10.p3.3.m3.1.2.cmml" xref="Thmproofx10.p3.3.m3.1.2"><eq id="Thmproofx10.p3.3.m3.1.2.1.cmml" xref="Thmproofx10.p3.3.m3.1.2.1"></eq><apply id="Thmproofx10.p3.3.m3.1.2.2.1.cmml" xref="Thmproofx10.p3.3.m3.1.2.2.2"><abs id="Thmproofx10.p3.3.m3.1.2.2.1.1.cmml" xref="Thmproofx10.p3.3.m3.1.2.2.2.1"></abs><ci id="Thmproofx10.p3.3.m3.1.1.cmml" xref="Thmproofx10.p3.3.m3.1.1">𝑓</ci></apply><apply id="Thmproofx10.p3.3.m3.1.2.3.cmml" xref="Thmproofx10.p3.3.m3.1.2.3"><csymbol cd="ambiguous" id="Thmproofx10.p3.3.m3.1.2.3.1.cmml" xref="Thmproofx10.p3.3.m3.1.2.3">subscript</csymbol><ci id="Thmproofx10.p3.3.m3.1.2.3.2.cmml" xref="Thmproofx10.p3.3.m3.1.2.3.2">𝑚</ci><ci id="Thmproofx10.p3.3.m3.1.2.3.3.cmml" xref="Thmproofx10.p3.3.m3.1.2.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p3.3.m3.1c">|f|=m_{c}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p3.3.m3.1d">| italic_f | = italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="F_{n}(\sigma,\alpha)\geq\left|p_{min}\right|+k\alpha\beta(\sigma)\geq\left|p_{% min}\right|+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha" class="ltx_Math" display="inline" id="Thmproofx10.p3.4.m4.7"><semantics id="Thmproofx10.p3.4.m4.7a"><mrow id="Thmproofx10.p3.4.m4.7.7" xref="Thmproofx10.p3.4.m4.7.7.cmml"><mrow id="Thmproofx10.p3.4.m4.7.7.4" xref="Thmproofx10.p3.4.m4.7.7.4.cmml"><msub id="Thmproofx10.p3.4.m4.7.7.4.2" xref="Thmproofx10.p3.4.m4.7.7.4.2.cmml"><mi id="Thmproofx10.p3.4.m4.7.7.4.2.2" xref="Thmproofx10.p3.4.m4.7.7.4.2.2.cmml">F</mi><mi id="Thmproofx10.p3.4.m4.7.7.4.2.3" xref="Thmproofx10.p3.4.m4.7.7.4.2.3.cmml">n</mi></msub><mo id="Thmproofx10.p3.4.m4.7.7.4.1" xref="Thmproofx10.p3.4.m4.7.7.4.1.cmml">⁢</mo><mrow id="Thmproofx10.p3.4.m4.7.7.4.3.2" xref="Thmproofx10.p3.4.m4.7.7.4.3.1.cmml"><mo id="Thmproofx10.p3.4.m4.7.7.4.3.2.1" stretchy="false" xref="Thmproofx10.p3.4.m4.7.7.4.3.1.cmml">(</mo><mi id="Thmproofx10.p3.4.m4.3.3" xref="Thmproofx10.p3.4.m4.3.3.cmml">σ</mi><mo id="Thmproofx10.p3.4.m4.7.7.4.3.2.2" xref="Thmproofx10.p3.4.m4.7.7.4.3.1.cmml">,</mo><mi id="Thmproofx10.p3.4.m4.4.4" xref="Thmproofx10.p3.4.m4.4.4.cmml">α</mi><mo id="Thmproofx10.p3.4.m4.7.7.4.3.2.3" stretchy="false" xref="Thmproofx10.p3.4.m4.7.7.4.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx10.p3.4.m4.7.7.5" xref="Thmproofx10.p3.4.m4.7.7.5.cmml">≥</mo><mrow id="Thmproofx10.p3.4.m4.6.6.1" xref="Thmproofx10.p3.4.m4.6.6.1.cmml"><mrow id="Thmproofx10.p3.4.m4.6.6.1.1.1" xref="Thmproofx10.p3.4.m4.6.6.1.1.2.cmml"><mo id="Thmproofx10.p3.4.m4.6.6.1.1.1.2" xref="Thmproofx10.p3.4.m4.6.6.1.1.2.1.cmml">|</mo><msub id="Thmproofx10.p3.4.m4.6.6.1.1.1.1" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.cmml"><mi id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.2" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.cmml"><mi id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.2" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.1" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.3" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.3.cmml">i</mi><mo id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.1a" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.4" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx10.p3.4.m4.6.6.1.1.1.3" xref="Thmproofx10.p3.4.m4.6.6.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx10.p3.4.m4.6.6.1.2" xref="Thmproofx10.p3.4.m4.6.6.1.2.cmml">+</mo><mrow id="Thmproofx10.p3.4.m4.6.6.1.3" xref="Thmproofx10.p3.4.m4.6.6.1.3.cmml"><mi id="Thmproofx10.p3.4.m4.6.6.1.3.2" xref="Thmproofx10.p3.4.m4.6.6.1.3.2.cmml">k</mi><mo id="Thmproofx10.p3.4.m4.6.6.1.3.1" xref="Thmproofx10.p3.4.m4.6.6.1.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.6.6.1.3.3" xref="Thmproofx10.p3.4.m4.6.6.1.3.3.cmml">α</mi><mo id="Thmproofx10.p3.4.m4.6.6.1.3.1a" xref="Thmproofx10.p3.4.m4.6.6.1.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.6.6.1.3.4" xref="Thmproofx10.p3.4.m4.6.6.1.3.4.cmml">β</mi><mo id="Thmproofx10.p3.4.m4.6.6.1.3.1b" xref="Thmproofx10.p3.4.m4.6.6.1.3.1.cmml">⁢</mo><mrow id="Thmproofx10.p3.4.m4.6.6.1.3.5.2" xref="Thmproofx10.p3.4.m4.6.6.1.3.cmml"><mo id="Thmproofx10.p3.4.m4.6.6.1.3.5.2.1" stretchy="false" xref="Thmproofx10.p3.4.m4.6.6.1.3.cmml">(</mo><mi id="Thmproofx10.p3.4.m4.5.5" xref="Thmproofx10.p3.4.m4.5.5.cmml">σ</mi><mo id="Thmproofx10.p3.4.m4.6.6.1.3.5.2.2" stretchy="false" xref="Thmproofx10.p3.4.m4.6.6.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx10.p3.4.m4.7.7.6" xref="Thmproofx10.p3.4.m4.7.7.6.cmml">≥</mo><mrow id="Thmproofx10.p3.4.m4.7.7.2" xref="Thmproofx10.p3.4.m4.7.7.2.cmml"><mrow id="Thmproofx10.p3.4.m4.7.7.2.1.1" xref="Thmproofx10.p3.4.m4.7.7.2.1.2.cmml"><mo id="Thmproofx10.p3.4.m4.7.7.2.1.1.2" xref="Thmproofx10.p3.4.m4.7.7.2.1.2.1.cmml">|</mo><msub id="Thmproofx10.p3.4.m4.7.7.2.1.1.1" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.cmml"><mi id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.2" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.2.cmml">p</mi><mrow id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.cmml"><mi id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.2" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.1" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.3" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.3.cmml">i</mi><mo id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.1a" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.4" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx10.p3.4.m4.7.7.2.1.1.3" xref="Thmproofx10.p3.4.m4.7.7.2.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx10.p3.4.m4.7.7.2.2" xref="Thmproofx10.p3.4.m4.7.7.2.2.cmml">+</mo><mrow id="Thmproofx10.p3.4.m4.7.7.2.3" xref="Thmproofx10.p3.4.m4.7.7.2.3.cmml"><mfrac id="Thmproofx10.p3.4.m4.2.2" xref="Thmproofx10.p3.4.m4.2.2.cmml"><msub id="Thmproofx10.p3.4.m4.2.2.2" xref="Thmproofx10.p3.4.m4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx10.p3.4.m4.2.2.2.4" xref="Thmproofx10.p3.4.m4.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx10.p3.4.m4.2.2.2.2.2.2" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.3.cmml"><msub id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.cmml"><mi id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.2" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.2.cmml">m</mi><mi id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.3" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.3.cmml">c</mi></msub><mo id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.2" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx10.p3.4.m4.1.1.1.1.1.1" xref="Thmproofx10.p3.4.m4.1.1.1.1.1.1.cmml">n</mi></mrow></msub><msub id="Thmproofx10.p3.4.m4.2.2.4" xref="Thmproofx10.p3.4.m4.2.2.4.cmml"><mi id="Thmproofx10.p3.4.m4.2.2.4.2" xref="Thmproofx10.p3.4.m4.2.2.4.2.cmml">m</mi><mi id="Thmproofx10.p3.4.m4.2.2.4.3" xref="Thmproofx10.p3.4.m4.2.2.4.3.cmml">c</mi></msub></mfrac><mo id="Thmproofx10.p3.4.m4.7.7.2.3.1" xref="Thmproofx10.p3.4.m4.7.7.2.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.7.7.2.3.2" xref="Thmproofx10.p3.4.m4.7.7.2.3.2.cmml">k</mi><mo id="Thmproofx10.p3.4.m4.7.7.2.3.1a" xref="Thmproofx10.p3.4.m4.7.7.2.3.1.cmml">⁢</mo><mi id="Thmproofx10.p3.4.m4.7.7.2.3.3" xref="Thmproofx10.p3.4.m4.7.7.2.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx10.p3.4.m4.7b"><apply id="Thmproofx10.p3.4.m4.7.7.cmml" xref="Thmproofx10.p3.4.m4.7.7"><and id="Thmproofx10.p3.4.m4.7.7a.cmml" xref="Thmproofx10.p3.4.m4.7.7"></and><apply id="Thmproofx10.p3.4.m4.7.7b.cmml" xref="Thmproofx10.p3.4.m4.7.7"><geq id="Thmproofx10.p3.4.m4.7.7.5.cmml" xref="Thmproofx10.p3.4.m4.7.7.5"></geq><apply id="Thmproofx10.p3.4.m4.7.7.4.cmml" xref="Thmproofx10.p3.4.m4.7.7.4"><times id="Thmproofx10.p3.4.m4.7.7.4.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.4.1"></times><apply id="Thmproofx10.p3.4.m4.7.7.4.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.4.2"><csymbol cd="ambiguous" id="Thmproofx10.p3.4.m4.7.7.4.2.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.4.2">subscript</csymbol><ci id="Thmproofx10.p3.4.m4.7.7.4.2.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.4.2.2">𝐹</ci><ci id="Thmproofx10.p3.4.m4.7.7.4.2.3.cmml" xref="Thmproofx10.p3.4.m4.7.7.4.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx10.p3.4.m4.7.7.4.3.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.4.3.2"><ci id="Thmproofx10.p3.4.m4.3.3.cmml" xref="Thmproofx10.p3.4.m4.3.3">𝜎</ci><ci id="Thmproofx10.p3.4.m4.4.4.cmml" xref="Thmproofx10.p3.4.m4.4.4">𝛼</ci></interval></apply><apply id="Thmproofx10.p3.4.m4.6.6.1.cmml" xref="Thmproofx10.p3.4.m4.6.6.1"><plus id="Thmproofx10.p3.4.m4.6.6.1.2.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.2"></plus><apply id="Thmproofx10.p3.4.m4.6.6.1.1.2.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1"><abs id="Thmproofx10.p3.4.m4.6.6.1.1.2.1.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.2"></abs><apply id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.1.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1">subscript</csymbol><ci id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.2.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.2">𝑝</ci><apply id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3"><times id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.1.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.1"></times><ci id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.2.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.3.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.3">𝑖</ci><ci id="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.4.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Thmproofx10.p3.4.m4.6.6.1.3.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.3"><times id="Thmproofx10.p3.4.m4.6.6.1.3.1.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.3.1"></times><ci id="Thmproofx10.p3.4.m4.6.6.1.3.2.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.3.2">𝑘</ci><ci id="Thmproofx10.p3.4.m4.6.6.1.3.3.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.3.3">𝛼</ci><ci id="Thmproofx10.p3.4.m4.6.6.1.3.4.cmml" xref="Thmproofx10.p3.4.m4.6.6.1.3.4">𝛽</ci><ci id="Thmproofx10.p3.4.m4.5.5.cmml" xref="Thmproofx10.p3.4.m4.5.5">𝜎</ci></apply></apply></apply><apply id="Thmproofx10.p3.4.m4.7.7c.cmml" xref="Thmproofx10.p3.4.m4.7.7"><geq id="Thmproofx10.p3.4.m4.7.7.6.cmml" xref="Thmproofx10.p3.4.m4.7.7.6"></geq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx10.p3.4.m4.6.6.1.cmml" id="Thmproofx10.p3.4.m4.7.7d.cmml" xref="Thmproofx10.p3.4.m4.7.7"></share><apply id="Thmproofx10.p3.4.m4.7.7.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.2"><plus id="Thmproofx10.p3.4.m4.7.7.2.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.2"></plus><apply id="Thmproofx10.p3.4.m4.7.7.2.1.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1"><abs id="Thmproofx10.p3.4.m4.7.7.2.1.2.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.2"></abs><apply id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1">subscript</csymbol><ci id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.2">𝑝</ci><apply id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3"><times id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.1"></times><ci id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.2">𝑚</ci><ci id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.3.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.3">𝑖</ci><ci id="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.4.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Thmproofx10.p3.4.m4.7.7.2.3.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.3"><times id="Thmproofx10.p3.4.m4.7.7.2.3.1.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.3.1"></times><apply id="Thmproofx10.p3.4.m4.2.2.cmml" xref="Thmproofx10.p3.4.m4.2.2"><divide id="Thmproofx10.p3.4.m4.2.2.3.cmml" xref="Thmproofx10.p3.4.m4.2.2"></divide><apply id="Thmproofx10.p3.4.m4.2.2.2.cmml" xref="Thmproofx10.p3.4.m4.2.2.2"><csymbol cd="ambiguous" id="Thmproofx10.p3.4.m4.2.2.2.3.cmml" xref="Thmproofx10.p3.4.m4.2.2.2">subscript</csymbol><ci id="Thmproofx10.p3.4.m4.2.2.2.4.cmml" xref="Thmproofx10.p3.4.m4.2.2.2.4">ℰ</ci><list id="Thmproofx10.p3.4.m4.2.2.2.2.2.3.cmml" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2"><apply id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.cmml" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.1.cmml" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1">subscript</csymbol><ci id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.2.cmml" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.2">𝑚</ci><ci id="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.3.cmml" xref="Thmproofx10.p3.4.m4.2.2.2.2.2.2.1.3">𝑐</ci></apply><ci id="Thmproofx10.p3.4.m4.1.1.1.1.1.1.cmml" xref="Thmproofx10.p3.4.m4.1.1.1.1.1.1">𝑛</ci></list></apply><apply id="Thmproofx10.p3.4.m4.2.2.4.cmml" xref="Thmproofx10.p3.4.m4.2.2.4"><csymbol cd="ambiguous" id="Thmproofx10.p3.4.m4.2.2.4.1.cmml" xref="Thmproofx10.p3.4.m4.2.2.4">subscript</csymbol><ci id="Thmproofx10.p3.4.m4.2.2.4.2.cmml" xref="Thmproofx10.p3.4.m4.2.2.4.2">𝑚</ci><ci id="Thmproofx10.p3.4.m4.2.2.4.3.cmml" xref="Thmproofx10.p3.4.m4.2.2.4.3">𝑐</ci></apply></apply><ci id="Thmproofx10.p3.4.m4.7.7.2.3.2.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.3.2">𝑘</ci><ci id="Thmproofx10.p3.4.m4.7.7.2.3.3.cmml" xref="Thmproofx10.p3.4.m4.7.7.2.3.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p3.4.m4.7c">F_{n}(\sigma,\alpha)\geq\left|p_{min}\right|+k\alpha\beta(\sigma)\geq\left|p_{% min}\right|+\frac{\mathcal{E}_{m_{c},n}}{m_{c}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p3.4.m4.7d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≥ | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + italic_k italic_α italic_β ( italic_σ ) ≥ | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. In total we have for <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx10.p3.5.m5.1"><semantics id="Thmproofx10.p3.5.m5.1a"><mi id="Thmproofx10.p3.5.m5.1.1" xref="Thmproofx10.p3.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx10.p3.5.m5.1b"><ci id="Thmproofx10.p3.5.m5.1.1.cmml" xref="Thmproofx10.p3.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p3.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p3.5.m5.1d">italic_α</annotation></semantics></math> as in the statement that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx31"> <tbody id="Pt0.A3.Ex42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(G,n,k,\alpha)\geq\frac{\left|p_{min}\right|+\frac{\mathcal{E}_% {m_{c},n}}{m_{c}}k\alpha}{n\left|p_{max}\right|+\frac{k\alpha}{m_{c}}}\geq(1-% \varepsilon)\mathcal{E}_{m_{c},n}" class="ltx_Math" display="inline" id="Pt0.A3.Ex42.m1.11"><semantics id="Pt0.A3.Ex42.m1.11a"><mrow id="Pt0.A3.Ex42.m1.11.11" xref="Pt0.A3.Ex42.m1.11.11.cmml"><mrow id="Pt0.A3.Ex42.m1.11.11.3" xref="Pt0.A3.Ex42.m1.11.11.3.cmml"><mi id="Pt0.A3.Ex42.m1.11.11.3.2" xref="Pt0.A3.Ex42.m1.11.11.3.2.cmml">r</mi><mo id="Pt0.A3.Ex42.m1.11.11.3.1" xref="Pt0.A3.Ex42.m1.11.11.3.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex42.m1.11.11.3.3.2" xref="Pt0.A3.Ex42.m1.11.11.3.3.1.cmml"><mo id="Pt0.A3.Ex42.m1.11.11.3.3.2.1" stretchy="false" xref="Pt0.A3.Ex42.m1.11.11.3.3.1.cmml">(</mo><mi id="Pt0.A3.Ex42.m1.7.7" xref="Pt0.A3.Ex42.m1.7.7.cmml">G</mi><mo id="Pt0.A3.Ex42.m1.11.11.3.3.2.2" xref="Pt0.A3.Ex42.m1.11.11.3.3.1.cmml">,</mo><mi id="Pt0.A3.Ex42.m1.8.8" xref="Pt0.A3.Ex42.m1.8.8.cmml">n</mi><mo id="Pt0.A3.Ex42.m1.11.11.3.3.2.3" xref="Pt0.A3.Ex42.m1.11.11.3.3.1.cmml">,</mo><mi id="Pt0.A3.Ex42.m1.9.9" xref="Pt0.A3.Ex42.m1.9.9.cmml">k</mi><mo id="Pt0.A3.Ex42.m1.11.11.3.3.2.4" xref="Pt0.A3.Ex42.m1.11.11.3.3.1.cmml">,</mo><mi id="Pt0.A3.Ex42.m1.10.10" xref="Pt0.A3.Ex42.m1.10.10.cmml">α</mi><mo id="Pt0.A3.Ex42.m1.11.11.3.3.2.5" stretchy="false" xref="Pt0.A3.Ex42.m1.11.11.3.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A3.Ex42.m1.11.11.4" xref="Pt0.A3.Ex42.m1.11.11.4.cmml">≥</mo><mstyle displaystyle="true" id="Pt0.A3.Ex42.m1.4.4" xref="Pt0.A3.Ex42.m1.4.4.cmml"><mfrac id="Pt0.A3.Ex42.m1.4.4a" xref="Pt0.A3.Ex42.m1.4.4.cmml"><mrow id="Pt0.A3.Ex42.m1.3.3.3" xref="Pt0.A3.Ex42.m1.3.3.3.cmml"><mrow id="Pt0.A3.Ex42.m1.3.3.3.3.1" xref="Pt0.A3.Ex42.m1.3.3.3.3.2.cmml"><mo id="Pt0.A3.Ex42.m1.3.3.3.3.1.2" xref="Pt0.A3.Ex42.m1.3.3.3.3.2.1.cmml">|</mo><msub id="Pt0.A3.Ex42.m1.3.3.3.3.1.1" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.cmml"><mi id="Pt0.A3.Ex42.m1.3.3.3.3.1.1.2" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.2.cmml">p</mi><mrow id="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.cmml"><mi id="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.2" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.2.cmml">m</mi><mo id="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.1" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.3" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.3.cmml">i</mi><mo id="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.1a" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.4" xref="Pt0.A3.Ex42.m1.3.3.3.3.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Pt0.A3.Ex42.m1.3.3.3.3.1.3" xref="Pt0.A3.Ex42.m1.3.3.3.3.2.1.cmml">|</mo></mrow><mo id="Pt0.A3.Ex42.m1.3.3.3.4" xref="Pt0.A3.Ex42.m1.3.3.3.4.cmml">+</mo><mrow id="Pt0.A3.Ex42.m1.3.3.3.5" xref="Pt0.A3.Ex42.m1.3.3.3.5.cmml"><mfrac id="Pt0.A3.Ex42.m1.2.2.2.2" xref="Pt0.A3.Ex42.m1.2.2.2.2.cmml"><msub id="Pt0.A3.Ex42.m1.2.2.2.2.2" xref="Pt0.A3.Ex42.m1.2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A3.Ex42.m1.2.2.2.2.2.4" xref="Pt0.A3.Ex42.m1.2.2.2.2.2.4.cmml">ℰ</mi><mrow id="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2" xref="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.3.cmml"><msub id="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2.1" xref="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2.1.cmml"><mi id="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2.1.2" xref="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2.1.2.cmml">m</mi><mi id="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2.1.3" xref="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2.1.3.cmml">c</mi></msub><mo id="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.2.2" xref="Pt0.A3.Ex42.m1.2.2.2.2.2.2.2.3.cmml">,</mo><mi id="Pt0.A3.Ex42.m1.1.1.1.1.1.1.1.1" xref="Pt0.A3.Ex42.m1.1.1.1.1.1.1.1.1.cmml">n</mi></mrow></msub><msub id="Pt0.A3.Ex42.m1.2.2.2.2.4" xref="Pt0.A3.Ex42.m1.2.2.2.2.4.cmml"><mi id="Pt0.A3.Ex42.m1.2.2.2.2.4.2" xref="Pt0.A3.Ex42.m1.2.2.2.2.4.2.cmml">m</mi><mi id="Pt0.A3.Ex42.m1.2.2.2.2.4.3" xref="Pt0.A3.Ex42.m1.2.2.2.2.4.3.cmml">c</mi></msub></mfrac><mo id="Pt0.A3.Ex42.m1.3.3.3.5.1" xref="Pt0.A3.Ex42.m1.3.3.3.5.1.cmml">⁢</mo><mi id="Pt0.A3.Ex42.m1.3.3.3.5.2" xref="Pt0.A3.Ex42.m1.3.3.3.5.2.cmml">k</mi><mo id="Pt0.A3.Ex42.m1.3.3.3.5.1a" xref="Pt0.A3.Ex42.m1.3.3.3.5.1.cmml">⁢</mo><mi id="Pt0.A3.Ex42.m1.3.3.3.5.3" xref="Pt0.A3.Ex42.m1.3.3.3.5.3.cmml">α</mi></mrow></mrow><mrow id="Pt0.A3.Ex42.m1.4.4.4" xref="Pt0.A3.Ex42.m1.4.4.4.cmml"><mrow id="Pt0.A3.Ex42.m1.4.4.4.1" xref="Pt0.A3.Ex42.m1.4.4.4.1.cmml"><mi id="Pt0.A3.Ex42.m1.4.4.4.1.3" xref="Pt0.A3.Ex42.m1.4.4.4.1.3.cmml">n</mi><mo id="Pt0.A3.Ex42.m1.4.4.4.1.2" xref="Pt0.A3.Ex42.m1.4.4.4.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex42.m1.4.4.4.1.1.1" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.2.cmml"><mo id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.2" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.2.1.cmml">|</mo><msub id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.cmml"><mi id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.2" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.2" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.2.cmml">m</mi><mo id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.1" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.3" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.3.cmml">a</mi><mo id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.1a" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.4" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="Pt0.A3.Ex42.m1.4.4.4.1.1.1.3" xref="Pt0.A3.Ex42.m1.4.4.4.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="Pt0.A3.Ex42.m1.4.4.4.2" xref="Pt0.A3.Ex42.m1.4.4.4.2.cmml">+</mo><mfrac id="Pt0.A3.Ex42.m1.4.4.4.3" xref="Pt0.A3.Ex42.m1.4.4.4.3.cmml"><mrow id="Pt0.A3.Ex42.m1.4.4.4.3.2" xref="Pt0.A3.Ex42.m1.4.4.4.3.2.cmml"><mi id="Pt0.A3.Ex42.m1.4.4.4.3.2.2" xref="Pt0.A3.Ex42.m1.4.4.4.3.2.2.cmml">k</mi><mo id="Pt0.A3.Ex42.m1.4.4.4.3.2.1" xref="Pt0.A3.Ex42.m1.4.4.4.3.2.1.cmml">⁢</mo><mi id="Pt0.A3.Ex42.m1.4.4.4.3.2.3" xref="Pt0.A3.Ex42.m1.4.4.4.3.2.3.cmml">α</mi></mrow><msub id="Pt0.A3.Ex42.m1.4.4.4.3.3" xref="Pt0.A3.Ex42.m1.4.4.4.3.3.cmml"><mi id="Pt0.A3.Ex42.m1.4.4.4.3.3.2" xref="Pt0.A3.Ex42.m1.4.4.4.3.3.2.cmml">m</mi><mi id="Pt0.A3.Ex42.m1.4.4.4.3.3.3" xref="Pt0.A3.Ex42.m1.4.4.4.3.3.3.cmml">c</mi></msub></mfrac></mrow></mfrac></mstyle><mo id="Pt0.A3.Ex42.m1.11.11.5" xref="Pt0.A3.Ex42.m1.11.11.5.cmml">≥</mo><mrow id="Pt0.A3.Ex42.m1.11.11.1" xref="Pt0.A3.Ex42.m1.11.11.1.cmml"><mrow id="Pt0.A3.Ex42.m1.11.11.1.1.1" xref="Pt0.A3.Ex42.m1.11.11.1.1.1.1.cmml"><mo id="Pt0.A3.Ex42.m1.11.11.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex42.m1.11.11.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex42.m1.11.11.1.1.1.1" xref="Pt0.A3.Ex42.m1.11.11.1.1.1.1.cmml"><mn id="Pt0.A3.Ex42.m1.11.11.1.1.1.1.2" xref="Pt0.A3.Ex42.m1.11.11.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A3.Ex42.m1.11.11.1.1.1.1.1" xref="Pt0.A3.Ex42.m1.11.11.1.1.1.1.1.cmml">−</mo><mi id="Pt0.A3.Ex42.m1.11.11.1.1.1.1.3" xref="Pt0.A3.Ex42.m1.11.11.1.1.1.1.3.cmml">ε</mi></mrow><mo id="Pt0.A3.Ex42.m1.11.11.1.1.1.3" stretchy="false" xref="Pt0.A3.Ex42.m1.11.11.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A3.Ex42.m1.11.11.1.2" xref="Pt0.A3.Ex42.m1.11.11.1.2.cmml">⁢</mo><msub id="Pt0.A3.Ex42.m1.11.11.1.3" xref="Pt0.A3.Ex42.m1.11.11.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A3.Ex42.m1.11.11.1.3.2" xref="Pt0.A3.Ex42.m1.11.11.1.3.2.cmml">ℰ</mi><mrow id="Pt0.A3.Ex42.m1.6.6.2.2" xref="Pt0.A3.Ex42.m1.6.6.2.3.cmml"><msub id="Pt0.A3.Ex42.m1.6.6.2.2.1" xref="Pt0.A3.Ex42.m1.6.6.2.2.1.cmml"><mi id="Pt0.A3.Ex42.m1.6.6.2.2.1.2" xref="Pt0.A3.Ex42.m1.6.6.2.2.1.2.cmml">m</mi><mi id="Pt0.A3.Ex42.m1.6.6.2.2.1.3" xref="Pt0.A3.Ex42.m1.6.6.2.2.1.3.cmml">c</mi></msub><mo id="Pt0.A3.Ex42.m1.6.6.2.2.2" xref="Pt0.A3.Ex42.m1.6.6.2.3.cmml">,</mo><mi id="Pt0.A3.Ex42.m1.5.5.1.1" xref="Pt0.A3.Ex42.m1.5.5.1.1.cmml">n</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex42.m1.11b"><apply id="Pt0.A3.Ex42.m1.11.11.cmml" xref="Pt0.A3.Ex42.m1.11.11"><and id="Pt0.A3.Ex42.m1.11.11a.cmml" xref="Pt0.A3.Ex42.m1.11.11"></and><apply id="Pt0.A3.Ex42.m1.11.11b.cmml" xref="Pt0.A3.Ex42.m1.11.11"><geq id="Pt0.A3.Ex42.m1.11.11.4.cmml" xref="Pt0.A3.Ex42.m1.11.11.4"></geq><apply id="Pt0.A3.Ex42.m1.11.11.3.cmml" xref="Pt0.A3.Ex42.m1.11.11.3"><times id="Pt0.A3.Ex42.m1.11.11.3.1.cmml" xref="Pt0.A3.Ex42.m1.11.11.3.1"></times><ci id="Pt0.A3.Ex42.m1.11.11.3.2.cmml" xref="Pt0.A3.Ex42.m1.11.11.3.2">𝑟</ci><vector id="Pt0.A3.Ex42.m1.11.11.3.3.1.cmml" xref="Pt0.A3.Ex42.m1.11.11.3.3.2"><ci id="Pt0.A3.Ex42.m1.7.7.cmml" xref="Pt0.A3.Ex42.m1.7.7">𝐺</ci><ci id="Pt0.A3.Ex42.m1.8.8.cmml" xref="Pt0.A3.Ex42.m1.8.8">𝑛</ci><ci id="Pt0.A3.Ex42.m1.9.9.cmml" xref="Pt0.A3.Ex42.m1.9.9">𝑘</ci><ci id="Pt0.A3.Ex42.m1.10.10.cmml" xref="Pt0.A3.Ex42.m1.10.10">𝛼</ci></vector></apply><apply id="Pt0.A3.Ex42.m1.4.4.cmml" xref="Pt0.A3.Ex42.m1.4.4"><divide id="Pt0.A3.Ex42.m1.4.4.5.cmml" xref="Pt0.A3.Ex42.m1.4.4"></divide><apply id="Pt0.A3.Ex42.m1.3.3.3.cmml" xref="Pt0.A3.Ex42.m1.3.3.3"><plus id="Pt0.A3.Ex42.m1.3.3.3.4.cmml" xref="Pt0.A3.Ex42.m1.3.3.3.4"></plus><apply 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xref="Pt0.A3.Ex42.m1.11.11.1.3.2">ℰ</ci><list id="Pt0.A3.Ex42.m1.6.6.2.3.cmml" xref="Pt0.A3.Ex42.m1.6.6.2.2"><apply id="Pt0.A3.Ex42.m1.6.6.2.2.1.cmml" xref="Pt0.A3.Ex42.m1.6.6.2.2.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex42.m1.6.6.2.2.1.1.cmml" xref="Pt0.A3.Ex42.m1.6.6.2.2.1">subscript</csymbol><ci id="Pt0.A3.Ex42.m1.6.6.2.2.1.2.cmml" xref="Pt0.A3.Ex42.m1.6.6.2.2.1.2">𝑚</ci><ci id="Pt0.A3.Ex42.m1.6.6.2.2.1.3.cmml" xref="Pt0.A3.Ex42.m1.6.6.2.2.1.3">𝑐</ci></apply><ci id="Pt0.A3.Ex42.m1.5.5.1.1.cmml" xref="Pt0.A3.Ex42.m1.5.5.1.1">𝑛</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex42.m1.11c">\displaystyle r(G,n,k,\alpha)\geq\frac{\left|p_{min}\right|+\frac{\mathcal{E}_% {m_{c},n}}{m_{c}}k\alpha}{n\left|p_{max}\right|+\frac{k\alpha}{m_{c}}}\geq(1-% \varepsilon)\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex42.m1.11d">italic_r ( italic_G , italic_n , italic_k , italic_α ) ≥ divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG italic_k italic_α end_ARG start_ARG italic_n | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + divide start_ARG italic_k italic_α end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG end_ARG ≥ ( 1 - italic_ε ) caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="Thmproofx10.p4"> <p class="ltx_p" id="Thmproofx10.p4.2">Item 4 follows from Items 2 and 3 because <math alttext="\varepsilon\to 0" class="ltx_Math" display="inline" id="Thmproofx10.p4.1.m1.1"><semantics id="Thmproofx10.p4.1.m1.1a"><mrow id="Thmproofx10.p4.1.m1.1.1" xref="Thmproofx10.p4.1.m1.1.1.cmml"><mi id="Thmproofx10.p4.1.m1.1.1.2" xref="Thmproofx10.p4.1.m1.1.1.2.cmml">ε</mi><mo id="Thmproofx10.p4.1.m1.1.1.1" stretchy="false" xref="Thmproofx10.p4.1.m1.1.1.1.cmml">→</mo><mn id="Thmproofx10.p4.1.m1.1.1.3" xref="Thmproofx10.p4.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx10.p4.1.m1.1b"><apply id="Thmproofx10.p4.1.m1.1.1.cmml" xref="Thmproofx10.p4.1.m1.1.1"><ci id="Thmproofx10.p4.1.m1.1.1.1.cmml" xref="Thmproofx10.p4.1.m1.1.1.1">→</ci><ci id="Thmproofx10.p4.1.m1.1.1.2.cmml" xref="Thmproofx10.p4.1.m1.1.1.2">𝜀</ci><cn id="Thmproofx10.p4.1.m1.1.1.3.cmml" type="integer" xref="Thmproofx10.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p4.1.m1.1c">\varepsilon\to 0</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p4.1.m1.1d">italic_ε → 0</annotation></semantics></math> as <math alttext="\alpha\to\infty" class="ltx_Math" display="inline" id="Thmproofx10.p4.2.m2.1"><semantics id="Thmproofx10.p4.2.m2.1a"><mrow id="Thmproofx10.p4.2.m2.1.1" xref="Thmproofx10.p4.2.m2.1.1.cmml"><mi id="Thmproofx10.p4.2.m2.1.1.2" xref="Thmproofx10.p4.2.m2.1.1.2.cmml">α</mi><mo id="Thmproofx10.p4.2.m2.1.1.1" stretchy="false" xref="Thmproofx10.p4.2.m2.1.1.1.cmml">→</mo><mi id="Thmproofx10.p4.2.m2.1.1.3" mathvariant="normal" xref="Thmproofx10.p4.2.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx10.p4.2.m2.1b"><apply id="Thmproofx10.p4.2.m2.1.1.cmml" xref="Thmproofx10.p4.2.m2.1.1"><ci id="Thmproofx10.p4.2.m2.1.1.1.cmml" xref="Thmproofx10.p4.2.m2.1.1.1">→</ci><ci id="Thmproofx10.p4.2.m2.1.1.2.cmml" xref="Thmproofx10.p4.2.m2.1.1.2">𝛼</ci><infinity id="Thmproofx10.p4.2.m2.1.1.3.cmml" xref="Thmproofx10.p4.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx10.p4.2.m2.1c">\alpha\to\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproofx10.p4.2.m2.1d">italic_α → ∞</annotation></semantics></math>, completing the proof. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx11"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem4" title="Theorem 4.4 ‣ Properties of 𝑟_𝑈 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.4</span></a>)</h6> <div class="ltx_para" id="Thmproofx11.p1"> <p class="ltx_p" id="Thmproofx11.p1.1">The idea is that <math alttext="r_{U}" class="ltx_Math" display="inline" id="Thmproofx11.p1.1.m1.1"><semantics id="Thmproofx11.p1.1.m1.1a"><msub id="Thmproofx11.p1.1.m1.1.1" xref="Thmproofx11.p1.1.m1.1.1.cmml"><mi id="Thmproofx11.p1.1.m1.1.1.2" xref="Thmproofx11.p1.1.m1.1.1.2.cmml">r</mi><mi id="Thmproofx11.p1.1.m1.1.1.3" xref="Thmproofx11.p1.1.m1.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.1.m1.1b"><apply id="Thmproofx11.p1.1.m1.1.1.cmml" xref="Thmproofx11.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.1.m1.1.1.1.cmml" xref="Thmproofx11.p1.1.m1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.1.m1.1.1.2.cmml" xref="Thmproofx11.p1.1.m1.1.1.2">𝑟</ci><ci id="Thmproofx11.p1.1.m1.1.1.3.cmml" xref="Thmproofx11.p1.1.m1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.1.m1.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.1.m1.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> is a ratio of two minimums of sets of affine functions. Define:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx32"> <tbody id="Pt0.A3.Ex43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\varphi(\alpha,n)=\min_{U^{n}\in\mathcal{U}_{n}}F_{n}(U^{n},% \alpha)=\min_{i\in[m_{c}]}\left(\frac{\mathcal{E}_{i,n}}{i}\left(k\alpha+S_{i}% \right)\right)=:\min_{i\in[m_{c}]}h_{i}(\alpha,n)" class="ltx_math_unparsed" display="inline" id="Pt0.A3.Ex43.m1.7"><semantics id="Pt0.A3.Ex43.m1.7a"><mrow id="Pt0.A3.Ex43.m1.7b"><mi id="Pt0.A3.Ex43.m1.7.8">φ</mi><mrow id="Pt0.A3.Ex43.m1.7.9"><mo id="Pt0.A3.Ex43.m1.7.9.1" stretchy="false">(</mo><mi id="Pt0.A3.Ex43.m1.5.5">α</mi><mo id="Pt0.A3.Ex43.m1.7.9.2">,</mo><mi id="Pt0.A3.Ex43.m1.6.6">n</mi><mo id="Pt0.A3.Ex43.m1.7.9.3" stretchy="false">)</mo></mrow><mo id="Pt0.A3.Ex43.m1.7.10">=</mo><munder id="Pt0.A3.Ex43.m1.7.11"><mi id="Pt0.A3.Ex43.m1.7.11.2">min</mi><mrow id="Pt0.A3.Ex43.m1.7.11.3"><msup id="Pt0.A3.Ex43.m1.7.11.3.2"><mi id="Pt0.A3.Ex43.m1.7.11.3.2.2">U</mi><mi id="Pt0.A3.Ex43.m1.7.11.3.2.3">n</mi></msup><mo id="Pt0.A3.Ex43.m1.7.11.3.1">∈</mo><msub id="Pt0.A3.Ex43.m1.7.11.3.3"><mi class="ltx_font_mathcaligraphic" id="Pt0.A3.Ex43.m1.7.11.3.3.2">𝒰</mi><mi id="Pt0.A3.Ex43.m1.7.11.3.3.3">n</mi></msub></mrow></munder><msub id="Pt0.A3.Ex43.m1.7.12"><mi id="Pt0.A3.Ex43.m1.7.12.2">F</mi><mi id="Pt0.A3.Ex43.m1.7.12.3">n</mi></msub><mrow id="Pt0.A3.Ex43.m1.7.13"><mo id="Pt0.A3.Ex43.m1.7.13.1" stretchy="false">(</mo><msup id="Pt0.A3.Ex43.m1.7.13.2"><mi id="Pt0.A3.Ex43.m1.7.13.2.2">U</mi><mi id="Pt0.A3.Ex43.m1.7.13.2.3">n</mi></msup><mo id="Pt0.A3.Ex43.m1.7.13.3">,</mo><mi id="Pt0.A3.Ex43.m1.7.7">α</mi><mo id="Pt0.A3.Ex43.m1.7.13.4" stretchy="false">)</mo></mrow><mo id="Pt0.A3.Ex43.m1.7.14">=</mo><munder id="Pt0.A3.Ex43.m1.7.15"><mi id="Pt0.A3.Ex43.m1.7.15.2">min</mi><mrow id="Pt0.A3.Ex43.m1.1.1.1"><mi id="Pt0.A3.Ex43.m1.1.1.1.3">i</mi><mo id="Pt0.A3.Ex43.m1.1.1.1.2">∈</mo><mrow id="Pt0.A3.Ex43.m1.1.1.1.1.1"><mo id="Pt0.A3.Ex43.m1.1.1.1.1.1.2" stretchy="false">[</mo><msub id="Pt0.A3.Ex43.m1.1.1.1.1.1.1"><mi id="Pt0.A3.Ex43.m1.1.1.1.1.1.1.2">m</mi><mi id="Pt0.A3.Ex43.m1.1.1.1.1.1.1.3">c</mi></msub><mo id="Pt0.A3.Ex43.m1.1.1.1.1.1.3" stretchy="false">]</mo></mrow></mrow></munder><mrow id="Pt0.A3.Ex43.m1.7.16"><mo id="Pt0.A3.Ex43.m1.7.16.1">(</mo><mstyle displaystyle="true" id="Pt0.A3.Ex43.m1.3.3"><mfrac id="Pt0.A3.Ex43.m1.3.3a"><msub id="Pt0.A3.Ex43.m1.3.3.2"><mi class="ltx_font_mathcaligraphic" id="Pt0.A3.Ex43.m1.3.3.2.4">ℰ</mi><mrow id="Pt0.A3.Ex43.m1.3.3.2.2.2.4"><mi id="Pt0.A3.Ex43.m1.2.2.1.1.1.1">i</mi><mo id="Pt0.A3.Ex43.m1.3.3.2.2.2.4.1">,</mo><mi id="Pt0.A3.Ex43.m1.3.3.2.2.2.2">n</mi></mrow></msub><mi id="Pt0.A3.Ex43.m1.3.3.4">i</mi></mfrac></mstyle><mrow id="Pt0.A3.Ex43.m1.7.16.2"><mo id="Pt0.A3.Ex43.m1.7.16.2.1">(</mo><mi id="Pt0.A3.Ex43.m1.7.16.2.2">k</mi><mi id="Pt0.A3.Ex43.m1.7.16.2.3">α</mi><mo id="Pt0.A3.Ex43.m1.7.16.2.4">+</mo><msub id="Pt0.A3.Ex43.m1.7.16.2.5"><mi id="Pt0.A3.Ex43.m1.7.16.2.5.2">S</mi><mi id="Pt0.A3.Ex43.m1.7.16.2.5.3">i</mi></msub><mo id="Pt0.A3.Ex43.m1.7.16.2.6">)</mo></mrow><mo id="Pt0.A3.Ex43.m1.7.16.3">)</mo></mrow><mo id="Pt0.A3.Ex43.m1.7.17" rspace="0em">=</mo><mo id="Pt0.A3.Ex43.m1.7.18" rspace="0.278em">:</mo><munder id="Pt0.A3.Ex43.m1.7.19"><mi id="Pt0.A3.Ex43.m1.7.19.2">min</mi><mrow id="Pt0.A3.Ex43.m1.4.4.1"><mi id="Pt0.A3.Ex43.m1.4.4.1.3">i</mi><mo id="Pt0.A3.Ex43.m1.4.4.1.2">∈</mo><mrow id="Pt0.A3.Ex43.m1.4.4.1.1.1"><mo id="Pt0.A3.Ex43.m1.4.4.1.1.1.2" stretchy="false">[</mo><msub id="Pt0.A3.Ex43.m1.4.4.1.1.1.1"><mi id="Pt0.A3.Ex43.m1.4.4.1.1.1.1.2">m</mi><mi id="Pt0.A3.Ex43.m1.4.4.1.1.1.1.3">c</mi></msub><mo id="Pt0.A3.Ex43.m1.4.4.1.1.1.3" stretchy="false">]</mo></mrow></mrow></munder><msub id="Pt0.A3.Ex43.m1.7.20"><mi id="Pt0.A3.Ex43.m1.7.20.2">h</mi><mi id="Pt0.A3.Ex43.m1.7.20.3">i</mi></msub><mrow id="Pt0.A3.Ex43.m1.7.21"><mo id="Pt0.A3.Ex43.m1.7.21.1" stretchy="false">(</mo><mi id="Pt0.A3.Ex43.m1.7.21.2">α</mi><mo id="Pt0.A3.Ex43.m1.7.21.3">,</mo><mi id="Pt0.A3.Ex43.m1.7.21.4">n</mi><mo id="Pt0.A3.Ex43.m1.7.21.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Pt0.A3.Ex43.m1.7c">\displaystyle\varphi(\alpha,n)=\min_{U^{n}\in\mathcal{U}_{n}}F_{n}(U^{n},% \alpha)=\min_{i\in[m_{c}]}\left(\frac{\mathcal{E}_{i,n}}{i}\left(k\alpha+S_{i}% \right)\right)=:\min_{i\in[m_{c}]}h_{i}(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex43.m1.7d">italic_φ ( italic_α , italic_n ) = roman_min start_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = roman_min start_POSTSUBSCRIPT italic_i ∈ [ italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT ( divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG ( italic_k italic_α + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) = : roman_min start_POSTSUBSCRIPT italic_i ∈ [ italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_α , italic_n )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx11.p1.30">Note that <math alttext="h_{i}" class="ltx_Math" display="inline" id="Thmproofx11.p1.2.m1.1"><semantics id="Thmproofx11.p1.2.m1.1a"><msub id="Thmproofx11.p1.2.m1.1.1" xref="Thmproofx11.p1.2.m1.1.1.cmml"><mi id="Thmproofx11.p1.2.m1.1.1.2" xref="Thmproofx11.p1.2.m1.1.1.2.cmml">h</mi><mi id="Thmproofx11.p1.2.m1.1.1.3" xref="Thmproofx11.p1.2.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.2.m1.1b"><apply id="Thmproofx11.p1.2.m1.1.1.cmml" xref="Thmproofx11.p1.2.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.2.m1.1.1.1.cmml" xref="Thmproofx11.p1.2.m1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.2.m1.1.1.2.cmml" xref="Thmproofx11.p1.2.m1.1.1.2">ℎ</ci><ci id="Thmproofx11.p1.2.m1.1.1.3.cmml" xref="Thmproofx11.p1.2.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.2.m1.1c">h_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.2.m1.1d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is an affine function in <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx11.p1.3.m2.1"><semantics id="Thmproofx11.p1.3.m2.1a"><mi id="Thmproofx11.p1.3.m2.1.1" xref="Thmproofx11.p1.3.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.3.m2.1b"><ci id="Thmproofx11.p1.3.m2.1.1.cmml" xref="Thmproofx11.p1.3.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.3.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.3.m2.1d">italic_α</annotation></semantics></math>, and that <math alttext="r_{U}(G,n,k,\alpha)=\frac{\varphi(\alpha,n)}{\varphi(\alpha,1)}" class="ltx_Math" display="inline" id="Thmproofx11.p1.4.m3.8"><semantics id="Thmproofx11.p1.4.m3.8a"><mrow id="Thmproofx11.p1.4.m3.8.9" xref="Thmproofx11.p1.4.m3.8.9.cmml"><mrow id="Thmproofx11.p1.4.m3.8.9.2" xref="Thmproofx11.p1.4.m3.8.9.2.cmml"><msub id="Thmproofx11.p1.4.m3.8.9.2.2" xref="Thmproofx11.p1.4.m3.8.9.2.2.cmml"><mi id="Thmproofx11.p1.4.m3.8.9.2.2.2" xref="Thmproofx11.p1.4.m3.8.9.2.2.2.cmml">r</mi><mi id="Thmproofx11.p1.4.m3.8.9.2.2.3" xref="Thmproofx11.p1.4.m3.8.9.2.2.3.cmml">U</mi></msub><mo id="Thmproofx11.p1.4.m3.8.9.2.1" xref="Thmproofx11.p1.4.m3.8.9.2.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.4.m3.8.9.2.3.2" xref="Thmproofx11.p1.4.m3.8.9.2.3.1.cmml"><mo id="Thmproofx11.p1.4.m3.8.9.2.3.2.1" stretchy="false" xref="Thmproofx11.p1.4.m3.8.9.2.3.1.cmml">(</mo><mi id="Thmproofx11.p1.4.m3.5.5" xref="Thmproofx11.p1.4.m3.5.5.cmml">G</mi><mo id="Thmproofx11.p1.4.m3.8.9.2.3.2.2" xref="Thmproofx11.p1.4.m3.8.9.2.3.1.cmml">,</mo><mi id="Thmproofx11.p1.4.m3.6.6" xref="Thmproofx11.p1.4.m3.6.6.cmml">n</mi><mo id="Thmproofx11.p1.4.m3.8.9.2.3.2.3" xref="Thmproofx11.p1.4.m3.8.9.2.3.1.cmml">,</mo><mi id="Thmproofx11.p1.4.m3.7.7" xref="Thmproofx11.p1.4.m3.7.7.cmml">k</mi><mo id="Thmproofx11.p1.4.m3.8.9.2.3.2.4" xref="Thmproofx11.p1.4.m3.8.9.2.3.1.cmml">,</mo><mi id="Thmproofx11.p1.4.m3.8.8" xref="Thmproofx11.p1.4.m3.8.8.cmml">α</mi><mo id="Thmproofx11.p1.4.m3.8.9.2.3.2.5" stretchy="false" xref="Thmproofx11.p1.4.m3.8.9.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx11.p1.4.m3.8.9.1" xref="Thmproofx11.p1.4.m3.8.9.1.cmml">=</mo><mfrac id="Thmproofx11.p1.4.m3.4.4" xref="Thmproofx11.p1.4.m3.4.4.cmml"><mrow id="Thmproofx11.p1.4.m3.2.2.2" xref="Thmproofx11.p1.4.m3.2.2.2.cmml"><mi id="Thmproofx11.p1.4.m3.2.2.2.4" xref="Thmproofx11.p1.4.m3.2.2.2.4.cmml">φ</mi><mo id="Thmproofx11.p1.4.m3.2.2.2.3" xref="Thmproofx11.p1.4.m3.2.2.2.3.cmml">⁢</mo><mrow id="Thmproofx11.p1.4.m3.2.2.2.5.2" xref="Thmproofx11.p1.4.m3.2.2.2.5.1.cmml"><mo id="Thmproofx11.p1.4.m3.2.2.2.5.2.1" stretchy="false" xref="Thmproofx11.p1.4.m3.2.2.2.5.1.cmml">(</mo><mi id="Thmproofx11.p1.4.m3.1.1.1.1" xref="Thmproofx11.p1.4.m3.1.1.1.1.cmml">α</mi><mo id="Thmproofx11.p1.4.m3.2.2.2.5.2.2" xref="Thmproofx11.p1.4.m3.2.2.2.5.1.cmml">,</mo><mi id="Thmproofx11.p1.4.m3.2.2.2.2" xref="Thmproofx11.p1.4.m3.2.2.2.2.cmml">n</mi><mo id="Thmproofx11.p1.4.m3.2.2.2.5.2.3" stretchy="false" xref="Thmproofx11.p1.4.m3.2.2.2.5.1.cmml">)</mo></mrow></mrow><mrow id="Thmproofx11.p1.4.m3.4.4.4" xref="Thmproofx11.p1.4.m3.4.4.4.cmml"><mi id="Thmproofx11.p1.4.m3.4.4.4.4" xref="Thmproofx11.p1.4.m3.4.4.4.4.cmml">φ</mi><mo id="Thmproofx11.p1.4.m3.4.4.4.3" xref="Thmproofx11.p1.4.m3.4.4.4.3.cmml">⁢</mo><mrow id="Thmproofx11.p1.4.m3.4.4.4.5.2" xref="Thmproofx11.p1.4.m3.4.4.4.5.1.cmml"><mo id="Thmproofx11.p1.4.m3.4.4.4.5.2.1" stretchy="false" xref="Thmproofx11.p1.4.m3.4.4.4.5.1.cmml">(</mo><mi id="Thmproofx11.p1.4.m3.3.3.3.1" xref="Thmproofx11.p1.4.m3.3.3.3.1.cmml">α</mi><mo id="Thmproofx11.p1.4.m3.4.4.4.5.2.2" xref="Thmproofx11.p1.4.m3.4.4.4.5.1.cmml">,</mo><mn id="Thmproofx11.p1.4.m3.4.4.4.2" xref="Thmproofx11.p1.4.m3.4.4.4.2.cmml">1</mn><mo id="Thmproofx11.p1.4.m3.4.4.4.5.2.3" stretchy="false" xref="Thmproofx11.p1.4.m3.4.4.4.5.1.cmml">)</mo></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.4.m3.8b"><apply id="Thmproofx11.p1.4.m3.8.9.cmml" xref="Thmproofx11.p1.4.m3.8.9"><eq id="Thmproofx11.p1.4.m3.8.9.1.cmml" xref="Thmproofx11.p1.4.m3.8.9.1"></eq><apply id="Thmproofx11.p1.4.m3.8.9.2.cmml" xref="Thmproofx11.p1.4.m3.8.9.2"><times id="Thmproofx11.p1.4.m3.8.9.2.1.cmml" xref="Thmproofx11.p1.4.m3.8.9.2.1"></times><apply id="Thmproofx11.p1.4.m3.8.9.2.2.cmml" xref="Thmproofx11.p1.4.m3.8.9.2.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.4.m3.8.9.2.2.1.cmml" xref="Thmproofx11.p1.4.m3.8.9.2.2">subscript</csymbol><ci id="Thmproofx11.p1.4.m3.8.9.2.2.2.cmml" xref="Thmproofx11.p1.4.m3.8.9.2.2.2">𝑟</ci><ci id="Thmproofx11.p1.4.m3.8.9.2.2.3.cmml" xref="Thmproofx11.p1.4.m3.8.9.2.2.3">𝑈</ci></apply><vector id="Thmproofx11.p1.4.m3.8.9.2.3.1.cmml" xref="Thmproofx11.p1.4.m3.8.9.2.3.2"><ci id="Thmproofx11.p1.4.m3.5.5.cmml" xref="Thmproofx11.p1.4.m3.5.5">𝐺</ci><ci id="Thmproofx11.p1.4.m3.6.6.cmml" xref="Thmproofx11.p1.4.m3.6.6">𝑛</ci><ci id="Thmproofx11.p1.4.m3.7.7.cmml" xref="Thmproofx11.p1.4.m3.7.7">𝑘</ci><ci id="Thmproofx11.p1.4.m3.8.8.cmml" xref="Thmproofx11.p1.4.m3.8.8">𝛼</ci></vector></apply><apply id="Thmproofx11.p1.4.m3.4.4.cmml" xref="Thmproofx11.p1.4.m3.4.4"><divide id="Thmproofx11.p1.4.m3.4.4.5.cmml" xref="Thmproofx11.p1.4.m3.4.4"></divide><apply id="Thmproofx11.p1.4.m3.2.2.2.cmml" xref="Thmproofx11.p1.4.m3.2.2.2"><times id="Thmproofx11.p1.4.m3.2.2.2.3.cmml" xref="Thmproofx11.p1.4.m3.2.2.2.3"></times><ci id="Thmproofx11.p1.4.m3.2.2.2.4.cmml" xref="Thmproofx11.p1.4.m3.2.2.2.4">𝜑</ci><interval closure="open" id="Thmproofx11.p1.4.m3.2.2.2.5.1.cmml" xref="Thmproofx11.p1.4.m3.2.2.2.5.2"><ci id="Thmproofx11.p1.4.m3.1.1.1.1.cmml" xref="Thmproofx11.p1.4.m3.1.1.1.1">𝛼</ci><ci id="Thmproofx11.p1.4.m3.2.2.2.2.cmml" xref="Thmproofx11.p1.4.m3.2.2.2.2">𝑛</ci></interval></apply><apply id="Thmproofx11.p1.4.m3.4.4.4.cmml" xref="Thmproofx11.p1.4.m3.4.4.4"><times id="Thmproofx11.p1.4.m3.4.4.4.3.cmml" xref="Thmproofx11.p1.4.m3.4.4.4.3"></times><ci id="Thmproofx11.p1.4.m3.4.4.4.4.cmml" xref="Thmproofx11.p1.4.m3.4.4.4.4">𝜑</ci><interval closure="open" id="Thmproofx11.p1.4.m3.4.4.4.5.1.cmml" xref="Thmproofx11.p1.4.m3.4.4.4.5.2"><ci id="Thmproofx11.p1.4.m3.3.3.3.1.cmml" xref="Thmproofx11.p1.4.m3.3.3.3.1">𝛼</ci><cn id="Thmproofx11.p1.4.m3.4.4.4.2.cmml" type="integer" xref="Thmproofx11.p1.4.m3.4.4.4.2">1</cn></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.4.m3.8c">r_{U}(G,n,k,\alpha)=\frac{\varphi(\alpha,n)}{\varphi(\alpha,1)}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.4.m3.8d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α ) = divide start_ARG italic_φ ( italic_α , italic_n ) end_ARG start_ARG italic_φ ( italic_α , 1 ) end_ARG</annotation></semantics></math>. Therefore <math alttext="r_{U}" class="ltx_Math" display="inline" id="Thmproofx11.p1.5.m4.1"><semantics id="Thmproofx11.p1.5.m4.1a"><msub id="Thmproofx11.p1.5.m4.1.1" xref="Thmproofx11.p1.5.m4.1.1.cmml"><mi id="Thmproofx11.p1.5.m4.1.1.2" xref="Thmproofx11.p1.5.m4.1.1.2.cmml">r</mi><mi id="Thmproofx11.p1.5.m4.1.1.3" xref="Thmproofx11.p1.5.m4.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.5.m4.1b"><apply id="Thmproofx11.p1.5.m4.1.1.cmml" xref="Thmproofx11.p1.5.m4.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.5.m4.1.1.1.cmml" xref="Thmproofx11.p1.5.m4.1.1">subscript</csymbol><ci id="Thmproofx11.p1.5.m4.1.1.2.cmml" xref="Thmproofx11.p1.5.m4.1.1.2">𝑟</ci><ci id="Thmproofx11.p1.5.m4.1.1.3.cmml" xref="Thmproofx11.p1.5.m4.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.5.m4.1c">r_{U}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.5.m4.1d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> is smooth a.s. and continuous. We show that <math alttext="\frac{\varphi(\alpha,n)}{\varphi(\alpha,1)}" class="ltx_Math" display="inline" id="Thmproofx11.p1.6.m5.4"><semantics id="Thmproofx11.p1.6.m5.4a"><mfrac id="Thmproofx11.p1.6.m5.4.4" xref="Thmproofx11.p1.6.m5.4.4.cmml"><mrow id="Thmproofx11.p1.6.m5.2.2.2" xref="Thmproofx11.p1.6.m5.2.2.2.cmml"><mi id="Thmproofx11.p1.6.m5.2.2.2.4" xref="Thmproofx11.p1.6.m5.2.2.2.4.cmml">φ</mi><mo id="Thmproofx11.p1.6.m5.2.2.2.3" xref="Thmproofx11.p1.6.m5.2.2.2.3.cmml">⁢</mo><mrow id="Thmproofx11.p1.6.m5.2.2.2.5.2" xref="Thmproofx11.p1.6.m5.2.2.2.5.1.cmml"><mo id="Thmproofx11.p1.6.m5.2.2.2.5.2.1" stretchy="false" xref="Thmproofx11.p1.6.m5.2.2.2.5.1.cmml">(</mo><mi id="Thmproofx11.p1.6.m5.1.1.1.1" xref="Thmproofx11.p1.6.m5.1.1.1.1.cmml">α</mi><mo id="Thmproofx11.p1.6.m5.2.2.2.5.2.2" xref="Thmproofx11.p1.6.m5.2.2.2.5.1.cmml">,</mo><mi id="Thmproofx11.p1.6.m5.2.2.2.2" xref="Thmproofx11.p1.6.m5.2.2.2.2.cmml">n</mi><mo id="Thmproofx11.p1.6.m5.2.2.2.5.2.3" stretchy="false" xref="Thmproofx11.p1.6.m5.2.2.2.5.1.cmml">)</mo></mrow></mrow><mrow id="Thmproofx11.p1.6.m5.4.4.4" xref="Thmproofx11.p1.6.m5.4.4.4.cmml"><mi id="Thmproofx11.p1.6.m5.4.4.4.4" xref="Thmproofx11.p1.6.m5.4.4.4.4.cmml">φ</mi><mo id="Thmproofx11.p1.6.m5.4.4.4.3" xref="Thmproofx11.p1.6.m5.4.4.4.3.cmml">⁢</mo><mrow id="Thmproofx11.p1.6.m5.4.4.4.5.2" xref="Thmproofx11.p1.6.m5.4.4.4.5.1.cmml"><mo id="Thmproofx11.p1.6.m5.4.4.4.5.2.1" stretchy="false" xref="Thmproofx11.p1.6.m5.4.4.4.5.1.cmml">(</mo><mi id="Thmproofx11.p1.6.m5.3.3.3.1" xref="Thmproofx11.p1.6.m5.3.3.3.1.cmml">α</mi><mo id="Thmproofx11.p1.6.m5.4.4.4.5.2.2" xref="Thmproofx11.p1.6.m5.4.4.4.5.1.cmml">,</mo><mn id="Thmproofx11.p1.6.m5.4.4.4.2" xref="Thmproofx11.p1.6.m5.4.4.4.2.cmml">1</mn><mo id="Thmproofx11.p1.6.m5.4.4.4.5.2.3" stretchy="false" xref="Thmproofx11.p1.6.m5.4.4.4.5.1.cmml">)</mo></mrow></mrow></mfrac><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.6.m5.4b"><apply id="Thmproofx11.p1.6.m5.4.4.cmml" xref="Thmproofx11.p1.6.m5.4.4"><divide id="Thmproofx11.p1.6.m5.4.4.5.cmml" xref="Thmproofx11.p1.6.m5.4.4"></divide><apply id="Thmproofx11.p1.6.m5.2.2.2.cmml" xref="Thmproofx11.p1.6.m5.2.2.2"><times id="Thmproofx11.p1.6.m5.2.2.2.3.cmml" xref="Thmproofx11.p1.6.m5.2.2.2.3"></times><ci id="Thmproofx11.p1.6.m5.2.2.2.4.cmml" xref="Thmproofx11.p1.6.m5.2.2.2.4">𝜑</ci><interval closure="open" id="Thmproofx11.p1.6.m5.2.2.2.5.1.cmml" xref="Thmproofx11.p1.6.m5.2.2.2.5.2"><ci id="Thmproofx11.p1.6.m5.1.1.1.1.cmml" xref="Thmproofx11.p1.6.m5.1.1.1.1">𝛼</ci><ci id="Thmproofx11.p1.6.m5.2.2.2.2.cmml" xref="Thmproofx11.p1.6.m5.2.2.2.2">𝑛</ci></interval></apply><apply id="Thmproofx11.p1.6.m5.4.4.4.cmml" xref="Thmproofx11.p1.6.m5.4.4.4"><times id="Thmproofx11.p1.6.m5.4.4.4.3.cmml" xref="Thmproofx11.p1.6.m5.4.4.4.3"></times><ci id="Thmproofx11.p1.6.m5.4.4.4.4.cmml" xref="Thmproofx11.p1.6.m5.4.4.4.4">𝜑</ci><interval closure="open" id="Thmproofx11.p1.6.m5.4.4.4.5.1.cmml" xref="Thmproofx11.p1.6.m5.4.4.4.5.2"><ci id="Thmproofx11.p1.6.m5.3.3.3.1.cmml" xref="Thmproofx11.p1.6.m5.3.3.3.1">𝛼</ci><cn id="Thmproofx11.p1.6.m5.4.4.4.2.cmml" type="integer" xref="Thmproofx11.p1.6.m5.4.4.4.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.6.m5.4c">\frac{\varphi(\alpha,n)}{\varphi(\alpha,1)}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.6.m5.4d">divide start_ARG italic_φ ( italic_α , italic_n ) end_ARG start_ARG italic_φ ( italic_α , 1 ) end_ARG</annotation></semantics></math> is strictly increasing in <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx11.p1.7.m6.1"><semantics id="Thmproofx11.p1.7.m6.1a"><mi id="Thmproofx11.p1.7.m6.1.1" xref="Thmproofx11.p1.7.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.7.m6.1b"><ci id="Thmproofx11.p1.7.m6.1.1.cmml" xref="Thmproofx11.p1.7.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.7.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.7.m6.1d">italic_α</annotation></semantics></math>. Fix <math alttext="\alpha_{0}" class="ltx_Math" display="inline" id="Thmproofx11.p1.8.m7.1"><semantics id="Thmproofx11.p1.8.m7.1a"><msub id="Thmproofx11.p1.8.m7.1.1" xref="Thmproofx11.p1.8.m7.1.1.cmml"><mi id="Thmproofx11.p1.8.m7.1.1.2" xref="Thmproofx11.p1.8.m7.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.8.m7.1.1.3" xref="Thmproofx11.p1.8.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.8.m7.1b"><apply id="Thmproofx11.p1.8.m7.1.1.cmml" xref="Thmproofx11.p1.8.m7.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.8.m7.1.1.1.cmml" xref="Thmproofx11.p1.8.m7.1.1">subscript</csymbol><ci id="Thmproofx11.p1.8.m7.1.1.2.cmml" xref="Thmproofx11.p1.8.m7.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.8.m7.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.8.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.8.m7.1c">\alpha_{0}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.8.m7.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. We ignore the countable set of points where more than one <math alttext="h_{i}" class="ltx_Math" display="inline" id="Thmproofx11.p1.9.m8.1"><semantics id="Thmproofx11.p1.9.m8.1a"><msub id="Thmproofx11.p1.9.m8.1.1" xref="Thmproofx11.p1.9.m8.1.1.cmml"><mi id="Thmproofx11.p1.9.m8.1.1.2" xref="Thmproofx11.p1.9.m8.1.1.2.cmml">h</mi><mi id="Thmproofx11.p1.9.m8.1.1.3" xref="Thmproofx11.p1.9.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.9.m8.1b"><apply id="Thmproofx11.p1.9.m8.1.1.cmml" xref="Thmproofx11.p1.9.m8.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.9.m8.1.1.1.cmml" xref="Thmproofx11.p1.9.m8.1.1">subscript</csymbol><ci id="Thmproofx11.p1.9.m8.1.1.2.cmml" xref="Thmproofx11.p1.9.m8.1.1.2">ℎ</ci><ci id="Thmproofx11.p1.9.m8.1.1.3.cmml" xref="Thmproofx11.p1.9.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.9.m8.1c">h_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.9.m8.1d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> attains <math alttext="\varphi(\alpha,n)" class="ltx_Math" display="inline" id="Thmproofx11.p1.10.m9.2"><semantics id="Thmproofx11.p1.10.m9.2a"><mrow id="Thmproofx11.p1.10.m9.2.3" xref="Thmproofx11.p1.10.m9.2.3.cmml"><mi id="Thmproofx11.p1.10.m9.2.3.2" xref="Thmproofx11.p1.10.m9.2.3.2.cmml">φ</mi><mo id="Thmproofx11.p1.10.m9.2.3.1" xref="Thmproofx11.p1.10.m9.2.3.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.10.m9.2.3.3.2" xref="Thmproofx11.p1.10.m9.2.3.3.1.cmml"><mo id="Thmproofx11.p1.10.m9.2.3.3.2.1" stretchy="false" xref="Thmproofx11.p1.10.m9.2.3.3.1.cmml">(</mo><mi id="Thmproofx11.p1.10.m9.1.1" xref="Thmproofx11.p1.10.m9.1.1.cmml">α</mi><mo id="Thmproofx11.p1.10.m9.2.3.3.2.2" xref="Thmproofx11.p1.10.m9.2.3.3.1.cmml">,</mo><mi id="Thmproofx11.p1.10.m9.2.2" xref="Thmproofx11.p1.10.m9.2.2.cmml">n</mi><mo id="Thmproofx11.p1.10.m9.2.3.3.2.3" stretchy="false" xref="Thmproofx11.p1.10.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.10.m9.2b"><apply id="Thmproofx11.p1.10.m9.2.3.cmml" xref="Thmproofx11.p1.10.m9.2.3"><times id="Thmproofx11.p1.10.m9.2.3.1.cmml" xref="Thmproofx11.p1.10.m9.2.3.1"></times><ci id="Thmproofx11.p1.10.m9.2.3.2.cmml" xref="Thmproofx11.p1.10.m9.2.3.2">𝜑</ci><interval closure="open" id="Thmproofx11.p1.10.m9.2.3.3.1.cmml" xref="Thmproofx11.p1.10.m9.2.3.3.2"><ci id="Thmproofx11.p1.10.m9.1.1.cmml" xref="Thmproofx11.p1.10.m9.1.1">𝛼</ci><ci id="Thmproofx11.p1.10.m9.2.2.cmml" xref="Thmproofx11.p1.10.m9.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.10.m9.2c">\varphi(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.10.m9.2d">italic_φ ( italic_α , italic_n )</annotation></semantics></math> for a fixed <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx11.p1.11.m10.1"><semantics id="Thmproofx11.p1.11.m10.1a"><mi id="Thmproofx11.p1.11.m10.1.1" xref="Thmproofx11.p1.11.m10.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.11.m10.1b"><ci id="Thmproofx11.p1.11.m10.1.1.cmml" xref="Thmproofx11.p1.11.m10.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.11.m10.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.11.m10.1d">italic_α</annotation></semantics></math>, because they are all isolated and the PoU is a continuous function in <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx11.p1.12.m11.1"><semantics id="Thmproofx11.p1.12.m11.1a"><mi id="Thmproofx11.p1.12.m11.1.1" xref="Thmproofx11.p1.12.m11.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.12.m11.1b"><ci id="Thmproofx11.p1.12.m11.1.1.cmml" xref="Thmproofx11.p1.12.m11.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.12.m11.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.12.m11.1d">italic_α</annotation></semantics></math>. Suppose <math alttext="\varphi(\alpha_{0},1)=h_{i}(\alpha_{0},1)" class="ltx_Math" display="inline" id="Thmproofx11.p1.13.m12.4"><semantics id="Thmproofx11.p1.13.m12.4a"><mrow id="Thmproofx11.p1.13.m12.4.4" xref="Thmproofx11.p1.13.m12.4.4.cmml"><mrow id="Thmproofx11.p1.13.m12.3.3.1" xref="Thmproofx11.p1.13.m12.3.3.1.cmml"><mi id="Thmproofx11.p1.13.m12.3.3.1.3" xref="Thmproofx11.p1.13.m12.3.3.1.3.cmml">φ</mi><mo id="Thmproofx11.p1.13.m12.3.3.1.2" xref="Thmproofx11.p1.13.m12.3.3.1.2.cmml">⁢</mo><mrow id="Thmproofx11.p1.13.m12.3.3.1.1.1" xref="Thmproofx11.p1.13.m12.3.3.1.1.2.cmml"><mo id="Thmproofx11.p1.13.m12.3.3.1.1.1.2" stretchy="false" xref="Thmproofx11.p1.13.m12.3.3.1.1.2.cmml">(</mo><msub id="Thmproofx11.p1.13.m12.3.3.1.1.1.1" xref="Thmproofx11.p1.13.m12.3.3.1.1.1.1.cmml"><mi id="Thmproofx11.p1.13.m12.3.3.1.1.1.1.2" xref="Thmproofx11.p1.13.m12.3.3.1.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.13.m12.3.3.1.1.1.1.3" xref="Thmproofx11.p1.13.m12.3.3.1.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p1.13.m12.3.3.1.1.1.3" xref="Thmproofx11.p1.13.m12.3.3.1.1.2.cmml">,</mo><mn id="Thmproofx11.p1.13.m12.1.1" xref="Thmproofx11.p1.13.m12.1.1.cmml">1</mn><mo id="Thmproofx11.p1.13.m12.3.3.1.1.1.4" stretchy="false" xref="Thmproofx11.p1.13.m12.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx11.p1.13.m12.4.4.3" xref="Thmproofx11.p1.13.m12.4.4.3.cmml">=</mo><mrow id="Thmproofx11.p1.13.m12.4.4.2" xref="Thmproofx11.p1.13.m12.4.4.2.cmml"><msub id="Thmproofx11.p1.13.m12.4.4.2.3" xref="Thmproofx11.p1.13.m12.4.4.2.3.cmml"><mi id="Thmproofx11.p1.13.m12.4.4.2.3.2" xref="Thmproofx11.p1.13.m12.4.4.2.3.2.cmml">h</mi><mi id="Thmproofx11.p1.13.m12.4.4.2.3.3" xref="Thmproofx11.p1.13.m12.4.4.2.3.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.13.m12.4.4.2.2" xref="Thmproofx11.p1.13.m12.4.4.2.2.cmml">⁢</mo><mrow id="Thmproofx11.p1.13.m12.4.4.2.1.1" xref="Thmproofx11.p1.13.m12.4.4.2.1.2.cmml"><mo id="Thmproofx11.p1.13.m12.4.4.2.1.1.2" stretchy="false" xref="Thmproofx11.p1.13.m12.4.4.2.1.2.cmml">(</mo><msub id="Thmproofx11.p1.13.m12.4.4.2.1.1.1" xref="Thmproofx11.p1.13.m12.4.4.2.1.1.1.cmml"><mi id="Thmproofx11.p1.13.m12.4.4.2.1.1.1.2" xref="Thmproofx11.p1.13.m12.4.4.2.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.13.m12.4.4.2.1.1.1.3" xref="Thmproofx11.p1.13.m12.4.4.2.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p1.13.m12.4.4.2.1.1.3" xref="Thmproofx11.p1.13.m12.4.4.2.1.2.cmml">,</mo><mn id="Thmproofx11.p1.13.m12.2.2" xref="Thmproofx11.p1.13.m12.2.2.cmml">1</mn><mo id="Thmproofx11.p1.13.m12.4.4.2.1.1.4" stretchy="false" xref="Thmproofx11.p1.13.m12.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.13.m12.4b"><apply id="Thmproofx11.p1.13.m12.4.4.cmml" xref="Thmproofx11.p1.13.m12.4.4"><eq id="Thmproofx11.p1.13.m12.4.4.3.cmml" xref="Thmproofx11.p1.13.m12.4.4.3"></eq><apply id="Thmproofx11.p1.13.m12.3.3.1.cmml" xref="Thmproofx11.p1.13.m12.3.3.1"><times id="Thmproofx11.p1.13.m12.3.3.1.2.cmml" xref="Thmproofx11.p1.13.m12.3.3.1.2"></times><ci id="Thmproofx11.p1.13.m12.3.3.1.3.cmml" xref="Thmproofx11.p1.13.m12.3.3.1.3">𝜑</ci><interval closure="open" id="Thmproofx11.p1.13.m12.3.3.1.1.2.cmml" xref="Thmproofx11.p1.13.m12.3.3.1.1.1"><apply id="Thmproofx11.p1.13.m12.3.3.1.1.1.1.cmml" xref="Thmproofx11.p1.13.m12.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.13.m12.3.3.1.1.1.1.1.cmml" xref="Thmproofx11.p1.13.m12.3.3.1.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.13.m12.3.3.1.1.1.1.2.cmml" xref="Thmproofx11.p1.13.m12.3.3.1.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.13.m12.3.3.1.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.13.m12.3.3.1.1.1.1.3">0</cn></apply><cn id="Thmproofx11.p1.13.m12.1.1.cmml" type="integer" xref="Thmproofx11.p1.13.m12.1.1">1</cn></interval></apply><apply id="Thmproofx11.p1.13.m12.4.4.2.cmml" xref="Thmproofx11.p1.13.m12.4.4.2"><times id="Thmproofx11.p1.13.m12.4.4.2.2.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.2"></times><apply id="Thmproofx11.p1.13.m12.4.4.2.3.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.3"><csymbol cd="ambiguous" id="Thmproofx11.p1.13.m12.4.4.2.3.1.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.3">subscript</csymbol><ci id="Thmproofx11.p1.13.m12.4.4.2.3.2.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.3.2">ℎ</ci><ci id="Thmproofx11.p1.13.m12.4.4.2.3.3.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.3.3">𝑖</ci></apply><interval closure="open" id="Thmproofx11.p1.13.m12.4.4.2.1.2.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.1.1"><apply id="Thmproofx11.p1.13.m12.4.4.2.1.1.1.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.13.m12.4.4.2.1.1.1.1.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.13.m12.4.4.2.1.1.1.2.cmml" xref="Thmproofx11.p1.13.m12.4.4.2.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.13.m12.4.4.2.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.13.m12.4.4.2.1.1.1.3">0</cn></apply><cn id="Thmproofx11.p1.13.m12.2.2.cmml" type="integer" xref="Thmproofx11.p1.13.m12.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.13.m12.4c">\varphi(\alpha_{0},1)=h_{i}(\alpha_{0},1)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.13.m12.4d">italic_φ ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) = italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 )</annotation></semantics></math> and let <math alttext="j>i" class="ltx_Math" display="inline" id="Thmproofx11.p1.14.m13.1"><semantics id="Thmproofx11.p1.14.m13.1a"><mrow id="Thmproofx11.p1.14.m13.1.1" xref="Thmproofx11.p1.14.m13.1.1.cmml"><mi id="Thmproofx11.p1.14.m13.1.1.2" xref="Thmproofx11.p1.14.m13.1.1.2.cmml">j</mi><mo id="Thmproofx11.p1.14.m13.1.1.1" xref="Thmproofx11.p1.14.m13.1.1.1.cmml">></mo><mi id="Thmproofx11.p1.14.m13.1.1.3" xref="Thmproofx11.p1.14.m13.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.14.m13.1b"><apply id="Thmproofx11.p1.14.m13.1.1.cmml" xref="Thmproofx11.p1.14.m13.1.1"><gt id="Thmproofx11.p1.14.m13.1.1.1.cmml" xref="Thmproofx11.p1.14.m13.1.1.1"></gt><ci id="Thmproofx11.p1.14.m13.1.1.2.cmml" xref="Thmproofx11.p1.14.m13.1.1.2">𝑗</ci><ci id="Thmproofx11.p1.14.m13.1.1.3.cmml" xref="Thmproofx11.p1.14.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.14.m13.1c">j>i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.14.m13.1d">italic_j > italic_i</annotation></semantics></math>. Then <math alttext="h_{i}(\alpha_{0},1)<h_{j}(\alpha_{0},1)" class="ltx_Math" display="inline" id="Thmproofx11.p1.15.m14.4"><semantics id="Thmproofx11.p1.15.m14.4a"><mrow id="Thmproofx11.p1.15.m14.4.4" xref="Thmproofx11.p1.15.m14.4.4.cmml"><mrow id="Thmproofx11.p1.15.m14.3.3.1" xref="Thmproofx11.p1.15.m14.3.3.1.cmml"><msub id="Thmproofx11.p1.15.m14.3.3.1.3" xref="Thmproofx11.p1.15.m14.3.3.1.3.cmml"><mi id="Thmproofx11.p1.15.m14.3.3.1.3.2" xref="Thmproofx11.p1.15.m14.3.3.1.3.2.cmml">h</mi><mi id="Thmproofx11.p1.15.m14.3.3.1.3.3" xref="Thmproofx11.p1.15.m14.3.3.1.3.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.15.m14.3.3.1.2" xref="Thmproofx11.p1.15.m14.3.3.1.2.cmml">⁢</mo><mrow id="Thmproofx11.p1.15.m14.3.3.1.1.1" xref="Thmproofx11.p1.15.m14.3.3.1.1.2.cmml"><mo id="Thmproofx11.p1.15.m14.3.3.1.1.1.2" stretchy="false" xref="Thmproofx11.p1.15.m14.3.3.1.1.2.cmml">(</mo><msub id="Thmproofx11.p1.15.m14.3.3.1.1.1.1" xref="Thmproofx11.p1.15.m14.3.3.1.1.1.1.cmml"><mi id="Thmproofx11.p1.15.m14.3.3.1.1.1.1.2" xref="Thmproofx11.p1.15.m14.3.3.1.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.15.m14.3.3.1.1.1.1.3" xref="Thmproofx11.p1.15.m14.3.3.1.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p1.15.m14.3.3.1.1.1.3" xref="Thmproofx11.p1.15.m14.3.3.1.1.2.cmml">,</mo><mn id="Thmproofx11.p1.15.m14.1.1" xref="Thmproofx11.p1.15.m14.1.1.cmml">1</mn><mo id="Thmproofx11.p1.15.m14.3.3.1.1.1.4" stretchy="false" xref="Thmproofx11.p1.15.m14.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx11.p1.15.m14.4.4.3" xref="Thmproofx11.p1.15.m14.4.4.3.cmml"><</mo><mrow id="Thmproofx11.p1.15.m14.4.4.2" xref="Thmproofx11.p1.15.m14.4.4.2.cmml"><msub id="Thmproofx11.p1.15.m14.4.4.2.3" xref="Thmproofx11.p1.15.m14.4.4.2.3.cmml"><mi id="Thmproofx11.p1.15.m14.4.4.2.3.2" xref="Thmproofx11.p1.15.m14.4.4.2.3.2.cmml">h</mi><mi id="Thmproofx11.p1.15.m14.4.4.2.3.3" xref="Thmproofx11.p1.15.m14.4.4.2.3.3.cmml">j</mi></msub><mo id="Thmproofx11.p1.15.m14.4.4.2.2" xref="Thmproofx11.p1.15.m14.4.4.2.2.cmml">⁢</mo><mrow id="Thmproofx11.p1.15.m14.4.4.2.1.1" xref="Thmproofx11.p1.15.m14.4.4.2.1.2.cmml"><mo id="Thmproofx11.p1.15.m14.4.4.2.1.1.2" stretchy="false" xref="Thmproofx11.p1.15.m14.4.4.2.1.2.cmml">(</mo><msub id="Thmproofx11.p1.15.m14.4.4.2.1.1.1" xref="Thmproofx11.p1.15.m14.4.4.2.1.1.1.cmml"><mi id="Thmproofx11.p1.15.m14.4.4.2.1.1.1.2" xref="Thmproofx11.p1.15.m14.4.4.2.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.15.m14.4.4.2.1.1.1.3" xref="Thmproofx11.p1.15.m14.4.4.2.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p1.15.m14.4.4.2.1.1.3" xref="Thmproofx11.p1.15.m14.4.4.2.1.2.cmml">,</mo><mn id="Thmproofx11.p1.15.m14.2.2" xref="Thmproofx11.p1.15.m14.2.2.cmml">1</mn><mo id="Thmproofx11.p1.15.m14.4.4.2.1.1.4" stretchy="false" xref="Thmproofx11.p1.15.m14.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.15.m14.4b"><apply id="Thmproofx11.p1.15.m14.4.4.cmml" xref="Thmproofx11.p1.15.m14.4.4"><lt id="Thmproofx11.p1.15.m14.4.4.3.cmml" xref="Thmproofx11.p1.15.m14.4.4.3"></lt><apply id="Thmproofx11.p1.15.m14.3.3.1.cmml" xref="Thmproofx11.p1.15.m14.3.3.1"><times id="Thmproofx11.p1.15.m14.3.3.1.2.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.2"></times><apply id="Thmproofx11.p1.15.m14.3.3.1.3.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p1.15.m14.3.3.1.3.1.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.3">subscript</csymbol><ci id="Thmproofx11.p1.15.m14.3.3.1.3.2.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.3.2">ℎ</ci><ci id="Thmproofx11.p1.15.m14.3.3.1.3.3.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.3.3">𝑖</ci></apply><interval closure="open" id="Thmproofx11.p1.15.m14.3.3.1.1.2.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.1.1"><apply id="Thmproofx11.p1.15.m14.3.3.1.1.1.1.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.15.m14.3.3.1.1.1.1.1.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.15.m14.3.3.1.1.1.1.2.cmml" xref="Thmproofx11.p1.15.m14.3.3.1.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.15.m14.3.3.1.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.15.m14.3.3.1.1.1.1.3">0</cn></apply><cn id="Thmproofx11.p1.15.m14.1.1.cmml" type="integer" xref="Thmproofx11.p1.15.m14.1.1">1</cn></interval></apply><apply id="Thmproofx11.p1.15.m14.4.4.2.cmml" xref="Thmproofx11.p1.15.m14.4.4.2"><times id="Thmproofx11.p1.15.m14.4.4.2.2.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.2"></times><apply id="Thmproofx11.p1.15.m14.4.4.2.3.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.3"><csymbol cd="ambiguous" id="Thmproofx11.p1.15.m14.4.4.2.3.1.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.3">subscript</csymbol><ci id="Thmproofx11.p1.15.m14.4.4.2.3.2.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.3.2">ℎ</ci><ci id="Thmproofx11.p1.15.m14.4.4.2.3.3.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.3.3">𝑗</ci></apply><interval closure="open" id="Thmproofx11.p1.15.m14.4.4.2.1.2.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.1.1"><apply id="Thmproofx11.p1.15.m14.4.4.2.1.1.1.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.15.m14.4.4.2.1.1.1.1.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.15.m14.4.4.2.1.1.1.2.cmml" xref="Thmproofx11.p1.15.m14.4.4.2.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.15.m14.4.4.2.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.15.m14.4.4.2.1.1.1.3">0</cn></apply><cn id="Thmproofx11.p1.15.m14.2.2.cmml" type="integer" xref="Thmproofx11.p1.15.m14.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.15.m14.4c">h_{i}(\alpha_{0},1)<h_{j}(\alpha_{0},1)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.15.m14.4d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) < italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 )</annotation></semantics></math>. For every <math alttext="n" class="ltx_Math" display="inline" id="Thmproofx11.p1.16.m15.1"><semantics id="Thmproofx11.p1.16.m15.1a"><mi id="Thmproofx11.p1.16.m15.1.1" xref="Thmproofx11.p1.16.m15.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.16.m15.1b"><ci id="Thmproofx11.p1.16.m15.1.1.cmml" xref="Thmproofx11.p1.16.m15.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.16.m15.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.16.m15.1d">italic_n</annotation></semantics></math>, the coefficient of <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx11.p1.17.m16.1"><semantics id="Thmproofx11.p1.17.m16.1a"><mi id="Thmproofx11.p1.17.m16.1.1" xref="Thmproofx11.p1.17.m16.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.17.m16.1b"><ci id="Thmproofx11.p1.17.m16.1.1.cmml" xref="Thmproofx11.p1.17.m16.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.17.m16.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.17.m16.1d">italic_α</annotation></semantics></math> in <math alttext="h_{i}(\alpha,n)" class="ltx_Math" display="inline" id="Thmproofx11.p1.18.m17.2"><semantics id="Thmproofx11.p1.18.m17.2a"><mrow id="Thmproofx11.p1.18.m17.2.3" xref="Thmproofx11.p1.18.m17.2.3.cmml"><msub id="Thmproofx11.p1.18.m17.2.3.2" xref="Thmproofx11.p1.18.m17.2.3.2.cmml"><mi id="Thmproofx11.p1.18.m17.2.3.2.2" xref="Thmproofx11.p1.18.m17.2.3.2.2.cmml">h</mi><mi id="Thmproofx11.p1.18.m17.2.3.2.3" xref="Thmproofx11.p1.18.m17.2.3.2.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.18.m17.2.3.1" xref="Thmproofx11.p1.18.m17.2.3.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.18.m17.2.3.3.2" xref="Thmproofx11.p1.18.m17.2.3.3.1.cmml"><mo id="Thmproofx11.p1.18.m17.2.3.3.2.1" stretchy="false" xref="Thmproofx11.p1.18.m17.2.3.3.1.cmml">(</mo><mi id="Thmproofx11.p1.18.m17.1.1" xref="Thmproofx11.p1.18.m17.1.1.cmml">α</mi><mo id="Thmproofx11.p1.18.m17.2.3.3.2.2" xref="Thmproofx11.p1.18.m17.2.3.3.1.cmml">,</mo><mi id="Thmproofx11.p1.18.m17.2.2" xref="Thmproofx11.p1.18.m17.2.2.cmml">n</mi><mo id="Thmproofx11.p1.18.m17.2.3.3.2.3" stretchy="false" xref="Thmproofx11.p1.18.m17.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.18.m17.2b"><apply id="Thmproofx11.p1.18.m17.2.3.cmml" xref="Thmproofx11.p1.18.m17.2.3"><times id="Thmproofx11.p1.18.m17.2.3.1.cmml" xref="Thmproofx11.p1.18.m17.2.3.1"></times><apply id="Thmproofx11.p1.18.m17.2.3.2.cmml" xref="Thmproofx11.p1.18.m17.2.3.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.18.m17.2.3.2.1.cmml" xref="Thmproofx11.p1.18.m17.2.3.2">subscript</csymbol><ci id="Thmproofx11.p1.18.m17.2.3.2.2.cmml" xref="Thmproofx11.p1.18.m17.2.3.2.2">ℎ</ci><ci id="Thmproofx11.p1.18.m17.2.3.2.3.cmml" xref="Thmproofx11.p1.18.m17.2.3.2.3">𝑖</ci></apply><interval closure="open" id="Thmproofx11.p1.18.m17.2.3.3.1.cmml" xref="Thmproofx11.p1.18.m17.2.3.3.2"><ci id="Thmproofx11.p1.18.m17.1.1.cmml" xref="Thmproofx11.p1.18.m17.1.1">𝛼</ci><ci id="Thmproofx11.p1.18.m17.2.2.cmml" xref="Thmproofx11.p1.18.m17.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.18.m17.2c">h_{i}(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.18.m17.2d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_α , italic_n )</annotation></semantics></math> is larger than that in <math alttext="h_{j}(\alpha,n)" class="ltx_Math" display="inline" id="Thmproofx11.p1.19.m18.2"><semantics id="Thmproofx11.p1.19.m18.2a"><mrow id="Thmproofx11.p1.19.m18.2.3" xref="Thmproofx11.p1.19.m18.2.3.cmml"><msub id="Thmproofx11.p1.19.m18.2.3.2" xref="Thmproofx11.p1.19.m18.2.3.2.cmml"><mi id="Thmproofx11.p1.19.m18.2.3.2.2" xref="Thmproofx11.p1.19.m18.2.3.2.2.cmml">h</mi><mi id="Thmproofx11.p1.19.m18.2.3.2.3" xref="Thmproofx11.p1.19.m18.2.3.2.3.cmml">j</mi></msub><mo id="Thmproofx11.p1.19.m18.2.3.1" xref="Thmproofx11.p1.19.m18.2.3.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.19.m18.2.3.3.2" xref="Thmproofx11.p1.19.m18.2.3.3.1.cmml"><mo id="Thmproofx11.p1.19.m18.2.3.3.2.1" stretchy="false" xref="Thmproofx11.p1.19.m18.2.3.3.1.cmml">(</mo><mi id="Thmproofx11.p1.19.m18.1.1" xref="Thmproofx11.p1.19.m18.1.1.cmml">α</mi><mo id="Thmproofx11.p1.19.m18.2.3.3.2.2" xref="Thmproofx11.p1.19.m18.2.3.3.1.cmml">,</mo><mi id="Thmproofx11.p1.19.m18.2.2" xref="Thmproofx11.p1.19.m18.2.2.cmml">n</mi><mo id="Thmproofx11.p1.19.m18.2.3.3.2.3" stretchy="false" xref="Thmproofx11.p1.19.m18.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.19.m18.2b"><apply id="Thmproofx11.p1.19.m18.2.3.cmml" xref="Thmproofx11.p1.19.m18.2.3"><times id="Thmproofx11.p1.19.m18.2.3.1.cmml" xref="Thmproofx11.p1.19.m18.2.3.1"></times><apply id="Thmproofx11.p1.19.m18.2.3.2.cmml" xref="Thmproofx11.p1.19.m18.2.3.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.19.m18.2.3.2.1.cmml" xref="Thmproofx11.p1.19.m18.2.3.2">subscript</csymbol><ci id="Thmproofx11.p1.19.m18.2.3.2.2.cmml" xref="Thmproofx11.p1.19.m18.2.3.2.2">ℎ</ci><ci id="Thmproofx11.p1.19.m18.2.3.2.3.cmml" xref="Thmproofx11.p1.19.m18.2.3.2.3">𝑗</ci></apply><interval closure="open" id="Thmproofx11.p1.19.m18.2.3.3.1.cmml" xref="Thmproofx11.p1.19.m18.2.3.3.2"><ci id="Thmproofx11.p1.19.m18.1.1.cmml" xref="Thmproofx11.p1.19.m18.1.1">𝛼</ci><ci id="Thmproofx11.p1.19.m18.2.2.cmml" xref="Thmproofx11.p1.19.m18.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.19.m18.2c">h_{j}(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.19.m18.2d">italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_α , italic_n )</annotation></semantics></math>, so after the second argument of <math alttext="h_{i},h_{j}" class="ltx_Math" display="inline" id="Thmproofx11.p1.20.m19.2"><semantics id="Thmproofx11.p1.20.m19.2a"><mrow id="Thmproofx11.p1.20.m19.2.2.2" xref="Thmproofx11.p1.20.m19.2.2.3.cmml"><msub id="Thmproofx11.p1.20.m19.1.1.1.1" xref="Thmproofx11.p1.20.m19.1.1.1.1.cmml"><mi id="Thmproofx11.p1.20.m19.1.1.1.1.2" xref="Thmproofx11.p1.20.m19.1.1.1.1.2.cmml">h</mi><mi id="Thmproofx11.p1.20.m19.1.1.1.1.3" xref="Thmproofx11.p1.20.m19.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.20.m19.2.2.2.3" xref="Thmproofx11.p1.20.m19.2.2.3.cmml">,</mo><msub id="Thmproofx11.p1.20.m19.2.2.2.2" xref="Thmproofx11.p1.20.m19.2.2.2.2.cmml"><mi id="Thmproofx11.p1.20.m19.2.2.2.2.2" xref="Thmproofx11.p1.20.m19.2.2.2.2.2.cmml">h</mi><mi id="Thmproofx11.p1.20.m19.2.2.2.2.3" xref="Thmproofx11.p1.20.m19.2.2.2.2.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.20.m19.2b"><list id="Thmproofx11.p1.20.m19.2.2.3.cmml" xref="Thmproofx11.p1.20.m19.2.2.2"><apply id="Thmproofx11.p1.20.m19.1.1.1.1.cmml" xref="Thmproofx11.p1.20.m19.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.20.m19.1.1.1.1.1.cmml" xref="Thmproofx11.p1.20.m19.1.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.20.m19.1.1.1.1.2.cmml" xref="Thmproofx11.p1.20.m19.1.1.1.1.2">ℎ</ci><ci id="Thmproofx11.p1.20.m19.1.1.1.1.3.cmml" xref="Thmproofx11.p1.20.m19.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx11.p1.20.m19.2.2.2.2.cmml" xref="Thmproofx11.p1.20.m19.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.20.m19.2.2.2.2.1.cmml" xref="Thmproofx11.p1.20.m19.2.2.2.2">subscript</csymbol><ci id="Thmproofx11.p1.20.m19.2.2.2.2.2.cmml" xref="Thmproofx11.p1.20.m19.2.2.2.2.2">ℎ</ci><ci id="Thmproofx11.p1.20.m19.2.2.2.2.3.cmml" xref="Thmproofx11.p1.20.m19.2.2.2.2.3">𝑗</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.20.m19.2c">h_{i},h_{j}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.20.m19.2d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is fixed to 1, the unique intersection point <math alttext="\alpha_{1}" class="ltx_Math" display="inline" id="Thmproofx11.p1.21.m20.1"><semantics id="Thmproofx11.p1.21.m20.1a"><msub id="Thmproofx11.p1.21.m20.1.1" xref="Thmproofx11.p1.21.m20.1.1.cmml"><mi id="Thmproofx11.p1.21.m20.1.1.2" xref="Thmproofx11.p1.21.m20.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.21.m20.1.1.3" xref="Thmproofx11.p1.21.m20.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.21.m20.1b"><apply id="Thmproofx11.p1.21.m20.1.1.cmml" xref="Thmproofx11.p1.21.m20.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.21.m20.1.1.1.cmml" xref="Thmproofx11.p1.21.m20.1.1">subscript</csymbol><ci id="Thmproofx11.p1.21.m20.1.1.2.cmml" xref="Thmproofx11.p1.21.m20.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.21.m20.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.21.m20.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.21.m20.1c">\alpha_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.21.m20.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> of the functions <math alttext="h_{i}(\alpha,1),h_{j}(\alpha,1)" class="ltx_Math" display="inline" id="Thmproofx11.p1.22.m21.6"><semantics id="Thmproofx11.p1.22.m21.6a"><mrow id="Thmproofx11.p1.22.m21.6.6.2" xref="Thmproofx11.p1.22.m21.6.6.3.cmml"><mrow id="Thmproofx11.p1.22.m21.5.5.1.1" xref="Thmproofx11.p1.22.m21.5.5.1.1.cmml"><msub id="Thmproofx11.p1.22.m21.5.5.1.1.2" xref="Thmproofx11.p1.22.m21.5.5.1.1.2.cmml"><mi id="Thmproofx11.p1.22.m21.5.5.1.1.2.2" xref="Thmproofx11.p1.22.m21.5.5.1.1.2.2.cmml">h</mi><mi id="Thmproofx11.p1.22.m21.5.5.1.1.2.3" xref="Thmproofx11.p1.22.m21.5.5.1.1.2.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.22.m21.5.5.1.1.1" xref="Thmproofx11.p1.22.m21.5.5.1.1.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.22.m21.5.5.1.1.3.2" xref="Thmproofx11.p1.22.m21.5.5.1.1.3.1.cmml"><mo id="Thmproofx11.p1.22.m21.5.5.1.1.3.2.1" stretchy="false" xref="Thmproofx11.p1.22.m21.5.5.1.1.3.1.cmml">(</mo><mi id="Thmproofx11.p1.22.m21.1.1" xref="Thmproofx11.p1.22.m21.1.1.cmml">α</mi><mo id="Thmproofx11.p1.22.m21.5.5.1.1.3.2.2" xref="Thmproofx11.p1.22.m21.5.5.1.1.3.1.cmml">,</mo><mn id="Thmproofx11.p1.22.m21.2.2" xref="Thmproofx11.p1.22.m21.2.2.cmml">1</mn><mo id="Thmproofx11.p1.22.m21.5.5.1.1.3.2.3" stretchy="false" xref="Thmproofx11.p1.22.m21.5.5.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx11.p1.22.m21.6.6.2.3" xref="Thmproofx11.p1.22.m21.6.6.3.cmml">,</mo><mrow id="Thmproofx11.p1.22.m21.6.6.2.2" xref="Thmproofx11.p1.22.m21.6.6.2.2.cmml"><msub id="Thmproofx11.p1.22.m21.6.6.2.2.2" xref="Thmproofx11.p1.22.m21.6.6.2.2.2.cmml"><mi id="Thmproofx11.p1.22.m21.6.6.2.2.2.2" xref="Thmproofx11.p1.22.m21.6.6.2.2.2.2.cmml">h</mi><mi id="Thmproofx11.p1.22.m21.6.6.2.2.2.3" xref="Thmproofx11.p1.22.m21.6.6.2.2.2.3.cmml">j</mi></msub><mo id="Thmproofx11.p1.22.m21.6.6.2.2.1" xref="Thmproofx11.p1.22.m21.6.6.2.2.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.22.m21.6.6.2.2.3.2" xref="Thmproofx11.p1.22.m21.6.6.2.2.3.1.cmml"><mo id="Thmproofx11.p1.22.m21.6.6.2.2.3.2.1" stretchy="false" xref="Thmproofx11.p1.22.m21.6.6.2.2.3.1.cmml">(</mo><mi id="Thmproofx11.p1.22.m21.3.3" xref="Thmproofx11.p1.22.m21.3.3.cmml">α</mi><mo id="Thmproofx11.p1.22.m21.6.6.2.2.3.2.2" xref="Thmproofx11.p1.22.m21.6.6.2.2.3.1.cmml">,</mo><mn id="Thmproofx11.p1.22.m21.4.4" xref="Thmproofx11.p1.22.m21.4.4.cmml">1</mn><mo id="Thmproofx11.p1.22.m21.6.6.2.2.3.2.3" stretchy="false" xref="Thmproofx11.p1.22.m21.6.6.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.22.m21.6b"><list id="Thmproofx11.p1.22.m21.6.6.3.cmml" xref="Thmproofx11.p1.22.m21.6.6.2"><apply id="Thmproofx11.p1.22.m21.5.5.1.1.cmml" xref="Thmproofx11.p1.22.m21.5.5.1.1"><times id="Thmproofx11.p1.22.m21.5.5.1.1.1.cmml" xref="Thmproofx11.p1.22.m21.5.5.1.1.1"></times><apply id="Thmproofx11.p1.22.m21.5.5.1.1.2.cmml" xref="Thmproofx11.p1.22.m21.5.5.1.1.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.22.m21.5.5.1.1.2.1.cmml" xref="Thmproofx11.p1.22.m21.5.5.1.1.2">subscript</csymbol><ci id="Thmproofx11.p1.22.m21.5.5.1.1.2.2.cmml" xref="Thmproofx11.p1.22.m21.5.5.1.1.2.2">ℎ</ci><ci id="Thmproofx11.p1.22.m21.5.5.1.1.2.3.cmml" xref="Thmproofx11.p1.22.m21.5.5.1.1.2.3">𝑖</ci></apply><interval closure="open" id="Thmproofx11.p1.22.m21.5.5.1.1.3.1.cmml" xref="Thmproofx11.p1.22.m21.5.5.1.1.3.2"><ci id="Thmproofx11.p1.22.m21.1.1.cmml" xref="Thmproofx11.p1.22.m21.1.1">𝛼</ci><cn id="Thmproofx11.p1.22.m21.2.2.cmml" type="integer" xref="Thmproofx11.p1.22.m21.2.2">1</cn></interval></apply><apply id="Thmproofx11.p1.22.m21.6.6.2.2.cmml" xref="Thmproofx11.p1.22.m21.6.6.2.2"><times id="Thmproofx11.p1.22.m21.6.6.2.2.1.cmml" xref="Thmproofx11.p1.22.m21.6.6.2.2.1"></times><apply id="Thmproofx11.p1.22.m21.6.6.2.2.2.cmml" xref="Thmproofx11.p1.22.m21.6.6.2.2.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.22.m21.6.6.2.2.2.1.cmml" xref="Thmproofx11.p1.22.m21.6.6.2.2.2">subscript</csymbol><ci id="Thmproofx11.p1.22.m21.6.6.2.2.2.2.cmml" xref="Thmproofx11.p1.22.m21.6.6.2.2.2.2">ℎ</ci><ci id="Thmproofx11.p1.22.m21.6.6.2.2.2.3.cmml" xref="Thmproofx11.p1.22.m21.6.6.2.2.2.3">𝑗</ci></apply><interval closure="open" id="Thmproofx11.p1.22.m21.6.6.2.2.3.1.cmml" xref="Thmproofx11.p1.22.m21.6.6.2.2.3.2"><ci id="Thmproofx11.p1.22.m21.3.3.cmml" xref="Thmproofx11.p1.22.m21.3.3">𝛼</ci><cn id="Thmproofx11.p1.22.m21.4.4.cmml" type="integer" xref="Thmproofx11.p1.22.m21.4.4">1</cn></interval></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.22.m21.6c">h_{i}(\alpha,1),h_{j}(\alpha,1)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.22.m21.6d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_α , 1 ) , italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_α , 1 )</annotation></semantics></math> satisfies <math alttext="\alpha_{1}>\alpha_{0}" class="ltx_Math" display="inline" id="Thmproofx11.p1.23.m22.1"><semantics id="Thmproofx11.p1.23.m22.1a"><mrow id="Thmproofx11.p1.23.m22.1.1" xref="Thmproofx11.p1.23.m22.1.1.cmml"><msub id="Thmproofx11.p1.23.m22.1.1.2" xref="Thmproofx11.p1.23.m22.1.1.2.cmml"><mi id="Thmproofx11.p1.23.m22.1.1.2.2" xref="Thmproofx11.p1.23.m22.1.1.2.2.cmml">α</mi><mn id="Thmproofx11.p1.23.m22.1.1.2.3" xref="Thmproofx11.p1.23.m22.1.1.2.3.cmml">1</mn></msub><mo id="Thmproofx11.p1.23.m22.1.1.1" xref="Thmproofx11.p1.23.m22.1.1.1.cmml">></mo><msub id="Thmproofx11.p1.23.m22.1.1.3" xref="Thmproofx11.p1.23.m22.1.1.3.cmml"><mi id="Thmproofx11.p1.23.m22.1.1.3.2" xref="Thmproofx11.p1.23.m22.1.1.3.2.cmml">α</mi><mn id="Thmproofx11.p1.23.m22.1.1.3.3" xref="Thmproofx11.p1.23.m22.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.23.m22.1b"><apply id="Thmproofx11.p1.23.m22.1.1.cmml" xref="Thmproofx11.p1.23.m22.1.1"><gt id="Thmproofx11.p1.23.m22.1.1.1.cmml" xref="Thmproofx11.p1.23.m22.1.1.1"></gt><apply id="Thmproofx11.p1.23.m22.1.1.2.cmml" xref="Thmproofx11.p1.23.m22.1.1.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.23.m22.1.1.2.1.cmml" xref="Thmproofx11.p1.23.m22.1.1.2">subscript</csymbol><ci id="Thmproofx11.p1.23.m22.1.1.2.2.cmml" xref="Thmproofx11.p1.23.m22.1.1.2.2">𝛼</ci><cn id="Thmproofx11.p1.23.m22.1.1.2.3.cmml" type="integer" xref="Thmproofx11.p1.23.m22.1.1.2.3">1</cn></apply><apply id="Thmproofx11.p1.23.m22.1.1.3.cmml" xref="Thmproofx11.p1.23.m22.1.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p1.23.m22.1.1.3.1.cmml" xref="Thmproofx11.p1.23.m22.1.1.3">subscript</csymbol><ci id="Thmproofx11.p1.23.m22.1.1.3.2.cmml" xref="Thmproofx11.p1.23.m22.1.1.3.2">𝛼</ci><cn id="Thmproofx11.p1.23.m22.1.1.3.3.cmml" type="integer" xref="Thmproofx11.p1.23.m22.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.23.m22.1c">\alpha_{1}>\alpha_{0}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.23.m22.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Fix the second argument to <math alttext="n>1" class="ltx_Math" display="inline" id="Thmproofx11.p1.24.m23.1"><semantics id="Thmproofx11.p1.24.m23.1a"><mrow id="Thmproofx11.p1.24.m23.1.1" xref="Thmproofx11.p1.24.m23.1.1.cmml"><mi id="Thmproofx11.p1.24.m23.1.1.2" xref="Thmproofx11.p1.24.m23.1.1.2.cmml">n</mi><mo id="Thmproofx11.p1.24.m23.1.1.1" xref="Thmproofx11.p1.24.m23.1.1.1.cmml">></mo><mn id="Thmproofx11.p1.24.m23.1.1.3" xref="Thmproofx11.p1.24.m23.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.24.m23.1b"><apply id="Thmproofx11.p1.24.m23.1.1.cmml" xref="Thmproofx11.p1.24.m23.1.1"><gt id="Thmproofx11.p1.24.m23.1.1.1.cmml" xref="Thmproofx11.p1.24.m23.1.1.1"></gt><ci id="Thmproofx11.p1.24.m23.1.1.2.cmml" xref="Thmproofx11.p1.24.m23.1.1.2">𝑛</ci><cn id="Thmproofx11.p1.24.m23.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.24.m23.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.24.m23.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.24.m23.1d">italic_n > 1</annotation></semantics></math> now, and let <math alttext="\alpha_{n}" class="ltx_Math" display="inline" id="Thmproofx11.p1.25.m24.1"><semantics id="Thmproofx11.p1.25.m24.1a"><msub id="Thmproofx11.p1.25.m24.1.1" xref="Thmproofx11.p1.25.m24.1.1.cmml"><mi id="Thmproofx11.p1.25.m24.1.1.2" xref="Thmproofx11.p1.25.m24.1.1.2.cmml">α</mi><mi id="Thmproofx11.p1.25.m24.1.1.3" xref="Thmproofx11.p1.25.m24.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.25.m24.1b"><apply id="Thmproofx11.p1.25.m24.1.1.cmml" xref="Thmproofx11.p1.25.m24.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.25.m24.1.1.1.cmml" xref="Thmproofx11.p1.25.m24.1.1">subscript</csymbol><ci id="Thmproofx11.p1.25.m24.1.1.2.cmml" xref="Thmproofx11.p1.25.m24.1.1.2">𝛼</ci><ci id="Thmproofx11.p1.25.m24.1.1.3.cmml" xref="Thmproofx11.p1.25.m24.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.25.m24.1c">\alpha_{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.25.m24.1d">italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> be the unique intersection point of <math alttext="h_{i}(\alpha,n),h_{j}(\alpha,n)" class="ltx_Math" display="inline" id="Thmproofx11.p1.26.m25.6"><semantics id="Thmproofx11.p1.26.m25.6a"><mrow id="Thmproofx11.p1.26.m25.6.6.2" xref="Thmproofx11.p1.26.m25.6.6.3.cmml"><mrow id="Thmproofx11.p1.26.m25.5.5.1.1" xref="Thmproofx11.p1.26.m25.5.5.1.1.cmml"><msub id="Thmproofx11.p1.26.m25.5.5.1.1.2" xref="Thmproofx11.p1.26.m25.5.5.1.1.2.cmml"><mi id="Thmproofx11.p1.26.m25.5.5.1.1.2.2" xref="Thmproofx11.p1.26.m25.5.5.1.1.2.2.cmml">h</mi><mi id="Thmproofx11.p1.26.m25.5.5.1.1.2.3" xref="Thmproofx11.p1.26.m25.5.5.1.1.2.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.26.m25.5.5.1.1.1" xref="Thmproofx11.p1.26.m25.5.5.1.1.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.26.m25.5.5.1.1.3.2" xref="Thmproofx11.p1.26.m25.5.5.1.1.3.1.cmml"><mo id="Thmproofx11.p1.26.m25.5.5.1.1.3.2.1" stretchy="false" xref="Thmproofx11.p1.26.m25.5.5.1.1.3.1.cmml">(</mo><mi id="Thmproofx11.p1.26.m25.1.1" xref="Thmproofx11.p1.26.m25.1.1.cmml">α</mi><mo id="Thmproofx11.p1.26.m25.5.5.1.1.3.2.2" xref="Thmproofx11.p1.26.m25.5.5.1.1.3.1.cmml">,</mo><mi id="Thmproofx11.p1.26.m25.2.2" xref="Thmproofx11.p1.26.m25.2.2.cmml">n</mi><mo id="Thmproofx11.p1.26.m25.5.5.1.1.3.2.3" stretchy="false" xref="Thmproofx11.p1.26.m25.5.5.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx11.p1.26.m25.6.6.2.3" xref="Thmproofx11.p1.26.m25.6.6.3.cmml">,</mo><mrow id="Thmproofx11.p1.26.m25.6.6.2.2" xref="Thmproofx11.p1.26.m25.6.6.2.2.cmml"><msub id="Thmproofx11.p1.26.m25.6.6.2.2.2" xref="Thmproofx11.p1.26.m25.6.6.2.2.2.cmml"><mi id="Thmproofx11.p1.26.m25.6.6.2.2.2.2" xref="Thmproofx11.p1.26.m25.6.6.2.2.2.2.cmml">h</mi><mi id="Thmproofx11.p1.26.m25.6.6.2.2.2.3" xref="Thmproofx11.p1.26.m25.6.6.2.2.2.3.cmml">j</mi></msub><mo id="Thmproofx11.p1.26.m25.6.6.2.2.1" xref="Thmproofx11.p1.26.m25.6.6.2.2.1.cmml">⁢</mo><mrow id="Thmproofx11.p1.26.m25.6.6.2.2.3.2" xref="Thmproofx11.p1.26.m25.6.6.2.2.3.1.cmml"><mo id="Thmproofx11.p1.26.m25.6.6.2.2.3.2.1" stretchy="false" xref="Thmproofx11.p1.26.m25.6.6.2.2.3.1.cmml">(</mo><mi id="Thmproofx11.p1.26.m25.3.3" xref="Thmproofx11.p1.26.m25.3.3.cmml">α</mi><mo id="Thmproofx11.p1.26.m25.6.6.2.2.3.2.2" xref="Thmproofx11.p1.26.m25.6.6.2.2.3.1.cmml">,</mo><mi id="Thmproofx11.p1.26.m25.4.4" xref="Thmproofx11.p1.26.m25.4.4.cmml">n</mi><mo id="Thmproofx11.p1.26.m25.6.6.2.2.3.2.3" stretchy="false" xref="Thmproofx11.p1.26.m25.6.6.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.26.m25.6b"><list id="Thmproofx11.p1.26.m25.6.6.3.cmml" xref="Thmproofx11.p1.26.m25.6.6.2"><apply id="Thmproofx11.p1.26.m25.5.5.1.1.cmml" xref="Thmproofx11.p1.26.m25.5.5.1.1"><times id="Thmproofx11.p1.26.m25.5.5.1.1.1.cmml" xref="Thmproofx11.p1.26.m25.5.5.1.1.1"></times><apply id="Thmproofx11.p1.26.m25.5.5.1.1.2.cmml" xref="Thmproofx11.p1.26.m25.5.5.1.1.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.26.m25.5.5.1.1.2.1.cmml" xref="Thmproofx11.p1.26.m25.5.5.1.1.2">subscript</csymbol><ci id="Thmproofx11.p1.26.m25.5.5.1.1.2.2.cmml" xref="Thmproofx11.p1.26.m25.5.5.1.1.2.2">ℎ</ci><ci id="Thmproofx11.p1.26.m25.5.5.1.1.2.3.cmml" xref="Thmproofx11.p1.26.m25.5.5.1.1.2.3">𝑖</ci></apply><interval closure="open" id="Thmproofx11.p1.26.m25.5.5.1.1.3.1.cmml" xref="Thmproofx11.p1.26.m25.5.5.1.1.3.2"><ci id="Thmproofx11.p1.26.m25.1.1.cmml" xref="Thmproofx11.p1.26.m25.1.1">𝛼</ci><ci id="Thmproofx11.p1.26.m25.2.2.cmml" xref="Thmproofx11.p1.26.m25.2.2">𝑛</ci></interval></apply><apply id="Thmproofx11.p1.26.m25.6.6.2.2.cmml" xref="Thmproofx11.p1.26.m25.6.6.2.2"><times id="Thmproofx11.p1.26.m25.6.6.2.2.1.cmml" xref="Thmproofx11.p1.26.m25.6.6.2.2.1"></times><apply id="Thmproofx11.p1.26.m25.6.6.2.2.2.cmml" xref="Thmproofx11.p1.26.m25.6.6.2.2.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.26.m25.6.6.2.2.2.1.cmml" xref="Thmproofx11.p1.26.m25.6.6.2.2.2">subscript</csymbol><ci id="Thmproofx11.p1.26.m25.6.6.2.2.2.2.cmml" xref="Thmproofx11.p1.26.m25.6.6.2.2.2.2">ℎ</ci><ci id="Thmproofx11.p1.26.m25.6.6.2.2.2.3.cmml" xref="Thmproofx11.p1.26.m25.6.6.2.2.2.3">𝑗</ci></apply><interval closure="open" id="Thmproofx11.p1.26.m25.6.6.2.2.3.1.cmml" xref="Thmproofx11.p1.26.m25.6.6.2.2.3.2"><ci id="Thmproofx11.p1.26.m25.3.3.cmml" xref="Thmproofx11.p1.26.m25.3.3">𝛼</ci><ci id="Thmproofx11.p1.26.m25.4.4.cmml" xref="Thmproofx11.p1.26.m25.4.4">𝑛</ci></interval></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.26.m25.6c">h_{i}(\alpha,n),h_{j}(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.26.m25.6d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_α , italic_n ) , italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_α , italic_n )</annotation></semantics></math>. We show that <math alttext="\alpha_{n}\geq\alpha_{1}" class="ltx_Math" display="inline" id="Thmproofx11.p1.27.m26.1"><semantics id="Thmproofx11.p1.27.m26.1a"><mrow id="Thmproofx11.p1.27.m26.1.1" xref="Thmproofx11.p1.27.m26.1.1.cmml"><msub id="Thmproofx11.p1.27.m26.1.1.2" xref="Thmproofx11.p1.27.m26.1.1.2.cmml"><mi id="Thmproofx11.p1.27.m26.1.1.2.2" xref="Thmproofx11.p1.27.m26.1.1.2.2.cmml">α</mi><mi id="Thmproofx11.p1.27.m26.1.1.2.3" xref="Thmproofx11.p1.27.m26.1.1.2.3.cmml">n</mi></msub><mo id="Thmproofx11.p1.27.m26.1.1.1" xref="Thmproofx11.p1.27.m26.1.1.1.cmml">≥</mo><msub id="Thmproofx11.p1.27.m26.1.1.3" xref="Thmproofx11.p1.27.m26.1.1.3.cmml"><mi id="Thmproofx11.p1.27.m26.1.1.3.2" xref="Thmproofx11.p1.27.m26.1.1.3.2.cmml">α</mi><mn id="Thmproofx11.p1.27.m26.1.1.3.3" xref="Thmproofx11.p1.27.m26.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.27.m26.1b"><apply id="Thmproofx11.p1.27.m26.1.1.cmml" xref="Thmproofx11.p1.27.m26.1.1"><geq id="Thmproofx11.p1.27.m26.1.1.1.cmml" xref="Thmproofx11.p1.27.m26.1.1.1"></geq><apply id="Thmproofx11.p1.27.m26.1.1.2.cmml" xref="Thmproofx11.p1.27.m26.1.1.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.27.m26.1.1.2.1.cmml" xref="Thmproofx11.p1.27.m26.1.1.2">subscript</csymbol><ci id="Thmproofx11.p1.27.m26.1.1.2.2.cmml" xref="Thmproofx11.p1.27.m26.1.1.2.2">𝛼</ci><ci id="Thmproofx11.p1.27.m26.1.1.2.3.cmml" xref="Thmproofx11.p1.27.m26.1.1.2.3">𝑛</ci></apply><apply id="Thmproofx11.p1.27.m26.1.1.3.cmml" xref="Thmproofx11.p1.27.m26.1.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p1.27.m26.1.1.3.1.cmml" xref="Thmproofx11.p1.27.m26.1.1.3">subscript</csymbol><ci id="Thmproofx11.p1.27.m26.1.1.3.2.cmml" xref="Thmproofx11.p1.27.m26.1.1.3.2">𝛼</ci><cn id="Thmproofx11.p1.27.m26.1.1.3.3.cmml" type="integer" xref="Thmproofx11.p1.27.m26.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.27.m26.1c">\alpha_{n}\geq\alpha_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.27.m26.1d">italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≥ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, which implies by <math alttext="h_{i},h_{j}" class="ltx_Math" display="inline" id="Thmproofx11.p1.28.m27.2"><semantics id="Thmproofx11.p1.28.m27.2a"><mrow id="Thmproofx11.p1.28.m27.2.2.2" xref="Thmproofx11.p1.28.m27.2.2.3.cmml"><msub id="Thmproofx11.p1.28.m27.1.1.1.1" xref="Thmproofx11.p1.28.m27.1.1.1.1.cmml"><mi id="Thmproofx11.p1.28.m27.1.1.1.1.2" xref="Thmproofx11.p1.28.m27.1.1.1.1.2.cmml">h</mi><mi id="Thmproofx11.p1.28.m27.1.1.1.1.3" xref="Thmproofx11.p1.28.m27.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.28.m27.2.2.2.3" xref="Thmproofx11.p1.28.m27.2.2.3.cmml">,</mo><msub id="Thmproofx11.p1.28.m27.2.2.2.2" xref="Thmproofx11.p1.28.m27.2.2.2.2.cmml"><mi id="Thmproofx11.p1.28.m27.2.2.2.2.2" xref="Thmproofx11.p1.28.m27.2.2.2.2.2.cmml">h</mi><mi id="Thmproofx11.p1.28.m27.2.2.2.2.3" xref="Thmproofx11.p1.28.m27.2.2.2.2.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.28.m27.2b"><list id="Thmproofx11.p1.28.m27.2.2.3.cmml" xref="Thmproofx11.p1.28.m27.2.2.2"><apply id="Thmproofx11.p1.28.m27.1.1.1.1.cmml" xref="Thmproofx11.p1.28.m27.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.28.m27.1.1.1.1.1.cmml" xref="Thmproofx11.p1.28.m27.1.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.28.m27.1.1.1.1.2.cmml" xref="Thmproofx11.p1.28.m27.1.1.1.1.2">ℎ</ci><ci id="Thmproofx11.p1.28.m27.1.1.1.1.3.cmml" xref="Thmproofx11.p1.28.m27.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx11.p1.28.m27.2.2.2.2.cmml" xref="Thmproofx11.p1.28.m27.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx11.p1.28.m27.2.2.2.2.1.cmml" xref="Thmproofx11.p1.28.m27.2.2.2.2">subscript</csymbol><ci id="Thmproofx11.p1.28.m27.2.2.2.2.2.cmml" xref="Thmproofx11.p1.28.m27.2.2.2.2.2">ℎ</ci><ci id="Thmproofx11.p1.28.m27.2.2.2.2.3.cmml" xref="Thmproofx11.p1.28.m27.2.2.2.2.3">𝑗</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.28.m27.2c">h_{i},h_{j}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.28.m27.2d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> being affine in <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx11.p1.29.m28.1"><semantics id="Thmproofx11.p1.29.m28.1a"><mi id="Thmproofx11.p1.29.m28.1.1" xref="Thmproofx11.p1.29.m28.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.29.m28.1b"><ci id="Thmproofx11.p1.29.m28.1.1.cmml" xref="Thmproofx11.p1.29.m28.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.29.m28.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.29.m28.1d">italic_α</annotation></semantics></math> and the above that <math alttext="h_{i}(\alpha_{0},n)<h_{j}(\alpha_{0},n)" class="ltx_Math" display="inline" id="Thmproofx11.p1.30.m29.4"><semantics id="Thmproofx11.p1.30.m29.4a"><mrow id="Thmproofx11.p1.30.m29.4.4" xref="Thmproofx11.p1.30.m29.4.4.cmml"><mrow id="Thmproofx11.p1.30.m29.3.3.1" xref="Thmproofx11.p1.30.m29.3.3.1.cmml"><msub id="Thmproofx11.p1.30.m29.3.3.1.3" xref="Thmproofx11.p1.30.m29.3.3.1.3.cmml"><mi id="Thmproofx11.p1.30.m29.3.3.1.3.2" xref="Thmproofx11.p1.30.m29.3.3.1.3.2.cmml">h</mi><mi id="Thmproofx11.p1.30.m29.3.3.1.3.3" xref="Thmproofx11.p1.30.m29.3.3.1.3.3.cmml">i</mi></msub><mo id="Thmproofx11.p1.30.m29.3.3.1.2" xref="Thmproofx11.p1.30.m29.3.3.1.2.cmml">⁢</mo><mrow id="Thmproofx11.p1.30.m29.3.3.1.1.1" xref="Thmproofx11.p1.30.m29.3.3.1.1.2.cmml"><mo id="Thmproofx11.p1.30.m29.3.3.1.1.1.2" stretchy="false" xref="Thmproofx11.p1.30.m29.3.3.1.1.2.cmml">(</mo><msub id="Thmproofx11.p1.30.m29.3.3.1.1.1.1" xref="Thmproofx11.p1.30.m29.3.3.1.1.1.1.cmml"><mi id="Thmproofx11.p1.30.m29.3.3.1.1.1.1.2" xref="Thmproofx11.p1.30.m29.3.3.1.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.30.m29.3.3.1.1.1.1.3" xref="Thmproofx11.p1.30.m29.3.3.1.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p1.30.m29.3.3.1.1.1.3" xref="Thmproofx11.p1.30.m29.3.3.1.1.2.cmml">,</mo><mi id="Thmproofx11.p1.30.m29.1.1" xref="Thmproofx11.p1.30.m29.1.1.cmml">n</mi><mo id="Thmproofx11.p1.30.m29.3.3.1.1.1.4" stretchy="false" xref="Thmproofx11.p1.30.m29.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx11.p1.30.m29.4.4.3" xref="Thmproofx11.p1.30.m29.4.4.3.cmml"><</mo><mrow id="Thmproofx11.p1.30.m29.4.4.2" xref="Thmproofx11.p1.30.m29.4.4.2.cmml"><msub id="Thmproofx11.p1.30.m29.4.4.2.3" xref="Thmproofx11.p1.30.m29.4.4.2.3.cmml"><mi id="Thmproofx11.p1.30.m29.4.4.2.3.2" xref="Thmproofx11.p1.30.m29.4.4.2.3.2.cmml">h</mi><mi id="Thmproofx11.p1.30.m29.4.4.2.3.3" xref="Thmproofx11.p1.30.m29.4.4.2.3.3.cmml">j</mi></msub><mo id="Thmproofx11.p1.30.m29.4.4.2.2" xref="Thmproofx11.p1.30.m29.4.4.2.2.cmml">⁢</mo><mrow id="Thmproofx11.p1.30.m29.4.4.2.1.1" xref="Thmproofx11.p1.30.m29.4.4.2.1.2.cmml"><mo id="Thmproofx11.p1.30.m29.4.4.2.1.1.2" stretchy="false" xref="Thmproofx11.p1.30.m29.4.4.2.1.2.cmml">(</mo><msub id="Thmproofx11.p1.30.m29.4.4.2.1.1.1" xref="Thmproofx11.p1.30.m29.4.4.2.1.1.1.cmml"><mi id="Thmproofx11.p1.30.m29.4.4.2.1.1.1.2" xref="Thmproofx11.p1.30.m29.4.4.2.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p1.30.m29.4.4.2.1.1.1.3" xref="Thmproofx11.p1.30.m29.4.4.2.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p1.30.m29.4.4.2.1.1.3" xref="Thmproofx11.p1.30.m29.4.4.2.1.2.cmml">,</mo><mi id="Thmproofx11.p1.30.m29.2.2" xref="Thmproofx11.p1.30.m29.2.2.cmml">n</mi><mo id="Thmproofx11.p1.30.m29.4.4.2.1.1.4" stretchy="false" xref="Thmproofx11.p1.30.m29.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p1.30.m29.4b"><apply id="Thmproofx11.p1.30.m29.4.4.cmml" xref="Thmproofx11.p1.30.m29.4.4"><lt id="Thmproofx11.p1.30.m29.4.4.3.cmml" xref="Thmproofx11.p1.30.m29.4.4.3"></lt><apply id="Thmproofx11.p1.30.m29.3.3.1.cmml" xref="Thmproofx11.p1.30.m29.3.3.1"><times id="Thmproofx11.p1.30.m29.3.3.1.2.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.2"></times><apply id="Thmproofx11.p1.30.m29.3.3.1.3.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p1.30.m29.3.3.1.3.1.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.3">subscript</csymbol><ci id="Thmproofx11.p1.30.m29.3.3.1.3.2.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.3.2">ℎ</ci><ci id="Thmproofx11.p1.30.m29.3.3.1.3.3.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.3.3">𝑖</ci></apply><interval closure="open" id="Thmproofx11.p1.30.m29.3.3.1.1.2.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.1.1"><apply id="Thmproofx11.p1.30.m29.3.3.1.1.1.1.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.30.m29.3.3.1.1.1.1.1.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.30.m29.3.3.1.1.1.1.2.cmml" xref="Thmproofx11.p1.30.m29.3.3.1.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.30.m29.3.3.1.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.30.m29.3.3.1.1.1.1.3">0</cn></apply><ci id="Thmproofx11.p1.30.m29.1.1.cmml" xref="Thmproofx11.p1.30.m29.1.1">𝑛</ci></interval></apply><apply id="Thmproofx11.p1.30.m29.4.4.2.cmml" xref="Thmproofx11.p1.30.m29.4.4.2"><times id="Thmproofx11.p1.30.m29.4.4.2.2.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.2"></times><apply id="Thmproofx11.p1.30.m29.4.4.2.3.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.3"><csymbol cd="ambiguous" id="Thmproofx11.p1.30.m29.4.4.2.3.1.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.3">subscript</csymbol><ci id="Thmproofx11.p1.30.m29.4.4.2.3.2.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.3.2">ℎ</ci><ci id="Thmproofx11.p1.30.m29.4.4.2.3.3.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.3.3">𝑗</ci></apply><interval closure="open" id="Thmproofx11.p1.30.m29.4.4.2.1.2.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.1.1"><apply id="Thmproofx11.p1.30.m29.4.4.2.1.1.1.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p1.30.m29.4.4.2.1.1.1.1.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.1.1.1">subscript</csymbol><ci id="Thmproofx11.p1.30.m29.4.4.2.1.1.1.2.cmml" xref="Thmproofx11.p1.30.m29.4.4.2.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p1.30.m29.4.4.2.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p1.30.m29.4.4.2.1.1.1.3">0</cn></apply><ci id="Thmproofx11.p1.30.m29.2.2.cmml" xref="Thmproofx11.p1.30.m29.2.2">𝑛</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p1.30.m29.4c">h_{i}(\alpha_{0},n)<h_{j}(\alpha_{0},n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p1.30.m29.4d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) < italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx11.p2"> <p class="ltx_p" id="Thmproofx11.p2.3">Let <math alttext="E_{i}=\mathcal{E}_{i,n}" class="ltx_Math" display="inline" id="Thmproofx11.p2.1.m1.2"><semantics id="Thmproofx11.p2.1.m1.2a"><mrow id="Thmproofx11.p2.1.m1.2.3" xref="Thmproofx11.p2.1.m1.2.3.cmml"><msub id="Thmproofx11.p2.1.m1.2.3.2" xref="Thmproofx11.p2.1.m1.2.3.2.cmml"><mi id="Thmproofx11.p2.1.m1.2.3.2.2" xref="Thmproofx11.p2.1.m1.2.3.2.2.cmml">E</mi><mi id="Thmproofx11.p2.1.m1.2.3.2.3" xref="Thmproofx11.p2.1.m1.2.3.2.3.cmml">i</mi></msub><mo id="Thmproofx11.p2.1.m1.2.3.1" xref="Thmproofx11.p2.1.m1.2.3.1.cmml">=</mo><msub id="Thmproofx11.p2.1.m1.2.3.3" xref="Thmproofx11.p2.1.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx11.p2.1.m1.2.3.3.2" xref="Thmproofx11.p2.1.m1.2.3.3.2.cmml">ℰ</mi><mrow id="Thmproofx11.p2.1.m1.2.2.2.4" xref="Thmproofx11.p2.1.m1.2.2.2.3.cmml"><mi id="Thmproofx11.p2.1.m1.1.1.1.1" xref="Thmproofx11.p2.1.m1.1.1.1.1.cmml">i</mi><mo id="Thmproofx11.p2.1.m1.2.2.2.4.1" xref="Thmproofx11.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="Thmproofx11.p2.1.m1.2.2.2.2" xref="Thmproofx11.p2.1.m1.2.2.2.2.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.1.m1.2b"><apply id="Thmproofx11.p2.1.m1.2.3.cmml" xref="Thmproofx11.p2.1.m1.2.3"><eq id="Thmproofx11.p2.1.m1.2.3.1.cmml" xref="Thmproofx11.p2.1.m1.2.3.1"></eq><apply id="Thmproofx11.p2.1.m1.2.3.2.cmml" xref="Thmproofx11.p2.1.m1.2.3.2"><csymbol cd="ambiguous" id="Thmproofx11.p2.1.m1.2.3.2.1.cmml" xref="Thmproofx11.p2.1.m1.2.3.2">subscript</csymbol><ci id="Thmproofx11.p2.1.m1.2.3.2.2.cmml" xref="Thmproofx11.p2.1.m1.2.3.2.2">𝐸</ci><ci id="Thmproofx11.p2.1.m1.2.3.2.3.cmml" xref="Thmproofx11.p2.1.m1.2.3.2.3">𝑖</ci></apply><apply id="Thmproofx11.p2.1.m1.2.3.3.cmml" xref="Thmproofx11.p2.1.m1.2.3.3"><csymbol cd="ambiguous" id="Thmproofx11.p2.1.m1.2.3.3.1.cmml" xref="Thmproofx11.p2.1.m1.2.3.3">subscript</csymbol><ci id="Thmproofx11.p2.1.m1.2.3.3.2.cmml" xref="Thmproofx11.p2.1.m1.2.3.3.2">ℰ</ci><list id="Thmproofx11.p2.1.m1.2.2.2.3.cmml" xref="Thmproofx11.p2.1.m1.2.2.2.4"><ci id="Thmproofx11.p2.1.m1.1.1.1.1.cmml" xref="Thmproofx11.p2.1.m1.1.1.1.1">𝑖</ci><ci id="Thmproofx11.p2.1.m1.2.2.2.2.cmml" xref="Thmproofx11.p2.1.m1.2.2.2.2">𝑛</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.1.m1.2c">E_{i}=\mathcal{E}_{i,n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.1.m1.2d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="E_{j}=\mathcal{E}_{j,n}" class="ltx_Math" display="inline" id="Thmproofx11.p2.2.m2.2"><semantics id="Thmproofx11.p2.2.m2.2a"><mrow id="Thmproofx11.p2.2.m2.2.3" xref="Thmproofx11.p2.2.m2.2.3.cmml"><msub id="Thmproofx11.p2.2.m2.2.3.2" xref="Thmproofx11.p2.2.m2.2.3.2.cmml"><mi id="Thmproofx11.p2.2.m2.2.3.2.2" xref="Thmproofx11.p2.2.m2.2.3.2.2.cmml">E</mi><mi id="Thmproofx11.p2.2.m2.2.3.2.3" xref="Thmproofx11.p2.2.m2.2.3.2.3.cmml">j</mi></msub><mo id="Thmproofx11.p2.2.m2.2.3.1" xref="Thmproofx11.p2.2.m2.2.3.1.cmml">=</mo><msub id="Thmproofx11.p2.2.m2.2.3.3" xref="Thmproofx11.p2.2.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx11.p2.2.m2.2.3.3.2" xref="Thmproofx11.p2.2.m2.2.3.3.2.cmml">ℰ</mi><mrow id="Thmproofx11.p2.2.m2.2.2.2.4" xref="Thmproofx11.p2.2.m2.2.2.2.3.cmml"><mi id="Thmproofx11.p2.2.m2.1.1.1.1" xref="Thmproofx11.p2.2.m2.1.1.1.1.cmml">j</mi><mo id="Thmproofx11.p2.2.m2.2.2.2.4.1" xref="Thmproofx11.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="Thmproofx11.p2.2.m2.2.2.2.2" xref="Thmproofx11.p2.2.m2.2.2.2.2.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.2.m2.2b"><apply id="Thmproofx11.p2.2.m2.2.3.cmml" xref="Thmproofx11.p2.2.m2.2.3"><eq id="Thmproofx11.p2.2.m2.2.3.1.cmml" xref="Thmproofx11.p2.2.m2.2.3.1"></eq><apply id="Thmproofx11.p2.2.m2.2.3.2.cmml" xref="Thmproofx11.p2.2.m2.2.3.2"><csymbol cd="ambiguous" id="Thmproofx11.p2.2.m2.2.3.2.1.cmml" xref="Thmproofx11.p2.2.m2.2.3.2">subscript</csymbol><ci id="Thmproofx11.p2.2.m2.2.3.2.2.cmml" xref="Thmproofx11.p2.2.m2.2.3.2.2">𝐸</ci><ci id="Thmproofx11.p2.2.m2.2.3.2.3.cmml" xref="Thmproofx11.p2.2.m2.2.3.2.3">𝑗</ci></apply><apply id="Thmproofx11.p2.2.m2.2.3.3.cmml" xref="Thmproofx11.p2.2.m2.2.3.3"><csymbol cd="ambiguous" id="Thmproofx11.p2.2.m2.2.3.3.1.cmml" xref="Thmproofx11.p2.2.m2.2.3.3">subscript</csymbol><ci id="Thmproofx11.p2.2.m2.2.3.3.2.cmml" xref="Thmproofx11.p2.2.m2.2.3.3.2">ℰ</ci><list id="Thmproofx11.p2.2.m2.2.2.2.3.cmml" xref="Thmproofx11.p2.2.m2.2.2.2.4"><ci id="Thmproofx11.p2.2.m2.1.1.1.1.cmml" xref="Thmproofx11.p2.2.m2.1.1.1.1">𝑗</ci><ci id="Thmproofx11.p2.2.m2.2.2.2.2.cmml" xref="Thmproofx11.p2.2.m2.2.2.2.2">𝑛</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.2.m2.2c">E_{j}=\mathcal{E}_{j,n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.2.m2.2d">italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = caligraphic_E start_POSTSUBSCRIPT italic_j , italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\alpha_{n}" class="ltx_Math" display="inline" id="Thmproofx11.p2.3.m3.1"><semantics id="Thmproofx11.p2.3.m3.1a"><msub id="Thmproofx11.p2.3.m3.1.1" xref="Thmproofx11.p2.3.m3.1.1.cmml"><mi id="Thmproofx11.p2.3.m3.1.1.2" xref="Thmproofx11.p2.3.m3.1.1.2.cmml">α</mi><mi id="Thmproofx11.p2.3.m3.1.1.3" xref="Thmproofx11.p2.3.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.3.m3.1b"><apply id="Thmproofx11.p2.3.m3.1.1.cmml" xref="Thmproofx11.p2.3.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p2.3.m3.1.1.1.cmml" xref="Thmproofx11.p2.3.m3.1.1">subscript</csymbol><ci id="Thmproofx11.p2.3.m3.1.1.2.cmml" xref="Thmproofx11.p2.3.m3.1.1.2">𝛼</ci><ci id="Thmproofx11.p2.3.m3.1.1.3.cmml" xref="Thmproofx11.p2.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.3.m3.1c">\alpha_{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.3.m3.1d">italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> satisfies</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx33"> <tbody id="Pt0.A3.Ex44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{E_{j}}{j}(k\alpha_{n}+S_{j})" class="ltx_Math" display="inline" id="Pt0.A3.Ex44.m1.1"><semantics id="Pt0.A3.Ex44.m1.1a"><mrow id="Pt0.A3.Ex44.m1.1.1" xref="Pt0.A3.Ex44.m1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex44.m1.1.1.3" xref="Pt0.A3.Ex44.m1.1.1.3.cmml"><mfrac id="Pt0.A3.Ex44.m1.1.1.3a" xref="Pt0.A3.Ex44.m1.1.1.3.cmml"><msub id="Pt0.A3.Ex44.m1.1.1.3.2" xref="Pt0.A3.Ex44.m1.1.1.3.2.cmml"><mi id="Pt0.A3.Ex44.m1.1.1.3.2.2" xref="Pt0.A3.Ex44.m1.1.1.3.2.2.cmml">E</mi><mi id="Pt0.A3.Ex44.m1.1.1.3.2.3" xref="Pt0.A3.Ex44.m1.1.1.3.2.3.cmml">j</mi></msub><mi id="Pt0.A3.Ex44.m1.1.1.3.3" xref="Pt0.A3.Ex44.m1.1.1.3.3.cmml">j</mi></mfrac></mstyle><mo id="Pt0.A3.Ex44.m1.1.1.2" xref="Pt0.A3.Ex44.m1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex44.m1.1.1.1.1" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex44.m1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex44.m1.1.1.1.1.1" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.cmml"><mrow id="Pt0.A3.Ex44.m1.1.1.1.1.1.2" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.2.cmml"><mi id="Pt0.A3.Ex44.m1.1.1.1.1.1.2.2" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.2.2.cmml">k</mi><mo id="Pt0.A3.Ex44.m1.1.1.1.1.1.2.1" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex44.m1.1.1.1.1.1.2.3" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.2.3.cmml"><mi id="Pt0.A3.Ex44.m1.1.1.1.1.1.2.3.2" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.2.3.2.cmml">α</mi><mi id="Pt0.A3.Ex44.m1.1.1.1.1.1.2.3.3" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.2.3.3.cmml">n</mi></msub></mrow><mo id="Pt0.A3.Ex44.m1.1.1.1.1.1.1" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.1.cmml">+</mo><msub id="Pt0.A3.Ex44.m1.1.1.1.1.1.3" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex44.m1.1.1.1.1.1.3.2" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.3.2.cmml">S</mi><mi id="Pt0.A3.Ex44.m1.1.1.1.1.1.3.3" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.3.3.cmml">j</mi></msub></mrow><mo id="Pt0.A3.Ex44.m1.1.1.1.1.3" stretchy="false" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex44.m1.1b"><apply id="Pt0.A3.Ex44.m1.1.1.cmml" xref="Pt0.A3.Ex44.m1.1.1"><times id="Pt0.A3.Ex44.m1.1.1.2.cmml" xref="Pt0.A3.Ex44.m1.1.1.2"></times><apply id="Pt0.A3.Ex44.m1.1.1.3.cmml" xref="Pt0.A3.Ex44.m1.1.1.3"><divide id="Pt0.A3.Ex44.m1.1.1.3.1.cmml" xref="Pt0.A3.Ex44.m1.1.1.3"></divide><apply id="Pt0.A3.Ex44.m1.1.1.3.2.cmml" xref="Pt0.A3.Ex44.m1.1.1.3.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex44.m1.1.1.3.2.1.cmml" xref="Pt0.A3.Ex44.m1.1.1.3.2">subscript</csymbol><ci id="Pt0.A3.Ex44.m1.1.1.3.2.2.cmml" xref="Pt0.A3.Ex44.m1.1.1.3.2.2">𝐸</ci><ci 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id="Pt0.A3.Ex44.m1.1.1.1.1.1.3.cmml" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex44.m1.1.1.1.1.1.3.1.cmml" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex44.m1.1.1.1.1.1.3.2.cmml" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.3.2">𝑆</ci><ci id="Pt0.A3.Ex44.m1.1.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex44.m1.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex44.m1.1c">\displaystyle\frac{E_{j}}{j}(k\alpha_{n}+S_{j})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex44.m1.1d">divide start_ARG italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_j end_ARG ( italic_k italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{E_{i}}{i}(k\alpha_{n}+S_{i})\iff" class="ltx_Math" 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xref="Pt0.A3.Ex44.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex44.m2.1.1.1.1.1.1" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.cmml"><mrow id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.cmml"><mi id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.2" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.1" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.3" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.3.cmml"><mi id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.3.2" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.3.2.cmml">α</mi><mi id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.3.3" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.2.3.3.cmml">n</mi></msub></mrow><mo id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.1" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.1.cmml">+</mo><msub 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xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.3.2">𝑆</ci><ci id="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex44.m2.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply><csymbol cd="latexml" id="Pt0.A3.Ex44.m2.1.1.3.cmml" xref="Pt0.A3.Ex44.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex44.m2.1c">\displaystyle=\frac{E_{i}}{i}(k\alpha_{n}+S_{i})\iff</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex44.m2.1d">= divide start_ARG italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG ( italic_k italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ⇔</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td 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id="Pt0.A3.Ex45.m1.1.1.1.1.1.3.2.1.cmml" xref="Pt0.A3.Ex45.m1.1.1.1.1.1.3.2">subscript</csymbol><ci id="Pt0.A3.Ex45.m1.1.1.1.1.1.3.2.2.cmml" xref="Pt0.A3.Ex45.m1.1.1.1.1.1.3.2.2">𝐸</ci><ci id="Pt0.A3.Ex45.m1.1.1.1.1.1.3.2.3.cmml" xref="Pt0.A3.Ex45.m1.1.1.1.1.1.3.2.3">𝑗</ci></apply><ci id="Pt0.A3.Ex45.m1.1.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex45.m1.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex45.m1.1c">\displaystyle k\alpha_{n}\left(\frac{E_{i}}{i}-\frac{E_{j}}{j}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex45.m1.1d">italic_k italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( divide start_ARG italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG - divide start_ARG italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_j end_ARG )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=S_{j}\frac{E_{j}}{j}-S_{j}\frac{E_{i}}{i}\iff" class="ltx_Math" display="inline" id="Pt0.A3.Ex45.m2.1"><semantics id="Pt0.A3.Ex45.m2.1a"><mrow id="Pt0.A3.Ex45.m2.1.1" xref="Pt0.A3.Ex45.m2.1.1.cmml"><mrow id="Pt0.A3.Ex45.m2.1.1.2" xref="Pt0.A3.Ex45.m2.1.1.2.cmml"><mi id="Pt0.A3.Ex45.m2.1.1.2.2" xref="Pt0.A3.Ex45.m2.1.1.2.2.cmml"></mi><mo id="Pt0.A3.Ex45.m2.1.1.2.1" xref="Pt0.A3.Ex45.m2.1.1.2.1.cmml">=</mo><mrow id="Pt0.A3.Ex45.m2.1.1.2.3" xref="Pt0.A3.Ex45.m2.1.1.2.3.cmml"><mrow id="Pt0.A3.Ex45.m2.1.1.2.3.2" xref="Pt0.A3.Ex45.m2.1.1.2.3.2.cmml"><msub id="Pt0.A3.Ex45.m2.1.1.2.3.2.2" xref="Pt0.A3.Ex45.m2.1.1.2.3.2.2.cmml"><mi id="Pt0.A3.Ex45.m2.1.1.2.3.2.2.2" xref="Pt0.A3.Ex45.m2.1.1.2.3.2.2.2.cmml">S</mi><mi id="Pt0.A3.Ex45.m2.1.1.2.3.2.2.3" xref="Pt0.A3.Ex45.m2.1.1.2.3.2.2.3.cmml">j</mi></msub><mo id="Pt0.A3.Ex45.m2.1.1.2.3.2.1" xref="Pt0.A3.Ex45.m2.1.1.2.3.2.1.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A3.Ex45.m2.1.1.2.3.2.3" 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id="Pt0.A3.Ex45.m2.1c">\displaystyle=S_{j}\frac{E_{j}}{j}-S_{j}\frac{E_{i}}{i}\iff</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex45.m2.1d">= italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_j end_ARG - italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG ⇔</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\alpha_{n}" class="ltx_Math" display="inline" id="Pt0.A3.Ex46.m1.1"><semantics id="Pt0.A3.Ex46.m1.1a"><msub id="Pt0.A3.Ex46.m1.1.1" xref="Pt0.A3.Ex46.m1.1.1.cmml"><mi id="Pt0.A3.Ex46.m1.1.1.2" xref="Pt0.A3.Ex46.m1.1.1.2.cmml">α</mi><mi id="Pt0.A3.Ex46.m1.1.1.3" xref="Pt0.A3.Ex46.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex46.m1.1b"><apply id="Pt0.A3.Ex46.m1.1.1.cmml" xref="Pt0.A3.Ex46.m1.1.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex46.m1.1.1.1.cmml" xref="Pt0.A3.Ex46.m1.1.1">subscript</csymbol><ci id="Pt0.A3.Ex46.m1.1.1.2.cmml" xref="Pt0.A3.Ex46.m1.1.1.2">𝛼</ci><ci id="Pt0.A3.Ex46.m1.1.1.3.cmml" xref="Pt0.A3.Ex46.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex46.m1.1c">\displaystyle\alpha_{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex46.m1.1d">italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{k}\frac{S_{j}\frac{E_{j}}{j}-S_{j}\frac{E_{i}}{i}}{% 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xref="Pt0.A3.Ex46.m2.1.1.6.3.3.3.3.2">𝐸</ci><ci id="Pt0.A3.Ex46.m2.1.1.6.3.3.3.3.3.cmml" xref="Pt0.A3.Ex46.m2.1.1.6.3.3.3.3.3">𝑗</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex46.m2.1c">\displaystyle=\frac{1}{k}\frac{S_{j}\frac{E_{j}}{j}-S_{j}\frac{E_{i}}{i}}{% \frac{E_{i}}{i}-\frac{E_{j}}{j}}=\frac{1}{k}\frac{iS_{j}E_{j}-jS_{i}E_{i}}{jE_% {i}-iE_{j}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex46.m2.1d">= divide start_ARG 1 end_ARG start_ARG italic_k end_ARG divide start_ARG italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_j end_ARG - italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG end_ARG start_ARG divide start_ARG italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG - divide start_ARG italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_j end_ARG end_ARG = divide start_ARG 1 end_ARG start_ARG italic_k end_ARG divide start_ARG italic_i italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_j italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_j italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_i italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex47"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Rightarrow~{}\alpha_{1}" class="ltx_Math" display="inline" id="Pt0.A3.Ex47.m1.1"><semantics id="Pt0.A3.Ex47.m1.1a"><mrow id="Pt0.A3.Ex47.m1.1.1" xref="Pt0.A3.Ex47.m1.1.1.cmml"><mi id="Pt0.A3.Ex47.m1.1.1.2" xref="Pt0.A3.Ex47.m1.1.1.2.cmml"></mi><mo id="Pt0.A3.Ex47.m1.1.1.1" rspace="0.608em" stretchy="false" xref="Pt0.A3.Ex47.m1.1.1.1.cmml">⇒</mo><msub id="Pt0.A3.Ex47.m1.1.1.3" xref="Pt0.A3.Ex47.m1.1.1.3.cmml"><mi id="Pt0.A3.Ex47.m1.1.1.3.2" xref="Pt0.A3.Ex47.m1.1.1.3.2.cmml">α</mi><mn id="Pt0.A3.Ex47.m1.1.1.3.3" xref="Pt0.A3.Ex47.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex47.m1.1b"><apply id="Pt0.A3.Ex47.m1.1.1.cmml" xref="Pt0.A3.Ex47.m1.1.1"><ci id="Pt0.A3.Ex47.m1.1.1.1.cmml" xref="Pt0.A3.Ex47.m1.1.1.1">⇒</ci><csymbol cd="latexml" id="Pt0.A3.Ex47.m1.1.1.2.cmml" xref="Pt0.A3.Ex47.m1.1.1.2">absent</csymbol><apply id="Pt0.A3.Ex47.m1.1.1.3.cmml" xref="Pt0.A3.Ex47.m1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex47.m1.1.1.3.1.cmml" xref="Pt0.A3.Ex47.m1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex47.m1.1.1.3.2.cmml" xref="Pt0.A3.Ex47.m1.1.1.3.2">𝛼</ci><cn id="Pt0.A3.Ex47.m1.1.1.3.3.cmml" type="integer" xref="Pt0.A3.Ex47.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex47.m1.1c">\displaystyle\Rightarrow~{}\alpha_{1}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex47.m1.1d">⇒ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{k}\frac{iS_{j}-jS_{i}}{j-i}" class="ltx_Math" display="inline" id="Pt0.A3.Ex47.m2.1"><semantics id="Pt0.A3.Ex47.m2.1a"><mrow id="Pt0.A3.Ex47.m2.1.1" xref="Pt0.A3.Ex47.m2.1.1.cmml"><mi id="Pt0.A3.Ex47.m2.1.1.2" xref="Pt0.A3.Ex47.m2.1.1.2.cmml"></mi><mo id="Pt0.A3.Ex47.m2.1.1.1" xref="Pt0.A3.Ex47.m2.1.1.1.cmml">=</mo><mrow id="Pt0.A3.Ex47.m2.1.1.3" xref="Pt0.A3.Ex47.m2.1.1.3.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex47.m2.1.1.3.2" xref="Pt0.A3.Ex47.m2.1.1.3.2.cmml"><mfrac id="Pt0.A3.Ex47.m2.1.1.3.2a" xref="Pt0.A3.Ex47.m2.1.1.3.2.cmml"><mn id="Pt0.A3.Ex47.m2.1.1.3.2.2" xref="Pt0.A3.Ex47.m2.1.1.3.2.2.cmml">1</mn><mi id="Pt0.A3.Ex47.m2.1.1.3.2.3" xref="Pt0.A3.Ex47.m2.1.1.3.2.3.cmml">k</mi></mfrac></mstyle><mo id="Pt0.A3.Ex47.m2.1.1.3.1" xref="Pt0.A3.Ex47.m2.1.1.3.1.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A3.Ex47.m2.1.1.3.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.cmml"><mfrac id="Pt0.A3.Ex47.m2.1.1.3.3a" xref="Pt0.A3.Ex47.m2.1.1.3.3.cmml"><mrow id="Pt0.A3.Ex47.m2.1.1.3.3.2" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.cmml"><mrow id="Pt0.A3.Ex47.m2.1.1.3.3.2.2" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.cmml"><mi id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.2" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.2.cmml">i</mi><mo id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.1" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.cmml"><mi id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.2" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.2.cmml">S</mi><mi id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.3.cmml">j</mi></msub></mrow><mo id="Pt0.A3.Ex47.m2.1.1.3.3.2.1" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.1.cmml">−</mo><mrow id="Pt0.A3.Ex47.m2.1.1.3.3.2.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.cmml"><mi id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.2" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.2.cmml">j</mi><mo id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.1" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.1.cmml">⁢</mo><msub id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.cmml"><mi id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.2" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.2.cmml">S</mi><mi id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.3.cmml">i</mi></msub></mrow></mrow><mrow id="Pt0.A3.Ex47.m2.1.1.3.3.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.3.cmml"><mi id="Pt0.A3.Ex47.m2.1.1.3.3.3.2" xref="Pt0.A3.Ex47.m2.1.1.3.3.3.2.cmml">j</mi><mo id="Pt0.A3.Ex47.m2.1.1.3.3.3.1" xref="Pt0.A3.Ex47.m2.1.1.3.3.3.1.cmml">−</mo><mi id="Pt0.A3.Ex47.m2.1.1.3.3.3.3" xref="Pt0.A3.Ex47.m2.1.1.3.3.3.3.cmml">i</mi></mrow></mfrac></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex47.m2.1b"><apply id="Pt0.A3.Ex47.m2.1.1.cmml" xref="Pt0.A3.Ex47.m2.1.1"><eq id="Pt0.A3.Ex47.m2.1.1.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.1"></eq><csymbol cd="latexml" id="Pt0.A3.Ex47.m2.1.1.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.2">absent</csymbol><apply id="Pt0.A3.Ex47.m2.1.1.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3"><times id="Pt0.A3.Ex47.m2.1.1.3.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.1"></times><apply id="Pt0.A3.Ex47.m2.1.1.3.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.2"><divide id="Pt0.A3.Ex47.m2.1.1.3.2.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.2"></divide><cn id="Pt0.A3.Ex47.m2.1.1.3.2.2.cmml" type="integer" xref="Pt0.A3.Ex47.m2.1.1.3.2.2">1</cn><ci id="Pt0.A3.Ex47.m2.1.1.3.2.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.2.3">𝑘</ci></apply><apply id="Pt0.A3.Ex47.m2.1.1.3.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3"><divide id="Pt0.A3.Ex47.m2.1.1.3.3.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3"></divide><apply id="Pt0.A3.Ex47.m2.1.1.3.3.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2"><minus id="Pt0.A3.Ex47.m2.1.1.3.3.2.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.1"></minus><apply id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2"><times id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.1"></times><ci id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.2">𝑖</ci><apply id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3">subscript</csymbol><ci id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.2">𝑆</ci><ci id="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.2.3.3">𝑗</ci></apply></apply><apply id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3"><times id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.1"></times><ci id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.2">𝑗</ci><apply id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3">subscript</csymbol><ci id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.2">𝑆</ci><ci id="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.2.3.3.3">𝑖</ci></apply></apply></apply><apply id="Pt0.A3.Ex47.m2.1.1.3.3.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.3"><minus id="Pt0.A3.Ex47.m2.1.1.3.3.3.1.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.3.1"></minus><ci id="Pt0.A3.Ex47.m2.1.1.3.3.3.2.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.3.2">𝑗</ci><ci id="Pt0.A3.Ex47.m2.1.1.3.3.3.3.cmml" xref="Pt0.A3.Ex47.m2.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex47.m2.1c">\displaystyle=\frac{1}{k}\frac{iS_{j}-jS_{i}}{j-i}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex47.m2.1d">= divide start_ARG 1 end_ARG start_ARG italic_k end_ARG divide start_ARG italic_i italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_j italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_j - italic_i end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx11.p2.15">and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx34"> <tbody id="Pt0.A3.Ex48"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\alpha_{n}" class="ltx_Math" display="inline" id="Pt0.A3.Ex48.m1.1"><semantics id="Pt0.A3.Ex48.m1.1a"><msub id="Pt0.A3.Ex48.m1.1.1" xref="Pt0.A3.Ex48.m1.1.1.cmml"><mi id="Pt0.A3.Ex48.m1.1.1.2" xref="Pt0.A3.Ex48.m1.1.1.2.cmml">α</mi><mi id="Pt0.A3.Ex48.m1.1.1.3" xref="Pt0.A3.Ex48.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex48.m1.1b"><apply id="Pt0.A3.Ex48.m1.1.1.cmml" xref="Pt0.A3.Ex48.m1.1.1"><csymbol cd="ambiguous" id="Pt0.A3.Ex48.m1.1.1.1.cmml" xref="Pt0.A3.Ex48.m1.1.1">subscript</csymbol><ci id="Pt0.A3.Ex48.m1.1.1.2.cmml" xref="Pt0.A3.Ex48.m1.1.1.2">𝛼</ci><ci id="Pt0.A3.Ex48.m1.1.1.3.cmml" xref="Pt0.A3.Ex48.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex48.m1.1c">\displaystyle\alpha_{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex48.m1.1d">italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\alpha_{1}\iff" class="ltx_Math" display="inline" id="Pt0.A3.Ex48.m2.1"><semantics id="Pt0.A3.Ex48.m2.1a"><mrow id="Pt0.A3.Ex48.m2.1.1" xref="Pt0.A3.Ex48.m2.1.1.cmml"><mrow id="Pt0.A3.Ex48.m2.1.1.2" xref="Pt0.A3.Ex48.m2.1.1.2.cmml"><mi id="Pt0.A3.Ex48.m2.1.1.2.2" xref="Pt0.A3.Ex48.m2.1.1.2.2.cmml"></mi><mo id="Pt0.A3.Ex48.m2.1.1.2.1" xref="Pt0.A3.Ex48.m2.1.1.2.1.cmml">≥</mo><msub id="Pt0.A3.Ex48.m2.1.1.2.3" xref="Pt0.A3.Ex48.m2.1.1.2.3.cmml"><mi id="Pt0.A3.Ex48.m2.1.1.2.3.2" xref="Pt0.A3.Ex48.m2.1.1.2.3.2.cmml">α</mi><mn id="Pt0.A3.Ex48.m2.1.1.2.3.3" xref="Pt0.A3.Ex48.m2.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="Pt0.A3.Ex48.m2.1.1.1" stretchy="false" xref="Pt0.A3.Ex48.m2.1.1.1.cmml">⇔</mo><mi id="Pt0.A3.Ex48.m2.1.1.3" xref="Pt0.A3.Ex48.m2.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex48.m2.1b"><apply id="Pt0.A3.Ex48.m2.1.1.cmml" xref="Pt0.A3.Ex48.m2.1.1"><csymbol cd="latexml" id="Pt0.A3.Ex48.m2.1.1.1.cmml" xref="Pt0.A3.Ex48.m2.1.1.1">iff</csymbol><apply id="Pt0.A3.Ex48.m2.1.1.2.cmml" xref="Pt0.A3.Ex48.m2.1.1.2"><geq id="Pt0.A3.Ex48.m2.1.1.2.1.cmml" xref="Pt0.A3.Ex48.m2.1.1.2.1"></geq><csymbol cd="latexml" id="Pt0.A3.Ex48.m2.1.1.2.2.cmml" xref="Pt0.A3.Ex48.m2.1.1.2.2">absent</csymbol><apply id="Pt0.A3.Ex48.m2.1.1.2.3.cmml" xref="Pt0.A3.Ex48.m2.1.1.2.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex48.m2.1.1.2.3.1.cmml" xref="Pt0.A3.Ex48.m2.1.1.2.3">subscript</csymbol><ci id="Pt0.A3.Ex48.m2.1.1.2.3.2.cmml" xref="Pt0.A3.Ex48.m2.1.1.2.3.2">𝛼</ci><cn id="Pt0.A3.Ex48.m2.1.1.2.3.3.cmml" type="integer" xref="Pt0.A3.Ex48.m2.1.1.2.3.3">1</cn></apply></apply><csymbol cd="latexml" id="Pt0.A3.Ex48.m2.1.1.3.cmml" xref="Pt0.A3.Ex48.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex48.m2.1c">\displaystyle\geq\alpha_{1}\iff</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex48.m2.1d">≥ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⇔</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody 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xref="Pt0.A3.Ex49.m1.1.1.2.2.3.3.cmml">j</mi></msub><mo id="Pt0.A3.Ex49.m1.1.1.2.2.1a" xref="Pt0.A3.Ex49.m1.1.1.2.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex49.m1.1.1.2.2.4" xref="Pt0.A3.Ex49.m1.1.1.2.2.4.cmml"><mi id="Pt0.A3.Ex49.m1.1.1.2.2.4.2" xref="Pt0.A3.Ex49.m1.1.1.2.2.4.2.cmml">E</mi><mi id="Pt0.A3.Ex49.m1.1.1.2.2.4.3" xref="Pt0.A3.Ex49.m1.1.1.2.2.4.3.cmml">j</mi></msub></mrow><mo id="Pt0.A3.Ex49.m1.1.1.2.1" xref="Pt0.A3.Ex49.m1.1.1.2.1.cmml">−</mo><mrow id="Pt0.A3.Ex49.m1.1.1.2.3" xref="Pt0.A3.Ex49.m1.1.1.2.3.cmml"><mi id="Pt0.A3.Ex49.m1.1.1.2.3.2" xref="Pt0.A3.Ex49.m1.1.1.2.3.2.cmml">j</mi><mo id="Pt0.A3.Ex49.m1.1.1.2.3.1" xref="Pt0.A3.Ex49.m1.1.1.2.3.1.cmml">⁢</mo><msub id="Pt0.A3.Ex49.m1.1.1.2.3.3" xref="Pt0.A3.Ex49.m1.1.1.2.3.3.cmml"><mi id="Pt0.A3.Ex49.m1.1.1.2.3.3.2" xref="Pt0.A3.Ex49.m1.1.1.2.3.3.2.cmml">S</mi><mi id="Pt0.A3.Ex49.m1.1.1.2.3.3.3" xref="Pt0.A3.Ex49.m1.1.1.2.3.3.3.cmml">i</mi></msub><mo id="Pt0.A3.Ex49.m1.1.1.2.3.1a" xref="Pt0.A3.Ex49.m1.1.1.2.3.1.cmml">⁢</mo><msub 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id="Pt0.A3.Ex49.m2.1.1.2.3.2.2.1.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.2.1"></times><ci id="Pt0.A3.Ex49.m2.1.1.2.3.2.2.2.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.2.2">𝑖</ci><apply id="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3.1.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3">subscript</csymbol><ci id="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3.2.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3.2">𝑆</ci><ci id="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.2.3.3">𝑗</ci></apply></apply><apply id="Pt0.A3.Ex49.m2.1.1.2.3.2.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.3"><times id="Pt0.A3.Ex49.m2.1.1.2.3.2.3.1.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.3.1"></times><ci id="Pt0.A3.Ex49.m2.1.1.2.3.2.3.2.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.3.2">𝑗</ci><apply id="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3.1.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3">subscript</csymbol><ci id="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3.2.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3.2">𝑆</ci><ci id="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.2.3.3.3">𝑖</ci></apply></apply></apply><apply id="Pt0.A3.Ex49.m2.1.1.2.3.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.3"><minus id="Pt0.A3.Ex49.m2.1.1.2.3.3.1.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.3.1"></minus><ci id="Pt0.A3.Ex49.m2.1.1.2.3.3.2.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.3.2">𝑗</ci><ci id="Pt0.A3.Ex49.m2.1.1.2.3.3.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.2.3.3.3">𝑖</ci></apply></apply></apply><csymbol cd="latexml" id="Pt0.A3.Ex49.m2.1.1.3.cmml" xref="Pt0.A3.Ex49.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex49.m2.1c">\displaystyle\geq\frac{iS_{j}-jS_{i}}{j-i}\iff</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex49.m2.1d">≥ divide start_ARG italic_i italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_j italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_j - italic_i end_ARG ⇔</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex50"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(j-i)(iS_{j}E_{j}-jS_{i}E_{i})" class="ltx_Math" display="inline" id="Pt0.A3.Ex50.m1.2"><semantics id="Pt0.A3.Ex50.m1.2a"><mrow id="Pt0.A3.Ex50.m1.2.2" xref="Pt0.A3.Ex50.m1.2.2.cmml"><mrow id="Pt0.A3.Ex50.m1.1.1.1.1" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex50.m1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex50.m1.1.1.1.1.1" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.cmml"><mi id="Pt0.A3.Ex50.m1.1.1.1.1.1.2" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.2.cmml">j</mi><mo id="Pt0.A3.Ex50.m1.1.1.1.1.1.1" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.1.cmml">−</mo><mi id="Pt0.A3.Ex50.m1.1.1.1.1.1.3" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.3.cmml">i</mi></mrow><mo id="Pt0.A3.Ex50.m1.1.1.1.1.3" stretchy="false" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A3.Ex50.m1.2.2.3" xref="Pt0.A3.Ex50.m1.2.2.3.cmml">⁢</mo><mrow id="Pt0.A3.Ex50.m1.2.2.2.1" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.cmml"><mo id="Pt0.A3.Ex50.m1.2.2.2.1.2" stretchy="false" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex50.m1.2.2.2.1.1" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.cmml"><mrow id="Pt0.A3.Ex50.m1.2.2.2.1.1.2" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.cmml"><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.2" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.2.cmml">i</mi><mo id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.1" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.cmml"><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.2" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.2.cmml">S</mi><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.3" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.3.cmml">j</mi></msub><mo id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.1a" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.cmml"><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.2" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.2.cmml">E</mi><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.3" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.3.cmml">j</mi></msub></mrow><mo id="Pt0.A3.Ex50.m1.2.2.2.1.1.1" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.1.cmml">−</mo><mrow id="Pt0.A3.Ex50.m1.2.2.2.1.1.3" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.cmml"><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.2" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.2.cmml">j</mi><mo id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.1" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.1.cmml">⁢</mo><msub id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.cmml"><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.2" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.2.cmml">S</mi><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.3" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.3.cmml">i</mi></msub><mo id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.1a" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.1.cmml">⁢</mo><msub id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.cmml"><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.2" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.2.cmml">E</mi><mi id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.3" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.3.cmml">i</mi></msub></mrow></mrow><mo id="Pt0.A3.Ex50.m1.2.2.2.1.3" stretchy="false" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex50.m1.2b"><apply id="Pt0.A3.Ex50.m1.2.2.cmml" xref="Pt0.A3.Ex50.m1.2.2"><times id="Pt0.A3.Ex50.m1.2.2.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.3"></times><apply id="Pt0.A3.Ex50.m1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex50.m1.1.1.1.1"><minus id="Pt0.A3.Ex50.m1.1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.1"></minus><ci id="Pt0.A3.Ex50.m1.1.1.1.1.1.2.cmml" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.2">𝑗</ci><ci id="Pt0.A3.Ex50.m1.1.1.1.1.1.3.cmml" xref="Pt0.A3.Ex50.m1.1.1.1.1.1.3">𝑖</ci></apply><apply id="Pt0.A3.Ex50.m1.2.2.2.1.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1"><minus id="Pt0.A3.Ex50.m1.2.2.2.1.1.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.1"></minus><apply id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2"><times id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.1"></times><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.2.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.2">𝑖</ci><apply id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3">subscript</csymbol><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.2.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.2">𝑆</ci><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.3.3">𝑗</ci></apply><apply id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4"><csymbol cd="ambiguous" id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4">subscript</csymbol><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.2.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.2">𝐸</ci><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.2.4.3">𝑗</ci></apply></apply><apply id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3"><times id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.1"></times><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.2.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.2">𝑗</ci><apply id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3">subscript</csymbol><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.2.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.2">𝑆</ci><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.3.3">𝑖</ci></apply><apply id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4"><csymbol cd="ambiguous" id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.1.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4">subscript</csymbol><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.2.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.2">𝐸</ci><ci id="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.3.cmml" xref="Pt0.A3.Ex50.m1.2.2.2.1.1.3.4.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex50.m1.2c">\displaystyle(j-i)(iS_{j}E_{j}-jS_{i}E_{i})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex50.m1.2d">( italic_j - italic_i ) ( italic_i italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_j italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq(jE_{i}-iE_{j})(iS_{j}-jS_{i})\iff" 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id="Pt0.A3.Ex50.m2.2c">\displaystyle\geq(jE_{i}-iE_{j})(iS_{j}-jS_{i})\iff</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex50.m2.2d">≥ ( italic_j italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_i italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ( italic_i italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_j italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ⇔</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex51"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle ij(S_{j}E_{j}+S_{i}E_{i})-i^{2}S_{j}E_{j}-j^{2}S_{i}E_{i}" class="ltx_Math" display="inline" id="Pt0.A3.Ex51.m1.1"><semantics id="Pt0.A3.Ex51.m1.1a"><mrow id="Pt0.A3.Ex51.m1.1.1" xref="Pt0.A3.Ex51.m1.1.1.cmml"><mrow id="Pt0.A3.Ex51.m1.1.1.1" 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xref="Pt0.A3.Ex51.m1.1.1.4.3.3">𝑖</ci></apply><apply id="Pt0.A3.Ex51.m1.1.1.4.4.cmml" xref="Pt0.A3.Ex51.m1.1.1.4.4"><csymbol cd="ambiguous" id="Pt0.A3.Ex51.m1.1.1.4.4.1.cmml" xref="Pt0.A3.Ex51.m1.1.1.4.4">subscript</csymbol><ci id="Pt0.A3.Ex51.m1.1.1.4.4.2.cmml" xref="Pt0.A3.Ex51.m1.1.1.4.4.2">𝐸</ci><ci id="Pt0.A3.Ex51.m1.1.1.4.4.3.cmml" xref="Pt0.A3.Ex51.m1.1.1.4.4.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex51.m1.1c">\displaystyle ij(S_{j}E_{j}+S_{i}E_{i})-i^{2}S_{j}E_{j}-j^{2}S_{i}E_{i}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex51.m1.1d">italic_i italic_j ( italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_i start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_j start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq ij(S_{j}E_{i}+S_{i}E_{j})-i^{2}S_{j}E_{j}-j^{2}S_{i}E_{i}\iff" class="ltx_Math" display="inline" id="Pt0.A3.Ex51.m2.1"><semantics id="Pt0.A3.Ex51.m2.1a"><mrow id="Pt0.A3.Ex51.m2.1.1" xref="Pt0.A3.Ex51.m2.1.1.cmml"><mrow id="Pt0.A3.Ex51.m2.1.1.1" xref="Pt0.A3.Ex51.m2.1.1.1.cmml"><mi id="Pt0.A3.Ex51.m2.1.1.1.3" xref="Pt0.A3.Ex51.m2.1.1.1.3.cmml"></mi><mo id="Pt0.A3.Ex51.m2.1.1.1.2" xref="Pt0.A3.Ex51.m2.1.1.1.2.cmml">≥</mo><mrow id="Pt0.A3.Ex51.m2.1.1.1.1" xref="Pt0.A3.Ex51.m2.1.1.1.1.cmml"><mrow id="Pt0.A3.Ex51.m2.1.1.1.1.1" xref="Pt0.A3.Ex51.m2.1.1.1.1.1.cmml"><mi id="Pt0.A3.Ex51.m2.1.1.1.1.1.3" xref="Pt0.A3.Ex51.m2.1.1.1.1.1.3.cmml">i</mi><mo 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ij(S_{j}E_{i}+S_{i}E_{j})-i^{2}S_{j}E_{j}-j^{2}S_{i}E_{i}\iff</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex51.m2.1d">≥ italic_i italic_j ( italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) - italic_i start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_j start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⇔</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle S_{j}E_{j}+S_{i}E_{i}" class="ltx_Math" display="inline" id="Pt0.A3.Ex52.m1.1"><semantics id="Pt0.A3.Ex52.m1.1a"><mrow id="Pt0.A3.Ex52.m1.1.1" xref="Pt0.A3.Ex52.m1.1.1.cmml"><mrow id="Pt0.A3.Ex52.m1.1.1.2" xref="Pt0.A3.Ex52.m1.1.1.2.cmml"><msub id="Pt0.A3.Ex52.m1.1.1.2.2" xref="Pt0.A3.Ex52.m1.1.1.2.2.cmml"><mi id="Pt0.A3.Ex52.m1.1.1.2.2.2" xref="Pt0.A3.Ex52.m1.1.1.2.2.2.cmml">S</mi><mi id="Pt0.A3.Ex52.m1.1.1.2.2.3" xref="Pt0.A3.Ex52.m1.1.1.2.2.3.cmml">j</mi></msub><mo id="Pt0.A3.Ex52.m1.1.1.2.1" xref="Pt0.A3.Ex52.m1.1.1.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex52.m1.1.1.2.3" xref="Pt0.A3.Ex52.m1.1.1.2.3.cmml"><mi id="Pt0.A3.Ex52.m1.1.1.2.3.2" xref="Pt0.A3.Ex52.m1.1.1.2.3.2.cmml">E</mi><mi id="Pt0.A3.Ex52.m1.1.1.2.3.3" xref="Pt0.A3.Ex52.m1.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="Pt0.A3.Ex52.m1.1.1.1" xref="Pt0.A3.Ex52.m1.1.1.1.cmml">+</mo><mrow id="Pt0.A3.Ex52.m1.1.1.3" xref="Pt0.A3.Ex52.m1.1.1.3.cmml"><msub 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id="Pt0.A3.Ex52.m1.1.1.2.2.1.cmml" xref="Pt0.A3.Ex52.m1.1.1.2.2">subscript</csymbol><ci id="Pt0.A3.Ex52.m1.1.1.2.2.2.cmml" xref="Pt0.A3.Ex52.m1.1.1.2.2.2">𝑆</ci><ci id="Pt0.A3.Ex52.m1.1.1.2.2.3.cmml" xref="Pt0.A3.Ex52.m1.1.1.2.2.3">𝑗</ci></apply><apply id="Pt0.A3.Ex52.m1.1.1.2.3.cmml" xref="Pt0.A3.Ex52.m1.1.1.2.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex52.m1.1.1.2.3.1.cmml" xref="Pt0.A3.Ex52.m1.1.1.2.3">subscript</csymbol><ci id="Pt0.A3.Ex52.m1.1.1.2.3.2.cmml" xref="Pt0.A3.Ex52.m1.1.1.2.3.2">𝐸</ci><ci id="Pt0.A3.Ex52.m1.1.1.2.3.3.cmml" xref="Pt0.A3.Ex52.m1.1.1.2.3.3">𝑗</ci></apply></apply><apply id="Pt0.A3.Ex52.m1.1.1.3.cmml" xref="Pt0.A3.Ex52.m1.1.1.3"><times id="Pt0.A3.Ex52.m1.1.1.3.1.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.1"></times><apply id="Pt0.A3.Ex52.m1.1.1.3.2.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex52.m1.1.1.3.2.1.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.2">subscript</csymbol><ci id="Pt0.A3.Ex52.m1.1.1.3.2.2.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.2.2">𝑆</ci><ci id="Pt0.A3.Ex52.m1.1.1.3.2.3.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.2.3">𝑖</ci></apply><apply id="Pt0.A3.Ex52.m1.1.1.3.3.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex52.m1.1.1.3.3.1.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.3">subscript</csymbol><ci id="Pt0.A3.Ex52.m1.1.1.3.3.2.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.3.2">𝐸</ci><ci id="Pt0.A3.Ex52.m1.1.1.3.3.3.cmml" xref="Pt0.A3.Ex52.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex52.m1.1c">\displaystyle S_{j}E_{j}+S_{i}E_{i}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex52.m1.1d">italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq 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xref="Pt0.A3.Ex52.m2.1.1.2.3.2.2">subscript</csymbol><ci id="Pt0.A3.Ex52.m2.1.1.2.3.2.2.2.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.2.2.2">𝑆</ci><ci id="Pt0.A3.Ex52.m2.1.1.2.3.2.2.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.2.2.3">𝑗</ci></apply><apply id="Pt0.A3.Ex52.m2.1.1.2.3.2.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.2.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex52.m2.1.1.2.3.2.3.1.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.2.3">subscript</csymbol><ci id="Pt0.A3.Ex52.m2.1.1.2.3.2.3.2.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.2.3.2">𝐸</ci><ci id="Pt0.A3.Ex52.m2.1.1.2.3.2.3.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.2.3.3">𝑖</ci></apply></apply><apply id="Pt0.A3.Ex52.m2.1.1.2.3.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3"><times id="Pt0.A3.Ex52.m2.1.1.2.3.3.1.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.1"></times><apply id="Pt0.A3.Ex52.m2.1.1.2.3.3.2.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex52.m2.1.1.2.3.3.2.1.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.2">subscript</csymbol><ci id="Pt0.A3.Ex52.m2.1.1.2.3.3.2.2.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.2.2">𝑆</ci><ci id="Pt0.A3.Ex52.m2.1.1.2.3.3.2.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.2.3">𝑖</ci></apply><apply id="Pt0.A3.Ex52.m2.1.1.2.3.3.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex52.m2.1.1.2.3.3.3.1.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.3">subscript</csymbol><ci id="Pt0.A3.Ex52.m2.1.1.2.3.3.3.2.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.3.2">𝐸</ci><ci id="Pt0.A3.Ex52.m2.1.1.2.3.3.3.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.2.3.3.3.3">𝑗</ci></apply></apply></apply></apply><csymbol cd="latexml" id="Pt0.A3.Ex52.m2.1.1.3.cmml" xref="Pt0.A3.Ex52.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex52.m2.1c">\displaystyle\geq S_{j}E_{i}+S_{i}E_{j}\iff</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex52.m2.1d">≥ italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⇔</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex53"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle S_{j}(E_{j}-E_{i})" class="ltx_Math" display="inline" id="Pt0.A3.Ex53.m1.1"><semantics id="Pt0.A3.Ex53.m1.1a"><mrow id="Pt0.A3.Ex53.m1.1.1" xref="Pt0.A3.Ex53.m1.1.1.cmml"><msub id="Pt0.A3.Ex53.m1.1.1.3" xref="Pt0.A3.Ex53.m1.1.1.3.cmml"><mi id="Pt0.A3.Ex53.m1.1.1.3.2" xref="Pt0.A3.Ex53.m1.1.1.3.2.cmml">S</mi><mi id="Pt0.A3.Ex53.m1.1.1.3.3" xref="Pt0.A3.Ex53.m1.1.1.3.3.cmml">j</mi></msub><mo id="Pt0.A3.Ex53.m1.1.1.2" xref="Pt0.A3.Ex53.m1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex53.m1.1.1.1.1" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex53.m1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex53.m1.1.1.1.1.1" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.cmml"><msub id="Pt0.A3.Ex53.m1.1.1.1.1.1.2" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.2.cmml"><mi id="Pt0.A3.Ex53.m1.1.1.1.1.1.2.2" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.2.2.cmml">E</mi><mi id="Pt0.A3.Ex53.m1.1.1.1.1.1.2.3" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.2.3.cmml">j</mi></msub><mo id="Pt0.A3.Ex53.m1.1.1.1.1.1.1" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.1.cmml">−</mo><msub id="Pt0.A3.Ex53.m1.1.1.1.1.1.3" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex53.m1.1.1.1.1.1.3.2" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.3.2.cmml">E</mi><mi id="Pt0.A3.Ex53.m1.1.1.1.1.1.3.3" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.3.3.cmml">i</mi></msub></mrow><mo id="Pt0.A3.Ex53.m1.1.1.1.1.3" stretchy="false" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex53.m1.1b"><apply id="Pt0.A3.Ex53.m1.1.1.cmml" xref="Pt0.A3.Ex53.m1.1.1"><times id="Pt0.A3.Ex53.m1.1.1.2.cmml" xref="Pt0.A3.Ex53.m1.1.1.2"></times><apply id="Pt0.A3.Ex53.m1.1.1.3.cmml" xref="Pt0.A3.Ex53.m1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex53.m1.1.1.3.1.cmml" xref="Pt0.A3.Ex53.m1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex53.m1.1.1.3.2.cmml" xref="Pt0.A3.Ex53.m1.1.1.3.2">𝑆</ci><ci id="Pt0.A3.Ex53.m1.1.1.3.3.cmml" xref="Pt0.A3.Ex53.m1.1.1.3.3">𝑗</ci></apply><apply id="Pt0.A3.Ex53.m1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1"><minus id="Pt0.A3.Ex53.m1.1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.1"></minus><apply id="Pt0.A3.Ex53.m1.1.1.1.1.1.2.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex53.m1.1.1.1.1.1.2.1.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.2">subscript</csymbol><ci id="Pt0.A3.Ex53.m1.1.1.1.1.1.2.2.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.2.2">𝐸</ci><ci id="Pt0.A3.Ex53.m1.1.1.1.1.1.2.3.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.2.3">𝑗</ci></apply><apply id="Pt0.A3.Ex53.m1.1.1.1.1.1.3.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex53.m1.1.1.1.1.1.3.1.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex53.m1.1.1.1.1.1.3.2.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.3.2">𝐸</ci><ci id="Pt0.A3.Ex53.m1.1.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex53.m1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex53.m1.1c">\displaystyle S_{j}(E_{j}-E_{i})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex53.m1.1d">italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq S_{i}(E_{j}-E_{i})" class="ltx_Math" display="inline" id="Pt0.A3.Ex53.m2.1"><semantics id="Pt0.A3.Ex53.m2.1a"><mrow id="Pt0.A3.Ex53.m2.1.1" xref="Pt0.A3.Ex53.m2.1.1.cmml"><mi id="Pt0.A3.Ex53.m2.1.1.3" xref="Pt0.A3.Ex53.m2.1.1.3.cmml"></mi><mo id="Pt0.A3.Ex53.m2.1.1.2" xref="Pt0.A3.Ex53.m2.1.1.2.cmml">≥</mo><mrow id="Pt0.A3.Ex53.m2.1.1.1" xref="Pt0.A3.Ex53.m2.1.1.1.cmml"><msub id="Pt0.A3.Ex53.m2.1.1.1.3" xref="Pt0.A3.Ex53.m2.1.1.1.3.cmml"><mi id="Pt0.A3.Ex53.m2.1.1.1.3.2" xref="Pt0.A3.Ex53.m2.1.1.1.3.2.cmml">S</mi><mi id="Pt0.A3.Ex53.m2.1.1.1.3.3" xref="Pt0.A3.Ex53.m2.1.1.1.3.3.cmml">i</mi></msub><mo id="Pt0.A3.Ex53.m2.1.1.1.2" xref="Pt0.A3.Ex53.m2.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex53.m2.1.1.1.1.1" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex53.m2.1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex53.m2.1.1.1.1.1.1" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.cmml"><msub id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.cmml"><mi id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.2" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.2.cmml">E</mi><mi id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.3" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.3.cmml">j</mi></msub><mo id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.1" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.1.cmml">−</mo><msub id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.2" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.2.cmml">E</mi><mi id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.3" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.3.cmml">i</mi></msub></mrow><mo id="Pt0.A3.Ex53.m2.1.1.1.1.1.3" stretchy="false" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex53.m2.1b"><apply id="Pt0.A3.Ex53.m2.1.1.cmml" xref="Pt0.A3.Ex53.m2.1.1"><geq id="Pt0.A3.Ex53.m2.1.1.2.cmml" xref="Pt0.A3.Ex53.m2.1.1.2"></geq><csymbol cd="latexml" id="Pt0.A3.Ex53.m2.1.1.3.cmml" xref="Pt0.A3.Ex53.m2.1.1.3">absent</csymbol><apply id="Pt0.A3.Ex53.m2.1.1.1.cmml" xref="Pt0.A3.Ex53.m2.1.1.1"><times id="Pt0.A3.Ex53.m2.1.1.1.2.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.2"></times><apply id="Pt0.A3.Ex53.m2.1.1.1.3.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex53.m2.1.1.1.3.1.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex53.m2.1.1.1.3.2.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.3.2">𝑆</ci><ci id="Pt0.A3.Ex53.m2.1.1.1.3.3.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.3.3">𝑖</ci></apply><apply id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1"><minus id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.1.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.1"></minus><apply id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.1.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2">subscript</csymbol><ci id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.2.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.2">𝐸</ci><ci id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.3.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.2.3">𝑗</ci></apply><apply id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.2">𝐸</ci><ci id="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex53.m2.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex53.m2.1c">\displaystyle\geq S_{i}(E_{j}-E_{i})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex53.m2.1d">≥ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx11.p2.14">and this holds because <math alttext="S_{j}>S_{i}" class="ltx_Math" display="inline" id="Thmproofx11.p2.4.m1.1"><semantics id="Thmproofx11.p2.4.m1.1a"><mrow id="Thmproofx11.p2.4.m1.1.1" xref="Thmproofx11.p2.4.m1.1.1.cmml"><msub id="Thmproofx11.p2.4.m1.1.1.2" xref="Thmproofx11.p2.4.m1.1.1.2.cmml"><mi id="Thmproofx11.p2.4.m1.1.1.2.2" xref="Thmproofx11.p2.4.m1.1.1.2.2.cmml">S</mi><mi id="Thmproofx11.p2.4.m1.1.1.2.3" xref="Thmproofx11.p2.4.m1.1.1.2.3.cmml">j</mi></msub><mo id="Thmproofx11.p2.4.m1.1.1.1" xref="Thmproofx11.p2.4.m1.1.1.1.cmml">></mo><msub id="Thmproofx11.p2.4.m1.1.1.3" xref="Thmproofx11.p2.4.m1.1.1.3.cmml"><mi id="Thmproofx11.p2.4.m1.1.1.3.2" xref="Thmproofx11.p2.4.m1.1.1.3.2.cmml">S</mi><mi id="Thmproofx11.p2.4.m1.1.1.3.3" xref="Thmproofx11.p2.4.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.4.m1.1b"><apply id="Thmproofx11.p2.4.m1.1.1.cmml" xref="Thmproofx11.p2.4.m1.1.1"><gt id="Thmproofx11.p2.4.m1.1.1.1.cmml" xref="Thmproofx11.p2.4.m1.1.1.1"></gt><apply id="Thmproofx11.p2.4.m1.1.1.2.cmml" xref="Thmproofx11.p2.4.m1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx11.p2.4.m1.1.1.2.1.cmml" xref="Thmproofx11.p2.4.m1.1.1.2">subscript</csymbol><ci id="Thmproofx11.p2.4.m1.1.1.2.2.cmml" xref="Thmproofx11.p2.4.m1.1.1.2.2">𝑆</ci><ci id="Thmproofx11.p2.4.m1.1.1.2.3.cmml" xref="Thmproofx11.p2.4.m1.1.1.2.3">𝑗</ci></apply><apply id="Thmproofx11.p2.4.m1.1.1.3.cmml" xref="Thmproofx11.p2.4.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p2.4.m1.1.1.3.1.cmml" xref="Thmproofx11.p2.4.m1.1.1.3">subscript</csymbol><ci id="Thmproofx11.p2.4.m1.1.1.3.2.cmml" xref="Thmproofx11.p2.4.m1.1.1.3.2">𝑆</ci><ci id="Thmproofx11.p2.4.m1.1.1.3.3.cmml" xref="Thmproofx11.p2.4.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.4.m1.1c">S_{j}>S_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.4.m1.1d">italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT > italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="E_{j}>E_{i}" class="ltx_Math" display="inline" id="Thmproofx11.p2.5.m2.1"><semantics id="Thmproofx11.p2.5.m2.1a"><mrow id="Thmproofx11.p2.5.m2.1.1" xref="Thmproofx11.p2.5.m2.1.1.cmml"><msub id="Thmproofx11.p2.5.m2.1.1.2" xref="Thmproofx11.p2.5.m2.1.1.2.cmml"><mi id="Thmproofx11.p2.5.m2.1.1.2.2" xref="Thmproofx11.p2.5.m2.1.1.2.2.cmml">E</mi><mi id="Thmproofx11.p2.5.m2.1.1.2.3" xref="Thmproofx11.p2.5.m2.1.1.2.3.cmml">j</mi></msub><mo id="Thmproofx11.p2.5.m2.1.1.1" xref="Thmproofx11.p2.5.m2.1.1.1.cmml">></mo><msub id="Thmproofx11.p2.5.m2.1.1.3" xref="Thmproofx11.p2.5.m2.1.1.3.cmml"><mi id="Thmproofx11.p2.5.m2.1.1.3.2" xref="Thmproofx11.p2.5.m2.1.1.3.2.cmml">E</mi><mi id="Thmproofx11.p2.5.m2.1.1.3.3" xref="Thmproofx11.p2.5.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.5.m2.1b"><apply id="Thmproofx11.p2.5.m2.1.1.cmml" xref="Thmproofx11.p2.5.m2.1.1"><gt id="Thmproofx11.p2.5.m2.1.1.1.cmml" xref="Thmproofx11.p2.5.m2.1.1.1"></gt><apply id="Thmproofx11.p2.5.m2.1.1.2.cmml" xref="Thmproofx11.p2.5.m2.1.1.2"><csymbol cd="ambiguous" id="Thmproofx11.p2.5.m2.1.1.2.1.cmml" xref="Thmproofx11.p2.5.m2.1.1.2">subscript</csymbol><ci id="Thmproofx11.p2.5.m2.1.1.2.2.cmml" xref="Thmproofx11.p2.5.m2.1.1.2.2">𝐸</ci><ci id="Thmproofx11.p2.5.m2.1.1.2.3.cmml" xref="Thmproofx11.p2.5.m2.1.1.2.3">𝑗</ci></apply><apply id="Thmproofx11.p2.5.m2.1.1.3.cmml" xref="Thmproofx11.p2.5.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p2.5.m2.1.1.3.1.cmml" xref="Thmproofx11.p2.5.m2.1.1.3">subscript</csymbol><ci id="Thmproofx11.p2.5.m2.1.1.3.2.cmml" xref="Thmproofx11.p2.5.m2.1.1.3.2">𝐸</ci><ci id="Thmproofx11.p2.5.m2.1.1.3.3.cmml" xref="Thmproofx11.p2.5.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.5.m2.1c">E_{j}>E_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.5.m2.1d">italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT > italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Hence, the coefficient of <math alttext="\alpha_{0}" class="ltx_Math" display="inline" id="Thmproofx11.p2.6.m3.1"><semantics id="Thmproofx11.p2.6.m3.1a"><msub id="Thmproofx11.p2.6.m3.1.1" xref="Thmproofx11.p2.6.m3.1.1.cmml"><mi id="Thmproofx11.p2.6.m3.1.1.2" xref="Thmproofx11.p2.6.m3.1.1.2.cmml">α</mi><mn id="Thmproofx11.p2.6.m3.1.1.3" xref="Thmproofx11.p2.6.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.6.m3.1b"><apply id="Thmproofx11.p2.6.m3.1.1.cmml" xref="Thmproofx11.p2.6.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p2.6.m3.1.1.1.cmml" xref="Thmproofx11.p2.6.m3.1.1">subscript</csymbol><ci id="Thmproofx11.p2.6.m3.1.1.2.cmml" xref="Thmproofx11.p2.6.m3.1.1.2">𝛼</ci><cn id="Thmproofx11.p2.6.m3.1.1.3.cmml" type="integer" xref="Thmproofx11.p2.6.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.6.m3.1c">\alpha_{0}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.6.m3.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\varphi(\alpha_{0},1)" class="ltx_Math" display="inline" id="Thmproofx11.p2.7.m4.2"><semantics id="Thmproofx11.p2.7.m4.2a"><mrow id="Thmproofx11.p2.7.m4.2.2" xref="Thmproofx11.p2.7.m4.2.2.cmml"><mi id="Thmproofx11.p2.7.m4.2.2.3" xref="Thmproofx11.p2.7.m4.2.2.3.cmml">φ</mi><mo id="Thmproofx11.p2.7.m4.2.2.2" xref="Thmproofx11.p2.7.m4.2.2.2.cmml">⁢</mo><mrow id="Thmproofx11.p2.7.m4.2.2.1.1" xref="Thmproofx11.p2.7.m4.2.2.1.2.cmml"><mo id="Thmproofx11.p2.7.m4.2.2.1.1.2" stretchy="false" xref="Thmproofx11.p2.7.m4.2.2.1.2.cmml">(</mo><msub id="Thmproofx11.p2.7.m4.2.2.1.1.1" xref="Thmproofx11.p2.7.m4.2.2.1.1.1.cmml"><mi id="Thmproofx11.p2.7.m4.2.2.1.1.1.2" xref="Thmproofx11.p2.7.m4.2.2.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p2.7.m4.2.2.1.1.1.3" xref="Thmproofx11.p2.7.m4.2.2.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p2.7.m4.2.2.1.1.3" xref="Thmproofx11.p2.7.m4.2.2.1.2.cmml">,</mo><mn id="Thmproofx11.p2.7.m4.1.1" xref="Thmproofx11.p2.7.m4.1.1.cmml">1</mn><mo id="Thmproofx11.p2.7.m4.2.2.1.1.4" stretchy="false" xref="Thmproofx11.p2.7.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.7.m4.2b"><apply id="Thmproofx11.p2.7.m4.2.2.cmml" xref="Thmproofx11.p2.7.m4.2.2"><times id="Thmproofx11.p2.7.m4.2.2.2.cmml" xref="Thmproofx11.p2.7.m4.2.2.2"></times><ci id="Thmproofx11.p2.7.m4.2.2.3.cmml" xref="Thmproofx11.p2.7.m4.2.2.3">𝜑</ci><interval closure="open" id="Thmproofx11.p2.7.m4.2.2.1.2.cmml" xref="Thmproofx11.p2.7.m4.2.2.1.1"><apply id="Thmproofx11.p2.7.m4.2.2.1.1.1.cmml" xref="Thmproofx11.p2.7.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p2.7.m4.2.2.1.1.1.1.cmml" xref="Thmproofx11.p2.7.m4.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx11.p2.7.m4.2.2.1.1.1.2.cmml" xref="Thmproofx11.p2.7.m4.2.2.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p2.7.m4.2.2.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p2.7.m4.2.2.1.1.1.3">0</cn></apply><cn id="Thmproofx11.p2.7.m4.1.1.cmml" type="integer" xref="Thmproofx11.p2.7.m4.1.1">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.7.m4.2c">\varphi(\alpha_{0},1)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.7.m4.2d">italic_φ ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 )</annotation></semantics></math> is <math alttext="\frac{k}{i}" class="ltx_Math" display="inline" id="Thmproofx11.p2.8.m5.1"><semantics id="Thmproofx11.p2.8.m5.1a"><mfrac id="Thmproofx11.p2.8.m5.1.1" xref="Thmproofx11.p2.8.m5.1.1.cmml"><mi id="Thmproofx11.p2.8.m5.1.1.2" xref="Thmproofx11.p2.8.m5.1.1.2.cmml">k</mi><mi id="Thmproofx11.p2.8.m5.1.1.3" xref="Thmproofx11.p2.8.m5.1.1.3.cmml">i</mi></mfrac><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.8.m5.1b"><apply id="Thmproofx11.p2.8.m5.1.1.cmml" xref="Thmproofx11.p2.8.m5.1.1"><divide id="Thmproofx11.p2.8.m5.1.1.1.cmml" xref="Thmproofx11.p2.8.m5.1.1"></divide><ci id="Thmproofx11.p2.8.m5.1.1.2.cmml" xref="Thmproofx11.p2.8.m5.1.1.2">𝑘</ci><ci id="Thmproofx11.p2.8.m5.1.1.3.cmml" xref="Thmproofx11.p2.8.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.8.m5.1c">\frac{k}{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.8.m5.1d">divide start_ARG italic_k end_ARG start_ARG italic_i end_ARG</annotation></semantics></math>, and in <math alttext="\varphi(\alpha_{0},n)" class="ltx_Math" display="inline" id="Thmproofx11.p2.9.m6.2"><semantics id="Thmproofx11.p2.9.m6.2a"><mrow id="Thmproofx11.p2.9.m6.2.2" xref="Thmproofx11.p2.9.m6.2.2.cmml"><mi id="Thmproofx11.p2.9.m6.2.2.3" xref="Thmproofx11.p2.9.m6.2.2.3.cmml">φ</mi><mo id="Thmproofx11.p2.9.m6.2.2.2" xref="Thmproofx11.p2.9.m6.2.2.2.cmml">⁢</mo><mrow id="Thmproofx11.p2.9.m6.2.2.1.1" xref="Thmproofx11.p2.9.m6.2.2.1.2.cmml"><mo id="Thmproofx11.p2.9.m6.2.2.1.1.2" stretchy="false" xref="Thmproofx11.p2.9.m6.2.2.1.2.cmml">(</mo><msub id="Thmproofx11.p2.9.m6.2.2.1.1.1" xref="Thmproofx11.p2.9.m6.2.2.1.1.1.cmml"><mi id="Thmproofx11.p2.9.m6.2.2.1.1.1.2" xref="Thmproofx11.p2.9.m6.2.2.1.1.1.2.cmml">α</mi><mn id="Thmproofx11.p2.9.m6.2.2.1.1.1.3" xref="Thmproofx11.p2.9.m6.2.2.1.1.1.3.cmml">0</mn></msub><mo id="Thmproofx11.p2.9.m6.2.2.1.1.3" xref="Thmproofx11.p2.9.m6.2.2.1.2.cmml">,</mo><mi id="Thmproofx11.p2.9.m6.1.1" xref="Thmproofx11.p2.9.m6.1.1.cmml">n</mi><mo id="Thmproofx11.p2.9.m6.2.2.1.1.4" stretchy="false" xref="Thmproofx11.p2.9.m6.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.9.m6.2b"><apply id="Thmproofx11.p2.9.m6.2.2.cmml" xref="Thmproofx11.p2.9.m6.2.2"><times id="Thmproofx11.p2.9.m6.2.2.2.cmml" xref="Thmproofx11.p2.9.m6.2.2.2"></times><ci id="Thmproofx11.p2.9.m6.2.2.3.cmml" xref="Thmproofx11.p2.9.m6.2.2.3">𝜑</ci><interval closure="open" id="Thmproofx11.p2.9.m6.2.2.1.2.cmml" xref="Thmproofx11.p2.9.m6.2.2.1.1"><apply id="Thmproofx11.p2.9.m6.2.2.1.1.1.cmml" xref="Thmproofx11.p2.9.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p2.9.m6.2.2.1.1.1.1.cmml" xref="Thmproofx11.p2.9.m6.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx11.p2.9.m6.2.2.1.1.1.2.cmml" xref="Thmproofx11.p2.9.m6.2.2.1.1.1.2">𝛼</ci><cn id="Thmproofx11.p2.9.m6.2.2.1.1.1.3.cmml" type="integer" xref="Thmproofx11.p2.9.m6.2.2.1.1.1.3">0</cn></apply><ci id="Thmproofx11.p2.9.m6.1.1.cmml" xref="Thmproofx11.p2.9.m6.1.1">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.9.m6.2c">\varphi(\alpha_{0},n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.9.m6.2d">italic_φ ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n )</annotation></semantics></math> it is at least <math alttext="\frac{kE_{i}}{i}" class="ltx_Math" display="inline" id="Thmproofx11.p2.10.m7.1"><semantics id="Thmproofx11.p2.10.m7.1a"><mfrac id="Thmproofx11.p2.10.m7.1.1" xref="Thmproofx11.p2.10.m7.1.1.cmml"><mrow id="Thmproofx11.p2.10.m7.1.1.2" xref="Thmproofx11.p2.10.m7.1.1.2.cmml"><mi id="Thmproofx11.p2.10.m7.1.1.2.2" xref="Thmproofx11.p2.10.m7.1.1.2.2.cmml">k</mi><mo id="Thmproofx11.p2.10.m7.1.1.2.1" xref="Thmproofx11.p2.10.m7.1.1.2.1.cmml">⁢</mo><msub id="Thmproofx11.p2.10.m7.1.1.2.3" xref="Thmproofx11.p2.10.m7.1.1.2.3.cmml"><mi id="Thmproofx11.p2.10.m7.1.1.2.3.2" xref="Thmproofx11.p2.10.m7.1.1.2.3.2.cmml">E</mi><mi id="Thmproofx11.p2.10.m7.1.1.2.3.3" xref="Thmproofx11.p2.10.m7.1.1.2.3.3.cmml">i</mi></msub></mrow><mi id="Thmproofx11.p2.10.m7.1.1.3" xref="Thmproofx11.p2.10.m7.1.1.3.cmml">i</mi></mfrac><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.10.m7.1b"><apply id="Thmproofx11.p2.10.m7.1.1.cmml" xref="Thmproofx11.p2.10.m7.1.1"><divide id="Thmproofx11.p2.10.m7.1.1.1.cmml" xref="Thmproofx11.p2.10.m7.1.1"></divide><apply id="Thmproofx11.p2.10.m7.1.1.2.cmml" xref="Thmproofx11.p2.10.m7.1.1.2"><times id="Thmproofx11.p2.10.m7.1.1.2.1.cmml" xref="Thmproofx11.p2.10.m7.1.1.2.1"></times><ci id="Thmproofx11.p2.10.m7.1.1.2.2.cmml" xref="Thmproofx11.p2.10.m7.1.1.2.2">𝑘</ci><apply id="Thmproofx11.p2.10.m7.1.1.2.3.cmml" xref="Thmproofx11.p2.10.m7.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx11.p2.10.m7.1.1.2.3.1.cmml" xref="Thmproofx11.p2.10.m7.1.1.2.3">subscript</csymbol><ci id="Thmproofx11.p2.10.m7.1.1.2.3.2.cmml" xref="Thmproofx11.p2.10.m7.1.1.2.3.2">𝐸</ci><ci id="Thmproofx11.p2.10.m7.1.1.2.3.3.cmml" xref="Thmproofx11.p2.10.m7.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Thmproofx11.p2.10.m7.1.1.3.cmml" xref="Thmproofx11.p2.10.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.10.m7.1c">\frac{kE_{i}}{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.10.m7.1d">divide start_ARG italic_k italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG</annotation></semantics></math>. Since <math alttext="E_{i}>1" class="ltx_Math" display="inline" id="Thmproofx11.p2.11.m8.1"><semantics id="Thmproofx11.p2.11.m8.1a"><mrow id="Thmproofx11.p2.11.m8.1.1" xref="Thmproofx11.p2.11.m8.1.1.cmml"><msub id="Thmproofx11.p2.11.m8.1.1.2" xref="Thmproofx11.p2.11.m8.1.1.2.cmml"><mi id="Thmproofx11.p2.11.m8.1.1.2.2" xref="Thmproofx11.p2.11.m8.1.1.2.2.cmml">E</mi><mi id="Thmproofx11.p2.11.m8.1.1.2.3" xref="Thmproofx11.p2.11.m8.1.1.2.3.cmml">i</mi></msub><mo id="Thmproofx11.p2.11.m8.1.1.1" xref="Thmproofx11.p2.11.m8.1.1.1.cmml">></mo><mn id="Thmproofx11.p2.11.m8.1.1.3" xref="Thmproofx11.p2.11.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.11.m8.1b"><apply id="Thmproofx11.p2.11.m8.1.1.cmml" xref="Thmproofx11.p2.11.m8.1.1"><gt id="Thmproofx11.p2.11.m8.1.1.1.cmml" xref="Thmproofx11.p2.11.m8.1.1.1"></gt><apply id="Thmproofx11.p2.11.m8.1.1.2.cmml" xref="Thmproofx11.p2.11.m8.1.1.2"><csymbol cd="ambiguous" id="Thmproofx11.p2.11.m8.1.1.2.1.cmml" xref="Thmproofx11.p2.11.m8.1.1.2">subscript</csymbol><ci id="Thmproofx11.p2.11.m8.1.1.2.2.cmml" xref="Thmproofx11.p2.11.m8.1.1.2.2">𝐸</ci><ci id="Thmproofx11.p2.11.m8.1.1.2.3.cmml" xref="Thmproofx11.p2.11.m8.1.1.2.3">𝑖</ci></apply><cn id="Thmproofx11.p2.11.m8.1.1.3.cmml" type="integer" xref="Thmproofx11.p2.11.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.11.m8.1c">E_{i}>1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.11.m8.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 1</annotation></semantics></math>, in total <math alttext="\frac{d}{d\alpha}\varphi(\alpha,n)\mid_{\alpha=\alpha_{0}}>\frac{d}{d\alpha}% \varphi(\alpha,1)\mid_{\alpha=\alpha_{0}}" class="ltx_Math" display="inline" id="Thmproofx11.p2.12.m9.8"><semantics id="Thmproofx11.p2.12.m9.8a"><mrow id="Thmproofx11.p2.12.m9.8.8" xref="Thmproofx11.p2.12.m9.8.8.cmml"><msub id="Thmproofx11.p2.12.m9.7.7.1.1" xref="Thmproofx11.p2.12.m9.7.7.1.2.cmml"><mrow id="Thmproofx11.p2.12.m9.7.7.1.1.1" xref="Thmproofx11.p2.12.m9.7.7.1.2.cmml"><mrow id="Thmproofx11.p2.12.m9.7.7.1.1.1.1" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.cmml"><mfrac id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.cmml"><mi id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.2" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.2.cmml">d</mi><mrow id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.cmml"><mi id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.2" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.2.cmml">d</mi><mo id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.1" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.3" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.3.cmml">α</mi></mrow></mfrac><mo id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.1" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.1.cmml">⁢</mo><mi id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.3" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.3.cmml">φ</mi><mo id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.1a" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.2" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.1.cmml"><mo id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.2.1" stretchy="false" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.1.cmml">(</mo><mi id="Thmproofx11.p2.12.m9.1.1" xref="Thmproofx11.p2.12.m9.1.1.cmml">α</mi><mo id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.2.2" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.1.cmml">,</mo><mi id="Thmproofx11.p2.12.m9.2.2" xref="Thmproofx11.p2.12.m9.2.2.cmml">n</mi><mo id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.2.3" stretchy="false" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.1.cmml">)</mo></mrow></mrow><mo fence="true" id="Thmproofx11.p2.12.m9.7.7.1.1.1.2" lspace="0em" xref="Thmproofx11.p2.12.m9.7.7.1.2.1.cmml">∣</mo></mrow><mrow id="Thmproofx11.p2.12.m9.3.3.1" xref="Thmproofx11.p2.12.m9.3.3.1.cmml"><mi id="Thmproofx11.p2.12.m9.3.3.1.2" xref="Thmproofx11.p2.12.m9.3.3.1.2.cmml">α</mi><mo id="Thmproofx11.p2.12.m9.3.3.1.1" xref="Thmproofx11.p2.12.m9.3.3.1.1.cmml">=</mo><msub id="Thmproofx11.p2.12.m9.3.3.1.3" xref="Thmproofx11.p2.12.m9.3.3.1.3.cmml"><mi id="Thmproofx11.p2.12.m9.3.3.1.3.2" xref="Thmproofx11.p2.12.m9.3.3.1.3.2.cmml">α</mi><mn id="Thmproofx11.p2.12.m9.3.3.1.3.3" xref="Thmproofx11.p2.12.m9.3.3.1.3.3.cmml">0</mn></msub></mrow></msub><mo id="Thmproofx11.p2.12.m9.8.8.3" xref="Thmproofx11.p2.12.m9.8.8.3.cmml">></mo><msub id="Thmproofx11.p2.12.m9.8.8.2.1" xref="Thmproofx11.p2.12.m9.8.8.2.2.cmml"><mrow id="Thmproofx11.p2.12.m9.8.8.2.1.1" xref="Thmproofx11.p2.12.m9.8.8.2.2.cmml"><mrow id="Thmproofx11.p2.12.m9.8.8.2.1.1.1" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.cmml"><mfrac id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.cmml"><mi id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.2" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.2.cmml">d</mi><mrow id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.cmml"><mi id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.2" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.2.cmml">d</mi><mo id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.1" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.1.cmml">⁢</mo><mi id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.3" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.3.cmml">α</mi></mrow></mfrac><mo id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.1" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.1.cmml">⁢</mo><mi id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.3" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.3.cmml">φ</mi><mo id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.1a" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.1.cmml">⁢</mo><mrow id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.2" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.1.cmml"><mo id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.2.1" stretchy="false" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.1.cmml">(</mo><mi id="Thmproofx11.p2.12.m9.4.4" xref="Thmproofx11.p2.12.m9.4.4.cmml">α</mi><mo id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.2.2" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.1.cmml">,</mo><mn id="Thmproofx11.p2.12.m9.5.5" xref="Thmproofx11.p2.12.m9.5.5.cmml">1</mn><mo id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.2.3" stretchy="false" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.1.cmml">)</mo></mrow></mrow><mo fence="true" id="Thmproofx11.p2.12.m9.8.8.2.1.1.2" lspace="0em" xref="Thmproofx11.p2.12.m9.8.8.2.2.1.cmml">∣</mo></mrow><mrow id="Thmproofx11.p2.12.m9.6.6.1" xref="Thmproofx11.p2.12.m9.6.6.1.cmml"><mi id="Thmproofx11.p2.12.m9.6.6.1.2" xref="Thmproofx11.p2.12.m9.6.6.1.2.cmml">α</mi><mo id="Thmproofx11.p2.12.m9.6.6.1.1" xref="Thmproofx11.p2.12.m9.6.6.1.1.cmml">=</mo><msub id="Thmproofx11.p2.12.m9.6.6.1.3" xref="Thmproofx11.p2.12.m9.6.6.1.3.cmml"><mi id="Thmproofx11.p2.12.m9.6.6.1.3.2" xref="Thmproofx11.p2.12.m9.6.6.1.3.2.cmml">α</mi><mn id="Thmproofx11.p2.12.m9.6.6.1.3.3" xref="Thmproofx11.p2.12.m9.6.6.1.3.3.cmml">0</mn></msub></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.12.m9.8b"><apply id="Thmproofx11.p2.12.m9.8.8.cmml" xref="Thmproofx11.p2.12.m9.8.8"><gt id="Thmproofx11.p2.12.m9.8.8.3.cmml" xref="Thmproofx11.p2.12.m9.8.8.3"></gt><apply id="Thmproofx11.p2.12.m9.7.7.1.2.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1"><csymbol cd="latexml" id="Thmproofx11.p2.12.m9.7.7.1.2.1.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.2">evaluated-at</csymbol><apply id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1"><times id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.1.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.1"></times><apply id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2"><divide id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.1.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2"></divide><ci id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.2.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.2">𝑑</ci><apply id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3"><times id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.1.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.1"></times><ci id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.2.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.2">𝑑</ci><ci id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.3.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.2.3.3">𝛼</ci></apply></apply><ci id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.3.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.3">𝜑</ci><interval closure="open" id="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.1.cmml" xref="Thmproofx11.p2.12.m9.7.7.1.1.1.1.4.2"><ci id="Thmproofx11.p2.12.m9.1.1.cmml" xref="Thmproofx11.p2.12.m9.1.1">𝛼</ci><ci id="Thmproofx11.p2.12.m9.2.2.cmml" xref="Thmproofx11.p2.12.m9.2.2">𝑛</ci></interval></apply><apply id="Thmproofx11.p2.12.m9.3.3.1.cmml" xref="Thmproofx11.p2.12.m9.3.3.1"><eq id="Thmproofx11.p2.12.m9.3.3.1.1.cmml" xref="Thmproofx11.p2.12.m9.3.3.1.1"></eq><ci id="Thmproofx11.p2.12.m9.3.3.1.2.cmml" xref="Thmproofx11.p2.12.m9.3.3.1.2">𝛼</ci><apply id="Thmproofx11.p2.12.m9.3.3.1.3.cmml" xref="Thmproofx11.p2.12.m9.3.3.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p2.12.m9.3.3.1.3.1.cmml" xref="Thmproofx11.p2.12.m9.3.3.1.3">subscript</csymbol><ci id="Thmproofx11.p2.12.m9.3.3.1.3.2.cmml" xref="Thmproofx11.p2.12.m9.3.3.1.3.2">𝛼</ci><cn id="Thmproofx11.p2.12.m9.3.3.1.3.3.cmml" type="integer" xref="Thmproofx11.p2.12.m9.3.3.1.3.3">0</cn></apply></apply></apply><apply id="Thmproofx11.p2.12.m9.8.8.2.2.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1"><csymbol cd="latexml" id="Thmproofx11.p2.12.m9.8.8.2.2.1.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.2">evaluated-at</csymbol><apply id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1"><times id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.1.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.1"></times><apply id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2"><divide id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.1.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2"></divide><ci id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.2.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.2">𝑑</ci><apply id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3"><times id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.1.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.1"></times><ci id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.2.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.2">𝑑</ci><ci id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.3.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.2.3.3">𝛼</ci></apply></apply><ci id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.3.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.3">𝜑</ci><interval closure="open" id="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.1.cmml" xref="Thmproofx11.p2.12.m9.8.8.2.1.1.1.4.2"><ci id="Thmproofx11.p2.12.m9.4.4.cmml" xref="Thmproofx11.p2.12.m9.4.4">𝛼</ci><cn id="Thmproofx11.p2.12.m9.5.5.cmml" type="integer" xref="Thmproofx11.p2.12.m9.5.5">1</cn></interval></apply><apply id="Thmproofx11.p2.12.m9.6.6.1.cmml" xref="Thmproofx11.p2.12.m9.6.6.1"><eq id="Thmproofx11.p2.12.m9.6.6.1.1.cmml" xref="Thmproofx11.p2.12.m9.6.6.1.1"></eq><ci id="Thmproofx11.p2.12.m9.6.6.1.2.cmml" xref="Thmproofx11.p2.12.m9.6.6.1.2">𝛼</ci><apply id="Thmproofx11.p2.12.m9.6.6.1.3.cmml" xref="Thmproofx11.p2.12.m9.6.6.1.3"><csymbol cd="ambiguous" id="Thmproofx11.p2.12.m9.6.6.1.3.1.cmml" xref="Thmproofx11.p2.12.m9.6.6.1.3">subscript</csymbol><ci id="Thmproofx11.p2.12.m9.6.6.1.3.2.cmml" xref="Thmproofx11.p2.12.m9.6.6.1.3.2">𝛼</ci><cn id="Thmproofx11.p2.12.m9.6.6.1.3.3.cmml" type="integer" xref="Thmproofx11.p2.12.m9.6.6.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.12.m9.8c">\frac{d}{d\alpha}\varphi(\alpha,n)\mid_{\alpha=\alpha_{0}}>\frac{d}{d\alpha}% \varphi(\alpha,1)\mid_{\alpha=\alpha_{0}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.12.m9.8d">divide start_ARG italic_d end_ARG start_ARG italic_d italic_α end_ARG italic_φ ( italic_α , italic_n ) ∣ start_POSTSUBSCRIPT italic_α = italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT > divide start_ARG italic_d end_ARG start_ARG italic_d italic_α end_ARG italic_φ ( italic_α , 1 ) ∣ start_POSTSUBSCRIPT italic_α = italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, so by continuity and smoothness a.s., the ratio <math alttext="\frac{\varphi(\alpha,n)}{\varphi(\alpha,1)}" class="ltx_Math" display="inline" id="Thmproofx11.p2.13.m10.4"><semantics id="Thmproofx11.p2.13.m10.4a"><mfrac id="Thmproofx11.p2.13.m10.4.4" xref="Thmproofx11.p2.13.m10.4.4.cmml"><mrow id="Thmproofx11.p2.13.m10.2.2.2" xref="Thmproofx11.p2.13.m10.2.2.2.cmml"><mi id="Thmproofx11.p2.13.m10.2.2.2.4" xref="Thmproofx11.p2.13.m10.2.2.2.4.cmml">φ</mi><mo id="Thmproofx11.p2.13.m10.2.2.2.3" xref="Thmproofx11.p2.13.m10.2.2.2.3.cmml">⁢</mo><mrow id="Thmproofx11.p2.13.m10.2.2.2.5.2" xref="Thmproofx11.p2.13.m10.2.2.2.5.1.cmml"><mo id="Thmproofx11.p2.13.m10.2.2.2.5.2.1" stretchy="false" xref="Thmproofx11.p2.13.m10.2.2.2.5.1.cmml">(</mo><mi id="Thmproofx11.p2.13.m10.1.1.1.1" xref="Thmproofx11.p2.13.m10.1.1.1.1.cmml">α</mi><mo id="Thmproofx11.p2.13.m10.2.2.2.5.2.2" xref="Thmproofx11.p2.13.m10.2.2.2.5.1.cmml">,</mo><mi id="Thmproofx11.p2.13.m10.2.2.2.2" xref="Thmproofx11.p2.13.m10.2.2.2.2.cmml">n</mi><mo id="Thmproofx11.p2.13.m10.2.2.2.5.2.3" stretchy="false" xref="Thmproofx11.p2.13.m10.2.2.2.5.1.cmml">)</mo></mrow></mrow><mrow id="Thmproofx11.p2.13.m10.4.4.4" xref="Thmproofx11.p2.13.m10.4.4.4.cmml"><mi id="Thmproofx11.p2.13.m10.4.4.4.4" xref="Thmproofx11.p2.13.m10.4.4.4.4.cmml">φ</mi><mo id="Thmproofx11.p2.13.m10.4.4.4.3" xref="Thmproofx11.p2.13.m10.4.4.4.3.cmml">⁢</mo><mrow id="Thmproofx11.p2.13.m10.4.4.4.5.2" xref="Thmproofx11.p2.13.m10.4.4.4.5.1.cmml"><mo id="Thmproofx11.p2.13.m10.4.4.4.5.2.1" stretchy="false" xref="Thmproofx11.p2.13.m10.4.4.4.5.1.cmml">(</mo><mi id="Thmproofx11.p2.13.m10.3.3.3.1" xref="Thmproofx11.p2.13.m10.3.3.3.1.cmml">α</mi><mo id="Thmproofx11.p2.13.m10.4.4.4.5.2.2" xref="Thmproofx11.p2.13.m10.4.4.4.5.1.cmml">,</mo><mn id="Thmproofx11.p2.13.m10.4.4.4.2" xref="Thmproofx11.p2.13.m10.4.4.4.2.cmml">1</mn><mo id="Thmproofx11.p2.13.m10.4.4.4.5.2.3" stretchy="false" xref="Thmproofx11.p2.13.m10.4.4.4.5.1.cmml">)</mo></mrow></mrow></mfrac><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.13.m10.4b"><apply id="Thmproofx11.p2.13.m10.4.4.cmml" xref="Thmproofx11.p2.13.m10.4.4"><divide id="Thmproofx11.p2.13.m10.4.4.5.cmml" xref="Thmproofx11.p2.13.m10.4.4"></divide><apply id="Thmproofx11.p2.13.m10.2.2.2.cmml" xref="Thmproofx11.p2.13.m10.2.2.2"><times id="Thmproofx11.p2.13.m10.2.2.2.3.cmml" xref="Thmproofx11.p2.13.m10.2.2.2.3"></times><ci id="Thmproofx11.p2.13.m10.2.2.2.4.cmml" xref="Thmproofx11.p2.13.m10.2.2.2.4">𝜑</ci><interval closure="open" id="Thmproofx11.p2.13.m10.2.2.2.5.1.cmml" xref="Thmproofx11.p2.13.m10.2.2.2.5.2"><ci id="Thmproofx11.p2.13.m10.1.1.1.1.cmml" xref="Thmproofx11.p2.13.m10.1.1.1.1">𝛼</ci><ci id="Thmproofx11.p2.13.m10.2.2.2.2.cmml" xref="Thmproofx11.p2.13.m10.2.2.2.2">𝑛</ci></interval></apply><apply id="Thmproofx11.p2.13.m10.4.4.4.cmml" xref="Thmproofx11.p2.13.m10.4.4.4"><times id="Thmproofx11.p2.13.m10.4.4.4.3.cmml" xref="Thmproofx11.p2.13.m10.4.4.4.3"></times><ci id="Thmproofx11.p2.13.m10.4.4.4.4.cmml" xref="Thmproofx11.p2.13.m10.4.4.4.4">𝜑</ci><interval closure="open" id="Thmproofx11.p2.13.m10.4.4.4.5.1.cmml" xref="Thmproofx11.p2.13.m10.4.4.4.5.2"><ci id="Thmproofx11.p2.13.m10.3.3.3.1.cmml" xref="Thmproofx11.p2.13.m10.3.3.3.1">𝛼</ci><cn id="Thmproofx11.p2.13.m10.4.4.4.2.cmml" type="integer" xref="Thmproofx11.p2.13.m10.4.4.4.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.13.m10.4c">\frac{\varphi(\alpha,n)}{\varphi(\alpha,1)}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.13.m10.4d">divide start_ARG italic_φ ( italic_α , italic_n ) end_ARG start_ARG italic_φ ( italic_α , 1 ) end_ARG</annotation></semantics></math> is strictly increasing for every <math alttext="\alpha>0" class="ltx_Math" display="inline" id="Thmproofx11.p2.14.m11.1"><semantics id="Thmproofx11.p2.14.m11.1a"><mrow id="Thmproofx11.p2.14.m11.1.1" xref="Thmproofx11.p2.14.m11.1.1.cmml"><mi id="Thmproofx11.p2.14.m11.1.1.2" xref="Thmproofx11.p2.14.m11.1.1.2.cmml">α</mi><mo id="Thmproofx11.p2.14.m11.1.1.1" xref="Thmproofx11.p2.14.m11.1.1.1.cmml">></mo><mn id="Thmproofx11.p2.14.m11.1.1.3" xref="Thmproofx11.p2.14.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p2.14.m11.1b"><apply id="Thmproofx11.p2.14.m11.1.1.cmml" xref="Thmproofx11.p2.14.m11.1.1"><gt id="Thmproofx11.p2.14.m11.1.1.1.cmml" xref="Thmproofx11.p2.14.m11.1.1.1"></gt><ci id="Thmproofx11.p2.14.m11.1.1.2.cmml" xref="Thmproofx11.p2.14.m11.1.1.2">𝛼</ci><cn id="Thmproofx11.p2.14.m11.1.1.3.cmml" type="integer" xref="Thmproofx11.p2.14.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p2.14.m11.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p2.14.m11.1d">italic_α > 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx11.p3"> <p class="ltx_p" id="Thmproofx11.p3.3">Therefore for sufficiently large <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx11.p3.1.m1.1"><semantics id="Thmproofx11.p3.1.m1.1a"><mi id="Thmproofx11.p3.1.m1.1.1" xref="Thmproofx11.p3.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx11.p3.1.m1.1b"><ci id="Thmproofx11.p3.1.m1.1.1.cmml" xref="Thmproofx11.p3.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p3.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p3.1.m1.1d">italic_α</annotation></semantics></math>, <math alttext="\varphi(\alpha,n)" class="ltx_Math" display="inline" id="Thmproofx11.p3.2.m2.2"><semantics id="Thmproofx11.p3.2.m2.2a"><mrow id="Thmproofx11.p3.2.m2.2.3" xref="Thmproofx11.p3.2.m2.2.3.cmml"><mi id="Thmproofx11.p3.2.m2.2.3.2" xref="Thmproofx11.p3.2.m2.2.3.2.cmml">φ</mi><mo id="Thmproofx11.p3.2.m2.2.3.1" xref="Thmproofx11.p3.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmproofx11.p3.2.m2.2.3.3.2" xref="Thmproofx11.p3.2.m2.2.3.3.1.cmml"><mo id="Thmproofx11.p3.2.m2.2.3.3.2.1" stretchy="false" xref="Thmproofx11.p3.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmproofx11.p3.2.m2.1.1" xref="Thmproofx11.p3.2.m2.1.1.cmml">α</mi><mo id="Thmproofx11.p3.2.m2.2.3.3.2.2" xref="Thmproofx11.p3.2.m2.2.3.3.1.cmml">,</mo><mi id="Thmproofx11.p3.2.m2.2.2" xref="Thmproofx11.p3.2.m2.2.2.cmml">n</mi><mo id="Thmproofx11.p3.2.m2.2.3.3.2.3" stretchy="false" xref="Thmproofx11.p3.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p3.2.m2.2b"><apply id="Thmproofx11.p3.2.m2.2.3.cmml" xref="Thmproofx11.p3.2.m2.2.3"><times id="Thmproofx11.p3.2.m2.2.3.1.cmml" xref="Thmproofx11.p3.2.m2.2.3.1"></times><ci id="Thmproofx11.p3.2.m2.2.3.2.cmml" xref="Thmproofx11.p3.2.m2.2.3.2">𝜑</ci><interval closure="open" id="Thmproofx11.p3.2.m2.2.3.3.1.cmml" xref="Thmproofx11.p3.2.m2.2.3.3.2"><ci id="Thmproofx11.p3.2.m2.1.1.cmml" xref="Thmproofx11.p3.2.m2.1.1">𝛼</ci><ci id="Thmproofx11.p3.2.m2.2.2.cmml" xref="Thmproofx11.p3.2.m2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p3.2.m2.2c">\varphi(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p3.2.m2.2d">italic_φ ( italic_α , italic_n )</annotation></semantics></math> equals the affine function with the smallest leading coefficient: there exists <math alttext="A(G,n,k)" class="ltx_Math" display="inline" id="Thmproofx11.p3.3.m3.3"><semantics id="Thmproofx11.p3.3.m3.3a"><mrow id="Thmproofx11.p3.3.m3.3.4" xref="Thmproofx11.p3.3.m3.3.4.cmml"><mi id="Thmproofx11.p3.3.m3.3.4.2" xref="Thmproofx11.p3.3.m3.3.4.2.cmml">A</mi><mo id="Thmproofx11.p3.3.m3.3.4.1" xref="Thmproofx11.p3.3.m3.3.4.1.cmml">⁢</mo><mrow id="Thmproofx11.p3.3.m3.3.4.3.2" xref="Thmproofx11.p3.3.m3.3.4.3.1.cmml"><mo id="Thmproofx11.p3.3.m3.3.4.3.2.1" stretchy="false" xref="Thmproofx11.p3.3.m3.3.4.3.1.cmml">(</mo><mi id="Thmproofx11.p3.3.m3.1.1" xref="Thmproofx11.p3.3.m3.1.1.cmml">G</mi><mo id="Thmproofx11.p3.3.m3.3.4.3.2.2" xref="Thmproofx11.p3.3.m3.3.4.3.1.cmml">,</mo><mi id="Thmproofx11.p3.3.m3.2.2" xref="Thmproofx11.p3.3.m3.2.2.cmml">n</mi><mo id="Thmproofx11.p3.3.m3.3.4.3.2.3" xref="Thmproofx11.p3.3.m3.3.4.3.1.cmml">,</mo><mi id="Thmproofx11.p3.3.m3.3.3" xref="Thmproofx11.p3.3.m3.3.3.cmml">k</mi><mo id="Thmproofx11.p3.3.m3.3.4.3.2.4" stretchy="false" xref="Thmproofx11.p3.3.m3.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p3.3.m3.3b"><apply id="Thmproofx11.p3.3.m3.3.4.cmml" xref="Thmproofx11.p3.3.m3.3.4"><times id="Thmproofx11.p3.3.m3.3.4.1.cmml" xref="Thmproofx11.p3.3.m3.3.4.1"></times><ci id="Thmproofx11.p3.3.m3.3.4.2.cmml" xref="Thmproofx11.p3.3.m3.3.4.2">𝐴</ci><vector id="Thmproofx11.p3.3.m3.3.4.3.1.cmml" xref="Thmproofx11.p3.3.m3.3.4.3.2"><ci id="Thmproofx11.p3.3.m3.1.1.cmml" xref="Thmproofx11.p3.3.m3.1.1">𝐺</ci><ci id="Thmproofx11.p3.3.m3.2.2.cmml" xref="Thmproofx11.p3.3.m3.2.2">𝑛</ci><ci id="Thmproofx11.p3.3.m3.3.3.cmml" xref="Thmproofx11.p3.3.m3.3.3">𝑘</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p3.3.m3.3c">A(G,n,k)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p3.3.m3.3d">italic_A ( italic_G , italic_n , italic_k )</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx35"> <tbody id="Pt0.A3.Ex54"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall\alpha\geq A(G,n,k)~{}\forall n^{\prime}\leq n~{}~{}~{}% \varphi(\alpha,n^{\prime})=\frac{\mathcal{E}_{m_{c},n^{\prime}}}{m_{c}}\left(k% \alpha+S_{m_{c}}\right)" class="ltx_Math" display="inline" id="Pt0.A3.Ex54.m1.8"><semantics id="Pt0.A3.Ex54.m1.8a"><mrow id="Pt0.A3.Ex54.m1.8.8" xref="Pt0.A3.Ex54.m1.8.8.cmml"><mrow id="Pt0.A3.Ex54.m1.8.8.4" xref="Pt0.A3.Ex54.m1.8.8.4.cmml"><mo id="Pt0.A3.Ex54.m1.8.8.4.1" rspace="0.167em" xref="Pt0.A3.Ex54.m1.8.8.4.1.cmml">∀</mo><mi id="Pt0.A3.Ex54.m1.8.8.4.2" xref="Pt0.A3.Ex54.m1.8.8.4.2.cmml">α</mi></mrow><mo id="Pt0.A3.Ex54.m1.8.8.5" xref="Pt0.A3.Ex54.m1.8.8.5.cmml">≥</mo><mrow id="Pt0.A3.Ex54.m1.8.8.6" xref="Pt0.A3.Ex54.m1.8.8.6.cmml"><mi id="Pt0.A3.Ex54.m1.8.8.6.2" xref="Pt0.A3.Ex54.m1.8.8.6.2.cmml">A</mi><mo id="Pt0.A3.Ex54.m1.8.8.6.1" xref="Pt0.A3.Ex54.m1.8.8.6.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex54.m1.8.8.6.3.2" xref="Pt0.A3.Ex54.m1.8.8.6.3.1.cmml"><mo id="Pt0.A3.Ex54.m1.8.8.6.3.2.1" stretchy="false" xref="Pt0.A3.Ex54.m1.8.8.6.3.1.cmml">(</mo><mi id="Pt0.A3.Ex54.m1.3.3" xref="Pt0.A3.Ex54.m1.3.3.cmml">G</mi><mo id="Pt0.A3.Ex54.m1.8.8.6.3.2.2" xref="Pt0.A3.Ex54.m1.8.8.6.3.1.cmml">,</mo><mi id="Pt0.A3.Ex54.m1.4.4" xref="Pt0.A3.Ex54.m1.4.4.cmml">n</mi><mo id="Pt0.A3.Ex54.m1.8.8.6.3.2.3" xref="Pt0.A3.Ex54.m1.8.8.6.3.1.cmml">,</mo><mi id="Pt0.A3.Ex54.m1.5.5" xref="Pt0.A3.Ex54.m1.5.5.cmml">k</mi><mo id="Pt0.A3.Ex54.m1.8.8.6.3.2.4" stretchy="false" xref="Pt0.A3.Ex54.m1.8.8.6.3.1.cmml">)</mo></mrow><mo id="Pt0.A3.Ex54.m1.8.8.6.1a" lspace="0.497em" xref="Pt0.A3.Ex54.m1.8.8.6.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex54.m1.8.8.6.4" xref="Pt0.A3.Ex54.m1.8.8.6.4.cmml"><mo id="Pt0.A3.Ex54.m1.8.8.6.4.1" rspace="0.167em" xref="Pt0.A3.Ex54.m1.8.8.6.4.1.cmml">∀</mo><msup 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id="Pt0.A3.Ex55.m1.6.6.2.2.1.1.cmml" xref="Pt0.A3.Ex55.m1.6.6.2.2.1">subscript</csymbol><ci id="Pt0.A3.Ex55.m1.6.6.2.2.1.2.cmml" xref="Pt0.A3.Ex55.m1.6.6.2.2.1.2">𝑚</ci><ci id="Pt0.A3.Ex55.m1.6.6.2.2.1.3.cmml" xref="Pt0.A3.Ex55.m1.6.6.2.2.1.3">𝑐</ci></apply><ci id="Pt0.A3.Ex55.m1.5.5.1.1.cmml" xref="Pt0.A3.Ex55.m1.5.5.1.1">𝑛</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex55.m1.13c">\displaystyle\forall\alpha\geq A(G,n,k)~{}~{}~{}r_{U}(G,n,k,\alpha)=\frac{% \varphi(\alpha,n)}{\varphi(\alpha,1)}=\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex55.m1.13d">∀ italic_α ≥ italic_A ( italic_G , italic_n , italic_k ) italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α ) = divide start_ARG italic_φ ( italic_α , italic_n ) end_ARG start_ARG italic_φ ( italic_α , 1 ) end_ARG = caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx11.p3.5">Now let <math alttext="\alpha<A(G,n,k)" class="ltx_Math" display="inline" id="Thmproofx11.p3.4.m1.3"><semantics id="Thmproofx11.p3.4.m1.3a"><mrow id="Thmproofx11.p3.4.m1.3.4" xref="Thmproofx11.p3.4.m1.3.4.cmml"><mi id="Thmproofx11.p3.4.m1.3.4.2" xref="Thmproofx11.p3.4.m1.3.4.2.cmml">α</mi><mo id="Thmproofx11.p3.4.m1.3.4.1" xref="Thmproofx11.p3.4.m1.3.4.1.cmml"><</mo><mrow id="Thmproofx11.p3.4.m1.3.4.3" xref="Thmproofx11.p3.4.m1.3.4.3.cmml"><mi id="Thmproofx11.p3.4.m1.3.4.3.2" xref="Thmproofx11.p3.4.m1.3.4.3.2.cmml">A</mi><mo id="Thmproofx11.p3.4.m1.3.4.3.1" xref="Thmproofx11.p3.4.m1.3.4.3.1.cmml">⁢</mo><mrow id="Thmproofx11.p3.4.m1.3.4.3.3.2" xref="Thmproofx11.p3.4.m1.3.4.3.3.1.cmml"><mo id="Thmproofx11.p3.4.m1.3.4.3.3.2.1" stretchy="false" xref="Thmproofx11.p3.4.m1.3.4.3.3.1.cmml">(</mo><mi id="Thmproofx11.p3.4.m1.1.1" xref="Thmproofx11.p3.4.m1.1.1.cmml">G</mi><mo id="Thmproofx11.p3.4.m1.3.4.3.3.2.2" xref="Thmproofx11.p3.4.m1.3.4.3.3.1.cmml">,</mo><mi id="Thmproofx11.p3.4.m1.2.2" xref="Thmproofx11.p3.4.m1.2.2.cmml">n</mi><mo id="Thmproofx11.p3.4.m1.3.4.3.3.2.3" xref="Thmproofx11.p3.4.m1.3.4.3.3.1.cmml">,</mo><mi id="Thmproofx11.p3.4.m1.3.3" xref="Thmproofx11.p3.4.m1.3.3.cmml">k</mi><mo id="Thmproofx11.p3.4.m1.3.4.3.3.2.4" stretchy="false" xref="Thmproofx11.p3.4.m1.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p3.4.m1.3b"><apply id="Thmproofx11.p3.4.m1.3.4.cmml" xref="Thmproofx11.p3.4.m1.3.4"><lt id="Thmproofx11.p3.4.m1.3.4.1.cmml" xref="Thmproofx11.p3.4.m1.3.4.1"></lt><ci id="Thmproofx11.p3.4.m1.3.4.2.cmml" xref="Thmproofx11.p3.4.m1.3.4.2">𝛼</ci><apply id="Thmproofx11.p3.4.m1.3.4.3.cmml" xref="Thmproofx11.p3.4.m1.3.4.3"><times id="Thmproofx11.p3.4.m1.3.4.3.1.cmml" xref="Thmproofx11.p3.4.m1.3.4.3.1"></times><ci id="Thmproofx11.p3.4.m1.3.4.3.2.cmml" xref="Thmproofx11.p3.4.m1.3.4.3.2">𝐴</ci><vector id="Thmproofx11.p3.4.m1.3.4.3.3.1.cmml" xref="Thmproofx11.p3.4.m1.3.4.3.3.2"><ci id="Thmproofx11.p3.4.m1.1.1.cmml" xref="Thmproofx11.p3.4.m1.1.1">𝐺</ci><ci id="Thmproofx11.p3.4.m1.2.2.cmml" xref="Thmproofx11.p3.4.m1.2.2">𝑛</ci><ci id="Thmproofx11.p3.4.m1.3.3.cmml" xref="Thmproofx11.p3.4.m1.3.3">𝑘</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p3.4.m1.3c">\alpha<A(G,n,k)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p3.4.m1.3d">italic_α < italic_A ( italic_G , italic_n , italic_k )</annotation></semantics></math>. Observe that for some <math alttext="i\in[m_{c}]" class="ltx_Math" display="inline" id="Thmproofx11.p3.5.m2.1"><semantics id="Thmproofx11.p3.5.m2.1a"><mrow id="Thmproofx11.p3.5.m2.1.1" xref="Thmproofx11.p3.5.m2.1.1.cmml"><mi id="Thmproofx11.p3.5.m2.1.1.3" xref="Thmproofx11.p3.5.m2.1.1.3.cmml">i</mi><mo id="Thmproofx11.p3.5.m2.1.1.2" xref="Thmproofx11.p3.5.m2.1.1.2.cmml">∈</mo><mrow id="Thmproofx11.p3.5.m2.1.1.1.1" xref="Thmproofx11.p3.5.m2.1.1.1.2.cmml"><mo id="Thmproofx11.p3.5.m2.1.1.1.1.2" stretchy="false" xref="Thmproofx11.p3.5.m2.1.1.1.2.1.cmml">[</mo><msub id="Thmproofx11.p3.5.m2.1.1.1.1.1" xref="Thmproofx11.p3.5.m2.1.1.1.1.1.cmml"><mi id="Thmproofx11.p3.5.m2.1.1.1.1.1.2" xref="Thmproofx11.p3.5.m2.1.1.1.1.1.2.cmml">m</mi><mi id="Thmproofx11.p3.5.m2.1.1.1.1.1.3" xref="Thmproofx11.p3.5.m2.1.1.1.1.1.3.cmml">c</mi></msub><mo id="Thmproofx11.p3.5.m2.1.1.1.1.3" stretchy="false" xref="Thmproofx11.p3.5.m2.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p3.5.m2.1b"><apply id="Thmproofx11.p3.5.m2.1.1.cmml" xref="Thmproofx11.p3.5.m2.1.1"><in id="Thmproofx11.p3.5.m2.1.1.2.cmml" xref="Thmproofx11.p3.5.m2.1.1.2"></in><ci id="Thmproofx11.p3.5.m2.1.1.3.cmml" xref="Thmproofx11.p3.5.m2.1.1.3">𝑖</ci><apply id="Thmproofx11.p3.5.m2.1.1.1.2.cmml" xref="Thmproofx11.p3.5.m2.1.1.1.1"><csymbol cd="latexml" id="Thmproofx11.p3.5.m2.1.1.1.2.1.cmml" xref="Thmproofx11.p3.5.m2.1.1.1.1.2">delimited-[]</csymbol><apply id="Thmproofx11.p3.5.m2.1.1.1.1.1.cmml" xref="Thmproofx11.p3.5.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx11.p3.5.m2.1.1.1.1.1.1.cmml" xref="Thmproofx11.p3.5.m2.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx11.p3.5.m2.1.1.1.1.1.2.cmml" xref="Thmproofx11.p3.5.m2.1.1.1.1.1.2">𝑚</ci><ci id="Thmproofx11.p3.5.m2.1.1.1.1.1.3.cmml" xref="Thmproofx11.p3.5.m2.1.1.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p3.5.m2.1c">i\in[m_{c}]</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p3.5.m2.1d">italic_i ∈ [ italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ]</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx37"> <tbody id="Pt0.A3.Ex56"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{E}_{m_{c},n}\varphi(\alpha,1)" class="ltx_Math" display="inline" id="Pt0.A3.Ex56.m1.4"><semantics id="Pt0.A3.Ex56.m1.4a"><mrow id="Pt0.A3.Ex56.m1.4.5" xref="Pt0.A3.Ex56.m1.4.5.cmml"><msub id="Pt0.A3.Ex56.m1.4.5.2" xref="Pt0.A3.Ex56.m1.4.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A3.Ex56.m1.4.5.2.2" xref="Pt0.A3.Ex56.m1.4.5.2.2.cmml">ℰ</mi><mrow id="Pt0.A3.Ex56.m1.2.2.2.2" xref="Pt0.A3.Ex56.m1.2.2.2.3.cmml"><msub id="Pt0.A3.Ex56.m1.2.2.2.2.1" xref="Pt0.A3.Ex56.m1.2.2.2.2.1.cmml"><mi id="Pt0.A3.Ex56.m1.2.2.2.2.1.2" xref="Pt0.A3.Ex56.m1.2.2.2.2.1.2.cmml">m</mi><mi id="Pt0.A3.Ex56.m1.2.2.2.2.1.3" xref="Pt0.A3.Ex56.m1.2.2.2.2.1.3.cmml">c</mi></msub><mo id="Pt0.A3.Ex56.m1.2.2.2.2.2" xref="Pt0.A3.Ex56.m1.2.2.2.3.cmml">,</mo><mi id="Pt0.A3.Ex56.m1.1.1.1.1" xref="Pt0.A3.Ex56.m1.1.1.1.1.cmml">n</mi></mrow></msub><mo id="Pt0.A3.Ex56.m1.4.5.1" xref="Pt0.A3.Ex56.m1.4.5.1.cmml">⁢</mo><mi id="Pt0.A3.Ex56.m1.4.5.3" xref="Pt0.A3.Ex56.m1.4.5.3.cmml">φ</mi><mo id="Pt0.A3.Ex56.m1.4.5.1a" xref="Pt0.A3.Ex56.m1.4.5.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex56.m1.4.5.4.2" xref="Pt0.A3.Ex56.m1.4.5.4.1.cmml"><mo id="Pt0.A3.Ex56.m1.4.5.4.2.1" stretchy="false" xref="Pt0.A3.Ex56.m1.4.5.4.1.cmml">(</mo><mi id="Pt0.A3.Ex56.m1.3.3" xref="Pt0.A3.Ex56.m1.3.3.cmml">α</mi><mo id="Pt0.A3.Ex56.m1.4.5.4.2.2" xref="Pt0.A3.Ex56.m1.4.5.4.1.cmml">,</mo><mn id="Pt0.A3.Ex56.m1.4.4" xref="Pt0.A3.Ex56.m1.4.4.cmml">1</mn><mo id="Pt0.A3.Ex56.m1.4.5.4.2.3" stretchy="false" xref="Pt0.A3.Ex56.m1.4.5.4.1.cmml">)</mo></mrow></mrow><annotation-xml 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xref="Pt0.A3.Ex56.m1.4.5.3">𝜑</ci><interval closure="open" id="Pt0.A3.Ex56.m1.4.5.4.1.cmml" xref="Pt0.A3.Ex56.m1.4.5.4.2"><ci id="Pt0.A3.Ex56.m1.3.3.cmml" xref="Pt0.A3.Ex56.m1.3.3">𝛼</ci><cn id="Pt0.A3.Ex56.m1.4.4.cmml" type="integer" xref="Pt0.A3.Ex56.m1.4.4">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex56.m1.4c">\displaystyle\mathcal{E}_{m_{c},n}\varphi(\alpha,1)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex56.m1.4d">caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT italic_φ ( italic_α , 1 )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{E}_{m_{c},n}\frac{\mathcal{E}_{i,1}}{i}\left(k\alpha+S_% {i}\right)=\frac{\mathcal{E}_{m_{c},n}}{i}\left(k\alpha+S_{i}\right)" class="ltx_Math" display="inline" id="Pt0.A3.Ex56.m2.8"><semantics id="Pt0.A3.Ex56.m2.8a"><mrow 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xref="Pt0.A3.Ex56.m2.8.8.2.1.1.1.2.3">𝛼</ci></apply><apply id="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3.cmml" xref="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3.1.cmml" xref="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3.2.cmml" xref="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3.2">𝑆</ci><ci id="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3.3.cmml" xref="Pt0.A3.Ex56.m2.8.8.2.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex56.m2.8c">\displaystyle=\mathcal{E}_{m_{c},n}\frac{\mathcal{E}_{i,1}}{i}\left(k\alpha+S_% {i}\right)=\frac{\mathcal{E}_{m_{c},n}}{i}\left(k\alpha+S_{i}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex56.m2.8d">= caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG ( italic_k italic_α + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG ( italic_k italic_α + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex57"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\stackrel{{\scriptstyle i,m_{c}\geq 1}}{{\geq}}\frac{\mathcal{E}_% {i,n}}{i}\left(k\alpha+S_{i}\right)\geq\varphi(\alpha,n)" class="ltx_Math" display="inline" id="Pt0.A3.Ex57.m1.7"><semantics id="Pt0.A3.Ex57.m1.7a"><mrow id="Pt0.A3.Ex57.m1.7.7" xref="Pt0.A3.Ex57.m1.7.7.cmml"><mi 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id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.1.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.1"></plus><apply id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2"><times id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2.1.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2.1"></times><ci id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2.2.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2.2">𝑘</ci><ci id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2.3.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.2.3">𝛼</ci></apply><apply id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3.1.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3.2.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3.2">𝑆</ci><ci id="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex57.m1.7.7.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply><apply id="Pt0.A3.Ex57.m1.7.7c.cmml" xref="Pt0.A3.Ex57.m1.7.7"><geq id="Pt0.A3.Ex57.m1.7.7.4.cmml" xref="Pt0.A3.Ex57.m1.7.7.4"></geq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A3.Ex57.m1.7.7.1.cmml" id="Pt0.A3.Ex57.m1.7.7d.cmml" xref="Pt0.A3.Ex57.m1.7.7"></share><apply id="Pt0.A3.Ex57.m1.7.7.5.cmml" xref="Pt0.A3.Ex57.m1.7.7.5"><times id="Pt0.A3.Ex57.m1.7.7.5.1.cmml" xref="Pt0.A3.Ex57.m1.7.7.5.1"></times><ci id="Pt0.A3.Ex57.m1.7.7.5.2.cmml" xref="Pt0.A3.Ex57.m1.7.7.5.2">𝜑</ci><interval closure="open" id="Pt0.A3.Ex57.m1.7.7.5.3.1.cmml" xref="Pt0.A3.Ex57.m1.7.7.5.3.2"><ci id="Pt0.A3.Ex57.m1.5.5.cmml" xref="Pt0.A3.Ex57.m1.5.5">𝛼</ci><ci id="Pt0.A3.Ex57.m1.6.6.cmml" xref="Pt0.A3.Ex57.m1.6.6">𝑛</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex57.m1.7c">\displaystyle\stackrel{{\scriptstyle i,m_{c}\geq 1}}{{\geq}}\frac{\mathcal{E}_% {i,n}}{i}\left(k\alpha+S_{i}\right)\geq\varphi(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex57.m1.7d">start_RELOP SUPERSCRIPTOP start_ARG ≥ end_ARG start_ARG italic_i , italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ≥ 1 end_ARG end_RELOP divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG ( italic_k italic_α + italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ italic_φ ( italic_α , italic_n )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx11.p3.6">therefore <math alttext="r_{U}(G,n,k,\alpha)\leq\mathcal{E}_{m_{c},n}" class="ltx_Math" display="inline" id="Thmproofx11.p3.6.m1.6"><semantics id="Thmproofx11.p3.6.m1.6a"><mrow id="Thmproofx11.p3.6.m1.6.7" xref="Thmproofx11.p3.6.m1.6.7.cmml"><mrow id="Thmproofx11.p3.6.m1.6.7.2" xref="Thmproofx11.p3.6.m1.6.7.2.cmml"><msub id="Thmproofx11.p3.6.m1.6.7.2.2" xref="Thmproofx11.p3.6.m1.6.7.2.2.cmml"><mi id="Thmproofx11.p3.6.m1.6.7.2.2.2" xref="Thmproofx11.p3.6.m1.6.7.2.2.2.cmml">r</mi><mi id="Thmproofx11.p3.6.m1.6.7.2.2.3" xref="Thmproofx11.p3.6.m1.6.7.2.2.3.cmml">U</mi></msub><mo id="Thmproofx11.p3.6.m1.6.7.2.1" xref="Thmproofx11.p3.6.m1.6.7.2.1.cmml">⁢</mo><mrow id="Thmproofx11.p3.6.m1.6.7.2.3.2" xref="Thmproofx11.p3.6.m1.6.7.2.3.1.cmml"><mo id="Thmproofx11.p3.6.m1.6.7.2.3.2.1" stretchy="false" xref="Thmproofx11.p3.6.m1.6.7.2.3.1.cmml">(</mo><mi id="Thmproofx11.p3.6.m1.3.3" xref="Thmproofx11.p3.6.m1.3.3.cmml">G</mi><mo id="Thmproofx11.p3.6.m1.6.7.2.3.2.2" xref="Thmproofx11.p3.6.m1.6.7.2.3.1.cmml">,</mo><mi id="Thmproofx11.p3.6.m1.4.4" xref="Thmproofx11.p3.6.m1.4.4.cmml">n</mi><mo id="Thmproofx11.p3.6.m1.6.7.2.3.2.3" xref="Thmproofx11.p3.6.m1.6.7.2.3.1.cmml">,</mo><mi id="Thmproofx11.p3.6.m1.5.5" xref="Thmproofx11.p3.6.m1.5.5.cmml">k</mi><mo id="Thmproofx11.p3.6.m1.6.7.2.3.2.4" xref="Thmproofx11.p3.6.m1.6.7.2.3.1.cmml">,</mo><mi id="Thmproofx11.p3.6.m1.6.6" xref="Thmproofx11.p3.6.m1.6.6.cmml">α</mi><mo id="Thmproofx11.p3.6.m1.6.7.2.3.2.5" stretchy="false" xref="Thmproofx11.p3.6.m1.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx11.p3.6.m1.6.7.1" xref="Thmproofx11.p3.6.m1.6.7.1.cmml">≤</mo><msub id="Thmproofx11.p3.6.m1.6.7.3" xref="Thmproofx11.p3.6.m1.6.7.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx11.p3.6.m1.6.7.3.2" xref="Thmproofx11.p3.6.m1.6.7.3.2.cmml">ℰ</mi><mrow id="Thmproofx11.p3.6.m1.2.2.2.2" xref="Thmproofx11.p3.6.m1.2.2.2.3.cmml"><msub id="Thmproofx11.p3.6.m1.2.2.2.2.1" xref="Thmproofx11.p3.6.m1.2.2.2.2.1.cmml"><mi id="Thmproofx11.p3.6.m1.2.2.2.2.1.2" xref="Thmproofx11.p3.6.m1.2.2.2.2.1.2.cmml">m</mi><mi id="Thmproofx11.p3.6.m1.2.2.2.2.1.3" xref="Thmproofx11.p3.6.m1.2.2.2.2.1.3.cmml">c</mi></msub><mo id="Thmproofx11.p3.6.m1.2.2.2.2.2" xref="Thmproofx11.p3.6.m1.2.2.2.3.cmml">,</mo><mi id="Thmproofx11.p3.6.m1.1.1.1.1" xref="Thmproofx11.p3.6.m1.1.1.1.1.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx11.p3.6.m1.6b"><apply id="Thmproofx11.p3.6.m1.6.7.cmml" xref="Thmproofx11.p3.6.m1.6.7"><leq id="Thmproofx11.p3.6.m1.6.7.1.cmml" xref="Thmproofx11.p3.6.m1.6.7.1"></leq><apply id="Thmproofx11.p3.6.m1.6.7.2.cmml" xref="Thmproofx11.p3.6.m1.6.7.2"><times id="Thmproofx11.p3.6.m1.6.7.2.1.cmml" xref="Thmproofx11.p3.6.m1.6.7.2.1"></times><apply id="Thmproofx11.p3.6.m1.6.7.2.2.cmml" xref="Thmproofx11.p3.6.m1.6.7.2.2"><csymbol cd="ambiguous" id="Thmproofx11.p3.6.m1.6.7.2.2.1.cmml" xref="Thmproofx11.p3.6.m1.6.7.2.2">subscript</csymbol><ci id="Thmproofx11.p3.6.m1.6.7.2.2.2.cmml" xref="Thmproofx11.p3.6.m1.6.7.2.2.2">𝑟</ci><ci id="Thmproofx11.p3.6.m1.6.7.2.2.3.cmml" xref="Thmproofx11.p3.6.m1.6.7.2.2.3">𝑈</ci></apply><vector id="Thmproofx11.p3.6.m1.6.7.2.3.1.cmml" xref="Thmproofx11.p3.6.m1.6.7.2.3.2"><ci id="Thmproofx11.p3.6.m1.3.3.cmml" xref="Thmproofx11.p3.6.m1.3.3">𝐺</ci><ci id="Thmproofx11.p3.6.m1.4.4.cmml" xref="Thmproofx11.p3.6.m1.4.4">𝑛</ci><ci id="Thmproofx11.p3.6.m1.5.5.cmml" xref="Thmproofx11.p3.6.m1.5.5">𝑘</ci><ci id="Thmproofx11.p3.6.m1.6.6.cmml" xref="Thmproofx11.p3.6.m1.6.6">𝛼</ci></vector></apply><apply id="Thmproofx11.p3.6.m1.6.7.3.cmml" xref="Thmproofx11.p3.6.m1.6.7.3"><csymbol cd="ambiguous" id="Thmproofx11.p3.6.m1.6.7.3.1.cmml" xref="Thmproofx11.p3.6.m1.6.7.3">subscript</csymbol><ci id="Thmproofx11.p3.6.m1.6.7.3.2.cmml" xref="Thmproofx11.p3.6.m1.6.7.3.2">ℰ</ci><list id="Thmproofx11.p3.6.m1.2.2.2.3.cmml" xref="Thmproofx11.p3.6.m1.2.2.2.2"><apply id="Thmproofx11.p3.6.m1.2.2.2.2.1.cmml" xref="Thmproofx11.p3.6.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmproofx11.p3.6.m1.2.2.2.2.1.1.cmml" xref="Thmproofx11.p3.6.m1.2.2.2.2.1">subscript</csymbol><ci id="Thmproofx11.p3.6.m1.2.2.2.2.1.2.cmml" xref="Thmproofx11.p3.6.m1.2.2.2.2.1.2">𝑚</ci><ci id="Thmproofx11.p3.6.m1.2.2.2.2.1.3.cmml" xref="Thmproofx11.p3.6.m1.2.2.2.2.1.3">𝑐</ci></apply><ci id="Thmproofx11.p3.6.m1.1.1.1.1.cmml" xref="Thmproofx11.p3.6.m1.1.1.1.1">𝑛</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx11.p3.6.m1.6c">r_{U}(G,n,k,\alpha)\leq\mathcal{E}_{m_{c},n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx11.p3.6.m1.6d">italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_G , italic_n , italic_k , italic_α ) ≤ caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, completing the proof. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx12"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Corollary <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmcorollary1" title="Corollary 1 ‣ Properties of 𝑟_𝑈 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">1</span></a>)</h6> <div class="ltx_para" id="Thmproofx12.p1"> <p class="ltx_p" id="Thmproofx12.p1.7">Note that finding each <math alttext="P^{*}_{i}" class="ltx_Math" display="inline" id="Thmproofx12.p1.1.m1.1"><semantics id="Thmproofx12.p1.1.m1.1a"><msubsup id="Thmproofx12.p1.1.m1.1.1" xref="Thmproofx12.p1.1.m1.1.1.cmml"><mi id="Thmproofx12.p1.1.m1.1.1.2.2" xref="Thmproofx12.p1.1.m1.1.1.2.2.cmml">P</mi><mi id="Thmproofx12.p1.1.m1.1.1.3" xref="Thmproofx12.p1.1.m1.1.1.3.cmml">i</mi><mo id="Thmproofx12.p1.1.m1.1.1.2.3" xref="Thmproofx12.p1.1.m1.1.1.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx12.p1.1.m1.1b"><apply id="Thmproofx12.p1.1.m1.1.1.cmml" xref="Thmproofx12.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx12.p1.1.m1.1.1.1.cmml" xref="Thmproofx12.p1.1.m1.1.1">subscript</csymbol><apply id="Thmproofx12.p1.1.m1.1.1.2.cmml" xref="Thmproofx12.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx12.p1.1.m1.1.1.2.1.cmml" xref="Thmproofx12.p1.1.m1.1.1">superscript</csymbol><ci id="Thmproofx12.p1.1.m1.1.1.2.2.cmml" xref="Thmproofx12.p1.1.m1.1.1.2.2">𝑃</ci><times id="Thmproofx12.p1.1.m1.1.1.2.3.cmml" xref="Thmproofx12.p1.1.m1.1.1.2.3"></times></apply><ci id="Thmproofx12.p1.1.m1.1.1.3.cmml" xref="Thmproofx12.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx12.p1.1.m1.1c">P^{*}_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx12.p1.1.m1.1d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> requires solving a mincost flow problem: the flow network is <math alttext="G" class="ltx_Math" display="inline" id="Thmproofx12.p1.2.m2.1"><semantics id="Thmproofx12.p1.2.m2.1a"><mi id="Thmproofx12.p1.2.m2.1.1" xref="Thmproofx12.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmproofx12.p1.2.m2.1b"><ci id="Thmproofx12.p1.2.m2.1.1.cmml" xref="Thmproofx12.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx12.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmproofx12.p1.2.m2.1d">italic_G</annotation></semantics></math>, the edge costs are all 1, and the demands are <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx12.p1.3.m3.1"><semantics id="Thmproofx12.p1.3.m3.1a"><mi id="Thmproofx12.p1.3.m3.1.1" xref="Thmproofx12.p1.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx12.p1.3.m3.1b"><ci id="Thmproofx12.p1.3.m3.1.1.cmml" xref="Thmproofx12.p1.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx12.p1.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx12.p1.3.m3.1d">italic_i</annotation></semantics></math> for <math alttext="s" class="ltx_Math" display="inline" id="Thmproofx12.p1.4.m4.1"><semantics id="Thmproofx12.p1.4.m4.1a"><mi id="Thmproofx12.p1.4.m4.1.1" xref="Thmproofx12.p1.4.m4.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="Thmproofx12.p1.4.m4.1b"><ci id="Thmproofx12.p1.4.m4.1.1.cmml" xref="Thmproofx12.p1.4.m4.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx12.p1.4.m4.1c">s</annotation><annotation encoding="application/x-llamapun" id="Thmproofx12.p1.4.m4.1d">italic_s</annotation></semantics></math>, <math alttext="-i" class="ltx_Math" display="inline" id="Thmproofx12.p1.5.m5.1"><semantics id="Thmproofx12.p1.5.m5.1a"><mrow id="Thmproofx12.p1.5.m5.1.1" xref="Thmproofx12.p1.5.m5.1.1.cmml"><mo id="Thmproofx12.p1.5.m5.1.1a" xref="Thmproofx12.p1.5.m5.1.1.cmml">−</mo><mi id="Thmproofx12.p1.5.m5.1.1.2" xref="Thmproofx12.p1.5.m5.1.1.2.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx12.p1.5.m5.1b"><apply id="Thmproofx12.p1.5.m5.1.1.cmml" xref="Thmproofx12.p1.5.m5.1.1"><minus id="Thmproofx12.p1.5.m5.1.1.1.cmml" xref="Thmproofx12.p1.5.m5.1.1"></minus><ci id="Thmproofx12.p1.5.m5.1.1.2.cmml" xref="Thmproofx12.p1.5.m5.1.1.2">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx12.p1.5.m5.1c">-i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx12.p1.5.m5.1d">- italic_i</annotation></semantics></math> for <math alttext="t" class="ltx_Math" display="inline" id="Thmproofx12.p1.6.m6.1"><semantics id="Thmproofx12.p1.6.m6.1a"><mi id="Thmproofx12.p1.6.m6.1.1" xref="Thmproofx12.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="Thmproofx12.p1.6.m6.1b"><ci id="Thmproofx12.p1.6.m6.1.1.cmml" xref="Thmproofx12.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx12.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="Thmproofx12.p1.6.m6.1d">italic_t</annotation></semantics></math> and 0 for all other <math alttext="v\in V" class="ltx_Math" display="inline" id="Thmproofx12.p1.7.m7.1"><semantics id="Thmproofx12.p1.7.m7.1a"><mrow id="Thmproofx12.p1.7.m7.1.1" xref="Thmproofx12.p1.7.m7.1.1.cmml"><mi id="Thmproofx12.p1.7.m7.1.1.2" xref="Thmproofx12.p1.7.m7.1.1.2.cmml">v</mi><mo id="Thmproofx12.p1.7.m7.1.1.1" xref="Thmproofx12.p1.7.m7.1.1.1.cmml">∈</mo><mi id="Thmproofx12.p1.7.m7.1.1.3" xref="Thmproofx12.p1.7.m7.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx12.p1.7.m7.1b"><apply id="Thmproofx12.p1.7.m7.1.1.cmml" xref="Thmproofx12.p1.7.m7.1.1"><in id="Thmproofx12.p1.7.m7.1.1.1.cmml" xref="Thmproofx12.p1.7.m7.1.1.1"></in><ci id="Thmproofx12.p1.7.m7.1.1.2.cmml" xref="Thmproofx12.p1.7.m7.1.1.2">𝑣</ci><ci id="Thmproofx12.p1.7.m7.1.1.3.cmml" xref="Thmproofx12.p1.7.m7.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx12.p1.7.m7.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="Thmproofx12.p1.7.m7.1d">italic_v ∈ italic_V</annotation></semantics></math>. Therefore algorithms such as those of <span class="ltx_ERROR undefined" id="Thmproofx12.p1.7.1">\citet</span>chen2022maximum and <span class="ltx_ERROR undefined" id="Thmproofx12.p1.7.2">\citet</span>orlin1988faster yield the above running times. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx13"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem5" title="Theorem 4.5 ‣ Properties of 𝑟_𝑈 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.5</span></a>)</h6> <div class="ltx_para" id="Thmproofx13.p1"> <p class="ltx_p" id="Thmproofx13.p1.3">We reuse the notations from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem4" title="Theorem 4.4 ‣ Properties of 𝑟_𝑈 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.4</span></a>. By the proof of the latter, <math alttext="\alpha\geq A(G,n,k)" class="ltx_Math" display="inline" id="Thmproofx13.p1.1.m1.3"><semantics id="Thmproofx13.p1.1.m1.3a"><mrow id="Thmproofx13.p1.1.m1.3.4" xref="Thmproofx13.p1.1.m1.3.4.cmml"><mi id="Thmproofx13.p1.1.m1.3.4.2" xref="Thmproofx13.p1.1.m1.3.4.2.cmml">α</mi><mo id="Thmproofx13.p1.1.m1.3.4.1" xref="Thmproofx13.p1.1.m1.3.4.1.cmml">≥</mo><mrow id="Thmproofx13.p1.1.m1.3.4.3" xref="Thmproofx13.p1.1.m1.3.4.3.cmml"><mi id="Thmproofx13.p1.1.m1.3.4.3.2" xref="Thmproofx13.p1.1.m1.3.4.3.2.cmml">A</mi><mo id="Thmproofx13.p1.1.m1.3.4.3.1" xref="Thmproofx13.p1.1.m1.3.4.3.1.cmml">⁢</mo><mrow id="Thmproofx13.p1.1.m1.3.4.3.3.2" xref="Thmproofx13.p1.1.m1.3.4.3.3.1.cmml"><mo id="Thmproofx13.p1.1.m1.3.4.3.3.2.1" stretchy="false" xref="Thmproofx13.p1.1.m1.3.4.3.3.1.cmml">(</mo><mi id="Thmproofx13.p1.1.m1.1.1" xref="Thmproofx13.p1.1.m1.1.1.cmml">G</mi><mo id="Thmproofx13.p1.1.m1.3.4.3.3.2.2" xref="Thmproofx13.p1.1.m1.3.4.3.3.1.cmml">,</mo><mi id="Thmproofx13.p1.1.m1.2.2" xref="Thmproofx13.p1.1.m1.2.2.cmml">n</mi><mo id="Thmproofx13.p1.1.m1.3.4.3.3.2.3" xref="Thmproofx13.p1.1.m1.3.4.3.3.1.cmml">,</mo><mi id="Thmproofx13.p1.1.m1.3.3" xref="Thmproofx13.p1.1.m1.3.3.cmml">k</mi><mo id="Thmproofx13.p1.1.m1.3.4.3.3.2.4" stretchy="false" xref="Thmproofx13.p1.1.m1.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx13.p1.1.m1.3b"><apply id="Thmproofx13.p1.1.m1.3.4.cmml" xref="Thmproofx13.p1.1.m1.3.4"><geq id="Thmproofx13.p1.1.m1.3.4.1.cmml" xref="Thmproofx13.p1.1.m1.3.4.1"></geq><ci id="Thmproofx13.p1.1.m1.3.4.2.cmml" xref="Thmproofx13.p1.1.m1.3.4.2">𝛼</ci><apply id="Thmproofx13.p1.1.m1.3.4.3.cmml" xref="Thmproofx13.p1.1.m1.3.4.3"><times id="Thmproofx13.p1.1.m1.3.4.3.1.cmml" xref="Thmproofx13.p1.1.m1.3.4.3.1"></times><ci id="Thmproofx13.p1.1.m1.3.4.3.2.cmml" xref="Thmproofx13.p1.1.m1.3.4.3.2">𝐴</ci><vector id="Thmproofx13.p1.1.m1.3.4.3.3.1.cmml" xref="Thmproofx13.p1.1.m1.3.4.3.3.2"><ci id="Thmproofx13.p1.1.m1.1.1.cmml" xref="Thmproofx13.p1.1.m1.1.1">𝐺</ci><ci id="Thmproofx13.p1.1.m1.2.2.cmml" xref="Thmproofx13.p1.1.m1.2.2">𝑛</ci><ci id="Thmproofx13.p1.1.m1.3.3.cmml" xref="Thmproofx13.p1.1.m1.3.3">𝑘</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx13.p1.1.m1.3c">\alpha\geq A(G,n,k)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx13.p1.1.m1.3d">italic_α ≥ italic_A ( italic_G , italic_n , italic_k )</annotation></semantics></math> when <math alttext="\varphi(\alpha,n)=h_{m_{c}}(\alpha,n)" class="ltx_Math" display="inline" id="Thmproofx13.p1.2.m2.4"><semantics id="Thmproofx13.p1.2.m2.4a"><mrow id="Thmproofx13.p1.2.m2.4.5" xref="Thmproofx13.p1.2.m2.4.5.cmml"><mrow id="Thmproofx13.p1.2.m2.4.5.2" xref="Thmproofx13.p1.2.m2.4.5.2.cmml"><mi id="Thmproofx13.p1.2.m2.4.5.2.2" xref="Thmproofx13.p1.2.m2.4.5.2.2.cmml">φ</mi><mo id="Thmproofx13.p1.2.m2.4.5.2.1" xref="Thmproofx13.p1.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx13.p1.2.m2.4.5.2.3.2" xref="Thmproofx13.p1.2.m2.4.5.2.3.1.cmml"><mo id="Thmproofx13.p1.2.m2.4.5.2.3.2.1" stretchy="false" xref="Thmproofx13.p1.2.m2.4.5.2.3.1.cmml">(</mo><mi id="Thmproofx13.p1.2.m2.1.1" xref="Thmproofx13.p1.2.m2.1.1.cmml">α</mi><mo id="Thmproofx13.p1.2.m2.4.5.2.3.2.2" xref="Thmproofx13.p1.2.m2.4.5.2.3.1.cmml">,</mo><mi id="Thmproofx13.p1.2.m2.2.2" xref="Thmproofx13.p1.2.m2.2.2.cmml">n</mi><mo id="Thmproofx13.p1.2.m2.4.5.2.3.2.3" stretchy="false" xref="Thmproofx13.p1.2.m2.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx13.p1.2.m2.4.5.1" xref="Thmproofx13.p1.2.m2.4.5.1.cmml">=</mo><mrow id="Thmproofx13.p1.2.m2.4.5.3" xref="Thmproofx13.p1.2.m2.4.5.3.cmml"><msub id="Thmproofx13.p1.2.m2.4.5.3.2" xref="Thmproofx13.p1.2.m2.4.5.3.2.cmml"><mi id="Thmproofx13.p1.2.m2.4.5.3.2.2" xref="Thmproofx13.p1.2.m2.4.5.3.2.2.cmml">h</mi><msub id="Thmproofx13.p1.2.m2.4.5.3.2.3" xref="Thmproofx13.p1.2.m2.4.5.3.2.3.cmml"><mi id="Thmproofx13.p1.2.m2.4.5.3.2.3.2" xref="Thmproofx13.p1.2.m2.4.5.3.2.3.2.cmml">m</mi><mi id="Thmproofx13.p1.2.m2.4.5.3.2.3.3" xref="Thmproofx13.p1.2.m2.4.5.3.2.3.3.cmml">c</mi></msub></msub><mo id="Thmproofx13.p1.2.m2.4.5.3.1" xref="Thmproofx13.p1.2.m2.4.5.3.1.cmml">⁢</mo><mrow id="Thmproofx13.p1.2.m2.4.5.3.3.2" xref="Thmproofx13.p1.2.m2.4.5.3.3.1.cmml"><mo id="Thmproofx13.p1.2.m2.4.5.3.3.2.1" stretchy="false" xref="Thmproofx13.p1.2.m2.4.5.3.3.1.cmml">(</mo><mi id="Thmproofx13.p1.2.m2.3.3" xref="Thmproofx13.p1.2.m2.3.3.cmml">α</mi><mo id="Thmproofx13.p1.2.m2.4.5.3.3.2.2" xref="Thmproofx13.p1.2.m2.4.5.3.3.1.cmml">,</mo><mi id="Thmproofx13.p1.2.m2.4.4" xref="Thmproofx13.p1.2.m2.4.4.cmml">n</mi><mo id="Thmproofx13.p1.2.m2.4.5.3.3.2.3" stretchy="false" xref="Thmproofx13.p1.2.m2.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx13.p1.2.m2.4b"><apply id="Thmproofx13.p1.2.m2.4.5.cmml" xref="Thmproofx13.p1.2.m2.4.5"><eq id="Thmproofx13.p1.2.m2.4.5.1.cmml" xref="Thmproofx13.p1.2.m2.4.5.1"></eq><apply id="Thmproofx13.p1.2.m2.4.5.2.cmml" xref="Thmproofx13.p1.2.m2.4.5.2"><times id="Thmproofx13.p1.2.m2.4.5.2.1.cmml" xref="Thmproofx13.p1.2.m2.4.5.2.1"></times><ci id="Thmproofx13.p1.2.m2.4.5.2.2.cmml" xref="Thmproofx13.p1.2.m2.4.5.2.2">𝜑</ci><interval closure="open" id="Thmproofx13.p1.2.m2.4.5.2.3.1.cmml" xref="Thmproofx13.p1.2.m2.4.5.2.3.2"><ci id="Thmproofx13.p1.2.m2.1.1.cmml" xref="Thmproofx13.p1.2.m2.1.1">𝛼</ci><ci id="Thmproofx13.p1.2.m2.2.2.cmml" xref="Thmproofx13.p1.2.m2.2.2">𝑛</ci></interval></apply><apply id="Thmproofx13.p1.2.m2.4.5.3.cmml" xref="Thmproofx13.p1.2.m2.4.5.3"><times id="Thmproofx13.p1.2.m2.4.5.3.1.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.1"></times><apply id="Thmproofx13.p1.2.m2.4.5.3.2.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.2"><csymbol cd="ambiguous" id="Thmproofx13.p1.2.m2.4.5.3.2.1.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.2">subscript</csymbol><ci id="Thmproofx13.p1.2.m2.4.5.3.2.2.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.2.2">ℎ</ci><apply id="Thmproofx13.p1.2.m2.4.5.3.2.3.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.2.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.2.m2.4.5.3.2.3.1.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.2.3">subscript</csymbol><ci id="Thmproofx13.p1.2.m2.4.5.3.2.3.2.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.2.3.2">𝑚</ci><ci id="Thmproofx13.p1.2.m2.4.5.3.2.3.3.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.2.3.3">𝑐</ci></apply></apply><interval closure="open" id="Thmproofx13.p1.2.m2.4.5.3.3.1.cmml" xref="Thmproofx13.p1.2.m2.4.5.3.3.2"><ci id="Thmproofx13.p1.2.m2.3.3.cmml" xref="Thmproofx13.p1.2.m2.3.3">𝛼</ci><ci id="Thmproofx13.p1.2.m2.4.4.cmml" xref="Thmproofx13.p1.2.m2.4.4">𝑛</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx13.p1.2.m2.4c">\varphi(\alpha,n)=h_{m_{c}}(\alpha,n)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx13.p1.2.m2.4d">italic_φ ( italic_α , italic_n ) = italic_h start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_α , italic_n )</annotation></semantics></math>. This holds iff for every <math alttext="i<m_{c}" class="ltx_Math" display="inline" id="Thmproofx13.p1.3.m3.1"><semantics id="Thmproofx13.p1.3.m3.1a"><mrow id="Thmproofx13.p1.3.m3.1.1" xref="Thmproofx13.p1.3.m3.1.1.cmml"><mi id="Thmproofx13.p1.3.m3.1.1.2" xref="Thmproofx13.p1.3.m3.1.1.2.cmml">i</mi><mo id="Thmproofx13.p1.3.m3.1.1.1" xref="Thmproofx13.p1.3.m3.1.1.1.cmml"><</mo><msub id="Thmproofx13.p1.3.m3.1.1.3" xref="Thmproofx13.p1.3.m3.1.1.3.cmml"><mi id="Thmproofx13.p1.3.m3.1.1.3.2" xref="Thmproofx13.p1.3.m3.1.1.3.2.cmml">m</mi><mi id="Thmproofx13.p1.3.m3.1.1.3.3" xref="Thmproofx13.p1.3.m3.1.1.3.3.cmml">c</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx13.p1.3.m3.1b"><apply id="Thmproofx13.p1.3.m3.1.1.cmml" xref="Thmproofx13.p1.3.m3.1.1"><lt id="Thmproofx13.p1.3.m3.1.1.1.cmml" xref="Thmproofx13.p1.3.m3.1.1.1"></lt><ci id="Thmproofx13.p1.3.m3.1.1.2.cmml" xref="Thmproofx13.p1.3.m3.1.1.2">𝑖</ci><apply id="Thmproofx13.p1.3.m3.1.1.3.cmml" xref="Thmproofx13.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.3.m3.1.1.3.1.cmml" xref="Thmproofx13.p1.3.m3.1.1.3">subscript</csymbol><ci id="Thmproofx13.p1.3.m3.1.1.3.2.cmml" xref="Thmproofx13.p1.3.m3.1.1.3.2">𝑚</ci><ci id="Thmproofx13.p1.3.m3.1.1.3.3.cmml" xref="Thmproofx13.p1.3.m3.1.1.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx13.p1.3.m3.1c">i<m_{c}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx13.p1.3.m3.1d">italic_i < italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx38"> <tbody id="Pt0.A3.Ex58"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\alpha" class="ltx_Math" display="inline" id="Pt0.A3.Ex58.m1.1"><semantics id="Pt0.A3.Ex58.m1.1a"><mi id="Pt0.A3.Ex58.m1.1.1" xref="Pt0.A3.Ex58.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex58.m1.1b"><ci id="Pt0.A3.Ex58.m1.1.1.cmml" xref="Pt0.A3.Ex58.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex58.m1.1c">\displaystyle\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex58.m1.1d">italic_α</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\frac{1}{k}\frac{iS_{m_{c}}E_{m_{c}}-m_{c}S_{i}E_{i}}{m_{c}E_% {i}-iE_{m_{c}}}=\frac{1}{k}\frac{S_{m_{c}}\left(1-\left(1-\frac{1}{m_{c}}% \right)^{n}\right)-S_{i}\left(1-\left(1-\frac{1}{i}\right)^{n}\right)}{\left(1% -\frac{1}{m_{c}}\right)^{n}-\left(1-\frac{1}{i}\right)^{n}}" class="ltx_Math" display="inline" id="Pt0.A3.Ex58.m2.4"><semantics id="Pt0.A3.Ex58.m2.4a"><mrow id="Pt0.A3.Ex58.m2.4.5" xref="Pt0.A3.Ex58.m2.4.5.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.2" xref="Pt0.A3.Ex58.m2.4.5.2.cmml"></mi><mo id="Pt0.A3.Ex58.m2.4.5.3" xref="Pt0.A3.Ex58.m2.4.5.3.cmml">≥</mo><mrow id="Pt0.A3.Ex58.m2.4.5.4" xref="Pt0.A3.Ex58.m2.4.5.4.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex58.m2.4.5.4.2" xref="Pt0.A3.Ex58.m2.4.5.4.2.cmml"><mfrac id="Pt0.A3.Ex58.m2.4.5.4.2a" xref="Pt0.A3.Ex58.m2.4.5.4.2.cmml"><mn id="Pt0.A3.Ex58.m2.4.5.4.2.2" xref="Pt0.A3.Ex58.m2.4.5.4.2.2.cmml">1</mn><mi id="Pt0.A3.Ex58.m2.4.5.4.2.3" xref="Pt0.A3.Ex58.m2.4.5.4.2.3.cmml">k</mi></mfrac></mstyle><mo id="Pt0.A3.Ex58.m2.4.5.4.1" xref="Pt0.A3.Ex58.m2.4.5.4.1.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A3.Ex58.m2.4.5.4.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.cmml"><mfrac id="Pt0.A3.Ex58.m2.4.5.4.3a" xref="Pt0.A3.Ex58.m2.4.5.4.3.cmml"><mrow id="Pt0.A3.Ex58.m2.4.5.4.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.cmml"><mrow id="Pt0.A3.Ex58.m2.4.5.4.3.2.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.2.cmml">i</mi><mo id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.1" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.2.cmml">S</mi><msub id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.3.2.cmml">m</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.3.3.3.cmml">c</mi></msub></msub><mo id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.1a" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.2.cmml">E</mi><msub id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.3.2.cmml">m</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.2.4.3.3.cmml">c</mi></msub></msub></mrow><mo id="Pt0.A3.Ex58.m2.4.5.4.3.2.1" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.1.cmml">−</mo><mrow id="Pt0.A3.Ex58.m2.4.5.4.3.2.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.cmml"><msub id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.2.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.2.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.2.2.cmml">m</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.2.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.2.3.cmml">c</mi></msub><mo id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.1" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.1.cmml">⁢</mo><msub id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.3.2.cmml">S</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.3.3.cmml">i</mi></msub><mo id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.1a" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.1.cmml">⁢</mo><msub id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.4" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.4.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.4.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.4.2.cmml">E</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.2.3.4.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.2.3.4.3.cmml">i</mi></msub></mrow></mrow><mrow id="Pt0.A3.Ex58.m2.4.5.4.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.cmml"><mrow id="Pt0.A3.Ex58.m2.4.5.4.3.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.cmml"><msub id="Pt0.A3.Ex58.m2.4.5.4.3.3.2.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.2.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.2.2.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.2.2.cmml">m</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.2.2.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.2.3.cmml">c</mi></msub><mo id="Pt0.A3.Ex58.m2.4.5.4.3.3.2.1" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.1.cmml">⁢</mo><msub id="Pt0.A3.Ex58.m2.4.5.4.3.3.2.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.2.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.3.2.cmml">E</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.2.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.2.3.3.cmml">i</mi></msub></mrow><mo id="Pt0.A3.Ex58.m2.4.5.4.3.3.1" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.1.cmml">−</mo><mrow id="Pt0.A3.Ex58.m2.4.5.4.3.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.2.cmml">i</mi><mo id="Pt0.A3.Ex58.m2.4.5.4.3.3.3.1" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.1.cmml">⁢</mo><msub id="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.2.cmml">E</mi><msub id="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.3.cmml"><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.3.2" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.3.2.cmml">m</mi><mi id="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.3.3" xref="Pt0.A3.Ex58.m2.4.5.4.3.3.3.3.3.3.cmml">c</mi></msub></msub></mrow></mrow></mfrac></mstyle></mrow><mo id="Pt0.A3.Ex58.m2.4.5.5" xref="Pt0.A3.Ex58.m2.4.5.5.cmml">=</mo><mrow id="Pt0.A3.Ex58.m2.4.5.6" xref="Pt0.A3.Ex58.m2.4.5.6.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex58.m2.4.5.6.2" xref="Pt0.A3.Ex58.m2.4.5.6.2.cmml"><mfrac id="Pt0.A3.Ex58.m2.4.5.6.2a" xref="Pt0.A3.Ex58.m2.4.5.6.2.cmml"><mn id="Pt0.A3.Ex58.m2.4.5.6.2.2" xref="Pt0.A3.Ex58.m2.4.5.6.2.2.cmml">1</mn><mi id="Pt0.A3.Ex58.m2.4.5.6.2.3" xref="Pt0.A3.Ex58.m2.4.5.6.2.3.cmml">k</mi></mfrac></mstyle><mo id="Pt0.A3.Ex58.m2.4.5.6.1" xref="Pt0.A3.Ex58.m2.4.5.6.1.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A3.Ex58.m2.4.4" xref="Pt0.A3.Ex58.m2.4.4.cmml"><mfrac id="Pt0.A3.Ex58.m2.4.4a" xref="Pt0.A3.Ex58.m2.4.4.cmml"><mrow id="Pt0.A3.Ex58.m2.2.2.2" xref="Pt0.A3.Ex58.m2.2.2.2.cmml"><mrow id="Pt0.A3.Ex58.m2.1.1.1.1" xref="Pt0.A3.Ex58.m2.1.1.1.1.cmml"><msub id="Pt0.A3.Ex58.m2.1.1.1.1.3" xref="Pt0.A3.Ex58.m2.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex58.m2.1.1.1.1.3.2" xref="Pt0.A3.Ex58.m2.1.1.1.1.3.2.cmml">S</mi><msub id="Pt0.A3.Ex58.m2.1.1.1.1.3.3" xref="Pt0.A3.Ex58.m2.1.1.1.1.3.3.cmml"><mi id="Pt0.A3.Ex58.m2.1.1.1.1.3.3.2" 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xref="Pt0.A3.Ex58.m2.3.3.3.1.1.1.1.3.3.3">𝑐</ci></apply></apply></apply><ci id="Pt0.A3.Ex58.m2.3.3.3.1.3.cmml" xref="Pt0.A3.Ex58.m2.3.3.3.1.3">𝑛</ci></apply><apply id="Pt0.A3.Ex58.m2.4.4.4.2.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2"><csymbol cd="ambiguous" id="Pt0.A3.Ex58.m2.4.4.4.2.2.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2">superscript</csymbol><apply id="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2.1.1"><minus id="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.1.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.1"></minus><cn id="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.2.cmml" type="integer" xref="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.2">1</cn><apply id="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3"><divide id="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3.1.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3"></divide><cn id="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3.2.cmml" type="integer" xref="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3.2">1</cn><ci id="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3.3.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2.1.1.1.3.3">𝑖</ci></apply></apply><ci id="Pt0.A3.Ex58.m2.4.4.4.2.3.cmml" xref="Pt0.A3.Ex58.m2.4.4.4.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex58.m2.4c">\displaystyle\geq\frac{1}{k}\frac{iS_{m_{c}}E_{m_{c}}-m_{c}S_{i}E_{i}}{m_{c}E_% {i}-iE_{m_{c}}}=\frac{1}{k}\frac{S_{m_{c}}\left(1-\left(1-\frac{1}{m_{c}}% \right)^{n}\right)-S_{i}\left(1-\left(1-\frac{1}{i}\right)^{n}\right)}{\left(1% -\frac{1}{m_{c}}\right)^{n}-\left(1-\frac{1}{i}\right)^{n}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex58.m2.4d">≥ divide start_ARG 1 end_ARG start_ARG italic_k end_ARG divide start_ARG italic_i italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_i italic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG italic_k end_ARG divide start_ARG italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_i end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_ARG start_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_i end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx13.p1.5">Rhs is lower-bounded by <math alttext="\frac{1}{k}\frac{S_{m_{c}}\left(1-\left(1-\frac{1}{m_{c}}\right)^{n}\right)-S_% {m_{c}-1}}{\left(1-\frac{1}{m_{c}}\right)^{n}}" class="ltx_Math" display="inline" id="Thmproofx13.p1.4.m1.2"><semantics id="Thmproofx13.p1.4.m1.2a"><mrow id="Thmproofx13.p1.4.m1.2.3" xref="Thmproofx13.p1.4.m1.2.3.cmml"><mfrac id="Thmproofx13.p1.4.m1.2.3.2" xref="Thmproofx13.p1.4.m1.2.3.2.cmml"><mn id="Thmproofx13.p1.4.m1.2.3.2.2" xref="Thmproofx13.p1.4.m1.2.3.2.2.cmml">1</mn><mi id="Thmproofx13.p1.4.m1.2.3.2.3" xref="Thmproofx13.p1.4.m1.2.3.2.3.cmml">k</mi></mfrac><mo id="Thmproofx13.p1.4.m1.2.3.1" 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xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx13.p1.4.m1.2.2.2.1.1.1" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.cmml"><mn id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.2" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.2.cmml">1</mn><mo id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.1" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.cmml"><mn id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.2" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.2.cmml">1</mn><msub id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.cmml"><mi id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.2" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.2.cmml">m</mi><mi id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.3" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.3.cmml">c</mi></msub></mfrac></mrow><mo id="Thmproofx13.p1.4.m1.2.2.2.1.1.3" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx13.p1.4.m1.2.2.2.3" xref="Thmproofx13.p1.4.m1.2.2.2.3.cmml">n</mi></msup></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx13.p1.4.m1.2b"><apply id="Thmproofx13.p1.4.m1.2.3.cmml" xref="Thmproofx13.p1.4.m1.2.3"><times id="Thmproofx13.p1.4.m1.2.3.1.cmml" xref="Thmproofx13.p1.4.m1.2.3.1"></times><apply id="Thmproofx13.p1.4.m1.2.3.2.cmml" xref="Thmproofx13.p1.4.m1.2.3.2"><divide id="Thmproofx13.p1.4.m1.2.3.2.1.cmml" xref="Thmproofx13.p1.4.m1.2.3.2"></divide><cn id="Thmproofx13.p1.4.m1.2.3.2.2.cmml" type="integer" xref="Thmproofx13.p1.4.m1.2.3.2.2">1</cn><ci id="Thmproofx13.p1.4.m1.2.3.2.3.cmml" xref="Thmproofx13.p1.4.m1.2.3.2.3">𝑘</ci></apply><apply id="Thmproofx13.p1.4.m1.2.2.cmml" xref="Thmproofx13.p1.4.m1.2.2"><divide id="Thmproofx13.p1.4.m1.2.2.3.cmml" xref="Thmproofx13.p1.4.m1.2.2"></divide><apply id="Thmproofx13.p1.4.m1.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1"><minus id="Thmproofx13.p1.4.m1.1.1.1.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.2"></minus><apply id="Thmproofx13.p1.4.m1.1.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1"><times id="Thmproofx13.p1.4.m1.1.1.1.1.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.2"></times><apply id="Thmproofx13.p1.4.m1.1.1.1.1.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.1.1.1.1.3.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.3">subscript</csymbol><ci id="Thmproofx13.p1.4.m1.1.1.1.1.3.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.3.2">𝑆</ci><apply id="Thmproofx13.p1.4.m1.1.1.1.1.3.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.1.1.1.1.3.3.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.3.3">subscript</csymbol><ci id="Thmproofx13.p1.4.m1.1.1.1.1.3.3.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.3.3.2">𝑚</ci><ci id="Thmproofx13.p1.4.m1.1.1.1.1.3.3.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.3.3.3">𝑐</ci></apply></apply><apply id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1"><minus id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.2"></minus><cn id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.3">1</cn><apply id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1"><minus id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.1"></minus><cn id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3"><divide id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3"></divide><cn id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.2">1</cn><apply id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.2">𝑚</ci><ci id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.3">𝑐</ci></apply></apply></apply><ci id="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmproofx13.p1.4.m1.1.1.1.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.1.1.1.3.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3">subscript</csymbol><ci id="Thmproofx13.p1.4.m1.1.1.1.3.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3.2">𝑆</ci><apply id="Thmproofx13.p1.4.m1.1.1.1.3.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3.3"><minus id="Thmproofx13.p1.4.m1.1.1.1.3.3.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3.3.1"></minus><apply id="Thmproofx13.p1.4.m1.1.1.1.3.3.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.1.1.1.3.3.2.1.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3.3.2">subscript</csymbol><ci id="Thmproofx13.p1.4.m1.1.1.1.3.3.2.2.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3.3.2.2">𝑚</ci><ci id="Thmproofx13.p1.4.m1.1.1.1.3.3.2.3.cmml" xref="Thmproofx13.p1.4.m1.1.1.1.3.3.2.3">𝑐</ci></apply><cn id="Thmproofx13.p1.4.m1.1.1.1.3.3.3.cmml" type="integer" xref="Thmproofx13.p1.4.m1.1.1.1.3.3.3">1</cn></apply></apply></apply><apply id="Thmproofx13.p1.4.m1.2.2.2.cmml" xref="Thmproofx13.p1.4.m1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.2.2.2.2.cmml" xref="Thmproofx13.p1.4.m1.2.2.2">superscript</csymbol><apply id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1"><minus id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.1.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.1"></minus><cn id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.2.cmml" type="integer" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.2">1</cn><apply id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3"><divide id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3"></divide><cn id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.2.cmml" type="integer" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.2">1</cn><apply id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.1.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3">subscript</csymbol><ci id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.2.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.2">𝑚</ci><ci id="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.3.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.1.1.1.3.3.3">𝑐</ci></apply></apply></apply><ci id="Thmproofx13.p1.4.m1.2.2.2.3.cmml" xref="Thmproofx13.p1.4.m1.2.2.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx13.p1.4.m1.2c">\frac{1}{k}\frac{S_{m_{c}}\left(1-\left(1-\frac{1}{m_{c}}\right)^{n}\right)-S_% {m_{c}-1}}{\left(1-\frac{1}{m_{c}}\right)^{n}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx13.p1.4.m1.2d">divide start_ARG 1 end_ARG start_ARG italic_k end_ARG divide start_ARG italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( 1 - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT end_ARG start_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>. Using the well-known equality <math alttext="\left(1-\frac{1}{m_{c}}\right)^{n}=e^{-\Theta(n/m_{c})}" class="ltx_Math" display="inline" id="Thmproofx13.p1.5.m2.2"><semantics id="Thmproofx13.p1.5.m2.2a"><mrow id="Thmproofx13.p1.5.m2.2.2" xref="Thmproofx13.p1.5.m2.2.2.cmml"><msup id="Thmproofx13.p1.5.m2.2.2.1" xref="Thmproofx13.p1.5.m2.2.2.1.cmml"><mrow id="Thmproofx13.p1.5.m2.2.2.1.1.1" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.cmml"><mo id="Thmproofx13.p1.5.m2.2.2.1.1.1.2" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.cmml">(</mo><mrow id="Thmproofx13.p1.5.m2.2.2.1.1.1.1" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.cmml"><mn id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.2" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.1" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.cmml"><mn id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.2" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.2.cmml">1</mn><msub id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.cmml"><mi id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.2" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.2.cmml">m</mi><mi id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.3" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.3.cmml">c</mi></msub></mfrac></mrow><mo id="Thmproofx13.p1.5.m2.2.2.1.1.1.3" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx13.p1.5.m2.2.2.1.3" xref="Thmproofx13.p1.5.m2.2.2.1.3.cmml">n</mi></msup><mo id="Thmproofx13.p1.5.m2.2.2.2" xref="Thmproofx13.p1.5.m2.2.2.2.cmml">=</mo><msup id="Thmproofx13.p1.5.m2.2.2.3" xref="Thmproofx13.p1.5.m2.2.2.3.cmml"><mi id="Thmproofx13.p1.5.m2.2.2.3.2" xref="Thmproofx13.p1.5.m2.2.2.3.2.cmml">e</mi><mrow id="Thmproofx13.p1.5.m2.1.1.1" xref="Thmproofx13.p1.5.m2.1.1.1.cmml"><mo id="Thmproofx13.p1.5.m2.1.1.1a" xref="Thmproofx13.p1.5.m2.1.1.1.cmml">−</mo><mrow id="Thmproofx13.p1.5.m2.1.1.1.1" xref="Thmproofx13.p1.5.m2.1.1.1.1.cmml"><mi id="Thmproofx13.p1.5.m2.1.1.1.1.3" mathvariant="normal" xref="Thmproofx13.p1.5.m2.1.1.1.1.3.cmml">Θ</mi><mo id="Thmproofx13.p1.5.m2.1.1.1.1.2" xref="Thmproofx13.p1.5.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx13.p1.5.m2.1.1.1.1.1.1" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.2" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.2.cmml">n</mi><mo id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.1" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.1.cmml">/</mo><msub id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.2" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.2.cmml">m</mi><mi id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.3" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.3.cmml">c</mi></msub></mrow><mo id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx13.p1.5.m2.2b"><apply id="Thmproofx13.p1.5.m2.2.2.cmml" xref="Thmproofx13.p1.5.m2.2.2"><eq id="Thmproofx13.p1.5.m2.2.2.2.cmml" xref="Thmproofx13.p1.5.m2.2.2.2"></eq><apply id="Thmproofx13.p1.5.m2.2.2.1.cmml" xref="Thmproofx13.p1.5.m2.2.2.1"><csymbol cd="ambiguous" id="Thmproofx13.p1.5.m2.2.2.1.2.cmml" xref="Thmproofx13.p1.5.m2.2.2.1">superscript</csymbol><apply id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1"><minus id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.1.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.1"></minus><cn id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.2.cmml" type="integer" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.2">1</cn><apply id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3"><divide id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.1.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3"></divide><cn id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.2">1</cn><apply id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.1.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3">subscript</csymbol><ci id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.2.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.2">𝑚</ci><ci id="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.3.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.1.1.1.3.3.3">𝑐</ci></apply></apply></apply><ci id="Thmproofx13.p1.5.m2.2.2.1.3.cmml" xref="Thmproofx13.p1.5.m2.2.2.1.3">𝑛</ci></apply><apply id="Thmproofx13.p1.5.m2.2.2.3.cmml" xref="Thmproofx13.p1.5.m2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.5.m2.2.2.3.1.cmml" xref="Thmproofx13.p1.5.m2.2.2.3">superscript</csymbol><ci id="Thmproofx13.p1.5.m2.2.2.3.2.cmml" xref="Thmproofx13.p1.5.m2.2.2.3.2">𝑒</ci><apply id="Thmproofx13.p1.5.m2.1.1.1.cmml" xref="Thmproofx13.p1.5.m2.1.1.1"><minus id="Thmproofx13.p1.5.m2.1.1.1.2.cmml" xref="Thmproofx13.p1.5.m2.1.1.1"></minus><apply id="Thmproofx13.p1.5.m2.1.1.1.1.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1"><times id="Thmproofx13.p1.5.m2.1.1.1.1.2.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.2"></times><ci id="Thmproofx13.p1.5.m2.1.1.1.1.3.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.3">Θ</ci><apply id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1"><divide id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.1"></divide><ci id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.2">𝑛</ci><apply id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx13.p1.5.m2.1.1.1.1.1.1.1.3.3">𝑐</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx13.p1.5.m2.2c">\left(1-\frac{1}{m_{c}}\right)^{n}=e^{-\Theta(n/m_{c})}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx13.p1.5.m2.2d">( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT - roman_Θ ( italic_n / italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math> we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx39"> <tbody id="Pt0.A3.Ex59"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{k}\frac{S_{m_{c}}\left(1-e^{-\Theta(n/m_{c})}\right)-S_% {m_{c}-1}}{e^{-\Theta(n/m_{c})}}" class="ltx_Math" display="inline" id="Pt0.A3.Ex59.m1.3"><semantics id="Pt0.A3.Ex59.m1.3a"><mrow id="Pt0.A3.Ex59.m1.3.4" xref="Pt0.A3.Ex59.m1.3.4.cmml"><mi id="Pt0.A3.Ex59.m1.3.4.2" xref="Pt0.A3.Ex59.m1.3.4.2.cmml"></mi><mo id="Pt0.A3.Ex59.m1.3.4.1" xref="Pt0.A3.Ex59.m1.3.4.1.cmml">=</mo><mrow id="Pt0.A3.Ex59.m1.3.4.3" xref="Pt0.A3.Ex59.m1.3.4.3.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex59.m1.3.4.3.2" xref="Pt0.A3.Ex59.m1.3.4.3.2.cmml"><mfrac id="Pt0.A3.Ex59.m1.3.4.3.2a" xref="Pt0.A3.Ex59.m1.3.4.3.2.cmml"><mn id="Pt0.A3.Ex59.m1.3.4.3.2.2" xref="Pt0.A3.Ex59.m1.3.4.3.2.2.cmml">1</mn><mi id="Pt0.A3.Ex59.m1.3.4.3.2.3" xref="Pt0.A3.Ex59.m1.3.4.3.2.3.cmml">k</mi></mfrac></mstyle><mo id="Pt0.A3.Ex59.m1.3.4.3.1" xref="Pt0.A3.Ex59.m1.3.4.3.1.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A3.Ex59.m1.3.3" xref="Pt0.A3.Ex59.m1.3.3.cmml"><mfrac id="Pt0.A3.Ex59.m1.3.3a" xref="Pt0.A3.Ex59.m1.3.3.cmml"><mrow id="Pt0.A3.Ex59.m1.2.2.2" xref="Pt0.A3.Ex59.m1.2.2.2.cmml"><mrow id="Pt0.A3.Ex59.m1.2.2.2.2" xref="Pt0.A3.Ex59.m1.2.2.2.2.cmml"><msub id="Pt0.A3.Ex59.m1.2.2.2.2.3" xref="Pt0.A3.Ex59.m1.2.2.2.2.3.cmml"><mi id="Pt0.A3.Ex59.m1.2.2.2.2.3.2" xref="Pt0.A3.Ex59.m1.2.2.2.2.3.2.cmml">S</mi><msub id="Pt0.A3.Ex59.m1.2.2.2.2.3.3" xref="Pt0.A3.Ex59.m1.2.2.2.2.3.3.cmml"><mi id="Pt0.A3.Ex59.m1.2.2.2.2.3.3.2" xref="Pt0.A3.Ex59.m1.2.2.2.2.3.3.2.cmml">m</mi><mi id="Pt0.A3.Ex59.m1.2.2.2.2.3.3.3" xref="Pt0.A3.Ex59.m1.2.2.2.2.3.3.3.cmml">c</mi></msub></msub><mo id="Pt0.A3.Ex59.m1.2.2.2.2.2" xref="Pt0.A3.Ex59.m1.2.2.2.2.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex59.m1.2.2.2.2.1.1" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.cmml"><mo id="Pt0.A3.Ex59.m1.2.2.2.2.1.1.2" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.cmml"><mn id="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.2" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.1" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.1.cmml">−</mo><msup id="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.3" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.3.cmml"><mi id="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.3.2" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.3.2.cmml">e</mi><mrow id="Pt0.A3.Ex59.m1.1.1.1.1.1" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex59.m1.1.1.1.1.1a" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.cmml">−</mo><mrow id="Pt0.A3.Ex59.m1.1.1.1.1.1.1" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.cmml"><mi id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.3" mathvariant="normal" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.3.cmml">Θ</mi><mo id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.2" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.2.cmml">n</mi><mo id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.1.cmml">/</mo><msub id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">m</mi><mi id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">c</mi></msub></mrow><mo id="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A3.Ex59.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="Pt0.A3.Ex59.m1.2.2.2.2.1.1.3" xref="Pt0.A3.Ex59.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A3.Ex59.m1.2.2.2.3" xref="Pt0.A3.Ex59.m1.2.2.2.3.cmml">−</mo><msub id="Pt0.A3.Ex59.m1.2.2.2.4" xref="Pt0.A3.Ex59.m1.2.2.2.4.cmml"><mi id="Pt0.A3.Ex59.m1.2.2.2.4.2" xref="Pt0.A3.Ex59.m1.2.2.2.4.2.cmml">S</mi><mrow id="Pt0.A3.Ex59.m1.2.2.2.4.3" xref="Pt0.A3.Ex59.m1.2.2.2.4.3.cmml"><msub id="Pt0.A3.Ex59.m1.2.2.2.4.3.2" xref="Pt0.A3.Ex59.m1.2.2.2.4.3.2.cmml"><mi id="Pt0.A3.Ex59.m1.2.2.2.4.3.2.2" xref="Pt0.A3.Ex59.m1.2.2.2.4.3.2.2.cmml">m</mi><mi id="Pt0.A3.Ex59.m1.2.2.2.4.3.2.3" xref="Pt0.A3.Ex59.m1.2.2.2.4.3.2.3.cmml">c</mi></msub><mo id="Pt0.A3.Ex59.m1.2.2.2.4.3.1" xref="Pt0.A3.Ex59.m1.2.2.2.4.3.1.cmml">−</mo><mn id="Pt0.A3.Ex59.m1.2.2.2.4.3.3" xref="Pt0.A3.Ex59.m1.2.2.2.4.3.3.cmml">1</mn></mrow></msub></mrow><msup id="Pt0.A3.Ex59.m1.3.3.3" xref="Pt0.A3.Ex59.m1.3.3.3.cmml"><mi id="Pt0.A3.Ex59.m1.3.3.3.3" xref="Pt0.A3.Ex59.m1.3.3.3.3.cmml">e</mi><mrow id="Pt0.A3.Ex59.m1.3.3.3.1.1" xref="Pt0.A3.Ex59.m1.3.3.3.1.1.cmml"><mo id="Pt0.A3.Ex59.m1.3.3.3.1.1a" xref="Pt0.A3.Ex59.m1.3.3.3.1.1.cmml">−</mo><mrow id="Pt0.A3.Ex59.m1.3.3.3.1.1.1" xref="Pt0.A3.Ex59.m1.3.3.3.1.1.1.cmml"><mi id="Pt0.A3.Ex59.m1.3.3.3.1.1.1.3" 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id="Pt0.A3.Ex59.m1.3.3.3.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex59.m1.3.3.3.1.1.1.1.1.1.3.3">𝑐</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex59.m1.3c">\displaystyle=\frac{1}{k}\frac{S_{m_{c}}\left(1-e^{-\Theta(n/m_{c})}\right)-S_% {m_{c}-1}}{e^{-\Theta(n/m_{c})}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex59.m1.3d">= divide start_ARG 1 end_ARG start_ARG italic_k end_ARG divide start_ARG italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( italic_n / italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_e start_POSTSUPERSCRIPT - roman_Θ ( italic_n / italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) 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xref="Pt0.A3.Ex60.m1.2.2.1.1.1.1.2">𝑛</ci><apply id="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3.cmml" xref="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3.1.cmml" xref="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3">subscript</csymbol><ci id="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3.2.cmml" xref="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3.2">𝑚</ci><ci id="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3.3.cmml" xref="Pt0.A3.Ex60.m1.2.2.1.1.1.1.3.3">𝑐</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex60.m1.3c">\displaystyle=\frac{1}{k}\left(S_{m_{c}}\left(e^{\Theta(n/m_{c})}-1\right)-S_{% m_{c}-1}e^{\Theta(n/m_{c})}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex60.m1.3d">= divide start_ARG 1 end_ARG start_ARG italic_k end_ARG ( italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT roman_Θ ( italic_n / italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT - 1 ) - italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT roman_Θ ( italic_n / italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex61"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{e^{\Theta(n/m_{c})}\left(S_{m_{c}}-S_{m_{c}-1}\right)-S_{m% _{c}}}{k}" class="ltx_Math" display="inline" id="Pt0.A3.Ex61.m1.2"><semantics id="Pt0.A3.Ex61.m1.2a"><mrow id="Pt0.A3.Ex61.m1.2.3" xref="Pt0.A3.Ex61.m1.2.3.cmml"><mi id="Pt0.A3.Ex61.m1.2.3.2" xref="Pt0.A3.Ex61.m1.2.3.2.cmml"></mi><mo id="Pt0.A3.Ex61.m1.2.3.1" xref="Pt0.A3.Ex61.m1.2.3.1.cmml">=</mo><mstyle displaystyle="true" id="Pt0.A3.Ex61.m1.2.2" xref="Pt0.A3.Ex61.m1.2.2.cmml"><mfrac id="Pt0.A3.Ex61.m1.2.2a" xref="Pt0.A3.Ex61.m1.2.2.cmml"><mrow id="Pt0.A3.Ex61.m1.2.2.2" xref="Pt0.A3.Ex61.m1.2.2.2.cmml"><mrow id="Pt0.A3.Ex61.m1.2.2.2.2" xref="Pt0.A3.Ex61.m1.2.2.2.2.cmml"><msup id="Pt0.A3.Ex61.m1.2.2.2.2.3" xref="Pt0.A3.Ex61.m1.2.2.2.2.3.cmml"><mi id="Pt0.A3.Ex61.m1.2.2.2.2.3.2" xref="Pt0.A3.Ex61.m1.2.2.2.2.3.2.cmml">e</mi><mrow id="Pt0.A3.Ex61.m1.1.1.1.1.1" xref="Pt0.A3.Ex61.m1.1.1.1.1.1.cmml"><mi id="Pt0.A3.Ex61.m1.1.1.1.1.1.3" mathvariant="normal" xref="Pt0.A3.Ex61.m1.1.1.1.1.1.3.cmml">Θ</mi><mo id="Pt0.A3.Ex61.m1.1.1.1.1.1.2" xref="Pt0.A3.Ex61.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex61.m1.1.1.1.1.1.1.1" xref="Pt0.A3.Ex61.m1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex61.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex61.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex61.m1.1.1.1.1.1.1.1.1" 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xref="Pt0.A3.Ex61.m1.2.2.4.cmml">k</mi></mfrac></mstyle></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A3.Ex61.m1.2b"><apply id="Pt0.A3.Ex61.m1.2.3.cmml" xref="Pt0.A3.Ex61.m1.2.3"><eq id="Pt0.A3.Ex61.m1.2.3.1.cmml" xref="Pt0.A3.Ex61.m1.2.3.1"></eq><csymbol cd="latexml" id="Pt0.A3.Ex61.m1.2.3.2.cmml" xref="Pt0.A3.Ex61.m1.2.3.2">absent</csymbol><apply id="Pt0.A3.Ex61.m1.2.2.cmml" xref="Pt0.A3.Ex61.m1.2.2"><divide id="Pt0.A3.Ex61.m1.2.2.3.cmml" xref="Pt0.A3.Ex61.m1.2.2"></divide><apply id="Pt0.A3.Ex61.m1.2.2.2.cmml" xref="Pt0.A3.Ex61.m1.2.2.2"><minus id="Pt0.A3.Ex61.m1.2.2.2.3.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.3"></minus><apply id="Pt0.A3.Ex61.m1.2.2.2.2.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.2"><times id="Pt0.A3.Ex61.m1.2.2.2.2.2.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.2.2"></times><apply id="Pt0.A3.Ex61.m1.2.2.2.2.3.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.2.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex61.m1.2.2.2.2.3.1.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.2.3">superscript</csymbol><ci 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xref="Pt0.A3.Ex61.m1.2.2.2.2.1.1.1.3.3.3">1</cn></apply></apply></apply></apply><apply id="Pt0.A3.Ex61.m1.2.2.2.4.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.4"><csymbol cd="ambiguous" id="Pt0.A3.Ex61.m1.2.2.2.4.1.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.4">subscript</csymbol><ci id="Pt0.A3.Ex61.m1.2.2.2.4.2.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.4.2">𝑆</ci><apply id="Pt0.A3.Ex61.m1.2.2.2.4.3.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.4.3"><csymbol cd="ambiguous" id="Pt0.A3.Ex61.m1.2.2.2.4.3.1.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.4.3">subscript</csymbol><ci id="Pt0.A3.Ex61.m1.2.2.2.4.3.2.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.4.3.2">𝑚</ci><ci id="Pt0.A3.Ex61.m1.2.2.2.4.3.3.cmml" xref="Pt0.A3.Ex61.m1.2.2.2.4.3.3">𝑐</ci></apply></apply></apply><ci id="Pt0.A3.Ex61.m1.2.2.4.cmml" xref="Pt0.A3.Ex61.m1.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex61.m1.2c">\displaystyle=\frac{e^{\Theta(n/m_{c})}\left(S_{m_{c}}-S_{m_{c}-1}\right)-S_{m% _{c}}}{k}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex61.m1.2d">= divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Θ ( italic_n / italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx13.p1.6">∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx14"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma9" title="Lemma 9 ‣ 4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">9</span></a>)</h6> <div class="ltx_para" id="Thmproofx14.p1"> <p class="ltx_p" id="Thmproofx14.p1.11">Let <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmproofx14.p1.1.m1.1"><semantics id="Thmproofx14.p1.1.m1.1a"><mi id="Thmproofx14.p1.1.m1.1.1" xref="Thmproofx14.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.1.m1.1b"><ci id="Thmproofx14.p1.1.m1.1.1.cmml" xref="Thmproofx14.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.1.m1.1d">italic_σ</annotation></semantics></math> and <math alttext="|p_{i}|<|p_{i^{\prime}}|" class="ltx_Math" display="inline" id="Thmproofx14.p1.2.m2.2"><semantics id="Thmproofx14.p1.2.m2.2a"><mrow id="Thmproofx14.p1.2.m2.2.2" xref="Thmproofx14.p1.2.m2.2.2.cmml"><mrow id="Thmproofx14.p1.2.m2.1.1.1.1" xref="Thmproofx14.p1.2.m2.1.1.1.2.cmml"><mo id="Thmproofx14.p1.2.m2.1.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.2.m2.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx14.p1.2.m2.1.1.1.1.1" xref="Thmproofx14.p1.2.m2.1.1.1.1.1.cmml"><mi id="Thmproofx14.p1.2.m2.1.1.1.1.1.2" xref="Thmproofx14.p1.2.m2.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.2.m2.1.1.1.1.1.3" xref="Thmproofx14.p1.2.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.2.m2.1.1.1.1.3" stretchy="false" xref="Thmproofx14.p1.2.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx14.p1.2.m2.2.2.3" xref="Thmproofx14.p1.2.m2.2.2.3.cmml"><</mo><mrow id="Thmproofx14.p1.2.m2.2.2.2.1" xref="Thmproofx14.p1.2.m2.2.2.2.2.cmml"><mo id="Thmproofx14.p1.2.m2.2.2.2.1.2" stretchy="false" xref="Thmproofx14.p1.2.m2.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx14.p1.2.m2.2.2.2.1.1" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.cmml"><mi id="Thmproofx14.p1.2.m2.2.2.2.1.1.2" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.2.cmml">p</mi><msup id="Thmproofx14.p1.2.m2.2.2.2.1.1.3" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.3.cmml"><mi id="Thmproofx14.p1.2.m2.2.2.2.1.1.3.2" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.3.2.cmml">i</mi><mo id="Thmproofx14.p1.2.m2.2.2.2.1.1.3.3" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx14.p1.2.m2.2.2.2.1.3" stretchy="false" xref="Thmproofx14.p1.2.m2.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.2.m2.2b"><apply id="Thmproofx14.p1.2.m2.2.2.cmml" xref="Thmproofx14.p1.2.m2.2.2"><lt id="Thmproofx14.p1.2.m2.2.2.3.cmml" xref="Thmproofx14.p1.2.m2.2.2.3"></lt><apply id="Thmproofx14.p1.2.m2.1.1.1.2.cmml" xref="Thmproofx14.p1.2.m2.1.1.1.1"><abs id="Thmproofx14.p1.2.m2.1.1.1.2.1.cmml" xref="Thmproofx14.p1.2.m2.1.1.1.1.2"></abs><apply id="Thmproofx14.p1.2.m2.1.1.1.1.1.cmml" xref="Thmproofx14.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.2.m2.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.2.m2.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.2.m2.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.2.m2.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.2.m2.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx14.p1.2.m2.2.2.2.2.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1"><abs id="Thmproofx14.p1.2.m2.2.2.2.2.1.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.2"></abs><apply id="Thmproofx14.p1.2.m2.2.2.2.1.1.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.2.m2.2.2.2.1.1.1.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx14.p1.2.m2.2.2.2.1.1.2.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.2.m2.2.2.2.1.1.3.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.2.m2.2.2.2.1.1.3.1.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.2.m2.2.2.2.1.1.3.2.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.2.m2.2.2.2.1.1.3.3.cmml" xref="Thmproofx14.p1.2.m2.2.2.2.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.2.m2.2c">|p_{i}|<|p_{i^{\prime}}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.2.m2.2d">| italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math>. For the distribution <math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="Thmproofx14.p1.3.m3.1"><semantics id="Thmproofx14.p1.3.m3.1a"><msup id="Thmproofx14.p1.3.m3.1.1" xref="Thmproofx14.p1.3.m3.1.1.cmml"><mi id="Thmproofx14.p1.3.m3.1.1.2" xref="Thmproofx14.p1.3.m3.1.1.2.cmml">σ</mi><mo id="Thmproofx14.p1.3.m3.1.1.3" xref="Thmproofx14.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.3.m3.1b"><apply id="Thmproofx14.p1.3.m3.1.1.cmml" xref="Thmproofx14.p1.3.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.3.m3.1.1.1.cmml" xref="Thmproofx14.p1.3.m3.1.1">superscript</csymbol><ci id="Thmproofx14.p1.3.m3.1.1.2.cmml" xref="Thmproofx14.p1.3.m3.1.1.2">𝜎</ci><ci id="Thmproofx14.p1.3.m3.1.1.3.cmml" xref="Thmproofx14.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.3.m3.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.3.m3.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> that results from <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmproofx14.p1.4.m4.1"><semantics id="Thmproofx14.p1.4.m4.1a"><mi id="Thmproofx14.p1.4.m4.1.1" xref="Thmproofx14.p1.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.4.m4.1b"><ci id="Thmproofx14.p1.4.m4.1.1.cmml" xref="Thmproofx14.p1.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.4.m4.1d">italic_σ</annotation></semantics></math> by swapping the weights on <math alttext="p_{i},p_{i}" class="ltx_Math" display="inline" id="Thmproofx14.p1.5.m5.2"><semantics id="Thmproofx14.p1.5.m5.2a"><mrow id="Thmproofx14.p1.5.m5.2.2.2" xref="Thmproofx14.p1.5.m5.2.2.3.cmml"><msub id="Thmproofx14.p1.5.m5.1.1.1.1" xref="Thmproofx14.p1.5.m5.1.1.1.1.cmml"><mi id="Thmproofx14.p1.5.m5.1.1.1.1.2" xref="Thmproofx14.p1.5.m5.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.5.m5.1.1.1.1.3" xref="Thmproofx14.p1.5.m5.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.5.m5.2.2.2.3" xref="Thmproofx14.p1.5.m5.2.2.3.cmml">,</mo><msub id="Thmproofx14.p1.5.m5.2.2.2.2" xref="Thmproofx14.p1.5.m5.2.2.2.2.cmml"><mi id="Thmproofx14.p1.5.m5.2.2.2.2.2" xref="Thmproofx14.p1.5.m5.2.2.2.2.2.cmml">p</mi><mi id="Thmproofx14.p1.5.m5.2.2.2.2.3" xref="Thmproofx14.p1.5.m5.2.2.2.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.5.m5.2b"><list id="Thmproofx14.p1.5.m5.2.2.3.cmml" xref="Thmproofx14.p1.5.m5.2.2.2"><apply id="Thmproofx14.p1.5.m5.1.1.1.1.cmml" xref="Thmproofx14.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.5.m5.1.1.1.1.1.cmml" xref="Thmproofx14.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.5.m5.1.1.1.1.2.cmml" xref="Thmproofx14.p1.5.m5.1.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.5.m5.1.1.1.1.3.cmml" xref="Thmproofx14.p1.5.m5.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx14.p1.5.m5.2.2.2.2.cmml" xref="Thmproofx14.p1.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx14.p1.5.m5.2.2.2.2.1.cmml" xref="Thmproofx14.p1.5.m5.2.2.2.2">subscript</csymbol><ci id="Thmproofx14.p1.5.m5.2.2.2.2.2.cmml" xref="Thmproofx14.p1.5.m5.2.2.2.2.2">𝑝</ci><ci id="Thmproofx14.p1.5.m5.2.2.2.2.3.cmml" xref="Thmproofx14.p1.5.m5.2.2.2.2.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.5.m5.2c">p_{i},p_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.5.m5.2d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> we have <math alttext="F_{n}(\sigma,\alpha)-F_{n}(\sigma^{\prime},\alpha)=\sigma(p_{i})(|p_{i}|-|p_{i% ^{\prime}}|)+\sigma(p_{i^{\prime}})(|p_{i^{\prime}}|-|p_{i}|)=(|p_{i^{\prime}}% |-|p_{i}|)(\sigma(p_{i^{\prime}})-\sigma(p_{i}))" class="ltx_Math" display="inline" id="Thmproofx14.p1.6.m6.10"><semantics id="Thmproofx14.p1.6.m6.10a"><mrow id="Thmproofx14.p1.6.m6.10.10" xref="Thmproofx14.p1.6.m6.10.10.cmml"><mrow id="Thmproofx14.p1.6.m6.4.4.1" xref="Thmproofx14.p1.6.m6.4.4.1.cmml"><mrow id="Thmproofx14.p1.6.m6.4.4.1.3" xref="Thmproofx14.p1.6.m6.4.4.1.3.cmml"><msub id="Thmproofx14.p1.6.m6.4.4.1.3.2" xref="Thmproofx14.p1.6.m6.4.4.1.3.2.cmml"><mi id="Thmproofx14.p1.6.m6.4.4.1.3.2.2" xref="Thmproofx14.p1.6.m6.4.4.1.3.2.2.cmml">F</mi><mi id="Thmproofx14.p1.6.m6.4.4.1.3.2.3" xref="Thmproofx14.p1.6.m6.4.4.1.3.2.3.cmml">n</mi></msub><mo id="Thmproofx14.p1.6.m6.4.4.1.3.1" xref="Thmproofx14.p1.6.m6.4.4.1.3.1.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.4.4.1.3.3.2" xref="Thmproofx14.p1.6.m6.4.4.1.3.3.1.cmml"><mo id="Thmproofx14.p1.6.m6.4.4.1.3.3.2.1" stretchy="false" xref="Thmproofx14.p1.6.m6.4.4.1.3.3.1.cmml">(</mo><mi id="Thmproofx14.p1.6.m6.1.1" xref="Thmproofx14.p1.6.m6.1.1.cmml">σ</mi><mo id="Thmproofx14.p1.6.m6.4.4.1.3.3.2.2" xref="Thmproofx14.p1.6.m6.4.4.1.3.3.1.cmml">,</mo><mi id="Thmproofx14.p1.6.m6.2.2" xref="Thmproofx14.p1.6.m6.2.2.cmml">α</mi><mo id="Thmproofx14.p1.6.m6.4.4.1.3.3.2.3" stretchy="false" xref="Thmproofx14.p1.6.m6.4.4.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx14.p1.6.m6.4.4.1.2" xref="Thmproofx14.p1.6.m6.4.4.1.2.cmml">−</mo><mrow id="Thmproofx14.p1.6.m6.4.4.1.1" xref="Thmproofx14.p1.6.m6.4.4.1.1.cmml"><msub id="Thmproofx14.p1.6.m6.4.4.1.1.3" xref="Thmproofx14.p1.6.m6.4.4.1.1.3.cmml"><mi id="Thmproofx14.p1.6.m6.4.4.1.1.3.2" xref="Thmproofx14.p1.6.m6.4.4.1.1.3.2.cmml">F</mi><mi id="Thmproofx14.p1.6.m6.4.4.1.1.3.3" xref="Thmproofx14.p1.6.m6.4.4.1.1.3.3.cmml">n</mi></msub><mo id="Thmproofx14.p1.6.m6.4.4.1.1.2" xref="Thmproofx14.p1.6.m6.4.4.1.1.2.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.4.4.1.1.1.1" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.2.cmml"><mo id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.2.cmml">(</mo><msup id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.2" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.2.cmml">σ</mi><mo id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.3" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.3" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.2.cmml">,</mo><mi id="Thmproofx14.p1.6.m6.3.3" xref="Thmproofx14.p1.6.m6.3.3.cmml">α</mi><mo id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.4" stretchy="false" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx14.p1.6.m6.10.10.9" xref="Thmproofx14.p1.6.m6.10.10.9.cmml">=</mo><mrow id="Thmproofx14.p1.6.m6.8.8.5" xref="Thmproofx14.p1.6.m6.8.8.5.cmml"><mrow id="Thmproofx14.p1.6.m6.6.6.3.2" xref="Thmproofx14.p1.6.m6.6.6.3.2.cmml"><mi id="Thmproofx14.p1.6.m6.6.6.3.2.4" xref="Thmproofx14.p1.6.m6.6.6.3.2.4.cmml">σ</mi><mo id="Thmproofx14.p1.6.m6.6.6.3.2.3" xref="Thmproofx14.p1.6.m6.6.6.3.2.3.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.5.5.2.1.1.1" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.cmml">(</mo><msub id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.2" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.3" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.cmml">)</mo></mrow><mo id="Thmproofx14.p1.6.m6.6.6.3.2.3a" xref="Thmproofx14.p1.6.m6.6.6.3.2.3.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.6.6.3.2.2.1" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.cmml">(</mo><mrow id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.cmml"><mrow id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.2.cmml"><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.2" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.3" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.3" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.3.cmml">−</mo><mrow id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.2.cmml"><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.2.1.cmml">|</mo><msub id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.2" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.2.cmml">p</mi><msup id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.cmml"><mi id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.2" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.2.cmml">i</mi><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.3" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx14.p1.6.m6.8.8.5.5" xref="Thmproofx14.p1.6.m6.8.8.5.5.cmml">+</mo><mrow id="Thmproofx14.p1.6.m6.8.8.5.4" xref="Thmproofx14.p1.6.m6.8.8.5.4.cmml"><mi id="Thmproofx14.p1.6.m6.8.8.5.4.4" xref="Thmproofx14.p1.6.m6.8.8.5.4.4.cmml">σ</mi><mo id="Thmproofx14.p1.6.m6.8.8.5.4.3" xref="Thmproofx14.p1.6.m6.8.8.5.4.3.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.7.7.4.3.1.1" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.cmml">(</mo><msub id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.2" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.2.cmml">p</mi><msup id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.cmml"><mi id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.2" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.3" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.cmml">)</mo></mrow><mo id="Thmproofx14.p1.6.m6.8.8.5.4.3a" xref="Thmproofx14.p1.6.m6.8.8.5.4.3.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.8.8.5.4.2.1" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.cmml">(</mo><mrow id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.cmml"><mrow id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.2.cmml"><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.2" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.2.cmml">p</mi><msup id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.cmml"><mi id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.2" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.3" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.3" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.3.cmml">−</mo><mrow id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.2.cmml"><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.2.1.cmml">|</mo><msub id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.2" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.3" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx14.p1.6.m6.10.10.10" xref="Thmproofx14.p1.6.m6.10.10.10.cmml">=</mo><mrow id="Thmproofx14.p1.6.m6.10.10.7" xref="Thmproofx14.p1.6.m6.10.10.7.cmml"><mrow id="Thmproofx14.p1.6.m6.9.9.6.1.1" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.9.9.6.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.cmml">(</mo><mrow id="Thmproofx14.p1.6.m6.9.9.6.1.1.1" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.cmml"><mrow id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1" 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xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.2.cmml"><mo id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.2.1.cmml">|</mo><msub id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.2" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.3" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Thmproofx14.p1.6.m6.9.9.6.1.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.cmml">)</mo></mrow><mo id="Thmproofx14.p1.6.m6.10.10.7.3" xref="Thmproofx14.p1.6.m6.10.10.7.3.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.10.10.7.2.1" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.cmml">(</mo><mrow id="Thmproofx14.p1.6.m6.10.10.7.2.1.1" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.cmml"><mrow id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.3" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.3.cmml">σ</mi><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.2" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.2" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.2.cmml">p</mi><msup id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.2" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.3" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.3" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.3.cmml">−</mo><mrow id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.cmml"><mi id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.3" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.3.cmml">σ</mi><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.2" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.2.cmml">⁢</mo><mrow id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.cmml"><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.2" stretchy="false" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.cmml">(</mo><msub id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.cmml"><mi id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.2" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.3" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx14.p1.6.m6.10.10.7.2.1.3" stretchy="false" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.6.m6.10b"><apply id="Thmproofx14.p1.6.m6.10.10.cmml" xref="Thmproofx14.p1.6.m6.10.10"><and id="Thmproofx14.p1.6.m6.10.10a.cmml" xref="Thmproofx14.p1.6.m6.10.10"></and><apply id="Thmproofx14.p1.6.m6.10.10b.cmml" xref="Thmproofx14.p1.6.m6.10.10"><eq id="Thmproofx14.p1.6.m6.10.10.9.cmml" xref="Thmproofx14.p1.6.m6.10.10.9"></eq><apply id="Thmproofx14.p1.6.m6.4.4.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1"><minus id="Thmproofx14.p1.6.m6.4.4.1.2.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.2"></minus><apply id="Thmproofx14.p1.6.m6.4.4.1.3.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.3"><times id="Thmproofx14.p1.6.m6.4.4.1.3.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.3.1"></times><apply id="Thmproofx14.p1.6.m6.4.4.1.3.2.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.3.2"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.4.4.1.3.2.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.3.2">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.4.4.1.3.2.2.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.3.2.2">𝐹</ci><ci id="Thmproofx14.p1.6.m6.4.4.1.3.2.3.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx14.p1.6.m6.4.4.1.3.3.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.3.3.2"><ci id="Thmproofx14.p1.6.m6.1.1.cmml" xref="Thmproofx14.p1.6.m6.1.1">𝜎</ci><ci id="Thmproofx14.p1.6.m6.2.2.cmml" xref="Thmproofx14.p1.6.m6.2.2">𝛼</ci></interval></apply><apply id="Thmproofx14.p1.6.m6.4.4.1.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1"><times id="Thmproofx14.p1.6.m6.4.4.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.2"></times><apply id="Thmproofx14.p1.6.m6.4.4.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.4.4.1.1.3.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.3">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.4.4.1.1.3.2.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.3.2">𝐹</ci><ci id="Thmproofx14.p1.6.m6.4.4.1.1.3.3.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.3.3">𝑛</ci></apply><interval closure="open" id="Thmproofx14.p1.6.m6.4.4.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1"><apply id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.2">𝜎</ci><ci id="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.4.4.1.1.1.1.1.3">′</ci></apply><ci id="Thmproofx14.p1.6.m6.3.3.cmml" xref="Thmproofx14.p1.6.m6.3.3">𝛼</ci></interval></apply></apply><apply id="Thmproofx14.p1.6.m6.8.8.5.cmml" xref="Thmproofx14.p1.6.m6.8.8.5"><plus id="Thmproofx14.p1.6.m6.8.8.5.5.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.5"></plus><apply id="Thmproofx14.p1.6.m6.6.6.3.2.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2"><times id="Thmproofx14.p1.6.m6.6.6.3.2.3.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.3"></times><ci id="Thmproofx14.p1.6.m6.6.6.3.2.4.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.4">𝜎</ci><apply id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.5.5.2.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1"><minus id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.3"></minus><apply id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1"><abs id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.2.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.2"></abs><apply id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.2.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1"><abs id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.2.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.2"></abs><apply id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.1.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.2.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.3.cmml" xref="Thmproofx14.p1.6.m6.6.6.3.2.2.1.1.2.1.1.3.3">′</ci></apply></apply></apply></apply></apply><apply id="Thmproofx14.p1.6.m6.8.8.5.4.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4"><times id="Thmproofx14.p1.6.m6.8.8.5.4.3.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.3"></times><ci id="Thmproofx14.p1.6.m6.8.8.5.4.4.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.4">𝜎</ci><apply id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.1.cmml" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.2.cmml" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.3.cmml" xref="Thmproofx14.p1.6.m6.7.7.4.3.1.1.1.3.3">′</ci></apply></apply><apply id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1"><minus id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.3"></minus><apply id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1"><abs id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.2.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.2"></abs><apply id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.2.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.3.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.1.1.1.3.3">′</ci></apply></apply></apply><apply id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.2.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1"><abs id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.2.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.2"></abs><apply id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.8.8.5.4.2.1.1.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></apply><apply id="Thmproofx14.p1.6.m6.10.10c.cmml" xref="Thmproofx14.p1.6.m6.10.10"><eq id="Thmproofx14.p1.6.m6.10.10.10.cmml" xref="Thmproofx14.p1.6.m6.10.10.10"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx14.p1.6.m6.8.8.5.cmml" id="Thmproofx14.p1.6.m6.10.10d.cmml" xref="Thmproofx14.p1.6.m6.10.10"></share><apply id="Thmproofx14.p1.6.m6.10.10.7.cmml" xref="Thmproofx14.p1.6.m6.10.10.7"><times id="Thmproofx14.p1.6.m6.10.10.7.3.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.3"></times><apply id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1"><minus id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.3"></minus><apply id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1"><abs id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.2.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.2"></abs><apply id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply><apply id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.2.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1"><abs id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.2.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.2"></abs><apply id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.9.9.6.1.1.1.2.1.1.3">𝑖</ci></apply></apply></apply><apply id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1"><minus id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.3"></minus><apply id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1"><times id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.2"></times><ci id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.3">𝜎</ci><apply id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply><apply id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2"><times id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.2.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.2"></times><ci id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.3.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.3">𝜎</ci><apply id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.1.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1">subscript</csymbol><ci id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.2.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.3.cmml" xref="Thmproofx14.p1.6.m6.10.10.7.2.1.1.2.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.6.m6.10c">F_{n}(\sigma,\alpha)-F_{n}(\sigma^{\prime},\alpha)=\sigma(p_{i})(|p_{i}|-|p_{i% ^{\prime}}|)+\sigma(p_{i^{\prime}})(|p_{i^{\prime}}|-|p_{i}|)=(|p_{i^{\prime}}% |-|p_{i}|)(\sigma(p_{i^{\prime}})-\sigma(p_{i}))</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.6.m6.10d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) - italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_α ) = italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | ) + italic_σ ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ( | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) = ( | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) ( italic_σ ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) )</annotation></semantics></math> by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma8" title="Lemma 8 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">8</span></a>. This is <math alttext="\leq 0" class="ltx_Math" display="inline" id="Thmproofx14.p1.7.m7.1"><semantics id="Thmproofx14.p1.7.m7.1a"><mrow id="Thmproofx14.p1.7.m7.1.1" xref="Thmproofx14.p1.7.m7.1.1.cmml"><mi id="Thmproofx14.p1.7.m7.1.1.2" xref="Thmproofx14.p1.7.m7.1.1.2.cmml"></mi><mo id="Thmproofx14.p1.7.m7.1.1.1" xref="Thmproofx14.p1.7.m7.1.1.1.cmml">≤</mo><mn id="Thmproofx14.p1.7.m7.1.1.3" xref="Thmproofx14.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.7.m7.1b"><apply id="Thmproofx14.p1.7.m7.1.1.cmml" xref="Thmproofx14.p1.7.m7.1.1"><leq id="Thmproofx14.p1.7.m7.1.1.1.cmml" xref="Thmproofx14.p1.7.m7.1.1.1"></leq><csymbol cd="latexml" id="Thmproofx14.p1.7.m7.1.1.2.cmml" xref="Thmproofx14.p1.7.m7.1.1.2">absent</csymbol><cn id="Thmproofx14.p1.7.m7.1.1.3.cmml" type="integer" xref="Thmproofx14.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.7.m7.1c">\leq 0</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.7.m7.1d">≤ 0</annotation></semantics></math> iff <math alttext="\sigma(p_{i})\geq\sigma(p_{i^{\prime}})" class="ltx_Math" display="inline" id="Thmproofx14.p1.8.m8.2"><semantics id="Thmproofx14.p1.8.m8.2a"><mrow id="Thmproofx14.p1.8.m8.2.2" xref="Thmproofx14.p1.8.m8.2.2.cmml"><mrow id="Thmproofx14.p1.8.m8.1.1.1" xref="Thmproofx14.p1.8.m8.1.1.1.cmml"><mi id="Thmproofx14.p1.8.m8.1.1.1.3" xref="Thmproofx14.p1.8.m8.1.1.1.3.cmml">σ</mi><mo id="Thmproofx14.p1.8.m8.1.1.1.2" xref="Thmproofx14.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx14.p1.8.m8.1.1.1.1.1" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="Thmproofx14.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx14.p1.8.m8.1.1.1.1.1.1" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="Thmproofx14.p1.8.m8.1.1.1.1.1.1.2" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.8.m8.1.1.1.1.1.1.3" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx14.p1.8.m8.2.2.3" xref="Thmproofx14.p1.8.m8.2.2.3.cmml">≥</mo><mrow id="Thmproofx14.p1.8.m8.2.2.2" xref="Thmproofx14.p1.8.m8.2.2.2.cmml"><mi id="Thmproofx14.p1.8.m8.2.2.2.3" xref="Thmproofx14.p1.8.m8.2.2.2.3.cmml">σ</mi><mo id="Thmproofx14.p1.8.m8.2.2.2.2" xref="Thmproofx14.p1.8.m8.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx14.p1.8.m8.2.2.2.1.1" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.cmml"><mo id="Thmproofx14.p1.8.m8.2.2.2.1.1.2" stretchy="false" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx14.p1.8.m8.2.2.2.1.1.1" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.cmml"><mi id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.2" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.2.cmml">p</mi><msup id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.cmml"><mi id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.2" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.3" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx14.p1.8.m8.2.2.2.1.1.3" stretchy="false" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.8.m8.2b"><apply id="Thmproofx14.p1.8.m8.2.2.cmml" xref="Thmproofx14.p1.8.m8.2.2"><geq id="Thmproofx14.p1.8.m8.2.2.3.cmml" xref="Thmproofx14.p1.8.m8.2.2.3"></geq><apply id="Thmproofx14.p1.8.m8.1.1.1.cmml" xref="Thmproofx14.p1.8.m8.1.1.1"><times id="Thmproofx14.p1.8.m8.1.1.1.2.cmml" xref="Thmproofx14.p1.8.m8.1.1.1.2"></times><ci id="Thmproofx14.p1.8.m8.1.1.1.3.cmml" xref="Thmproofx14.p1.8.m8.1.1.1.3">𝜎</ci><apply id="Thmproofx14.p1.8.m8.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.8.m8.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx14.p1.8.m8.2.2.2.cmml" xref="Thmproofx14.p1.8.m8.2.2.2"><times id="Thmproofx14.p1.8.m8.2.2.2.2.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.2"></times><ci id="Thmproofx14.p1.8.m8.2.2.2.3.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.3">𝜎</ci><apply id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.1.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.2.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.2.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx14.p1.8.m8.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.8.m8.2c">\sigma(p_{i})\geq\sigma(p_{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.8.m8.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ italic_σ ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>. When <math alttext="|p_{i}|=|p_{i^{\prime}}|" class="ltx_Math" display="inline" id="Thmproofx14.p1.9.m9.2"><semantics id="Thmproofx14.p1.9.m9.2a"><mrow id="Thmproofx14.p1.9.m9.2.2" xref="Thmproofx14.p1.9.m9.2.2.cmml"><mrow id="Thmproofx14.p1.9.m9.1.1.1.1" xref="Thmproofx14.p1.9.m9.1.1.1.2.cmml"><mo id="Thmproofx14.p1.9.m9.1.1.1.1.2" stretchy="false" xref="Thmproofx14.p1.9.m9.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx14.p1.9.m9.1.1.1.1.1" xref="Thmproofx14.p1.9.m9.1.1.1.1.1.cmml"><mi id="Thmproofx14.p1.9.m9.1.1.1.1.1.2" xref="Thmproofx14.p1.9.m9.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.9.m9.1.1.1.1.1.3" xref="Thmproofx14.p1.9.m9.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.9.m9.1.1.1.1.3" stretchy="false" xref="Thmproofx14.p1.9.m9.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx14.p1.9.m9.2.2.3" xref="Thmproofx14.p1.9.m9.2.2.3.cmml">=</mo><mrow id="Thmproofx14.p1.9.m9.2.2.2.1" xref="Thmproofx14.p1.9.m9.2.2.2.2.cmml"><mo id="Thmproofx14.p1.9.m9.2.2.2.1.2" stretchy="false" xref="Thmproofx14.p1.9.m9.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx14.p1.9.m9.2.2.2.1.1" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.cmml"><mi id="Thmproofx14.p1.9.m9.2.2.2.1.1.2" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.2.cmml">p</mi><msup id="Thmproofx14.p1.9.m9.2.2.2.1.1.3" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.3.cmml"><mi id="Thmproofx14.p1.9.m9.2.2.2.1.1.3.2" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.3.2.cmml">i</mi><mo id="Thmproofx14.p1.9.m9.2.2.2.1.1.3.3" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx14.p1.9.m9.2.2.2.1.3" stretchy="false" xref="Thmproofx14.p1.9.m9.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.9.m9.2b"><apply id="Thmproofx14.p1.9.m9.2.2.cmml" xref="Thmproofx14.p1.9.m9.2.2"><eq id="Thmproofx14.p1.9.m9.2.2.3.cmml" xref="Thmproofx14.p1.9.m9.2.2.3"></eq><apply id="Thmproofx14.p1.9.m9.1.1.1.2.cmml" xref="Thmproofx14.p1.9.m9.1.1.1.1"><abs id="Thmproofx14.p1.9.m9.1.1.1.2.1.cmml" xref="Thmproofx14.p1.9.m9.1.1.1.1.2"></abs><apply id="Thmproofx14.p1.9.m9.1.1.1.1.1.cmml" xref="Thmproofx14.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.9.m9.1.1.1.1.1.1.cmml" xref="Thmproofx14.p1.9.m9.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.9.m9.1.1.1.1.1.2.cmml" xref="Thmproofx14.p1.9.m9.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.9.m9.1.1.1.1.1.3.cmml" xref="Thmproofx14.p1.9.m9.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx14.p1.9.m9.2.2.2.2.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1"><abs id="Thmproofx14.p1.9.m9.2.2.2.2.1.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.2"></abs><apply id="Thmproofx14.p1.9.m9.2.2.2.1.1.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.9.m9.2.2.2.1.1.1.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx14.p1.9.m9.2.2.2.1.1.2.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.2">𝑝</ci><apply id="Thmproofx14.p1.9.m9.2.2.2.1.1.3.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.9.m9.2.2.2.1.1.3.1.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.3">superscript</csymbol><ci id="Thmproofx14.p1.9.m9.2.2.2.1.1.3.2.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.3.2">𝑖</ci><ci id="Thmproofx14.p1.9.m9.2.2.2.1.1.3.3.cmml" xref="Thmproofx14.p1.9.m9.2.2.2.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.9.m9.2c">|p_{i}|=|p_{i^{\prime}}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.9.m9.2d">| italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math>, the difference above is 0 so we reorder the weights of <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmproofx14.p1.10.m10.1"><semantics id="Thmproofx14.p1.10.m10.1a"><mi id="Thmproofx14.p1.10.m10.1.1" xref="Thmproofx14.p1.10.m10.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.10.m10.1b"><ci id="Thmproofx14.p1.10.m10.1.1.cmml" xref="Thmproofx14.p1.10.m10.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.10.m10.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.10.m10.1d">italic_σ</annotation></semantics></math> on <math alttext="p_{i},p_{i^{\prime}}" class="ltx_Math" display="inline" id="Thmproofx14.p1.11.m11.2"><semantics id="Thmproofx14.p1.11.m11.2a"><mrow id="Thmproofx14.p1.11.m11.2.2.2" xref="Thmproofx14.p1.11.m11.2.2.3.cmml"><msub id="Thmproofx14.p1.11.m11.1.1.1.1" xref="Thmproofx14.p1.11.m11.1.1.1.1.cmml"><mi id="Thmproofx14.p1.11.m11.1.1.1.1.2" xref="Thmproofx14.p1.11.m11.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx14.p1.11.m11.1.1.1.1.3" xref="Thmproofx14.p1.11.m11.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx14.p1.11.m11.2.2.2.3" xref="Thmproofx14.p1.11.m11.2.2.3.cmml">,</mo><msub id="Thmproofx14.p1.11.m11.2.2.2.2" xref="Thmproofx14.p1.11.m11.2.2.2.2.cmml"><mi id="Thmproofx14.p1.11.m11.2.2.2.2.2" xref="Thmproofx14.p1.11.m11.2.2.2.2.2.cmml">p</mi><msup id="Thmproofx14.p1.11.m11.2.2.2.2.3" xref="Thmproofx14.p1.11.m11.2.2.2.2.3.cmml"><mi id="Thmproofx14.p1.11.m11.2.2.2.2.3.2" xref="Thmproofx14.p1.11.m11.2.2.2.2.3.2.cmml">i</mi><mo id="Thmproofx14.p1.11.m11.2.2.2.2.3.3" xref="Thmproofx14.p1.11.m11.2.2.2.2.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx14.p1.11.m11.2b"><list id="Thmproofx14.p1.11.m11.2.2.3.cmml" xref="Thmproofx14.p1.11.m11.2.2.2"><apply id="Thmproofx14.p1.11.m11.1.1.1.1.cmml" xref="Thmproofx14.p1.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx14.p1.11.m11.1.1.1.1.1.cmml" xref="Thmproofx14.p1.11.m11.1.1.1.1">subscript</csymbol><ci id="Thmproofx14.p1.11.m11.1.1.1.1.2.cmml" xref="Thmproofx14.p1.11.m11.1.1.1.1.2">𝑝</ci><ci id="Thmproofx14.p1.11.m11.1.1.1.1.3.cmml" xref="Thmproofx14.p1.11.m11.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx14.p1.11.m11.2.2.2.2.cmml" xref="Thmproofx14.p1.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx14.p1.11.m11.2.2.2.2.1.cmml" xref="Thmproofx14.p1.11.m11.2.2.2.2">subscript</csymbol><ci id="Thmproofx14.p1.11.m11.2.2.2.2.2.cmml" xref="Thmproofx14.p1.11.m11.2.2.2.2.2">𝑝</ci><apply id="Thmproofx14.p1.11.m11.2.2.2.2.3.cmml" xref="Thmproofx14.p1.11.m11.2.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx14.p1.11.m11.2.2.2.2.3.1.cmml" xref="Thmproofx14.p1.11.m11.2.2.2.2.3">superscript</csymbol><ci id="Thmproofx14.p1.11.m11.2.2.2.2.3.2.cmml" xref="Thmproofx14.p1.11.m11.2.2.2.2.3.2">𝑖</ci><ci id="Thmproofx14.p1.11.m11.2.2.2.2.3.3.cmml" xref="Thmproofx14.p1.11.m11.2.2.2.2.3.3">′</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx14.p1.11.m11.2c">p_{i},p_{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx14.p1.11.m11.2d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> to make the property in the lemma statement hold. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx15"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma10" title="Lemma 10 ‣ 4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">10</span></a>)</h6> <div class="ltx_para" id="Thmproofx15.p1"> <p class="ltx_p" id="Thmproofx15.p1.4">Let <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmproofx15.p1.1.m1.1"><semantics id="Thmproofx15.p1.1.m1.1a"><mi id="Thmproofx15.p1.1.m1.1.1" xref="Thmproofx15.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx15.p1.1.m1.1b"><ci id="Thmproofx15.p1.1.m1.1.1.cmml" xref="Thmproofx15.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx15.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx15.p1.1.m1.1d">italic_σ</annotation></semantics></math>. Take the distribution <math alttext="X=\left(\frac{\sum_{j\in[n]}\sigma_{ij}}{n}\right)_{i\in[m]}" class="ltx_Math" display="inline" id="Thmproofx15.p1.2.m2.2"><semantics id="Thmproofx15.p1.2.m2.2a"><mrow id="Thmproofx15.p1.2.m2.2.3" xref="Thmproofx15.p1.2.m2.2.3.cmml"><mi id="Thmproofx15.p1.2.m2.2.3.2" xref="Thmproofx15.p1.2.m2.2.3.2.cmml">X</mi><mo id="Thmproofx15.p1.2.m2.2.3.1" xref="Thmproofx15.p1.2.m2.2.3.1.cmml">=</mo><msub id="Thmproofx15.p1.2.m2.2.3.3" xref="Thmproofx15.p1.2.m2.2.3.3.cmml"><mrow id="Thmproofx15.p1.2.m2.2.3.3.2.2" xref="Thmproofx15.p1.2.m2.1.1.cmml"><mo id="Thmproofx15.p1.2.m2.2.3.3.2.2.1" xref="Thmproofx15.p1.2.m2.1.1.cmml">(</mo><mfrac id="Thmproofx15.p1.2.m2.1.1" xref="Thmproofx15.p1.2.m2.1.1.cmml"><mrow id="Thmproofx15.p1.2.m2.1.1.1" xref="Thmproofx15.p1.2.m2.1.1.1.cmml"><mstyle displaystyle="false" id="Thmproofx15.p1.2.m2.1.1.1.2" xref="Thmproofx15.p1.2.m2.1.1.1.2.cmml"><msub id="Thmproofx15.p1.2.m2.1.1.1.2a" xref="Thmproofx15.p1.2.m2.1.1.1.2.cmml"><mo id="Thmproofx15.p1.2.m2.1.1.1.2.2" xref="Thmproofx15.p1.2.m2.1.1.1.2.2.cmml">∑</mo><mrow id="Thmproofx15.p1.2.m2.1.1.1.1.1" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.cmml"><mi id="Thmproofx15.p1.2.m2.1.1.1.1.1.3" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.3.cmml">j</mi><mo id="Thmproofx15.p1.2.m2.1.1.1.1.1.2" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.2.cmml">∈</mo><mrow id="Thmproofx15.p1.2.m2.1.1.1.1.1.4.2" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.4.1.cmml"><mo id="Thmproofx15.p1.2.m2.1.1.1.1.1.4.2.1" stretchy="false" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.4.1.1.cmml">[</mo><mi id="Thmproofx15.p1.2.m2.1.1.1.1.1.1" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.1.cmml">n</mi><mo id="Thmproofx15.p1.2.m2.1.1.1.1.1.4.2.2" stretchy="false" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub></mstyle><msub id="Thmproofx15.p1.2.m2.1.1.1.3" xref="Thmproofx15.p1.2.m2.1.1.1.3.cmml"><mi id="Thmproofx15.p1.2.m2.1.1.1.3.2" xref="Thmproofx15.p1.2.m2.1.1.1.3.2.cmml">σ</mi><mrow id="Thmproofx15.p1.2.m2.1.1.1.3.3" xref="Thmproofx15.p1.2.m2.1.1.1.3.3.cmml"><mi id="Thmproofx15.p1.2.m2.1.1.1.3.3.2" xref="Thmproofx15.p1.2.m2.1.1.1.3.3.2.cmml">i</mi><mo id="Thmproofx15.p1.2.m2.1.1.1.3.3.1" xref="Thmproofx15.p1.2.m2.1.1.1.3.3.1.cmml">⁢</mo><mi id="Thmproofx15.p1.2.m2.1.1.1.3.3.3" xref="Thmproofx15.p1.2.m2.1.1.1.3.3.3.cmml">j</mi></mrow></msub></mrow><mi id="Thmproofx15.p1.2.m2.1.1.3" xref="Thmproofx15.p1.2.m2.1.1.3.cmml">n</mi></mfrac><mo id="Thmproofx15.p1.2.m2.2.3.3.2.2.2" xref="Thmproofx15.p1.2.m2.1.1.cmml">)</mo></mrow><mrow id="Thmproofx15.p1.2.m2.2.2.1" xref="Thmproofx15.p1.2.m2.2.2.1.cmml"><mi id="Thmproofx15.p1.2.m2.2.2.1.3" xref="Thmproofx15.p1.2.m2.2.2.1.3.cmml">i</mi><mo id="Thmproofx15.p1.2.m2.2.2.1.2" xref="Thmproofx15.p1.2.m2.2.2.1.2.cmml">∈</mo><mrow id="Thmproofx15.p1.2.m2.2.2.1.4.2" xref="Thmproofx15.p1.2.m2.2.2.1.4.1.cmml"><mo id="Thmproofx15.p1.2.m2.2.2.1.4.2.1" stretchy="false" xref="Thmproofx15.p1.2.m2.2.2.1.4.1.1.cmml">[</mo><mi id="Thmproofx15.p1.2.m2.2.2.1.1" xref="Thmproofx15.p1.2.m2.2.2.1.1.cmml">m</mi><mo id="Thmproofx15.p1.2.m2.2.2.1.4.2.2" stretchy="false" xref="Thmproofx15.p1.2.m2.2.2.1.4.1.1.cmml">]</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx15.p1.2.m2.2b"><apply id="Thmproofx15.p1.2.m2.2.3.cmml" xref="Thmproofx15.p1.2.m2.2.3"><eq id="Thmproofx15.p1.2.m2.2.3.1.cmml" xref="Thmproofx15.p1.2.m2.2.3.1"></eq><ci id="Thmproofx15.p1.2.m2.2.3.2.cmml" xref="Thmproofx15.p1.2.m2.2.3.2">𝑋</ci><apply id="Thmproofx15.p1.2.m2.2.3.3.cmml" xref="Thmproofx15.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="Thmproofx15.p1.2.m2.2.3.3.1.cmml" xref="Thmproofx15.p1.2.m2.2.3.3">subscript</csymbol><apply id="Thmproofx15.p1.2.m2.1.1.cmml" xref="Thmproofx15.p1.2.m2.2.3.3.2.2"><divide id="Thmproofx15.p1.2.m2.1.1.2.cmml" xref="Thmproofx15.p1.2.m2.2.3.3.2.2"></divide><apply id="Thmproofx15.p1.2.m2.1.1.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1"><apply id="Thmproofx15.p1.2.m2.1.1.1.2.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx15.p1.2.m2.1.1.1.2.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.2">subscript</csymbol><sum id="Thmproofx15.p1.2.m2.1.1.1.2.2.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.2.2"></sum><apply id="Thmproofx15.p1.2.m2.1.1.1.1.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.1.1"><in id="Thmproofx15.p1.2.m2.1.1.1.1.1.2.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.2"></in><ci id="Thmproofx15.p1.2.m2.1.1.1.1.1.3.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.3">𝑗</ci><apply id="Thmproofx15.p1.2.m2.1.1.1.1.1.4.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.4.2"><csymbol cd="latexml" id="Thmproofx15.p1.2.m2.1.1.1.1.1.4.1.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Thmproofx15.p1.2.m2.1.1.1.1.1.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.1.1.1">𝑛</ci></apply></apply></apply><apply id="Thmproofx15.p1.2.m2.1.1.1.3.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx15.p1.2.m2.1.1.1.3.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.3">subscript</csymbol><ci id="Thmproofx15.p1.2.m2.1.1.1.3.2.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.3.2">𝜎</ci><apply id="Thmproofx15.p1.2.m2.1.1.1.3.3.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.3.3"><times id="Thmproofx15.p1.2.m2.1.1.1.3.3.1.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.3.3.1"></times><ci id="Thmproofx15.p1.2.m2.1.1.1.3.3.2.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.3.3.2">𝑖</ci><ci id="Thmproofx15.p1.2.m2.1.1.1.3.3.3.cmml" xref="Thmproofx15.p1.2.m2.1.1.1.3.3.3">𝑗</ci></apply></apply></apply><ci id="Thmproofx15.p1.2.m2.1.1.3.cmml" xref="Thmproofx15.p1.2.m2.1.1.3">𝑛</ci></apply><apply id="Thmproofx15.p1.2.m2.2.2.1.cmml" xref="Thmproofx15.p1.2.m2.2.2.1"><in id="Thmproofx15.p1.2.m2.2.2.1.2.cmml" xref="Thmproofx15.p1.2.m2.2.2.1.2"></in><ci id="Thmproofx15.p1.2.m2.2.2.1.3.cmml" xref="Thmproofx15.p1.2.m2.2.2.1.3">𝑖</ci><apply id="Thmproofx15.p1.2.m2.2.2.1.4.1.cmml" xref="Thmproofx15.p1.2.m2.2.2.1.4.2"><csymbol cd="latexml" id="Thmproofx15.p1.2.m2.2.2.1.4.1.1.cmml" xref="Thmproofx15.p1.2.m2.2.2.1.4.2.1">delimited-[]</csymbol><ci id="Thmproofx15.p1.2.m2.2.2.1.1.cmml" xref="Thmproofx15.p1.2.m2.2.2.1.1">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx15.p1.2.m2.2c">X=\left(\frac{\sum_{j\in[n]}\sigma_{ij}}{n}\right)_{i\in[m]}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx15.p1.2.m2.2d">italic_X = ( divide start_ARG ∑ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_n end_ARG ) start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\mu=X^{n}" class="ltx_Math" display="inline" id="Thmproofx15.p1.3.m3.1"><semantics id="Thmproofx15.p1.3.m3.1a"><mrow id="Thmproofx15.p1.3.m3.1.1" xref="Thmproofx15.p1.3.m3.1.1.cmml"><mi id="Thmproofx15.p1.3.m3.1.1.2" xref="Thmproofx15.p1.3.m3.1.1.2.cmml">μ</mi><mo id="Thmproofx15.p1.3.m3.1.1.1" xref="Thmproofx15.p1.3.m3.1.1.1.cmml">=</mo><msup id="Thmproofx15.p1.3.m3.1.1.3" xref="Thmproofx15.p1.3.m3.1.1.3.cmml"><mi id="Thmproofx15.p1.3.m3.1.1.3.2" xref="Thmproofx15.p1.3.m3.1.1.3.2.cmml">X</mi><mi id="Thmproofx15.p1.3.m3.1.1.3.3" xref="Thmproofx15.p1.3.m3.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx15.p1.3.m3.1b"><apply id="Thmproofx15.p1.3.m3.1.1.cmml" xref="Thmproofx15.p1.3.m3.1.1"><eq id="Thmproofx15.p1.3.m3.1.1.1.cmml" xref="Thmproofx15.p1.3.m3.1.1.1"></eq><ci id="Thmproofx15.p1.3.m3.1.1.2.cmml" xref="Thmproofx15.p1.3.m3.1.1.2">𝜇</ci><apply id="Thmproofx15.p1.3.m3.1.1.3.cmml" xref="Thmproofx15.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="Thmproofx15.p1.3.m3.1.1.3.1.cmml" xref="Thmproofx15.p1.3.m3.1.1.3">superscript</csymbol><ci id="Thmproofx15.p1.3.m3.1.1.3.2.cmml" xref="Thmproofx15.p1.3.m3.1.1.3.2">𝑋</ci><ci id="Thmproofx15.p1.3.m3.1.1.3.3.cmml" xref="Thmproofx15.p1.3.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx15.p1.3.m3.1c">\mu=X^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx15.p1.3.m3.1d">italic_μ = italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Then for every path <math alttext="p_{i}" class="ltx_Math" display="inline" id="Thmproofx15.p1.4.m4.1"><semantics id="Thmproofx15.p1.4.m4.1a"><msub id="Thmproofx15.p1.4.m4.1.1" xref="Thmproofx15.p1.4.m4.1.1.cmml"><mi id="Thmproofx15.p1.4.m4.1.1.2" xref="Thmproofx15.p1.4.m4.1.1.2.cmml">p</mi><mi id="Thmproofx15.p1.4.m4.1.1.3" xref="Thmproofx15.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx15.p1.4.m4.1b"><apply id="Thmproofx15.p1.4.m4.1.1.cmml" xref="Thmproofx15.p1.4.m4.1.1"><csymbol cd="ambiguous" id="Thmproofx15.p1.4.m4.1.1.1.cmml" xref="Thmproofx15.p1.4.m4.1.1">subscript</csymbol><ci id="Thmproofx15.p1.4.m4.1.1.2.cmml" xref="Thmproofx15.p1.4.m4.1.1.2">𝑝</ci><ci id="Thmproofx15.p1.4.m4.1.1.3.cmml" xref="Thmproofx15.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx15.p1.4.m4.1c">p_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx15.p1.4.m4.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx40"> <tbody id="Pt0.A3.Ex62"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sigma(p_{i})\stackrel{{\scriptstyle\text{def}}}{{=}}1-\prod_{j}(% 1-\sigma_{ij})" class="ltx_Math" display="inline" id="Pt0.A3.Ex62.m1.2"><semantics id="Pt0.A3.Ex62.m1.2a"><mrow id="Pt0.A3.Ex62.m1.2.2" xref="Pt0.A3.Ex62.m1.2.2.cmml"><mrow id="Pt0.A3.Ex62.m1.1.1.1" xref="Pt0.A3.Ex62.m1.1.1.1.cmml"><mi id="Pt0.A3.Ex62.m1.1.1.1.3" xref="Pt0.A3.Ex62.m1.1.1.1.3.cmml">σ</mi><mo id="Pt0.A3.Ex62.m1.1.1.1.2" xref="Pt0.A3.Ex62.m1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A3.Ex62.m1.1.1.1.1.1" xref="Pt0.A3.Ex62.m1.1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex62.m1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex62.m1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A3.Ex62.m1.1.1.1.1.1.1" xref="Pt0.A3.Ex62.m1.1.1.1.1.1.1.cmml"><mi id="Pt0.A3.Ex62.m1.1.1.1.1.1.1.2" xref="Pt0.A3.Ex62.m1.1.1.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A3.Ex62.m1.1.1.1.1.1.1.3" xref="Pt0.A3.Ex62.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Pt0.A3.Ex62.m1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A3.Ex62.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mover id="Pt0.A3.Ex62.m1.2.2.3" xref="Pt0.A3.Ex62.m1.2.2.3.cmml"><mo id="Pt0.A3.Ex62.m1.2.2.3.2" xref="Pt0.A3.Ex62.m1.2.2.3.2.cmml">=</mo><mtext id="Pt0.A3.Ex62.m1.2.2.3.3" xref="Pt0.A3.Ex62.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="Pt0.A3.Ex62.m1.2.2.2" xref="Pt0.A3.Ex62.m1.2.2.2.cmml"><mn id="Pt0.A3.Ex62.m1.2.2.2.3" xref="Pt0.A3.Ex62.m1.2.2.2.3.cmml">1</mn><mo id="Pt0.A3.Ex62.m1.2.2.2.2" xref="Pt0.A3.Ex62.m1.2.2.2.2.cmml">−</mo><mrow id="Pt0.A3.Ex62.m1.2.2.2.1" xref="Pt0.A3.Ex62.m1.2.2.2.1.cmml"><mstyle displaystyle="true" id="Pt0.A3.Ex62.m1.2.2.2.1.2" xref="Pt0.A3.Ex62.m1.2.2.2.1.2.cmml"><munder id="Pt0.A3.Ex62.m1.2.2.2.1.2a" xref="Pt0.A3.Ex62.m1.2.2.2.1.2.cmml"><mo id="Pt0.A3.Ex62.m1.2.2.2.1.2.2" movablelimits="false" xref="Pt0.A3.Ex62.m1.2.2.2.1.2.2.cmml">∏</mo><mi id="Pt0.A3.Ex62.m1.2.2.2.1.2.3" xref="Pt0.A3.Ex62.m1.2.2.2.1.2.3.cmml">j</mi></munder></mstyle><mrow id="Pt0.A3.Ex62.m1.2.2.2.1.1.1" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.cmml"><mo id="Pt0.A3.Ex62.m1.2.2.2.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.cmml"><mn id="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.2" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.1" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.1.cmml">−</mo><msub id="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.3" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.3.cmml"><mi id="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.3.2" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.3.2.cmml">σ</mi><mrow id="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.3.3" xref="Pt0.A3.Ex62.m1.2.2.2.1.1.1.1.3.3.cmml"><mi 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end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_n end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A3.Ex64"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\iff\sqrt[n]{\prod_{j}(1-\sigma_{ij})}" class="ltx_Math" display="inline" id="Pt0.A3.Ex64.m1.1"><semantics id="Pt0.A3.Ex64.m1.1a"><mrow id="Pt0.A3.Ex64.m1.1.2" xref="Pt0.A3.Ex64.m1.1.2.cmml"><mi id="Pt0.A3.Ex64.m1.1.2.2" xref="Pt0.A3.Ex64.m1.1.2.2.cmml"></mi><mo id="Pt0.A3.Ex64.m1.1.2.1" stretchy="false" xref="Pt0.A3.Ex64.m1.1.2.1.cmml">⇔</mo><mroot id="Pt0.A3.Ex64.m1.1.1" xref="Pt0.A3.Ex64.m1.1.1.cmml"><mrow id="Pt0.A3.Ex64.m1.1.1.1" xref="Pt0.A3.Ex64.m1.1.1.1.cmml"><mstyle displaystyle="true" 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id="Pt0.A3.Ex64.m2.1.1.1.3.2.cmml" xref="Pt0.A3.Ex64.m2.1.1.1.3.2">𝜎</ci><apply id="Pt0.A3.Ex64.m2.1.1.1.3.3.cmml" xref="Pt0.A3.Ex64.m2.1.1.1.3.3"><times id="Pt0.A3.Ex64.m2.1.1.1.3.3.1.cmml" xref="Pt0.A3.Ex64.m2.1.1.1.3.3.1"></times><ci id="Pt0.A3.Ex64.m2.1.1.1.3.3.2.cmml" xref="Pt0.A3.Ex64.m2.1.1.1.3.3.2">𝑖</ci><ci id="Pt0.A3.Ex64.m2.1.1.1.3.3.3.cmml" xref="Pt0.A3.Ex64.m2.1.1.1.3.3.3">𝑗</ci></apply></apply></apply><ci id="Pt0.A3.Ex64.m2.1.1.3.cmml" xref="Pt0.A3.Ex64.m2.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A3.Ex64.m2.1c">\displaystyle\leq 1-\frac{\sum_{j\in[n]}\sigma_{ij}}{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A3.Ex64.m2.1d">≤ 1 - divide start_ARG ∑ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_n end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx15.p1.7">The last inequality follows from the AMGM inequality (with equality iff <math alttext="\{\sigma_{ij}\}_{j}" class="ltx_Math" display="inline" id="Thmproofx15.p1.5.m1.1"><semantics id="Thmproofx15.p1.5.m1.1a"><msub id="Thmproofx15.p1.5.m1.1.1" xref="Thmproofx15.p1.5.m1.1.1.cmml"><mrow id="Thmproofx15.p1.5.m1.1.1.1.1" xref="Thmproofx15.p1.5.m1.1.1.1.2.cmml"><mo id="Thmproofx15.p1.5.m1.1.1.1.1.2" stretchy="false" xref="Thmproofx15.p1.5.m1.1.1.1.2.cmml">{</mo><msub id="Thmproofx15.p1.5.m1.1.1.1.1.1" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.cmml"><mi id="Thmproofx15.p1.5.m1.1.1.1.1.1.2" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.2.cmml">σ</mi><mrow id="Thmproofx15.p1.5.m1.1.1.1.1.1.3" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3.cmml"><mi id="Thmproofx15.p1.5.m1.1.1.1.1.1.3.2" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx15.p1.5.m1.1.1.1.1.1.3.1" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx15.p1.5.m1.1.1.1.1.1.3.3" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="Thmproofx15.p1.5.m1.1.1.1.1.3" stretchy="false" xref="Thmproofx15.p1.5.m1.1.1.1.2.cmml">}</mo></mrow><mi id="Thmproofx15.p1.5.m1.1.1.3" xref="Thmproofx15.p1.5.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx15.p1.5.m1.1b"><apply id="Thmproofx15.p1.5.m1.1.1.cmml" xref="Thmproofx15.p1.5.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx15.p1.5.m1.1.1.2.cmml" xref="Thmproofx15.p1.5.m1.1.1">subscript</csymbol><set id="Thmproofx15.p1.5.m1.1.1.1.2.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1"><apply id="Thmproofx15.p1.5.m1.1.1.1.1.1.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx15.p1.5.m1.1.1.1.1.1.1.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx15.p1.5.m1.1.1.1.1.1.2.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.2">𝜎</ci><apply id="Thmproofx15.p1.5.m1.1.1.1.1.1.3.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3"><times id="Thmproofx15.p1.5.m1.1.1.1.1.1.3.1.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3.1"></times><ci id="Thmproofx15.p1.5.m1.1.1.1.1.1.3.2.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx15.p1.5.m1.1.1.1.1.1.3.3.cmml" xref="Thmproofx15.p1.5.m1.1.1.1.1.1.3.3">𝑗</ci></apply></apply></set><ci id="Thmproofx15.p1.5.m1.1.1.3.cmml" xref="Thmproofx15.p1.5.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx15.p1.5.m1.1c">\{\sigma_{ij}\}_{j}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx15.p1.5.m1.1d">{ italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are all equal). Therefore by monotonicity with respect to the probabilities <math alttext="\sigma(p_{i})" class="ltx_Math" display="inline" id="Thmproofx15.p1.6.m2.1"><semantics id="Thmproofx15.p1.6.m2.1a"><mrow id="Thmproofx15.p1.6.m2.1.1" xref="Thmproofx15.p1.6.m2.1.1.cmml"><mi id="Thmproofx15.p1.6.m2.1.1.3" xref="Thmproofx15.p1.6.m2.1.1.3.cmml">σ</mi><mo id="Thmproofx15.p1.6.m2.1.1.2" xref="Thmproofx15.p1.6.m2.1.1.2.cmml">⁢</mo><mrow id="Thmproofx15.p1.6.m2.1.1.1.1" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.cmml"><mo id="Thmproofx15.p1.6.m2.1.1.1.1.2" stretchy="false" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx15.p1.6.m2.1.1.1.1.1" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.cmml"><mi id="Thmproofx15.p1.6.m2.1.1.1.1.1.2" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx15.p1.6.m2.1.1.1.1.1.3" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx15.p1.6.m2.1.1.1.1.3" stretchy="false" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx15.p1.6.m2.1b"><apply id="Thmproofx15.p1.6.m2.1.1.cmml" xref="Thmproofx15.p1.6.m2.1.1"><times id="Thmproofx15.p1.6.m2.1.1.2.cmml" xref="Thmproofx15.p1.6.m2.1.1.2"></times><ci id="Thmproofx15.p1.6.m2.1.1.3.cmml" xref="Thmproofx15.p1.6.m2.1.1.3">𝜎</ci><apply id="Thmproofx15.p1.6.m2.1.1.1.1.1.cmml" xref="Thmproofx15.p1.6.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx15.p1.6.m2.1.1.1.1.1.1.cmml" xref="Thmproofx15.p1.6.m2.1.1.1.1">subscript</csymbol><ci id="Thmproofx15.p1.6.m2.1.1.1.1.1.2.cmml" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx15.p1.6.m2.1.1.1.1.1.3.cmml" xref="Thmproofx15.p1.6.m2.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx15.p1.6.m2.1c">\sigma(p_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx15.p1.6.m2.1d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="F_{n}(\sigma,\alpha)\geq F_{n}(\mu,\alpha)" class="ltx_Math" display="inline" id="Thmproofx15.p1.7.m3.4"><semantics id="Thmproofx15.p1.7.m3.4a"><mrow id="Thmproofx15.p1.7.m3.4.5" xref="Thmproofx15.p1.7.m3.4.5.cmml"><mrow id="Thmproofx15.p1.7.m3.4.5.2" xref="Thmproofx15.p1.7.m3.4.5.2.cmml"><msub id="Thmproofx15.p1.7.m3.4.5.2.2" xref="Thmproofx15.p1.7.m3.4.5.2.2.cmml"><mi id="Thmproofx15.p1.7.m3.4.5.2.2.2" xref="Thmproofx15.p1.7.m3.4.5.2.2.2.cmml">F</mi><mi id="Thmproofx15.p1.7.m3.4.5.2.2.3" xref="Thmproofx15.p1.7.m3.4.5.2.2.3.cmml">n</mi></msub><mo id="Thmproofx15.p1.7.m3.4.5.2.1" xref="Thmproofx15.p1.7.m3.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx15.p1.7.m3.4.5.2.3.2" xref="Thmproofx15.p1.7.m3.4.5.2.3.1.cmml"><mo id="Thmproofx15.p1.7.m3.4.5.2.3.2.1" stretchy="false" xref="Thmproofx15.p1.7.m3.4.5.2.3.1.cmml">(</mo><mi id="Thmproofx15.p1.7.m3.1.1" xref="Thmproofx15.p1.7.m3.1.1.cmml">σ</mi><mo 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xref="Thmproofx15.p1.7.m3.3.3.cmml">μ</mi><mo id="Thmproofx15.p1.7.m3.4.5.3.3.2.2" xref="Thmproofx15.p1.7.m3.4.5.3.3.1.cmml">,</mo><mi id="Thmproofx15.p1.7.m3.4.4" xref="Thmproofx15.p1.7.m3.4.4.cmml">α</mi><mo id="Thmproofx15.p1.7.m3.4.5.3.3.2.3" stretchy="false" xref="Thmproofx15.p1.7.m3.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx15.p1.7.m3.4b"><apply id="Thmproofx15.p1.7.m3.4.5.cmml" xref="Thmproofx15.p1.7.m3.4.5"><geq id="Thmproofx15.p1.7.m3.4.5.1.cmml" xref="Thmproofx15.p1.7.m3.4.5.1"></geq><apply id="Thmproofx15.p1.7.m3.4.5.2.cmml" xref="Thmproofx15.p1.7.m3.4.5.2"><times id="Thmproofx15.p1.7.m3.4.5.2.1.cmml" xref="Thmproofx15.p1.7.m3.4.5.2.1"></times><apply id="Thmproofx15.p1.7.m3.4.5.2.2.cmml" xref="Thmproofx15.p1.7.m3.4.5.2.2"><csymbol cd="ambiguous" id="Thmproofx15.p1.7.m3.4.5.2.2.1.cmml" xref="Thmproofx15.p1.7.m3.4.5.2.2">subscript</csymbol><ci id="Thmproofx15.p1.7.m3.4.5.2.2.2.cmml" xref="Thmproofx15.p1.7.m3.4.5.2.2.2">𝐹</ci><ci id="Thmproofx15.p1.7.m3.4.5.2.2.3.cmml" xref="Thmproofx15.p1.7.m3.4.5.2.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx15.p1.7.m3.4.5.2.3.1.cmml" xref="Thmproofx15.p1.7.m3.4.5.2.3.2"><ci id="Thmproofx15.p1.7.m3.1.1.cmml" xref="Thmproofx15.p1.7.m3.1.1">𝜎</ci><ci id="Thmproofx15.p1.7.m3.2.2.cmml" xref="Thmproofx15.p1.7.m3.2.2">𝛼</ci></interval></apply><apply id="Thmproofx15.p1.7.m3.4.5.3.cmml" xref="Thmproofx15.p1.7.m3.4.5.3"><times id="Thmproofx15.p1.7.m3.4.5.3.1.cmml" xref="Thmproofx15.p1.7.m3.4.5.3.1"></times><apply id="Thmproofx15.p1.7.m3.4.5.3.2.cmml" xref="Thmproofx15.p1.7.m3.4.5.3.2"><csymbol cd="ambiguous" id="Thmproofx15.p1.7.m3.4.5.3.2.1.cmml" xref="Thmproofx15.p1.7.m3.4.5.3.2">subscript</csymbol><ci id="Thmproofx15.p1.7.m3.4.5.3.2.2.cmml" xref="Thmproofx15.p1.7.m3.4.5.3.2.2">𝐹</ci><ci id="Thmproofx15.p1.7.m3.4.5.3.2.3.cmml" xref="Thmproofx15.p1.7.m3.4.5.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx15.p1.7.m3.4.5.3.3.1.cmml" xref="Thmproofx15.p1.7.m3.4.5.3.3.2"><ci id="Thmproofx15.p1.7.m3.3.3.cmml" xref="Thmproofx15.p1.7.m3.3.3">𝜇</ci><ci id="Thmproofx15.p1.7.m3.4.4.cmml" xref="Thmproofx15.p1.7.m3.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx15.p1.7.m3.4c">F_{n}(\sigma,\alpha)\geq F_{n}(\mu,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx15.p1.7.m3.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≥ italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_μ , italic_α )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx16"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S4.Thmtheorem6" title="Theorem 4.6 ‣ 4.2 Disjoint paths ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">4.6</span></a>)</h6> <div class="ltx_para" id="Thmproofx16.p1"> <p class="ltx_p" id="Thmproofx16.p1.3">For every <math alttext="X\in\operatorname{conv}(\mathcal{U}_{1})" class="ltx_Math" display="inline" id="Thmproofx16.p1.1.m1.2"><semantics id="Thmproofx16.p1.1.m1.2a"><mrow id="Thmproofx16.p1.1.m1.2.2" xref="Thmproofx16.p1.1.m1.2.2.cmml"><mi id="Thmproofx16.p1.1.m1.2.2.3" xref="Thmproofx16.p1.1.m1.2.2.3.cmml">X</mi><mo id="Thmproofx16.p1.1.m1.2.2.2" xref="Thmproofx16.p1.1.m1.2.2.2.cmml">∈</mo><mrow id="Thmproofx16.p1.1.m1.2.2.1.1" xref="Thmproofx16.p1.1.m1.2.2.1.2.cmml"><mi id="Thmproofx16.p1.1.m1.1.1" xref="Thmproofx16.p1.1.m1.1.1.cmml">conv</mi><mo id="Thmproofx16.p1.1.m1.2.2.1.1a" xref="Thmproofx16.p1.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="Thmproofx16.p1.1.m1.2.2.1.1.1" xref="Thmproofx16.p1.1.m1.2.2.1.2.cmml"><mo id="Thmproofx16.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="Thmproofx16.p1.1.m1.2.2.1.2.cmml">(</mo><msub id="Thmproofx16.p1.1.m1.2.2.1.1.1.1" xref="Thmproofx16.p1.1.m1.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx16.p1.1.m1.2.2.1.1.1.1.2" xref="Thmproofx16.p1.1.m1.2.2.1.1.1.1.2.cmml">𝒰</mi><mn id="Thmproofx16.p1.1.m1.2.2.1.1.1.1.3" xref="Thmproofx16.p1.1.m1.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx16.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="Thmproofx16.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx16.p1.1.m1.2b"><apply id="Thmproofx16.p1.1.m1.2.2.cmml" xref="Thmproofx16.p1.1.m1.2.2"><in id="Thmproofx16.p1.1.m1.2.2.2.cmml" xref="Thmproofx16.p1.1.m1.2.2.2"></in><ci id="Thmproofx16.p1.1.m1.2.2.3.cmml" xref="Thmproofx16.p1.1.m1.2.2.3">𝑋</ci><apply id="Thmproofx16.p1.1.m1.2.2.1.2.cmml" xref="Thmproofx16.p1.1.m1.2.2.1.1"><ci id="Thmproofx16.p1.1.m1.1.1.cmml" xref="Thmproofx16.p1.1.m1.1.1">conv</ci><apply id="Thmproofx16.p1.1.m1.2.2.1.1.1.1.cmml" xref="Thmproofx16.p1.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx16.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="Thmproofx16.p1.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="Thmproofx16.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="Thmproofx16.p1.1.m1.2.2.1.1.1.1.2">𝒰</ci><cn id="Thmproofx16.p1.1.m1.2.2.1.1.1.1.3.cmml" type="integer" xref="Thmproofx16.p1.1.m1.2.2.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx16.p1.1.m1.2c">X\in\operatorname{conv}(\mathcal{U}_{1})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx16.p1.1.m1.2d">italic_X ∈ roman_conv ( caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, by concavity of <math alttext="F^{1}_{n}" class="ltx_Math" display="inline" id="Thmproofx16.p1.2.m2.1"><semantics id="Thmproofx16.p1.2.m2.1a"><msubsup id="Thmproofx16.p1.2.m2.1.1" xref="Thmproofx16.p1.2.m2.1.1.cmml"><mi id="Thmproofx16.p1.2.m2.1.1.2.2" xref="Thmproofx16.p1.2.m2.1.1.2.2.cmml">F</mi><mi id="Thmproofx16.p1.2.m2.1.1.3" xref="Thmproofx16.p1.2.m2.1.1.3.cmml">n</mi><mn id="Thmproofx16.p1.2.m2.1.1.2.3" xref="Thmproofx16.p1.2.m2.1.1.2.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx16.p1.2.m2.1b"><apply id="Thmproofx16.p1.2.m2.1.1.cmml" xref="Thmproofx16.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmproofx16.p1.2.m2.1.1.1.cmml" xref="Thmproofx16.p1.2.m2.1.1">subscript</csymbol><apply id="Thmproofx16.p1.2.m2.1.1.2.cmml" xref="Thmproofx16.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmproofx16.p1.2.m2.1.1.2.1.cmml" xref="Thmproofx16.p1.2.m2.1.1">superscript</csymbol><ci id="Thmproofx16.p1.2.m2.1.1.2.2.cmml" xref="Thmproofx16.p1.2.m2.1.1.2.2">𝐹</ci><cn id="Thmproofx16.p1.2.m2.1.1.2.3.cmml" type="integer" xref="Thmproofx16.p1.2.m2.1.1.2.3">1</cn></apply><ci id="Thmproofx16.p1.2.m2.1.1.3.cmml" xref="Thmproofx16.p1.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx16.p1.2.m2.1c">F^{1}_{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx16.p1.2.m2.1d">italic_F start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> there holds <math alttext="F^{1}_{n}(X)\geq\min_{X\in\mathcal{U}_{1}}F^{1}_{n}\left(X\right)" class="ltx_Math" display="inline" id="Thmproofx16.p1.3.m3.2"><semantics id="Thmproofx16.p1.3.m3.2a"><mrow id="Thmproofx16.p1.3.m3.2.3" xref="Thmproofx16.p1.3.m3.2.3.cmml"><mrow id="Thmproofx16.p1.3.m3.2.3.2" xref="Thmproofx16.p1.3.m3.2.3.2.cmml"><msubsup id="Thmproofx16.p1.3.m3.2.3.2.2" xref="Thmproofx16.p1.3.m3.2.3.2.2.cmml"><mi id="Thmproofx16.p1.3.m3.2.3.2.2.2.2" xref="Thmproofx16.p1.3.m3.2.3.2.2.2.2.cmml">F</mi><mi id="Thmproofx16.p1.3.m3.2.3.2.2.3" xref="Thmproofx16.p1.3.m3.2.3.2.2.3.cmml">n</mi><mn id="Thmproofx16.p1.3.m3.2.3.2.2.2.3" xref="Thmproofx16.p1.3.m3.2.3.2.2.2.3.cmml">1</mn></msubsup><mo id="Thmproofx16.p1.3.m3.2.3.2.1" xref="Thmproofx16.p1.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="Thmproofx16.p1.3.m3.2.3.2.3.2" xref="Thmproofx16.p1.3.m3.2.3.2.cmml"><mo id="Thmproofx16.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="Thmproofx16.p1.3.m3.2.3.2.cmml">(</mo><mi id="Thmproofx16.p1.3.m3.1.1" xref="Thmproofx16.p1.3.m3.1.1.cmml">X</mi><mo id="Thmproofx16.p1.3.m3.2.3.2.3.2.2" stretchy="false" xref="Thmproofx16.p1.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx16.p1.3.m3.2.3.1" xref="Thmproofx16.p1.3.m3.2.3.1.cmml">≥</mo><mrow id="Thmproofx16.p1.3.m3.2.3.3" xref="Thmproofx16.p1.3.m3.2.3.3.cmml"><mrow id="Thmproofx16.p1.3.m3.2.3.3.2" xref="Thmproofx16.p1.3.m3.2.3.3.2.cmml"><msub id="Thmproofx16.p1.3.m3.2.3.3.2.1" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.cmml"><mi id="Thmproofx16.p1.3.m3.2.3.3.2.1.2" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.2.cmml">min</mi><mrow id="Thmproofx16.p1.3.m3.2.3.3.2.1.3" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.cmml"><mi id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.2" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.2.cmml">X</mi><mo id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.1" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.1.cmml">∈</mo><msub id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.2" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.2.cmml">𝒰</mi><mn id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.3" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.3.cmml">1</mn></msub></mrow></msub><mo id="Thmproofx16.p1.3.m3.2.3.3.2a" lspace="0.167em" xref="Thmproofx16.p1.3.m3.2.3.3.2.cmml">⁡</mo><msubsup id="Thmproofx16.p1.3.m3.2.3.3.2.2" xref="Thmproofx16.p1.3.m3.2.3.3.2.2.cmml"><mi id="Thmproofx16.p1.3.m3.2.3.3.2.2.2.2" xref="Thmproofx16.p1.3.m3.2.3.3.2.2.2.2.cmml">F</mi><mi id="Thmproofx16.p1.3.m3.2.3.3.2.2.3" xref="Thmproofx16.p1.3.m3.2.3.3.2.2.3.cmml">n</mi><mn id="Thmproofx16.p1.3.m3.2.3.3.2.2.2.3" xref="Thmproofx16.p1.3.m3.2.3.3.2.2.2.3.cmml">1</mn></msubsup></mrow><mo id="Thmproofx16.p1.3.m3.2.3.3.1" xref="Thmproofx16.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="Thmproofx16.p1.3.m3.2.3.3.3.2" xref="Thmproofx16.p1.3.m3.2.3.3.cmml"><mo id="Thmproofx16.p1.3.m3.2.3.3.3.2.1" xref="Thmproofx16.p1.3.m3.2.3.3.cmml">(</mo><mi id="Thmproofx16.p1.3.m3.2.2" xref="Thmproofx16.p1.3.m3.2.2.cmml">X</mi><mo id="Thmproofx16.p1.3.m3.2.3.3.3.2.2" xref="Thmproofx16.p1.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx16.p1.3.m3.2b"><apply id="Thmproofx16.p1.3.m3.2.3.cmml" xref="Thmproofx16.p1.3.m3.2.3"><geq id="Thmproofx16.p1.3.m3.2.3.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.1"></geq><apply id="Thmproofx16.p1.3.m3.2.3.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.2"><times id="Thmproofx16.p1.3.m3.2.3.2.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.2.1"></times><apply id="Thmproofx16.p1.3.m3.2.3.2.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="Thmproofx16.p1.3.m3.2.3.2.2.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.2.2">subscript</csymbol><apply id="Thmproofx16.p1.3.m3.2.3.2.2.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="Thmproofx16.p1.3.m3.2.3.2.2.2.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.2.2">superscript</csymbol><ci id="Thmproofx16.p1.3.m3.2.3.2.2.2.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.2.2.2.2">𝐹</ci><cn id="Thmproofx16.p1.3.m3.2.3.2.2.2.3.cmml" type="integer" xref="Thmproofx16.p1.3.m3.2.3.2.2.2.3">1</cn></apply><ci id="Thmproofx16.p1.3.m3.2.3.2.2.3.cmml" xref="Thmproofx16.p1.3.m3.2.3.2.2.3">𝑛</ci></apply><ci id="Thmproofx16.p1.3.m3.1.1.cmml" xref="Thmproofx16.p1.3.m3.1.1">𝑋</ci></apply><apply id="Thmproofx16.p1.3.m3.2.3.3.cmml" xref="Thmproofx16.p1.3.m3.2.3.3"><times id="Thmproofx16.p1.3.m3.2.3.3.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.1"></times><apply id="Thmproofx16.p1.3.m3.2.3.3.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2"><apply id="Thmproofx16.p1.3.m3.2.3.3.2.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1"><csymbol cd="ambiguous" id="Thmproofx16.p1.3.m3.2.3.3.2.1.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1">subscript</csymbol><min id="Thmproofx16.p1.3.m3.2.3.3.2.1.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.2"></min><apply id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3"><in id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.1"></in><ci id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.2">𝑋</ci><apply id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3"><csymbol cd="ambiguous" id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3">subscript</csymbol><ci id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.2">𝒰</ci><cn id="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.3.cmml" type="integer" xref="Thmproofx16.p1.3.m3.2.3.3.2.1.3.3.3">1</cn></apply></apply></apply><apply id="Thmproofx16.p1.3.m3.2.3.3.2.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.2"><csymbol cd="ambiguous" id="Thmproofx16.p1.3.m3.2.3.3.2.2.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.2">subscript</csymbol><apply id="Thmproofx16.p1.3.m3.2.3.3.2.2.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.2"><csymbol cd="ambiguous" id="Thmproofx16.p1.3.m3.2.3.3.2.2.2.1.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.2">superscript</csymbol><ci id="Thmproofx16.p1.3.m3.2.3.3.2.2.2.2.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.2.2.2">𝐹</ci><cn id="Thmproofx16.p1.3.m3.2.3.3.2.2.2.3.cmml" type="integer" xref="Thmproofx16.p1.3.m3.2.3.3.2.2.2.3">1</cn></apply><ci id="Thmproofx16.p1.3.m3.2.3.3.2.2.3.cmml" xref="Thmproofx16.p1.3.m3.2.3.3.2.2.3">𝑛</ci></apply></apply><ci id="Thmproofx16.p1.3.m3.2.2.cmml" xref="Thmproofx16.p1.3.m3.2.2">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx16.p1.3.m3.2c">F^{1}_{n}(X)\geq\min_{X\in\mathcal{U}_{1}}F^{1}_{n}\left(X\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx16.p1.3.m3.2d">italic_F start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_X ) ≥ roman_min start_POSTSUBSCRIPT italic_X ∈ caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math>. Then, by the lemmas above,</p> <table class="ltx_equation ltx_eqn_table" id="Pt0.A3.Ex65"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\min_{X\in\mathcal{U}_{1}}F^{1}_{n}\left(X\right)=\min_{\sigma\in\mathcal{U}_{% n}}F_{n}(\sigma,\alpha)=\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="block" id="Pt0.A3.Ex65.m1.6"><semantics id="Pt0.A3.Ex65.m1.6a"><mrow id="Pt0.A3.Ex65.m1.6.7" xref="Pt0.A3.Ex65.m1.6.7.cmml"><mrow id="Pt0.A3.Ex65.m1.6.7.2" xref="Pt0.A3.Ex65.m1.6.7.2.cmml"><mrow id="Pt0.A3.Ex65.m1.6.7.2.2" xref="Pt0.A3.Ex65.m1.6.7.2.2.cmml"><munder id="Pt0.A3.Ex65.m1.6.7.2.2.1" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.cmml"><mi id="Pt0.A3.Ex65.m1.6.7.2.2.1.2" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.2.cmml">min</mi><mrow id="Pt0.A3.Ex65.m1.6.7.2.2.1.3" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.3.cmml"><mi id="Pt0.A3.Ex65.m1.6.7.2.2.1.3.2" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.3.2.cmml">X</mi><mo id="Pt0.A3.Ex65.m1.6.7.2.2.1.3.1" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.3.1.cmml">∈</mo><msub id="Pt0.A3.Ex65.m1.6.7.2.2.1.3.3" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A3.Ex65.m1.6.7.2.2.1.3.3.2" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.3.3.2.cmml">𝒰</mi><mn id="Pt0.A3.Ex65.m1.6.7.2.2.1.3.3.3" xref="Pt0.A3.Ex65.m1.6.7.2.2.1.3.3.3.cmml">1</mn></msub></mrow></munder><mo id="Pt0.A3.Ex65.m1.6.7.2.2a" lspace="0.167em" xref="Pt0.A3.Ex65.m1.6.7.2.2.cmml">⁡</mo><msubsup id="Pt0.A3.Ex65.m1.6.7.2.2.2" xref="Pt0.A3.Ex65.m1.6.7.2.2.2.cmml"><mi id="Pt0.A3.Ex65.m1.6.7.2.2.2.2.2" xref="Pt0.A3.Ex65.m1.6.7.2.2.2.2.2.cmml">F</mi><mi id="Pt0.A3.Ex65.m1.6.7.2.2.2.3" xref="Pt0.A3.Ex65.m1.6.7.2.2.2.3.cmml">n</mi><mn id="Pt0.A3.Ex65.m1.6.7.2.2.2.2.3" xref="Pt0.A3.Ex65.m1.6.7.2.2.2.2.3.cmml">1</mn></msubsup></mrow><mo id="Pt0.A3.Ex65.m1.6.7.2.1" xref="Pt0.A3.Ex65.m1.6.7.2.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex65.m1.6.7.2.3.2" xref="Pt0.A3.Ex65.m1.6.7.2.cmml"><mo id="Pt0.A3.Ex65.m1.6.7.2.3.2.1" xref="Pt0.A3.Ex65.m1.6.7.2.cmml">(</mo><mi id="Pt0.A3.Ex65.m1.2.2" xref="Pt0.A3.Ex65.m1.2.2.cmml">X</mi><mo id="Pt0.A3.Ex65.m1.6.7.2.3.2.2" xref="Pt0.A3.Ex65.m1.6.7.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A3.Ex65.m1.6.7.3" xref="Pt0.A3.Ex65.m1.6.7.3.cmml">=</mo><mrow id="Pt0.A3.Ex65.m1.6.7.4" xref="Pt0.A3.Ex65.m1.6.7.4.cmml"><mrow id="Pt0.A3.Ex65.m1.6.7.4.2" xref="Pt0.A3.Ex65.m1.6.7.4.2.cmml"><munder id="Pt0.A3.Ex65.m1.6.7.4.2.1" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.cmml"><mi id="Pt0.A3.Ex65.m1.6.7.4.2.1.2" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.2.cmml">min</mi><mrow id="Pt0.A3.Ex65.m1.6.7.4.2.1.3" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.3.cmml"><mi id="Pt0.A3.Ex65.m1.6.7.4.2.1.3.2" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.3.2.cmml">σ</mi><mo id="Pt0.A3.Ex65.m1.6.7.4.2.1.3.1" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.3.1.cmml">∈</mo><msub id="Pt0.A3.Ex65.m1.6.7.4.2.1.3.3" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A3.Ex65.m1.6.7.4.2.1.3.3.2" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.3.3.2.cmml">𝒰</mi><mi id="Pt0.A3.Ex65.m1.6.7.4.2.1.3.3.3" xref="Pt0.A3.Ex65.m1.6.7.4.2.1.3.3.3.cmml">n</mi></msub></mrow></munder><mo id="Pt0.A3.Ex65.m1.6.7.4.2a" lspace="0.167em" xref="Pt0.A3.Ex65.m1.6.7.4.2.cmml">⁡</mo><msub id="Pt0.A3.Ex65.m1.6.7.4.2.2" xref="Pt0.A3.Ex65.m1.6.7.4.2.2.cmml"><mi id="Pt0.A3.Ex65.m1.6.7.4.2.2.2" xref="Pt0.A3.Ex65.m1.6.7.4.2.2.2.cmml">F</mi><mi id="Pt0.A3.Ex65.m1.6.7.4.2.2.3" xref="Pt0.A3.Ex65.m1.6.7.4.2.2.3.cmml">n</mi></msub></mrow><mo id="Pt0.A3.Ex65.m1.6.7.4.1" xref="Pt0.A3.Ex65.m1.6.7.4.1.cmml">⁢</mo><mrow id="Pt0.A3.Ex65.m1.6.7.4.3.2" xref="Pt0.A3.Ex65.m1.6.7.4.3.1.cmml"><mo id="Pt0.A3.Ex65.m1.6.7.4.3.2.1" stretchy="false" xref="Pt0.A3.Ex65.m1.6.7.4.3.1.cmml">(</mo><mi id="Pt0.A3.Ex65.m1.3.3" xref="Pt0.A3.Ex65.m1.3.3.cmml">σ</mi><mo id="Pt0.A3.Ex65.m1.6.7.4.3.2.2" xref="Pt0.A3.Ex65.m1.6.7.4.3.1.cmml">,</mo><mi id="Pt0.A3.Ex65.m1.4.4" xref="Pt0.A3.Ex65.m1.4.4.cmml">α</mi><mo id="Pt0.A3.Ex65.m1.6.7.4.3.2.3" stretchy="false" xref="Pt0.A3.Ex65.m1.6.7.4.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A3.Ex65.m1.6.7.5" xref="Pt0.A3.Ex65.m1.6.7.5.cmml">=</mo><mrow id="Pt0.A3.Ex65.m1.6.7.6" xref="Pt0.A3.Ex65.m1.6.7.6.cmml"><mrow id="Pt0.A3.Ex65.m1.6.7.6.2" xref="Pt0.A3.Ex65.m1.6.7.6.2.cmml"><munder id="Pt0.A3.Ex65.m1.6.7.6.2.1" xref="Pt0.A3.Ex65.m1.6.7.6.2.1.cmml"><mi id="Pt0.A3.Ex65.m1.6.7.6.2.1.2" xref="Pt0.A3.Ex65.m1.6.7.6.2.1.2.cmml">min</mi><mrow id="Pt0.A3.Ex65.m1.1.1.1" xref="Pt0.A3.Ex65.m1.1.1.1.cmml"><mi id="Pt0.A3.Ex65.m1.1.1.1.3" xref="Pt0.A3.Ex65.m1.1.1.1.3.cmml">σ</mi><mo id="Pt0.A3.Ex65.m1.1.1.1.2" xref="Pt0.A3.Ex65.m1.1.1.1.2.cmml">∈</mo><msup id="Pt0.A3.Ex65.m1.1.1.1.1" xref="Pt0.A3.Ex65.m1.1.1.1.1.cmml"><mrow id="Pt0.A3.Ex65.m1.1.1.1.1.1.1" xref="Pt0.A3.Ex65.m1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A3.Ex65.m1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A3.Ex65.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A3.Ex65.m1.1.1.1.1.1.1.1" 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start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx16.p1.4"><math alttext="r=r_{U}" class="ltx_Math" display="inline" id="Thmproofx16.p1.4.m1.1"><semantics id="Thmproofx16.p1.4.m1.1a"><mrow id="Thmproofx16.p1.4.m1.1.1" xref="Thmproofx16.p1.4.m1.1.1.cmml"><mi id="Thmproofx16.p1.4.m1.1.1.2" xref="Thmproofx16.p1.4.m1.1.1.2.cmml">r</mi><mo id="Thmproofx16.p1.4.m1.1.1.1" xref="Thmproofx16.p1.4.m1.1.1.1.cmml">=</mo><msub id="Thmproofx16.p1.4.m1.1.1.3" xref="Thmproofx16.p1.4.m1.1.1.3.cmml"><mi id="Thmproofx16.p1.4.m1.1.1.3.2" 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id="Thmproofx16.p1.4.m1.1d">italic_r = italic_r start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> follows by definition. ∎</p> </div> </div> </section> <section class="ltx_appendix" id="Pt0.A4"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix 0.D </span>Omitted Details in Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a> </h2> <section class="ltx_subsection" id="Pt0.A4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">0.D.1 </span>Illustrating example</h3> <div class="ltx_theorem ltx_theorem_example" id="Thmexample2"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 2</span></h6> <div class="ltx_para" id="Thmexample2.p1"> <p class="ltx_p" id="Thmexample2.p1.1">Consider the graph in Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.F2" title="Figure 2 ‣ Example 2 ‣ 0.D.1 Illustrating example ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">2</span></a>:</p> </div> <figure class="ltx_figure" id="Pt0.A4.F2"><svg class="ltx_picture ltx_centering" height="77.83" id="Pt0.A4.F2.pic1" overflow="visible" version="1.1" width="180.31"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,77.83) matrix(1 0 0 -1 0 0) translate(11.21,0) translate(0,32.87)"><path d="M 10.93 0 C 10.93 6.04 6.04 10.93 0 10.93 C -6.04 10.93 -10.93 6.04 -10.93 0 C -10.93 -6.04 -6.04 -10.93 0 -10.93 C 6.04 -10.93 10.93 -6.04 10.93 0 Z M 0 0" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 -3.24 -2.98)"><foreignobject height="5.96" overflow="visible" 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id="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1" mathvariant="normal" xref="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.F2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1d">roman_ℓ</annotation></semantics></math></foreignobject></g><path d="M 9.7 -5.6 C 56.17 -32.52 100.86 -32.59 144.53 -7.47" style="fill:none"></path><g transform="matrix(0.86679 0.49867 -0.49867 0.86679 144.53 -7.47)"><path d="M 3.32 0 C 1.94 0.28 -0.55 0.83 -2.21 2.08 C -0.83 0.55 -0.83 -0.55 -2.21 -2.08 C -0.55 -0.83 1.94 -0.28 3.32 0 Z" style="stroke:none"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 75.07 -20.96)"><foreignobject height="8.92" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.92"><math alttext="1" class="ltx_Math" display="inline" id="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1a"><mn id="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1.1" xref="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1b"><cn id="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" type="integer" xref="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.F2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.1d">1</annotation></semantics></math></foreignobject></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Illustrating example of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem2" title="Theorem 5.2 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.2</span></a></figcaption> </figure> <div class="ltx_para" id="Thmexample2.p2"> <p class="ltx_p" id="Thmexample2.p2.14">where <math alttext="1<\ell\in\mathbb{N}" class="ltx_Math" display="inline" id="Thmexample2.p2.1.m1.1"><semantics id="Thmexample2.p2.1.m1.1a"><mrow id="Thmexample2.p2.1.m1.1.1" xref="Thmexample2.p2.1.m1.1.1.cmml"><mn id="Thmexample2.p2.1.m1.1.1.2" xref="Thmexample2.p2.1.m1.1.1.2.cmml">1</mn><mo id="Thmexample2.p2.1.m1.1.1.3" xref="Thmexample2.p2.1.m1.1.1.3.cmml"><</mo><mi id="Thmexample2.p2.1.m1.1.1.4" mathvariant="normal" xref="Thmexample2.p2.1.m1.1.1.4.cmml">ℓ</mi><mo id="Thmexample2.p2.1.m1.1.1.5" xref="Thmexample2.p2.1.m1.1.1.5.cmml">∈</mo><mi id="Thmexample2.p2.1.m1.1.1.6" xref="Thmexample2.p2.1.m1.1.1.6.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.1.m1.1b"><apply id="Thmexample2.p2.1.m1.1.1.cmml" xref="Thmexample2.p2.1.m1.1.1"><and id="Thmexample2.p2.1.m1.1.1a.cmml" xref="Thmexample2.p2.1.m1.1.1"></and><apply id="Thmexample2.p2.1.m1.1.1b.cmml" xref="Thmexample2.p2.1.m1.1.1"><lt id="Thmexample2.p2.1.m1.1.1.3.cmml" xref="Thmexample2.p2.1.m1.1.1.3"></lt><cn id="Thmexample2.p2.1.m1.1.1.2.cmml" type="integer" xref="Thmexample2.p2.1.m1.1.1.2">1</cn><ci id="Thmexample2.p2.1.m1.1.1.4.cmml" xref="Thmexample2.p2.1.m1.1.1.4">ℓ</ci></apply><apply id="Thmexample2.p2.1.m1.1.1c.cmml" xref="Thmexample2.p2.1.m1.1.1"><in id="Thmexample2.p2.1.m1.1.1.5.cmml" xref="Thmexample2.p2.1.m1.1.1.5"></in><share href="https://arxiv.org/html/2308.08047v3#Thmexample2.p2.1.m1.1.1.4.cmml" id="Thmexample2.p2.1.m1.1.1d.cmml" xref="Thmexample2.p2.1.m1.1.1"></share><ci id="Thmexample2.p2.1.m1.1.1.6.cmml" xref="Thmexample2.p2.1.m1.1.1.6">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.1.m1.1c">1<\ell\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.1.m1.1d">1 < roman_ℓ ∈ blackboard_N</annotation></semantics></math> are such that the 2 edges represent paths of the corresponding lengths. Let <math alttext="n>k=1" class="ltx_Math" display="inline" id="Thmexample2.p2.2.m2.1"><semantics id="Thmexample2.p2.2.m2.1a"><mrow id="Thmexample2.p2.2.m2.1.1" xref="Thmexample2.p2.2.m2.1.1.cmml"><mi id="Thmexample2.p2.2.m2.1.1.2" xref="Thmexample2.p2.2.m2.1.1.2.cmml">n</mi><mo id="Thmexample2.p2.2.m2.1.1.3" xref="Thmexample2.p2.2.m2.1.1.3.cmml">></mo><mi id="Thmexample2.p2.2.m2.1.1.4" xref="Thmexample2.p2.2.m2.1.1.4.cmml">k</mi><mo id="Thmexample2.p2.2.m2.1.1.5" xref="Thmexample2.p2.2.m2.1.1.5.cmml">=</mo><mn id="Thmexample2.p2.2.m2.1.1.6" xref="Thmexample2.p2.2.m2.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.2.m2.1b"><apply id="Thmexample2.p2.2.m2.1.1.cmml" xref="Thmexample2.p2.2.m2.1.1"><and id="Thmexample2.p2.2.m2.1.1a.cmml" xref="Thmexample2.p2.2.m2.1.1"></and><apply id="Thmexample2.p2.2.m2.1.1b.cmml" xref="Thmexample2.p2.2.m2.1.1"><gt id="Thmexample2.p2.2.m2.1.1.3.cmml" xref="Thmexample2.p2.2.m2.1.1.3"></gt><ci id="Thmexample2.p2.2.m2.1.1.2.cmml" xref="Thmexample2.p2.2.m2.1.1.2">𝑛</ci><ci id="Thmexample2.p2.2.m2.1.1.4.cmml" xref="Thmexample2.p2.2.m2.1.1.4">𝑘</ci></apply><apply id="Thmexample2.p2.2.m2.1.1c.cmml" xref="Thmexample2.p2.2.m2.1.1"><eq id="Thmexample2.p2.2.m2.1.1.5.cmml" xref="Thmexample2.p2.2.m2.1.1.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmexample2.p2.2.m2.1.1.4.cmml" id="Thmexample2.p2.2.m2.1.1d.cmml" xref="Thmexample2.p2.2.m2.1.1"></share><cn id="Thmexample2.p2.2.m2.1.1.6.cmml" type="integer" xref="Thmexample2.p2.2.m2.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.2.m2.1c">n>k=1</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.2.m2.1d">italic_n > italic_k = 1</annotation></semantics></math> and <math alttext="\alpha\geq\ell-1" class="ltx_Math" display="inline" id="Thmexample2.p2.3.m3.1"><semantics id="Thmexample2.p2.3.m3.1a"><mrow id="Thmexample2.p2.3.m3.1.1" xref="Thmexample2.p2.3.m3.1.1.cmml"><mi id="Thmexample2.p2.3.m3.1.1.2" xref="Thmexample2.p2.3.m3.1.1.2.cmml">α</mi><mo id="Thmexample2.p2.3.m3.1.1.1" xref="Thmexample2.p2.3.m3.1.1.1.cmml">≥</mo><mrow id="Thmexample2.p2.3.m3.1.1.3" xref="Thmexample2.p2.3.m3.1.1.3.cmml"><mi id="Thmexample2.p2.3.m3.1.1.3.2" mathvariant="normal" xref="Thmexample2.p2.3.m3.1.1.3.2.cmml">ℓ</mi><mo id="Thmexample2.p2.3.m3.1.1.3.1" xref="Thmexample2.p2.3.m3.1.1.3.1.cmml">−</mo><mn id="Thmexample2.p2.3.m3.1.1.3.3" xref="Thmexample2.p2.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.3.m3.1b"><apply id="Thmexample2.p2.3.m3.1.1.cmml" xref="Thmexample2.p2.3.m3.1.1"><geq id="Thmexample2.p2.3.m3.1.1.1.cmml" xref="Thmexample2.p2.3.m3.1.1.1"></geq><ci id="Thmexample2.p2.3.m3.1.1.2.cmml" xref="Thmexample2.p2.3.m3.1.1.2">𝛼</ci><apply id="Thmexample2.p2.3.m3.1.1.3.cmml" xref="Thmexample2.p2.3.m3.1.1.3"><minus id="Thmexample2.p2.3.m3.1.1.3.1.cmml" xref="Thmexample2.p2.3.m3.1.1.3.1"></minus><ci id="Thmexample2.p2.3.m3.1.1.3.2.cmml" xref="Thmexample2.p2.3.m3.1.1.3.2">ℓ</ci><cn id="Thmexample2.p2.3.m3.1.1.3.3.cmml" type="integer" xref="Thmexample2.p2.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.3.m3.1c">\alpha\geq\ell-1</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.3.m3.1d">italic_α ≥ roman_ℓ - 1</annotation></semantics></math>. Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma13" title="Lemma 13 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">13</span></a> will show that we can assume for minimization that the agents’ strategies are iid. Then we can identify their iid strategies with <math alttext="x\in[0,1]" class="ltx_Math" display="inline" id="Thmexample2.p2.4.m4.2"><semantics id="Thmexample2.p2.4.m4.2a"><mrow id="Thmexample2.p2.4.m4.2.3" xref="Thmexample2.p2.4.m4.2.3.cmml"><mi id="Thmexample2.p2.4.m4.2.3.2" xref="Thmexample2.p2.4.m4.2.3.2.cmml">x</mi><mo id="Thmexample2.p2.4.m4.2.3.1" xref="Thmexample2.p2.4.m4.2.3.1.cmml">∈</mo><mrow id="Thmexample2.p2.4.m4.2.3.3.2" xref="Thmexample2.p2.4.m4.2.3.3.1.cmml"><mo id="Thmexample2.p2.4.m4.2.3.3.2.1" stretchy="false" xref="Thmexample2.p2.4.m4.2.3.3.1.cmml">[</mo><mn id="Thmexample2.p2.4.m4.1.1" xref="Thmexample2.p2.4.m4.1.1.cmml">0</mn><mo id="Thmexample2.p2.4.m4.2.3.3.2.2" xref="Thmexample2.p2.4.m4.2.3.3.1.cmml">,</mo><mn id="Thmexample2.p2.4.m4.2.2" xref="Thmexample2.p2.4.m4.2.2.cmml">1</mn><mo id="Thmexample2.p2.4.m4.2.3.3.2.3" stretchy="false" xref="Thmexample2.p2.4.m4.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.4.m4.2b"><apply id="Thmexample2.p2.4.m4.2.3.cmml" xref="Thmexample2.p2.4.m4.2.3"><in id="Thmexample2.p2.4.m4.2.3.1.cmml" xref="Thmexample2.p2.4.m4.2.3.1"></in><ci id="Thmexample2.p2.4.m4.2.3.2.cmml" xref="Thmexample2.p2.4.m4.2.3.2">𝑥</ci><interval closure="closed" id="Thmexample2.p2.4.m4.2.3.3.1.cmml" xref="Thmexample2.p2.4.m4.2.3.3.2"><cn id="Thmexample2.p2.4.m4.1.1.cmml" type="integer" xref="Thmexample2.p2.4.m4.1.1">0</cn><cn id="Thmexample2.p2.4.m4.2.2.cmml" type="integer" xref="Thmexample2.p2.4.m4.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.4.m4.2c">x\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.4.m4.2d">italic_x ∈ [ 0 , 1 ]</annotation></semantics></math>, where <math alttext="x" class="ltx_Math" display="inline" id="Thmexample2.p2.5.m5.1"><semantics id="Thmexample2.p2.5.m5.1a"><mi id="Thmexample2.p2.5.m5.1.1" xref="Thmexample2.p2.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.5.m5.1b"><ci id="Thmexample2.p2.5.m5.1.1.cmml" xref="Thmexample2.p2.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.5.m5.1d">italic_x</annotation></semantics></math> is the probability that a agent chooses the short (bottom) path. Similarly, because the paths are disjoint, we identify the interceptor’s strategy with <math alttext="y\in[0,1]" class="ltx_Math" display="inline" id="Thmexample2.p2.6.m6.2"><semantics id="Thmexample2.p2.6.m6.2a"><mrow id="Thmexample2.p2.6.m6.2.3" xref="Thmexample2.p2.6.m6.2.3.cmml"><mi id="Thmexample2.p2.6.m6.2.3.2" xref="Thmexample2.p2.6.m6.2.3.2.cmml">y</mi><mo id="Thmexample2.p2.6.m6.2.3.1" xref="Thmexample2.p2.6.m6.2.3.1.cmml">∈</mo><mrow id="Thmexample2.p2.6.m6.2.3.3.2" xref="Thmexample2.p2.6.m6.2.3.3.1.cmml"><mo id="Thmexample2.p2.6.m6.2.3.3.2.1" stretchy="false" xref="Thmexample2.p2.6.m6.2.3.3.1.cmml">[</mo><mn id="Thmexample2.p2.6.m6.1.1" xref="Thmexample2.p2.6.m6.1.1.cmml">0</mn><mo id="Thmexample2.p2.6.m6.2.3.3.2.2" xref="Thmexample2.p2.6.m6.2.3.3.1.cmml">,</mo><mn id="Thmexample2.p2.6.m6.2.2" xref="Thmexample2.p2.6.m6.2.2.cmml">1</mn><mo id="Thmexample2.p2.6.m6.2.3.3.2.3" stretchy="false" xref="Thmexample2.p2.6.m6.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.6.m6.2b"><apply id="Thmexample2.p2.6.m6.2.3.cmml" xref="Thmexample2.p2.6.m6.2.3"><in id="Thmexample2.p2.6.m6.2.3.1.cmml" xref="Thmexample2.p2.6.m6.2.3.1"></in><ci id="Thmexample2.p2.6.m6.2.3.2.cmml" xref="Thmexample2.p2.6.m6.2.3.2">𝑦</ci><interval closure="closed" id="Thmexample2.p2.6.m6.2.3.3.1.cmml" xref="Thmexample2.p2.6.m6.2.3.3.2"><cn id="Thmexample2.p2.6.m6.1.1.cmml" type="integer" xref="Thmexample2.p2.6.m6.1.1">0</cn><cn id="Thmexample2.p2.6.m6.2.2.cmml" type="integer" xref="Thmexample2.p2.6.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.6.m6.2c">y\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.6.m6.2d">italic_y ∈ [ 0 , 1 ]</annotation></semantics></math>. Now we can write in short <math alttext="C(x,y,\alpha)" class="ltx_Math" display="inline" id="Thmexample2.p2.7.m7.3"><semantics id="Thmexample2.p2.7.m7.3a"><mrow id="Thmexample2.p2.7.m7.3.4" xref="Thmexample2.p2.7.m7.3.4.cmml"><mi id="Thmexample2.p2.7.m7.3.4.2" xref="Thmexample2.p2.7.m7.3.4.2.cmml">C</mi><mo id="Thmexample2.p2.7.m7.3.4.1" xref="Thmexample2.p2.7.m7.3.4.1.cmml">⁢</mo><mrow id="Thmexample2.p2.7.m7.3.4.3.2" xref="Thmexample2.p2.7.m7.3.4.3.1.cmml"><mo id="Thmexample2.p2.7.m7.3.4.3.2.1" stretchy="false" xref="Thmexample2.p2.7.m7.3.4.3.1.cmml">(</mo><mi id="Thmexample2.p2.7.m7.1.1" xref="Thmexample2.p2.7.m7.1.1.cmml">x</mi><mo id="Thmexample2.p2.7.m7.3.4.3.2.2" xref="Thmexample2.p2.7.m7.3.4.3.1.cmml">,</mo><mi id="Thmexample2.p2.7.m7.2.2" xref="Thmexample2.p2.7.m7.2.2.cmml">y</mi><mo id="Thmexample2.p2.7.m7.3.4.3.2.3" xref="Thmexample2.p2.7.m7.3.4.3.1.cmml">,</mo><mi id="Thmexample2.p2.7.m7.3.3" xref="Thmexample2.p2.7.m7.3.3.cmml">α</mi><mo id="Thmexample2.p2.7.m7.3.4.3.2.4" stretchy="false" xref="Thmexample2.p2.7.m7.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.7.m7.3b"><apply id="Thmexample2.p2.7.m7.3.4.cmml" xref="Thmexample2.p2.7.m7.3.4"><times id="Thmexample2.p2.7.m7.3.4.1.cmml" xref="Thmexample2.p2.7.m7.3.4.1"></times><ci id="Thmexample2.p2.7.m7.3.4.2.cmml" xref="Thmexample2.p2.7.m7.3.4.2">𝐶</ci><vector id="Thmexample2.p2.7.m7.3.4.3.1.cmml" xref="Thmexample2.p2.7.m7.3.4.3.2"><ci id="Thmexample2.p2.7.m7.1.1.cmml" xref="Thmexample2.p2.7.m7.1.1">𝑥</ci><ci id="Thmexample2.p2.7.m7.2.2.cmml" xref="Thmexample2.p2.7.m7.2.2">𝑦</ci><ci id="Thmexample2.p2.7.m7.3.3.cmml" xref="Thmexample2.p2.7.m7.3.3">𝛼</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.7.m7.3c">C(x,y,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.7.m7.3d">italic_C ( italic_x , italic_y , italic_α )</annotation></semantics></math> for the corresponding cost. We show that for every such <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmexample2.p2.8.m8.1"><semantics id="Thmexample2.p2.8.m8.1a"><mi id="Thmexample2.p2.8.m8.1.1" xref="Thmexample2.p2.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.8.m8.1b"><ci id="Thmexample2.p2.8.m8.1.1.cmml" xref="Thmexample2.p2.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.8.m8.1d">italic_α</annotation></semantics></math>, <math alttext="x^{*}:=\operatorname*{arg\,min}_{x}\max_{y}C(x,y,\alpha)\in\left(2^{-1},2^{-1/% n}\right]" class="ltx_Math" display="inline" id="Thmexample2.p2.9.m9.5"><semantics id="Thmexample2.p2.9.m9.5a"><mrow id="Thmexample2.p2.9.m9.5.5" xref="Thmexample2.p2.9.m9.5.5.cmml"><msup id="Thmexample2.p2.9.m9.5.5.4" xref="Thmexample2.p2.9.m9.5.5.4.cmml"><mi id="Thmexample2.p2.9.m9.5.5.4.2" xref="Thmexample2.p2.9.m9.5.5.4.2.cmml">x</mi><mo id="Thmexample2.p2.9.m9.5.5.4.3" xref="Thmexample2.p2.9.m9.5.5.4.3.cmml">∗</mo></msup><mo id="Thmexample2.p2.9.m9.5.5.5" lspace="0.278em" rspace="0.278em" xref="Thmexample2.p2.9.m9.5.5.5.cmml">:=</mo><mrow id="Thmexample2.p2.9.m9.5.5.6" xref="Thmexample2.p2.9.m9.5.5.6.cmml"><mrow id="Thmexample2.p2.9.m9.5.5.6.2" xref="Thmexample2.p2.9.m9.5.5.6.2.cmml"><mrow id="Thmexample2.p2.9.m9.5.5.6.2.1" xref="Thmexample2.p2.9.m9.5.5.6.2.1.cmml"><msub id="Thmexample2.p2.9.m9.5.5.6.2.1.1" xref="Thmexample2.p2.9.m9.5.5.6.2.1.1.cmml"><mrow id="Thmexample2.p2.9.m9.5.5.6.2.1.1.2" xref="Thmexample2.p2.9.m9.5.5.6.2.1.1.2.cmml"><mi id="Thmexample2.p2.9.m9.5.5.6.2.1.1.2.2" xref="Thmexample2.p2.9.m9.5.5.6.2.1.1.2.2.cmml">arg</mi><mo id="Thmexample2.p2.9.m9.5.5.6.2.1.1.2.1" lspace="0.170em" xref="Thmexample2.p2.9.m9.5.5.6.2.1.1.2.1.cmml">⁢</mo><mi id="Thmexample2.p2.9.m9.5.5.6.2.1.1.2.3" xref="Thmexample2.p2.9.m9.5.5.6.2.1.1.2.3.cmml">min</mi></mrow><mi id="Thmexample2.p2.9.m9.5.5.6.2.1.1.3" xref="Thmexample2.p2.9.m9.5.5.6.2.1.1.3.cmml">x</mi></msub><mo id="Thmexample2.p2.9.m9.5.5.6.2.1a" lspace="0.167em" xref="Thmexample2.p2.9.m9.5.5.6.2.1.cmml">⁡</mo><msub id="Thmexample2.p2.9.m9.5.5.6.2.1.2" xref="Thmexample2.p2.9.m9.5.5.6.2.1.2.cmml"><mi id="Thmexample2.p2.9.m9.5.5.6.2.1.2.2" xref="Thmexample2.p2.9.m9.5.5.6.2.1.2.2.cmml">max</mi><mi id="Thmexample2.p2.9.m9.5.5.6.2.1.2.3" xref="Thmexample2.p2.9.m9.5.5.6.2.1.2.3.cmml">y</mi></msub></mrow><mo id="Thmexample2.p2.9.m9.5.5.6.2a" lspace="0.167em" xref="Thmexample2.p2.9.m9.5.5.6.2.cmml">⁡</mo><mi id="Thmexample2.p2.9.m9.5.5.6.2.2" xref="Thmexample2.p2.9.m9.5.5.6.2.2.cmml">C</mi></mrow><mo id="Thmexample2.p2.9.m9.5.5.6.1" xref="Thmexample2.p2.9.m9.5.5.6.1.cmml">⁢</mo><mrow id="Thmexample2.p2.9.m9.5.5.6.3.2" xref="Thmexample2.p2.9.m9.5.5.6.3.1.cmml"><mo id="Thmexample2.p2.9.m9.5.5.6.3.2.1" stretchy="false" xref="Thmexample2.p2.9.m9.5.5.6.3.1.cmml">(</mo><mi id="Thmexample2.p2.9.m9.1.1" xref="Thmexample2.p2.9.m9.1.1.cmml">x</mi><mo id="Thmexample2.p2.9.m9.5.5.6.3.2.2" xref="Thmexample2.p2.9.m9.5.5.6.3.1.cmml">,</mo><mi id="Thmexample2.p2.9.m9.2.2" xref="Thmexample2.p2.9.m9.2.2.cmml">y</mi><mo id="Thmexample2.p2.9.m9.5.5.6.3.2.3" xref="Thmexample2.p2.9.m9.5.5.6.3.1.cmml">,</mo><mi id="Thmexample2.p2.9.m9.3.3" xref="Thmexample2.p2.9.m9.3.3.cmml">α</mi><mo id="Thmexample2.p2.9.m9.5.5.6.3.2.4" stretchy="false" xref="Thmexample2.p2.9.m9.5.5.6.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p2.9.m9.5.5.7" xref="Thmexample2.p2.9.m9.5.5.7.cmml">∈</mo><mrow id="Thmexample2.p2.9.m9.5.5.2.2" xref="Thmexample2.p2.9.m9.5.5.2.3.cmml"><mo id="Thmexample2.p2.9.m9.5.5.2.2.3" xref="Thmexample2.p2.9.m9.5.5.2.3.cmml">(</mo><msup id="Thmexample2.p2.9.m9.4.4.1.1.1" xref="Thmexample2.p2.9.m9.4.4.1.1.1.cmml"><mn id="Thmexample2.p2.9.m9.4.4.1.1.1.2" xref="Thmexample2.p2.9.m9.4.4.1.1.1.2.cmml">2</mn><mrow id="Thmexample2.p2.9.m9.4.4.1.1.1.3" xref="Thmexample2.p2.9.m9.4.4.1.1.1.3.cmml"><mo 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id="Thmexample2.p2.10.m10.1.1.3.3.1.cmml" xref="Thmexample2.p2.10.m10.1.1.3.3"></minus><apply id="Thmexample2.p2.10.m10.1.1.3.3.2.cmml" xref="Thmexample2.p2.10.m10.1.1.3.3.2"><divide id="Thmexample2.p2.10.m10.1.1.3.3.2.1.cmml" xref="Thmexample2.p2.10.m10.1.1.3.3.2.1"></divide><cn id="Thmexample2.p2.10.m10.1.1.3.3.2.2.cmml" type="integer" xref="Thmexample2.p2.10.m10.1.1.3.3.2.2">1</cn><ci id="Thmexample2.p2.10.m10.1.1.3.3.2.3.cmml" xref="Thmexample2.p2.10.m10.1.1.3.3.2.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.10.m10.1c">x^{*}=2^{-1/n}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.10.m10.1d">italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 2 start_POSTSUPERSCRIPT - 1 / italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="\alpha=\ell-1" class="ltx_Math" display="inline" id="Thmexample2.p2.11.m11.1"><semantics id="Thmexample2.p2.11.m11.1a"><mrow id="Thmexample2.p2.11.m11.1.1" xref="Thmexample2.p2.11.m11.1.1.cmml"><mi id="Thmexample2.p2.11.m11.1.1.2" xref="Thmexample2.p2.11.m11.1.1.2.cmml">α</mi><mo id="Thmexample2.p2.11.m11.1.1.1" xref="Thmexample2.p2.11.m11.1.1.1.cmml">=</mo><mrow id="Thmexample2.p2.11.m11.1.1.3" xref="Thmexample2.p2.11.m11.1.1.3.cmml"><mi id="Thmexample2.p2.11.m11.1.1.3.2" mathvariant="normal" xref="Thmexample2.p2.11.m11.1.1.3.2.cmml">ℓ</mi><mo id="Thmexample2.p2.11.m11.1.1.3.1" xref="Thmexample2.p2.11.m11.1.1.3.1.cmml">−</mo><mn id="Thmexample2.p2.11.m11.1.1.3.3" xref="Thmexample2.p2.11.m11.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.11.m11.1b"><apply id="Thmexample2.p2.11.m11.1.1.cmml" xref="Thmexample2.p2.11.m11.1.1"><eq id="Thmexample2.p2.11.m11.1.1.1.cmml" xref="Thmexample2.p2.11.m11.1.1.1"></eq><ci id="Thmexample2.p2.11.m11.1.1.2.cmml" xref="Thmexample2.p2.11.m11.1.1.2">𝛼</ci><apply id="Thmexample2.p2.11.m11.1.1.3.cmml" xref="Thmexample2.p2.11.m11.1.1.3"><minus id="Thmexample2.p2.11.m11.1.1.3.1.cmml" xref="Thmexample2.p2.11.m11.1.1.3.1"></minus><ci id="Thmexample2.p2.11.m11.1.1.3.2.cmml" xref="Thmexample2.p2.11.m11.1.1.3.2">ℓ</ci><cn id="Thmexample2.p2.11.m11.1.1.3.3.cmml" type="integer" xref="Thmexample2.p2.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.11.m11.1c">\alpha=\ell-1</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.11.m11.1d">italic_α = roman_ℓ - 1</annotation></semantics></math>, and <math alttext="x^{*}" class="ltx_Math" display="inline" id="Thmexample2.p2.12.m12.1"><semantics id="Thmexample2.p2.12.m12.1a"><msup id="Thmexample2.p2.12.m12.1.1" xref="Thmexample2.p2.12.m12.1.1.cmml"><mi id="Thmexample2.p2.12.m12.1.1.2" xref="Thmexample2.p2.12.m12.1.1.2.cmml">x</mi><mo id="Thmexample2.p2.12.m12.1.1.3" xref="Thmexample2.p2.12.m12.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.12.m12.1b"><apply id="Thmexample2.p2.12.m12.1.1.cmml" xref="Thmexample2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="Thmexample2.p2.12.m12.1.1.1.cmml" xref="Thmexample2.p2.12.m12.1.1">superscript</csymbol><ci id="Thmexample2.p2.12.m12.1.1.2.cmml" xref="Thmexample2.p2.12.m12.1.1.2">𝑥</ci><times id="Thmexample2.p2.12.m12.1.1.3.cmml" xref="Thmexample2.p2.12.m12.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.12.m12.1c">x^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.12.m12.1d">italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> strictly decreases as <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmexample2.p2.13.m13.1"><semantics id="Thmexample2.p2.13.m13.1a"><mi id="Thmexample2.p2.13.m13.1.1" xref="Thmexample2.p2.13.m13.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmexample2.p2.13.m13.1b"><ci id="Thmexample2.p2.13.m13.1.1.cmml" xref="Thmexample2.p2.13.m13.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p2.13.m13.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.13.m13.1d">italic_α</annotation></semantics></math> increases. In particular, <math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}\max_{\tau}C(\sigma,\tau,\alpha)=:C(x^{*},% y^{*},\alpha)<\min_{\sigma\in\mathcal{U}_{n}}\max_{\tau}C(\sigma,\tau,\alpha)" class="ltx_math_unparsed" display="inline" id="Thmexample2.p2.14.m14.4"><semantics id="Thmexample2.p2.14.m14.4a"><mrow id="Thmexample2.p2.14.m14.4b"><msub id="Thmexample2.p2.14.m14.4.5"><mi id="Thmexample2.p2.14.m14.4.5.2">min</mi><mrow id="Thmexample2.p2.14.m14.1.1.1"><mi id="Thmexample2.p2.14.m14.1.1.1.3">σ</mi><mo id="Thmexample2.p2.14.m14.1.1.1.2">∈</mo><msup id="Thmexample2.p2.14.m14.1.1.1.1"><mrow id="Thmexample2.p2.14.m14.1.1.1.1.1.1"><mo id="Thmexample2.p2.14.m14.1.1.1.1.1.1.2" stretchy="false">(</mo><mrow id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1"><mi id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.3" mathvariant="normal">Δ</mi><mo id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.2">⁢</mo><mrow id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1"><mo id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.2" stretchy="false">(</mo><msub id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.1"><mi id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.1.2">P</mi><mrow id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.1.3"><mi id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.1.3.2">s</mi><mo id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.1.3.1">⁢</mo><mi id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.1.3.3">t</mi></mrow></msub><mo id="Thmexample2.p2.14.m14.1.1.1.1.1.1.1.1.1.3" stretchy="false">)</mo></mrow></mrow><mo id="Thmexample2.p2.14.m14.1.1.1.1.1.1.3" stretchy="false">)</mo></mrow><mi id="Thmexample2.p2.14.m14.1.1.1.1.3">n</mi></msup></mrow></msub><msub id="Thmexample2.p2.14.m14.4.6"><mi id="Thmexample2.p2.14.m14.4.6.2">max</mi><mi id="Thmexample2.p2.14.m14.4.6.3">τ</mi></msub><mi id="Thmexample2.p2.14.m14.4.7">C</mi><mrow id="Thmexample2.p2.14.m14.4.8"><mo id="Thmexample2.p2.14.m14.4.8.1" stretchy="false">(</mo><mi id="Thmexample2.p2.14.m14.2.2">σ</mi><mo id="Thmexample2.p2.14.m14.4.8.2">,</mo><mi id="Thmexample2.p2.14.m14.3.3">τ</mi><mo id="Thmexample2.p2.14.m14.4.8.3">,</mo><mi id="Thmexample2.p2.14.m14.4.4">α</mi><mo id="Thmexample2.p2.14.m14.4.8.4" stretchy="false">)</mo></mrow><mo id="Thmexample2.p2.14.m14.4.9" rspace="0em">=</mo><mo id="Thmexample2.p2.14.m14.4.10" rspace="0.278em">:</mo><mi id="Thmexample2.p2.14.m14.4.11">C</mi><mrow id="Thmexample2.p2.14.m14.4.12"><mo id="Thmexample2.p2.14.m14.4.12.1" stretchy="false">(</mo><msup id="Thmexample2.p2.14.m14.4.12.2"><mi id="Thmexample2.p2.14.m14.4.12.2.2">x</mi><mo id="Thmexample2.p2.14.m14.4.12.2.3">∗</mo></msup><mo id="Thmexample2.p2.14.m14.4.12.3">,</mo><msup id="Thmexample2.p2.14.m14.4.12.4"><mi id="Thmexample2.p2.14.m14.4.12.4.2">y</mi><mo id="Thmexample2.p2.14.m14.4.12.4.3">∗</mo></msup><mo id="Thmexample2.p2.14.m14.4.12.5">,</mo><mi id="Thmexample2.p2.14.m14.4.12.6">α</mi><mo id="Thmexample2.p2.14.m14.4.12.7" stretchy="false">)</mo></mrow><mo id="Thmexample2.p2.14.m14.4.13"><</mo><msub id="Thmexample2.p2.14.m14.4.14"><mi id="Thmexample2.p2.14.m14.4.14.2">min</mi><mrow id="Thmexample2.p2.14.m14.4.14.3"><mi id="Thmexample2.p2.14.m14.4.14.3.2">σ</mi><mo id="Thmexample2.p2.14.m14.4.14.3.1">∈</mo><msub id="Thmexample2.p2.14.m14.4.14.3.3"><mi class="ltx_font_mathcaligraphic" id="Thmexample2.p2.14.m14.4.14.3.3.2">𝒰</mi><mi id="Thmexample2.p2.14.m14.4.14.3.3.3">n</mi></msub></mrow></msub><msub id="Thmexample2.p2.14.m14.4.15"><mi id="Thmexample2.p2.14.m14.4.15.2">max</mi><mi id="Thmexample2.p2.14.m14.4.15.3">τ</mi></msub><mi id="Thmexample2.p2.14.m14.4.16">C</mi><mrow id="Thmexample2.p2.14.m14.4.17"><mo id="Thmexample2.p2.14.m14.4.17.1" stretchy="false">(</mo><mi id="Thmexample2.p2.14.m14.4.17.2">σ</mi><mo id="Thmexample2.p2.14.m14.4.17.3">,</mo><mi id="Thmexample2.p2.14.m14.4.17.4">τ</mi><mo id="Thmexample2.p2.14.m14.4.17.5">,</mo><mi id="Thmexample2.p2.14.m14.4.17.6">α</mi><mo id="Thmexample2.p2.14.m14.4.17.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmexample2.p2.14.m14.4c">\min_{\sigma\in(\Delta(P_{st}))^{n}}\max_{\tau}C(\sigma,\tau,\alpha)=:C(x^{*},% y^{*},\alpha)<\min_{\sigma\in\mathcal{U}_{n}}\max_{\tau}C(\sigma,\tau,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p2.14.m14.4d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT italic_C ( italic_σ , italic_τ , italic_α ) = : italic_C ( italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_α ) < roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT italic_C ( italic_σ , italic_τ , italic_α )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmexample2.p3"> <p class="ltx_p" id="Thmexample2.p3.1">For <math alttext="x\in[0,1]" class="ltx_Math" display="inline" id="Thmexample2.p3.1.m1.2"><semantics id="Thmexample2.p3.1.m1.2a"><mrow id="Thmexample2.p3.1.m1.2.3" xref="Thmexample2.p3.1.m1.2.3.cmml"><mi id="Thmexample2.p3.1.m1.2.3.2" xref="Thmexample2.p3.1.m1.2.3.2.cmml">x</mi><mo id="Thmexample2.p3.1.m1.2.3.1" xref="Thmexample2.p3.1.m1.2.3.1.cmml">∈</mo><mrow id="Thmexample2.p3.1.m1.2.3.3.2" xref="Thmexample2.p3.1.m1.2.3.3.1.cmml"><mo id="Thmexample2.p3.1.m1.2.3.3.2.1" stretchy="false" xref="Thmexample2.p3.1.m1.2.3.3.1.cmml">[</mo><mn id="Thmexample2.p3.1.m1.1.1" xref="Thmexample2.p3.1.m1.1.1.cmml">0</mn><mo id="Thmexample2.p3.1.m1.2.3.3.2.2" xref="Thmexample2.p3.1.m1.2.3.3.1.cmml">,</mo><mn id="Thmexample2.p3.1.m1.2.2" xref="Thmexample2.p3.1.m1.2.2.cmml">1</mn><mo id="Thmexample2.p3.1.m1.2.3.3.2.3" stretchy="false" xref="Thmexample2.p3.1.m1.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.1.m1.2b"><apply id="Thmexample2.p3.1.m1.2.3.cmml" xref="Thmexample2.p3.1.m1.2.3"><in id="Thmexample2.p3.1.m1.2.3.1.cmml" xref="Thmexample2.p3.1.m1.2.3.1"></in><ci id="Thmexample2.p3.1.m1.2.3.2.cmml" xref="Thmexample2.p3.1.m1.2.3.2">𝑥</ci><interval closure="closed" id="Thmexample2.p3.1.m1.2.3.3.1.cmml" xref="Thmexample2.p3.1.m1.2.3.3.2"><cn id="Thmexample2.p3.1.m1.1.1.cmml" type="integer" xref="Thmexample2.p3.1.m1.1.1">0</cn><cn id="Thmexample2.p3.1.m1.2.2.cmml" type="integer" xref="Thmexample2.p3.1.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.1.m1.2c">x\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.1.m1.2d">italic_x ∈ [ 0 , 1 ]</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx41"> <tbody id="Pt0.A4.Ex66"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{1}(x,\alpha)=C(x,1,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex66.m1.5"><semantics id="Pt0.A4.Ex66.m1.5a"><mrow id="Pt0.A4.Ex66.m1.5.6" xref="Pt0.A4.Ex66.m1.5.6.cmml"><mrow id="Pt0.A4.Ex66.m1.5.6.2" xref="Pt0.A4.Ex66.m1.5.6.2.cmml"><msub id="Pt0.A4.Ex66.m1.5.6.2.2" xref="Pt0.A4.Ex66.m1.5.6.2.2.cmml"><mi id="Pt0.A4.Ex66.m1.5.6.2.2.2" xref="Pt0.A4.Ex66.m1.5.6.2.2.2.cmml">f</mi><mn id="Pt0.A4.Ex66.m1.5.6.2.2.3" xref="Pt0.A4.Ex66.m1.5.6.2.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex66.m1.5.6.2.1" xref="Pt0.A4.Ex66.m1.5.6.2.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex66.m1.5.6.2.3.2" xref="Pt0.A4.Ex66.m1.5.6.2.3.1.cmml"><mo id="Pt0.A4.Ex66.m1.5.6.2.3.2.1" stretchy="false" xref="Pt0.A4.Ex66.m1.5.6.2.3.1.cmml">(</mo><mi id="Pt0.A4.Ex66.m1.1.1" xref="Pt0.A4.Ex66.m1.1.1.cmml">x</mi><mo id="Pt0.A4.Ex66.m1.5.6.2.3.2.2" xref="Pt0.A4.Ex66.m1.5.6.2.3.1.cmml">,</mo><mi id="Pt0.A4.Ex66.m1.2.2" xref="Pt0.A4.Ex66.m1.2.2.cmml">α</mi><mo id="Pt0.A4.Ex66.m1.5.6.2.3.2.3" stretchy="false" xref="Pt0.A4.Ex66.m1.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex66.m1.5.6.1" xref="Pt0.A4.Ex66.m1.5.6.1.cmml">=</mo><mrow id="Pt0.A4.Ex66.m1.5.6.3" xref="Pt0.A4.Ex66.m1.5.6.3.cmml"><mi id="Pt0.A4.Ex66.m1.5.6.3.2" xref="Pt0.A4.Ex66.m1.5.6.3.2.cmml">C</mi><mo id="Pt0.A4.Ex66.m1.5.6.3.1" xref="Pt0.A4.Ex66.m1.5.6.3.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex66.m1.5.6.3.3.2" xref="Pt0.A4.Ex66.m1.5.6.3.3.1.cmml"><mo id="Pt0.A4.Ex66.m1.5.6.3.3.2.1" stretchy="false" xref="Pt0.A4.Ex66.m1.5.6.3.3.1.cmml">(</mo><mi id="Pt0.A4.Ex66.m1.3.3" xref="Pt0.A4.Ex66.m1.3.3.cmml">x</mi><mo id="Pt0.A4.Ex66.m1.5.6.3.3.2.2" xref="Pt0.A4.Ex66.m1.5.6.3.3.1.cmml">,</mo><mn id="Pt0.A4.Ex66.m1.4.4" xref="Pt0.A4.Ex66.m1.4.4.cmml">1</mn><mo id="Pt0.A4.Ex66.m1.5.6.3.3.2.3" xref="Pt0.A4.Ex66.m1.5.6.3.3.1.cmml">,</mo><mi id="Pt0.A4.Ex66.m1.5.5" xref="Pt0.A4.Ex66.m1.5.5.cmml">α</mi><mo id="Pt0.A4.Ex66.m1.5.6.3.3.2.4" stretchy="false" xref="Pt0.A4.Ex66.m1.5.6.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex66.m1.5b"><apply id="Pt0.A4.Ex66.m1.5.6.cmml" xref="Pt0.A4.Ex66.m1.5.6"><eq id="Pt0.A4.Ex66.m1.5.6.1.cmml" xref="Pt0.A4.Ex66.m1.5.6.1"></eq><apply id="Pt0.A4.Ex66.m1.5.6.2.cmml" xref="Pt0.A4.Ex66.m1.5.6.2"><times id="Pt0.A4.Ex66.m1.5.6.2.1.cmml" xref="Pt0.A4.Ex66.m1.5.6.2.1"></times><apply id="Pt0.A4.Ex66.m1.5.6.2.2.cmml" xref="Pt0.A4.Ex66.m1.5.6.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex66.m1.5.6.2.2.1.cmml" xref="Pt0.A4.Ex66.m1.5.6.2.2">subscript</csymbol><ci id="Pt0.A4.Ex66.m1.5.6.2.2.2.cmml" xref="Pt0.A4.Ex66.m1.5.6.2.2.2">𝑓</ci><cn id="Pt0.A4.Ex66.m1.5.6.2.2.3.cmml" type="integer" xref="Pt0.A4.Ex66.m1.5.6.2.2.3">1</cn></apply><interval closure="open" id="Pt0.A4.Ex66.m1.5.6.2.3.1.cmml" xref="Pt0.A4.Ex66.m1.5.6.2.3.2"><ci id="Pt0.A4.Ex66.m1.1.1.cmml" xref="Pt0.A4.Ex66.m1.1.1">𝑥</ci><ci id="Pt0.A4.Ex66.m1.2.2.cmml" xref="Pt0.A4.Ex66.m1.2.2">𝛼</ci></interval></apply><apply id="Pt0.A4.Ex66.m1.5.6.3.cmml" xref="Pt0.A4.Ex66.m1.5.6.3"><times id="Pt0.A4.Ex66.m1.5.6.3.1.cmml" xref="Pt0.A4.Ex66.m1.5.6.3.1"></times><ci id="Pt0.A4.Ex66.m1.5.6.3.2.cmml" xref="Pt0.A4.Ex66.m1.5.6.3.2">𝐶</ci><vector id="Pt0.A4.Ex66.m1.5.6.3.3.1.cmml" xref="Pt0.A4.Ex66.m1.5.6.3.3.2"><ci id="Pt0.A4.Ex66.m1.3.3.cmml" xref="Pt0.A4.Ex66.m1.3.3">𝑥</ci><cn id="Pt0.A4.Ex66.m1.4.4.cmml" type="integer" xref="Pt0.A4.Ex66.m1.4.4">1</cn><ci id="Pt0.A4.Ex66.m1.5.5.cmml" xref="Pt0.A4.Ex66.m1.5.5">𝛼</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex66.m1.5c">\displaystyle f_{1}(x,\alpha)=C(x,1,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex66.m1.5d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_α ) = italic_C ( italic_x , 1 , italic_α )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="Pt0.A4.Ex66.m2.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="Pt0.A4.Ex66.m2.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex66.m2.1.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="Pt0.A4.Ex66.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex66.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex66.m2.1.1.1.1.1.3" xref="Pt0.A4.Ex66.m2.1.1.1.1.1.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex66.m2.1.1.1.1.2" xref="Pt0.A4.Ex66.m2.1.1.1.1.2.cmml">⁢</mo><mi id="Pt0.A4.Ex66.m2.1.1.1.1.3" mathvariant="normal" xref="Pt0.A4.Ex66.m2.1.1.1.1.3.cmml">ℓ</mi></mrow><mo id="Pt0.A4.Ex66.m2.3.3.3.4" xref="Pt0.A4.Ex66.m2.3.3.3.4.cmml">+</mo><mrow id="Pt0.A4.Ex66.m2.3.3.3.3" xref="Pt0.A4.Ex66.m2.3.3.3.3.cmml"><mrow id="Pt0.A4.Ex66.m2.2.2.2.2.1.1" xref="Pt0.A4.Ex66.m2.2.2.2.2.1.1.1.cmml"><mo id="Pt0.A4.Ex66.m2.2.2.2.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex66.m2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex66.m2.2.2.2.2.1.1.1" xref="Pt0.A4.Ex66.m2.2.2.2.2.1.1.1.cmml"><mn id="Pt0.A4.Ex66.m2.2.2.2.2.1.1.1.3" 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id="Pt0.A4.Ex66.m2.3c">\displaystyle=(1-x)^{n}\ell+(1-(1-x)^{n})(1+\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex66.m2.3d">= ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_ℓ + ( 1 - ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ( 1 + italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex67"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{2}(x,\alpha)=C(x,0,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex67.m1.5"><semantics id="Pt0.A4.Ex67.m1.5a"><mrow id="Pt0.A4.Ex67.m1.5.6" xref="Pt0.A4.Ex67.m1.5.6.cmml"><mrow id="Pt0.A4.Ex67.m1.5.6.2" xref="Pt0.A4.Ex67.m1.5.6.2.cmml"><msub id="Pt0.A4.Ex67.m1.5.6.2.2" xref="Pt0.A4.Ex67.m1.5.6.2.2.cmml"><mi id="Pt0.A4.Ex67.m1.5.6.2.2.2" xref="Pt0.A4.Ex67.m1.5.6.2.2.2.cmml">f</mi><mn id="Pt0.A4.Ex67.m1.5.6.2.2.3" xref="Pt0.A4.Ex67.m1.5.6.2.2.3.cmml">2</mn></msub><mo id="Pt0.A4.Ex67.m1.5.6.2.1" xref="Pt0.A4.Ex67.m1.5.6.2.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex67.m1.5.6.2.3.2" xref="Pt0.A4.Ex67.m1.5.6.2.3.1.cmml"><mo id="Pt0.A4.Ex67.m1.5.6.2.3.2.1" stretchy="false" xref="Pt0.A4.Ex67.m1.5.6.2.3.1.cmml">(</mo><mi id="Pt0.A4.Ex67.m1.1.1" xref="Pt0.A4.Ex67.m1.1.1.cmml">x</mi><mo id="Pt0.A4.Ex67.m1.5.6.2.3.2.2" xref="Pt0.A4.Ex67.m1.5.6.2.3.1.cmml">,</mo><mi id="Pt0.A4.Ex67.m1.2.2" xref="Pt0.A4.Ex67.m1.2.2.cmml">α</mi><mo id="Pt0.A4.Ex67.m1.5.6.2.3.2.3" stretchy="false" xref="Pt0.A4.Ex67.m1.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex67.m1.5.6.1" xref="Pt0.A4.Ex67.m1.5.6.1.cmml">=</mo><mrow id="Pt0.A4.Ex67.m1.5.6.3" xref="Pt0.A4.Ex67.m1.5.6.3.cmml"><mi id="Pt0.A4.Ex67.m1.5.6.3.2" xref="Pt0.A4.Ex67.m1.5.6.3.2.cmml">C</mi><mo id="Pt0.A4.Ex67.m1.5.6.3.1" xref="Pt0.A4.Ex67.m1.5.6.3.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex67.m1.5.6.3.3.2" xref="Pt0.A4.Ex67.m1.5.6.3.3.1.cmml"><mo id="Pt0.A4.Ex67.m1.5.6.3.3.2.1" stretchy="false" xref="Pt0.A4.Ex67.m1.5.6.3.3.1.cmml">(</mo><mi id="Pt0.A4.Ex67.m1.3.3" xref="Pt0.A4.Ex67.m1.3.3.cmml">x</mi><mo id="Pt0.A4.Ex67.m1.5.6.3.3.2.2" xref="Pt0.A4.Ex67.m1.5.6.3.3.1.cmml">,</mo><mn id="Pt0.A4.Ex67.m1.4.4" xref="Pt0.A4.Ex67.m1.4.4.cmml">0</mn><mo id="Pt0.A4.Ex67.m1.5.6.3.3.2.3" xref="Pt0.A4.Ex67.m1.5.6.3.3.1.cmml">,</mo><mi id="Pt0.A4.Ex67.m1.5.5" xref="Pt0.A4.Ex67.m1.5.5.cmml">α</mi><mo id="Pt0.A4.Ex67.m1.5.6.3.3.2.4" stretchy="false" xref="Pt0.A4.Ex67.m1.5.6.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex67.m1.5b"><apply id="Pt0.A4.Ex67.m1.5.6.cmml" xref="Pt0.A4.Ex67.m1.5.6"><eq id="Pt0.A4.Ex67.m1.5.6.1.cmml" xref="Pt0.A4.Ex67.m1.5.6.1"></eq><apply id="Pt0.A4.Ex67.m1.5.6.2.cmml" xref="Pt0.A4.Ex67.m1.5.6.2"><times id="Pt0.A4.Ex67.m1.5.6.2.1.cmml" xref="Pt0.A4.Ex67.m1.5.6.2.1"></times><apply id="Pt0.A4.Ex67.m1.5.6.2.2.cmml" xref="Pt0.A4.Ex67.m1.5.6.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex67.m1.5.6.2.2.1.cmml" xref="Pt0.A4.Ex67.m1.5.6.2.2">subscript</csymbol><ci id="Pt0.A4.Ex67.m1.5.6.2.2.2.cmml" xref="Pt0.A4.Ex67.m1.5.6.2.2.2">𝑓</ci><cn id="Pt0.A4.Ex67.m1.5.6.2.2.3.cmml" type="integer" xref="Pt0.A4.Ex67.m1.5.6.2.2.3">2</cn></apply><interval closure="open" id="Pt0.A4.Ex67.m1.5.6.2.3.1.cmml" xref="Pt0.A4.Ex67.m1.5.6.2.3.2"><ci id="Pt0.A4.Ex67.m1.1.1.cmml" xref="Pt0.A4.Ex67.m1.1.1">𝑥</ci><ci id="Pt0.A4.Ex67.m1.2.2.cmml" xref="Pt0.A4.Ex67.m1.2.2">𝛼</ci></interval></apply><apply id="Pt0.A4.Ex67.m1.5.6.3.cmml" xref="Pt0.A4.Ex67.m1.5.6.3"><times id="Pt0.A4.Ex67.m1.5.6.3.1.cmml" xref="Pt0.A4.Ex67.m1.5.6.3.1"></times><ci id="Pt0.A4.Ex67.m1.5.6.3.2.cmml" xref="Pt0.A4.Ex67.m1.5.6.3.2">𝐶</ci><vector id="Pt0.A4.Ex67.m1.5.6.3.3.1.cmml" xref="Pt0.A4.Ex67.m1.5.6.3.3.2"><ci id="Pt0.A4.Ex67.m1.3.3.cmml" xref="Pt0.A4.Ex67.m1.3.3">𝑥</ci><cn id="Pt0.A4.Ex67.m1.4.4.cmml" type="integer" xref="Pt0.A4.Ex67.m1.4.4">0</cn><ci id="Pt0.A4.Ex67.m1.5.5.cmml" xref="Pt0.A4.Ex67.m1.5.5">𝛼</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex67.m1.5c">\displaystyle f_{2}(x,\alpha)=C(x,0,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex67.m1.5d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_α ) = italic_C ( italic_x , 0 , italic_α )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=x^{n}+(1-x^{n})(\ell+\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex67.m2.2"><semantics id="Pt0.A4.Ex67.m2.2a"><mrow id="Pt0.A4.Ex67.m2.2.2" xref="Pt0.A4.Ex67.m2.2.2.cmml"><mi id="Pt0.A4.Ex67.m2.2.2.4" xref="Pt0.A4.Ex67.m2.2.2.4.cmml"></mi><mo id="Pt0.A4.Ex67.m2.2.2.3" xref="Pt0.A4.Ex67.m2.2.2.3.cmml">=</mo><mrow id="Pt0.A4.Ex67.m2.2.2.2" xref="Pt0.A4.Ex67.m2.2.2.2.cmml"><msup id="Pt0.A4.Ex67.m2.2.2.2.4" xref="Pt0.A4.Ex67.m2.2.2.2.4.cmml"><mi id="Pt0.A4.Ex67.m2.2.2.2.4.2" xref="Pt0.A4.Ex67.m2.2.2.2.4.2.cmml">x</mi><mi id="Pt0.A4.Ex67.m2.2.2.2.4.3" xref="Pt0.A4.Ex67.m2.2.2.2.4.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex67.m2.2.2.2.3" xref="Pt0.A4.Ex67.m2.2.2.2.3.cmml">+</mo><mrow id="Pt0.A4.Ex67.m2.2.2.2.2" xref="Pt0.A4.Ex67.m2.2.2.2.2.cmml"><mrow id="Pt0.A4.Ex67.m2.1.1.1.1.1.1" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.cmml"><mn id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.1.cmml">−</mo><msup id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.2.cmml">x</mi><mi id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.3.cmml">n</mi></msup></mrow><mo id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex67.m2.2.2.2.2.3" xref="Pt0.A4.Ex67.m2.2.2.2.2.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex67.m2.2.2.2.2.2.1" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.cmml"><mo id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.2" stretchy="false" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.cmml"><mi id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.2" mathvariant="normal" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.2.cmml">ℓ</mi><mo id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.1" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.1.cmml">+</mo><mi id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.3" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.3.cmml">α</mi></mrow><mo id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.3" stretchy="false" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex67.m2.2b"><apply id="Pt0.A4.Ex67.m2.2.2.cmml" xref="Pt0.A4.Ex67.m2.2.2"><eq id="Pt0.A4.Ex67.m2.2.2.3.cmml" xref="Pt0.A4.Ex67.m2.2.2.3"></eq><csymbol cd="latexml" id="Pt0.A4.Ex67.m2.2.2.4.cmml" xref="Pt0.A4.Ex67.m2.2.2.4">absent</csymbol><apply id="Pt0.A4.Ex67.m2.2.2.2.cmml" xref="Pt0.A4.Ex67.m2.2.2.2"><plus id="Pt0.A4.Ex67.m2.2.2.2.3.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.3"></plus><apply id="Pt0.A4.Ex67.m2.2.2.2.4.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.4"><csymbol cd="ambiguous" id="Pt0.A4.Ex67.m2.2.2.2.4.1.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.4">superscript</csymbol><ci id="Pt0.A4.Ex67.m2.2.2.2.4.2.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.4.2">𝑥</ci><ci id="Pt0.A4.Ex67.m2.2.2.2.4.3.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.4.3">𝑛</ci></apply><apply id="Pt0.A4.Ex67.m2.2.2.2.2.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.2"><times id="Pt0.A4.Ex67.m2.2.2.2.2.3.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.2.3"></times><apply id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1"><minus id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.1"></minus><cn id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.2">1</cn><apply id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.2">𝑥</ci><ci id="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex67.m2.1.1.1.1.1.1.1.3.3">𝑛</ci></apply></apply><apply id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1"><plus id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.1.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.1"></plus><ci id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.2.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.2">ℓ</ci><ci id="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.3.cmml" xref="Pt0.A4.Ex67.m2.2.2.2.2.2.1.1.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex67.m2.2c">\displaystyle=x^{n}+(1-x^{n})(\ell+\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex67.m2.2d">= italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + ( 1 - italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ( roman_ℓ + italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmexample2.p3.10"><math alttext="f_{1}" class="ltx_Math" display="inline" id="Thmexample2.p3.2.m1.1"><semantics id="Thmexample2.p3.2.m1.1a"><msub id="Thmexample2.p3.2.m1.1.1" xref="Thmexample2.p3.2.m1.1.1.cmml"><mi id="Thmexample2.p3.2.m1.1.1.2" xref="Thmexample2.p3.2.m1.1.1.2.cmml">f</mi><mn id="Thmexample2.p3.2.m1.1.1.3" xref="Thmexample2.p3.2.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.2.m1.1b"><apply id="Thmexample2.p3.2.m1.1.1.cmml" xref="Thmexample2.p3.2.m1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p3.2.m1.1.1.1.cmml" xref="Thmexample2.p3.2.m1.1.1">subscript</csymbol><ci id="Thmexample2.p3.2.m1.1.1.2.cmml" xref="Thmexample2.p3.2.m1.1.1.2">𝑓</ci><cn id="Thmexample2.p3.2.m1.1.1.3.cmml" type="integer" xref="Thmexample2.p3.2.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.2.m1.1c">f_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.2.m1.1d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is increasing in <math alttext="x" class="ltx_Math" display="inline" id="Thmexample2.p3.3.m2.1"><semantics id="Thmexample2.p3.3.m2.1a"><mi id="Thmexample2.p3.3.m2.1.1" xref="Thmexample2.p3.3.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.3.m2.1b"><ci id="Thmexample2.p3.3.m2.1.1.cmml" xref="Thmexample2.p3.3.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.3.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.3.m2.1d">italic_x</annotation></semantics></math>, and <math alttext="f_{2}" class="ltx_Math" display="inline" id="Thmexample2.p3.4.m3.1"><semantics id="Thmexample2.p3.4.m3.1a"><msub id="Thmexample2.p3.4.m3.1.1" xref="Thmexample2.p3.4.m3.1.1.cmml"><mi id="Thmexample2.p3.4.m3.1.1.2" xref="Thmexample2.p3.4.m3.1.1.2.cmml">f</mi><mn id="Thmexample2.p3.4.m3.1.1.3" xref="Thmexample2.p3.4.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.4.m3.1b"><apply id="Thmexample2.p3.4.m3.1.1.cmml" xref="Thmexample2.p3.4.m3.1.1"><csymbol cd="ambiguous" id="Thmexample2.p3.4.m3.1.1.1.cmml" xref="Thmexample2.p3.4.m3.1.1">subscript</csymbol><ci id="Thmexample2.p3.4.m3.1.1.2.cmml" xref="Thmexample2.p3.4.m3.1.1.2">𝑓</ci><cn id="Thmexample2.p3.4.m3.1.1.3.cmml" type="integer" xref="Thmexample2.p3.4.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.4.m3.1c">f_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.4.m3.1d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is decreasing in <math alttext="x" class="ltx_Math" display="inline" id="Thmexample2.p3.5.m4.1"><semantics id="Thmexample2.p3.5.m4.1a"><mi id="Thmexample2.p3.5.m4.1.1" xref="Thmexample2.p3.5.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.5.m4.1b"><ci id="Thmexample2.p3.5.m4.1.1.cmml" xref="Thmexample2.p3.5.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.5.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.5.m4.1d">italic_x</annotation></semantics></math>, so they intersect at at most one <math alttext="x" class="ltx_Math" display="inline" id="Thmexample2.p3.6.m5.1"><semantics id="Thmexample2.p3.6.m5.1a"><mi id="Thmexample2.p3.6.m5.1.1" xref="Thmexample2.p3.6.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.6.m5.1b"><ci id="Thmexample2.p3.6.m5.1.1.cmml" xref="Thmexample2.p3.6.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.6.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.6.m5.1d">italic_x</annotation></semantics></math>. Also, <math alttext="\max_{y}C(x,y,\alpha)=\max(f_{1}(x,\alpha),f_{2}(x,\alpha))" class="ltx_Math" display="inline" id="Thmexample2.p3.7.m6.10"><semantics id="Thmexample2.p3.7.m6.10a"><mrow id="Thmexample2.p3.7.m6.10.10" xref="Thmexample2.p3.7.m6.10.10.cmml"><mrow id="Thmexample2.p3.7.m6.10.10.4" xref="Thmexample2.p3.7.m6.10.10.4.cmml"><mrow id="Thmexample2.p3.7.m6.10.10.4.2" xref="Thmexample2.p3.7.m6.10.10.4.2.cmml"><msub id="Thmexample2.p3.7.m6.10.10.4.2.1" xref="Thmexample2.p3.7.m6.10.10.4.2.1.cmml"><mi id="Thmexample2.p3.7.m6.10.10.4.2.1.2" xref="Thmexample2.p3.7.m6.10.10.4.2.1.2.cmml">max</mi><mi id="Thmexample2.p3.7.m6.10.10.4.2.1.3" xref="Thmexample2.p3.7.m6.10.10.4.2.1.3.cmml">y</mi></msub><mo id="Thmexample2.p3.7.m6.10.10.4.2a" lspace="0.167em" xref="Thmexample2.p3.7.m6.10.10.4.2.cmml">⁡</mo><mi id="Thmexample2.p3.7.m6.10.10.4.2.2" xref="Thmexample2.p3.7.m6.10.10.4.2.2.cmml">C</mi></mrow><mo id="Thmexample2.p3.7.m6.10.10.4.1" xref="Thmexample2.p3.7.m6.10.10.4.1.cmml">⁢</mo><mrow id="Thmexample2.p3.7.m6.10.10.4.3.2" xref="Thmexample2.p3.7.m6.10.10.4.3.1.cmml"><mo id="Thmexample2.p3.7.m6.10.10.4.3.2.1" stretchy="false" xref="Thmexample2.p3.7.m6.10.10.4.3.1.cmml">(</mo><mi id="Thmexample2.p3.7.m6.1.1" xref="Thmexample2.p3.7.m6.1.1.cmml">x</mi><mo id="Thmexample2.p3.7.m6.10.10.4.3.2.2" xref="Thmexample2.p3.7.m6.10.10.4.3.1.cmml">,</mo><mi id="Thmexample2.p3.7.m6.2.2" xref="Thmexample2.p3.7.m6.2.2.cmml">y</mi><mo id="Thmexample2.p3.7.m6.10.10.4.3.2.3" xref="Thmexample2.p3.7.m6.10.10.4.3.1.cmml">,</mo><mi id="Thmexample2.p3.7.m6.3.3" xref="Thmexample2.p3.7.m6.3.3.cmml">α</mi><mo id="Thmexample2.p3.7.m6.10.10.4.3.2.4" stretchy="false" xref="Thmexample2.p3.7.m6.10.10.4.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.7.m6.10.10.3" xref="Thmexample2.p3.7.m6.10.10.3.cmml">=</mo><mrow id="Thmexample2.p3.7.m6.10.10.2.2" xref="Thmexample2.p3.7.m6.10.10.2.3.cmml"><mi id="Thmexample2.p3.7.m6.8.8" xref="Thmexample2.p3.7.m6.8.8.cmml">max</mi><mo id="Thmexample2.p3.7.m6.10.10.2.2a" xref="Thmexample2.p3.7.m6.10.10.2.3.cmml">⁡</mo><mrow id="Thmexample2.p3.7.m6.10.10.2.2.2" xref="Thmexample2.p3.7.m6.10.10.2.3.cmml"><mo id="Thmexample2.p3.7.m6.10.10.2.2.2.3" stretchy="false" xref="Thmexample2.p3.7.m6.10.10.2.3.cmml">(</mo><mrow id="Thmexample2.p3.7.m6.9.9.1.1.1.1" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.cmml"><msub id="Thmexample2.p3.7.m6.9.9.1.1.1.1.2" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.cmml"><mi id="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.2" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.2.cmml">f</mi><mn id="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.3" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample2.p3.7.m6.9.9.1.1.1.1.1" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.2" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.1.cmml"><mo id="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.2.1" stretchy="false" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.1.cmml">(</mo><mi id="Thmexample2.p3.7.m6.4.4" xref="Thmexample2.p3.7.m6.4.4.cmml">x</mi><mo id="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.2.2" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.1.cmml">,</mo><mi id="Thmexample2.p3.7.m6.5.5" xref="Thmexample2.p3.7.m6.5.5.cmml">α</mi><mo id="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.2.3" stretchy="false" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.7.m6.10.10.2.2.2.4" xref="Thmexample2.p3.7.m6.10.10.2.3.cmml">,</mo><mrow id="Thmexample2.p3.7.m6.10.10.2.2.2.2" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.cmml"><msub id="Thmexample2.p3.7.m6.10.10.2.2.2.2.2" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.cmml"><mi id="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.2" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.2.cmml">f</mi><mn id="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.3" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.3.cmml">2</mn></msub><mo id="Thmexample2.p3.7.m6.10.10.2.2.2.2.1" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.1.cmml">⁢</mo><mrow id="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.2" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.1.cmml"><mo id="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.2.1" stretchy="false" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.1.cmml">(</mo><mi id="Thmexample2.p3.7.m6.6.6" xref="Thmexample2.p3.7.m6.6.6.cmml">x</mi><mo id="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.2.2" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.1.cmml">,</mo><mi id="Thmexample2.p3.7.m6.7.7" xref="Thmexample2.p3.7.m6.7.7.cmml">α</mi><mo id="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.2.3" stretchy="false" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.7.m6.10.10.2.2.2.5" stretchy="false" xref="Thmexample2.p3.7.m6.10.10.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.7.m6.10b"><apply id="Thmexample2.p3.7.m6.10.10.cmml" xref="Thmexample2.p3.7.m6.10.10"><eq id="Thmexample2.p3.7.m6.10.10.3.cmml" xref="Thmexample2.p3.7.m6.10.10.3"></eq><apply id="Thmexample2.p3.7.m6.10.10.4.cmml" xref="Thmexample2.p3.7.m6.10.10.4"><times id="Thmexample2.p3.7.m6.10.10.4.1.cmml" xref="Thmexample2.p3.7.m6.10.10.4.1"></times><apply id="Thmexample2.p3.7.m6.10.10.4.2.cmml" xref="Thmexample2.p3.7.m6.10.10.4.2"><apply id="Thmexample2.p3.7.m6.10.10.4.2.1.cmml" xref="Thmexample2.p3.7.m6.10.10.4.2.1"><csymbol cd="ambiguous" id="Thmexample2.p3.7.m6.10.10.4.2.1.1.cmml" xref="Thmexample2.p3.7.m6.10.10.4.2.1">subscript</csymbol><max id="Thmexample2.p3.7.m6.10.10.4.2.1.2.cmml" xref="Thmexample2.p3.7.m6.10.10.4.2.1.2"></max><ci id="Thmexample2.p3.7.m6.10.10.4.2.1.3.cmml" xref="Thmexample2.p3.7.m6.10.10.4.2.1.3">𝑦</ci></apply><ci id="Thmexample2.p3.7.m6.10.10.4.2.2.cmml" xref="Thmexample2.p3.7.m6.10.10.4.2.2">𝐶</ci></apply><vector id="Thmexample2.p3.7.m6.10.10.4.3.1.cmml" xref="Thmexample2.p3.7.m6.10.10.4.3.2"><ci id="Thmexample2.p3.7.m6.1.1.cmml" xref="Thmexample2.p3.7.m6.1.1">𝑥</ci><ci id="Thmexample2.p3.7.m6.2.2.cmml" xref="Thmexample2.p3.7.m6.2.2">𝑦</ci><ci id="Thmexample2.p3.7.m6.3.3.cmml" xref="Thmexample2.p3.7.m6.3.3">𝛼</ci></vector></apply><apply id="Thmexample2.p3.7.m6.10.10.2.3.cmml" xref="Thmexample2.p3.7.m6.10.10.2.2"><max id="Thmexample2.p3.7.m6.8.8.cmml" xref="Thmexample2.p3.7.m6.8.8"></max><apply id="Thmexample2.p3.7.m6.9.9.1.1.1.1.cmml" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1"><times id="Thmexample2.p3.7.m6.9.9.1.1.1.1.1.cmml" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.1"></times><apply id="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.cmml" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.1.cmml" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.2">subscript</csymbol><ci id="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.2.cmml" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.2">𝑓</ci><cn id="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.3.cmml" type="integer" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.2.3">1</cn></apply><interval closure="open" id="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.1.cmml" xref="Thmexample2.p3.7.m6.9.9.1.1.1.1.3.2"><ci id="Thmexample2.p3.7.m6.4.4.cmml" xref="Thmexample2.p3.7.m6.4.4">𝑥</ci><ci id="Thmexample2.p3.7.m6.5.5.cmml" xref="Thmexample2.p3.7.m6.5.5">𝛼</ci></interval></apply><apply id="Thmexample2.p3.7.m6.10.10.2.2.2.2.cmml" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2"><times id="Thmexample2.p3.7.m6.10.10.2.2.2.2.1.cmml" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.1"></times><apply id="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.cmml" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.1.cmml" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.2">subscript</csymbol><ci id="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.2.cmml" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.2">𝑓</ci><cn id="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.3.cmml" type="integer" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.2.3">2</cn></apply><interval closure="open" id="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.1.cmml" xref="Thmexample2.p3.7.m6.10.10.2.2.2.2.3.2"><ci id="Thmexample2.p3.7.m6.6.6.cmml" xref="Thmexample2.p3.7.m6.6.6">𝑥</ci><ci id="Thmexample2.p3.7.m6.7.7.cmml" xref="Thmexample2.p3.7.m6.7.7">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.7.m6.10c">\max_{y}C(x,y,\alpha)=\max(f_{1}(x,\alpha),f_{2}(x,\alpha))</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.7.m6.10d">roman_max start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT italic_C ( italic_x , italic_y , italic_α ) = roman_max ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_α ) , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_α ) )</annotation></semantics></math>. So <math alttext="\min_{x}\max(f_{1}(x,\alpha),f_{2}(x,\alpha))" class="ltx_Math" display="inline" id="Thmexample2.p3.8.m7.7"><semantics id="Thmexample2.p3.8.m7.7a"><mrow id="Thmexample2.p3.8.m7.7.7" xref="Thmexample2.p3.8.m7.7.7.cmml"><msub id="Thmexample2.p3.8.m7.7.7.3" xref="Thmexample2.p3.8.m7.7.7.3.cmml"><mi id="Thmexample2.p3.8.m7.7.7.3.2" xref="Thmexample2.p3.8.m7.7.7.3.2.cmml">min</mi><mi id="Thmexample2.p3.8.m7.7.7.3.3" xref="Thmexample2.p3.8.m7.7.7.3.3.cmml">x</mi></msub><mo id="Thmexample2.p3.8.m7.7.7a" lspace="0.167em" xref="Thmexample2.p3.8.m7.7.7.cmml">⁡</mo><mrow id="Thmexample2.p3.8.m7.7.7.2.2" xref="Thmexample2.p3.8.m7.7.7.2.3.cmml"><mi id="Thmexample2.p3.8.m7.5.5" xref="Thmexample2.p3.8.m7.5.5.cmml">max</mi><mo id="Thmexample2.p3.8.m7.7.7.2.2a" xref="Thmexample2.p3.8.m7.7.7.2.3.cmml">⁡</mo><mrow id="Thmexample2.p3.8.m7.7.7.2.2.2" xref="Thmexample2.p3.8.m7.7.7.2.3.cmml"><mo id="Thmexample2.p3.8.m7.7.7.2.2.2.3" stretchy="false" xref="Thmexample2.p3.8.m7.7.7.2.3.cmml">(</mo><mrow id="Thmexample2.p3.8.m7.6.6.1.1.1.1" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.cmml"><msub id="Thmexample2.p3.8.m7.6.6.1.1.1.1.2" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.cmml"><mi id="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.2" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.2.cmml">f</mi><mn id="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.3" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample2.p3.8.m7.6.6.1.1.1.1.1" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.2" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.1.cmml"><mo id="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.2.1" stretchy="false" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.1.cmml">(</mo><mi id="Thmexample2.p3.8.m7.1.1" xref="Thmexample2.p3.8.m7.1.1.cmml">x</mi><mo id="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.2.2" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.1.cmml">,</mo><mi id="Thmexample2.p3.8.m7.2.2" xref="Thmexample2.p3.8.m7.2.2.cmml">α</mi><mo id="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.2.3" stretchy="false" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.8.m7.7.7.2.2.2.4" xref="Thmexample2.p3.8.m7.7.7.2.3.cmml">,</mo><mrow id="Thmexample2.p3.8.m7.7.7.2.2.2.2" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.cmml"><msub id="Thmexample2.p3.8.m7.7.7.2.2.2.2.2" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.cmml"><mi id="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.2" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.2.cmml">f</mi><mn id="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.3" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.3.cmml">2</mn></msub><mo id="Thmexample2.p3.8.m7.7.7.2.2.2.2.1" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.1.cmml">⁢</mo><mrow id="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.2" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.1.cmml"><mo id="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.2.1" stretchy="false" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.1.cmml">(</mo><mi id="Thmexample2.p3.8.m7.3.3" xref="Thmexample2.p3.8.m7.3.3.cmml">x</mi><mo id="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.2.2" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.1.cmml">,</mo><mi id="Thmexample2.p3.8.m7.4.4" xref="Thmexample2.p3.8.m7.4.4.cmml">α</mi><mo id="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.2.3" stretchy="false" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.8.m7.7.7.2.2.2.5" stretchy="false" xref="Thmexample2.p3.8.m7.7.7.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.8.m7.7b"><apply id="Thmexample2.p3.8.m7.7.7.cmml" xref="Thmexample2.p3.8.m7.7.7"><apply id="Thmexample2.p3.8.m7.7.7.3.cmml" xref="Thmexample2.p3.8.m7.7.7.3"><csymbol cd="ambiguous" id="Thmexample2.p3.8.m7.7.7.3.1.cmml" xref="Thmexample2.p3.8.m7.7.7.3">subscript</csymbol><min id="Thmexample2.p3.8.m7.7.7.3.2.cmml" xref="Thmexample2.p3.8.m7.7.7.3.2"></min><ci id="Thmexample2.p3.8.m7.7.7.3.3.cmml" xref="Thmexample2.p3.8.m7.7.7.3.3">𝑥</ci></apply><apply id="Thmexample2.p3.8.m7.7.7.2.3.cmml" xref="Thmexample2.p3.8.m7.7.7.2.2"><max id="Thmexample2.p3.8.m7.5.5.cmml" xref="Thmexample2.p3.8.m7.5.5"></max><apply id="Thmexample2.p3.8.m7.6.6.1.1.1.1.cmml" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1"><times id="Thmexample2.p3.8.m7.6.6.1.1.1.1.1.cmml" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.1"></times><apply id="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.cmml" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.1.cmml" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.2">subscript</csymbol><ci id="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.2.cmml" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.2">𝑓</ci><cn id="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.3.cmml" type="integer" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.2.3">1</cn></apply><interval closure="open" id="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.1.cmml" xref="Thmexample2.p3.8.m7.6.6.1.1.1.1.3.2"><ci id="Thmexample2.p3.8.m7.1.1.cmml" xref="Thmexample2.p3.8.m7.1.1">𝑥</ci><ci id="Thmexample2.p3.8.m7.2.2.cmml" xref="Thmexample2.p3.8.m7.2.2">𝛼</ci></interval></apply><apply id="Thmexample2.p3.8.m7.7.7.2.2.2.2.cmml" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2"><times id="Thmexample2.p3.8.m7.7.7.2.2.2.2.1.cmml" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.1"></times><apply id="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.cmml" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.1.cmml" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.2">subscript</csymbol><ci id="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.2.cmml" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.2">𝑓</ci><cn id="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.3.cmml" type="integer" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.2.3">2</cn></apply><interval closure="open" id="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.1.cmml" xref="Thmexample2.p3.8.m7.7.7.2.2.2.2.3.2"><ci id="Thmexample2.p3.8.m7.3.3.cmml" xref="Thmexample2.p3.8.m7.3.3">𝑥</ci><ci id="Thmexample2.p3.8.m7.4.4.cmml" xref="Thmexample2.p3.8.m7.4.4">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.8.m7.7c">\min_{x}\max(f_{1}(x,\alpha),f_{2}(x,\alpha))</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.8.m7.7d">roman_min start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_α ) , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_α ) )</annotation></semantics></math> is obtained when <math alttext="f_{1}(x,\alpha)=f_{2}(x,\alpha)" class="ltx_Math" display="inline" id="Thmexample2.p3.9.m8.4"><semantics id="Thmexample2.p3.9.m8.4a"><mrow id="Thmexample2.p3.9.m8.4.5" xref="Thmexample2.p3.9.m8.4.5.cmml"><mrow id="Thmexample2.p3.9.m8.4.5.2" xref="Thmexample2.p3.9.m8.4.5.2.cmml"><msub id="Thmexample2.p3.9.m8.4.5.2.2" xref="Thmexample2.p3.9.m8.4.5.2.2.cmml"><mi id="Thmexample2.p3.9.m8.4.5.2.2.2" xref="Thmexample2.p3.9.m8.4.5.2.2.2.cmml">f</mi><mn id="Thmexample2.p3.9.m8.4.5.2.2.3" xref="Thmexample2.p3.9.m8.4.5.2.2.3.cmml">1</mn></msub><mo id="Thmexample2.p3.9.m8.4.5.2.1" xref="Thmexample2.p3.9.m8.4.5.2.1.cmml">⁢</mo><mrow id="Thmexample2.p3.9.m8.4.5.2.3.2" xref="Thmexample2.p3.9.m8.4.5.2.3.1.cmml"><mo id="Thmexample2.p3.9.m8.4.5.2.3.2.1" stretchy="false" xref="Thmexample2.p3.9.m8.4.5.2.3.1.cmml">(</mo><mi id="Thmexample2.p3.9.m8.1.1" xref="Thmexample2.p3.9.m8.1.1.cmml">x</mi><mo id="Thmexample2.p3.9.m8.4.5.2.3.2.2" xref="Thmexample2.p3.9.m8.4.5.2.3.1.cmml">,</mo><mi id="Thmexample2.p3.9.m8.2.2" xref="Thmexample2.p3.9.m8.2.2.cmml">α</mi><mo id="Thmexample2.p3.9.m8.4.5.2.3.2.3" stretchy="false" xref="Thmexample2.p3.9.m8.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.9.m8.4.5.1" xref="Thmexample2.p3.9.m8.4.5.1.cmml">=</mo><mrow id="Thmexample2.p3.9.m8.4.5.3" xref="Thmexample2.p3.9.m8.4.5.3.cmml"><msub id="Thmexample2.p3.9.m8.4.5.3.2" xref="Thmexample2.p3.9.m8.4.5.3.2.cmml"><mi id="Thmexample2.p3.9.m8.4.5.3.2.2" xref="Thmexample2.p3.9.m8.4.5.3.2.2.cmml">f</mi><mn id="Thmexample2.p3.9.m8.4.5.3.2.3" xref="Thmexample2.p3.9.m8.4.5.3.2.3.cmml">2</mn></msub><mo id="Thmexample2.p3.9.m8.4.5.3.1" xref="Thmexample2.p3.9.m8.4.5.3.1.cmml">⁢</mo><mrow id="Thmexample2.p3.9.m8.4.5.3.3.2" xref="Thmexample2.p3.9.m8.4.5.3.3.1.cmml"><mo id="Thmexample2.p3.9.m8.4.5.3.3.2.1" stretchy="false" xref="Thmexample2.p3.9.m8.4.5.3.3.1.cmml">(</mo><mi id="Thmexample2.p3.9.m8.3.3" xref="Thmexample2.p3.9.m8.3.3.cmml">x</mi><mo id="Thmexample2.p3.9.m8.4.5.3.3.2.2" xref="Thmexample2.p3.9.m8.4.5.3.3.1.cmml">,</mo><mi id="Thmexample2.p3.9.m8.4.4" xref="Thmexample2.p3.9.m8.4.4.cmml">α</mi><mo id="Thmexample2.p3.9.m8.4.5.3.3.2.3" stretchy="false" xref="Thmexample2.p3.9.m8.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.9.m8.4b"><apply id="Thmexample2.p3.9.m8.4.5.cmml" xref="Thmexample2.p3.9.m8.4.5"><eq id="Thmexample2.p3.9.m8.4.5.1.cmml" xref="Thmexample2.p3.9.m8.4.5.1"></eq><apply id="Thmexample2.p3.9.m8.4.5.2.cmml" xref="Thmexample2.p3.9.m8.4.5.2"><times id="Thmexample2.p3.9.m8.4.5.2.1.cmml" xref="Thmexample2.p3.9.m8.4.5.2.1"></times><apply id="Thmexample2.p3.9.m8.4.5.2.2.cmml" xref="Thmexample2.p3.9.m8.4.5.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.9.m8.4.5.2.2.1.cmml" xref="Thmexample2.p3.9.m8.4.5.2.2">subscript</csymbol><ci id="Thmexample2.p3.9.m8.4.5.2.2.2.cmml" xref="Thmexample2.p3.9.m8.4.5.2.2.2">𝑓</ci><cn id="Thmexample2.p3.9.m8.4.5.2.2.3.cmml" type="integer" xref="Thmexample2.p3.9.m8.4.5.2.2.3">1</cn></apply><interval closure="open" id="Thmexample2.p3.9.m8.4.5.2.3.1.cmml" xref="Thmexample2.p3.9.m8.4.5.2.3.2"><ci id="Thmexample2.p3.9.m8.1.1.cmml" xref="Thmexample2.p3.9.m8.1.1">𝑥</ci><ci id="Thmexample2.p3.9.m8.2.2.cmml" xref="Thmexample2.p3.9.m8.2.2">𝛼</ci></interval></apply><apply id="Thmexample2.p3.9.m8.4.5.3.cmml" xref="Thmexample2.p3.9.m8.4.5.3"><times id="Thmexample2.p3.9.m8.4.5.3.1.cmml" xref="Thmexample2.p3.9.m8.4.5.3.1"></times><apply id="Thmexample2.p3.9.m8.4.5.3.2.cmml" xref="Thmexample2.p3.9.m8.4.5.3.2"><csymbol cd="ambiguous" id="Thmexample2.p3.9.m8.4.5.3.2.1.cmml" xref="Thmexample2.p3.9.m8.4.5.3.2">subscript</csymbol><ci id="Thmexample2.p3.9.m8.4.5.3.2.2.cmml" xref="Thmexample2.p3.9.m8.4.5.3.2.2">𝑓</ci><cn id="Thmexample2.p3.9.m8.4.5.3.2.3.cmml" type="integer" xref="Thmexample2.p3.9.m8.4.5.3.2.3">2</cn></apply><interval closure="open" id="Thmexample2.p3.9.m8.4.5.3.3.1.cmml" xref="Thmexample2.p3.9.m8.4.5.3.3.2"><ci id="Thmexample2.p3.9.m8.3.3.cmml" xref="Thmexample2.p3.9.m8.3.3">𝑥</ci><ci id="Thmexample2.p3.9.m8.4.4.cmml" xref="Thmexample2.p3.9.m8.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.9.m8.4c">f_{1}(x,\alpha)=f_{2}(x,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.9.m8.4d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_α ) = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_α )</annotation></semantics></math>. At <math alttext="x=\frac{1}{2}" class="ltx_Math" display="inline" id="Thmexample2.p3.10.m9.1"><semantics id="Thmexample2.p3.10.m9.1a"><mrow id="Thmexample2.p3.10.m9.1.1" xref="Thmexample2.p3.10.m9.1.1.cmml"><mi id="Thmexample2.p3.10.m9.1.1.2" xref="Thmexample2.p3.10.m9.1.1.2.cmml">x</mi><mo id="Thmexample2.p3.10.m9.1.1.1" xref="Thmexample2.p3.10.m9.1.1.1.cmml">=</mo><mfrac id="Thmexample2.p3.10.m9.1.1.3" xref="Thmexample2.p3.10.m9.1.1.3.cmml"><mn id="Thmexample2.p3.10.m9.1.1.3.2" xref="Thmexample2.p3.10.m9.1.1.3.2.cmml">1</mn><mn id="Thmexample2.p3.10.m9.1.1.3.3" xref="Thmexample2.p3.10.m9.1.1.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.10.m9.1b"><apply id="Thmexample2.p3.10.m9.1.1.cmml" xref="Thmexample2.p3.10.m9.1.1"><eq id="Thmexample2.p3.10.m9.1.1.1.cmml" xref="Thmexample2.p3.10.m9.1.1.1"></eq><ci id="Thmexample2.p3.10.m9.1.1.2.cmml" xref="Thmexample2.p3.10.m9.1.1.2">𝑥</ci><apply id="Thmexample2.p3.10.m9.1.1.3.cmml" xref="Thmexample2.p3.10.m9.1.1.3"><divide id="Thmexample2.p3.10.m9.1.1.3.1.cmml" xref="Thmexample2.p3.10.m9.1.1.3"></divide><cn id="Thmexample2.p3.10.m9.1.1.3.2.cmml" type="integer" xref="Thmexample2.p3.10.m9.1.1.3.2">1</cn><cn id="Thmexample2.p3.10.m9.1.1.3.3.cmml" type="integer" xref="Thmexample2.p3.10.m9.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.10.m9.1c">x=\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.10.m9.1d">italic_x = divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math> we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx42"> <tbody id="Pt0.A4.Ex68"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{1}\left(\frac{1}{2},\alpha\right)=2^{-n}\ell+(1-2^{-n})(1+% \alpha)<2^{-n}+(1-2^{-n})(\ell+\alpha)=f_{2}\left(\frac{1}{2},\alpha\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex68.m1.8"><semantics id="Pt0.A4.Ex68.m1.8a"><mrow id="Pt0.A4.Ex68.m1.8.8" xref="Pt0.A4.Ex68.m1.8.8.cmml"><mrow id="Pt0.A4.Ex68.m1.8.8.6" xref="Pt0.A4.Ex68.m1.8.8.6.cmml"><msub id="Pt0.A4.Ex68.m1.8.8.6.2" xref="Pt0.A4.Ex68.m1.8.8.6.2.cmml"><mi id="Pt0.A4.Ex68.m1.8.8.6.2.2" xref="Pt0.A4.Ex68.m1.8.8.6.2.2.cmml">f</mi><mn id="Pt0.A4.Ex68.m1.8.8.6.2.3" xref="Pt0.A4.Ex68.m1.8.8.6.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex68.m1.8.8.6.1" xref="Pt0.A4.Ex68.m1.8.8.6.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex68.m1.8.8.6.3.2" xref="Pt0.A4.Ex68.m1.8.8.6.3.1.cmml"><mo id="Pt0.A4.Ex68.m1.8.8.6.3.2.1" xref="Pt0.A4.Ex68.m1.8.8.6.3.1.cmml">(</mo><mstyle displaystyle="true" id="Pt0.A4.Ex68.m1.1.1" xref="Pt0.A4.Ex68.m1.1.1.cmml"><mfrac id="Pt0.A4.Ex68.m1.1.1a" xref="Pt0.A4.Ex68.m1.1.1.cmml"><mn id="Pt0.A4.Ex68.m1.1.1.2" xref="Pt0.A4.Ex68.m1.1.1.2.cmml">1</mn><mn id="Pt0.A4.Ex68.m1.1.1.3" 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xref="Pt0.A4.Ex68.m1.3.3"></divide><cn id="Pt0.A4.Ex68.m1.3.3.2.cmml" type="integer" xref="Pt0.A4.Ex68.m1.3.3.2">1</cn><cn id="Pt0.A4.Ex68.m1.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex68.m1.3.3.3">2</cn></apply><ci id="Pt0.A4.Ex68.m1.4.4.cmml" xref="Pt0.A4.Ex68.m1.4.4">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex68.m1.8c">\displaystyle f_{1}\left(\frac{1}{2},\alpha\right)=2^{-n}\ell+(1-2^{-n})(1+% \alpha)<2^{-n}+(1-2^{-n})(\ell+\alpha)=f_{2}\left(\frac{1}{2},\alpha\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex68.m1.8d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG , italic_α ) = 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT roman_ℓ + ( 1 - 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ) ( 1 + italic_α ) < 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT + ( 1 - 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ) ( roman_ℓ + italic_α ) = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmexample2.p3.14">iff <math alttext="(\ell-1)(1-2^{-n+1})>0" class="ltx_Math" display="inline" id="Thmexample2.p3.11.m1.2"><semantics id="Thmexample2.p3.11.m1.2a"><mrow id="Thmexample2.p3.11.m1.2.2" xref="Thmexample2.p3.11.m1.2.2.cmml"><mrow id="Thmexample2.p3.11.m1.2.2.2" xref="Thmexample2.p3.11.m1.2.2.2.cmml"><mrow id="Thmexample2.p3.11.m1.1.1.1.1.1" xref="Thmexample2.p3.11.m1.1.1.1.1.1.1.cmml"><mo id="Thmexample2.p3.11.m1.1.1.1.1.1.2" stretchy="false" xref="Thmexample2.p3.11.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample2.p3.11.m1.1.1.1.1.1.1" xref="Thmexample2.p3.11.m1.1.1.1.1.1.1.cmml"><mi id="Thmexample2.p3.11.m1.1.1.1.1.1.1.2" mathvariant="normal" 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xref="Thmexample2.p3.11.m1.2.2.2.2.1"><minus id="Thmexample2.p3.11.m1.2.2.2.2.1.1.1.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.1"></minus><cn id="Thmexample2.p3.11.m1.2.2.2.2.1.1.2.cmml" type="integer" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.2">1</cn><apply id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.1.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3">superscript</csymbol><cn id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.2.cmml" type="integer" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.2">2</cn><apply id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3"><plus id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.1.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.1"></plus><apply id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.2.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.2"><minus id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.2.1.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.2"></minus><ci id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.2.2.cmml" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.2.2">𝑛</ci></apply><cn id="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.3.cmml" type="integer" xref="Thmexample2.p3.11.m1.2.2.2.2.1.1.3.3.3">1</cn></apply></apply></apply></apply><cn id="Thmexample2.p3.11.m1.2.2.4.cmml" type="integer" xref="Thmexample2.p3.11.m1.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.11.m1.2c">(\ell-1)(1-2^{-n+1})>0</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.11.m1.2d">( roman_ℓ - 1 ) ( 1 - 2 start_POSTSUPERSCRIPT - italic_n + 1 end_POSTSUPERSCRIPT ) > 0</annotation></semantics></math>, which is the case here. Hence <math alttext="f_{1},f_{2}" class="ltx_Math" display="inline" id="Thmexample2.p3.12.m2.2"><semantics id="Thmexample2.p3.12.m2.2a"><mrow id="Thmexample2.p3.12.m2.2.2.2" xref="Thmexample2.p3.12.m2.2.2.3.cmml"><msub id="Thmexample2.p3.12.m2.1.1.1.1" xref="Thmexample2.p3.12.m2.1.1.1.1.cmml"><mi id="Thmexample2.p3.12.m2.1.1.1.1.2" xref="Thmexample2.p3.12.m2.1.1.1.1.2.cmml">f</mi><mn id="Thmexample2.p3.12.m2.1.1.1.1.3" xref="Thmexample2.p3.12.m2.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample2.p3.12.m2.2.2.2.3" xref="Thmexample2.p3.12.m2.2.2.3.cmml">,</mo><msub id="Thmexample2.p3.12.m2.2.2.2.2" xref="Thmexample2.p3.12.m2.2.2.2.2.cmml"><mi id="Thmexample2.p3.12.m2.2.2.2.2.2" xref="Thmexample2.p3.12.m2.2.2.2.2.2.cmml">f</mi><mn id="Thmexample2.p3.12.m2.2.2.2.2.3" xref="Thmexample2.p3.12.m2.2.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.12.m2.2b"><list id="Thmexample2.p3.12.m2.2.2.3.cmml" xref="Thmexample2.p3.12.m2.2.2.2"><apply id="Thmexample2.p3.12.m2.1.1.1.1.cmml" xref="Thmexample2.p3.12.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p3.12.m2.1.1.1.1.1.cmml" xref="Thmexample2.p3.12.m2.1.1.1.1">subscript</csymbol><ci id="Thmexample2.p3.12.m2.1.1.1.1.2.cmml" xref="Thmexample2.p3.12.m2.1.1.1.1.2">𝑓</ci><cn id="Thmexample2.p3.12.m2.1.1.1.1.3.cmml" type="integer" xref="Thmexample2.p3.12.m2.1.1.1.1.3">1</cn></apply><apply id="Thmexample2.p3.12.m2.2.2.2.2.cmml" xref="Thmexample2.p3.12.m2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.12.m2.2.2.2.2.1.cmml" xref="Thmexample2.p3.12.m2.2.2.2.2">subscript</csymbol><ci id="Thmexample2.p3.12.m2.2.2.2.2.2.cmml" xref="Thmexample2.p3.12.m2.2.2.2.2.2">𝑓</ci><cn id="Thmexample2.p3.12.m2.2.2.2.2.3.cmml" type="integer" xref="Thmexample2.p3.12.m2.2.2.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.12.m2.2c">f_{1},f_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.12.m2.2d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> intersect after <math alttext="x=\frac{1}{2}" class="ltx_Math" display="inline" id="Thmexample2.p3.13.m3.1"><semantics id="Thmexample2.p3.13.m3.1a"><mrow id="Thmexample2.p3.13.m3.1.1" xref="Thmexample2.p3.13.m3.1.1.cmml"><mi id="Thmexample2.p3.13.m3.1.1.2" xref="Thmexample2.p3.13.m3.1.1.2.cmml">x</mi><mo id="Thmexample2.p3.13.m3.1.1.1" xref="Thmexample2.p3.13.m3.1.1.1.cmml">=</mo><mfrac id="Thmexample2.p3.13.m3.1.1.3" xref="Thmexample2.p3.13.m3.1.1.3.cmml"><mn id="Thmexample2.p3.13.m3.1.1.3.2" xref="Thmexample2.p3.13.m3.1.1.3.2.cmml">1</mn><mn id="Thmexample2.p3.13.m3.1.1.3.3" xref="Thmexample2.p3.13.m3.1.1.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.13.m3.1b"><apply id="Thmexample2.p3.13.m3.1.1.cmml" xref="Thmexample2.p3.13.m3.1.1"><eq id="Thmexample2.p3.13.m3.1.1.1.cmml" xref="Thmexample2.p3.13.m3.1.1.1"></eq><ci id="Thmexample2.p3.13.m3.1.1.2.cmml" xref="Thmexample2.p3.13.m3.1.1.2">𝑥</ci><apply id="Thmexample2.p3.13.m3.1.1.3.cmml" xref="Thmexample2.p3.13.m3.1.1.3"><divide id="Thmexample2.p3.13.m3.1.1.3.1.cmml" xref="Thmexample2.p3.13.m3.1.1.3"></divide><cn id="Thmexample2.p3.13.m3.1.1.3.2.cmml" type="integer" xref="Thmexample2.p3.13.m3.1.1.3.2">1</cn><cn id="Thmexample2.p3.13.m3.1.1.3.3.cmml" type="integer" xref="Thmexample2.p3.13.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.13.m3.1c">x=\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.13.m3.1d">italic_x = divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. At <math alttext="x=1" class="ltx_Math" display="inline" id="Thmexample2.p3.14.m4.1"><semantics id="Thmexample2.p3.14.m4.1a"><mrow id="Thmexample2.p3.14.m4.1.1" xref="Thmexample2.p3.14.m4.1.1.cmml"><mi id="Thmexample2.p3.14.m4.1.1.2" xref="Thmexample2.p3.14.m4.1.1.2.cmml">x</mi><mo id="Thmexample2.p3.14.m4.1.1.1" xref="Thmexample2.p3.14.m4.1.1.1.cmml">=</mo><mn id="Thmexample2.p3.14.m4.1.1.3" xref="Thmexample2.p3.14.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.14.m4.1b"><apply id="Thmexample2.p3.14.m4.1.1.cmml" xref="Thmexample2.p3.14.m4.1.1"><eq id="Thmexample2.p3.14.m4.1.1.1.cmml" xref="Thmexample2.p3.14.m4.1.1.1"></eq><ci id="Thmexample2.p3.14.m4.1.1.2.cmml" xref="Thmexample2.p3.14.m4.1.1.2">𝑥</ci><cn id="Thmexample2.p3.14.m4.1.1.3.cmml" type="integer" xref="Thmexample2.p3.14.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.14.m4.1c">x=1</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.14.m4.1d">italic_x = 1</annotation></semantics></math> we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx43"> <tbody id="Pt0.A4.Ex69"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{1}(1,\alpha)=1+\alpha>1=f_{2}(1,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex69.m1.4"><semantics id="Pt0.A4.Ex69.m1.4a"><mrow id="Pt0.A4.Ex69.m1.4.5" xref="Pt0.A4.Ex69.m1.4.5.cmml"><mrow id="Pt0.A4.Ex69.m1.4.5.2" xref="Pt0.A4.Ex69.m1.4.5.2.cmml"><msub id="Pt0.A4.Ex69.m1.4.5.2.2" xref="Pt0.A4.Ex69.m1.4.5.2.2.cmml"><mi id="Pt0.A4.Ex69.m1.4.5.2.2.2" xref="Pt0.A4.Ex69.m1.4.5.2.2.2.cmml">f</mi><mn id="Pt0.A4.Ex69.m1.4.5.2.2.3" xref="Pt0.A4.Ex69.m1.4.5.2.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex69.m1.4.5.2.1" xref="Pt0.A4.Ex69.m1.4.5.2.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex69.m1.4.5.2.3.2" xref="Pt0.A4.Ex69.m1.4.5.2.3.1.cmml"><mo id="Pt0.A4.Ex69.m1.4.5.2.3.2.1" stretchy="false" xref="Pt0.A4.Ex69.m1.4.5.2.3.1.cmml">(</mo><mn id="Pt0.A4.Ex69.m1.1.1" xref="Pt0.A4.Ex69.m1.1.1.cmml">1</mn><mo id="Pt0.A4.Ex69.m1.4.5.2.3.2.2" xref="Pt0.A4.Ex69.m1.4.5.2.3.1.cmml">,</mo><mi id="Pt0.A4.Ex69.m1.2.2" xref="Pt0.A4.Ex69.m1.2.2.cmml">α</mi><mo id="Pt0.A4.Ex69.m1.4.5.2.3.2.3" stretchy="false" xref="Pt0.A4.Ex69.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex69.m1.4.5.3" xref="Pt0.A4.Ex69.m1.4.5.3.cmml">=</mo><mrow id="Pt0.A4.Ex69.m1.4.5.4" xref="Pt0.A4.Ex69.m1.4.5.4.cmml"><mn id="Pt0.A4.Ex69.m1.4.5.4.2" xref="Pt0.A4.Ex69.m1.4.5.4.2.cmml">1</mn><mo id="Pt0.A4.Ex69.m1.4.5.4.1" xref="Pt0.A4.Ex69.m1.4.5.4.1.cmml">+</mo><mi id="Pt0.A4.Ex69.m1.4.5.4.3" xref="Pt0.A4.Ex69.m1.4.5.4.3.cmml">α</mi></mrow><mo id="Pt0.A4.Ex69.m1.4.5.5" xref="Pt0.A4.Ex69.m1.4.5.5.cmml">></mo><mn id="Pt0.A4.Ex69.m1.4.5.6" xref="Pt0.A4.Ex69.m1.4.5.6.cmml">1</mn><mo id="Pt0.A4.Ex69.m1.4.5.7" xref="Pt0.A4.Ex69.m1.4.5.7.cmml">=</mo><mrow id="Pt0.A4.Ex69.m1.4.5.8" xref="Pt0.A4.Ex69.m1.4.5.8.cmml"><msub id="Pt0.A4.Ex69.m1.4.5.8.2" xref="Pt0.A4.Ex69.m1.4.5.8.2.cmml"><mi id="Pt0.A4.Ex69.m1.4.5.8.2.2" xref="Pt0.A4.Ex69.m1.4.5.8.2.2.cmml">f</mi><mn id="Pt0.A4.Ex69.m1.4.5.8.2.3" xref="Pt0.A4.Ex69.m1.4.5.8.2.3.cmml">2</mn></msub><mo id="Pt0.A4.Ex69.m1.4.5.8.1" xref="Pt0.A4.Ex69.m1.4.5.8.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex69.m1.4.5.8.3.2" xref="Pt0.A4.Ex69.m1.4.5.8.3.1.cmml"><mo id="Pt0.A4.Ex69.m1.4.5.8.3.2.1" stretchy="false" xref="Pt0.A4.Ex69.m1.4.5.8.3.1.cmml">(</mo><mn id="Pt0.A4.Ex69.m1.3.3" xref="Pt0.A4.Ex69.m1.3.3.cmml">1</mn><mo id="Pt0.A4.Ex69.m1.4.5.8.3.2.2" xref="Pt0.A4.Ex69.m1.4.5.8.3.1.cmml">,</mo><mi id="Pt0.A4.Ex69.m1.4.4" xref="Pt0.A4.Ex69.m1.4.4.cmml">α</mi><mo id="Pt0.A4.Ex69.m1.4.5.8.3.2.3" stretchy="false" xref="Pt0.A4.Ex69.m1.4.5.8.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex69.m1.4b"><apply id="Pt0.A4.Ex69.m1.4.5.cmml" xref="Pt0.A4.Ex69.m1.4.5"><and id="Pt0.A4.Ex69.m1.4.5a.cmml" xref="Pt0.A4.Ex69.m1.4.5"></and><apply id="Pt0.A4.Ex69.m1.4.5b.cmml" xref="Pt0.A4.Ex69.m1.4.5"><eq id="Pt0.A4.Ex69.m1.4.5.3.cmml" xref="Pt0.A4.Ex69.m1.4.5.3"></eq><apply id="Pt0.A4.Ex69.m1.4.5.2.cmml" xref="Pt0.A4.Ex69.m1.4.5.2"><times id="Pt0.A4.Ex69.m1.4.5.2.1.cmml" xref="Pt0.A4.Ex69.m1.4.5.2.1"></times><apply id="Pt0.A4.Ex69.m1.4.5.2.2.cmml" xref="Pt0.A4.Ex69.m1.4.5.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex69.m1.4.5.2.2.1.cmml" xref="Pt0.A4.Ex69.m1.4.5.2.2">subscript</csymbol><ci id="Pt0.A4.Ex69.m1.4.5.2.2.2.cmml" xref="Pt0.A4.Ex69.m1.4.5.2.2.2">𝑓</ci><cn id="Pt0.A4.Ex69.m1.4.5.2.2.3.cmml" type="integer" xref="Pt0.A4.Ex69.m1.4.5.2.2.3">1</cn></apply><interval closure="open" id="Pt0.A4.Ex69.m1.4.5.2.3.1.cmml" xref="Pt0.A4.Ex69.m1.4.5.2.3.2"><cn id="Pt0.A4.Ex69.m1.1.1.cmml" type="integer" xref="Pt0.A4.Ex69.m1.1.1">1</cn><ci id="Pt0.A4.Ex69.m1.2.2.cmml" xref="Pt0.A4.Ex69.m1.2.2">𝛼</ci></interval></apply><apply id="Pt0.A4.Ex69.m1.4.5.4.cmml" xref="Pt0.A4.Ex69.m1.4.5.4"><plus id="Pt0.A4.Ex69.m1.4.5.4.1.cmml" xref="Pt0.A4.Ex69.m1.4.5.4.1"></plus><cn id="Pt0.A4.Ex69.m1.4.5.4.2.cmml" type="integer" xref="Pt0.A4.Ex69.m1.4.5.4.2">1</cn><ci id="Pt0.A4.Ex69.m1.4.5.4.3.cmml" xref="Pt0.A4.Ex69.m1.4.5.4.3">𝛼</ci></apply></apply><apply id="Pt0.A4.Ex69.m1.4.5c.cmml" xref="Pt0.A4.Ex69.m1.4.5"><gt id="Pt0.A4.Ex69.m1.4.5.5.cmml" xref="Pt0.A4.Ex69.m1.4.5.5"></gt><share href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex69.m1.4.5.4.cmml" id="Pt0.A4.Ex69.m1.4.5d.cmml" xref="Pt0.A4.Ex69.m1.4.5"></share><cn id="Pt0.A4.Ex69.m1.4.5.6.cmml" type="integer" xref="Pt0.A4.Ex69.m1.4.5.6">1</cn></apply><apply id="Pt0.A4.Ex69.m1.4.5e.cmml" xref="Pt0.A4.Ex69.m1.4.5"><eq id="Pt0.A4.Ex69.m1.4.5.7.cmml" xref="Pt0.A4.Ex69.m1.4.5.7"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex69.m1.4.5.6.cmml" id="Pt0.A4.Ex69.m1.4.5f.cmml" xref="Pt0.A4.Ex69.m1.4.5"></share><apply id="Pt0.A4.Ex69.m1.4.5.8.cmml" xref="Pt0.A4.Ex69.m1.4.5.8"><times id="Pt0.A4.Ex69.m1.4.5.8.1.cmml" xref="Pt0.A4.Ex69.m1.4.5.8.1"></times><apply id="Pt0.A4.Ex69.m1.4.5.8.2.cmml" xref="Pt0.A4.Ex69.m1.4.5.8.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex69.m1.4.5.8.2.1.cmml" xref="Pt0.A4.Ex69.m1.4.5.8.2">subscript</csymbol><ci id="Pt0.A4.Ex69.m1.4.5.8.2.2.cmml" xref="Pt0.A4.Ex69.m1.4.5.8.2.2">𝑓</ci><cn id="Pt0.A4.Ex69.m1.4.5.8.2.3.cmml" type="integer" xref="Pt0.A4.Ex69.m1.4.5.8.2.3">2</cn></apply><interval closure="open" id="Pt0.A4.Ex69.m1.4.5.8.3.1.cmml" xref="Pt0.A4.Ex69.m1.4.5.8.3.2"><cn id="Pt0.A4.Ex69.m1.3.3.cmml" type="integer" xref="Pt0.A4.Ex69.m1.3.3">1</cn><ci id="Pt0.A4.Ex69.m1.4.4.cmml" xref="Pt0.A4.Ex69.m1.4.4">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex69.m1.4c">\displaystyle f_{1}(1,\alpha)=1+\alpha>1=f_{2}(1,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex69.m1.4d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 1 , italic_α ) = 1 + italic_α > 1 = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmexample2.p3.20">so <math alttext="f_{1},f_{2}" class="ltx_Math" display="inline" id="Thmexample2.p3.15.m1.2"><semantics id="Thmexample2.p3.15.m1.2a"><mrow id="Thmexample2.p3.15.m1.2.2.2" xref="Thmexample2.p3.15.m1.2.2.3.cmml"><msub id="Thmexample2.p3.15.m1.1.1.1.1" xref="Thmexample2.p3.15.m1.1.1.1.1.cmml"><mi id="Thmexample2.p3.15.m1.1.1.1.1.2" xref="Thmexample2.p3.15.m1.1.1.1.1.2.cmml">f</mi><mn id="Thmexample2.p3.15.m1.1.1.1.1.3" xref="Thmexample2.p3.15.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample2.p3.15.m1.2.2.2.3" xref="Thmexample2.p3.15.m1.2.2.3.cmml">,</mo><msub id="Thmexample2.p3.15.m1.2.2.2.2" xref="Thmexample2.p3.15.m1.2.2.2.2.cmml"><mi id="Thmexample2.p3.15.m1.2.2.2.2.2" xref="Thmexample2.p3.15.m1.2.2.2.2.2.cmml">f</mi><mn id="Thmexample2.p3.15.m1.2.2.2.2.3" xref="Thmexample2.p3.15.m1.2.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.15.m1.2b"><list id="Thmexample2.p3.15.m1.2.2.3.cmml" xref="Thmexample2.p3.15.m1.2.2.2"><apply id="Thmexample2.p3.15.m1.1.1.1.1.cmml" xref="Thmexample2.p3.15.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p3.15.m1.1.1.1.1.1.cmml" xref="Thmexample2.p3.15.m1.1.1.1.1">subscript</csymbol><ci id="Thmexample2.p3.15.m1.1.1.1.1.2.cmml" xref="Thmexample2.p3.15.m1.1.1.1.1.2">𝑓</ci><cn id="Thmexample2.p3.15.m1.1.1.1.1.3.cmml" type="integer" xref="Thmexample2.p3.15.m1.1.1.1.1.3">1</cn></apply><apply id="Thmexample2.p3.15.m1.2.2.2.2.cmml" xref="Thmexample2.p3.15.m1.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.15.m1.2.2.2.2.1.cmml" xref="Thmexample2.p3.15.m1.2.2.2.2">subscript</csymbol><ci id="Thmexample2.p3.15.m1.2.2.2.2.2.cmml" xref="Thmexample2.p3.15.m1.2.2.2.2.2">𝑓</ci><cn id="Thmexample2.p3.15.m1.2.2.2.2.3.cmml" type="integer" xref="Thmexample2.p3.15.m1.2.2.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.15.m1.2c">f_{1},f_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.15.m1.2d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> intersect in <math alttext="\left(\frac{1}{2},1\right)" class="ltx_Math" display="inline" id="Thmexample2.p3.16.m2.2"><semantics id="Thmexample2.p3.16.m2.2a"><mrow id="Thmexample2.p3.16.m2.2.3.2" xref="Thmexample2.p3.16.m2.2.3.1.cmml"><mo id="Thmexample2.p3.16.m2.2.3.2.1" xref="Thmexample2.p3.16.m2.2.3.1.cmml">(</mo><mfrac id="Thmexample2.p3.16.m2.1.1" xref="Thmexample2.p3.16.m2.1.1.cmml"><mn id="Thmexample2.p3.16.m2.1.1.2" xref="Thmexample2.p3.16.m2.1.1.2.cmml">1</mn><mn id="Thmexample2.p3.16.m2.1.1.3" xref="Thmexample2.p3.16.m2.1.1.3.cmml">2</mn></mfrac><mo id="Thmexample2.p3.16.m2.2.3.2.2" xref="Thmexample2.p3.16.m2.2.3.1.cmml">,</mo><mn id="Thmexample2.p3.16.m2.2.2" xref="Thmexample2.p3.16.m2.2.2.cmml">1</mn><mo id="Thmexample2.p3.16.m2.2.3.2.3" xref="Thmexample2.p3.16.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.16.m2.2b"><interval closure="open" id="Thmexample2.p3.16.m2.2.3.1.cmml" xref="Thmexample2.p3.16.m2.2.3.2"><apply id="Thmexample2.p3.16.m2.1.1.cmml" xref="Thmexample2.p3.16.m2.1.1"><divide id="Thmexample2.p3.16.m2.1.1.1.cmml" xref="Thmexample2.p3.16.m2.1.1"></divide><cn id="Thmexample2.p3.16.m2.1.1.2.cmml" type="integer" xref="Thmexample2.p3.16.m2.1.1.2">1</cn><cn id="Thmexample2.p3.16.m2.1.1.3.cmml" type="integer" xref="Thmexample2.p3.16.m2.1.1.3">2</cn></apply><cn id="Thmexample2.p3.16.m2.2.2.cmml" type="integer" xref="Thmexample2.p3.16.m2.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.16.m2.2c">\left(\frac{1}{2},1\right)</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.16.m2.2d">( divide start_ARG 1 end_ARG start_ARG 2 end_ARG , 1 )</annotation></semantics></math>, therefore <math alttext="\min_{x}\max(f_{1}(x,\alpha),f_{2}(x,\alpha))\in\left(\frac{1}{2},1\right)" class="ltx_Math" display="inline" id="Thmexample2.p3.17.m3.9"><semantics id="Thmexample2.p3.17.m3.9a"><mrow id="Thmexample2.p3.17.m3.9.9" xref="Thmexample2.p3.17.m3.9.9.cmml"><mrow id="Thmexample2.p3.17.m3.9.9.2" xref="Thmexample2.p3.17.m3.9.9.2.cmml"><msub id="Thmexample2.p3.17.m3.9.9.2.3" xref="Thmexample2.p3.17.m3.9.9.2.3.cmml"><mi id="Thmexample2.p3.17.m3.9.9.2.3.2" xref="Thmexample2.p3.17.m3.9.9.2.3.2.cmml">min</mi><mi id="Thmexample2.p3.17.m3.9.9.2.3.3" xref="Thmexample2.p3.17.m3.9.9.2.3.3.cmml">x</mi></msub><mo id="Thmexample2.p3.17.m3.9.9.2a" lspace="0.167em" xref="Thmexample2.p3.17.m3.9.9.2.cmml">⁡</mo><mrow id="Thmexample2.p3.17.m3.9.9.2.2.2" xref="Thmexample2.p3.17.m3.9.9.2.2.3.cmml"><mi id="Thmexample2.p3.17.m3.5.5" xref="Thmexample2.p3.17.m3.5.5.cmml">max</mi><mo id="Thmexample2.p3.17.m3.9.9.2.2.2a" xref="Thmexample2.p3.17.m3.9.9.2.2.3.cmml">⁡</mo><mrow id="Thmexample2.p3.17.m3.9.9.2.2.2.2" xref="Thmexample2.p3.17.m3.9.9.2.2.3.cmml"><mo id="Thmexample2.p3.17.m3.9.9.2.2.2.2.3" stretchy="false" xref="Thmexample2.p3.17.m3.9.9.2.2.3.cmml">(</mo><mrow id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.cmml"><msub id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.cmml"><mi id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.2" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.2.cmml">f</mi><mn id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.3" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.1" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.2" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.1.cmml"><mo id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.2.1" stretchy="false" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.1.cmml">(</mo><mi id="Thmexample2.p3.17.m3.1.1" xref="Thmexample2.p3.17.m3.1.1.cmml">x</mi><mo id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.2.2" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.1.cmml">,</mo><mi id="Thmexample2.p3.17.m3.2.2" xref="Thmexample2.p3.17.m3.2.2.cmml">α</mi><mo id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.2.3" stretchy="false" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.17.m3.9.9.2.2.2.2.4" xref="Thmexample2.p3.17.m3.9.9.2.2.3.cmml">,</mo><mrow id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.cmml"><msub id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.cmml"><mi id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.2" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.2.cmml">f</mi><mn id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.3" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.1" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.1.cmml">⁢</mo><mrow id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.2" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.1.cmml"><mo id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.2.1" stretchy="false" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.1.cmml">(</mo><mi id="Thmexample2.p3.17.m3.3.3" xref="Thmexample2.p3.17.m3.3.3.cmml">x</mi><mo id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.2.2" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.1.cmml">,</mo><mi id="Thmexample2.p3.17.m3.4.4" xref="Thmexample2.p3.17.m3.4.4.cmml">α</mi><mo id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.2.3" stretchy="false" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.17.m3.9.9.2.2.2.2.5" stretchy="false" xref="Thmexample2.p3.17.m3.9.9.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="Thmexample2.p3.17.m3.9.9.3" xref="Thmexample2.p3.17.m3.9.9.3.cmml">∈</mo><mrow id="Thmexample2.p3.17.m3.9.9.4.2" xref="Thmexample2.p3.17.m3.9.9.4.1.cmml"><mo id="Thmexample2.p3.17.m3.9.9.4.2.1" xref="Thmexample2.p3.17.m3.9.9.4.1.cmml">(</mo><mfrac id="Thmexample2.p3.17.m3.6.6" xref="Thmexample2.p3.17.m3.6.6.cmml"><mn id="Thmexample2.p3.17.m3.6.6.2" xref="Thmexample2.p3.17.m3.6.6.2.cmml">1</mn><mn id="Thmexample2.p3.17.m3.6.6.3" xref="Thmexample2.p3.17.m3.6.6.3.cmml">2</mn></mfrac><mo id="Thmexample2.p3.17.m3.9.9.4.2.2" xref="Thmexample2.p3.17.m3.9.9.4.1.cmml">,</mo><mn id="Thmexample2.p3.17.m3.7.7" xref="Thmexample2.p3.17.m3.7.7.cmml">1</mn><mo id="Thmexample2.p3.17.m3.9.9.4.2.3" xref="Thmexample2.p3.17.m3.9.9.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.17.m3.9b"><apply id="Thmexample2.p3.17.m3.9.9.cmml" xref="Thmexample2.p3.17.m3.9.9"><in id="Thmexample2.p3.17.m3.9.9.3.cmml" xref="Thmexample2.p3.17.m3.9.9.3"></in><apply id="Thmexample2.p3.17.m3.9.9.2.cmml" xref="Thmexample2.p3.17.m3.9.9.2"><apply id="Thmexample2.p3.17.m3.9.9.2.3.cmml" xref="Thmexample2.p3.17.m3.9.9.2.3"><csymbol cd="ambiguous" id="Thmexample2.p3.17.m3.9.9.2.3.1.cmml" xref="Thmexample2.p3.17.m3.9.9.2.3">subscript</csymbol><min id="Thmexample2.p3.17.m3.9.9.2.3.2.cmml" xref="Thmexample2.p3.17.m3.9.9.2.3.2"></min><ci id="Thmexample2.p3.17.m3.9.9.2.3.3.cmml" xref="Thmexample2.p3.17.m3.9.9.2.3.3">𝑥</ci></apply><apply id="Thmexample2.p3.17.m3.9.9.2.2.3.cmml" xref="Thmexample2.p3.17.m3.9.9.2.2.2"><max id="Thmexample2.p3.17.m3.5.5.cmml" xref="Thmexample2.p3.17.m3.5.5"></max><apply id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.cmml" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1"><times id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.1.cmml" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.1"></times><apply id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.cmml" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.1.cmml" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2">subscript</csymbol><ci id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.2.cmml" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.2">𝑓</ci><cn id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.3.cmml" type="integer" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.2.3">1</cn></apply><interval closure="open" id="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.1.cmml" xref="Thmexample2.p3.17.m3.8.8.1.1.1.1.1.3.2"><ci id="Thmexample2.p3.17.m3.1.1.cmml" xref="Thmexample2.p3.17.m3.1.1">𝑥</ci><ci id="Thmexample2.p3.17.m3.2.2.cmml" xref="Thmexample2.p3.17.m3.2.2">𝛼</ci></interval></apply><apply id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.cmml" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2"><times id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.1.cmml" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.1"></times><apply id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.cmml" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.1.cmml" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2">subscript</csymbol><ci id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.2.cmml" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.2">𝑓</ci><cn id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.3.cmml" type="integer" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.2.3">2</cn></apply><interval closure="open" id="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.1.cmml" xref="Thmexample2.p3.17.m3.9.9.2.2.2.2.2.3.2"><ci id="Thmexample2.p3.17.m3.3.3.cmml" xref="Thmexample2.p3.17.m3.3.3">𝑥</ci><ci id="Thmexample2.p3.17.m3.4.4.cmml" xref="Thmexample2.p3.17.m3.4.4">𝛼</ci></interval></apply></apply></apply><interval closure="open" id="Thmexample2.p3.17.m3.9.9.4.1.cmml" xref="Thmexample2.p3.17.m3.9.9.4.2"><apply id="Thmexample2.p3.17.m3.6.6.cmml" xref="Thmexample2.p3.17.m3.6.6"><divide id="Thmexample2.p3.17.m3.6.6.1.cmml" xref="Thmexample2.p3.17.m3.6.6"></divide><cn id="Thmexample2.p3.17.m3.6.6.2.cmml" type="integer" xref="Thmexample2.p3.17.m3.6.6.2">1</cn><cn id="Thmexample2.p3.17.m3.6.6.3.cmml" type="integer" xref="Thmexample2.p3.17.m3.6.6.3">2</cn></apply><cn id="Thmexample2.p3.17.m3.7.7.cmml" type="integer" xref="Thmexample2.p3.17.m3.7.7">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.17.m3.9c">\min_{x}\max(f_{1}(x,\alpha),f_{2}(x,\alpha))\in\left(\frac{1}{2},1\right)</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.17.m3.9d">roman_min start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_α ) , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_α ) ) ∈ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG , 1 )</annotation></semantics></math>. The exact range <math alttext="\left(2^{-1},2^{-1/n}\right]" class="ltx_Math" display="inline" id="Thmexample2.p3.18.m4.2"><semantics id="Thmexample2.p3.18.m4.2a"><mrow id="Thmexample2.p3.18.m4.2.2.2" xref="Thmexample2.p3.18.m4.2.2.3.cmml"><mo id="Thmexample2.p3.18.m4.2.2.2.3" xref="Thmexample2.p3.18.m4.2.2.3.cmml">(</mo><msup id="Thmexample2.p3.18.m4.1.1.1.1" xref="Thmexample2.p3.18.m4.1.1.1.1.cmml"><mn id="Thmexample2.p3.18.m4.1.1.1.1.2" xref="Thmexample2.p3.18.m4.1.1.1.1.2.cmml">2</mn><mrow id="Thmexample2.p3.18.m4.1.1.1.1.3" xref="Thmexample2.p3.18.m4.1.1.1.1.3.cmml"><mo id="Thmexample2.p3.18.m4.1.1.1.1.3a" xref="Thmexample2.p3.18.m4.1.1.1.1.3.cmml">−</mo><mn id="Thmexample2.p3.18.m4.1.1.1.1.3.2" xref="Thmexample2.p3.18.m4.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="Thmexample2.p3.18.m4.2.2.2.4" xref="Thmexample2.p3.18.m4.2.2.3.cmml">,</mo><msup id="Thmexample2.p3.18.m4.2.2.2.2" xref="Thmexample2.p3.18.m4.2.2.2.2.cmml"><mn id="Thmexample2.p3.18.m4.2.2.2.2.2" xref="Thmexample2.p3.18.m4.2.2.2.2.2.cmml">2</mn><mrow id="Thmexample2.p3.18.m4.2.2.2.2.3" xref="Thmexample2.p3.18.m4.2.2.2.2.3.cmml"><mo id="Thmexample2.p3.18.m4.2.2.2.2.3a" xref="Thmexample2.p3.18.m4.2.2.2.2.3.cmml">−</mo><mrow id="Thmexample2.p3.18.m4.2.2.2.2.3.2" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2.cmml"><mn id="Thmexample2.p3.18.m4.2.2.2.2.3.2.2" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2.2.cmml">1</mn><mo id="Thmexample2.p3.18.m4.2.2.2.2.3.2.1" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2.1.cmml">/</mo><mi id="Thmexample2.p3.18.m4.2.2.2.2.3.2.3" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2.3.cmml">n</mi></mrow></mrow></msup><mo id="Thmexample2.p3.18.m4.2.2.2.5" xref="Thmexample2.p3.18.m4.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.18.m4.2b"><interval closure="open-closed" id="Thmexample2.p3.18.m4.2.2.3.cmml" xref="Thmexample2.p3.18.m4.2.2.2"><apply id="Thmexample2.p3.18.m4.1.1.1.1.cmml" xref="Thmexample2.p3.18.m4.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p3.18.m4.1.1.1.1.1.cmml" xref="Thmexample2.p3.18.m4.1.1.1.1">superscript</csymbol><cn id="Thmexample2.p3.18.m4.1.1.1.1.2.cmml" type="integer" xref="Thmexample2.p3.18.m4.1.1.1.1.2">2</cn><apply id="Thmexample2.p3.18.m4.1.1.1.1.3.cmml" xref="Thmexample2.p3.18.m4.1.1.1.1.3"><minus id="Thmexample2.p3.18.m4.1.1.1.1.3.1.cmml" xref="Thmexample2.p3.18.m4.1.1.1.1.3"></minus><cn id="Thmexample2.p3.18.m4.1.1.1.1.3.2.cmml" type="integer" xref="Thmexample2.p3.18.m4.1.1.1.1.3.2">1</cn></apply></apply><apply id="Thmexample2.p3.18.m4.2.2.2.2.cmml" xref="Thmexample2.p3.18.m4.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.18.m4.2.2.2.2.1.cmml" xref="Thmexample2.p3.18.m4.2.2.2.2">superscript</csymbol><cn id="Thmexample2.p3.18.m4.2.2.2.2.2.cmml" type="integer" xref="Thmexample2.p3.18.m4.2.2.2.2.2">2</cn><apply id="Thmexample2.p3.18.m4.2.2.2.2.3.cmml" xref="Thmexample2.p3.18.m4.2.2.2.2.3"><minus id="Thmexample2.p3.18.m4.2.2.2.2.3.1.cmml" xref="Thmexample2.p3.18.m4.2.2.2.2.3"></minus><apply id="Thmexample2.p3.18.m4.2.2.2.2.3.2.cmml" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2"><divide id="Thmexample2.p3.18.m4.2.2.2.2.3.2.1.cmml" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2.1"></divide><cn id="Thmexample2.p3.18.m4.2.2.2.2.3.2.2.cmml" type="integer" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2.2">1</cn><ci id="Thmexample2.p3.18.m4.2.2.2.2.3.2.3.cmml" xref="Thmexample2.p3.18.m4.2.2.2.2.3.2.3">𝑛</ci></apply></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.18.m4.2c">\left(2^{-1},2^{-1/n}\right]</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.18.m4.2d">( 2 start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , 2 start_POSTSUPERSCRIPT - 1 / italic_n end_POSTSUPERSCRIPT ]</annotation></semantics></math> is obtained from solving <math alttext="f_{1}(x,\alpha)=f_{2}(x,\alpha)" class="ltx_Math" display="inline" id="Thmexample2.p3.19.m5.4"><semantics id="Thmexample2.p3.19.m5.4a"><mrow id="Thmexample2.p3.19.m5.4.5" xref="Thmexample2.p3.19.m5.4.5.cmml"><mrow id="Thmexample2.p3.19.m5.4.5.2" xref="Thmexample2.p3.19.m5.4.5.2.cmml"><msub id="Thmexample2.p3.19.m5.4.5.2.2" xref="Thmexample2.p3.19.m5.4.5.2.2.cmml"><mi id="Thmexample2.p3.19.m5.4.5.2.2.2" xref="Thmexample2.p3.19.m5.4.5.2.2.2.cmml">f</mi><mn id="Thmexample2.p3.19.m5.4.5.2.2.3" xref="Thmexample2.p3.19.m5.4.5.2.2.3.cmml">1</mn></msub><mo id="Thmexample2.p3.19.m5.4.5.2.1" xref="Thmexample2.p3.19.m5.4.5.2.1.cmml">⁢</mo><mrow id="Thmexample2.p3.19.m5.4.5.2.3.2" xref="Thmexample2.p3.19.m5.4.5.2.3.1.cmml"><mo id="Thmexample2.p3.19.m5.4.5.2.3.2.1" stretchy="false" xref="Thmexample2.p3.19.m5.4.5.2.3.1.cmml">(</mo><mi id="Thmexample2.p3.19.m5.1.1" xref="Thmexample2.p3.19.m5.1.1.cmml">x</mi><mo id="Thmexample2.p3.19.m5.4.5.2.3.2.2" xref="Thmexample2.p3.19.m5.4.5.2.3.1.cmml">,</mo><mi id="Thmexample2.p3.19.m5.2.2" xref="Thmexample2.p3.19.m5.2.2.cmml">α</mi><mo id="Thmexample2.p3.19.m5.4.5.2.3.2.3" stretchy="false" xref="Thmexample2.p3.19.m5.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmexample2.p3.19.m5.4.5.1" xref="Thmexample2.p3.19.m5.4.5.1.cmml">=</mo><mrow id="Thmexample2.p3.19.m5.4.5.3" xref="Thmexample2.p3.19.m5.4.5.3.cmml"><msub id="Thmexample2.p3.19.m5.4.5.3.2" xref="Thmexample2.p3.19.m5.4.5.3.2.cmml"><mi id="Thmexample2.p3.19.m5.4.5.3.2.2" xref="Thmexample2.p3.19.m5.4.5.3.2.2.cmml">f</mi><mn id="Thmexample2.p3.19.m5.4.5.3.2.3" xref="Thmexample2.p3.19.m5.4.5.3.2.3.cmml">2</mn></msub><mo id="Thmexample2.p3.19.m5.4.5.3.1" xref="Thmexample2.p3.19.m5.4.5.3.1.cmml">⁢</mo><mrow id="Thmexample2.p3.19.m5.4.5.3.3.2" xref="Thmexample2.p3.19.m5.4.5.3.3.1.cmml"><mo id="Thmexample2.p3.19.m5.4.5.3.3.2.1" stretchy="false" xref="Thmexample2.p3.19.m5.4.5.3.3.1.cmml">(</mo><mi id="Thmexample2.p3.19.m5.3.3" xref="Thmexample2.p3.19.m5.3.3.cmml">x</mi><mo id="Thmexample2.p3.19.m5.4.5.3.3.2.2" xref="Thmexample2.p3.19.m5.4.5.3.3.1.cmml">,</mo><mi id="Thmexample2.p3.19.m5.4.4" xref="Thmexample2.p3.19.m5.4.4.cmml">α</mi><mo id="Thmexample2.p3.19.m5.4.5.3.3.2.3" stretchy="false" xref="Thmexample2.p3.19.m5.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.19.m5.4b"><apply id="Thmexample2.p3.19.m5.4.5.cmml" xref="Thmexample2.p3.19.m5.4.5"><eq id="Thmexample2.p3.19.m5.4.5.1.cmml" xref="Thmexample2.p3.19.m5.4.5.1"></eq><apply id="Thmexample2.p3.19.m5.4.5.2.cmml" xref="Thmexample2.p3.19.m5.4.5.2"><times id="Thmexample2.p3.19.m5.4.5.2.1.cmml" xref="Thmexample2.p3.19.m5.4.5.2.1"></times><apply id="Thmexample2.p3.19.m5.4.5.2.2.cmml" xref="Thmexample2.p3.19.m5.4.5.2.2"><csymbol cd="ambiguous" id="Thmexample2.p3.19.m5.4.5.2.2.1.cmml" xref="Thmexample2.p3.19.m5.4.5.2.2">subscript</csymbol><ci id="Thmexample2.p3.19.m5.4.5.2.2.2.cmml" xref="Thmexample2.p3.19.m5.4.5.2.2.2">𝑓</ci><cn id="Thmexample2.p3.19.m5.4.5.2.2.3.cmml" type="integer" xref="Thmexample2.p3.19.m5.4.5.2.2.3">1</cn></apply><interval closure="open" id="Thmexample2.p3.19.m5.4.5.2.3.1.cmml" xref="Thmexample2.p3.19.m5.4.5.2.3.2"><ci id="Thmexample2.p3.19.m5.1.1.cmml" xref="Thmexample2.p3.19.m5.1.1">𝑥</ci><ci id="Thmexample2.p3.19.m5.2.2.cmml" xref="Thmexample2.p3.19.m5.2.2">𝛼</ci></interval></apply><apply id="Thmexample2.p3.19.m5.4.5.3.cmml" xref="Thmexample2.p3.19.m5.4.5.3"><times id="Thmexample2.p3.19.m5.4.5.3.1.cmml" xref="Thmexample2.p3.19.m5.4.5.3.1"></times><apply id="Thmexample2.p3.19.m5.4.5.3.2.cmml" xref="Thmexample2.p3.19.m5.4.5.3.2"><csymbol cd="ambiguous" id="Thmexample2.p3.19.m5.4.5.3.2.1.cmml" xref="Thmexample2.p3.19.m5.4.5.3.2">subscript</csymbol><ci id="Thmexample2.p3.19.m5.4.5.3.2.2.cmml" xref="Thmexample2.p3.19.m5.4.5.3.2.2">𝑓</ci><cn id="Thmexample2.p3.19.m5.4.5.3.2.3.cmml" type="integer" xref="Thmexample2.p3.19.m5.4.5.3.2.3">2</cn></apply><interval closure="open" id="Thmexample2.p3.19.m5.4.5.3.3.1.cmml" xref="Thmexample2.p3.19.m5.4.5.3.3.2"><ci id="Thmexample2.p3.19.m5.3.3.cmml" xref="Thmexample2.p3.19.m5.3.3">𝑥</ci><ci id="Thmexample2.p3.19.m5.4.4.cmml" xref="Thmexample2.p3.19.m5.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.19.m5.4c">f_{1}(x,\alpha)=f_{2}(x,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.19.m5.4d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_α ) = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_α )</annotation></semantics></math> for <math alttext="\alpha\geq\ell-1" class="ltx_Math" display="inline" id="Thmexample2.p3.20.m6.1"><semantics id="Thmexample2.p3.20.m6.1a"><mrow id="Thmexample2.p3.20.m6.1.1" xref="Thmexample2.p3.20.m6.1.1.cmml"><mi id="Thmexample2.p3.20.m6.1.1.2" xref="Thmexample2.p3.20.m6.1.1.2.cmml">α</mi><mo id="Thmexample2.p3.20.m6.1.1.1" xref="Thmexample2.p3.20.m6.1.1.1.cmml">≥</mo><mrow id="Thmexample2.p3.20.m6.1.1.3" xref="Thmexample2.p3.20.m6.1.1.3.cmml"><mi id="Thmexample2.p3.20.m6.1.1.3.2" mathvariant="normal" xref="Thmexample2.p3.20.m6.1.1.3.2.cmml">ℓ</mi><mo id="Thmexample2.p3.20.m6.1.1.3.1" xref="Thmexample2.p3.20.m6.1.1.3.1.cmml">−</mo><mn id="Thmexample2.p3.20.m6.1.1.3.3" xref="Thmexample2.p3.20.m6.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p3.20.m6.1b"><apply id="Thmexample2.p3.20.m6.1.1.cmml" xref="Thmexample2.p3.20.m6.1.1"><geq id="Thmexample2.p3.20.m6.1.1.1.cmml" xref="Thmexample2.p3.20.m6.1.1.1"></geq><ci id="Thmexample2.p3.20.m6.1.1.2.cmml" xref="Thmexample2.p3.20.m6.1.1.2">𝛼</ci><apply id="Thmexample2.p3.20.m6.1.1.3.cmml" xref="Thmexample2.p3.20.m6.1.1.3"><minus id="Thmexample2.p3.20.m6.1.1.3.1.cmml" xref="Thmexample2.p3.20.m6.1.1.3.1"></minus><ci id="Thmexample2.p3.20.m6.1.1.3.2.cmml" xref="Thmexample2.p3.20.m6.1.1.3.2">ℓ</ci><cn id="Thmexample2.p3.20.m6.1.1.3.3.cmml" type="integer" xref="Thmexample2.p3.20.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p3.20.m6.1c">\alpha\geq\ell-1</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p3.20.m6.1d">italic_α ≥ roman_ℓ - 1</annotation></semantics></math>. This finishes the proof.</p> </div> </div> </section> <section class="ltx_subsection" id="Pt0.A4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">0.D.2 </span>More definitions</h3> <div class="ltx_para" id="Pt0.A4.SS2.p1"> <p class="ltx_p" id="Pt0.A4.SS2.p1.25">We need more definitions to prove the results of Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a>. For <math alttext="\alpha" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.1.m1.1"><semantics id="Pt0.A4.SS2.p1.1.m1.1a"><mi id="Pt0.A4.SS2.p1.1.m1.1.1" xref="Pt0.A4.SS2.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.1.m1.1b"><ci id="Pt0.A4.SS2.p1.1.m1.1.1.cmml" xref="Pt0.A4.SS2.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.1.m1.1d">italic_α</annotation></semantics></math> and <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.2.m2.1"><semantics id="Pt0.A4.SS2.p1.2.m2.1a"><mrow id="Pt0.A4.SS2.p1.2.m2.1.1" xref="Pt0.A4.SS2.p1.2.m2.1.1.cmml"><mi id="Pt0.A4.SS2.p1.2.m2.1.1.2" xref="Pt0.A4.SS2.p1.2.m2.1.1.2.cmml">d</mi><mo id="Pt0.A4.SS2.p1.2.m2.1.1.1" xref="Pt0.A4.SS2.p1.2.m2.1.1.1.cmml">∈</mo><msup id="Pt0.A4.SS2.p1.2.m2.1.1.3" xref="Pt0.A4.SS2.p1.2.m2.1.1.3.cmml"><mi id="Pt0.A4.SS2.p1.2.m2.1.1.3.2" xref="Pt0.A4.SS2.p1.2.m2.1.1.3.2.cmml">E</mi><mi id="Pt0.A4.SS2.p1.2.m2.1.1.3.3" xref="Pt0.A4.SS2.p1.2.m2.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.2.m2.1b"><apply id="Pt0.A4.SS2.p1.2.m2.1.1.cmml" xref="Pt0.A4.SS2.p1.2.m2.1.1"><in id="Pt0.A4.SS2.p1.2.m2.1.1.1.cmml" xref="Pt0.A4.SS2.p1.2.m2.1.1.1"></in><ci id="Pt0.A4.SS2.p1.2.m2.1.1.2.cmml" xref="Pt0.A4.SS2.p1.2.m2.1.1.2">𝑑</ci><apply id="Pt0.A4.SS2.p1.2.m2.1.1.3.cmml" xref="Pt0.A4.SS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.2.m2.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.2.m2.1.1.3">superscript</csymbol><ci id="Pt0.A4.SS2.p1.2.m2.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.2.m2.1.1.3.2">𝐸</ci><ci id="Pt0.A4.SS2.p1.2.m2.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.2.m2.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.2.m2.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="c_{d,\alpha}:E\to\mathbb{R}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.3.m3.2"><semantics id="Pt0.A4.SS2.p1.3.m3.2a"><mrow id="Pt0.A4.SS2.p1.3.m3.2.3" xref="Pt0.A4.SS2.p1.3.m3.2.3.cmml"><msub id="Pt0.A4.SS2.p1.3.m3.2.3.2" xref="Pt0.A4.SS2.p1.3.m3.2.3.2.cmml"><mi id="Pt0.A4.SS2.p1.3.m3.2.3.2.2" xref="Pt0.A4.SS2.p1.3.m3.2.3.2.2.cmml">c</mi><mrow id="Pt0.A4.SS2.p1.3.m3.2.2.2.4" xref="Pt0.A4.SS2.p1.3.m3.2.2.2.3.cmml"><mi id="Pt0.A4.SS2.p1.3.m3.1.1.1.1" xref="Pt0.A4.SS2.p1.3.m3.1.1.1.1.cmml">d</mi><mo id="Pt0.A4.SS2.p1.3.m3.2.2.2.4.1" xref="Pt0.A4.SS2.p1.3.m3.2.2.2.3.cmml">,</mo><mi id="Pt0.A4.SS2.p1.3.m3.2.2.2.2" xref="Pt0.A4.SS2.p1.3.m3.2.2.2.2.cmml">α</mi></mrow></msub><mo id="Pt0.A4.SS2.p1.3.m3.2.3.1" lspace="0.278em" rspace="0.278em" xref="Pt0.A4.SS2.p1.3.m3.2.3.1.cmml">:</mo><mrow id="Pt0.A4.SS2.p1.3.m3.2.3.3" xref="Pt0.A4.SS2.p1.3.m3.2.3.3.cmml"><mi id="Pt0.A4.SS2.p1.3.m3.2.3.3.2" xref="Pt0.A4.SS2.p1.3.m3.2.3.3.2.cmml">E</mi><mo id="Pt0.A4.SS2.p1.3.m3.2.3.3.1" stretchy="false" xref="Pt0.A4.SS2.p1.3.m3.2.3.3.1.cmml">→</mo><mi id="Pt0.A4.SS2.p1.3.m3.2.3.3.3" xref="Pt0.A4.SS2.p1.3.m3.2.3.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.3.m3.2b"><apply id="Pt0.A4.SS2.p1.3.m3.2.3.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3"><ci id="Pt0.A4.SS2.p1.3.m3.2.3.1.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.1">:</ci><apply id="Pt0.A4.SS2.p1.3.m3.2.3.2.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.3.m3.2.3.2.1.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.2">subscript</csymbol><ci id="Pt0.A4.SS2.p1.3.m3.2.3.2.2.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.2.2">𝑐</ci><list id="Pt0.A4.SS2.p1.3.m3.2.2.2.3.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.2.2.4"><ci id="Pt0.A4.SS2.p1.3.m3.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.3.m3.1.1.1.1">𝑑</ci><ci id="Pt0.A4.SS2.p1.3.m3.2.2.2.2.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.2.2.2">𝛼</ci></list></apply><apply id="Pt0.A4.SS2.p1.3.m3.2.3.3.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.3"><ci id="Pt0.A4.SS2.p1.3.m3.2.3.3.1.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.3.1">→</ci><ci id="Pt0.A4.SS2.p1.3.m3.2.3.3.2.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.3.2">𝐸</ci><ci id="Pt0.A4.SS2.p1.3.m3.2.3.3.3.cmml" xref="Pt0.A4.SS2.p1.3.m3.2.3.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.3.m3.2c">c_{d,\alpha}:E\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.3.m3.2d">italic_c start_POSTSUBSCRIPT italic_d , italic_α end_POSTSUBSCRIPT : italic_E → blackboard_R</annotation></semantics></math> by defined by <math alttext="c_{d,\alpha}(e)=1+\alpha\left|\left\{i\in[k]\mid d_{i}=e\right\}\right|" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.4.m4.5"><semantics id="Pt0.A4.SS2.p1.4.m4.5a"><mrow id="Pt0.A4.SS2.p1.4.m4.5.5" xref="Pt0.A4.SS2.p1.4.m4.5.5.cmml"><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.3" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.cmml"><msub id="Pt0.A4.SS2.p1.4.m4.5.5.3.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.2.cmml"><mi id="Pt0.A4.SS2.p1.4.m4.5.5.3.2.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.2.2.cmml">c</mi><mrow id="Pt0.A4.SS2.p1.4.m4.2.2.2.4" xref="Pt0.A4.SS2.p1.4.m4.2.2.2.3.cmml"><mi id="Pt0.A4.SS2.p1.4.m4.1.1.1.1" xref="Pt0.A4.SS2.p1.4.m4.1.1.1.1.cmml">d</mi><mo id="Pt0.A4.SS2.p1.4.m4.2.2.2.4.1" xref="Pt0.A4.SS2.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="Pt0.A4.SS2.p1.4.m4.2.2.2.2" xref="Pt0.A4.SS2.p1.4.m4.2.2.2.2.cmml">α</mi></mrow></msub><mo id="Pt0.A4.SS2.p1.4.m4.5.5.3.1" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.1.cmml">⁢</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.3.3.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.cmml"><mo id="Pt0.A4.SS2.p1.4.m4.5.5.3.3.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.cmml">(</mo><mi id="Pt0.A4.SS2.p1.4.m4.3.3" xref="Pt0.A4.SS2.p1.4.m4.3.3.cmml">e</mi><mo id="Pt0.A4.SS2.p1.4.m4.5.5.3.3.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS2.p1.4.m4.5.5.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.2.cmml">=</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.1" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.cmml"><mn id="Pt0.A4.SS2.p1.4.m4.5.5.1.3" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.3.cmml">1</mn><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.2.cmml">+</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.1.1" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.cmml"><mi id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.3" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.3.cmml">α</mi><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.2.cmml"><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.2.1.cmml">|</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.3.cmml"><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.3" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.3.1.cmml">{</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.2.cmml">i</mi><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.1" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.1.cmml">∈</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.1.cmml"><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.1.1.cmml">[</mo><mi id="Pt0.A4.SS2.p1.4.m4.4.4" xref="Pt0.A4.SS2.p1.4.m4.4.4.cmml">k</mi><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.1.1.cmml">]</mo></mrow></mrow><mo fence="true" id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.4" lspace="0em" rspace="0em" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.3.1.cmml">∣</mo><mrow id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.cmml"><msub id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.cmml"><mi id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.2" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.2.cmml">d</mi><mi id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.3" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.3.cmml">i</mi></msub><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.1" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.1.cmml">=</mo><mi id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.3" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.3.cmml">e</mi></mrow><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.5" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.3.1.cmml">}</mo></mrow><mo id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.3" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.4.m4.5b"><apply id="Pt0.A4.SS2.p1.4.m4.5.5.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5"><eq id="Pt0.A4.SS2.p1.4.m4.5.5.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.2"></eq><apply id="Pt0.A4.SS2.p1.4.m4.5.5.3.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.3"><times id="Pt0.A4.SS2.p1.4.m4.5.5.3.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.1"></times><apply id="Pt0.A4.SS2.p1.4.m4.5.5.3.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.4.m4.5.5.3.2.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.2">subscript</csymbol><ci id="Pt0.A4.SS2.p1.4.m4.5.5.3.2.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.3.2.2">𝑐</ci><list id="Pt0.A4.SS2.p1.4.m4.2.2.2.3.cmml" xref="Pt0.A4.SS2.p1.4.m4.2.2.2.4"><ci id="Pt0.A4.SS2.p1.4.m4.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.1.1.1.1">𝑑</ci><ci id="Pt0.A4.SS2.p1.4.m4.2.2.2.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.2.2.2.2">𝛼</ci></list></apply><ci id="Pt0.A4.SS2.p1.4.m4.3.3.cmml" xref="Pt0.A4.SS2.p1.4.m4.3.3">𝑒</ci></apply><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1"><plus id="Pt0.A4.SS2.p1.4.m4.5.5.1.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.2"></plus><cn id="Pt0.A4.SS2.p1.4.m4.5.5.1.3.cmml" type="integer" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.3">1</cn><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1"><times id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.2"></times><ci id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.3.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.3">𝛼</ci><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1"><abs id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.2"></abs><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.3">conditional-set</csymbol><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1"><in id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.1"></in><ci id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.2">𝑖</ci><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.1.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.1.1.3.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS2.p1.4.m4.4.4.cmml" xref="Pt0.A4.SS2.p1.4.m4.4.4">𝑘</ci></apply></apply><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2"><eq id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.1"></eq><apply id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.1.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.2.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.2">𝑑</ci><ci id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.3.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.2.3">𝑖</ci></apply><ci id="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.3.cmml" xref="Pt0.A4.SS2.p1.4.m4.5.5.1.1.1.1.1.2.2.3">𝑒</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.4.m4.5c">c_{d,\alpha}(e)=1+\alpha\left|\left\{i\in[k]\mid d_{i}=e\right\}\right|</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.4.m4.5d">italic_c start_POSTSUBSCRIPT italic_d , italic_α end_POSTSUBSCRIPT ( italic_e ) = 1 + italic_α | { italic_i ∈ [ italic_k ] ∣ italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_e } |</annotation></semantics></math> i.e., the cost to unlock edge <math alttext="e" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.5.m5.1"><semantics id="Pt0.A4.SS2.p1.5.m5.1a"><mi id="Pt0.A4.SS2.p1.5.m5.1.1" xref="Pt0.A4.SS2.p1.5.m5.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.5.m5.1b"><ci id="Pt0.A4.SS2.p1.5.m5.1.1.cmml" xref="Pt0.A4.SS2.p1.5.m5.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.5.m5.1c">e</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.5.m5.1d">italic_e</annotation></semantics></math> given <math alttext="\alpha,d" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.6.m6.2"><semantics id="Pt0.A4.SS2.p1.6.m6.2a"><mrow id="Pt0.A4.SS2.p1.6.m6.2.3.2" xref="Pt0.A4.SS2.p1.6.m6.2.3.1.cmml"><mi id="Pt0.A4.SS2.p1.6.m6.1.1" xref="Pt0.A4.SS2.p1.6.m6.1.1.cmml">α</mi><mo id="Pt0.A4.SS2.p1.6.m6.2.3.2.1" xref="Pt0.A4.SS2.p1.6.m6.2.3.1.cmml">,</mo><mi id="Pt0.A4.SS2.p1.6.m6.2.2" xref="Pt0.A4.SS2.p1.6.m6.2.2.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.6.m6.2b"><list id="Pt0.A4.SS2.p1.6.m6.2.3.1.cmml" xref="Pt0.A4.SS2.p1.6.m6.2.3.2"><ci id="Pt0.A4.SS2.p1.6.m6.1.1.cmml" xref="Pt0.A4.SS2.p1.6.m6.1.1">𝛼</ci><ci id="Pt0.A4.SS2.p1.6.m6.2.2.cmml" xref="Pt0.A4.SS2.p1.6.m6.2.2">𝑑</ci></list></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.6.m6.2c">\alpha,d</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.6.m6.2d">italic_α , italic_d</annotation></semantics></math>. When <math alttext="\alpha" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.7.m7.1"><semantics id="Pt0.A4.SS2.p1.7.m7.1a"><mi id="Pt0.A4.SS2.p1.7.m7.1.1" xref="Pt0.A4.SS2.p1.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.7.m7.1b"><ci id="Pt0.A4.SS2.p1.7.m7.1.1.cmml" xref="Pt0.A4.SS2.p1.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.7.m7.1d">italic_α</annotation></semantics></math> is clear from context, we will write <math alttext="c_{d}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.8.m8.1"><semantics id="Pt0.A4.SS2.p1.8.m8.1a"><msub id="Pt0.A4.SS2.p1.8.m8.1.1" xref="Pt0.A4.SS2.p1.8.m8.1.1.cmml"><mi id="Pt0.A4.SS2.p1.8.m8.1.1.2" xref="Pt0.A4.SS2.p1.8.m8.1.1.2.cmml">c</mi><mi id="Pt0.A4.SS2.p1.8.m8.1.1.3" xref="Pt0.A4.SS2.p1.8.m8.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.8.m8.1b"><apply id="Pt0.A4.SS2.p1.8.m8.1.1.cmml" xref="Pt0.A4.SS2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.8.m8.1.1.1.cmml" xref="Pt0.A4.SS2.p1.8.m8.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.8.m8.1.1.2.cmml" xref="Pt0.A4.SS2.p1.8.m8.1.1.2">𝑐</ci><ci id="Pt0.A4.SS2.p1.8.m8.1.1.3.cmml" xref="Pt0.A4.SS2.p1.8.m8.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.8.m8.1c">c_{d}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.8.m8.1d">italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math>. We abuse notation and for <math alttext="p\in P_{st}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.9.m9.1"><semantics id="Pt0.A4.SS2.p1.9.m9.1a"><mrow id="Pt0.A4.SS2.p1.9.m9.1.1" xref="Pt0.A4.SS2.p1.9.m9.1.1.cmml"><mi id="Pt0.A4.SS2.p1.9.m9.1.1.2" xref="Pt0.A4.SS2.p1.9.m9.1.1.2.cmml">p</mi><mo id="Pt0.A4.SS2.p1.9.m9.1.1.1" xref="Pt0.A4.SS2.p1.9.m9.1.1.1.cmml">∈</mo><msub id="Pt0.A4.SS2.p1.9.m9.1.1.3" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.cmml"><mi id="Pt0.A4.SS2.p1.9.m9.1.1.3.2" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.2.cmml">P</mi><mrow id="Pt0.A4.SS2.p1.9.m9.1.1.3.3" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3.cmml"><mi id="Pt0.A4.SS2.p1.9.m9.1.1.3.3.2" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3.2.cmml">s</mi><mo id="Pt0.A4.SS2.p1.9.m9.1.1.3.3.1" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3.1.cmml">⁢</mo><mi id="Pt0.A4.SS2.p1.9.m9.1.1.3.3.3" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3.3.cmml">t</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.9.m9.1b"><apply id="Pt0.A4.SS2.p1.9.m9.1.1.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1"><in id="Pt0.A4.SS2.p1.9.m9.1.1.1.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.1"></in><ci id="Pt0.A4.SS2.p1.9.m9.1.1.2.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.2">𝑝</ci><apply id="Pt0.A4.SS2.p1.9.m9.1.1.3.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.9.m9.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.9.m9.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.2">𝑃</ci><apply id="Pt0.A4.SS2.p1.9.m9.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3"><times id="Pt0.A4.SS2.p1.9.m9.1.1.3.3.1.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3.1"></times><ci id="Pt0.A4.SS2.p1.9.m9.1.1.3.3.2.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3.2">𝑠</ci><ci id="Pt0.A4.SS2.p1.9.m9.1.1.3.3.3.cmml" xref="Pt0.A4.SS2.p1.9.m9.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.9.m9.1c">p\in P_{st}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.9.m9.1d">italic_p ∈ italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math> write <math alttext="c_{d}(p)=\sum_{e\in p}c_{d}(p)" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.10.m10.2"><semantics id="Pt0.A4.SS2.p1.10.m10.2a"><mrow id="Pt0.A4.SS2.p1.10.m10.2.3" xref="Pt0.A4.SS2.p1.10.m10.2.3.cmml"><mrow id="Pt0.A4.SS2.p1.10.m10.2.3.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.cmml"><msub id="Pt0.A4.SS2.p1.10.m10.2.3.2.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.2.cmml"><mi id="Pt0.A4.SS2.p1.10.m10.2.3.2.2.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.2.2.cmml">c</mi><mi id="Pt0.A4.SS2.p1.10.m10.2.3.2.2.3" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.2.3.cmml">d</mi></msub><mo id="Pt0.A4.SS2.p1.10.m10.2.3.2.1" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="Pt0.A4.SS2.p1.10.m10.2.3.2.3.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.cmml"><mo id="Pt0.A4.SS2.p1.10.m10.2.3.2.3.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.cmml">(</mo><mi id="Pt0.A4.SS2.p1.10.m10.1.1" xref="Pt0.A4.SS2.p1.10.m10.1.1.cmml">p</mi><mo id="Pt0.A4.SS2.p1.10.m10.2.3.2.3.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.10.m10.2.3.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS2.p1.10.m10.2.3.1" rspace="0.111em" xref="Pt0.A4.SS2.p1.10.m10.2.3.1.cmml">=</mo><mrow id="Pt0.A4.SS2.p1.10.m10.2.3.3" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.cmml"><msub id="Pt0.A4.SS2.p1.10.m10.2.3.3.1" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.1.cmml"><mo id="Pt0.A4.SS2.p1.10.m10.2.3.3.1.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.1.2.cmml">∑</mo><mrow id="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3.cmml"><mi id="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3.2.cmml">e</mi><mo id="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3.1" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3.1.cmml">∈</mo><mi id="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3.3" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.1.3.3.cmml">p</mi></mrow></msub><mrow id="Pt0.A4.SS2.p1.10.m10.2.3.3.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.cmml"><msub id="Pt0.A4.SS2.p1.10.m10.2.3.3.2.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.2.cmml"><mi id="Pt0.A4.SS2.p1.10.m10.2.3.3.2.2.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.2.2.cmml">c</mi><mi id="Pt0.A4.SS2.p1.10.m10.2.3.3.2.2.3" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.2.3.cmml">d</mi></msub><mo id="Pt0.A4.SS2.p1.10.m10.2.3.3.2.1" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.1.cmml">⁢</mo><mrow id="Pt0.A4.SS2.p1.10.m10.2.3.3.2.3.2" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.cmml"><mo id="Pt0.A4.SS2.p1.10.m10.2.3.3.2.3.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.cmml">(</mo><mi id="Pt0.A4.SS2.p1.10.m10.2.2" xref="Pt0.A4.SS2.p1.10.m10.2.2.cmml">p</mi><mo id="Pt0.A4.SS2.p1.10.m10.2.3.3.2.3.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.10.m10.2.3.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.10.m10.2b"><apply id="Pt0.A4.SS2.p1.10.m10.2.3.cmml" xref="Pt0.A4.SS2.p1.10.m10.2.3"><eq id="Pt0.A4.SS2.p1.10.m10.2.3.1.cmml" xref="Pt0.A4.SS2.p1.10.m10.2.3.1"></eq><apply 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xref="Pt0.A4.SS2.p1.11.m11.4.4.4.1"></times><apply id="Pt0.A4.SS2.p1.11.m11.4.4.4.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.4.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.11.m11.4.4.4.2.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.4.2">subscript</csymbol><ci id="Pt0.A4.SS2.p1.11.m11.4.4.4.2.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.4.2.2">𝑐</ci><ci id="Pt0.A4.SS2.p1.11.m11.4.4.4.2.3.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.4.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.11.m11.1.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.1.1">𝑎</ci></apply><vector id="Pt0.A4.SS2.p1.11.m11.4.4.2.3.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2"><apply id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1"><times id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.2"></times><apply id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3.2">𝑐</ci><ci id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.3.3">𝑑</ci></apply><apply id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1.1.2">𝑎</ci><cn id="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1.1.3.cmml" type="integer" xref="Pt0.A4.SS2.p1.11.m11.3.3.1.1.1.1.1.1.3">1</cn></apply></apply><ci id="Pt0.A4.SS2.p1.11.m11.2.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.2.2">…</ci><apply id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2"><times id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.2"></times><apply id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3.2">𝑐</ci><ci id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3.3.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.3.3">𝑑</ci></apply><apply id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1.1.2">𝑎</ci><ci id="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.11.m11.4.4.2.2.2.1.1.1.3">𝑛</ci></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.11.m11.4c">c_{d}(a)=(c_{d}(a_{1}),...,c_{d}(a_{n}))</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.11.m11.4d">italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) = ( italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) )</annotation></semantics></math> for <math alttext="a\in(P_{st})^{n}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.12.m12.1"><semantics id="Pt0.A4.SS2.p1.12.m12.1a"><mrow id="Pt0.A4.SS2.p1.12.m12.1.1" xref="Pt0.A4.SS2.p1.12.m12.1.1.cmml"><mi id="Pt0.A4.SS2.p1.12.m12.1.1.3" xref="Pt0.A4.SS2.p1.12.m12.1.1.3.cmml">a</mi><mo id="Pt0.A4.SS2.p1.12.m12.1.1.2" xref="Pt0.A4.SS2.p1.12.m12.1.1.2.cmml">∈</mo><msup id="Pt0.A4.SS2.p1.12.m12.1.1.1" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.cmml"><mrow id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.2" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.2" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.1" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.3" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.SS2.p1.12.m12.1.1.1.3" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.12.m12.1b"><apply id="Pt0.A4.SS2.p1.12.m12.1.1.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1"><in id="Pt0.A4.SS2.p1.12.m12.1.1.2.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.2"></in><ci id="Pt0.A4.SS2.p1.12.m12.1.1.3.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.3">𝑎</ci><apply id="Pt0.A4.SS2.p1.12.m12.1.1.1.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.12.m12.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1">superscript</csymbol><apply id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.2">𝑃</ci><apply id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3"><times id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.1"></times><ci id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><ci id="Pt0.A4.SS2.p1.12.m12.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.12.m12.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.12.m12.1c">a\in(P_{st})^{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.12.m12.1d">italic_a ∈ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="c_{0}=c_{d,0}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.13.m13.2"><semantics id="Pt0.A4.SS2.p1.13.m13.2a"><mrow id="Pt0.A4.SS2.p1.13.m13.2.3" xref="Pt0.A4.SS2.p1.13.m13.2.3.cmml"><msub id="Pt0.A4.SS2.p1.13.m13.2.3.2" xref="Pt0.A4.SS2.p1.13.m13.2.3.2.cmml"><mi id="Pt0.A4.SS2.p1.13.m13.2.3.2.2" xref="Pt0.A4.SS2.p1.13.m13.2.3.2.2.cmml">c</mi><mn id="Pt0.A4.SS2.p1.13.m13.2.3.2.3" xref="Pt0.A4.SS2.p1.13.m13.2.3.2.3.cmml">0</mn></msub><mo id="Pt0.A4.SS2.p1.13.m13.2.3.1" xref="Pt0.A4.SS2.p1.13.m13.2.3.1.cmml">=</mo><msub id="Pt0.A4.SS2.p1.13.m13.2.3.3" xref="Pt0.A4.SS2.p1.13.m13.2.3.3.cmml"><mi id="Pt0.A4.SS2.p1.13.m13.2.3.3.2" xref="Pt0.A4.SS2.p1.13.m13.2.3.3.2.cmml">c</mi><mrow id="Pt0.A4.SS2.p1.13.m13.2.2.2.4" xref="Pt0.A4.SS2.p1.13.m13.2.2.2.3.cmml"><mi id="Pt0.A4.SS2.p1.13.m13.1.1.1.1" xref="Pt0.A4.SS2.p1.13.m13.1.1.1.1.cmml">d</mi><mo id="Pt0.A4.SS2.p1.13.m13.2.2.2.4.1" xref="Pt0.A4.SS2.p1.13.m13.2.2.2.3.cmml">,</mo><mn id="Pt0.A4.SS2.p1.13.m13.2.2.2.2" xref="Pt0.A4.SS2.p1.13.m13.2.2.2.2.cmml">0</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.13.m13.2b"><apply id="Pt0.A4.SS2.p1.13.m13.2.3.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3"><eq id="Pt0.A4.SS2.p1.13.m13.2.3.1.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3.1"></eq><apply id="Pt0.A4.SS2.p1.13.m13.2.3.2.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.13.m13.2.3.2.1.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3.2">subscript</csymbol><ci id="Pt0.A4.SS2.p1.13.m13.2.3.2.2.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3.2.2">𝑐</ci><cn id="Pt0.A4.SS2.p1.13.m13.2.3.2.3.cmml" type="integer" xref="Pt0.A4.SS2.p1.13.m13.2.3.2.3">0</cn></apply><apply id="Pt0.A4.SS2.p1.13.m13.2.3.3.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.13.m13.2.3.3.1.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.13.m13.2.3.3.2.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.3.3.2">𝑐</ci><list id="Pt0.A4.SS2.p1.13.m13.2.2.2.3.cmml" xref="Pt0.A4.SS2.p1.13.m13.2.2.2.4"><ci id="Pt0.A4.SS2.p1.13.m13.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.13.m13.1.1.1.1">𝑑</ci><cn id="Pt0.A4.SS2.p1.13.m13.2.2.2.2.cmml" type="integer" xref="Pt0.A4.SS2.p1.13.m13.2.2.2.2">0</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.13.m13.2c">c_{0}=c_{d,0}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.13.m13.2d">italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_d , 0 end_POSTSUBSCRIPT</annotation></semantics></math> denote the default cost function, where there are no adversaries. Let <math alttext="P_{st}^{d}=\{p^{d}_{i}\}_{i\in[m]}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.14.m14.2"><semantics id="Pt0.A4.SS2.p1.14.m14.2a"><mrow id="Pt0.A4.SS2.p1.14.m14.2.2" xref="Pt0.A4.SS2.p1.14.m14.2.2.cmml"><msubsup id="Pt0.A4.SS2.p1.14.m14.2.2.3" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.cmml"><mi id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.2" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.2.cmml">P</mi><mrow id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.cmml"><mi id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.2" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.2.cmml">s</mi><mo id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.1" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.1.cmml">⁢</mo><mi id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.3" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.3.cmml">t</mi></mrow><mi id="Pt0.A4.SS2.p1.14.m14.2.2.3.3" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.3.cmml">d</mi></msubsup><mo id="Pt0.A4.SS2.p1.14.m14.2.2.2" xref="Pt0.A4.SS2.p1.14.m14.2.2.2.cmml">=</mo><msub id="Pt0.A4.SS2.p1.14.m14.2.2.1" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.cmml"><mrow id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.2.cmml"><mo id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.2" stretchy="false" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.2.cmml">{</mo><msubsup id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.2" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.3" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.3" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.3" stretchy="false" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.2.cmml">}</mo></mrow><mrow id="Pt0.A4.SS2.p1.14.m14.1.1.1" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.14.m14.1.1.1.3" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.3.cmml">i</mi><mo id="Pt0.A4.SS2.p1.14.m14.1.1.1.2" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.SS2.p1.14.m14.1.1.1.4.2" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.4.1.cmml"><mo id="Pt0.A4.SS2.p1.14.m14.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.SS2.p1.14.m14.1.1.1.1" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.SS2.p1.14.m14.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.14.m14.2b"><apply id="Pt0.A4.SS2.p1.14.m14.2.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2"><eq id="Pt0.A4.SS2.p1.14.m14.2.2.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.2"></eq><apply id="Pt0.A4.SS2.p1.14.m14.2.2.3.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.14.m14.2.2.3.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3">superscript</csymbol><apply id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.2">𝑃</ci><apply id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3"><times id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.1"></times><ci id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.2">𝑠</ci><ci id="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.3.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.2.3.3">𝑡</ci></apply></apply><ci id="Pt0.A4.SS2.p1.14.m14.2.2.3.3.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.3.3">𝑑</ci></apply><apply id="Pt0.A4.SS2.p1.14.m14.2.2.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.14.m14.2.2.1.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1">subscript</csymbol><set id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1"><apply id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1">subscript</csymbol><apply id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.2">𝑝</ci><ci id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.3.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.14.m14.2.2.1.1.1.1.3">𝑖</ci></apply></set><apply id="Pt0.A4.SS2.p1.14.m14.1.1.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.1.1.1"><in id="Pt0.A4.SS2.p1.14.m14.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.2"></in><ci id="Pt0.A4.SS2.p1.14.m14.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.3">𝑖</ci><apply id="Pt0.A4.SS2.p1.14.m14.1.1.1.4.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.4.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.14.m14.1.1.1.4.1.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS2.p1.14.m14.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.14.m14.1.1.1.1">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.14.m14.2c">P_{st}^{d}=\{p^{d}_{i}\}_{i\in[m]}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.14.m14.2d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT = { italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT</annotation></semantics></math> denote the paths <math alttext="P_{st}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.15.m15.1"><semantics id="Pt0.A4.SS2.p1.15.m15.1a"><msub id="Pt0.A4.SS2.p1.15.m15.1.1" xref="Pt0.A4.SS2.p1.15.m15.1.1.cmml"><mi id="Pt0.A4.SS2.p1.15.m15.1.1.2" xref="Pt0.A4.SS2.p1.15.m15.1.1.2.cmml">P</mi><mrow id="Pt0.A4.SS2.p1.15.m15.1.1.3" xref="Pt0.A4.SS2.p1.15.m15.1.1.3.cmml"><mi id="Pt0.A4.SS2.p1.15.m15.1.1.3.2" xref="Pt0.A4.SS2.p1.15.m15.1.1.3.2.cmml">s</mi><mo id="Pt0.A4.SS2.p1.15.m15.1.1.3.1" xref="Pt0.A4.SS2.p1.15.m15.1.1.3.1.cmml">⁢</mo><mi id="Pt0.A4.SS2.p1.15.m15.1.1.3.3" xref="Pt0.A4.SS2.p1.15.m15.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.15.m15.1b"><apply id="Pt0.A4.SS2.p1.15.m15.1.1.cmml" xref="Pt0.A4.SS2.p1.15.m15.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.15.m15.1.1.1.cmml" xref="Pt0.A4.SS2.p1.15.m15.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.15.m15.1.1.2.cmml" xref="Pt0.A4.SS2.p1.15.m15.1.1.2">𝑃</ci><apply id="Pt0.A4.SS2.p1.15.m15.1.1.3.cmml" xref="Pt0.A4.SS2.p1.15.m15.1.1.3"><times id="Pt0.A4.SS2.p1.15.m15.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.15.m15.1.1.3.1"></times><ci id="Pt0.A4.SS2.p1.15.m15.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.15.m15.1.1.3.2">𝑠</ci><ci id="Pt0.A4.SS2.p1.15.m15.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.15.m15.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.15.m15.1c">P_{st}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.15.m15.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT</annotation></semantics></math> where the paths indices are increasing with respect to <math alttext="c_{d}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.16.m16.1"><semantics id="Pt0.A4.SS2.p1.16.m16.1a"><msub id="Pt0.A4.SS2.p1.16.m16.1.1" xref="Pt0.A4.SS2.p1.16.m16.1.1.cmml"><mi id="Pt0.A4.SS2.p1.16.m16.1.1.2" xref="Pt0.A4.SS2.p1.16.m16.1.1.2.cmml">c</mi><mi id="Pt0.A4.SS2.p1.16.m16.1.1.3" xref="Pt0.A4.SS2.p1.16.m16.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.16.m16.1b"><apply id="Pt0.A4.SS2.p1.16.m16.1.1.cmml" xref="Pt0.A4.SS2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.16.m16.1.1.1.cmml" xref="Pt0.A4.SS2.p1.16.m16.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.16.m16.1.1.2.cmml" xref="Pt0.A4.SS2.p1.16.m16.1.1.2">𝑐</ci><ci id="Pt0.A4.SS2.p1.16.m16.1.1.3.cmml" xref="Pt0.A4.SS2.p1.16.m16.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.16.m16.1c">c_{d}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.16.m16.1d">italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="I_{d}:[m]\to[m]" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.17.m17.2"><semantics id="Pt0.A4.SS2.p1.17.m17.2a"><mrow id="Pt0.A4.SS2.p1.17.m17.2.3" xref="Pt0.A4.SS2.p1.17.m17.2.3.cmml"><msub id="Pt0.A4.SS2.p1.17.m17.2.3.2" xref="Pt0.A4.SS2.p1.17.m17.2.3.2.cmml"><mi id="Pt0.A4.SS2.p1.17.m17.2.3.2.2" xref="Pt0.A4.SS2.p1.17.m17.2.3.2.2.cmml">I</mi><mi id="Pt0.A4.SS2.p1.17.m17.2.3.2.3" xref="Pt0.A4.SS2.p1.17.m17.2.3.2.3.cmml">d</mi></msub><mo id="Pt0.A4.SS2.p1.17.m17.2.3.1" lspace="0.278em" rspace="0.278em" xref="Pt0.A4.SS2.p1.17.m17.2.3.1.cmml">:</mo><mrow id="Pt0.A4.SS2.p1.17.m17.2.3.3" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.cmml"><mrow id="Pt0.A4.SS2.p1.17.m17.2.3.3.2.2" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.2.1.cmml"><mo id="Pt0.A4.SS2.p1.17.m17.2.3.3.2.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.2.1.1.cmml">[</mo><mi id="Pt0.A4.SS2.p1.17.m17.1.1" xref="Pt0.A4.SS2.p1.17.m17.1.1.cmml">m</mi><mo id="Pt0.A4.SS2.p1.17.m17.2.3.3.2.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Pt0.A4.SS2.p1.17.m17.2.3.3.1" stretchy="false" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.1.cmml">→</mo><mrow id="Pt0.A4.SS2.p1.17.m17.2.3.3.3.2" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.3.1.cmml"><mo id="Pt0.A4.SS2.p1.17.m17.2.3.3.3.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.3.1.1.cmml">[</mo><mi id="Pt0.A4.SS2.p1.17.m17.2.2" xref="Pt0.A4.SS2.p1.17.m17.2.2.cmml">m</mi><mo id="Pt0.A4.SS2.p1.17.m17.2.3.3.3.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.17.m17.2b"><apply id="Pt0.A4.SS2.p1.17.m17.2.3.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3"><ci id="Pt0.A4.SS2.p1.17.m17.2.3.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.1">:</ci><apply id="Pt0.A4.SS2.p1.17.m17.2.3.2.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.17.m17.2.3.2.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.2">subscript</csymbol><ci id="Pt0.A4.SS2.p1.17.m17.2.3.2.2.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.2.2">𝐼</ci><ci id="Pt0.A4.SS2.p1.17.m17.2.3.2.3.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.2.3">𝑑</ci></apply><apply id="Pt0.A4.SS2.p1.17.m17.2.3.3.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.3"><ci id="Pt0.A4.SS2.p1.17.m17.2.3.3.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.1">→</ci><apply id="Pt0.A4.SS2.p1.17.m17.2.3.3.2.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.2.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.17.m17.2.3.3.2.1.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS2.p1.17.m17.1.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.1.1">𝑚</ci></apply><apply id="Pt0.A4.SS2.p1.17.m17.2.3.3.3.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.3.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.17.m17.2.3.3.3.1.1.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS2.p1.17.m17.2.2.cmml" xref="Pt0.A4.SS2.p1.17.m17.2.2">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.17.m17.2c">I_{d}:[m]\to[m]</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.17.m17.2d">italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT : [ italic_m ] → [ italic_m ]</annotation></semantics></math> be the corresponding mapping of indices from <math alttext="\{p^{d}_{i}\}_{i\in[m]}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.18.m18.2"><semantics id="Pt0.A4.SS2.p1.18.m18.2a"><msub id="Pt0.A4.SS2.p1.18.m18.2.2" xref="Pt0.A4.SS2.p1.18.m18.2.2.cmml"><mrow id="Pt0.A4.SS2.p1.18.m18.2.2.1.1" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.2.cmml"><mo id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.2" stretchy="false" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.2.cmml">{</mo><msubsup id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.2" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.3" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.3" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.3" stretchy="false" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.2.cmml">}</mo></mrow><mrow id="Pt0.A4.SS2.p1.18.m18.1.1.1" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.18.m18.1.1.1.3" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.3.cmml">i</mi><mo id="Pt0.A4.SS2.p1.18.m18.1.1.1.2" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.SS2.p1.18.m18.1.1.1.4.2" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.4.1.cmml"><mo id="Pt0.A4.SS2.p1.18.m18.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.SS2.p1.18.m18.1.1.1.1" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.SS2.p1.18.m18.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.18.m18.2b"><apply id="Pt0.A4.SS2.p1.18.m18.2.2.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.18.m18.2.2.2.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2">subscript</csymbol><set id="Pt0.A4.SS2.p1.18.m18.2.2.1.2.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1"><apply id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1">subscript</csymbol><apply id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.2">𝑝</ci><ci id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.3.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.18.m18.2.2.1.1.1.3">𝑖</ci></apply></set><apply id="Pt0.A4.SS2.p1.18.m18.1.1.1.cmml" xref="Pt0.A4.SS2.p1.18.m18.1.1.1"><in id="Pt0.A4.SS2.p1.18.m18.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.2"></in><ci id="Pt0.A4.SS2.p1.18.m18.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.3">𝑖</ci><apply id="Pt0.A4.SS2.p1.18.m18.1.1.1.4.1.cmml" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.4.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.18.m18.1.1.1.4.1.1.cmml" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS2.p1.18.m18.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.18.m18.1.1.1.1">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.18.m18.2c">\{p^{d}_{i}\}_{i\in[m]}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.18.m18.2d">{ italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\{p_{i}\}_{i\in[m]}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.19.m19.2"><semantics id="Pt0.A4.SS2.p1.19.m19.2a"><msub id="Pt0.A4.SS2.p1.19.m19.2.2" xref="Pt0.A4.SS2.p1.19.m19.2.2.cmml"><mrow id="Pt0.A4.SS2.p1.19.m19.2.2.1.1" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.2.cmml"><mo id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.2" stretchy="false" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.2.cmml">{</mo><msub id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.2" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.3" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.3" stretchy="false" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.2.cmml">}</mo></mrow><mrow id="Pt0.A4.SS2.p1.19.m19.1.1.1" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.19.m19.1.1.1.3" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.3.cmml">i</mi><mo id="Pt0.A4.SS2.p1.19.m19.1.1.1.2" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.SS2.p1.19.m19.1.1.1.4.2" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.4.1.cmml"><mo id="Pt0.A4.SS2.p1.19.m19.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.SS2.p1.19.m19.1.1.1.1" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.SS2.p1.19.m19.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.19.m19.2b"><apply id="Pt0.A4.SS2.p1.19.m19.2.2.cmml" xref="Pt0.A4.SS2.p1.19.m19.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.19.m19.2.2.2.cmml" xref="Pt0.A4.SS2.p1.19.m19.2.2">subscript</csymbol><set id="Pt0.A4.SS2.p1.19.m19.2.2.1.2.cmml" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1"><apply id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.cmml" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.2">𝑝</ci><ci id="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.19.m19.2.2.1.1.1.3">𝑖</ci></apply></set><apply id="Pt0.A4.SS2.p1.19.m19.1.1.1.cmml" xref="Pt0.A4.SS2.p1.19.m19.1.1.1"><in id="Pt0.A4.SS2.p1.19.m19.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.2"></in><ci id="Pt0.A4.SS2.p1.19.m19.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.3">𝑖</ci><apply id="Pt0.A4.SS2.p1.19.m19.1.1.1.4.1.cmml" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.4.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.19.m19.1.1.1.4.1.1.cmml" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS2.p1.19.m19.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.19.m19.1.1.1.1">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.19.m19.2c">\{p_{i}\}_{i\in[m]}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.19.m19.2d">{ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="k^{d}_{i}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.20.m20.1"><semantics id="Pt0.A4.SS2.p1.20.m20.1a"><msubsup id="Pt0.A4.SS2.p1.20.m20.1.1" xref="Pt0.A4.SS2.p1.20.m20.1.1.cmml"><mi id="Pt0.A4.SS2.p1.20.m20.1.1.2.2" xref="Pt0.A4.SS2.p1.20.m20.1.1.2.2.cmml">k</mi><mi id="Pt0.A4.SS2.p1.20.m20.1.1.3" xref="Pt0.A4.SS2.p1.20.m20.1.1.3.cmml">i</mi><mi id="Pt0.A4.SS2.p1.20.m20.1.1.2.3" xref="Pt0.A4.SS2.p1.20.m20.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.20.m20.1b"><apply id="Pt0.A4.SS2.p1.20.m20.1.1.cmml" xref="Pt0.A4.SS2.p1.20.m20.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.20.m20.1.1.1.cmml" xref="Pt0.A4.SS2.p1.20.m20.1.1">subscript</csymbol><apply id="Pt0.A4.SS2.p1.20.m20.1.1.2.cmml" xref="Pt0.A4.SS2.p1.20.m20.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.20.m20.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.20.m20.1.1">superscript</csymbol><ci id="Pt0.A4.SS2.p1.20.m20.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.20.m20.1.1.2.2">𝑘</ci><ci id="Pt0.A4.SS2.p1.20.m20.1.1.2.3.cmml" xref="Pt0.A4.SS2.p1.20.m20.1.1.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.20.m20.1.1.3.cmml" xref="Pt0.A4.SS2.p1.20.m20.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.20.m20.1c">k^{d}_{i}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.20.m20.1d">italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> be the number of interceptors who picked edges on path <math alttext="p^{d}_{i}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.21.m21.1"><semantics id="Pt0.A4.SS2.p1.21.m21.1a"><msubsup id="Pt0.A4.SS2.p1.21.m21.1.1" xref="Pt0.A4.SS2.p1.21.m21.1.1.cmml"><mi id="Pt0.A4.SS2.p1.21.m21.1.1.2.2" xref="Pt0.A4.SS2.p1.21.m21.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.SS2.p1.21.m21.1.1.3" xref="Pt0.A4.SS2.p1.21.m21.1.1.3.cmml">i</mi><mi id="Pt0.A4.SS2.p1.21.m21.1.1.2.3" xref="Pt0.A4.SS2.p1.21.m21.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.21.m21.1b"><apply id="Pt0.A4.SS2.p1.21.m21.1.1.cmml" xref="Pt0.A4.SS2.p1.21.m21.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.21.m21.1.1.1.cmml" xref="Pt0.A4.SS2.p1.21.m21.1.1">subscript</csymbol><apply id="Pt0.A4.SS2.p1.21.m21.1.1.2.cmml" xref="Pt0.A4.SS2.p1.21.m21.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.21.m21.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.21.m21.1.1">superscript</csymbol><ci id="Pt0.A4.SS2.p1.21.m21.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.21.m21.1.1.2.2">𝑝</ci><ci id="Pt0.A4.SS2.p1.21.m21.1.1.2.3.cmml" xref="Pt0.A4.SS2.p1.21.m21.1.1.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.21.m21.1.1.3.cmml" xref="Pt0.A4.SS2.p1.21.m21.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.21.m21.1c">p^{d}_{i}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.21.m21.1d">italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and define <math alttext="k^{d}_{0}=0" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.22.m22.1"><semantics id="Pt0.A4.SS2.p1.22.m22.1a"><mrow id="Pt0.A4.SS2.p1.22.m22.1.1" xref="Pt0.A4.SS2.p1.22.m22.1.1.cmml"><msubsup id="Pt0.A4.SS2.p1.22.m22.1.1.2" xref="Pt0.A4.SS2.p1.22.m22.1.1.2.cmml"><mi id="Pt0.A4.SS2.p1.22.m22.1.1.2.2.2" xref="Pt0.A4.SS2.p1.22.m22.1.1.2.2.2.cmml">k</mi><mn id="Pt0.A4.SS2.p1.22.m22.1.1.2.3" xref="Pt0.A4.SS2.p1.22.m22.1.1.2.3.cmml">0</mn><mi id="Pt0.A4.SS2.p1.22.m22.1.1.2.2.3" xref="Pt0.A4.SS2.p1.22.m22.1.1.2.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.SS2.p1.22.m22.1.1.1" xref="Pt0.A4.SS2.p1.22.m22.1.1.1.cmml">=</mo><mn id="Pt0.A4.SS2.p1.22.m22.1.1.3" xref="Pt0.A4.SS2.p1.22.m22.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.22.m22.1b"><apply id="Pt0.A4.SS2.p1.22.m22.1.1.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1"><eq id="Pt0.A4.SS2.p1.22.m22.1.1.1.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1.1"></eq><apply id="Pt0.A4.SS2.p1.22.m22.1.1.2.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.22.m22.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1.2">subscript</csymbol><apply id="Pt0.A4.SS2.p1.22.m22.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.22.m22.1.1.2.2.1.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1.2">superscript</csymbol><ci id="Pt0.A4.SS2.p1.22.m22.1.1.2.2.2.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1.2.2.2">𝑘</ci><ci id="Pt0.A4.SS2.p1.22.m22.1.1.2.2.3.cmml" xref="Pt0.A4.SS2.p1.22.m22.1.1.2.2.3">𝑑</ci></apply><cn id="Pt0.A4.SS2.p1.22.m22.1.1.2.3.cmml" type="integer" xref="Pt0.A4.SS2.p1.22.m22.1.1.2.3">0</cn></apply><cn id="Pt0.A4.SS2.p1.22.m22.1.1.3.cmml" type="integer" xref="Pt0.A4.SS2.p1.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.22.m22.1c">k^{d}_{0}=0</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.22.m22.1d">italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math>. By definition <math alttext="c_{d}(p^{d}_{i})=|p_{I_{d}(i)}|+k^{d}_{i}\alpha" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.23.m23.3"><semantics id="Pt0.A4.SS2.p1.23.m23.3a"><mrow id="Pt0.A4.SS2.p1.23.m23.3.3" xref="Pt0.A4.SS2.p1.23.m23.3.3.cmml"><mrow id="Pt0.A4.SS2.p1.23.m23.2.2.1" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.cmml"><msub id="Pt0.A4.SS2.p1.23.m23.2.2.1.3" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.3.cmml"><mi id="Pt0.A4.SS2.p1.23.m23.2.2.1.3.2" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.3.2.cmml">c</mi><mi id="Pt0.A4.SS2.p1.23.m23.2.2.1.3.3" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.3.3.cmml">d</mi></msub><mo id="Pt0.A4.SS2.p1.23.m23.2.2.1.2" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.cmml"><mo id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.2" stretchy="false" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.2" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.3" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.3" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.3" stretchy="false" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS2.p1.23.m23.3.3.3" xref="Pt0.A4.SS2.p1.23.m23.3.3.3.cmml">=</mo><mrow id="Pt0.A4.SS2.p1.23.m23.3.3.2" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.cmml"><mrow id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.2.cmml"><mo id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.2" stretchy="false" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.2.1.cmml">|</mo><msub id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1.2" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.SS2.p1.23.m23.1.1.1" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.cmml"><msub id="Pt0.A4.SS2.p1.23.m23.1.1.1.3" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.3.cmml"><mi id="Pt0.A4.SS2.p1.23.m23.1.1.1.3.2" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.3.2.cmml">I</mi><mi id="Pt0.A4.SS2.p1.23.m23.1.1.1.3.3" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.3.3.cmml">d</mi></msub><mo id="Pt0.A4.SS2.p1.23.m23.1.1.1.2" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS2.p1.23.m23.1.1.1.4.2" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.cmml"><mo id="Pt0.A4.SS2.p1.23.m23.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.cmml">(</mo><mi id="Pt0.A4.SS2.p1.23.m23.1.1.1.1" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.1.cmml">i</mi><mo id="Pt0.A4.SS2.p1.23.m23.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.3" stretchy="false" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.SS2.p1.23.m23.3.3.2.2" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.2.cmml">+</mo><mrow id="Pt0.A4.SS2.p1.23.m23.3.3.2.3" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.cmml"><msubsup id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.cmml"><mi id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.2" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.2.cmml">k</mi><mi id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.3" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.3.cmml">i</mi><mi id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.3" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.1" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.1.cmml">⁢</mo><mi id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.3" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.23.m23.3b"><apply id="Pt0.A4.SS2.p1.23.m23.3.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3"><eq id="Pt0.A4.SS2.p1.23.m23.3.3.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.3"></eq><apply id="Pt0.A4.SS2.p1.23.m23.2.2.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1"><times id="Pt0.A4.SS2.p1.23.m23.2.2.1.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.2"></times><apply id="Pt0.A4.SS2.p1.23.m23.2.2.1.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.23.m23.2.2.1.3.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.23.m23.2.2.1.3.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.3.2">𝑐</ci><ci id="Pt0.A4.SS2.p1.23.m23.2.2.1.3.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.3.3">𝑑</ci></apply><apply id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1">subscript</csymbol><apply id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.2">𝑝</ci><ci id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.2.2.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Pt0.A4.SS2.p1.23.m23.3.3.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2"><plus id="Pt0.A4.SS2.p1.23.m23.3.3.2.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.2"></plus><apply id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1"><abs id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.2.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.2"></abs><apply id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS2.p1.23.m23.1.1.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.1.1.1"><times id="Pt0.A4.SS2.p1.23.m23.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.2"></times><apply id="Pt0.A4.SS2.p1.23.m23.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.23.m23.1.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.23.m23.1.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.3.2">𝐼</ci><ci id="Pt0.A4.SS2.p1.23.m23.1.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.3.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.23.m23.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.1.1.1.1">𝑖</ci></apply></apply></apply><apply id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3"><times id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.1"></times><apply id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2">subscript</csymbol><apply id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.1.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2">superscript</csymbol><ci id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.2.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.2">𝑘</ci><ci id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.2.3">𝑖</ci></apply><ci id="Pt0.A4.SS2.p1.23.m23.3.3.2.3.3.cmml" xref="Pt0.A4.SS2.p1.23.m23.3.3.2.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.23.m23.3c">c_{d}(p^{d}_{i})=|p_{I_{d}(i)}|+k^{d}_{i}\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.23.m23.3d">italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT | + italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_α</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>Here we won’t consider powers of <math alttext="k" class="ltx_Math" display="inline" id="footnote3.m1.1"><semantics id="footnote3.m1.1b"><mi id="footnote3.m1.1.1" xref="footnote3.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="footnote3.m1.1c"><ci id="footnote3.m1.1.1.cmml" xref="footnote3.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="footnote3.m1.1e">italic_k</annotation></semantics></math> as an integer.</span></span></span> If <math alttext="k=1" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.24.m24.1"><semantics id="Pt0.A4.SS2.p1.24.m24.1a"><mrow id="Pt0.A4.SS2.p1.24.m24.1.1" xref="Pt0.A4.SS2.p1.24.m24.1.1.cmml"><mi id="Pt0.A4.SS2.p1.24.m24.1.1.2" xref="Pt0.A4.SS2.p1.24.m24.1.1.2.cmml">k</mi><mo id="Pt0.A4.SS2.p1.24.m24.1.1.1" xref="Pt0.A4.SS2.p1.24.m24.1.1.1.cmml">=</mo><mn id="Pt0.A4.SS2.p1.24.m24.1.1.3" xref="Pt0.A4.SS2.p1.24.m24.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.24.m24.1b"><apply id="Pt0.A4.SS2.p1.24.m24.1.1.cmml" xref="Pt0.A4.SS2.p1.24.m24.1.1"><eq id="Pt0.A4.SS2.p1.24.m24.1.1.1.cmml" xref="Pt0.A4.SS2.p1.24.m24.1.1.1"></eq><ci id="Pt0.A4.SS2.p1.24.m24.1.1.2.cmml" xref="Pt0.A4.SS2.p1.24.m24.1.1.2">𝑘</ci><cn id="Pt0.A4.SS2.p1.24.m24.1.1.3.cmml" type="integer" xref="Pt0.A4.SS2.p1.24.m24.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.24.m24.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.24.m24.1d">italic_k = 1</annotation></semantics></math> we simplify notation by writing <math alttext="c_{i},k^{i}_{i^{\prime}},I_{i},P_{st}^{i}=\{p^{i}_{i^{\prime}}\}_{i^{\prime}% \in[m]}" class="ltx_Math" display="inline" id="Pt0.A4.SS2.p1.25.m25.6"><semantics id="Pt0.A4.SS2.p1.25.m25.6a"><mrow id="Pt0.A4.SS2.p1.25.m25.6.6" xref="Pt0.A4.SS2.p1.25.m25.6.6.cmml"><mrow id="Pt0.A4.SS2.p1.25.m25.5.5.4.4" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.5.cmml"><msub id="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1" xref="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.2" xref="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.2.cmml">c</mi><mi id="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.3" xref="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.5" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.5.cmml">,</mo><msubsup id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.2" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.2.cmml">k</mi><msup id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.2" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.2.cmml">i</mi><mo id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.3" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.3.cmml">′</mo></msup><mi id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.3" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.3.cmml">i</mi></msubsup><mo id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.6" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.5.cmml">,</mo><msub id="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3" xref="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.2" xref="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.2.cmml">I</mi><mi id="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.3" xref="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.3.cmml">i</mi></msub><mo id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.7" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.5.cmml">,</mo><msubsup id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.2" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.2.cmml">P</mi><mrow id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.2" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.2.cmml">s</mi><mo id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.1" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.1.cmml">⁢</mo><mi id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.3" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.3.cmml">t</mi></mrow><mi id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.3" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.3.cmml">i</mi></msubsup></mrow><mo id="Pt0.A4.SS2.p1.25.m25.6.6.6" xref="Pt0.A4.SS2.p1.25.m25.6.6.6.cmml">=</mo><msub id="Pt0.A4.SS2.p1.25.m25.6.6.5" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.cmml"><mrow id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.2.cmml"><mo id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.2" stretchy="false" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.2.cmml">{</mo><msubsup id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.2" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.2.cmml">p</mi><msup id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.2" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.3" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.3.cmml">′</mo></msup><mi id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.3" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.3.cmml">i</mi></msubsup><mo id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.3" stretchy="false" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.2.cmml">}</mo></mrow><mrow id="Pt0.A4.SS2.p1.25.m25.1.1.1" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.cmml"><msup id="Pt0.A4.SS2.p1.25.m25.1.1.1.3" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.3.cmml"><mi id="Pt0.A4.SS2.p1.25.m25.1.1.1.3.2" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS2.p1.25.m25.1.1.1.3.3" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.3.3.cmml">′</mo></msup><mo id="Pt0.A4.SS2.p1.25.m25.1.1.1.2" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.SS2.p1.25.m25.1.1.1.4.2" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.4.1.cmml"><mo id="Pt0.A4.SS2.p1.25.m25.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.SS2.p1.25.m25.1.1.1.1" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.SS2.p1.25.m25.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS2.p1.25.m25.6b"><apply id="Pt0.A4.SS2.p1.25.m25.6.6.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6"><eq id="Pt0.A4.SS2.p1.25.m25.6.6.6.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.6"></eq><list id="Pt0.A4.SS2.p1.25.m25.5.5.4.5.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4"><apply id="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.2">𝑐</ci><ci id="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.2.2.1.1.1.3">𝑖</ci></apply><apply id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2">subscript</csymbol><apply id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2">superscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.2">𝑘</ci><ci id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.2.3">𝑖</ci></apply><apply id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3">superscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.2">𝑖</ci><ci id="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.3.3.2.2.2.3.3">′</ci></apply></apply><apply id="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3">subscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.2">𝐼</ci><ci id="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.4.4.3.3.3.3">𝑖</ci></apply><apply id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4">superscript</csymbol><apply id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4">subscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.2">𝑃</ci><apply id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3"><times id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.1"></times><ci id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.2">𝑠</ci><ci id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.2.3.3">𝑡</ci></apply></apply><ci id="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.5.5.4.4.4.3">𝑖</ci></apply></list><apply id="Pt0.A4.SS2.p1.25.m25.6.6.5.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.6.6.5.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5">subscript</csymbol><set id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1"><apply id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1">subscript</csymbol><apply id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.2">𝑝</ci><ci id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.2.3">𝑖</ci></apply><apply id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.6.6.5.1.1.1.3.3">′</ci></apply></apply></set><apply id="Pt0.A4.SS2.p1.25.m25.1.1.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1"><in id="Pt0.A4.SS2.p1.25.m25.1.1.1.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.2"></in><apply id="Pt0.A4.SS2.p1.25.m25.1.1.1.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS2.p1.25.m25.1.1.1.3.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.1.1.1.3.2.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.SS2.p1.25.m25.1.1.1.3.3.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.3.3">′</ci></apply><apply id="Pt0.A4.SS2.p1.25.m25.1.1.1.4.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.4.2"><csymbol cd="latexml" id="Pt0.A4.SS2.p1.25.m25.1.1.1.4.1.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS2.p1.25.m25.1.1.1.1.cmml" xref="Pt0.A4.SS2.p1.25.m25.1.1.1.1">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS2.p1.25.m25.6c">c_{i},k^{i}_{i^{\prime}},I_{i},P_{st}^{i}=\{p^{i}_{i^{\prime}}\}_{i^{\prime}% \in[m]}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS2.p1.25.m25.6d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_k start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = { italic_p start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ [ italic_m ] end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="Pt0.A4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">0.D.3 </span>Expressions for MAX cost</h3> <div class="ltx_para" id="Pt0.A4.SS3.p1"> <p class="ltx_p" id="Pt0.A4.SS3.p1.5">Here we derive some expressions for <math alttext="f_{d}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p1.1.m1.2"><semantics id="Pt0.A4.SS3.p1.1.m1.2a"><mrow id="Pt0.A4.SS3.p1.1.m1.2.3" xref="Pt0.A4.SS3.p1.1.m1.2.3.cmml"><msub id="Pt0.A4.SS3.p1.1.m1.2.3.2" xref="Pt0.A4.SS3.p1.1.m1.2.3.2.cmml"><mi id="Pt0.A4.SS3.p1.1.m1.2.3.2.2" xref="Pt0.A4.SS3.p1.1.m1.2.3.2.2.cmml">f</mi><mi id="Pt0.A4.SS3.p1.1.m1.2.3.2.3" xref="Pt0.A4.SS3.p1.1.m1.2.3.2.3.cmml">d</mi></msub><mo id="Pt0.A4.SS3.p1.1.m1.2.3.1" xref="Pt0.A4.SS3.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="Pt0.A4.SS3.p1.1.m1.2.3.3.2" xref="Pt0.A4.SS3.p1.1.m1.2.3.3.1.cmml"><mo id="Pt0.A4.SS3.p1.1.m1.2.3.3.2.1" stretchy="false" xref="Pt0.A4.SS3.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="Pt0.A4.SS3.p1.1.m1.1.1" xref="Pt0.A4.SS3.p1.1.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.SS3.p1.1.m1.2.3.3.2.2" xref="Pt0.A4.SS3.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="Pt0.A4.SS3.p1.1.m1.2.2" xref="Pt0.A4.SS3.p1.1.m1.2.2.cmml">α</mi><mo id="Pt0.A4.SS3.p1.1.m1.2.3.3.2.3" stretchy="false" xref="Pt0.A4.SS3.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p1.1.m1.2b"><apply id="Pt0.A4.SS3.p1.1.m1.2.3.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.3"><times id="Pt0.A4.SS3.p1.1.m1.2.3.1.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.3.1"></times><apply id="Pt0.A4.SS3.p1.1.m1.2.3.2.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.1.m1.2.3.2.1.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.3.2">subscript</csymbol><ci id="Pt0.A4.SS3.p1.1.m1.2.3.2.2.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.3.2.2">𝑓</ci><ci id="Pt0.A4.SS3.p1.1.m1.2.3.2.3.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.3.2.3">𝑑</ci></apply><interval closure="open" id="Pt0.A4.SS3.p1.1.m1.2.3.3.1.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.3.3.2"><ci id="Pt0.A4.SS3.p1.1.m1.1.1.cmml" xref="Pt0.A4.SS3.p1.1.m1.1.1">𝜎</ci><ci id="Pt0.A4.SS3.p1.1.m1.2.2.cmml" xref="Pt0.A4.SS3.p1.1.m1.2.2">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p1.1.m1.2c">f_{d}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p1.1.m1.2d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math> that will help in future proofs. For <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p1.2.m2.1"><semantics id="Pt0.A4.SS3.p1.2.m2.1a"><mrow id="Pt0.A4.SS3.p1.2.m2.1.1" xref="Pt0.A4.SS3.p1.2.m2.1.1.cmml"><mi id="Pt0.A4.SS3.p1.2.m2.1.1.2" xref="Pt0.A4.SS3.p1.2.m2.1.1.2.cmml">d</mi><mo id="Pt0.A4.SS3.p1.2.m2.1.1.1" xref="Pt0.A4.SS3.p1.2.m2.1.1.1.cmml">∈</mo><msup id="Pt0.A4.SS3.p1.2.m2.1.1.3" xref="Pt0.A4.SS3.p1.2.m2.1.1.3.cmml"><mi id="Pt0.A4.SS3.p1.2.m2.1.1.3.2" xref="Pt0.A4.SS3.p1.2.m2.1.1.3.2.cmml">E</mi><mi id="Pt0.A4.SS3.p1.2.m2.1.1.3.3" xref="Pt0.A4.SS3.p1.2.m2.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p1.2.m2.1b"><apply id="Pt0.A4.SS3.p1.2.m2.1.1.cmml" xref="Pt0.A4.SS3.p1.2.m2.1.1"><in id="Pt0.A4.SS3.p1.2.m2.1.1.1.cmml" xref="Pt0.A4.SS3.p1.2.m2.1.1.1"></in><ci id="Pt0.A4.SS3.p1.2.m2.1.1.2.cmml" xref="Pt0.A4.SS3.p1.2.m2.1.1.2">𝑑</ci><apply id="Pt0.A4.SS3.p1.2.m2.1.1.3.cmml" xref="Pt0.A4.SS3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.2.m2.1.1.3.1.cmml" xref="Pt0.A4.SS3.p1.2.m2.1.1.3">superscript</csymbol><ci id="Pt0.A4.SS3.p1.2.m2.1.1.3.2.cmml" xref="Pt0.A4.SS3.p1.2.m2.1.1.3.2">𝐸</ci><ci id="Pt0.A4.SS3.p1.2.m2.1.1.3.3.cmml" xref="Pt0.A4.SS3.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p1.2.m2.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p1.2.m2.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="c_{d},k^{d}_{i},p^{d}_{i}" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p1.3.m3.3"><semantics id="Pt0.A4.SS3.p1.3.m3.3a"><mrow id="Pt0.A4.SS3.p1.3.m3.3.3.3" xref="Pt0.A4.SS3.p1.3.m3.3.3.4.cmml"><msub id="Pt0.A4.SS3.p1.3.m3.1.1.1.1" xref="Pt0.A4.SS3.p1.3.m3.1.1.1.1.cmml"><mi id="Pt0.A4.SS3.p1.3.m3.1.1.1.1.2" xref="Pt0.A4.SS3.p1.3.m3.1.1.1.1.2.cmml">c</mi><mi id="Pt0.A4.SS3.p1.3.m3.1.1.1.1.3" xref="Pt0.A4.SS3.p1.3.m3.1.1.1.1.3.cmml">d</mi></msub><mo id="Pt0.A4.SS3.p1.3.m3.3.3.3.4" xref="Pt0.A4.SS3.p1.3.m3.3.3.4.cmml">,</mo><msubsup id="Pt0.A4.SS3.p1.3.m3.2.2.2.2" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2.cmml"><mi id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.2" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.2.cmml">k</mi><mi id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.3" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2.3.cmml">i</mi><mi id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.3" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.SS3.p1.3.m3.3.3.3.5" xref="Pt0.A4.SS3.p1.3.m3.3.3.4.cmml">,</mo><msubsup id="Pt0.A4.SS3.p1.3.m3.3.3.3.3" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3.cmml"><mi id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.2" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.2.cmml">p</mi><mi id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.3" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3.3.cmml">i</mi><mi id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.3" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p1.3.m3.3b"><list id="Pt0.A4.SS3.p1.3.m3.3.3.4.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3"><apply id="Pt0.A4.SS3.p1.3.m3.1.1.1.1.cmml" xref="Pt0.A4.SS3.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.3.m3.1.1.1.1.1.cmml" xref="Pt0.A4.SS3.p1.3.m3.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS3.p1.3.m3.1.1.1.1.2.cmml" xref="Pt0.A4.SS3.p1.3.m3.1.1.1.1.2">𝑐</ci><ci id="Pt0.A4.SS3.p1.3.m3.1.1.1.1.3.cmml" xref="Pt0.A4.SS3.p1.3.m3.1.1.1.1.3">𝑑</ci></apply><apply id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.cmml" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.1.cmml" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2">subscript</csymbol><apply id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.cmml" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.1.cmml" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2">superscript</csymbol><ci id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.2.cmml" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.2">𝑘</ci><ci id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.3.cmml" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS3.p1.3.m3.2.2.2.2.3.cmml" xref="Pt0.A4.SS3.p1.3.m3.2.2.2.2.3">𝑖</ci></apply><apply id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.1.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3">subscript</csymbol><apply id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.1.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3">superscript</csymbol><ci id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.2.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.2">𝑝</ci><ci id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.3.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3.2.3">𝑑</ci></apply><ci id="Pt0.A4.SS3.p1.3.m3.3.3.3.3.3.cmml" xref="Pt0.A4.SS3.p1.3.m3.3.3.3.3.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p1.3.m3.3c">c_{d},k^{d}_{i},p^{d}_{i}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p1.3.m3.3d">italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT , italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as defined in Section <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5" title="5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5</span></a>. Using the equality <math alttext="\operatorname*{\mathbb{E}}(X)=\sum_{x}\Pr(X\geq x)" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p1.4.m4.4"><semantics id="Pt0.A4.SS3.p1.4.m4.4a"><mrow id="Pt0.A4.SS3.p1.4.m4.4.4" xref="Pt0.A4.SS3.p1.4.m4.4.4.cmml"><mrow id="Pt0.A4.SS3.p1.4.m4.4.4.3.2" xref="Pt0.A4.SS3.p1.4.m4.4.4.3.1.cmml"><mo id="Pt0.A4.SS3.p1.4.m4.1.1" rspace="0em" xref="Pt0.A4.SS3.p1.4.m4.1.1.cmml">𝔼</mo><mrow id="Pt0.A4.SS3.p1.4.m4.4.4.3.2.1" xref="Pt0.A4.SS3.p1.4.m4.4.4.3.1.cmml"><mo id="Pt0.A4.SS3.p1.4.m4.4.4.3.2.1.1" stretchy="false" xref="Pt0.A4.SS3.p1.4.m4.4.4.3.1.cmml">(</mo><mi id="Pt0.A4.SS3.p1.4.m4.2.2" xref="Pt0.A4.SS3.p1.4.m4.2.2.cmml">X</mi><mo id="Pt0.A4.SS3.p1.4.m4.4.4.3.2.1.2" stretchy="false" xref="Pt0.A4.SS3.p1.4.m4.4.4.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS3.p1.4.m4.4.4.2" rspace="0.111em" xref="Pt0.A4.SS3.p1.4.m4.4.4.2.cmml">=</mo><mrow id="Pt0.A4.SS3.p1.4.m4.4.4.1" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.cmml"><msub id="Pt0.A4.SS3.p1.4.m4.4.4.1.2" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.2.cmml"><mo id="Pt0.A4.SS3.p1.4.m4.4.4.1.2.2" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.2.2.cmml">∑</mo><mi id="Pt0.A4.SS3.p1.4.m4.4.4.1.2.3" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.2.3.cmml">x</mi></msub><mrow id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.2.cmml"><mi id="Pt0.A4.SS3.p1.4.m4.3.3" xref="Pt0.A4.SS3.p1.4.m4.3.3.cmml">Pr</mi><mo id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1a" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.2.cmml">⁡</mo><mrow id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.2.cmml"><mo id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.2.cmml">(</mo><mrow id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.2" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.2.cmml">X</mi><mo id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.1" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.1.cmml">≥</mo><mi id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.3" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p1.4.m4.4b"><apply id="Pt0.A4.SS3.p1.4.m4.4.4.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4"><eq id="Pt0.A4.SS3.p1.4.m4.4.4.2.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.2"></eq><apply id="Pt0.A4.SS3.p1.4.m4.4.4.3.1.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.3.2"><ci id="Pt0.A4.SS3.p1.4.m4.1.1.cmml" xref="Pt0.A4.SS3.p1.4.m4.1.1">𝔼</ci><ci id="Pt0.A4.SS3.p1.4.m4.2.2.cmml" xref="Pt0.A4.SS3.p1.4.m4.2.2">𝑋</ci></apply><apply id="Pt0.A4.SS3.p1.4.m4.4.4.1.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1"><apply id="Pt0.A4.SS3.p1.4.m4.4.4.1.2.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS3.p1.4.m4.4.4.1.2.1.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.2">subscript</csymbol><sum id="Pt0.A4.SS3.p1.4.m4.4.4.1.2.2.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.2.2"></sum><ci id="Pt0.A4.SS3.p1.4.m4.4.4.1.2.3.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.2.3">𝑥</ci></apply><apply id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.2.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1"><ci id="Pt0.A4.SS3.p1.4.m4.3.3.cmml" xref="Pt0.A4.SS3.p1.4.m4.3.3">Pr</ci><apply id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1"><geq id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.1"></geq><ci id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.2">𝑋</ci><ci id="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS3.p1.4.m4.4.4.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p1.4.m4.4c">\operatorname*{\mathbb{E}}(X)=\sum_{x}\Pr(X\geq x)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p1.4.m4.4d">blackboard_E ( italic_X ) = ∑ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT roman_Pr ( italic_X ≥ italic_x )</annotation></semantics></math> for a random variable <math alttext="X" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p1.5.m5.1"><semantics id="Pt0.A4.SS3.p1.5.m5.1a"><mi id="Pt0.A4.SS3.p1.5.m5.1.1" xref="Pt0.A4.SS3.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p1.5.m5.1b"><ci id="Pt0.A4.SS3.p1.5.m5.1.1.cmml" xref="Pt0.A4.SS3.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p1.5.m5.1d">italic_X</annotation></semantics></math>, we derive</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx44"> <tbody id="Pt0.A4.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{d}(\sigma,\alpha)=" class="ltx_Math" display="inline" id="Pt0.A4.E6.m1.2"><semantics id="Pt0.A4.E6.m1.2a"><mrow id="Pt0.A4.E6.m1.2.3" xref="Pt0.A4.E6.m1.2.3.cmml"><mrow id="Pt0.A4.E6.m1.2.3.2" xref="Pt0.A4.E6.m1.2.3.2.cmml"><msub id="Pt0.A4.E6.m1.2.3.2.2" xref="Pt0.A4.E6.m1.2.3.2.2.cmml"><mi id="Pt0.A4.E6.m1.2.3.2.2.2" xref="Pt0.A4.E6.m1.2.3.2.2.2.cmml">f</mi><mi id="Pt0.A4.E6.m1.2.3.2.2.3" xref="Pt0.A4.E6.m1.2.3.2.2.3.cmml">d</mi></msub><mo id="Pt0.A4.E6.m1.2.3.2.1" xref="Pt0.A4.E6.m1.2.3.2.1.cmml">⁢</mo><mrow id="Pt0.A4.E6.m1.2.3.2.3.2" xref="Pt0.A4.E6.m1.2.3.2.3.1.cmml"><mo id="Pt0.A4.E6.m1.2.3.2.3.2.1" stretchy="false" xref="Pt0.A4.E6.m1.2.3.2.3.1.cmml">(</mo><mi id="Pt0.A4.E6.m1.1.1" xref="Pt0.A4.E6.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.E6.m1.2.3.2.3.2.2" xref="Pt0.A4.E6.m1.2.3.2.3.1.cmml">,</mo><mi id="Pt0.A4.E6.m1.2.2" xref="Pt0.A4.E6.m1.2.2.cmml">α</mi><mo id="Pt0.A4.E6.m1.2.3.2.3.2.3" stretchy="false" xref="Pt0.A4.E6.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.E6.m1.2.3.1" xref="Pt0.A4.E6.m1.2.3.1.cmml">=</mo><mi id="Pt0.A4.E6.m1.2.3.3" xref="Pt0.A4.E6.m1.2.3.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.E6.m1.2b"><apply id="Pt0.A4.E6.m1.2.3.cmml" xref="Pt0.A4.E6.m1.2.3"><eq id="Pt0.A4.E6.m1.2.3.1.cmml" xref="Pt0.A4.E6.m1.2.3.1"></eq><apply id="Pt0.A4.E6.m1.2.3.2.cmml" xref="Pt0.A4.E6.m1.2.3.2"><times id="Pt0.A4.E6.m1.2.3.2.1.cmml" xref="Pt0.A4.E6.m1.2.3.2.1"></times><apply id="Pt0.A4.E6.m1.2.3.2.2.cmml" xref="Pt0.A4.E6.m1.2.3.2.2"><csymbol cd="ambiguous" id="Pt0.A4.E6.m1.2.3.2.2.1.cmml" xref="Pt0.A4.E6.m1.2.3.2.2">subscript</csymbol><ci id="Pt0.A4.E6.m1.2.3.2.2.2.cmml" xref="Pt0.A4.E6.m1.2.3.2.2.2">𝑓</ci><ci id="Pt0.A4.E6.m1.2.3.2.2.3.cmml" xref="Pt0.A4.E6.m1.2.3.2.2.3">𝑑</ci></apply><interval closure="open" id="Pt0.A4.E6.m1.2.3.2.3.1.cmml" xref="Pt0.A4.E6.m1.2.3.2.3.2"><ci id="Pt0.A4.E6.m1.1.1.cmml" xref="Pt0.A4.E6.m1.1.1">𝜎</ci><ci id="Pt0.A4.E6.m1.2.2.cmml" xref="Pt0.A4.E6.m1.2.2">𝛼</ci></interval></apply><csymbol cd="latexml" id="Pt0.A4.E6.m1.2.3.3.cmml" xref="Pt0.A4.E6.m1.2.3.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.E6.m1.2c">\displaystyle f_{d}(\sigma,\alpha)=</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.E6.m1.2d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_σ , italic_α ) =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{i\in[m]}(c_{d}(p^{d}_{i})-c_{d}(p^{d}_{i-1}))\Pr_{a\sim% \sigma}\left(\left\lVert c_{d}(a)\right\rVert_{\infty}\geq c_{d}(p^{d}_{i})\right)" class="ltx_Math" display="inline" id="Pt0.A4.E6.m2.5"><semantics id="Pt0.A4.E6.m2.5a"><mrow id="Pt0.A4.E6.m2.5.5" xref="Pt0.A4.E6.m2.5.5.cmml"><mstyle displaystyle="true" id="Pt0.A4.E6.m2.5.5.4" xref="Pt0.A4.E6.m2.5.5.4.cmml"><munder id="Pt0.A4.E6.m2.5.5.4a" xref="Pt0.A4.E6.m2.5.5.4.cmml"><mo id="Pt0.A4.E6.m2.5.5.4.2" movablelimits="false" xref="Pt0.A4.E6.m2.5.5.4.2.cmml">∑</mo><mrow id="Pt0.A4.E6.m2.1.1.1" xref="Pt0.A4.E6.m2.1.1.1.cmml"><mi id="Pt0.A4.E6.m2.1.1.1.3" xref="Pt0.A4.E6.m2.1.1.1.3.cmml">i</mi><mo id="Pt0.A4.E6.m2.1.1.1.2" xref="Pt0.A4.E6.m2.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.E6.m2.1.1.1.4.2" xref="Pt0.A4.E6.m2.1.1.1.4.1.cmml"><mo id="Pt0.A4.E6.m2.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.E6.m2.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.E6.m2.1.1.1.1" xref="Pt0.A4.E6.m2.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.E6.m2.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.E6.m2.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.E6.m2.5.5.3" xref="Pt0.A4.E6.m2.5.5.3.cmml"><mrow id="Pt0.A4.E6.m2.3.3.1.1.1" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.cmml"><mo id="Pt0.A4.E6.m2.3.3.1.1.1.2" stretchy="false" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.E6.m2.3.3.1.1.1.1" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.cmml"><mrow id="Pt0.A4.E6.m2.3.3.1.1.1.1.1" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.cmml"><msub id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.3.2" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.3.2.cmml">c</mi><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.3.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.3.3.cmml">d</mi></msub><mo id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.2" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.2.2" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.2.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.E6.m2.3.3.1.1.1.1.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.E6.m2.3.3.1.1.1.1.2" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.cmml"><msub id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.3.cmml"><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.3.2" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.3.2.cmml">c</mi><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.3.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.3.3.cmml">d</mi></msub><mo id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.2" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.2.cmml">⁢</mo><mrow id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1.cmml"><mo id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.2" stretchy="false" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1.cmml"><mi id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1.2.2" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1.2.2.cmml">p</mi><mrow id="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1.3" xref="Pt0.A4.E6.m2.3.3.1.1.1.1.2.1.1.1.3.cmml"><mi 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xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3.cmml"><mi id="Pt0.A4.E6.m2.4.4.2.2.1.1.3.2" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3.2.cmml">a</mi><mo id="Pt0.A4.E6.m2.4.4.2.2.1.1.3.1" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3.1.cmml">∼</mo><mi id="Pt0.A4.E6.m2.4.4.2.2.1.1.3.3" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3.3.cmml">σ</mi></mrow></munder><mo id="Pt0.A4.E6.m2.5.5.3.3.2a" xref="Pt0.A4.E6.m2.5.5.3.3.3.cmml">⁡</mo><mrow id="Pt0.A4.E6.m2.5.5.3.3.2.2" xref="Pt0.A4.E6.m2.5.5.3.3.3.cmml"><mo id="Pt0.A4.E6.m2.5.5.3.3.2.2.2" xref="Pt0.A4.E6.m2.5.5.3.3.3.cmml">(</mo><mrow id="Pt0.A4.E6.m2.5.5.3.3.2.2.1" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.cmml"><msub id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.cmml"><mrow id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.1" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.2.cmml"><mo fence="true" id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.1.2" lspace="0em" rspace="0em" stretchy="true" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.2.1.cmml">∥</mo><mrow id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.1.1" 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xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.cmml"><mi id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.2.2" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.3" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.2.3" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.3" stretchy="false" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.E6.m2.5.5.3.3.2.2.3" xref="Pt0.A4.E6.m2.5.5.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.E6.m2.5b"><apply id="Pt0.A4.E6.m2.5.5.cmml" xref="Pt0.A4.E6.m2.5.5"><apply id="Pt0.A4.E6.m2.5.5.4.cmml" xref="Pt0.A4.E6.m2.5.5.4"><csymbol cd="ambiguous" id="Pt0.A4.E6.m2.5.5.4.1.cmml" xref="Pt0.A4.E6.m2.5.5.4">subscript</csymbol><sum id="Pt0.A4.E6.m2.5.5.4.2.cmml" xref="Pt0.A4.E6.m2.5.5.4.2"></sum><apply id="Pt0.A4.E6.m2.1.1.1.cmml" 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id="Pt0.A4.E6.m2.4.4.2.2.1.1.2.cmml" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.2">Pr</ci><apply id="Pt0.A4.E6.m2.4.4.2.2.1.1.3.cmml" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3"><csymbol cd="latexml" id="Pt0.A4.E6.m2.4.4.2.2.1.1.3.1.cmml" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3.1">similar-to</csymbol><ci id="Pt0.A4.E6.m2.4.4.2.2.1.1.3.2.cmml" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3.2">𝑎</ci><ci id="Pt0.A4.E6.m2.4.4.2.2.1.1.3.3.cmml" xref="Pt0.A4.E6.m2.4.4.2.2.1.1.3.3">𝜎</ci></apply></apply><apply id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.cmml" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1"><geq id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.3.cmml" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.3"></geq><apply id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.cmml" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1"><csymbol cd="ambiguous" id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.2.cmml" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1">subscript</csymbol><apply id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.2.cmml" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.1"><csymbol cd="latexml" id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.1.1.2.1.cmml" 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xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.2.2">𝑝</ci><ci id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.2.3.cmml" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.2.3">𝑑</ci></apply><ci id="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.3.cmml" xref="Pt0.A4.E6.m2.5.5.3.3.2.2.1.2.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.E6.m2.5c">\displaystyle\sum_{i\in[m]}(c_{d}(p^{d}_{i})-c_{d}(p^{d}_{i-1}))\Pr_{a\sim% \sigma}\left(\left\lVert c_{d}(a)\right\rVert_{\infty}\geq c_{d}(p^{d}_{i})\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.E6.m2.5d">∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ) roman_Pr start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( ∥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> <tbody id="Pt0.A4.Ex70"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="Pt0.A4.Ex70.m1.1"><semantics id="Pt0.A4.Ex70.m1.1a"><mo id="Pt0.A4.Ex70.m1.1.1" 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start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ) ( 1 - ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i - 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex71"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="Pt0.A4.Ex71.m1.1"><semantics id="Pt0.A4.Ex71.m1.1a"><mo id="Pt0.A4.Ex71.m1.1.1" xref="Pt0.A4.Ex71.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex71.m1.1b"><eq id="Pt0.A4.Ex71.m1.1.1.cmml" xref="Pt0.A4.Ex71.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex71.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex71.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{i\in[m]}c_{d}(p^{d}_{i})\left(\Pr_{a\sim\sigma}\left(\left% \lVert c_{d}(a)\right\rVert_{\infty}\geq c_{d}(p^{d}_{i})\right)-\Pr_{a\sim% \sigma}\left(\left\lVert c_{d}(a)\right\rVert_{\infty}>c_{d}(p^{d}_{i})\right)\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex71.m2.5"><semantics id="Pt0.A4.Ex71.m2.5a"><mrow id="Pt0.A4.Ex71.m2.5.5" xref="Pt0.A4.Ex71.m2.5.5.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex71.m2.5.5.3" xref="Pt0.A4.Ex71.m2.5.5.3.cmml"><munder id="Pt0.A4.Ex71.m2.5.5.3a" xref="Pt0.A4.Ex71.m2.5.5.3.cmml"><mo id="Pt0.A4.Ex71.m2.5.5.3.2" movablelimits="false" xref="Pt0.A4.Ex71.m2.5.5.3.2.cmml">∑</mo><mrow id="Pt0.A4.Ex71.m2.1.1.1" xref="Pt0.A4.Ex71.m2.1.1.1.cmml"><mi id="Pt0.A4.Ex71.m2.1.1.1.3" xref="Pt0.A4.Ex71.m2.1.1.1.3.cmml">i</mi><mo id="Pt0.A4.Ex71.m2.1.1.1.2" xref="Pt0.A4.Ex71.m2.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.Ex71.m2.1.1.1.4.2" xref="Pt0.A4.Ex71.m2.1.1.1.4.1.cmml"><mo id="Pt0.A4.Ex71.m2.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.Ex71.m2.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.Ex71.m2.1.1.1.1" xref="Pt0.A4.Ex71.m2.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.Ex71.m2.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.Ex71.m2.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex71.m2.5.5.2" xref="Pt0.A4.Ex71.m2.5.5.2.cmml"><msub id="Pt0.A4.Ex71.m2.5.5.2.4" xref="Pt0.A4.Ex71.m2.5.5.2.4.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.4.2" xref="Pt0.A4.Ex71.m2.5.5.2.4.2.cmml">c</mi><mi id="Pt0.A4.Ex71.m2.5.5.2.4.3" xref="Pt0.A4.Ex71.m2.5.5.2.4.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex71.m2.5.5.2.3" xref="Pt0.A4.Ex71.m2.5.5.2.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex71.m2.4.4.1.1.1" xref="Pt0.A4.Ex71.m2.4.4.1.1.1.1.cmml"><mo id="Pt0.A4.Ex71.m2.4.4.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex71.m2.4.4.1.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.Ex71.m2.4.4.1.1.1.1" xref="Pt0.A4.Ex71.m2.4.4.1.1.1.1.cmml"><mi id="Pt0.A4.Ex71.m2.4.4.1.1.1.1.2.2" xref="Pt0.A4.Ex71.m2.4.4.1.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.Ex71.m2.4.4.1.1.1.1.3" xref="Pt0.A4.Ex71.m2.4.4.1.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.Ex71.m2.4.4.1.1.1.1.2.3" xref="Pt0.A4.Ex71.m2.4.4.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex71.m2.4.4.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex71.m2.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex71.m2.5.5.2.3a" xref="Pt0.A4.Ex71.m2.5.5.2.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.cmml"><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.cmml"><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.3.cmml"><munder id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.2.cmml">Pr</mi><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3.2.cmml">a</mi><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3.1.cmml">∼</mo><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.1.1.1.3.3.cmml">σ</mi></mrow></munder><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2a" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.3.cmml">⁡</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.3.cmml"><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.3.cmml">(</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.cmml"><msub id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.cmml"><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.2.cmml"><mo fence="true" id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.2" lspace="0em" rspace="0em" stretchy="true" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.2.1.cmml">∥</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.cmml"><msub id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.2.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.2.2.cmml">c</mi><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.2.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.2.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.3.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.3.2.1" stretchy="false" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.cmml">(</mo><mi id="Pt0.A4.Ex71.m2.2.2" xref="Pt0.A4.Ex71.m2.2.2.cmml">a</mi><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.3.2.2" stretchy="false" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo fence="true" id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.1.3" lspace="0em" rspace="0.1389em" stretchy="true" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.1.2.1.cmml">∥</mo></mrow><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.3" mathvariant="normal" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.1.3.cmml">∞</mi></msub><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.3" lspace="0.1389em" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.3.cmml">≥</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.cmml"><msub id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.3.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.3.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.3.2.cmml">c</mi><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.3.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.3.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.cmml"><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.2.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.3" stretchy="false" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.2.2.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.5" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.5.cmml">−</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.3.cmml"><munder id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.2.cmml">Pr</mi><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3.2.cmml">a</mi><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3.1.cmml">∼</mo><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.3.1.1.3.3.cmml">σ</mi></mrow></munder><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2a" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.3.cmml">⁡</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.3.cmml"><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.3.cmml">(</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.cmml"><msub id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.cmml"><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.2.cmml"><mo fence="true" id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.2" lspace="0em" rspace="0em" stretchy="true" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.2.1.cmml">∥</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.cmml"><msub id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.2.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.2.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.2.2.cmml">c</mi><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.2.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.2.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.3.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.3.2.1" stretchy="false" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.cmml">(</mo><mi id="Pt0.A4.Ex71.m2.3.3" xref="Pt0.A4.Ex71.m2.3.3.cmml">a</mi><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.3.2.2" stretchy="false" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo fence="true" id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.1.3" lspace="0em" rspace="0.1389em" stretchy="true" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.1.2.1.cmml">∥</mo></mrow><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.3" mathvariant="normal" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.1.3.cmml">∞</mi></msub><mo id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.3" lspace="0.1389em" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.3.cmml">></mo><mrow id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.2.cmml"><msub id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.2.3" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.2.3.cmml"><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.2.3.2" xref="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.2.3.2.cmml">c</mi><mi id="Pt0.A4.Ex71.m2.5.5.2.2.1.1.4.2.2.1.2.3.3" 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id="Pt0.A4.Ex71.m2.5c">\displaystyle\sum_{i\in[m]}c_{d}(p^{d}_{i})\left(\Pr_{a\sim\sigma}\left(\left% \lVert c_{d}(a)\right\rVert_{\infty}\geq c_{d}(p^{d}_{i})\right)-\Pr_{a\sim% \sigma}\left(\left\lVert c_{d}(a)\right\rVert_{\infty}>c_{d}(p^{d}_{i})\right)\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex71.m2.5d">∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( roman_Pr start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( ∥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) - roman_Pr start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( ∥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT > italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex72"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="Pt0.A4.Ex72.m1.1"><semantics id="Pt0.A4.Ex72.m1.1a"><mo id="Pt0.A4.Ex72.m1.1.1" xref="Pt0.A4.Ex72.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex72.m1.1b"><eq id="Pt0.A4.Ex72.m1.1.1.cmml" xref="Pt0.A4.Ex72.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" 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xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.3">subscript</csymbol><ci id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.3.2.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.3.2">𝜎</ci><ci id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.3.3">𝑗</ci></apply><apply id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1">subscript</csymbol><apply id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.2.1.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1">superscript</csymbol><ci id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.2.2.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.2.2">𝑝</ci><ci id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.2.3.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.2.3">𝑑</ci></apply><apply id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex72.m2.5.5.2.2.1.1.2.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex72.m2.5c">\displaystyle\sum_{i\in[m]}c_{d}(p^{d}_{i})\left(\prod_{j\in[n]}\left(\sum_{i^% {\prime}\leq i}\sigma_{j}(p^{d}_{i^{\prime}})\right)-\prod_{j\in[n]}\left(\sum% _{i^{\prime}\leq i-1}\sigma_{j}(p^{d}_{i^{\prime}})\right)\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex72.m2.5d">∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) - ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i - 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="Pt0.A4.SS3.p2"> <p class="ltx_p" id="Pt0.A4.SS3.p2.2">We write Equation <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E6" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">6</span></a> more explicitly when <math alttext="k=1" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p2.1.m1.1"><semantics id="Pt0.A4.SS3.p2.1.m1.1a"><mrow id="Pt0.A4.SS3.p2.1.m1.1.1" xref="Pt0.A4.SS3.p2.1.m1.1.1.cmml"><mi id="Pt0.A4.SS3.p2.1.m1.1.1.2" xref="Pt0.A4.SS3.p2.1.m1.1.1.2.cmml">k</mi><mo id="Pt0.A4.SS3.p2.1.m1.1.1.1" xref="Pt0.A4.SS3.p2.1.m1.1.1.1.cmml">=</mo><mn id="Pt0.A4.SS3.p2.1.m1.1.1.3" xref="Pt0.A4.SS3.p2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p2.1.m1.1b"><apply id="Pt0.A4.SS3.p2.1.m1.1.1.cmml" xref="Pt0.A4.SS3.p2.1.m1.1.1"><eq id="Pt0.A4.SS3.p2.1.m1.1.1.1.cmml" xref="Pt0.A4.SS3.p2.1.m1.1.1.1"></eq><ci id="Pt0.A4.SS3.p2.1.m1.1.1.2.cmml" xref="Pt0.A4.SS3.p2.1.m1.1.1.2">𝑘</ci><cn id="Pt0.A4.SS3.p2.1.m1.1.1.3.cmml" type="integer" xref="Pt0.A4.SS3.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p2.1.m1.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p2.1.m1.1d">italic_k = 1</annotation></semantics></math>. For every <math alttext="i<m" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p2.2.m2.1"><semantics id="Pt0.A4.SS3.p2.2.m2.1a"><mrow id="Pt0.A4.SS3.p2.2.m2.1.1" xref="Pt0.A4.SS3.p2.2.m2.1.1.cmml"><mi id="Pt0.A4.SS3.p2.2.m2.1.1.2" xref="Pt0.A4.SS3.p2.2.m2.1.1.2.cmml">i</mi><mo id="Pt0.A4.SS3.p2.2.m2.1.1.1" xref="Pt0.A4.SS3.p2.2.m2.1.1.1.cmml"><</mo><mi id="Pt0.A4.SS3.p2.2.m2.1.1.3" xref="Pt0.A4.SS3.p2.2.m2.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p2.2.m2.1b"><apply id="Pt0.A4.SS3.p2.2.m2.1.1.cmml" xref="Pt0.A4.SS3.p2.2.m2.1.1"><lt id="Pt0.A4.SS3.p2.2.m2.1.1.1.cmml" xref="Pt0.A4.SS3.p2.2.m2.1.1.1"></lt><ci id="Pt0.A4.SS3.p2.2.m2.1.1.2.cmml" xref="Pt0.A4.SS3.p2.2.m2.1.1.2">𝑖</ci><ci id="Pt0.A4.SS3.p2.2.m2.1.1.3.cmml" xref="Pt0.A4.SS3.p2.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p2.2.m2.1c">i<m</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p2.2.m2.1d">italic_i < italic_m</annotation></semantics></math>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx45"> <tbody id="Pt0.A4.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{i}(\sigma,\alpha)=" class="ltx_Math" display="inline" id="Pt0.A4.E7.m1.2"><semantics id="Pt0.A4.E7.m1.2a"><mrow id="Pt0.A4.E7.m1.2.3" xref="Pt0.A4.E7.m1.2.3.cmml"><mrow id="Pt0.A4.E7.m1.2.3.2" xref="Pt0.A4.E7.m1.2.3.2.cmml"><msub id="Pt0.A4.E7.m1.2.3.2.2" xref="Pt0.A4.E7.m1.2.3.2.2.cmml"><mi id="Pt0.A4.E7.m1.2.3.2.2.2" xref="Pt0.A4.E7.m1.2.3.2.2.2.cmml">f</mi><mi id="Pt0.A4.E7.m1.2.3.2.2.3" xref="Pt0.A4.E7.m1.2.3.2.2.3.cmml">i</mi></msub><mo id="Pt0.A4.E7.m1.2.3.2.1" xref="Pt0.A4.E7.m1.2.3.2.1.cmml">⁢</mo><mrow id="Pt0.A4.E7.m1.2.3.2.3.2" xref="Pt0.A4.E7.m1.2.3.2.3.1.cmml"><mo id="Pt0.A4.E7.m1.2.3.2.3.2.1" stretchy="false" xref="Pt0.A4.E7.m1.2.3.2.3.1.cmml">(</mo><mi id="Pt0.A4.E7.m1.1.1" xref="Pt0.A4.E7.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.E7.m1.2.3.2.3.2.2" xref="Pt0.A4.E7.m1.2.3.2.3.1.cmml">,</mo><mi id="Pt0.A4.E7.m1.2.2" xref="Pt0.A4.E7.m1.2.2.cmml">α</mi><mo id="Pt0.A4.E7.m1.2.3.2.3.2.3" stretchy="false" xref="Pt0.A4.E7.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.E7.m1.2.3.1" xref="Pt0.A4.E7.m1.2.3.1.cmml">=</mo><mi id="Pt0.A4.E7.m1.2.3.3" xref="Pt0.A4.E7.m1.2.3.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.E7.m1.2b"><apply id="Pt0.A4.E7.m1.2.3.cmml" xref="Pt0.A4.E7.m1.2.3"><eq id="Pt0.A4.E7.m1.2.3.1.cmml" xref="Pt0.A4.E7.m1.2.3.1"></eq><apply id="Pt0.A4.E7.m1.2.3.2.cmml" xref="Pt0.A4.E7.m1.2.3.2"><times id="Pt0.A4.E7.m1.2.3.2.1.cmml" xref="Pt0.A4.E7.m1.2.3.2.1"></times><apply id="Pt0.A4.E7.m1.2.3.2.2.cmml" xref="Pt0.A4.E7.m1.2.3.2.2"><csymbol cd="ambiguous" id="Pt0.A4.E7.m1.2.3.2.2.1.cmml" xref="Pt0.A4.E7.m1.2.3.2.2">subscript</csymbol><ci id="Pt0.A4.E7.m1.2.3.2.2.2.cmml" xref="Pt0.A4.E7.m1.2.3.2.2.2">𝑓</ci><ci id="Pt0.A4.E7.m1.2.3.2.2.3.cmml" xref="Pt0.A4.E7.m1.2.3.2.2.3">𝑖</ci></apply><interval closure="open" id="Pt0.A4.E7.m1.2.3.2.3.1.cmml" xref="Pt0.A4.E7.m1.2.3.2.3.2"><ci id="Pt0.A4.E7.m1.1.1.cmml" xref="Pt0.A4.E7.m1.1.1">𝜎</ci><ci id="Pt0.A4.E7.m1.2.2.cmml" xref="Pt0.A4.E7.m1.2.2">𝛼</ci></interval></apply><csymbol cd="latexml" id="Pt0.A4.E7.m1.2.3.3.cmml" xref="Pt0.A4.E7.m1.2.3.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.E7.m1.2c">\displaystyle f_{i}(\sigma,\alpha)=</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.E7.m1.2d">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_σ , italic_α ) =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{i^{\prime}\in[m]}(c_{i}(p^{i}_{i^{\prime}})-c_{i}(p^{i}_{i^% {\prime}-1}))\Pr_{a\sim\sigma}\left(\left\lVert c_{i}(a)\right\rVert_{\infty}% \geq c_{i}(p^{i}_{i^{\prime}})\right)" class="ltx_Math" display="inline" id="Pt0.A4.E7.m2.5"><semantics id="Pt0.A4.E7.m2.5a"><mrow id="Pt0.A4.E7.m2.5.5" xref="Pt0.A4.E7.m2.5.5.cmml"><mstyle displaystyle="true" id="Pt0.A4.E7.m2.5.5.4" xref="Pt0.A4.E7.m2.5.5.4.cmml"><munder id="Pt0.A4.E7.m2.5.5.4a" xref="Pt0.A4.E7.m2.5.5.4.cmml"><mo id="Pt0.A4.E7.m2.5.5.4.2" movablelimits="false" xref="Pt0.A4.E7.m2.5.5.4.2.cmml">∑</mo><mrow id="Pt0.A4.E7.m2.1.1.1" xref="Pt0.A4.E7.m2.1.1.1.cmml"><msup id="Pt0.A4.E7.m2.1.1.1.3" xref="Pt0.A4.E7.m2.1.1.1.3.cmml"><mi id="Pt0.A4.E7.m2.1.1.1.3.2" xref="Pt0.A4.E7.m2.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.E7.m2.1.1.1.3.3" xref="Pt0.A4.E7.m2.1.1.1.3.3.cmml">′</mo></msup><mo id="Pt0.A4.E7.m2.1.1.1.2" xref="Pt0.A4.E7.m2.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.E7.m2.1.1.1.4.2" xref="Pt0.A4.E7.m2.1.1.1.4.1.cmml"><mo id="Pt0.A4.E7.m2.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.E7.m2.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.E7.m2.1.1.1.1" xref="Pt0.A4.E7.m2.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.E7.m2.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.E7.m2.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.E7.m2.5.5.3" xref="Pt0.A4.E7.m2.5.5.3.cmml"><mrow id="Pt0.A4.E7.m2.3.3.1.1.1" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.cmml"><mo id="Pt0.A4.E7.m2.3.3.1.1.1.2" stretchy="false" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.E7.m2.3.3.1.1.1.1" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.cmml"><mrow id="Pt0.A4.E7.m2.3.3.1.1.1.1.1" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.1.cmml"><msub id="Pt0.A4.E7.m2.3.3.1.1.1.1.1.3" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.E7.m2.3.3.1.1.1.1.1.3.2" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.1.3.2.cmml">c</mi><mi id="Pt0.A4.E7.m2.3.3.1.1.1.1.1.3.3" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.1.3.3.cmml">i</mi></msub><mo id="Pt0.A4.E7.m2.3.3.1.1.1.1.1.2" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.E7.m2.3.3.1.1.1.1.1.1.1" xref="Pt0.A4.E7.m2.3.3.1.1.1.1.1.1.1.1.cmml"><mo 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xref="Pt0.A4.E7.m2.4.4.2.2.1.1.2.cmml">Pr</mi><mrow id="Pt0.A4.E7.m2.4.4.2.2.1.1.3" xref="Pt0.A4.E7.m2.4.4.2.2.1.1.3.cmml"><mi id="Pt0.A4.E7.m2.4.4.2.2.1.1.3.2" xref="Pt0.A4.E7.m2.4.4.2.2.1.1.3.2.cmml">a</mi><mo id="Pt0.A4.E7.m2.4.4.2.2.1.1.3.1" xref="Pt0.A4.E7.m2.4.4.2.2.1.1.3.1.cmml">∼</mo><mi id="Pt0.A4.E7.m2.4.4.2.2.1.1.3.3" xref="Pt0.A4.E7.m2.4.4.2.2.1.1.3.3.cmml">σ</mi></mrow></munder><mo id="Pt0.A4.E7.m2.5.5.3.3.2a" xref="Pt0.A4.E7.m2.5.5.3.3.3.cmml">⁡</mo><mrow id="Pt0.A4.E7.m2.5.5.3.3.2.2" xref="Pt0.A4.E7.m2.5.5.3.3.3.cmml"><mo id="Pt0.A4.E7.m2.5.5.3.3.2.2.2" xref="Pt0.A4.E7.m2.5.5.3.3.3.cmml">(</mo><mrow id="Pt0.A4.E7.m2.5.5.3.3.2.2.1" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.cmml"><msub id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.1" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.1.cmml"><mrow id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.1.1.1" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.1.1.2.cmml"><mo fence="true" id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.1.1.1.2" lspace="0em" rspace="0em" stretchy="true" 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xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1"><csymbol cd="ambiguous" id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.1.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1">subscript</csymbol><apply id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.2.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1"><csymbol cd="ambiguous" id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.2.1.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1">superscript</csymbol><ci id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.2.2.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.2.2">𝑝</ci><ci id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.2.3.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.2.3">𝑖</ci></apply><apply id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3.1.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3.2.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3.3.cmml" xref="Pt0.A4.E7.m2.5.5.3.3.2.2.1.2.1.1.1.3.3">′</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.E7.m2.5c">\displaystyle\sum_{i^{\prime}\in[m]}(c_{i}(p^{i}_{i^{\prime}})-c_{i}(p^{i}_{i^% {\prime}-1}))\Pr_{a\sim\sigma}\left(\left\lVert c_{i}(a)\right\rVert_{\infty}% \geq c_{i}(p^{i}_{i^{\prime}})\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.E7.m2.5d">∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ [ italic_m ] end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 end_POSTSUBSCRIPT ) ) roman_Pr start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( ∥ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≥ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> <tbody id="Pt0.A4.Ex73"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" 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id="Pt0.A4.Ex73.m2.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex73.m2.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3">′′</ci></apply></apply></apply></apply></apply></apply></apply></apply><plus id="Pt0.A4.Ex73.m2.3.3.4.cmml" xref="Pt0.A4.Ex73.m2.3.3.4"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex73.m2.3c">\displaystyle\sum_{i^{\prime}<i}(|p_{i^{\prime}}|-|p_{i^{\prime}-1}|)\left(1-% \prod_{j\in[n]}\left(\sum_{i^{\prime\prime}<i^{\prime}}\sigma_{j}(p_{i^{\prime% \prime}})\right)\right)~{}+</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex73.m2.3d">∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_i end_POSTSUBSCRIPT ( | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 end_POSTSUBSCRIPT | ) ( 1 - ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) ) +</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex74"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(|p_{i+1}|-|p_{i-1}|)\left(1-\prod_{j\in[n]}\left(\sum_{i^{\prime% \prime}<i}\sigma_{j}(p_{i^{\prime\prime}})\right)\right)~{}+" class="ltx_Math" display="inline" id="Pt0.A4.Ex74.m1.3"><semantics id="Pt0.A4.Ex74.m1.3a"><mrow id="Pt0.A4.Ex74.m1.3.3" xref="Pt0.A4.Ex74.m1.3.3.cmml"><mrow 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xref="Pt0.A4.Ex75.m1.1.1.1.3.cmml">j</mi><mo id="Pt0.A4.Ex75.m1.1.1.1.2" xref="Pt0.A4.Ex75.m1.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.Ex75.m1.1.1.1.4.2" xref="Pt0.A4.Ex75.m1.1.1.1.4.1.cmml"><mo id="Pt0.A4.Ex75.m1.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.Ex75.m1.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.Ex75.m1.1.1.1.1" xref="Pt0.A4.Ex75.m1.1.1.1.1.cmml">n</mi><mo id="Pt0.A4.Ex75.m1.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.Ex75.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.2" 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xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.2.3.cmml">′′</mo></msup><mo id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.1" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.1.cmml"><</mo><msup id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.3" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.3.2" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.3.2.cmml">i</mi><mo id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.3.3" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.2.3.3.3.cmml">′</mo></msup></mrow></munder></mstyle><mrow id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">σ</mi><mi id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml">j</mi></msub><mo id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><msup id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2" 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xref="Pt0.A4.Ex75.m1.3.3.2.2.2.1.1.1.1.1.1.2.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply><plus id="Pt0.A4.Ex75.m1.3.3.4.cmml" xref="Pt0.A4.Ex75.m1.3.3.4"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex75.m1.3c">\displaystyle\sum_{i^{\prime}>i+1}(|p_{i^{\prime}}|-|p_{i^{\prime}-1}|)\left(1% -\prod_{j\in[n]}\left(\left(\sum_{i^{\prime\prime}<i^{\prime}}\sigma_{j}(p_{i^% {\prime\prime}})\right)-\sigma_{j}(p_{i})\right)\right)~{}+</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex75.m1.3d">∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_i + 1 end_POSTSUBSCRIPT ( | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 end_POSTSUBSCRIPT | ) ( 1 - ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( ( ∑ start_POSTSUBSCRIPT 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(p_{i})\right)\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex76.m1.3d">( | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | + italic_α - | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ) ( 1 - ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A4.SS3.p2.3">For <math alttext="i=m" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p2.3.m1.1"><semantics id="Pt0.A4.SS3.p2.3.m1.1a"><mrow id="Pt0.A4.SS3.p2.3.m1.1.1" xref="Pt0.A4.SS3.p2.3.m1.1.1.cmml"><mi id="Pt0.A4.SS3.p2.3.m1.1.1.2" xref="Pt0.A4.SS3.p2.3.m1.1.1.2.cmml">i</mi><mo id="Pt0.A4.SS3.p2.3.m1.1.1.1" xref="Pt0.A4.SS3.p2.3.m1.1.1.1.cmml">=</mo><mi id="Pt0.A4.SS3.p2.3.m1.1.1.3" xref="Pt0.A4.SS3.p2.3.m1.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS3.p2.3.m1.1b"><apply id="Pt0.A4.SS3.p2.3.m1.1.1.cmml" xref="Pt0.A4.SS3.p2.3.m1.1.1"><eq id="Pt0.A4.SS3.p2.3.m1.1.1.1.cmml" xref="Pt0.A4.SS3.p2.3.m1.1.1.1"></eq><ci id="Pt0.A4.SS3.p2.3.m1.1.1.2.cmml" xref="Pt0.A4.SS3.p2.3.m1.1.1.2">𝑖</ci><ci id="Pt0.A4.SS3.p2.3.m1.1.1.3.cmml" xref="Pt0.A4.SS3.p2.3.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS3.p2.3.m1.1c">i=m</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS3.p2.3.m1.1d">italic_i = italic_m</annotation></semantics></math>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx46"> <tbody id="Pt0.A4.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{m}(\sigma,\alpha)=" class="ltx_Math" display="inline" id="Pt0.A4.E8.m1.2"><semantics id="Pt0.A4.E8.m1.2a"><mrow id="Pt0.A4.E8.m1.2.3" xref="Pt0.A4.E8.m1.2.3.cmml"><mrow id="Pt0.A4.E8.m1.2.3.2" xref="Pt0.A4.E8.m1.2.3.2.cmml"><msub id="Pt0.A4.E8.m1.2.3.2.2" xref="Pt0.A4.E8.m1.2.3.2.2.cmml"><mi id="Pt0.A4.E8.m1.2.3.2.2.2" xref="Pt0.A4.E8.m1.2.3.2.2.2.cmml">f</mi><mi id="Pt0.A4.E8.m1.2.3.2.2.3" xref="Pt0.A4.E8.m1.2.3.2.2.3.cmml">m</mi></msub><mo id="Pt0.A4.E8.m1.2.3.2.1" xref="Pt0.A4.E8.m1.2.3.2.1.cmml">⁢</mo><mrow id="Pt0.A4.E8.m1.2.3.2.3.2" xref="Pt0.A4.E8.m1.2.3.2.3.1.cmml"><mo id="Pt0.A4.E8.m1.2.3.2.3.2.1" stretchy="false" xref="Pt0.A4.E8.m1.2.3.2.3.1.cmml">(</mo><mi id="Pt0.A4.E8.m1.1.1" xref="Pt0.A4.E8.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.E8.m1.2.3.2.3.2.2" xref="Pt0.A4.E8.m1.2.3.2.3.1.cmml">,</mo><mi id="Pt0.A4.E8.m1.2.2" xref="Pt0.A4.E8.m1.2.2.cmml">α</mi><mo id="Pt0.A4.E8.m1.2.3.2.3.2.3" stretchy="false" xref="Pt0.A4.E8.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.E8.m1.2.3.1" xref="Pt0.A4.E8.m1.2.3.1.cmml">=</mo><mi id="Pt0.A4.E8.m1.2.3.3" xref="Pt0.A4.E8.m1.2.3.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.E8.m1.2b"><apply id="Pt0.A4.E8.m1.2.3.cmml" xref="Pt0.A4.E8.m1.2.3"><eq id="Pt0.A4.E8.m1.2.3.1.cmml" xref="Pt0.A4.E8.m1.2.3.1"></eq><apply id="Pt0.A4.E8.m1.2.3.2.cmml" xref="Pt0.A4.E8.m1.2.3.2"><times id="Pt0.A4.E8.m1.2.3.2.1.cmml" xref="Pt0.A4.E8.m1.2.3.2.1"></times><apply id="Pt0.A4.E8.m1.2.3.2.2.cmml" xref="Pt0.A4.E8.m1.2.3.2.2"><csymbol cd="ambiguous" id="Pt0.A4.E8.m1.2.3.2.2.1.cmml" xref="Pt0.A4.E8.m1.2.3.2.2">subscript</csymbol><ci id="Pt0.A4.E8.m1.2.3.2.2.2.cmml" xref="Pt0.A4.E8.m1.2.3.2.2.2">𝑓</ci><ci id="Pt0.A4.E8.m1.2.3.2.2.3.cmml" xref="Pt0.A4.E8.m1.2.3.2.2.3">𝑚</ci></apply><interval closure="open" id="Pt0.A4.E8.m1.2.3.2.3.1.cmml" xref="Pt0.A4.E8.m1.2.3.2.3.2"><ci id="Pt0.A4.E8.m1.1.1.cmml" xref="Pt0.A4.E8.m1.1.1">𝜎</ci><ci id="Pt0.A4.E8.m1.2.2.cmml" xref="Pt0.A4.E8.m1.2.2">𝛼</ci></interval></apply><csymbol cd="latexml" id="Pt0.A4.E8.m1.2.3.3.cmml" xref="Pt0.A4.E8.m1.2.3.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.E8.m1.2c">\displaystyle f_{m}(\sigma,\alpha)=</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.E8.m1.2d">italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_σ , italic_α ) =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{i^{\prime}\in[m]}(c_{m}(p^{m}_{i^{\prime}})-c_{m}(p^{m}_{i^% {\prime}-1}))\Pr_{a\sim\sigma}\left(\left\lVert c_{m}(a)\right\rVert_{\infty}% \geq c_{m}(p^{m}_{i^{\prime}})\right)" class="ltx_Math" display="inline" id="Pt0.A4.E8.m2.5"><semantics id="Pt0.A4.E8.m2.5a"><mrow id="Pt0.A4.E8.m2.5.5" xref="Pt0.A4.E8.m2.5.5.cmml"><mstyle displaystyle="true" id="Pt0.A4.E8.m2.5.5.4" xref="Pt0.A4.E8.m2.5.5.4.cmml"><munder id="Pt0.A4.E8.m2.5.5.4a" xref="Pt0.A4.E8.m2.5.5.4.cmml"><mo id="Pt0.A4.E8.m2.5.5.4.2" 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xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.cmml"><mo id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.2" stretchy="false" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.cmml"><mi id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.2.2" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.2.2.cmml">p</mi><mrow id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.cmml"><msup id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.2" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.2.cmml"><mi id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.2.2" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.2.2.cmml">i</mi><mo id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.2.3" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.1" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.3" xref="Pt0.A4.E8.m2.3.3.1.1.1.1.2.1.1.1.3.3.cmml">1</mn></mrow><mi 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xref="Pt0.A4.E8.m2.4.4.2.2.1.1.3.3.cmml">σ</mi></mrow></munder><mo id="Pt0.A4.E8.m2.5.5.3.3.2a" xref="Pt0.A4.E8.m2.5.5.3.3.3.cmml">⁡</mo><mrow id="Pt0.A4.E8.m2.5.5.3.3.2.2" xref="Pt0.A4.E8.m2.5.5.3.3.3.cmml"><mo id="Pt0.A4.E8.m2.5.5.3.3.2.2.2" xref="Pt0.A4.E8.m2.5.5.3.3.3.cmml">(</mo><mrow id="Pt0.A4.E8.m2.5.5.3.3.2.2.1" xref="Pt0.A4.E8.m2.5.5.3.3.2.2.1.cmml"><msub id="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1" xref="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.cmml"><mrow id="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1" xref="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.2.cmml"><mo fence="true" id="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1.2" lspace="0em" rspace="0em" stretchy="true" xref="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.2.1.cmml">∥</mo><mrow id="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1.1" xref="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1.1.cmml"><msub id="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1.1.2" xref="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1.1.2.cmml"><mi id="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1.1.2.2" xref="Pt0.A4.E8.m2.5.5.3.3.2.2.1.1.1.1.1.2.2.cmml">c</mi><mi 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c_{m}(p^{m}_{i^{\prime}})\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.E8.m2.5d">∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ [ italic_m ] end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 end_POSTSUBSCRIPT ) ) roman_Pr start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( ∥ italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≥ italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> <tbody id="Pt0.A4.Ex77"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="Pt0.A4.Ex77.m1.1"><semantics id="Pt0.A4.Ex77.m1.1a"><mo id="Pt0.A4.Ex77.m1.1.1" xref="Pt0.A4.Ex77.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex77.m1.1b"><eq id="Pt0.A4.Ex77.m1.1.1.cmml" xref="Pt0.A4.Ex77.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex77.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" 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id="Pt0.A4.Ex77.m2.3d">∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_m end_POSTSUBSCRIPT ( | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 end_POSTSUBSCRIPT | ) ( 1 - ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) ) +</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex78"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td 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id="Pt0.A4.Ex78.m1.3d">( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | + italic_α - | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | ) ( 1 - ∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( 1 - italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A4.SS3.p2.4">So <math alttext="F_{n}(\sigma,\alpha)=\max_{i\in[m]}f_{i}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.SS3.p2.4.m1.5"><semantics id="Pt0.A4.SS3.p2.4.m1.5a"><mrow id="Pt0.A4.SS3.p2.4.m1.5.6" xref="Pt0.A4.SS3.p2.4.m1.5.6.cmml"><mrow id="Pt0.A4.SS3.p2.4.m1.5.6.2" xref="Pt0.A4.SS3.p2.4.m1.5.6.2.cmml"><msub id="Pt0.A4.SS3.p2.4.m1.5.6.2.2" xref="Pt0.A4.SS3.p2.4.m1.5.6.2.2.cmml"><mi id="Pt0.A4.SS3.p2.4.m1.5.6.2.2.2" xref="Pt0.A4.SS3.p2.4.m1.5.6.2.2.2.cmml">F</mi><mi 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italic_σ , italic_α )</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="Pt0.A4.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">0.D.4 </span>Omitted proofs</h3> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx17"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma12" title="Lemma 12 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">12</span></a>)</h6> <div class="ltx_para" id="Thmproofx17.p1"> <p class="ltx_p" id="Thmproofx17.p1.18">For the lower bound, take <math alttext="\sigma\in(\Delta(P_{st}))^{n}" class="ltx_Math" display="inline" id="Thmproofx17.p1.1.m1.1"><semantics id="Thmproofx17.p1.1.m1.1a"><mrow id="Thmproofx17.p1.1.m1.1.1" xref="Thmproofx17.p1.1.m1.1.1.cmml"><mi id="Thmproofx17.p1.1.m1.1.1.3" xref="Thmproofx17.p1.1.m1.1.1.3.cmml">σ</mi><mo id="Thmproofx17.p1.1.m1.1.1.2" xref="Thmproofx17.p1.1.m1.1.1.2.cmml">∈</mo><msup id="Thmproofx17.p1.1.m1.1.1.1" xref="Thmproofx17.p1.1.m1.1.1.1.cmml"><mrow id="Thmproofx17.p1.1.m1.1.1.1.1.1" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmproofx17.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx17.p1.1.m1.1.1.1.1.1.1" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.3" mathvariant="normal" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.2" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx17.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx17.p1.1.m1.1.1.1.3" xref="Thmproofx17.p1.1.m1.1.1.1.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.1.m1.1b"><apply id="Thmproofx17.p1.1.m1.1.1.cmml" xref="Thmproofx17.p1.1.m1.1.1"><in id="Thmproofx17.p1.1.m1.1.1.2.cmml" xref="Thmproofx17.p1.1.m1.1.1.2"></in><ci id="Thmproofx17.p1.1.m1.1.1.3.cmml" xref="Thmproofx17.p1.1.m1.1.1.3">𝜎</ci><apply id="Thmproofx17.p1.1.m1.1.1.1.cmml" xref="Thmproofx17.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.1.m1.1.1.1.2.cmml" xref="Thmproofx17.p1.1.m1.1.1.1">superscript</csymbol><apply id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1"><times id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.2"></times><ci id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.3">Δ</ci><apply id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3"><times id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx17.p1.1.m1.1.1.1.3.cmml" xref="Thmproofx17.p1.1.m1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.1.m1.1c">\sigma\in(\Delta(P_{st}))^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.1.m1.1d">italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. The interceptors can respond by having all pick an edge <math alttext="e\in\operatorname*{arg\,max}_{e^{\prime}}\sigma^{E}(e^{\prime})" class="ltx_Math" display="inline" id="Thmproofx17.p1.2.m2.1"><semantics id="Thmproofx17.p1.2.m2.1a"><mrow id="Thmproofx17.p1.2.m2.1.1" xref="Thmproofx17.p1.2.m2.1.1.cmml"><mi id="Thmproofx17.p1.2.m2.1.1.3" xref="Thmproofx17.p1.2.m2.1.1.3.cmml">e</mi><mo id="Thmproofx17.p1.2.m2.1.1.2" xref="Thmproofx17.p1.2.m2.1.1.2.cmml">∈</mo><mrow id="Thmproofx17.p1.2.m2.1.1.1" xref="Thmproofx17.p1.2.m2.1.1.1.cmml"><mrow id="Thmproofx17.p1.2.m2.1.1.1.3" xref="Thmproofx17.p1.2.m2.1.1.1.3.cmml"><msub id="Thmproofx17.p1.2.m2.1.1.1.3.1" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.cmml"><mrow id="Thmproofx17.p1.2.m2.1.1.1.3.1.2" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2.cmml"><mi id="Thmproofx17.p1.2.m2.1.1.1.3.1.2.2" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2.2.cmml">arg</mi><mo id="Thmproofx17.p1.2.m2.1.1.1.3.1.2.1" lspace="0.170em" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2.1.cmml">⁢</mo><mi id="Thmproofx17.p1.2.m2.1.1.1.3.1.2.3" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2.3.cmml">max</mi></mrow><msup id="Thmproofx17.p1.2.m2.1.1.1.3.1.3" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.3.cmml"><mi id="Thmproofx17.p1.2.m2.1.1.1.3.1.3.2" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.3.2.cmml">e</mi><mo id="Thmproofx17.p1.2.m2.1.1.1.3.1.3.3" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.3.3.cmml">′</mo></msup></msub><mo id="Thmproofx17.p1.2.m2.1.1.1.3a" xref="Thmproofx17.p1.2.m2.1.1.1.3.cmml">⁡</mo><msup id="Thmproofx17.p1.2.m2.1.1.1.3.2" xref="Thmproofx17.p1.2.m2.1.1.1.3.2.cmml"><mi id="Thmproofx17.p1.2.m2.1.1.1.3.2.2" xref="Thmproofx17.p1.2.m2.1.1.1.3.2.2.cmml">σ</mi><mi id="Thmproofx17.p1.2.m2.1.1.1.3.2.3" xref="Thmproofx17.p1.2.m2.1.1.1.3.2.3.cmml">E</mi></msup></mrow><mo id="Thmproofx17.p1.2.m2.1.1.1.2" xref="Thmproofx17.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx17.p1.2.m2.1.1.1.1.1" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="Thmproofx17.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="Thmproofx17.p1.2.m2.1.1.1.1.1.1" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="Thmproofx17.p1.2.m2.1.1.1.1.1.1.2" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.2.cmml">e</mi><mo id="Thmproofx17.p1.2.m2.1.1.1.1.1.1.3" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Thmproofx17.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.2.m2.1b"><apply id="Thmproofx17.p1.2.m2.1.1.cmml" xref="Thmproofx17.p1.2.m2.1.1"><in id="Thmproofx17.p1.2.m2.1.1.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.2"></in><ci id="Thmproofx17.p1.2.m2.1.1.3.cmml" xref="Thmproofx17.p1.2.m2.1.1.3">𝑒</ci><apply id="Thmproofx17.p1.2.m2.1.1.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1"><times id="Thmproofx17.p1.2.m2.1.1.1.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.2"></times><apply id="Thmproofx17.p1.2.m2.1.1.1.3.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3"><apply id="Thmproofx17.p1.2.m2.1.1.1.3.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.2.m2.1.1.1.3.1.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1">subscript</csymbol><apply id="Thmproofx17.p1.2.m2.1.1.1.3.1.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2"><times id="Thmproofx17.p1.2.m2.1.1.1.3.1.2.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2.1"></times><ci id="Thmproofx17.p1.2.m2.1.1.1.3.1.2.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2.2">arg</ci><ci id="Thmproofx17.p1.2.m2.1.1.1.3.1.2.3.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.2.3">max</ci></apply><apply id="Thmproofx17.p1.2.m2.1.1.1.3.1.3.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.3"><csymbol cd="ambiguous" id="Thmproofx17.p1.2.m2.1.1.1.3.1.3.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.3">superscript</csymbol><ci id="Thmproofx17.p1.2.m2.1.1.1.3.1.3.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.3.2">𝑒</ci><ci id="Thmproofx17.p1.2.m2.1.1.1.3.1.3.3.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.1.3.3">′</ci></apply></apply><apply id="Thmproofx17.p1.2.m2.1.1.1.3.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.2"><csymbol cd="ambiguous" id="Thmproofx17.p1.2.m2.1.1.1.3.2.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.2">superscript</csymbol><ci id="Thmproofx17.p1.2.m2.1.1.1.3.2.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.2.2">𝜎</ci><ci id="Thmproofx17.p1.2.m2.1.1.1.3.2.3.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.3.2.3">𝐸</ci></apply></apply><apply id="Thmproofx17.p1.2.m2.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx17.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.2">𝑒</ci><ci id="Thmproofx17.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="Thmproofx17.p1.2.m2.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.2.m2.1c">e\in\operatorname*{arg\,max}_{e^{\prime}}\sigma^{E}(e^{\prime})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.2.m2.1d">italic_e ∈ start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_E end_POSTSUPERSCRIPT ( italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> (the same <math alttext="e" class="ltx_Math" display="inline" id="Thmproofx17.p1.3.m3.1"><semantics id="Thmproofx17.p1.3.m3.1a"><mi id="Thmproofx17.p1.3.m3.1.1" xref="Thmproofx17.p1.3.m3.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.3.m3.1b"><ci id="Thmproofx17.p1.3.m3.1.1.cmml" xref="Thmproofx17.p1.3.m3.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.3.m3.1c">e</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.3.m3.1d">italic_e</annotation></semantics></math>). Since <math alttext="k\alpha>\left|p_{max}\right|-\left|p_{min}\right|" class="ltx_Math" display="inline" id="Thmproofx17.p1.4.m4.2"><semantics id="Thmproofx17.p1.4.m4.2a"><mrow id="Thmproofx17.p1.4.m4.2.2" xref="Thmproofx17.p1.4.m4.2.2.cmml"><mrow id="Thmproofx17.p1.4.m4.2.2.4" xref="Thmproofx17.p1.4.m4.2.2.4.cmml"><mi id="Thmproofx17.p1.4.m4.2.2.4.2" xref="Thmproofx17.p1.4.m4.2.2.4.2.cmml">k</mi><mo id="Thmproofx17.p1.4.m4.2.2.4.1" xref="Thmproofx17.p1.4.m4.2.2.4.1.cmml">⁢</mo><mi id="Thmproofx17.p1.4.m4.2.2.4.3" xref="Thmproofx17.p1.4.m4.2.2.4.3.cmml">α</mi></mrow><mo id="Thmproofx17.p1.4.m4.2.2.3" xref="Thmproofx17.p1.4.m4.2.2.3.cmml">></mo><mrow id="Thmproofx17.p1.4.m4.2.2.2" xref="Thmproofx17.p1.4.m4.2.2.2.cmml"><mrow id="Thmproofx17.p1.4.m4.1.1.1.1.1" xref="Thmproofx17.p1.4.m4.1.1.1.1.2.cmml"><mo id="Thmproofx17.p1.4.m4.1.1.1.1.1.2" xref="Thmproofx17.p1.4.m4.1.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx17.p1.4.m4.1.1.1.1.1.1" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.2" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.2" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.1" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.3" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.3.cmml">a</mi><mo id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.1a" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.4" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="Thmproofx17.p1.4.m4.1.1.1.1.1.3" xref="Thmproofx17.p1.4.m4.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx17.p1.4.m4.2.2.2.3" xref="Thmproofx17.p1.4.m4.2.2.2.3.cmml">−</mo><mrow id="Thmproofx17.p1.4.m4.2.2.2.2.1" xref="Thmproofx17.p1.4.m4.2.2.2.2.2.cmml"><mo id="Thmproofx17.p1.4.m4.2.2.2.2.1.2" xref="Thmproofx17.p1.4.m4.2.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx17.p1.4.m4.2.2.2.2.1.1" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.cmml"><mi id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.2" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.cmml"><mi id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.2" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.1" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.3" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.3.cmml">i</mi><mo id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.1a" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.4" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx17.p1.4.m4.2.2.2.2.1.3" xref="Thmproofx17.p1.4.m4.2.2.2.2.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.4.m4.2b"><apply id="Thmproofx17.p1.4.m4.2.2.cmml" xref="Thmproofx17.p1.4.m4.2.2"><gt id="Thmproofx17.p1.4.m4.2.2.3.cmml" xref="Thmproofx17.p1.4.m4.2.2.3"></gt><apply id="Thmproofx17.p1.4.m4.2.2.4.cmml" xref="Thmproofx17.p1.4.m4.2.2.4"><times id="Thmproofx17.p1.4.m4.2.2.4.1.cmml" xref="Thmproofx17.p1.4.m4.2.2.4.1"></times><ci id="Thmproofx17.p1.4.m4.2.2.4.2.cmml" xref="Thmproofx17.p1.4.m4.2.2.4.2">𝑘</ci><ci id="Thmproofx17.p1.4.m4.2.2.4.3.cmml" xref="Thmproofx17.p1.4.m4.2.2.4.3">𝛼</ci></apply><apply id="Thmproofx17.p1.4.m4.2.2.2.cmml" xref="Thmproofx17.p1.4.m4.2.2.2"><minus id="Thmproofx17.p1.4.m4.2.2.2.3.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.3"></minus><apply id="Thmproofx17.p1.4.m4.1.1.1.1.2.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1"><abs id="Thmproofx17.p1.4.m4.1.1.1.1.2.1.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.2"></abs><apply id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.2">𝑝</ci><apply id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3"><times id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.1"></times><ci id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.3">𝑎</ci><ci id="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.4.cmml" xref="Thmproofx17.p1.4.m4.1.1.1.1.1.1.3.4">𝑥</ci></apply></apply></apply><apply id="Thmproofx17.p1.4.m4.2.2.2.2.2.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1"><abs id="Thmproofx17.p1.4.m4.2.2.2.2.2.1.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.2"></abs><apply id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.1.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.2.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.2">𝑝</ci><apply id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3"><times id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.1.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.1"></times><ci id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.2.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.3.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.3">𝑖</ci><ci id="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.4.cmml" xref="Thmproofx17.p1.4.m4.2.2.2.2.1.1.3.4">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.4.m4.2c">k\alpha>\left|p_{max}\right|-\left|p_{min}\right|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.4.m4.2d">italic_k italic_α > | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT |</annotation></semantics></math>, with probability <math alttext="\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx17.p1.5.m5.1"><semantics id="Thmproofx17.p1.5.m5.1a"><mrow id="Thmproofx17.p1.5.m5.1.2" xref="Thmproofx17.p1.5.m5.1.2.cmml"><mi id="Thmproofx17.p1.5.m5.1.2.2" xref="Thmproofx17.p1.5.m5.1.2.2.cmml">β</mi><mo id="Thmproofx17.p1.5.m5.1.2.1" xref="Thmproofx17.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="Thmproofx17.p1.5.m5.1.2.3.2" xref="Thmproofx17.p1.5.m5.1.2.cmml"><mo id="Thmproofx17.p1.5.m5.1.2.3.2.1" stretchy="false" xref="Thmproofx17.p1.5.m5.1.2.cmml">(</mo><mi id="Thmproofx17.p1.5.m5.1.1" xref="Thmproofx17.p1.5.m5.1.1.cmml">σ</mi><mo id="Thmproofx17.p1.5.m5.1.2.3.2.2" stretchy="false" xref="Thmproofx17.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.5.m5.1b"><apply id="Thmproofx17.p1.5.m5.1.2.cmml" xref="Thmproofx17.p1.5.m5.1.2"><times id="Thmproofx17.p1.5.m5.1.2.1.cmml" xref="Thmproofx17.p1.5.m5.1.2.1"></times><ci id="Thmproofx17.p1.5.m5.1.2.2.cmml" xref="Thmproofx17.p1.5.m5.1.2.2">𝛽</ci><ci id="Thmproofx17.p1.5.m5.1.1.cmml" xref="Thmproofx17.p1.5.m5.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.5.m5.1c">\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.5.m5.1d">italic_β ( italic_σ )</annotation></semantics></math> over <math alttext="a\sim\sigma" class="ltx_Math" display="inline" id="Thmproofx17.p1.6.m6.1"><semantics id="Thmproofx17.p1.6.m6.1a"><mrow id="Thmproofx17.p1.6.m6.1.1" xref="Thmproofx17.p1.6.m6.1.1.cmml"><mi id="Thmproofx17.p1.6.m6.1.1.2" xref="Thmproofx17.p1.6.m6.1.1.2.cmml">a</mi><mo id="Thmproofx17.p1.6.m6.1.1.1" xref="Thmproofx17.p1.6.m6.1.1.1.cmml">∼</mo><mi id="Thmproofx17.p1.6.m6.1.1.3" xref="Thmproofx17.p1.6.m6.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.6.m6.1b"><apply id="Thmproofx17.p1.6.m6.1.1.cmml" xref="Thmproofx17.p1.6.m6.1.1"><csymbol cd="latexml" id="Thmproofx17.p1.6.m6.1.1.1.cmml" xref="Thmproofx17.p1.6.m6.1.1.1">similar-to</csymbol><ci id="Thmproofx17.p1.6.m6.1.1.2.cmml" xref="Thmproofx17.p1.6.m6.1.1.2">𝑎</ci><ci id="Thmproofx17.p1.6.m6.1.1.3.cmml" xref="Thmproofx17.p1.6.m6.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.6.m6.1c">a\sim\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.6.m6.1d">italic_a ∼ italic_σ</annotation></semantics></math> the cost <math alttext="C(a,e^{k},\alpha)" class="ltx_Math" display="inline" id="Thmproofx17.p1.7.m7.3"><semantics id="Thmproofx17.p1.7.m7.3a"><mrow id="Thmproofx17.p1.7.m7.3.3" xref="Thmproofx17.p1.7.m7.3.3.cmml"><mi id="Thmproofx17.p1.7.m7.3.3.3" xref="Thmproofx17.p1.7.m7.3.3.3.cmml">C</mi><mo id="Thmproofx17.p1.7.m7.3.3.2" xref="Thmproofx17.p1.7.m7.3.3.2.cmml">⁢</mo><mrow id="Thmproofx17.p1.7.m7.3.3.1.1" xref="Thmproofx17.p1.7.m7.3.3.1.2.cmml"><mo id="Thmproofx17.p1.7.m7.3.3.1.1.2" stretchy="false" xref="Thmproofx17.p1.7.m7.3.3.1.2.cmml">(</mo><mi id="Thmproofx17.p1.7.m7.1.1" xref="Thmproofx17.p1.7.m7.1.1.cmml">a</mi><mo id="Thmproofx17.p1.7.m7.3.3.1.1.3" xref="Thmproofx17.p1.7.m7.3.3.1.2.cmml">,</mo><msup id="Thmproofx17.p1.7.m7.3.3.1.1.1" xref="Thmproofx17.p1.7.m7.3.3.1.1.1.cmml"><mi id="Thmproofx17.p1.7.m7.3.3.1.1.1.2" xref="Thmproofx17.p1.7.m7.3.3.1.1.1.2.cmml">e</mi><mi id="Thmproofx17.p1.7.m7.3.3.1.1.1.3" xref="Thmproofx17.p1.7.m7.3.3.1.1.1.3.cmml">k</mi></msup><mo id="Thmproofx17.p1.7.m7.3.3.1.1.4" xref="Thmproofx17.p1.7.m7.3.3.1.2.cmml">,</mo><mi id="Thmproofx17.p1.7.m7.2.2" xref="Thmproofx17.p1.7.m7.2.2.cmml">α</mi><mo id="Thmproofx17.p1.7.m7.3.3.1.1.5" stretchy="false" xref="Thmproofx17.p1.7.m7.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.7.m7.3b"><apply id="Thmproofx17.p1.7.m7.3.3.cmml" xref="Thmproofx17.p1.7.m7.3.3"><times id="Thmproofx17.p1.7.m7.3.3.2.cmml" xref="Thmproofx17.p1.7.m7.3.3.2"></times><ci id="Thmproofx17.p1.7.m7.3.3.3.cmml" xref="Thmproofx17.p1.7.m7.3.3.3">𝐶</ci><vector id="Thmproofx17.p1.7.m7.3.3.1.2.cmml" xref="Thmproofx17.p1.7.m7.3.3.1.1"><ci id="Thmproofx17.p1.7.m7.1.1.cmml" xref="Thmproofx17.p1.7.m7.1.1">𝑎</ci><apply id="Thmproofx17.p1.7.m7.3.3.1.1.1.cmml" xref="Thmproofx17.p1.7.m7.3.3.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.7.m7.3.3.1.1.1.1.cmml" xref="Thmproofx17.p1.7.m7.3.3.1.1.1">superscript</csymbol><ci id="Thmproofx17.p1.7.m7.3.3.1.1.1.2.cmml" xref="Thmproofx17.p1.7.m7.3.3.1.1.1.2">𝑒</ci><ci id="Thmproofx17.p1.7.m7.3.3.1.1.1.3.cmml" xref="Thmproofx17.p1.7.m7.3.3.1.1.1.3">𝑘</ci></apply><ci id="Thmproofx17.p1.7.m7.2.2.cmml" xref="Thmproofx17.p1.7.m7.2.2">𝛼</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.7.m7.3c">C(a,e^{k},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.7.m7.3d">italic_C ( italic_a , italic_e start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math> will have the form <math alttext="|p|+k\alpha" class="ltx_Math" display="inline" id="Thmproofx17.p1.8.m8.1"><semantics id="Thmproofx17.p1.8.m8.1a"><mrow id="Thmproofx17.p1.8.m8.1.2" xref="Thmproofx17.p1.8.m8.1.2.cmml"><mrow id="Thmproofx17.p1.8.m8.1.2.2.2" xref="Thmproofx17.p1.8.m8.1.2.2.1.cmml"><mo id="Thmproofx17.p1.8.m8.1.2.2.2.1" stretchy="false" xref="Thmproofx17.p1.8.m8.1.2.2.1.1.cmml">|</mo><mi id="Thmproofx17.p1.8.m8.1.1" xref="Thmproofx17.p1.8.m8.1.1.cmml">p</mi><mo id="Thmproofx17.p1.8.m8.1.2.2.2.2" stretchy="false" xref="Thmproofx17.p1.8.m8.1.2.2.1.1.cmml">|</mo></mrow><mo id="Thmproofx17.p1.8.m8.1.2.1" xref="Thmproofx17.p1.8.m8.1.2.1.cmml">+</mo><mrow id="Thmproofx17.p1.8.m8.1.2.3" xref="Thmproofx17.p1.8.m8.1.2.3.cmml"><mi id="Thmproofx17.p1.8.m8.1.2.3.2" xref="Thmproofx17.p1.8.m8.1.2.3.2.cmml">k</mi><mo id="Thmproofx17.p1.8.m8.1.2.3.1" xref="Thmproofx17.p1.8.m8.1.2.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.8.m8.1.2.3.3" xref="Thmproofx17.p1.8.m8.1.2.3.3.cmml">α</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.8.m8.1b"><apply id="Thmproofx17.p1.8.m8.1.2.cmml" xref="Thmproofx17.p1.8.m8.1.2"><plus id="Thmproofx17.p1.8.m8.1.2.1.cmml" xref="Thmproofx17.p1.8.m8.1.2.1"></plus><apply id="Thmproofx17.p1.8.m8.1.2.2.1.cmml" xref="Thmproofx17.p1.8.m8.1.2.2.2"><abs id="Thmproofx17.p1.8.m8.1.2.2.1.1.cmml" xref="Thmproofx17.p1.8.m8.1.2.2.2.1"></abs><ci id="Thmproofx17.p1.8.m8.1.1.cmml" xref="Thmproofx17.p1.8.m8.1.1">𝑝</ci></apply><apply id="Thmproofx17.p1.8.m8.1.2.3.cmml" xref="Thmproofx17.p1.8.m8.1.2.3"><times id="Thmproofx17.p1.8.m8.1.2.3.1.cmml" xref="Thmproofx17.p1.8.m8.1.2.3.1"></times><ci id="Thmproofx17.p1.8.m8.1.2.3.2.cmml" xref="Thmproofx17.p1.8.m8.1.2.3.2">𝑘</ci><ci id="Thmproofx17.p1.8.m8.1.2.3.3.cmml" xref="Thmproofx17.p1.8.m8.1.2.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.8.m8.1c">|p|+k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.8.m8.1d">| italic_p | + italic_k italic_α</annotation></semantics></math> for <math alttext="p\ni e" class="ltx_Math" display="inline" id="Thmproofx17.p1.9.m9.1"><semantics id="Thmproofx17.p1.9.m9.1a"><mrow id="Thmproofx17.p1.9.m9.1.1" xref="Thmproofx17.p1.9.m9.1.1.cmml"><mi id="Thmproofx17.p1.9.m9.1.1.2" xref="Thmproofx17.p1.9.m9.1.1.2.cmml">p</mi><mo id="Thmproofx17.p1.9.m9.1.1.1" xref="Thmproofx17.p1.9.m9.1.1.cmml">∋</mo><mi id="Thmproofx17.p1.9.m9.1.1.3" xref="Thmproofx17.p1.9.m9.1.1.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.9.m9.1b"><apply id="Thmproofx17.p1.9.m9.1.1.cmml" xref="Thmproofx17.p1.9.m9.1.1"><in id="Thmproofx17.p1.9.m9.1.1a.cmml" xref="Thmproofx17.p1.9.m9.1.1"></in><ci id="Thmproofx17.p1.9.m9.1.1.3.cmml" xref="Thmproofx17.p1.9.m9.1.1.3">𝑒</ci><ci id="Thmproofx17.p1.9.m9.1.1.2.cmml" xref="Thmproofx17.p1.9.m9.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.9.m9.1c">p\ni e</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.9.m9.1d">italic_p ∋ italic_e</annotation></semantics></math>. So the cost is <math alttext="C(\sigma,e^{k},\alpha)\geq\operatorname*{\mathbb{E}}_{a\sim\sigma}(\left\lVert c% _{d}(a)\right\rVert_{\infty}-1_{e\in a}\cdot k\alpha)+k\alpha\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx17.p1.10.m10.7"><semantics id="Thmproofx17.p1.10.m10.7a"><mrow id="Thmproofx17.p1.10.m10.7.7" xref="Thmproofx17.p1.10.m10.7.7.cmml"><mrow id="Thmproofx17.p1.10.m10.5.5.1" xref="Thmproofx17.p1.10.m10.5.5.1.cmml"><mi id="Thmproofx17.p1.10.m10.5.5.1.3" xref="Thmproofx17.p1.10.m10.5.5.1.3.cmml">C</mi><mo id="Thmproofx17.p1.10.m10.5.5.1.2" xref="Thmproofx17.p1.10.m10.5.5.1.2.cmml">⁢</mo><mrow id="Thmproofx17.p1.10.m10.5.5.1.1.1" xref="Thmproofx17.p1.10.m10.5.5.1.1.2.cmml"><mo id="Thmproofx17.p1.10.m10.5.5.1.1.1.2" stretchy="false" xref="Thmproofx17.p1.10.m10.5.5.1.1.2.cmml">(</mo><mi id="Thmproofx17.p1.10.m10.1.1" xref="Thmproofx17.p1.10.m10.1.1.cmml">σ</mi><mo id="Thmproofx17.p1.10.m10.5.5.1.1.1.3" xref="Thmproofx17.p1.10.m10.5.5.1.1.2.cmml">,</mo><msup id="Thmproofx17.p1.10.m10.5.5.1.1.1.1" xref="Thmproofx17.p1.10.m10.5.5.1.1.1.1.cmml"><mi id="Thmproofx17.p1.10.m10.5.5.1.1.1.1.2" xref="Thmproofx17.p1.10.m10.5.5.1.1.1.1.2.cmml">e</mi><mi id="Thmproofx17.p1.10.m10.5.5.1.1.1.1.3" xref="Thmproofx17.p1.10.m10.5.5.1.1.1.1.3.cmml">k</mi></msup><mo id="Thmproofx17.p1.10.m10.5.5.1.1.1.4" xref="Thmproofx17.p1.10.m10.5.5.1.1.2.cmml">,</mo><mi id="Thmproofx17.p1.10.m10.2.2" xref="Thmproofx17.p1.10.m10.2.2.cmml">α</mi><mo id="Thmproofx17.p1.10.m10.5.5.1.1.1.5" stretchy="false" xref="Thmproofx17.p1.10.m10.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx17.p1.10.m10.7.7.4" rspace="0.1389em" xref="Thmproofx17.p1.10.m10.7.7.4.cmml">≥</mo><mrow id="Thmproofx17.p1.10.m10.7.7.3" xref="Thmproofx17.p1.10.m10.7.7.3.cmml"><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.3.cmml"><msub id="Thmproofx17.p1.10.m10.6.6.2.1.1.1" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.cmml"><mo id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.2" lspace="0.1389em" rspace="0em" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.2.cmml">𝔼</mo><mrow id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.cmml"><mi id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.2" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.2.cmml">a</mi><mo id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.1" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.1.cmml">∼</mo><mi id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.3" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.3.cmml">σ</mi></mrow></msub><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.3.cmml"><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.2" stretchy="false" xref="Thmproofx17.p1.10.m10.7.7.3.2.3.cmml">(</mo><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.cmml"><msub id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.cmml"><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.2.cmml"><mo fence="true" id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.2" lspace="0em" rspace="0em" stretchy="true" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.2.1.cmml">∥</mo><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.cmml"><msub id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.cmml"><mi id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.2.cmml">c</mi><mi id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.3" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.3.cmml">d</mi></msub><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.1" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.3.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.cmml"><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.3.2.1" stretchy="false" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.cmml">(</mo><mi id="Thmproofx17.p1.10.m10.3.3" xref="Thmproofx17.p1.10.m10.3.3.cmml">a</mi><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.3.2.2" stretchy="false" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo fence="true" id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.3" lspace="0em" rspace="0em" stretchy="true" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.2.1.cmml">∥</mo></mrow><mi id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.3" mathvariant="normal" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.3.cmml">∞</mi></msub><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.2.cmml">−</mo><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.cmml"><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.cmml"><msub id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.cmml"><mn id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.2.cmml">1</mn><mrow id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.cmml"><mi id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.2" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.2.cmml">e</mi><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.1" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.1.cmml">∈</mo><mi id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.3" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.3.cmml">a</mi></mrow></msub><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.1.cmml">⋅</mo><mi id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.3" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.3.cmml">k</mi></mrow><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.1" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.3" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.3.cmml">α</mi></mrow></mrow><mo id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.3" stretchy="false" xref="Thmproofx17.p1.10.m10.7.7.3.2.3.cmml">)</mo></mrow></mrow><mo id="Thmproofx17.p1.10.m10.7.7.3.3" xref="Thmproofx17.p1.10.m10.7.7.3.3.cmml">+</mo><mrow id="Thmproofx17.p1.10.m10.7.7.3.4" xref="Thmproofx17.p1.10.m10.7.7.3.4.cmml"><mi id="Thmproofx17.p1.10.m10.7.7.3.4.2" xref="Thmproofx17.p1.10.m10.7.7.3.4.2.cmml">k</mi><mo id="Thmproofx17.p1.10.m10.7.7.3.4.1" xref="Thmproofx17.p1.10.m10.7.7.3.4.1.cmml">⁢</mo><mi id="Thmproofx17.p1.10.m10.7.7.3.4.3" xref="Thmproofx17.p1.10.m10.7.7.3.4.3.cmml">α</mi><mo id="Thmproofx17.p1.10.m10.7.7.3.4.1a" xref="Thmproofx17.p1.10.m10.7.7.3.4.1.cmml">⁢</mo><mi id="Thmproofx17.p1.10.m10.7.7.3.4.4" xref="Thmproofx17.p1.10.m10.7.7.3.4.4.cmml">β</mi><mo id="Thmproofx17.p1.10.m10.7.7.3.4.1b" xref="Thmproofx17.p1.10.m10.7.7.3.4.1.cmml">⁢</mo><mrow id="Thmproofx17.p1.10.m10.7.7.3.4.5.2" xref="Thmproofx17.p1.10.m10.7.7.3.4.cmml"><mo id="Thmproofx17.p1.10.m10.7.7.3.4.5.2.1" stretchy="false" xref="Thmproofx17.p1.10.m10.7.7.3.4.cmml">(</mo><mi id="Thmproofx17.p1.10.m10.4.4" xref="Thmproofx17.p1.10.m10.4.4.cmml">σ</mi><mo id="Thmproofx17.p1.10.m10.7.7.3.4.5.2.2" stretchy="false" xref="Thmproofx17.p1.10.m10.7.7.3.4.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.10.m10.7b"><apply id="Thmproofx17.p1.10.m10.7.7.cmml" xref="Thmproofx17.p1.10.m10.7.7"><geq id="Thmproofx17.p1.10.m10.7.7.4.cmml" xref="Thmproofx17.p1.10.m10.7.7.4"></geq><apply id="Thmproofx17.p1.10.m10.5.5.1.cmml" xref="Thmproofx17.p1.10.m10.5.5.1"><times id="Thmproofx17.p1.10.m10.5.5.1.2.cmml" xref="Thmproofx17.p1.10.m10.5.5.1.2"></times><ci id="Thmproofx17.p1.10.m10.5.5.1.3.cmml" xref="Thmproofx17.p1.10.m10.5.5.1.3">𝐶</ci><vector id="Thmproofx17.p1.10.m10.5.5.1.1.2.cmml" xref="Thmproofx17.p1.10.m10.5.5.1.1.1"><ci id="Thmproofx17.p1.10.m10.1.1.cmml" xref="Thmproofx17.p1.10.m10.1.1">𝜎</ci><apply id="Thmproofx17.p1.10.m10.5.5.1.1.1.1.cmml" xref="Thmproofx17.p1.10.m10.5.5.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.10.m10.5.5.1.1.1.1.1.cmml" xref="Thmproofx17.p1.10.m10.5.5.1.1.1.1">superscript</csymbol><ci id="Thmproofx17.p1.10.m10.5.5.1.1.1.1.2.cmml" xref="Thmproofx17.p1.10.m10.5.5.1.1.1.1.2">𝑒</ci><ci id="Thmproofx17.p1.10.m10.5.5.1.1.1.1.3.cmml" xref="Thmproofx17.p1.10.m10.5.5.1.1.1.1.3">𝑘</ci></apply><ci id="Thmproofx17.p1.10.m10.2.2.cmml" xref="Thmproofx17.p1.10.m10.2.2">𝛼</ci></vector></apply><apply id="Thmproofx17.p1.10.m10.7.7.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3"><plus id="Thmproofx17.p1.10.m10.7.7.3.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.3"></plus><apply id="Thmproofx17.p1.10.m10.7.7.3.2.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2"><apply id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.cmml" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.1.cmml" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1">subscript</csymbol><ci id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.2.cmml" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.2">𝔼</ci><apply id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.cmml" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3"><csymbol cd="latexml" id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.1.cmml" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.1">similar-to</csymbol><ci id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.2.cmml" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.2">𝑎</ci><ci id="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.3.cmml" xref="Thmproofx17.p1.10.m10.6.6.2.1.1.1.3.3">𝜎</ci></apply></apply><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1"><minus id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.2"></minus><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1">subscript</csymbol><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1"><csymbol cd="latexml" id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.2.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.2">delimited-∥∥</csymbol><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1"><times id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.1"></times><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2">subscript</csymbol><ci id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.2">𝑐</ci><ci id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.1.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx17.p1.10.m10.3.3.cmml" xref="Thmproofx17.p1.10.m10.3.3">𝑎</ci></apply></apply><infinity id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.1.3"></infinity></apply><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3"><times id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.1"></times><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2"><ci id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.1">⋅</ci><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2"><csymbol cd="ambiguous" id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2">subscript</csymbol><cn id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.2.cmml" type="integer" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.2">1</cn><apply id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3"><in id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.1"></in><ci id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.2">𝑒</ci><ci id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.2.3.3">𝑎</ci></apply></apply><ci id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.2.3">𝑘</ci></apply><ci id="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.2.2.2.1.3.3">𝛼</ci></apply></apply></apply><apply id="Thmproofx17.p1.10.m10.7.7.3.4.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.4"><times id="Thmproofx17.p1.10.m10.7.7.3.4.1.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.4.1"></times><ci id="Thmproofx17.p1.10.m10.7.7.3.4.2.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.4.2">𝑘</ci><ci id="Thmproofx17.p1.10.m10.7.7.3.4.3.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.4.3">𝛼</ci><ci id="Thmproofx17.p1.10.m10.7.7.3.4.4.cmml" xref="Thmproofx17.p1.10.m10.7.7.3.4.4">𝛽</ci><ci id="Thmproofx17.p1.10.m10.4.4.cmml" xref="Thmproofx17.p1.10.m10.4.4">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.10.m10.7c">C(\sigma,e^{k},\alpha)\geq\operatorname*{\mathbb{E}}_{a\sim\sigma}(\left\lVert c% _{d}(a)\right\rVert_{\infty}-1_{e\in a}\cdot k\alpha)+k\alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.10.m10.7d">italic_C ( italic_σ , italic_e start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_α ) ≥ blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_σ end_POSTSUBSCRIPT ( ∥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT - 1 start_POSTSUBSCRIPT italic_e ∈ italic_a end_POSTSUBSCRIPT ⋅ italic_k italic_α ) + italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math>. Note that <math alttext="\left\lVert c_{d}(a)\right\rVert_{\infty}-1_{e\in a}\cdot k\alpha\in[\left|p_{% min}\right|,\left|p_{max}\right|]" class="ltx_Math" display="inline" id="Thmproofx17.p1.11.m11.4"><semantics id="Thmproofx17.p1.11.m11.4a"><mrow id="Thmproofx17.p1.11.m11.4.4" xref="Thmproofx17.p1.11.m11.4.4.cmml"><mrow id="Thmproofx17.p1.11.m11.2.2.1" xref="Thmproofx17.p1.11.m11.2.2.1.cmml"><msub id="Thmproofx17.p1.11.m11.2.2.1.1" xref="Thmproofx17.p1.11.m11.2.2.1.1.cmml"><mrow id="Thmproofx17.p1.11.m11.2.2.1.1.1.1" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.2.cmml"><mo fence="true" id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.2" rspace="0em" stretchy="true" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.2.1.cmml">∥</mo><mrow id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.cmml"><msub id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.cmml"><mi id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.2" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.2.cmml">c</mi><mi id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.3" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.3.cmml">d</mi></msub><mo id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.1" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.3.2" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.cmml"><mo id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.3.2.1" stretchy="false" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.cmml">(</mo><mi id="Thmproofx17.p1.11.m11.1.1" xref="Thmproofx17.p1.11.m11.1.1.cmml">a</mi><mo id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.3.2.2" stretchy="false" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo fence="true" id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.3" lspace="0em" stretchy="true" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.2.1.cmml">∥</mo></mrow><mi id="Thmproofx17.p1.11.m11.2.2.1.1.3" mathvariant="normal" xref="Thmproofx17.p1.11.m11.2.2.1.1.3.cmml">∞</mi></msub><mo id="Thmproofx17.p1.11.m11.2.2.1.2" xref="Thmproofx17.p1.11.m11.2.2.1.2.cmml">−</mo><mrow id="Thmproofx17.p1.11.m11.2.2.1.3" xref="Thmproofx17.p1.11.m11.2.2.1.3.cmml"><mrow id="Thmproofx17.p1.11.m11.2.2.1.3.2" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.cmml"><msub id="Thmproofx17.p1.11.m11.2.2.1.3.2.2" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.cmml"><mn id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.2" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.2.cmml">1</mn><mrow id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.cmml"><mi id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.2" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.2.cmml">e</mi><mo id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.1" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.1.cmml">∈</mo><mi id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.3" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.3.cmml">a</mi></mrow></msub><mo id="Thmproofx17.p1.11.m11.2.2.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.1.cmml">⋅</mo><mi id="Thmproofx17.p1.11.m11.2.2.1.3.2.3" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.3.cmml">k</mi></mrow><mo id="Thmproofx17.p1.11.m11.2.2.1.3.1" xref="Thmproofx17.p1.11.m11.2.2.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.11.m11.2.2.1.3.3" xref="Thmproofx17.p1.11.m11.2.2.1.3.3.cmml">α</mi></mrow></mrow><mo id="Thmproofx17.p1.11.m11.4.4.4" xref="Thmproofx17.p1.11.m11.4.4.4.cmml">∈</mo><mrow id="Thmproofx17.p1.11.m11.4.4.3.2" xref="Thmproofx17.p1.11.m11.4.4.3.3.cmml"><mo id="Thmproofx17.p1.11.m11.4.4.3.2.3" stretchy="false" xref="Thmproofx17.p1.11.m11.4.4.3.3.cmml">[</mo><mrow id="Thmproofx17.p1.11.m11.3.3.2.1.1.1" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.2.cmml"><mo id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.2" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.2.1.cmml">|</mo><msub id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.cmml"><mi id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.2" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.cmml"><mi id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.2" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.1" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.3" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.3.cmml">i</mi><mo id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.1a" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.4" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.3" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx17.p1.11.m11.4.4.3.2.4" xref="Thmproofx17.p1.11.m11.4.4.3.3.cmml">,</mo><mrow id="Thmproofx17.p1.11.m11.4.4.3.2.2.1" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.2.cmml"><mo id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.2" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.2.1.cmml">|</mo><msub id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.cmml"><mi id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.2" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.cmml"><mi id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.2" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.1" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.3" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.3.cmml">a</mi><mo id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.1a" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.4" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.4.cmml">x</mi></mrow></msub><mo id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.3" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.2.1.cmml">|</mo></mrow><mo id="Thmproofx17.p1.11.m11.4.4.3.2.5" stretchy="false" xref="Thmproofx17.p1.11.m11.4.4.3.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.11.m11.4b"><apply id="Thmproofx17.p1.11.m11.4.4.cmml" xref="Thmproofx17.p1.11.m11.4.4"><in id="Thmproofx17.p1.11.m11.4.4.4.cmml" xref="Thmproofx17.p1.11.m11.4.4.4"></in><apply id="Thmproofx17.p1.11.m11.2.2.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1"><minus id="Thmproofx17.p1.11.m11.2.2.1.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.2"></minus><apply id="Thmproofx17.p1.11.m11.2.2.1.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.11.m11.2.2.1.1.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1">subscript</csymbol><apply id="Thmproofx17.p1.11.m11.2.2.1.1.1.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1"><csymbol cd="latexml" id="Thmproofx17.p1.11.m11.2.2.1.1.1.2.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.2">delimited-∥∥</csymbol><apply id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1"><times id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.1"></times><apply id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2">subscript</csymbol><ci id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.2">𝑐</ci><ci id="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.3.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.1.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx17.p1.11.m11.1.1.cmml" xref="Thmproofx17.p1.11.m11.1.1">𝑎</ci></apply></apply><infinity id="Thmproofx17.p1.11.m11.2.2.1.1.3.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.1.3"></infinity></apply><apply id="Thmproofx17.p1.11.m11.2.2.1.3.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3"><times id="Thmproofx17.p1.11.m11.2.2.1.3.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.1"></times><apply id="Thmproofx17.p1.11.m11.2.2.1.3.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2"><ci id="Thmproofx17.p1.11.m11.2.2.1.3.2.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.1">⋅</ci><apply id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2"><csymbol cd="ambiguous" id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2">subscript</csymbol><cn id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.2.cmml" type="integer" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.2">1</cn><apply id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3"><in id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.1.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.1"></in><ci id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.2.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.2">𝑒</ci><ci id="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.3.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.2.3.3">𝑎</ci></apply></apply><ci id="Thmproofx17.p1.11.m11.2.2.1.3.2.3.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.2.3">𝑘</ci></apply><ci id="Thmproofx17.p1.11.m11.2.2.1.3.3.cmml" xref="Thmproofx17.p1.11.m11.2.2.1.3.3">𝛼</ci></apply></apply><interval closure="closed" id="Thmproofx17.p1.11.m11.4.4.3.3.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2"><apply id="Thmproofx17.p1.11.m11.3.3.2.1.1.2.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1"><abs id="Thmproofx17.p1.11.m11.3.3.2.1.1.2.1.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.2"></abs><apply id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.1.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1">subscript</csymbol><ci id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.2.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.2">𝑝</ci><apply id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3"><times id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.1.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.1"></times><ci id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.2.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.3.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.3">𝑖</ci><ci id="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.4.cmml" xref="Thmproofx17.p1.11.m11.3.3.2.1.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Thmproofx17.p1.11.m11.4.4.3.2.2.2.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1"><abs id="Thmproofx17.p1.11.m11.4.4.3.2.2.2.1.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.2"></abs><apply id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.1.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1">subscript</csymbol><ci id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.2.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.2">𝑝</ci><apply id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3"><times id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.1.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.1"></times><ci id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.2.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.3.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.3">𝑎</ci><ci id="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.4.cmml" xref="Thmproofx17.p1.11.m11.4.4.3.2.2.1.1.3.4">𝑥</ci></apply></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.11.m11.4c">\left\lVert c_{d}(a)\right\rVert_{\infty}-1_{e\in a}\cdot k\alpha\in[\left|p_{% min}\right|,\left|p_{max}\right|]</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.11.m11.4d">∥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT - 1 start_POSTSUBSCRIPT italic_e ∈ italic_a end_POSTSUBSCRIPT ⋅ italic_k italic_α ∈ [ | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | , | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | ]</annotation></semantics></math> for every <math alttext="a" class="ltx_Math" display="inline" id="Thmproofx17.p1.12.m12.1"><semantics id="Thmproofx17.p1.12.m12.1a"><mi id="Thmproofx17.p1.12.m12.1.1" xref="Thmproofx17.p1.12.m12.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.12.m12.1b"><ci id="Thmproofx17.p1.12.m12.1.1.cmml" xref="Thmproofx17.p1.12.m12.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.12.m12.1c">a</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.12.m12.1d">italic_a</annotation></semantics></math>, because this is the length of the longest path in <math alttext="a" class="ltx_Math" display="inline" id="Thmproofx17.p1.13.m13.1"><semantics id="Thmproofx17.p1.13.m13.1a"><mi id="Thmproofx17.p1.13.m13.1.1" xref="Thmproofx17.p1.13.m13.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.13.m13.1b"><ci id="Thmproofx17.p1.13.m13.1.1.cmml" xref="Thmproofx17.p1.13.m13.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.13.m13.1c">a</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.13.m13.1d">italic_a</annotation></semantics></math> according to <math alttext="c_{d}" class="ltx_Math" display="inline" id="Thmproofx17.p1.14.m14.1"><semantics id="Thmproofx17.p1.14.m14.1a"><msub id="Thmproofx17.p1.14.m14.1.1" xref="Thmproofx17.p1.14.m14.1.1.cmml"><mi id="Thmproofx17.p1.14.m14.1.1.2" xref="Thmproofx17.p1.14.m14.1.1.2.cmml">c</mi><mi id="Thmproofx17.p1.14.m14.1.1.3" xref="Thmproofx17.p1.14.m14.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.14.m14.1b"><apply id="Thmproofx17.p1.14.m14.1.1.cmml" xref="Thmproofx17.p1.14.m14.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.14.m14.1.1.1.cmml" xref="Thmproofx17.p1.14.m14.1.1">subscript</csymbol><ci id="Thmproofx17.p1.14.m14.1.1.2.cmml" xref="Thmproofx17.p1.14.m14.1.1.2">𝑐</ci><ci id="Thmproofx17.p1.14.m14.1.1.3.cmml" xref="Thmproofx17.p1.14.m14.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.14.m14.1c">c_{d}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.14.m14.1d">italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math>, with subtracting the additional <math alttext="k\alpha" class="ltx_Math" display="inline" id="Thmproofx17.p1.15.m15.1"><semantics id="Thmproofx17.p1.15.m15.1a"><mrow id="Thmproofx17.p1.15.m15.1.1" xref="Thmproofx17.p1.15.m15.1.1.cmml"><mi id="Thmproofx17.p1.15.m15.1.1.2" xref="Thmproofx17.p1.15.m15.1.1.2.cmml">k</mi><mo id="Thmproofx17.p1.15.m15.1.1.1" xref="Thmproofx17.p1.15.m15.1.1.1.cmml">⁢</mo><mi id="Thmproofx17.p1.15.m15.1.1.3" xref="Thmproofx17.p1.15.m15.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.15.m15.1b"><apply id="Thmproofx17.p1.15.m15.1.1.cmml" xref="Thmproofx17.p1.15.m15.1.1"><times id="Thmproofx17.p1.15.m15.1.1.1.cmml" xref="Thmproofx17.p1.15.m15.1.1.1"></times><ci id="Thmproofx17.p1.15.m15.1.1.2.cmml" xref="Thmproofx17.p1.15.m15.1.1.2">𝑘</ci><ci id="Thmproofx17.p1.15.m15.1.1.3.cmml" xref="Thmproofx17.p1.15.m15.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.15.m15.1c">k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.15.m15.1d">italic_k italic_α</annotation></semantics></math> term, if <math alttext="e" class="ltx_Math" display="inline" id="Thmproofx17.p1.16.m16.1"><semantics id="Thmproofx17.p1.16.m16.1a"><mi id="Thmproofx17.p1.16.m16.1.1" xref="Thmproofx17.p1.16.m16.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.16.m16.1b"><ci id="Thmproofx17.p1.16.m16.1.1.cmml" xref="Thmproofx17.p1.16.m16.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.16.m16.1c">e</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.16.m16.1d">italic_e</annotation></semantics></math> is contained in that path. Then, the cost to the agents is <math alttext="C(\sigma,e^{k},\alpha)\geq\left|p_{min}\right|+k\alpha\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx17.p1.17.m17.5"><semantics id="Thmproofx17.p1.17.m17.5a"><mrow id="Thmproofx17.p1.17.m17.5.5" xref="Thmproofx17.p1.17.m17.5.5.cmml"><mrow id="Thmproofx17.p1.17.m17.4.4.1" xref="Thmproofx17.p1.17.m17.4.4.1.cmml"><mi id="Thmproofx17.p1.17.m17.4.4.1.3" xref="Thmproofx17.p1.17.m17.4.4.1.3.cmml">C</mi><mo id="Thmproofx17.p1.17.m17.4.4.1.2" xref="Thmproofx17.p1.17.m17.4.4.1.2.cmml">⁢</mo><mrow id="Thmproofx17.p1.17.m17.4.4.1.1.1" xref="Thmproofx17.p1.17.m17.4.4.1.1.2.cmml"><mo id="Thmproofx17.p1.17.m17.4.4.1.1.1.2" stretchy="false" xref="Thmproofx17.p1.17.m17.4.4.1.1.2.cmml">(</mo><mi id="Thmproofx17.p1.17.m17.1.1" xref="Thmproofx17.p1.17.m17.1.1.cmml">σ</mi><mo id="Thmproofx17.p1.17.m17.4.4.1.1.1.3" xref="Thmproofx17.p1.17.m17.4.4.1.1.2.cmml">,</mo><msup id="Thmproofx17.p1.17.m17.4.4.1.1.1.1" xref="Thmproofx17.p1.17.m17.4.4.1.1.1.1.cmml"><mi id="Thmproofx17.p1.17.m17.4.4.1.1.1.1.2" xref="Thmproofx17.p1.17.m17.4.4.1.1.1.1.2.cmml">e</mi><mi id="Thmproofx17.p1.17.m17.4.4.1.1.1.1.3" xref="Thmproofx17.p1.17.m17.4.4.1.1.1.1.3.cmml">k</mi></msup><mo id="Thmproofx17.p1.17.m17.4.4.1.1.1.4" xref="Thmproofx17.p1.17.m17.4.4.1.1.2.cmml">,</mo><mi id="Thmproofx17.p1.17.m17.2.2" xref="Thmproofx17.p1.17.m17.2.2.cmml">α</mi><mo id="Thmproofx17.p1.17.m17.4.4.1.1.1.5" stretchy="false" xref="Thmproofx17.p1.17.m17.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx17.p1.17.m17.5.5.3" xref="Thmproofx17.p1.17.m17.5.5.3.cmml">≥</mo><mrow id="Thmproofx17.p1.17.m17.5.5.2" xref="Thmproofx17.p1.17.m17.5.5.2.cmml"><mrow id="Thmproofx17.p1.17.m17.5.5.2.1.1" xref="Thmproofx17.p1.17.m17.5.5.2.1.2.cmml"><mo id="Thmproofx17.p1.17.m17.5.5.2.1.1.2" xref="Thmproofx17.p1.17.m17.5.5.2.1.2.1.cmml">|</mo><msub id="Thmproofx17.p1.17.m17.5.5.2.1.1.1" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.cmml"><mi id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.2" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.cmml"><mi id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.2" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.1" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.3" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.3.cmml">i</mi><mo id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.1a" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.4" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx17.p1.17.m17.5.5.2.1.1.3" xref="Thmproofx17.p1.17.m17.5.5.2.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx17.p1.17.m17.5.5.2.2" xref="Thmproofx17.p1.17.m17.5.5.2.2.cmml">+</mo><mrow id="Thmproofx17.p1.17.m17.5.5.2.3" xref="Thmproofx17.p1.17.m17.5.5.2.3.cmml"><mi id="Thmproofx17.p1.17.m17.5.5.2.3.2" xref="Thmproofx17.p1.17.m17.5.5.2.3.2.cmml">k</mi><mo id="Thmproofx17.p1.17.m17.5.5.2.3.1" xref="Thmproofx17.p1.17.m17.5.5.2.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.17.m17.5.5.2.3.3" xref="Thmproofx17.p1.17.m17.5.5.2.3.3.cmml">α</mi><mo id="Thmproofx17.p1.17.m17.5.5.2.3.1a" xref="Thmproofx17.p1.17.m17.5.5.2.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.17.m17.5.5.2.3.4" xref="Thmproofx17.p1.17.m17.5.5.2.3.4.cmml">β</mi><mo id="Thmproofx17.p1.17.m17.5.5.2.3.1b" xref="Thmproofx17.p1.17.m17.5.5.2.3.1.cmml">⁢</mo><mrow id="Thmproofx17.p1.17.m17.5.5.2.3.5.2" xref="Thmproofx17.p1.17.m17.5.5.2.3.cmml"><mo id="Thmproofx17.p1.17.m17.5.5.2.3.5.2.1" stretchy="false" xref="Thmproofx17.p1.17.m17.5.5.2.3.cmml">(</mo><mi id="Thmproofx17.p1.17.m17.3.3" xref="Thmproofx17.p1.17.m17.3.3.cmml">σ</mi><mo id="Thmproofx17.p1.17.m17.5.5.2.3.5.2.2" stretchy="false" xref="Thmproofx17.p1.17.m17.5.5.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p1.17.m17.5b"><apply id="Thmproofx17.p1.17.m17.5.5.cmml" xref="Thmproofx17.p1.17.m17.5.5"><geq id="Thmproofx17.p1.17.m17.5.5.3.cmml" xref="Thmproofx17.p1.17.m17.5.5.3"></geq><apply id="Thmproofx17.p1.17.m17.4.4.1.cmml" xref="Thmproofx17.p1.17.m17.4.4.1"><times id="Thmproofx17.p1.17.m17.4.4.1.2.cmml" xref="Thmproofx17.p1.17.m17.4.4.1.2"></times><ci id="Thmproofx17.p1.17.m17.4.4.1.3.cmml" xref="Thmproofx17.p1.17.m17.4.4.1.3">𝐶</ci><vector id="Thmproofx17.p1.17.m17.4.4.1.1.2.cmml" xref="Thmproofx17.p1.17.m17.4.4.1.1.1"><ci id="Thmproofx17.p1.17.m17.1.1.cmml" xref="Thmproofx17.p1.17.m17.1.1">𝜎</ci><apply id="Thmproofx17.p1.17.m17.4.4.1.1.1.1.cmml" xref="Thmproofx17.p1.17.m17.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.17.m17.4.4.1.1.1.1.1.cmml" xref="Thmproofx17.p1.17.m17.4.4.1.1.1.1">superscript</csymbol><ci id="Thmproofx17.p1.17.m17.4.4.1.1.1.1.2.cmml" xref="Thmproofx17.p1.17.m17.4.4.1.1.1.1.2">𝑒</ci><ci id="Thmproofx17.p1.17.m17.4.4.1.1.1.1.3.cmml" xref="Thmproofx17.p1.17.m17.4.4.1.1.1.1.3">𝑘</ci></apply><ci id="Thmproofx17.p1.17.m17.2.2.cmml" xref="Thmproofx17.p1.17.m17.2.2">𝛼</ci></vector></apply><apply id="Thmproofx17.p1.17.m17.5.5.2.cmml" xref="Thmproofx17.p1.17.m17.5.5.2"><plus id="Thmproofx17.p1.17.m17.5.5.2.2.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.2"></plus><apply id="Thmproofx17.p1.17.m17.5.5.2.1.2.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1"><abs id="Thmproofx17.p1.17.m17.5.5.2.1.2.1.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.2"></abs><apply id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.1.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1">subscript</csymbol><ci id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.2.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.2">𝑝</ci><apply id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3"><times id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.1.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.1"></times><ci id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.2.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.3.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.3">𝑖</ci><ci id="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.4.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Thmproofx17.p1.17.m17.5.5.2.3.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.3"><times id="Thmproofx17.p1.17.m17.5.5.2.3.1.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.3.1"></times><ci id="Thmproofx17.p1.17.m17.5.5.2.3.2.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.3.2">𝑘</ci><ci id="Thmproofx17.p1.17.m17.5.5.2.3.3.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.3.3">𝛼</ci><ci id="Thmproofx17.p1.17.m17.5.5.2.3.4.cmml" xref="Thmproofx17.p1.17.m17.5.5.2.3.4">𝛽</ci><ci id="Thmproofx17.p1.17.m17.3.3.cmml" xref="Thmproofx17.p1.17.m17.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.17.m17.5c">C(\sigma,e^{k},\alpha)\geq\left|p_{min}\right|+k\alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.17.m17.5d">italic_C ( italic_σ , italic_e start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_α ) ≥ | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math>. Therefore <math alttext="F_{n}(\sigma,\alpha)\geq C(\sigma,e^{k},\alpha)\geq\left|p_{min}\right|+k% \alpha\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx17.p1.18.m18.7"><semantics id="Thmproofx17.p1.18.m18.7a"><mrow id="Thmproofx17.p1.18.m18.7.7" xref="Thmproofx17.p1.18.m18.7.7.cmml"><mrow id="Thmproofx17.p1.18.m18.7.7.4" xref="Thmproofx17.p1.18.m18.7.7.4.cmml"><msub id="Thmproofx17.p1.18.m18.7.7.4.2" xref="Thmproofx17.p1.18.m18.7.7.4.2.cmml"><mi id="Thmproofx17.p1.18.m18.7.7.4.2.2" xref="Thmproofx17.p1.18.m18.7.7.4.2.2.cmml">F</mi><mi id="Thmproofx17.p1.18.m18.7.7.4.2.3" xref="Thmproofx17.p1.18.m18.7.7.4.2.3.cmml">n</mi></msub><mo id="Thmproofx17.p1.18.m18.7.7.4.1" xref="Thmproofx17.p1.18.m18.7.7.4.1.cmml">⁢</mo><mrow id="Thmproofx17.p1.18.m18.7.7.4.3.2" xref="Thmproofx17.p1.18.m18.7.7.4.3.1.cmml"><mo id="Thmproofx17.p1.18.m18.7.7.4.3.2.1" stretchy="false" xref="Thmproofx17.p1.18.m18.7.7.4.3.1.cmml">(</mo><mi id="Thmproofx17.p1.18.m18.1.1" xref="Thmproofx17.p1.18.m18.1.1.cmml">σ</mi><mo id="Thmproofx17.p1.18.m18.7.7.4.3.2.2" xref="Thmproofx17.p1.18.m18.7.7.4.3.1.cmml">,</mo><mi id="Thmproofx17.p1.18.m18.2.2" xref="Thmproofx17.p1.18.m18.2.2.cmml">α</mi><mo id="Thmproofx17.p1.18.m18.7.7.4.3.2.3" stretchy="false" xref="Thmproofx17.p1.18.m18.7.7.4.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx17.p1.18.m18.7.7.5" xref="Thmproofx17.p1.18.m18.7.7.5.cmml">≥</mo><mrow id="Thmproofx17.p1.18.m18.6.6.1" xref="Thmproofx17.p1.18.m18.6.6.1.cmml"><mi id="Thmproofx17.p1.18.m18.6.6.1.3" xref="Thmproofx17.p1.18.m18.6.6.1.3.cmml">C</mi><mo id="Thmproofx17.p1.18.m18.6.6.1.2" xref="Thmproofx17.p1.18.m18.6.6.1.2.cmml">⁢</mo><mrow id="Thmproofx17.p1.18.m18.6.6.1.1.1" xref="Thmproofx17.p1.18.m18.6.6.1.1.2.cmml"><mo id="Thmproofx17.p1.18.m18.6.6.1.1.1.2" stretchy="false" xref="Thmproofx17.p1.18.m18.6.6.1.1.2.cmml">(</mo><mi id="Thmproofx17.p1.18.m18.3.3" xref="Thmproofx17.p1.18.m18.3.3.cmml">σ</mi><mo id="Thmproofx17.p1.18.m18.6.6.1.1.1.3" xref="Thmproofx17.p1.18.m18.6.6.1.1.2.cmml">,</mo><msup id="Thmproofx17.p1.18.m18.6.6.1.1.1.1" xref="Thmproofx17.p1.18.m18.6.6.1.1.1.1.cmml"><mi id="Thmproofx17.p1.18.m18.6.6.1.1.1.1.2" xref="Thmproofx17.p1.18.m18.6.6.1.1.1.1.2.cmml">e</mi><mi id="Thmproofx17.p1.18.m18.6.6.1.1.1.1.3" xref="Thmproofx17.p1.18.m18.6.6.1.1.1.1.3.cmml">k</mi></msup><mo id="Thmproofx17.p1.18.m18.6.6.1.1.1.4" xref="Thmproofx17.p1.18.m18.6.6.1.1.2.cmml">,</mo><mi id="Thmproofx17.p1.18.m18.4.4" xref="Thmproofx17.p1.18.m18.4.4.cmml">α</mi><mo id="Thmproofx17.p1.18.m18.6.6.1.1.1.5" stretchy="false" xref="Thmproofx17.p1.18.m18.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx17.p1.18.m18.7.7.6" xref="Thmproofx17.p1.18.m18.7.7.6.cmml">≥</mo><mrow id="Thmproofx17.p1.18.m18.7.7.2" xref="Thmproofx17.p1.18.m18.7.7.2.cmml"><mrow id="Thmproofx17.p1.18.m18.7.7.2.1.1" xref="Thmproofx17.p1.18.m18.7.7.2.1.2.cmml"><mo id="Thmproofx17.p1.18.m18.7.7.2.1.1.2" xref="Thmproofx17.p1.18.m18.7.7.2.1.2.1.cmml">|</mo><msub id="Thmproofx17.p1.18.m18.7.7.2.1.1.1" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.cmml"><mi id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.2" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.cmml"><mi id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.2" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.1" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.3" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.3.cmml">i</mi><mo id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.1a" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.4" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="Thmproofx17.p1.18.m18.7.7.2.1.1.3" xref="Thmproofx17.p1.18.m18.7.7.2.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx17.p1.18.m18.7.7.2.2" 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id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.1.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1">subscript</csymbol><ci id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.2.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.2">𝑝</ci><apply id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3"><times id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.1.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.1"></times><ci id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.2.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.3.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.3">𝑖</ci><ci id="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.4.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.1.1.1.3.4">𝑛</ci></apply></apply></apply><apply id="Thmproofx17.p1.18.m18.7.7.2.3.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.3"><times id="Thmproofx17.p1.18.m18.7.7.2.3.1.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.3.1"></times><ci id="Thmproofx17.p1.18.m18.7.7.2.3.2.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.3.2">𝑘</ci><ci id="Thmproofx17.p1.18.m18.7.7.2.3.3.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.3.3">𝛼</ci><ci id="Thmproofx17.p1.18.m18.7.7.2.3.4.cmml" xref="Thmproofx17.p1.18.m18.7.7.2.3.4">𝛽</ci><ci id="Thmproofx17.p1.18.m18.5.5.cmml" xref="Thmproofx17.p1.18.m18.5.5">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p1.18.m18.7c">F_{n}(\sigma,\alpha)\geq C(\sigma,e^{k},\alpha)\geq\left|p_{min}\right|+k% \alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p1.18.m18.7d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≥ italic_C ( italic_σ , italic_e start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_α ) ≥ | italic_p start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT | + italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx17.p2"> <p class="ltx_p" id="Thmproofx17.p2.7">For the upper bound: as MAX can be written as a norm, we can use the triangle inequality:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx47"> <tbody id="Pt0.A4.Ex79"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C(a,d,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex79.m1.3"><semantics id="Pt0.A4.Ex79.m1.3a"><mrow id="Pt0.A4.Ex79.m1.3.4" xref="Pt0.A4.Ex79.m1.3.4.cmml"><mi id="Pt0.A4.Ex79.m1.3.4.2" xref="Pt0.A4.Ex79.m1.3.4.2.cmml">C</mi><mo id="Pt0.A4.Ex79.m1.3.4.1" xref="Pt0.A4.Ex79.m1.3.4.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex79.m1.3.4.3.2" xref="Pt0.A4.Ex79.m1.3.4.3.1.cmml"><mo id="Pt0.A4.Ex79.m1.3.4.3.2.1" stretchy="false" xref="Pt0.A4.Ex79.m1.3.4.3.1.cmml">(</mo><mi id="Pt0.A4.Ex79.m1.1.1" xref="Pt0.A4.Ex79.m1.1.1.cmml">a</mi><mo id="Pt0.A4.Ex79.m1.3.4.3.2.2" xref="Pt0.A4.Ex79.m1.3.4.3.1.cmml">,</mo><mi id="Pt0.A4.Ex79.m1.2.2" xref="Pt0.A4.Ex79.m1.2.2.cmml">d</mi><mo id="Pt0.A4.Ex79.m1.3.4.3.2.3" xref="Pt0.A4.Ex79.m1.3.4.3.1.cmml">,</mo><mi id="Pt0.A4.Ex79.m1.3.3" xref="Pt0.A4.Ex79.m1.3.3.cmml">α</mi><mo id="Pt0.A4.Ex79.m1.3.4.3.2.4" stretchy="false" xref="Pt0.A4.Ex79.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex79.m1.3b"><apply id="Pt0.A4.Ex79.m1.3.4.cmml" xref="Pt0.A4.Ex79.m1.3.4"><times id="Pt0.A4.Ex79.m1.3.4.1.cmml" xref="Pt0.A4.Ex79.m1.3.4.1"></times><ci id="Pt0.A4.Ex79.m1.3.4.2.cmml" xref="Pt0.A4.Ex79.m1.3.4.2">𝐶</ci><vector id="Pt0.A4.Ex79.m1.3.4.3.1.cmml" xref="Pt0.A4.Ex79.m1.3.4.3.2"><ci id="Pt0.A4.Ex79.m1.1.1.cmml" xref="Pt0.A4.Ex79.m1.1.1">𝑎</ci><ci id="Pt0.A4.Ex79.m1.2.2.cmml" xref="Pt0.A4.Ex79.m1.2.2">𝑑</ci><ci id="Pt0.A4.Ex79.m1.3.3.cmml" xref="Pt0.A4.Ex79.m1.3.3">𝛼</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex79.m1.3c">\displaystyle C(a,d,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex79.m1.3d">italic_C ( italic_a , italic_d , italic_α )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\left\lVert c_{d}(a)\right\rVert_{\infty}\leq\left\lVert c_{0}(a% )\right\rVert_{\infty}+\left\lVert c_{d}(a)-c_{0}(a)\right\rVert_{\infty}" class="ltx_Math" display="inline" id="Pt0.A4.Ex79.m2.7"><semantics id="Pt0.A4.Ex79.m2.7a"><mrow id="Pt0.A4.Ex79.m2.7.7" xref="Pt0.A4.Ex79.m2.7.7.cmml"><mi id="Pt0.A4.Ex79.m2.7.7.5" xref="Pt0.A4.Ex79.m2.7.7.5.cmml"></mi><mo id="Pt0.A4.Ex79.m2.7.7.6" rspace="0.1389em" xref="Pt0.A4.Ex79.m2.7.7.6.cmml">=</mo><msub id="Pt0.A4.Ex79.m2.5.5.1" xref="Pt0.A4.Ex79.m2.5.5.1.cmml"><mrow id="Pt0.A4.Ex79.m2.5.5.1.1.1" xref="Pt0.A4.Ex79.m2.5.5.1.1.2.cmml"><mo fence="true" 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id="Pt0.A4.Ex79.m2.7.7.3.2.1.1.1.3.2.1.cmml" xref="Pt0.A4.Ex79.m2.7.7.3.2.1.1.1.3.2">subscript</csymbol><ci id="Pt0.A4.Ex79.m2.7.7.3.2.1.1.1.3.2.2.cmml" xref="Pt0.A4.Ex79.m2.7.7.3.2.1.1.1.3.2.2">𝑐</ci><cn id="Pt0.A4.Ex79.m2.7.7.3.2.1.1.1.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex79.m2.7.7.3.2.1.1.1.3.2.3">0</cn></apply><ci id="Pt0.A4.Ex79.m2.4.4.cmml" xref="Pt0.A4.Ex79.m2.4.4">𝑎</ci></apply></apply></apply><infinity id="Pt0.A4.Ex79.m2.7.7.3.2.3.cmml" xref="Pt0.A4.Ex79.m2.7.7.3.2.3"></infinity></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex79.m2.7c">\displaystyle=\left\lVert c_{d}(a)\right\rVert_{\infty}\leq\left\lVert c_{0}(a% )\right\rVert_{\infty}+\left\lVert c_{d}(a)-c_{0}(a)\right\rVert_{\infty}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex79.m2.7d">= ∥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≤ ∥ italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT + ∥ italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a ) - italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex80"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\left\lVert c_{0}(a)\right\rVert_{\infty}+\alpha\sum_{j\leq k% }1\left\{d_{j}\in\bigcup a\right\}=(*)" class="ltx_Math" display="inline" id="Pt0.A4.Ex80.m1.4"><semantics id="Pt0.A4.Ex80.m1.4a"><mrow id="Pt0.A4.Ex80.m1.4.4" xref="Pt0.A4.Ex80.m1.4.4.cmml"><mi id="Pt0.A4.Ex80.m1.4.4.4" xref="Pt0.A4.Ex80.m1.4.4.4.cmml"></mi><mo id="Pt0.A4.Ex80.m1.4.4.5" rspace="0.1389em" 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id="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.2.2.cmml" xref="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.2.2">𝑑</ci><ci id="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.2.3.cmml" xref="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.2.3">𝑗</ci></apply><apply id="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.3"><union id="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.3.1"></union><ci id="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.Ex80.m1.4.4.2.2.1.1.1.1.1.3.2">𝑎</ci></apply></apply></set></apply></apply></apply></apply></apply><apply id="Pt0.A4.Ex80.m1.4.4c.cmml" xref="Pt0.A4.Ex80.m1.4.4"><eq id="Pt0.A4.Ex80.m1.4.4.6.cmml" xref="Pt0.A4.Ex80.m1.4.4.6"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex80.m1.4.4.2.cmml" id="Pt0.A4.Ex80.m1.4.4d.cmml" xref="Pt0.A4.Ex80.m1.4.4"></share><times id="Pt0.A4.Ex80.m1.2.2.cmml" xref="Pt0.A4.Ex80.m1.2.2"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex80.m1.4c">\displaystyle\leq\left\lVert c_{0}(a)\right\rVert_{\infty}+\alpha\sum_{j\leq k% }1\left\{d_{j}\in\bigcup a\right\}=(*)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex80.m1.4d">≤ ∥ italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT + italic_α ∑ start_POSTSUBSCRIPT italic_j ≤ italic_k end_POSTSUBSCRIPT 1 { italic_d start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ ⋃ italic_a } = ( ∗ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx17.p2.6">The norm in <math alttext="(*)" class="ltx_Math" display="inline" id="Thmproofx17.p2.1.m1.1"><semantics id="Thmproofx17.p2.1.m1.1a"><mrow id="Thmproofx17.p2.1.m1.1.2.2"><mo id="Thmproofx17.p2.1.m1.1.2.2.1" stretchy="false">(</mo><mo id="Thmproofx17.p2.1.m1.1.1" lspace="0em" rspace="0em" xref="Thmproofx17.p2.1.m1.1.1.cmml">∗</mo><mo id="Thmproofx17.p2.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p2.1.m1.1b"><times id="Thmproofx17.p2.1.m1.1.1.cmml" xref="Thmproofx17.p2.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p2.1.m1.1c">(*)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p2.1.m1.1d">( ∗ )</annotation></semantics></math> is bounded by <math alttext="\left|p_{max}\right|" class="ltx_Math" display="inline" id="Thmproofx17.p2.2.m2.1"><semantics id="Thmproofx17.p2.2.m2.1a"><mrow id="Thmproofx17.p2.2.m2.1.1.1" xref="Thmproofx17.p2.2.m2.1.1.2.cmml"><mo id="Thmproofx17.p2.2.m2.1.1.1.2" xref="Thmproofx17.p2.2.m2.1.1.2.1.cmml">|</mo><msub id="Thmproofx17.p2.2.m2.1.1.1.1" xref="Thmproofx17.p2.2.m2.1.1.1.1.cmml"><mi id="Thmproofx17.p2.2.m2.1.1.1.1.2" xref="Thmproofx17.p2.2.m2.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p2.2.m2.1.1.1.1.3" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.cmml"><mi id="Thmproofx17.p2.2.m2.1.1.1.1.3.2" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p2.2.m2.1.1.1.1.3.1" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.2.m2.1.1.1.1.3.3" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.3.cmml">a</mi><mo id="Thmproofx17.p2.2.m2.1.1.1.1.3.1a" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.2.m2.1.1.1.1.3.4" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="Thmproofx17.p2.2.m2.1.1.1.3" xref="Thmproofx17.p2.2.m2.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p2.2.m2.1b"><apply id="Thmproofx17.p2.2.m2.1.1.2.cmml" xref="Thmproofx17.p2.2.m2.1.1.1"><abs id="Thmproofx17.p2.2.m2.1.1.2.1.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.2"></abs><apply id="Thmproofx17.p2.2.m2.1.1.1.1.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p2.2.m2.1.1.1.1.1.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1">subscript</csymbol><ci id="Thmproofx17.p2.2.m2.1.1.1.1.2.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1.2">𝑝</ci><apply id="Thmproofx17.p2.2.m2.1.1.1.1.3.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1.3"><times id="Thmproofx17.p2.2.m2.1.1.1.1.3.1.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.1"></times><ci id="Thmproofx17.p2.2.m2.1.1.1.1.3.2.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p2.2.m2.1.1.1.1.3.3.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.3">𝑎</ci><ci id="Thmproofx17.p2.2.m2.1.1.1.1.3.4.cmml" xref="Thmproofx17.p2.2.m2.1.1.1.1.3.4">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p2.2.m2.1c">\left|p_{max}\right|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p2.2.m2.1d">| italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT |</annotation></semantics></math>. In the proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma8" title="Lemma 8 ‣ 4.1 General 𝑠-𝑡 graphs ‣ 4 SUM Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">8</span></a> we show that the sum in <math alttext="(*)" class="ltx_Math" display="inline" id="Thmproofx17.p2.3.m3.1"><semantics id="Thmproofx17.p2.3.m3.1a"><mrow id="Thmproofx17.p2.3.m3.1.2.2"><mo id="Thmproofx17.p2.3.m3.1.2.2.1" stretchy="false">(</mo><mo id="Thmproofx17.p2.3.m3.1.1" lspace="0em" rspace="0em" xref="Thmproofx17.p2.3.m3.1.1.cmml">∗</mo><mo id="Thmproofx17.p2.3.m3.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p2.3.m3.1b"><times id="Thmproofx17.p2.3.m3.1.1.cmml" xref="Thmproofx17.p2.3.m3.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p2.3.m3.1c">(*)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p2.3.m3.1d">( ∗ )</annotation></semantics></math> is upper bounded by <math alttext="k\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx17.p2.4.m4.1"><semantics id="Thmproofx17.p2.4.m4.1a"><mrow id="Thmproofx17.p2.4.m4.1.2" xref="Thmproofx17.p2.4.m4.1.2.cmml"><mi id="Thmproofx17.p2.4.m4.1.2.2" xref="Thmproofx17.p2.4.m4.1.2.2.cmml">k</mi><mo id="Thmproofx17.p2.4.m4.1.2.1" xref="Thmproofx17.p2.4.m4.1.2.1.cmml">⁢</mo><mi id="Thmproofx17.p2.4.m4.1.2.3" xref="Thmproofx17.p2.4.m4.1.2.3.cmml">β</mi><mo id="Thmproofx17.p2.4.m4.1.2.1a" xref="Thmproofx17.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="Thmproofx17.p2.4.m4.1.2.4.2" xref="Thmproofx17.p2.4.m4.1.2.cmml"><mo id="Thmproofx17.p2.4.m4.1.2.4.2.1" stretchy="false" xref="Thmproofx17.p2.4.m4.1.2.cmml">(</mo><mi id="Thmproofx17.p2.4.m4.1.1" xref="Thmproofx17.p2.4.m4.1.1.cmml">σ</mi><mo id="Thmproofx17.p2.4.m4.1.2.4.2.2" stretchy="false" xref="Thmproofx17.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p2.4.m4.1b"><apply id="Thmproofx17.p2.4.m4.1.2.cmml" xref="Thmproofx17.p2.4.m4.1.2"><times id="Thmproofx17.p2.4.m4.1.2.1.cmml" xref="Thmproofx17.p2.4.m4.1.2.1"></times><ci id="Thmproofx17.p2.4.m4.1.2.2.cmml" xref="Thmproofx17.p2.4.m4.1.2.2">𝑘</ci><ci id="Thmproofx17.p2.4.m4.1.2.3.cmml" xref="Thmproofx17.p2.4.m4.1.2.3">𝛽</ci><ci id="Thmproofx17.p2.4.m4.1.1.cmml" xref="Thmproofx17.p2.4.m4.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p2.4.m4.1c">k\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p2.4.m4.1d">italic_k italic_β ( italic_σ )</annotation></semantics></math>. Therefore, <math alttext="(*)\leq\left|p_{max}\right|+k\alpha\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx17.p2.5.m5.3"><semantics id="Thmproofx17.p2.5.m5.3a"><mrow id="Thmproofx17.p2.5.m5.3.3" xref="Thmproofx17.p2.5.m5.3.3.cmml"><mrow id="Thmproofx17.p2.5.m5.3.3.3.2" xref="Thmproofx17.p2.5.m5.3.3.cmml"><mo id="Thmproofx17.p2.5.m5.3.3.3.2.1" stretchy="false" xref="Thmproofx17.p2.5.m5.3.3.cmml">(</mo><mo id="Thmproofx17.p2.5.m5.1.1" lspace="0em" rspace="0em" xref="Thmproofx17.p2.5.m5.1.1.cmml">∗</mo><mo id="Thmproofx17.p2.5.m5.3.3.3.2.2" stretchy="false" xref="Thmproofx17.p2.5.m5.3.3.cmml">)</mo></mrow><mo id="Thmproofx17.p2.5.m5.3.3.2" xref="Thmproofx17.p2.5.m5.3.3.2.cmml">≤</mo><mrow id="Thmproofx17.p2.5.m5.3.3.1" xref="Thmproofx17.p2.5.m5.3.3.1.cmml"><mrow id="Thmproofx17.p2.5.m5.3.3.1.1.1" xref="Thmproofx17.p2.5.m5.3.3.1.1.2.cmml"><mo id="Thmproofx17.p2.5.m5.3.3.1.1.1.2" xref="Thmproofx17.p2.5.m5.3.3.1.1.2.1.cmml">|</mo><msub id="Thmproofx17.p2.5.m5.3.3.1.1.1.1" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.cmml"><mi id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.2" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.cmml"><mi id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.2" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.1" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.3" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.3.cmml">a</mi><mo id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.1a" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.4" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="Thmproofx17.p2.5.m5.3.3.1.1.1.3" xref="Thmproofx17.p2.5.m5.3.3.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx17.p2.5.m5.3.3.1.2" xref="Thmproofx17.p2.5.m5.3.3.1.2.cmml">+</mo><mrow id="Thmproofx17.p2.5.m5.3.3.1.3" xref="Thmproofx17.p2.5.m5.3.3.1.3.cmml"><mi id="Thmproofx17.p2.5.m5.3.3.1.3.2" xref="Thmproofx17.p2.5.m5.3.3.1.3.2.cmml">k</mi><mo id="Thmproofx17.p2.5.m5.3.3.1.3.1" xref="Thmproofx17.p2.5.m5.3.3.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.5.m5.3.3.1.3.3" xref="Thmproofx17.p2.5.m5.3.3.1.3.3.cmml">α</mi><mo id="Thmproofx17.p2.5.m5.3.3.1.3.1a" xref="Thmproofx17.p2.5.m5.3.3.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.5.m5.3.3.1.3.4" xref="Thmproofx17.p2.5.m5.3.3.1.3.4.cmml">β</mi><mo id="Thmproofx17.p2.5.m5.3.3.1.3.1b" xref="Thmproofx17.p2.5.m5.3.3.1.3.1.cmml">⁢</mo><mrow id="Thmproofx17.p2.5.m5.3.3.1.3.5.2" xref="Thmproofx17.p2.5.m5.3.3.1.3.cmml"><mo id="Thmproofx17.p2.5.m5.3.3.1.3.5.2.1" stretchy="false" xref="Thmproofx17.p2.5.m5.3.3.1.3.cmml">(</mo><mi id="Thmproofx17.p2.5.m5.2.2" xref="Thmproofx17.p2.5.m5.2.2.cmml">σ</mi><mo id="Thmproofx17.p2.5.m5.3.3.1.3.5.2.2" stretchy="false" xref="Thmproofx17.p2.5.m5.3.3.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p2.5.m5.3b"><apply id="Thmproofx17.p2.5.m5.3.3.cmml" xref="Thmproofx17.p2.5.m5.3.3"><leq id="Thmproofx17.p2.5.m5.3.3.2.cmml" xref="Thmproofx17.p2.5.m5.3.3.2"></leq><times id="Thmproofx17.p2.5.m5.1.1.cmml" xref="Thmproofx17.p2.5.m5.1.1"></times><apply id="Thmproofx17.p2.5.m5.3.3.1.cmml" xref="Thmproofx17.p2.5.m5.3.3.1"><plus id="Thmproofx17.p2.5.m5.3.3.1.2.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.2"></plus><apply id="Thmproofx17.p2.5.m5.3.3.1.1.2.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1"><abs id="Thmproofx17.p2.5.m5.3.3.1.1.2.1.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.2"></abs><apply id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.1.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1">subscript</csymbol><ci id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.2.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.2">𝑝</ci><apply id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3"><times id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.1.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.1"></times><ci id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.2.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.3.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.3">𝑎</ci><ci id="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.4.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.1.1.1.3.4">𝑥</ci></apply></apply></apply><apply id="Thmproofx17.p2.5.m5.3.3.1.3.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.3"><times id="Thmproofx17.p2.5.m5.3.3.1.3.1.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.3.1"></times><ci id="Thmproofx17.p2.5.m5.3.3.1.3.2.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.3.2">𝑘</ci><ci id="Thmproofx17.p2.5.m5.3.3.1.3.3.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.3.3">𝛼</ci><ci id="Thmproofx17.p2.5.m5.3.3.1.3.4.cmml" xref="Thmproofx17.p2.5.m5.3.3.1.3.4">𝛽</ci><ci id="Thmproofx17.p2.5.m5.2.2.cmml" xref="Thmproofx17.p2.5.m5.2.2">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p2.5.m5.3c">(*)\leq\left|p_{max}\right|+k\alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p2.5.m5.3d">( ∗ ) ≤ | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math>. So <math alttext="F_{n}(\sigma,\alpha)\leq\left|p_{max}\right|+k\alpha\beta(\sigma)" class="ltx_Math" display="inline" id="Thmproofx17.p2.6.m6.4"><semantics id="Thmproofx17.p2.6.m6.4a"><mrow id="Thmproofx17.p2.6.m6.4.4" xref="Thmproofx17.p2.6.m6.4.4.cmml"><mrow id="Thmproofx17.p2.6.m6.4.4.3" xref="Thmproofx17.p2.6.m6.4.4.3.cmml"><msub id="Thmproofx17.p2.6.m6.4.4.3.2" xref="Thmproofx17.p2.6.m6.4.4.3.2.cmml"><mi id="Thmproofx17.p2.6.m6.4.4.3.2.2" xref="Thmproofx17.p2.6.m6.4.4.3.2.2.cmml">F</mi><mi id="Thmproofx17.p2.6.m6.4.4.3.2.3" xref="Thmproofx17.p2.6.m6.4.4.3.2.3.cmml">n</mi></msub><mo id="Thmproofx17.p2.6.m6.4.4.3.1" xref="Thmproofx17.p2.6.m6.4.4.3.1.cmml">⁢</mo><mrow id="Thmproofx17.p2.6.m6.4.4.3.3.2" xref="Thmproofx17.p2.6.m6.4.4.3.3.1.cmml"><mo id="Thmproofx17.p2.6.m6.4.4.3.3.2.1" stretchy="false" xref="Thmproofx17.p2.6.m6.4.4.3.3.1.cmml">(</mo><mi id="Thmproofx17.p2.6.m6.1.1" xref="Thmproofx17.p2.6.m6.1.1.cmml">σ</mi><mo id="Thmproofx17.p2.6.m6.4.4.3.3.2.2" xref="Thmproofx17.p2.6.m6.4.4.3.3.1.cmml">,</mo><mi id="Thmproofx17.p2.6.m6.2.2" xref="Thmproofx17.p2.6.m6.2.2.cmml">α</mi><mo id="Thmproofx17.p2.6.m6.4.4.3.3.2.3" stretchy="false" xref="Thmproofx17.p2.6.m6.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx17.p2.6.m6.4.4.2" xref="Thmproofx17.p2.6.m6.4.4.2.cmml">≤</mo><mrow id="Thmproofx17.p2.6.m6.4.4.1" xref="Thmproofx17.p2.6.m6.4.4.1.cmml"><mrow id="Thmproofx17.p2.6.m6.4.4.1.1.1" xref="Thmproofx17.p2.6.m6.4.4.1.1.2.cmml"><mo id="Thmproofx17.p2.6.m6.4.4.1.1.1.2" xref="Thmproofx17.p2.6.m6.4.4.1.1.2.1.cmml">|</mo><msub id="Thmproofx17.p2.6.m6.4.4.1.1.1.1" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.cmml"><mi id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.2" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.cmml"><mi id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.2" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.1" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.3" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.3.cmml">a</mi><mo id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.1a" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.4" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.4.cmml">x</mi></mrow></msub><mo id="Thmproofx17.p2.6.m6.4.4.1.1.1.3" xref="Thmproofx17.p2.6.m6.4.4.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx17.p2.6.m6.4.4.1.2" xref="Thmproofx17.p2.6.m6.4.4.1.2.cmml">+</mo><mrow id="Thmproofx17.p2.6.m6.4.4.1.3" xref="Thmproofx17.p2.6.m6.4.4.1.3.cmml"><mi id="Thmproofx17.p2.6.m6.4.4.1.3.2" xref="Thmproofx17.p2.6.m6.4.4.1.3.2.cmml">k</mi><mo id="Thmproofx17.p2.6.m6.4.4.1.3.1" xref="Thmproofx17.p2.6.m6.4.4.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.6.m6.4.4.1.3.3" xref="Thmproofx17.p2.6.m6.4.4.1.3.3.cmml">α</mi><mo id="Thmproofx17.p2.6.m6.4.4.1.3.1a" xref="Thmproofx17.p2.6.m6.4.4.1.3.1.cmml">⁢</mo><mi id="Thmproofx17.p2.6.m6.4.4.1.3.4" xref="Thmproofx17.p2.6.m6.4.4.1.3.4.cmml">β</mi><mo id="Thmproofx17.p2.6.m6.4.4.1.3.1b" xref="Thmproofx17.p2.6.m6.4.4.1.3.1.cmml">⁢</mo><mrow id="Thmproofx17.p2.6.m6.4.4.1.3.5.2" xref="Thmproofx17.p2.6.m6.4.4.1.3.cmml"><mo id="Thmproofx17.p2.6.m6.4.4.1.3.5.2.1" stretchy="false" xref="Thmproofx17.p2.6.m6.4.4.1.3.cmml">(</mo><mi id="Thmproofx17.p2.6.m6.3.3" xref="Thmproofx17.p2.6.m6.3.3.cmml">σ</mi><mo id="Thmproofx17.p2.6.m6.4.4.1.3.5.2.2" stretchy="false" xref="Thmproofx17.p2.6.m6.4.4.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx17.p2.6.m6.4b"><apply id="Thmproofx17.p2.6.m6.4.4.cmml" xref="Thmproofx17.p2.6.m6.4.4"><leq id="Thmproofx17.p2.6.m6.4.4.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.2"></leq><apply id="Thmproofx17.p2.6.m6.4.4.3.cmml" xref="Thmproofx17.p2.6.m6.4.4.3"><times id="Thmproofx17.p2.6.m6.4.4.3.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.3.1"></times><apply id="Thmproofx17.p2.6.m6.4.4.3.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.3.2"><csymbol cd="ambiguous" id="Thmproofx17.p2.6.m6.4.4.3.2.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.3.2">subscript</csymbol><ci id="Thmproofx17.p2.6.m6.4.4.3.2.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.3.2.2">𝐹</ci><ci id="Thmproofx17.p2.6.m6.4.4.3.2.3.cmml" xref="Thmproofx17.p2.6.m6.4.4.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx17.p2.6.m6.4.4.3.3.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.3.3.2"><ci id="Thmproofx17.p2.6.m6.1.1.cmml" xref="Thmproofx17.p2.6.m6.1.1">𝜎</ci><ci id="Thmproofx17.p2.6.m6.2.2.cmml" xref="Thmproofx17.p2.6.m6.2.2">𝛼</ci></interval></apply><apply id="Thmproofx17.p2.6.m6.4.4.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.1"><plus id="Thmproofx17.p2.6.m6.4.4.1.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.2"></plus><apply id="Thmproofx17.p2.6.m6.4.4.1.1.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1"><abs id="Thmproofx17.p2.6.m6.4.4.1.1.2.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.2"></abs><apply id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1">subscript</csymbol><ci id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.2">𝑝</ci><apply id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3"><times id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.1"></times><ci id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.2">𝑚</ci><ci id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.3.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.3">𝑎</ci><ci id="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.4.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.1.1.1.3.4">𝑥</ci></apply></apply></apply><apply id="Thmproofx17.p2.6.m6.4.4.1.3.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.3"><times id="Thmproofx17.p2.6.m6.4.4.1.3.1.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.3.1"></times><ci id="Thmproofx17.p2.6.m6.4.4.1.3.2.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.3.2">𝑘</ci><ci id="Thmproofx17.p2.6.m6.4.4.1.3.3.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.3.3">𝛼</ci><ci id="Thmproofx17.p2.6.m6.4.4.1.3.4.cmml" xref="Thmproofx17.p2.6.m6.4.4.1.3.4">𝛽</ci><ci id="Thmproofx17.p2.6.m6.3.3.cmml" xref="Thmproofx17.p2.6.m6.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx17.p2.6.m6.4c">F_{n}(\sigma,\alpha)\leq\left|p_{max}\right|+k\alpha\beta(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx17.p2.6.m6.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≤ | italic_p start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT | + italic_k italic_α italic_β ( italic_σ )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx18"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma13" title="Lemma 13 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">13</span></a>)</h6> <div class="ltx_para" id="Thmproofx18.p1"> <p class="ltx_p" id="Thmproofx18.p1.3">Let <math alttext="X(p^{d}_{i})=\left(\frac{\sum_{j\in[n]}\sigma_{j}(p^{d}_{i})}{n}\right)_{i\in[% m]}" class="ltx_Math" display="inline" id="Thmproofx18.p1.1.m1.4"><semantics id="Thmproofx18.p1.1.m1.4a"><mrow id="Thmproofx18.p1.1.m1.4.4" xref="Thmproofx18.p1.1.m1.4.4.cmml"><mrow id="Thmproofx18.p1.1.m1.4.4.1" xref="Thmproofx18.p1.1.m1.4.4.1.cmml"><mi id="Thmproofx18.p1.1.m1.4.4.1.3" xref="Thmproofx18.p1.1.m1.4.4.1.3.cmml">X</mi><mo id="Thmproofx18.p1.1.m1.4.4.1.2" xref="Thmproofx18.p1.1.m1.4.4.1.2.cmml">⁢</mo><mrow id="Thmproofx18.p1.1.m1.4.4.1.1.1" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.cmml"><mo id="Thmproofx18.p1.1.m1.4.4.1.1.1.2" stretchy="false" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.cmml">(</mo><msubsup id="Thmproofx18.p1.1.m1.4.4.1.1.1.1" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.cmml"><mi id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.2" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.2.cmml">p</mi><mi id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.3" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.3.cmml">i</mi><mi id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.3" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Thmproofx18.p1.1.m1.4.4.1.1.1.3" stretchy="false" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx18.p1.1.m1.4.4.2" xref="Thmproofx18.p1.1.m1.4.4.2.cmml">=</mo><msub id="Thmproofx18.p1.1.m1.4.4.3" xref="Thmproofx18.p1.1.m1.4.4.3.cmml"><mrow id="Thmproofx18.p1.1.m1.4.4.3.2.2" xref="Thmproofx18.p1.1.m1.2.2.cmml"><mo id="Thmproofx18.p1.1.m1.4.4.3.2.2.1" xref="Thmproofx18.p1.1.m1.2.2.cmml">(</mo><mfrac id="Thmproofx18.p1.1.m1.2.2" xref="Thmproofx18.p1.1.m1.2.2.cmml"><mrow id="Thmproofx18.p1.1.m1.2.2.2" xref="Thmproofx18.p1.1.m1.2.2.2.cmml"><mstyle displaystyle="false" id="Thmproofx18.p1.1.m1.2.2.2.3" xref="Thmproofx18.p1.1.m1.2.2.2.3.cmml"><msub id="Thmproofx18.p1.1.m1.2.2.2.3a" xref="Thmproofx18.p1.1.m1.2.2.2.3.cmml"><mo id="Thmproofx18.p1.1.m1.2.2.2.3.2" xref="Thmproofx18.p1.1.m1.2.2.2.3.2.cmml">∑</mo><mrow id="Thmproofx18.p1.1.m1.1.1.1.1.1" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.cmml"><mi id="Thmproofx18.p1.1.m1.1.1.1.1.1.3" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.3.cmml">j</mi><mo id="Thmproofx18.p1.1.m1.1.1.1.1.1.2" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.2.cmml">∈</mo><mrow id="Thmproofx18.p1.1.m1.1.1.1.1.1.4.2" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.4.1.cmml"><mo id="Thmproofx18.p1.1.m1.1.1.1.1.1.4.2.1" stretchy="false" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.4.1.1.cmml">[</mo><mi id="Thmproofx18.p1.1.m1.1.1.1.1.1.1" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.1.cmml">n</mi><mo id="Thmproofx18.p1.1.m1.1.1.1.1.1.4.2.2" stretchy="false" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub></mstyle><mrow id="Thmproofx18.p1.1.m1.2.2.2.2" xref="Thmproofx18.p1.1.m1.2.2.2.2.cmml"><msub id="Thmproofx18.p1.1.m1.2.2.2.2.3" xref="Thmproofx18.p1.1.m1.2.2.2.2.3.cmml"><mi id="Thmproofx18.p1.1.m1.2.2.2.2.3.2" xref="Thmproofx18.p1.1.m1.2.2.2.2.3.2.cmml">σ</mi><mi id="Thmproofx18.p1.1.m1.2.2.2.2.3.3" xref="Thmproofx18.p1.1.m1.2.2.2.2.3.3.cmml">j</mi></msub><mo id="Thmproofx18.p1.1.m1.2.2.2.2.2" xref="Thmproofx18.p1.1.m1.2.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx18.p1.1.m1.2.2.2.2.1.1" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.cmml"><mo id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.2" stretchy="false" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.cmml">(</mo><msubsup id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.cmml"><mi id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.2" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.2.cmml">p</mi><mi id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.3" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.3.cmml">i</mi><mi id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.3" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.3" stretchy="false" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mi id="Thmproofx18.p1.1.m1.2.2.4" xref="Thmproofx18.p1.1.m1.2.2.4.cmml">n</mi></mfrac><mo id="Thmproofx18.p1.1.m1.4.4.3.2.2.2" xref="Thmproofx18.p1.1.m1.2.2.cmml">)</mo></mrow><mrow id="Thmproofx18.p1.1.m1.3.3.1" xref="Thmproofx18.p1.1.m1.3.3.1.cmml"><mi id="Thmproofx18.p1.1.m1.3.3.1.3" xref="Thmproofx18.p1.1.m1.3.3.1.3.cmml">i</mi><mo id="Thmproofx18.p1.1.m1.3.3.1.2" xref="Thmproofx18.p1.1.m1.3.3.1.2.cmml">∈</mo><mrow id="Thmproofx18.p1.1.m1.3.3.1.4.2" xref="Thmproofx18.p1.1.m1.3.3.1.4.1.cmml"><mo id="Thmproofx18.p1.1.m1.3.3.1.4.2.1" stretchy="false" xref="Thmproofx18.p1.1.m1.3.3.1.4.1.1.cmml">[</mo><mi id="Thmproofx18.p1.1.m1.3.3.1.1" xref="Thmproofx18.p1.1.m1.3.3.1.1.cmml">m</mi><mo id="Thmproofx18.p1.1.m1.3.3.1.4.2.2" stretchy="false" xref="Thmproofx18.p1.1.m1.3.3.1.4.1.1.cmml">]</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx18.p1.1.m1.4b"><apply id="Thmproofx18.p1.1.m1.4.4.cmml" xref="Thmproofx18.p1.1.m1.4.4"><eq id="Thmproofx18.p1.1.m1.4.4.2.cmml" xref="Thmproofx18.p1.1.m1.4.4.2"></eq><apply id="Thmproofx18.p1.1.m1.4.4.1.cmml" xref="Thmproofx18.p1.1.m1.4.4.1"><times id="Thmproofx18.p1.1.m1.4.4.1.2.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.2"></times><ci id="Thmproofx18.p1.1.m1.4.4.1.3.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.3">𝑋</ci><apply id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.1.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.1.1">subscript</csymbol><apply id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.1.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.1.1">superscript</csymbol><ci id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.2.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.2">𝑝</ci><ci id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.3.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx18.p1.1.m1.4.4.1.1.1.1.3.cmml" xref="Thmproofx18.p1.1.m1.4.4.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx18.p1.1.m1.4.4.3.cmml" xref="Thmproofx18.p1.1.m1.4.4.3"><csymbol cd="ambiguous" id="Thmproofx18.p1.1.m1.4.4.3.1.cmml" xref="Thmproofx18.p1.1.m1.4.4.3">subscript</csymbol><apply id="Thmproofx18.p1.1.m1.2.2.cmml" xref="Thmproofx18.p1.1.m1.4.4.3.2.2"><divide id="Thmproofx18.p1.1.m1.2.2.3.cmml" xref="Thmproofx18.p1.1.m1.4.4.3.2.2"></divide><apply id="Thmproofx18.p1.1.m1.2.2.2.cmml" xref="Thmproofx18.p1.1.m1.2.2.2"><apply id="Thmproofx18.p1.1.m1.2.2.2.3.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx18.p1.1.m1.2.2.2.3.1.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.3">subscript</csymbol><sum id="Thmproofx18.p1.1.m1.2.2.2.3.2.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.3.2"></sum><apply id="Thmproofx18.p1.1.m1.1.1.1.1.1.cmml" xref="Thmproofx18.p1.1.m1.1.1.1.1.1"><in id="Thmproofx18.p1.1.m1.1.1.1.1.1.2.cmml" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.2"></in><ci id="Thmproofx18.p1.1.m1.1.1.1.1.1.3.cmml" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.3">𝑗</ci><apply id="Thmproofx18.p1.1.m1.1.1.1.1.1.4.1.cmml" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.4.2"><csymbol cd="latexml" id="Thmproofx18.p1.1.m1.1.1.1.1.1.4.1.1.cmml" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Thmproofx18.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmproofx18.p1.1.m1.1.1.1.1.1.1">𝑛</ci></apply></apply></apply><apply id="Thmproofx18.p1.1.m1.2.2.2.2.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2"><times id="Thmproofx18.p1.1.m1.2.2.2.2.2.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.2"></times><apply id="Thmproofx18.p1.1.m1.2.2.2.2.3.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx18.p1.1.m1.2.2.2.2.3.1.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.3">subscript</csymbol><ci id="Thmproofx18.p1.1.m1.2.2.2.2.3.2.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.3.2">𝜎</ci><ci id="Thmproofx18.p1.1.m1.2.2.2.2.3.3.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.3.3">𝑗</ci></apply><apply id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.1.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1">subscript</csymbol><apply id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.1.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1">superscript</csymbol><ci id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.2.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.2">𝑝</ci><ci id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.3.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.3.cmml" xref="Thmproofx18.p1.1.m1.2.2.2.2.1.1.1.3">𝑖</ci></apply></apply></apply><ci id="Thmproofx18.p1.1.m1.2.2.4.cmml" xref="Thmproofx18.p1.1.m1.2.2.4">𝑛</ci></apply><apply id="Thmproofx18.p1.1.m1.3.3.1.cmml" xref="Thmproofx18.p1.1.m1.3.3.1"><in id="Thmproofx18.p1.1.m1.3.3.1.2.cmml" xref="Thmproofx18.p1.1.m1.3.3.1.2"></in><ci id="Thmproofx18.p1.1.m1.3.3.1.3.cmml" xref="Thmproofx18.p1.1.m1.3.3.1.3">𝑖</ci><apply id="Thmproofx18.p1.1.m1.3.3.1.4.1.cmml" xref="Thmproofx18.p1.1.m1.3.3.1.4.2"><csymbol cd="latexml" id="Thmproofx18.p1.1.m1.3.3.1.4.1.1.cmml" xref="Thmproofx18.p1.1.m1.3.3.1.4.2.1">delimited-[]</csymbol><ci id="Thmproofx18.p1.1.m1.3.3.1.1.cmml" xref="Thmproofx18.p1.1.m1.3.3.1.1">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx18.p1.1.m1.4c">X(p^{d}_{i})=\left(\frac{\sum_{j\in[n]}\sigma_{j}(p^{d}_{i})}{n}\right)_{i\in[% m]}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx18.p1.1.m1.4d">italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = ( divide start_ARG ∑ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG italic_n end_ARG ) start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\mu=X^{n}" class="ltx_Math" display="inline" id="Thmproofx18.p1.2.m2.1"><semantics id="Thmproofx18.p1.2.m2.1a"><mrow id="Thmproofx18.p1.2.m2.1.1" xref="Thmproofx18.p1.2.m2.1.1.cmml"><mi id="Thmproofx18.p1.2.m2.1.1.2" xref="Thmproofx18.p1.2.m2.1.1.2.cmml">μ</mi><mo id="Thmproofx18.p1.2.m2.1.1.1" xref="Thmproofx18.p1.2.m2.1.1.1.cmml">=</mo><msup id="Thmproofx18.p1.2.m2.1.1.3" xref="Thmproofx18.p1.2.m2.1.1.3.cmml"><mi id="Thmproofx18.p1.2.m2.1.1.3.2" xref="Thmproofx18.p1.2.m2.1.1.3.2.cmml">X</mi><mi id="Thmproofx18.p1.2.m2.1.1.3.3" xref="Thmproofx18.p1.2.m2.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx18.p1.2.m2.1b"><apply id="Thmproofx18.p1.2.m2.1.1.cmml" xref="Thmproofx18.p1.2.m2.1.1"><eq id="Thmproofx18.p1.2.m2.1.1.1.cmml" xref="Thmproofx18.p1.2.m2.1.1.1"></eq><ci id="Thmproofx18.p1.2.m2.1.1.2.cmml" xref="Thmproofx18.p1.2.m2.1.1.2">𝜇</ci><apply id="Thmproofx18.p1.2.m2.1.1.3.cmml" xref="Thmproofx18.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx18.p1.2.m2.1.1.3.1.cmml" xref="Thmproofx18.p1.2.m2.1.1.3">superscript</csymbol><ci id="Thmproofx18.p1.2.m2.1.1.3.2.cmml" xref="Thmproofx18.p1.2.m2.1.1.3.2">𝑋</ci><ci id="Thmproofx18.p1.2.m2.1.1.3.3.cmml" xref="Thmproofx18.p1.2.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx18.p1.2.m2.1c">\mu=X^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx18.p1.2.m2.1d">italic_μ = italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Then for every <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx18.p1.3.m3.1"><semantics id="Thmproofx18.p1.3.m3.1a"><mi id="Thmproofx18.p1.3.m3.1.1" xref="Thmproofx18.p1.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx18.p1.3.m3.1b"><ci id="Thmproofx18.p1.3.m3.1.1.cmml" xref="Thmproofx18.p1.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx18.p1.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx18.p1.3.m3.1d">italic_i</annotation></semantics></math>, by the AMGM inequality:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx48"> <tbody id="Pt0.A4.Ex81"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\prod_{j\in[n]}\left(\sum_{i^{\prime}\leq i-1}\sigma_{j}(p^{d}_{i% ^{\prime}})\right)\leq\left(\frac{1}{n}\sum_{j\in[n]}\sum_{i^{\prime}\leq i-1}% \sigma_{j}(p^{d}_{i^{\prime}})\right)^{n}=\left(\sum_{i^{\prime}\leq i-1}X(p^{% d}_{i^{\prime}})\right)^{n}" class="ltx_Math" display="inline" id="Pt0.A4.Ex81.m1.5"><semantics id="Pt0.A4.Ex81.m1.5a"><mrow id="Pt0.A4.Ex81.m1.5.5" xref="Pt0.A4.Ex81.m1.5.5.cmml"><mrow id="Pt0.A4.Ex81.m1.3.3.1" xref="Pt0.A4.Ex81.m1.3.3.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex81.m1.3.3.1.2" xref="Pt0.A4.Ex81.m1.3.3.1.2.cmml"><munder id="Pt0.A4.Ex81.m1.3.3.1.2a" xref="Pt0.A4.Ex81.m1.3.3.1.2.cmml"><mo id="Pt0.A4.Ex81.m1.3.3.1.2.2" movablelimits="false" xref="Pt0.A4.Ex81.m1.3.3.1.2.2.cmml">∏</mo><mrow id="Pt0.A4.Ex81.m1.1.1.1" xref="Pt0.A4.Ex81.m1.1.1.1.cmml"><mi id="Pt0.A4.Ex81.m1.1.1.1.3" xref="Pt0.A4.Ex81.m1.1.1.1.3.cmml">j</mi><mo id="Pt0.A4.Ex81.m1.1.1.1.2" xref="Pt0.A4.Ex81.m1.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.Ex81.m1.1.1.1.4.2" xref="Pt0.A4.Ex81.m1.1.1.1.4.1.cmml"><mo id="Pt0.A4.Ex81.m1.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.Ex81.m1.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.Ex81.m1.1.1.1.1" xref="Pt0.A4.Ex81.m1.1.1.1.1.cmml">n</mi><mo id="Pt0.A4.Ex81.m1.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.Ex81.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex81.m1.3.3.1.1.1" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.cmml"><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex81.m1.3.3.1.1.1.1" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.cmml"><munder id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2a" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.2" movablelimits="false" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.2.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.2.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.1" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.1.cmml">≤</mo><mrow id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3.2.cmml">i</mi><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3.1" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3.1.cmml">−</mo><mn id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.cmml"><msub id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.3.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.3.2.cmml">σ</mi><mi id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.3.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.3.3.cmml">j</mi></msub><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.2.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.2.2.cmml">p</mi><msup id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup><mi id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex81.m1.3.3.1.1.1.3" xref="Pt0.A4.Ex81.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex81.m1.5.5.5" xref="Pt0.A4.Ex81.m1.5.5.5.cmml">≤</mo><msup id="Pt0.A4.Ex81.m1.4.4.2" xref="Pt0.A4.Ex81.m1.4.4.2.cmml"><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.cmml"><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3.cmml"><mfrac id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3a" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3.cmml"><mn id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3.2.cmml">1</mn><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.3.3.cmml">n</mi></mfrac></mstyle><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.2.cmml"><munder id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.2a" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.2.2" movablelimits="false" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex81.m1.2.2.1" xref="Pt0.A4.Ex81.m1.2.2.1.cmml"><mi id="Pt0.A4.Ex81.m1.2.2.1.3" xref="Pt0.A4.Ex81.m1.2.2.1.3.cmml">j</mi><mo id="Pt0.A4.Ex81.m1.2.2.1.2" xref="Pt0.A4.Ex81.m1.2.2.1.2.cmml">∈</mo><mrow id="Pt0.A4.Ex81.m1.2.2.1.4.2" xref="Pt0.A4.Ex81.m1.2.2.1.4.1.cmml"><mo id="Pt0.A4.Ex81.m1.2.2.1.4.2.1" stretchy="false" xref="Pt0.A4.Ex81.m1.2.2.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.Ex81.m1.2.2.1.1" xref="Pt0.A4.Ex81.m1.2.2.1.1.cmml">n</mi><mo id="Pt0.A4.Ex81.m1.2.2.1.4.2.2" stretchy="false" xref="Pt0.A4.Ex81.m1.2.2.1.4.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.cmml"><munder id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2a" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.2" movablelimits="false" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.2.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.2.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.1.cmml">≤</mo><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3.2.cmml">i</mi><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3.1.cmml">−</mo><mn id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.cmml"><msub id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.3.2.cmml">σ</mi><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.3.3.cmml">j</mi></msub><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.2.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.2.2.cmml">p</mi><msup id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup><mi id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex81.m1.4.4.2.1.1.1.1.1.1.1.1.3" stretchy="false" 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xref="Pt0.A4.Ex81.m1.5.5.3.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.Ex81.m1.5.5.3.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex81.m1.5.5.3.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply><ci id="Pt0.A4.Ex81.m1.5.5.3.3.cmml" xref="Pt0.A4.Ex81.m1.5.5.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex81.m1.5c">\displaystyle\prod_{j\in[n]}\left(\sum_{i^{\prime}\leq i-1}\sigma_{j}(p^{d}_{i% ^{\prime}})\right)\leq\left(\frac{1}{n}\sum_{j\in[n]}\sum_{i^{\prime}\leq i-1}% \sigma_{j}(p^{d}_{i^{\prime}})\right)^{n}=\left(\sum_{i^{\prime}\leq i-1}X(p^{% d}_{i^{\prime}})\right)^{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex81.m1.5d">∏ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i - 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) ≤ ( divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ∑ start_POSTSUBSCRIPT italic_j ∈ [ italic_n ] end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i - 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx18.p1.4">Therefore by Equation <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E6" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">6</span></a>, for every <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="Thmproofx18.p1.4.m1.1"><semantics id="Thmproofx18.p1.4.m1.1a"><mrow id="Thmproofx18.p1.4.m1.1.1" xref="Thmproofx18.p1.4.m1.1.1.cmml"><mi id="Thmproofx18.p1.4.m1.1.1.2" xref="Thmproofx18.p1.4.m1.1.1.2.cmml">d</mi><mo id="Thmproofx18.p1.4.m1.1.1.1" xref="Thmproofx18.p1.4.m1.1.1.1.cmml">∈</mo><msup id="Thmproofx18.p1.4.m1.1.1.3" xref="Thmproofx18.p1.4.m1.1.1.3.cmml"><mi id="Thmproofx18.p1.4.m1.1.1.3.2" xref="Thmproofx18.p1.4.m1.1.1.3.2.cmml">E</mi><mi id="Thmproofx18.p1.4.m1.1.1.3.3" xref="Thmproofx18.p1.4.m1.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx18.p1.4.m1.1b"><apply id="Thmproofx18.p1.4.m1.1.1.cmml" xref="Thmproofx18.p1.4.m1.1.1"><in id="Thmproofx18.p1.4.m1.1.1.1.cmml" xref="Thmproofx18.p1.4.m1.1.1.1"></in><ci id="Thmproofx18.p1.4.m1.1.1.2.cmml" xref="Thmproofx18.p1.4.m1.1.1.2">𝑑</ci><apply id="Thmproofx18.p1.4.m1.1.1.3.cmml" xref="Thmproofx18.p1.4.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx18.p1.4.m1.1.1.3.1.cmml" xref="Thmproofx18.p1.4.m1.1.1.3">superscript</csymbol><ci id="Thmproofx18.p1.4.m1.1.1.3.2.cmml" xref="Thmproofx18.p1.4.m1.1.1.3.2">𝐸</ci><ci id="Thmproofx18.p1.4.m1.1.1.3.3.cmml" xref="Thmproofx18.p1.4.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx18.p1.4.m1.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx18.p1.4.m1.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx49"> <tbody id="Pt0.A4.Ex82"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{d}(\sigma,\alpha)\geq\sum_{i\in[m]}(c_{d}(p^{d}_{i})-c_{d}(p^% {d}_{i-1}))\left(1-\left(\sum_{i^{\prime}\leq i-1}X(p^{d}_{i^{\prime}})\right)% ^{n}\right)\stackrel{{\scriptstyle\text{def}}}{{=}}f_{d}(\mu,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex82.m1.7"><semantics id="Pt0.A4.Ex82.m1.7a"><mrow id="Pt0.A4.Ex82.m1.7.7" xref="Pt0.A4.Ex82.m1.7.7.cmml"><mrow id="Pt0.A4.Ex82.m1.7.7.4" xref="Pt0.A4.Ex82.m1.7.7.4.cmml"><msub id="Pt0.A4.Ex82.m1.7.7.4.2" xref="Pt0.A4.Ex82.m1.7.7.4.2.cmml"><mi id="Pt0.A4.Ex82.m1.7.7.4.2.2" xref="Pt0.A4.Ex82.m1.7.7.4.2.2.cmml">f</mi><mi id="Pt0.A4.Ex82.m1.7.7.4.2.3" xref="Pt0.A4.Ex82.m1.7.7.4.2.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex82.m1.7.7.4.1" xref="Pt0.A4.Ex82.m1.7.7.4.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex82.m1.7.7.4.3.2" xref="Pt0.A4.Ex82.m1.7.7.4.3.1.cmml"><mo id="Pt0.A4.Ex82.m1.7.7.4.3.2.1" stretchy="false" xref="Pt0.A4.Ex82.m1.7.7.4.3.1.cmml">(</mo><mi id="Pt0.A4.Ex82.m1.2.2" xref="Pt0.A4.Ex82.m1.2.2.cmml">σ</mi><mo id="Pt0.A4.Ex82.m1.7.7.4.3.2.2" xref="Pt0.A4.Ex82.m1.7.7.4.3.1.cmml">,</mo><mi id="Pt0.A4.Ex82.m1.3.3" xref="Pt0.A4.Ex82.m1.3.3.cmml">α</mi><mo id="Pt0.A4.Ex82.m1.7.7.4.3.2.3" stretchy="false" xref="Pt0.A4.Ex82.m1.7.7.4.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex82.m1.7.7.5" xref="Pt0.A4.Ex82.m1.7.7.5.cmml">≥</mo><mrow id="Pt0.A4.Ex82.m1.7.7.2" xref="Pt0.A4.Ex82.m1.7.7.2.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex82.m1.7.7.2.3" xref="Pt0.A4.Ex82.m1.7.7.2.3.cmml"><munder id="Pt0.A4.Ex82.m1.7.7.2.3a" xref="Pt0.A4.Ex82.m1.7.7.2.3.cmml"><mo id="Pt0.A4.Ex82.m1.7.7.2.3.2" movablelimits="false" xref="Pt0.A4.Ex82.m1.7.7.2.3.2.cmml">∑</mo><mrow id="Pt0.A4.Ex82.m1.1.1.1" xref="Pt0.A4.Ex82.m1.1.1.1.cmml"><mi id="Pt0.A4.Ex82.m1.1.1.1.3" xref="Pt0.A4.Ex82.m1.1.1.1.3.cmml">i</mi><mo id="Pt0.A4.Ex82.m1.1.1.1.2" xref="Pt0.A4.Ex82.m1.1.1.1.2.cmml">∈</mo><mrow id="Pt0.A4.Ex82.m1.1.1.1.4.2" xref="Pt0.A4.Ex82.m1.1.1.1.4.1.cmml"><mo id="Pt0.A4.Ex82.m1.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.Ex82.m1.1.1.1.4.1.1.cmml">[</mo><mi id="Pt0.A4.Ex82.m1.1.1.1.1" xref="Pt0.A4.Ex82.m1.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.Ex82.m1.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.Ex82.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex82.m1.7.7.2.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.cmml"><mrow id="Pt0.A4.Ex82.m1.6.6.1.1.1.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.cmml"><msub id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.3.2.cmml">c</mi><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.3.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.2.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.2.2.cmml">p</mi><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.3.cmml">i</mi><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.cmml"><msub id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.3.cmml"><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.3.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.3.2.cmml">c</mi><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.3.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.3.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.cmml"><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.cmml"><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.2.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.2.2.cmml">p</mi><mrow id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3.cmml"><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3.2" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3.1" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.3.3.cmml">1</mn></mrow><mi id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.2.3" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.3" stretchy="false" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex82.m1.6.6.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex82.m1.6.6.1.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex82.m1.7.7.2.2.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.cmml"><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.cmml"><mn id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.3.cmml">1</mn><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.2.cmml">−</mo><msup id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.cmml"><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.cmml"><munder id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2a" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.2" movablelimits="false" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.2.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.2.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.1.cmml">≤</mo><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3.2.cmml">i</mi><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3.1.cmml">−</mo><mn id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.2.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.2.2.cmml">p</mi><msup id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup><mi id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo 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xref="Pt0.A4.Ex82.m1.7.7.2.2.2.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply><apply id="Pt0.A4.Ex82.m1.7.7c.cmml" xref="Pt0.A4.Ex82.m1.7.7"><apply id="Pt0.A4.Ex82.m1.7.7.6.cmml" xref="Pt0.A4.Ex82.m1.7.7.6"><csymbol cd="ambiguous" id="Pt0.A4.Ex82.m1.7.7.6.1.cmml" xref="Pt0.A4.Ex82.m1.7.7.6">superscript</csymbol><eq id="Pt0.A4.Ex82.m1.7.7.6.2.cmml" xref="Pt0.A4.Ex82.m1.7.7.6.2"></eq><ci id="Pt0.A4.Ex82.m1.7.7.6.3a.cmml" xref="Pt0.A4.Ex82.m1.7.7.6.3"><mtext id="Pt0.A4.Ex82.m1.7.7.6.3.cmml" mathsize="70%" xref="Pt0.A4.Ex82.m1.7.7.6.3">def</mtext></ci></apply><share href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex82.m1.7.7.2.cmml" id="Pt0.A4.Ex82.m1.7.7d.cmml" xref="Pt0.A4.Ex82.m1.7.7"></share><apply id="Pt0.A4.Ex82.m1.7.7.7.cmml" xref="Pt0.A4.Ex82.m1.7.7.7"><times id="Pt0.A4.Ex82.m1.7.7.7.1.cmml" xref="Pt0.A4.Ex82.m1.7.7.7.1"></times><apply id="Pt0.A4.Ex82.m1.7.7.7.2.cmml" xref="Pt0.A4.Ex82.m1.7.7.7.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex82.m1.7.7.7.2.1.cmml" xref="Pt0.A4.Ex82.m1.7.7.7.2">subscript</csymbol><ci id="Pt0.A4.Ex82.m1.7.7.7.2.2.cmml" xref="Pt0.A4.Ex82.m1.7.7.7.2.2">𝑓</ci><ci id="Pt0.A4.Ex82.m1.7.7.7.2.3.cmml" xref="Pt0.A4.Ex82.m1.7.7.7.2.3">𝑑</ci></apply><interval closure="open" id="Pt0.A4.Ex82.m1.7.7.7.3.1.cmml" xref="Pt0.A4.Ex82.m1.7.7.7.3.2"><ci id="Pt0.A4.Ex82.m1.4.4.cmml" xref="Pt0.A4.Ex82.m1.4.4">𝜇</ci><ci id="Pt0.A4.Ex82.m1.5.5.cmml" xref="Pt0.A4.Ex82.m1.5.5">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex82.m1.7c">\displaystyle f_{d}(\sigma,\alpha)\geq\sum_{i\in[m]}(c_{d}(p^{d}_{i})-c_{d}(p^% {d}_{i-1}))\left(1-\left(\sum_{i^{\prime}\leq i-1}X(p^{d}_{i^{\prime}})\right)% ^{n}\right)\stackrel{{\scriptstyle\text{def}}}{{=}}f_{d}(\mu,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex82.m1.7d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≥ ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ) ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_μ , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx18.p1.5">So <math alttext="F_{n}(\sigma,\alpha)\geq F_{n}(\mu,\alpha)" class="ltx_Math" display="inline" id="Thmproofx18.p1.5.m1.4"><semantics id="Thmproofx18.p1.5.m1.4a"><mrow id="Thmproofx18.p1.5.m1.4.5" xref="Thmproofx18.p1.5.m1.4.5.cmml"><mrow id="Thmproofx18.p1.5.m1.4.5.2" xref="Thmproofx18.p1.5.m1.4.5.2.cmml"><msub id="Thmproofx18.p1.5.m1.4.5.2.2" xref="Thmproofx18.p1.5.m1.4.5.2.2.cmml"><mi id="Thmproofx18.p1.5.m1.4.5.2.2.2" xref="Thmproofx18.p1.5.m1.4.5.2.2.2.cmml">F</mi><mi id="Thmproofx18.p1.5.m1.4.5.2.2.3" xref="Thmproofx18.p1.5.m1.4.5.2.2.3.cmml">n</mi></msub><mo id="Thmproofx18.p1.5.m1.4.5.2.1" xref="Thmproofx18.p1.5.m1.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx18.p1.5.m1.4.5.2.3.2" xref="Thmproofx18.p1.5.m1.4.5.2.3.1.cmml"><mo id="Thmproofx18.p1.5.m1.4.5.2.3.2.1" stretchy="false" xref="Thmproofx18.p1.5.m1.4.5.2.3.1.cmml">(</mo><mi id="Thmproofx18.p1.5.m1.1.1" xref="Thmproofx18.p1.5.m1.1.1.cmml">σ</mi><mo id="Thmproofx18.p1.5.m1.4.5.2.3.2.2" xref="Thmproofx18.p1.5.m1.4.5.2.3.1.cmml">,</mo><mi id="Thmproofx18.p1.5.m1.2.2" xref="Thmproofx18.p1.5.m1.2.2.cmml">α</mi><mo id="Thmproofx18.p1.5.m1.4.5.2.3.2.3" stretchy="false" xref="Thmproofx18.p1.5.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx18.p1.5.m1.4.5.1" xref="Thmproofx18.p1.5.m1.4.5.1.cmml">≥</mo><mrow id="Thmproofx18.p1.5.m1.4.5.3" xref="Thmproofx18.p1.5.m1.4.5.3.cmml"><msub id="Thmproofx18.p1.5.m1.4.5.3.2" xref="Thmproofx18.p1.5.m1.4.5.3.2.cmml"><mi id="Thmproofx18.p1.5.m1.4.5.3.2.2" xref="Thmproofx18.p1.5.m1.4.5.3.2.2.cmml">F</mi><mi id="Thmproofx18.p1.5.m1.4.5.3.2.3" xref="Thmproofx18.p1.5.m1.4.5.3.2.3.cmml">n</mi></msub><mo id="Thmproofx18.p1.5.m1.4.5.3.1" xref="Thmproofx18.p1.5.m1.4.5.3.1.cmml">⁢</mo><mrow id="Thmproofx18.p1.5.m1.4.5.3.3.2" xref="Thmproofx18.p1.5.m1.4.5.3.3.1.cmml"><mo id="Thmproofx18.p1.5.m1.4.5.3.3.2.1" stretchy="false" xref="Thmproofx18.p1.5.m1.4.5.3.3.1.cmml">(</mo><mi id="Thmproofx18.p1.5.m1.3.3" xref="Thmproofx18.p1.5.m1.3.3.cmml">μ</mi><mo id="Thmproofx18.p1.5.m1.4.5.3.3.2.2" xref="Thmproofx18.p1.5.m1.4.5.3.3.1.cmml">,</mo><mi id="Thmproofx18.p1.5.m1.4.4" xref="Thmproofx18.p1.5.m1.4.4.cmml">α</mi><mo id="Thmproofx18.p1.5.m1.4.5.3.3.2.3" stretchy="false" xref="Thmproofx18.p1.5.m1.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx18.p1.5.m1.4b"><apply id="Thmproofx18.p1.5.m1.4.5.cmml" xref="Thmproofx18.p1.5.m1.4.5"><geq id="Thmproofx18.p1.5.m1.4.5.1.cmml" xref="Thmproofx18.p1.5.m1.4.5.1"></geq><apply id="Thmproofx18.p1.5.m1.4.5.2.cmml" xref="Thmproofx18.p1.5.m1.4.5.2"><times id="Thmproofx18.p1.5.m1.4.5.2.1.cmml" xref="Thmproofx18.p1.5.m1.4.5.2.1"></times><apply id="Thmproofx18.p1.5.m1.4.5.2.2.cmml" xref="Thmproofx18.p1.5.m1.4.5.2.2"><csymbol cd="ambiguous" id="Thmproofx18.p1.5.m1.4.5.2.2.1.cmml" xref="Thmproofx18.p1.5.m1.4.5.2.2">subscript</csymbol><ci id="Thmproofx18.p1.5.m1.4.5.2.2.2.cmml" xref="Thmproofx18.p1.5.m1.4.5.2.2.2">𝐹</ci><ci id="Thmproofx18.p1.5.m1.4.5.2.2.3.cmml" xref="Thmproofx18.p1.5.m1.4.5.2.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx18.p1.5.m1.4.5.2.3.1.cmml" xref="Thmproofx18.p1.5.m1.4.5.2.3.2"><ci id="Thmproofx18.p1.5.m1.1.1.cmml" xref="Thmproofx18.p1.5.m1.1.1">𝜎</ci><ci id="Thmproofx18.p1.5.m1.2.2.cmml" xref="Thmproofx18.p1.5.m1.2.2">𝛼</ci></interval></apply><apply id="Thmproofx18.p1.5.m1.4.5.3.cmml" xref="Thmproofx18.p1.5.m1.4.5.3"><times id="Thmproofx18.p1.5.m1.4.5.3.1.cmml" xref="Thmproofx18.p1.5.m1.4.5.3.1"></times><apply id="Thmproofx18.p1.5.m1.4.5.3.2.cmml" xref="Thmproofx18.p1.5.m1.4.5.3.2"><csymbol cd="ambiguous" id="Thmproofx18.p1.5.m1.4.5.3.2.1.cmml" xref="Thmproofx18.p1.5.m1.4.5.3.2">subscript</csymbol><ci id="Thmproofx18.p1.5.m1.4.5.3.2.2.cmml" xref="Thmproofx18.p1.5.m1.4.5.3.2.2">𝐹</ci><ci id="Thmproofx18.p1.5.m1.4.5.3.2.3.cmml" xref="Thmproofx18.p1.5.m1.4.5.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx18.p1.5.m1.4.5.3.3.1.cmml" xref="Thmproofx18.p1.5.m1.4.5.3.3.2"><ci id="Thmproofx18.p1.5.m1.3.3.cmml" xref="Thmproofx18.p1.5.m1.3.3">𝜇</ci><ci id="Thmproofx18.p1.5.m1.4.4.cmml" xref="Thmproofx18.p1.5.m1.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx18.p1.5.m1.4c">F_{n}(\sigma,\alpha)\geq F_{n}(\mu,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx18.p1.5.m1.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≥ italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_μ , italic_α )</annotation></semantics></math>, proving the lemma. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx19"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma14" title="Lemma 14 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">14</span></a>)</h6> <div class="ltx_para" id="Thmproofx19.p1"> <p class="ltx_p" id="Thmproofx19.p1.4">Let <math alttext="\sigma=U_{P_{st}}^{n}" class="ltx_Math" display="inline" id="Thmproofx19.p1.1.m1.1"><semantics id="Thmproofx19.p1.1.m1.1a"><mrow id="Thmproofx19.p1.1.m1.1.1" xref="Thmproofx19.p1.1.m1.1.1.cmml"><mi id="Thmproofx19.p1.1.m1.1.1.2" xref="Thmproofx19.p1.1.m1.1.1.2.cmml">σ</mi><mo id="Thmproofx19.p1.1.m1.1.1.1" xref="Thmproofx19.p1.1.m1.1.1.1.cmml">=</mo><msubsup id="Thmproofx19.p1.1.m1.1.1.3" xref="Thmproofx19.p1.1.m1.1.1.3.cmml"><mi id="Thmproofx19.p1.1.m1.1.1.3.2.2" xref="Thmproofx19.p1.1.m1.1.1.3.2.2.cmml">U</mi><msub id="Thmproofx19.p1.1.m1.1.1.3.2.3" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.cmml"><mi id="Thmproofx19.p1.1.m1.1.1.3.2.3.2" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.2.cmml">P</mi><mrow id="Thmproofx19.p1.1.m1.1.1.3.2.3.3" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3.cmml"><mi id="Thmproofx19.p1.1.m1.1.1.3.2.3.3.2" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3.2.cmml">s</mi><mo id="Thmproofx19.p1.1.m1.1.1.3.2.3.3.1" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx19.p1.1.m1.1.1.3.2.3.3.3" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx19.p1.1.m1.1.1.3.3" xref="Thmproofx19.p1.1.m1.1.1.3.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.1.m1.1b"><apply id="Thmproofx19.p1.1.m1.1.1.cmml" xref="Thmproofx19.p1.1.m1.1.1"><eq id="Thmproofx19.p1.1.m1.1.1.1.cmml" xref="Thmproofx19.p1.1.m1.1.1.1"></eq><ci id="Thmproofx19.p1.1.m1.1.1.2.cmml" xref="Thmproofx19.p1.1.m1.1.1.2">𝜎</ci><apply id="Thmproofx19.p1.1.m1.1.1.3.cmml" xref="Thmproofx19.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.1.m1.1.1.3.1.cmml" xref="Thmproofx19.p1.1.m1.1.1.3">superscript</csymbol><apply id="Thmproofx19.p1.1.m1.1.1.3.2.cmml" xref="Thmproofx19.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.1.m1.1.1.3.2.1.cmml" xref="Thmproofx19.p1.1.m1.1.1.3">subscript</csymbol><ci id="Thmproofx19.p1.1.m1.1.1.3.2.2.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.2">𝑈</ci><apply id="Thmproofx19.p1.1.m1.1.1.3.2.3.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.1.m1.1.1.3.2.3.1.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.3">subscript</csymbol><ci id="Thmproofx19.p1.1.m1.1.1.3.2.3.2.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.2">𝑃</ci><apply id="Thmproofx19.p1.1.m1.1.1.3.2.3.3.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3"><times id="Thmproofx19.p1.1.m1.1.1.3.2.3.3.1.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3.1"></times><ci id="Thmproofx19.p1.1.m1.1.1.3.2.3.3.2.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3.2">𝑠</ci><ci id="Thmproofx19.p1.1.m1.1.1.3.2.3.3.3.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx19.p1.1.m1.1.1.3.3.cmml" xref="Thmproofx19.p1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.1.m1.1c">\sigma=U_{P_{st}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.1.m1.1d">italic_σ = italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, and let <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="Thmproofx19.p1.2.m2.1"><semantics id="Thmproofx19.p1.2.m2.1a"><mrow id="Thmproofx19.p1.2.m2.1.1" xref="Thmproofx19.p1.2.m2.1.1.cmml"><mi id="Thmproofx19.p1.2.m2.1.1.2" xref="Thmproofx19.p1.2.m2.1.1.2.cmml">d</mi><mo id="Thmproofx19.p1.2.m2.1.1.1" xref="Thmproofx19.p1.2.m2.1.1.1.cmml">∈</mo><msup id="Thmproofx19.p1.2.m2.1.1.3" xref="Thmproofx19.p1.2.m2.1.1.3.cmml"><mi id="Thmproofx19.p1.2.m2.1.1.3.2" xref="Thmproofx19.p1.2.m2.1.1.3.2.cmml">E</mi><mi id="Thmproofx19.p1.2.m2.1.1.3.3" xref="Thmproofx19.p1.2.m2.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.2.m2.1b"><apply id="Thmproofx19.p1.2.m2.1.1.cmml" xref="Thmproofx19.p1.2.m2.1.1"><in id="Thmproofx19.p1.2.m2.1.1.1.cmml" xref="Thmproofx19.p1.2.m2.1.1.1"></in><ci id="Thmproofx19.p1.2.m2.1.1.2.cmml" xref="Thmproofx19.p1.2.m2.1.1.2">𝑑</ci><apply id="Thmproofx19.p1.2.m2.1.1.3.cmml" xref="Thmproofx19.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.2.m2.1.1.3.1.cmml" xref="Thmproofx19.p1.2.m2.1.1.3">superscript</csymbol><ci id="Thmproofx19.p1.2.m2.1.1.3.2.cmml" xref="Thmproofx19.p1.2.m2.1.1.3.2">𝐸</ci><ci id="Thmproofx19.p1.2.m2.1.1.3.3.cmml" xref="Thmproofx19.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.2.m2.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.2.m2.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. Consider the resulting cost <math alttext="c_{d}" class="ltx_Math" display="inline" id="Thmproofx19.p1.3.m3.1"><semantics id="Thmproofx19.p1.3.m3.1a"><msub id="Thmproofx19.p1.3.m3.1.1" xref="Thmproofx19.p1.3.m3.1.1.cmml"><mi id="Thmproofx19.p1.3.m3.1.1.2" xref="Thmproofx19.p1.3.m3.1.1.2.cmml">c</mi><mi id="Thmproofx19.p1.3.m3.1.1.3" xref="Thmproofx19.p1.3.m3.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.3.m3.1b"><apply id="Thmproofx19.p1.3.m3.1.1.cmml" xref="Thmproofx19.p1.3.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.3.m3.1.1.1.cmml" xref="Thmproofx19.p1.3.m3.1.1">subscript</csymbol><ci id="Thmproofx19.p1.3.m3.1.1.2.cmml" xref="Thmproofx19.p1.3.m3.1.1.2">𝑐</ci><ci id="Thmproofx19.p1.3.m3.1.1.3.cmml" xref="Thmproofx19.p1.3.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.3.m3.1c">c_{d}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.3.m3.1d">italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> and the paths <math alttext="P_{st}^{d}" class="ltx_Math" display="inline" id="Thmproofx19.p1.4.m4.1"><semantics id="Thmproofx19.p1.4.m4.1a"><msubsup id="Thmproofx19.p1.4.m4.1.1" xref="Thmproofx19.p1.4.m4.1.1.cmml"><mi id="Thmproofx19.p1.4.m4.1.1.2.2" xref="Thmproofx19.p1.4.m4.1.1.2.2.cmml">P</mi><mrow id="Thmproofx19.p1.4.m4.1.1.2.3" xref="Thmproofx19.p1.4.m4.1.1.2.3.cmml"><mi id="Thmproofx19.p1.4.m4.1.1.2.3.2" xref="Thmproofx19.p1.4.m4.1.1.2.3.2.cmml">s</mi><mo id="Thmproofx19.p1.4.m4.1.1.2.3.1" xref="Thmproofx19.p1.4.m4.1.1.2.3.1.cmml">⁢</mo><mi id="Thmproofx19.p1.4.m4.1.1.2.3.3" xref="Thmproofx19.p1.4.m4.1.1.2.3.3.cmml">t</mi></mrow><mi id="Thmproofx19.p1.4.m4.1.1.3" xref="Thmproofx19.p1.4.m4.1.1.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.4.m4.1b"><apply id="Thmproofx19.p1.4.m4.1.1.cmml" xref="Thmproofx19.p1.4.m4.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.4.m4.1.1.1.cmml" xref="Thmproofx19.p1.4.m4.1.1">superscript</csymbol><apply id="Thmproofx19.p1.4.m4.1.1.2.cmml" xref="Thmproofx19.p1.4.m4.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.4.m4.1.1.2.1.cmml" xref="Thmproofx19.p1.4.m4.1.1">subscript</csymbol><ci id="Thmproofx19.p1.4.m4.1.1.2.2.cmml" xref="Thmproofx19.p1.4.m4.1.1.2.2">𝑃</ci><apply id="Thmproofx19.p1.4.m4.1.1.2.3.cmml" xref="Thmproofx19.p1.4.m4.1.1.2.3"><times id="Thmproofx19.p1.4.m4.1.1.2.3.1.cmml" xref="Thmproofx19.p1.4.m4.1.1.2.3.1"></times><ci id="Thmproofx19.p1.4.m4.1.1.2.3.2.cmml" xref="Thmproofx19.p1.4.m4.1.1.2.3.2">𝑠</ci><ci id="Thmproofx19.p1.4.m4.1.1.2.3.3.cmml" xref="Thmproofx19.p1.4.m4.1.1.2.3.3">𝑡</ci></apply></apply><ci id="Thmproofx19.p1.4.m4.1.1.3.cmml" xref="Thmproofx19.p1.4.m4.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.4.m4.1c">P_{st}^{d}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. Applying Equation <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E6" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">6</span></a>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx50"> <tbody id="Pt0.A4.Ex83"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{d}\left(U_{P_{st}}^{n},\alpha\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex83.m1.2"><semantics id="Pt0.A4.Ex83.m1.2a"><mrow id="Pt0.A4.Ex83.m1.2.2" xref="Pt0.A4.Ex83.m1.2.2.cmml"><msub id="Pt0.A4.Ex83.m1.2.2.3" xref="Pt0.A4.Ex83.m1.2.2.3.cmml"><mi id="Pt0.A4.Ex83.m1.2.2.3.2" xref="Pt0.A4.Ex83.m1.2.2.3.2.cmml">f</mi><mi id="Pt0.A4.Ex83.m1.2.2.3.3" xref="Pt0.A4.Ex83.m1.2.2.3.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex83.m1.2.2.2" xref="Pt0.A4.Ex83.m1.2.2.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex83.m1.2.2.1.1" xref="Pt0.A4.Ex83.m1.2.2.1.2.cmml"><mo id="Pt0.A4.Ex83.m1.2.2.1.1.2" xref="Pt0.A4.Ex83.m1.2.2.1.2.cmml">(</mo><msubsup id="Pt0.A4.Ex83.m1.2.2.1.1.1" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.cmml"><mi id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.2" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.2.cmml">U</mi><msub id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.cmml"><mi id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.2" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.2.cmml">P</mi><mrow id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.2" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.2.cmml">s</mi><mo id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.1" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.3" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Pt0.A4.Ex83.m1.2.2.1.1.1.3" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Pt0.A4.Ex83.m1.2.2.1.1.3" xref="Pt0.A4.Ex83.m1.2.2.1.2.cmml">,</mo><mi id="Pt0.A4.Ex83.m1.1.1" xref="Pt0.A4.Ex83.m1.1.1.cmml">α</mi><mo id="Pt0.A4.Ex83.m1.2.2.1.1.4" xref="Pt0.A4.Ex83.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex83.m1.2b"><apply id="Pt0.A4.Ex83.m1.2.2.cmml" xref="Pt0.A4.Ex83.m1.2.2"><times id="Pt0.A4.Ex83.m1.2.2.2.cmml" xref="Pt0.A4.Ex83.m1.2.2.2"></times><apply id="Pt0.A4.Ex83.m1.2.2.3.cmml" xref="Pt0.A4.Ex83.m1.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex83.m1.2.2.3.1.cmml" xref="Pt0.A4.Ex83.m1.2.2.3">subscript</csymbol><ci id="Pt0.A4.Ex83.m1.2.2.3.2.cmml" xref="Pt0.A4.Ex83.m1.2.2.3.2">𝑓</ci><ci id="Pt0.A4.Ex83.m1.2.2.3.3.cmml" xref="Pt0.A4.Ex83.m1.2.2.3.3">𝑑</ci></apply><interval closure="open" id="Pt0.A4.Ex83.m1.2.2.1.2.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1"><apply id="Pt0.A4.Ex83.m1.2.2.1.1.1.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex83.m1.2.2.1.1.1.1.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.1.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.2.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.2">𝑈</ci><apply id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.1.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3">subscript</csymbol><ci id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.2.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.2">𝑃</ci><apply id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3"><times id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.1.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.1"></times><ci id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.2.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.2">𝑠</ci><ci id="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.3.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Pt0.A4.Ex83.m1.2.2.1.1.1.3.cmml" xref="Pt0.A4.Ex83.m1.2.2.1.1.1.3">𝑛</ci></apply><ci id="Pt0.A4.Ex83.m1.1.1.cmml" xref="Pt0.A4.Ex83.m1.1.1">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex83.m1.2c">\displaystyle f_{d}\left(U_{P_{st}}^{n},\alpha\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex83.m1.2d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i=1}^{m}(c_{d}(p^{d}_{i})-c_{d}(p^{d}_{i-1}))\left(1-\left% (\frac{i-1}{m}\right)^{n}\right)" class="ltx_Math" display="inline" 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xref="Pt0.A4.Ex83.m2.1.1.3">𝑚</ci></apply><ci id="Pt0.A4.Ex83.m2.3.3.2.2.2.1.1.3.3.cmml" xref="Pt0.A4.Ex83.m2.3.3.2.2.2.1.1.3.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex83.m2.3c">\displaystyle=\sum_{i=1}^{m}(c_{d}(p^{d}_{i})-c_{d}(p^{d}_{i-1}))\left(1-\left% (\frac{i-1}{m}\right)^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex83.m2.3d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ) ( 1 - ( divide start_ARG italic_i - 1 end_ARG start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex84"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i=1}^{m}(|p_{I_{d}(i)}|-|p_{I_{d}(i-1)}|+(k^{d}_{i}-k^{d}_% {i-1})\alpha)\left(1-\left(\frac{i-1}{m}\right)^{n}\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex84.m1.5"><semantics id="Pt0.A4.Ex84.m1.5a"><mrow id="Pt0.A4.Ex84.m1.5.5" xref="Pt0.A4.Ex84.m1.5.5.cmml"><mi id="Pt0.A4.Ex84.m1.5.5.4" xref="Pt0.A4.Ex84.m1.5.5.4.cmml"></mi><mo id="Pt0.A4.Ex84.m1.5.5.3" xref="Pt0.A4.Ex84.m1.5.5.3.cmml">=</mo><mrow id="Pt0.A4.Ex84.m1.5.5.2" xref="Pt0.A4.Ex84.m1.5.5.2.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex84.m1.5.5.2.3" xref="Pt0.A4.Ex84.m1.5.5.2.3.cmml"><munderover id="Pt0.A4.Ex84.m1.5.5.2.3a" 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xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1"><minus id="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.1.cmml" xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.1"></minus><cn id="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.2.cmml" type="integer" xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.2">1</cn><apply id="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3.cmml" xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3.1.cmml" xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3">superscript</csymbol><apply id="Pt0.A4.Ex84.m1.3.3.cmml" xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3.2.2"><divide id="Pt0.A4.Ex84.m1.3.3.1.cmml" xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3.2.2"></divide><apply id="Pt0.A4.Ex84.m1.3.3.2.cmml" xref="Pt0.A4.Ex84.m1.3.3.2"><minus id="Pt0.A4.Ex84.m1.3.3.2.1.cmml" xref="Pt0.A4.Ex84.m1.3.3.2.1"></minus><ci id="Pt0.A4.Ex84.m1.3.3.2.2.cmml" xref="Pt0.A4.Ex84.m1.3.3.2.2">𝑖</ci><cn id="Pt0.A4.Ex84.m1.3.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex84.m1.3.3.2.3">1</cn></apply><ci id="Pt0.A4.Ex84.m1.3.3.3.cmml" xref="Pt0.A4.Ex84.m1.3.3.3">𝑚</ci></apply><ci id="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3.3.cmml" xref="Pt0.A4.Ex84.m1.5.5.2.2.2.1.1.3.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex84.m1.5c">\displaystyle=\sum_{i=1}^{m}(|p_{I_{d}(i)}|-|p_{I_{d}(i-1)}|+(k^{d}_{i}-k^{d}_% {i-1})\alpha)\left(1-\left(\frac{i-1}{m}\right)^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex84.m1.5d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i - 1 ) end_POSTSUBSCRIPT | + ( italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) italic_α ) ( 1 - ( divide start_ARG italic_i - 1 end_ARG start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\alpha\sum_{i=1}^{m}k^{d}_{i}\frac{i^{n}-(i-1)^{n}}{m^{n}}+\sum_% {i=1}^{m}|p_{I_{d}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}" class="ltx_Math" display="inline" id="Pt0.A4.E9.m1.4"><semantics id="Pt0.A4.E9.m1.4a"><mrow id="Pt0.A4.E9.m1.4.4" xref="Pt0.A4.E9.m1.4.4.cmml"><mi id="Pt0.A4.E9.m1.4.4.3" xref="Pt0.A4.E9.m1.4.4.3.cmml"></mi><mo id="Pt0.A4.E9.m1.4.4.2" xref="Pt0.A4.E9.m1.4.4.2.cmml">=</mo><mrow id="Pt0.A4.E9.m1.4.4.1" 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id="Pt0.A4.E9.m1.3.3.1.1.1.1.1.2.cmml" xref="Pt0.A4.E9.m1.3.3.1.1.1.1.1.2">𝑖</ci><cn id="Pt0.A4.E9.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="Pt0.A4.E9.m1.3.3.1.1.1.1.1.3">1</cn></apply><ci id="Pt0.A4.E9.m1.3.3.1.1.3.cmml" xref="Pt0.A4.E9.m1.3.3.1.1.3">𝑛</ci></apply></apply><apply id="Pt0.A4.E9.m1.3.3.3.cmml" xref="Pt0.A4.E9.m1.3.3.3"><csymbol cd="ambiguous" id="Pt0.A4.E9.m1.3.3.3.1.cmml" xref="Pt0.A4.E9.m1.3.3.3">superscript</csymbol><ci id="Pt0.A4.E9.m1.3.3.3.2.cmml" xref="Pt0.A4.E9.m1.3.3.3.2">𝑚</ci><ci id="Pt0.A4.E9.m1.3.3.3.3.cmml" xref="Pt0.A4.E9.m1.3.3.3.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.E9.m1.4c">\displaystyle=\alpha\sum_{i=1}^{m}k^{d}_{i}\frac{i^{n}-(i-1)^{n}}{m^{n}}+\sum_% {i=1}^{m}|p_{I_{d}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.E9.m1.4d">= italic_α ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT | divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx19.p1.10">We prove the lemma by showing that (<a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex83" title="Proof (Proof of Lemma 14) ‣ 0.D.4 Omitted proofs ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.D.4</span></a>) attains its strict maximum over <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="Thmproofx19.p1.5.m1.1"><semantics id="Thmproofx19.p1.5.m1.1a"><mrow id="Thmproofx19.p1.5.m1.1.1" xref="Thmproofx19.p1.5.m1.1.1.cmml"><mi id="Thmproofx19.p1.5.m1.1.1.2" xref="Thmproofx19.p1.5.m1.1.1.2.cmml">d</mi><mo id="Thmproofx19.p1.5.m1.1.1.1" xref="Thmproofx19.p1.5.m1.1.1.1.cmml">∈</mo><msup id="Thmproofx19.p1.5.m1.1.1.3" xref="Thmproofx19.p1.5.m1.1.1.3.cmml"><mi id="Thmproofx19.p1.5.m1.1.1.3.2" xref="Thmproofx19.p1.5.m1.1.1.3.2.cmml">E</mi><mi id="Thmproofx19.p1.5.m1.1.1.3.3" xref="Thmproofx19.p1.5.m1.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.5.m1.1b"><apply id="Thmproofx19.p1.5.m1.1.1.cmml" xref="Thmproofx19.p1.5.m1.1.1"><in id="Thmproofx19.p1.5.m1.1.1.1.cmml" xref="Thmproofx19.p1.5.m1.1.1.1"></in><ci id="Thmproofx19.p1.5.m1.1.1.2.cmml" xref="Thmproofx19.p1.5.m1.1.1.2">𝑑</ci><apply id="Thmproofx19.p1.5.m1.1.1.3.cmml" xref="Thmproofx19.p1.5.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.5.m1.1.1.3.1.cmml" xref="Thmproofx19.p1.5.m1.1.1.3">superscript</csymbol><ci id="Thmproofx19.p1.5.m1.1.1.3.2.cmml" xref="Thmproofx19.p1.5.m1.1.1.3.2">𝐸</ci><ci id="Thmproofx19.p1.5.m1.1.1.3.3.cmml" xref="Thmproofx19.p1.5.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.5.m1.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.5.m1.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> when <math alttext="d=(m,...,m)" class="ltx_Math" display="inline" id="Thmproofx19.p1.6.m2.3"><semantics id="Thmproofx19.p1.6.m2.3a"><mrow id="Thmproofx19.p1.6.m2.3.4" xref="Thmproofx19.p1.6.m2.3.4.cmml"><mi id="Thmproofx19.p1.6.m2.3.4.2" xref="Thmproofx19.p1.6.m2.3.4.2.cmml">d</mi><mo id="Thmproofx19.p1.6.m2.3.4.1" xref="Thmproofx19.p1.6.m2.3.4.1.cmml">=</mo><mrow id="Thmproofx19.p1.6.m2.3.4.3.2" xref="Thmproofx19.p1.6.m2.3.4.3.1.cmml"><mo id="Thmproofx19.p1.6.m2.3.4.3.2.1" stretchy="false" xref="Thmproofx19.p1.6.m2.3.4.3.1.cmml">(</mo><mi id="Thmproofx19.p1.6.m2.1.1" xref="Thmproofx19.p1.6.m2.1.1.cmml">m</mi><mo id="Thmproofx19.p1.6.m2.3.4.3.2.2" xref="Thmproofx19.p1.6.m2.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.6.m2.2.2" mathvariant="normal" xref="Thmproofx19.p1.6.m2.2.2.cmml">…</mi><mo id="Thmproofx19.p1.6.m2.3.4.3.2.3" xref="Thmproofx19.p1.6.m2.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.6.m2.3.3" xref="Thmproofx19.p1.6.m2.3.3.cmml">m</mi><mo id="Thmproofx19.p1.6.m2.3.4.3.2.4" stretchy="false" xref="Thmproofx19.p1.6.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.6.m2.3b"><apply id="Thmproofx19.p1.6.m2.3.4.cmml" xref="Thmproofx19.p1.6.m2.3.4"><eq id="Thmproofx19.p1.6.m2.3.4.1.cmml" xref="Thmproofx19.p1.6.m2.3.4.1"></eq><ci id="Thmproofx19.p1.6.m2.3.4.2.cmml" xref="Thmproofx19.p1.6.m2.3.4.2">𝑑</ci><vector id="Thmproofx19.p1.6.m2.3.4.3.1.cmml" xref="Thmproofx19.p1.6.m2.3.4.3.2"><ci id="Thmproofx19.p1.6.m2.1.1.cmml" xref="Thmproofx19.p1.6.m2.1.1">𝑚</ci><ci id="Thmproofx19.p1.6.m2.2.2.cmml" xref="Thmproofx19.p1.6.m2.2.2">…</ci><ci id="Thmproofx19.p1.6.m2.3.3.cmml" xref="Thmproofx19.p1.6.m2.3.3">𝑚</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.6.m2.3c">d=(m,...,m)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.6.m2.3d">italic_d = ( italic_m , … , italic_m )</annotation></semantics></math>. (<a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex83" title="Proof (Proof of Lemma 14) ‣ 0.D.4 Omitted proofs ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.D.4</span></a>) is an affine function in <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx19.p1.7.m3.1"><semantics id="Thmproofx19.p1.7.m3.1a"><mi id="Thmproofx19.p1.7.m3.1.1" xref="Thmproofx19.p1.7.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.7.m3.1b"><ci id="Thmproofx19.p1.7.m3.1.1.cmml" xref="Thmproofx19.p1.7.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.7.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.7.m3.1d">italic_α</annotation></semantics></math>. <math alttext="m,n>1" class="ltx_Math" display="inline" id="Thmproofx19.p1.8.m4.2"><semantics id="Thmproofx19.p1.8.m4.2a"><mrow id="Thmproofx19.p1.8.m4.2.3" xref="Thmproofx19.p1.8.m4.2.3.cmml"><mrow id="Thmproofx19.p1.8.m4.2.3.2.2" xref="Thmproofx19.p1.8.m4.2.3.2.1.cmml"><mi id="Thmproofx19.p1.8.m4.1.1" xref="Thmproofx19.p1.8.m4.1.1.cmml">m</mi><mo id="Thmproofx19.p1.8.m4.2.3.2.2.1" xref="Thmproofx19.p1.8.m4.2.3.2.1.cmml">,</mo><mi id="Thmproofx19.p1.8.m4.2.2" xref="Thmproofx19.p1.8.m4.2.2.cmml">n</mi></mrow><mo id="Thmproofx19.p1.8.m4.2.3.1" xref="Thmproofx19.p1.8.m4.2.3.1.cmml">></mo><mn id="Thmproofx19.p1.8.m4.2.3.3" xref="Thmproofx19.p1.8.m4.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.8.m4.2b"><apply id="Thmproofx19.p1.8.m4.2.3.cmml" xref="Thmproofx19.p1.8.m4.2.3"><gt id="Thmproofx19.p1.8.m4.2.3.1.cmml" xref="Thmproofx19.p1.8.m4.2.3.1"></gt><list id="Thmproofx19.p1.8.m4.2.3.2.1.cmml" xref="Thmproofx19.p1.8.m4.2.3.2.2"><ci id="Thmproofx19.p1.8.m4.1.1.cmml" xref="Thmproofx19.p1.8.m4.1.1">𝑚</ci><ci id="Thmproofx19.p1.8.m4.2.2.cmml" xref="Thmproofx19.p1.8.m4.2.2">𝑛</ci></list><cn id="Thmproofx19.p1.8.m4.2.3.3.cmml" type="integer" xref="Thmproofx19.p1.8.m4.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.8.m4.2c">m,n>1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.8.m4.2d">italic_m , italic_n > 1</annotation></semantics></math> so <math alttext="\frac{i^{n}-(i-1)^{n}}{m^{n}}" class="ltx_Math" display="inline" id="Thmproofx19.p1.9.m5.1"><semantics id="Thmproofx19.p1.9.m5.1a"><mfrac id="Thmproofx19.p1.9.m5.1.1" xref="Thmproofx19.p1.9.m5.1.1.cmml"><mrow id="Thmproofx19.p1.9.m5.1.1.1" xref="Thmproofx19.p1.9.m5.1.1.1.cmml"><msup id="Thmproofx19.p1.9.m5.1.1.1.3" xref="Thmproofx19.p1.9.m5.1.1.1.3.cmml"><mi id="Thmproofx19.p1.9.m5.1.1.1.3.2" xref="Thmproofx19.p1.9.m5.1.1.1.3.2.cmml">i</mi><mi id="Thmproofx19.p1.9.m5.1.1.1.3.3" xref="Thmproofx19.p1.9.m5.1.1.1.3.3.cmml">n</mi></msup><mo id="Thmproofx19.p1.9.m5.1.1.1.2" xref="Thmproofx19.p1.9.m5.1.1.1.2.cmml">−</mo><msup id="Thmproofx19.p1.9.m5.1.1.1.1" xref="Thmproofx19.p1.9.m5.1.1.1.1.cmml"><mrow id="Thmproofx19.p1.9.m5.1.1.1.1.1.1" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.2" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.2.cmml">i</mi><mo id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.1" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.3" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx19.p1.9.m5.1.1.1.1.3" xref="Thmproofx19.p1.9.m5.1.1.1.1.3.cmml">n</mi></msup></mrow><msup id="Thmproofx19.p1.9.m5.1.1.3" xref="Thmproofx19.p1.9.m5.1.1.3.cmml"><mi id="Thmproofx19.p1.9.m5.1.1.3.2" xref="Thmproofx19.p1.9.m5.1.1.3.2.cmml">m</mi><mi id="Thmproofx19.p1.9.m5.1.1.3.3" xref="Thmproofx19.p1.9.m5.1.1.3.3.cmml">n</mi></msup></mfrac><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.9.m5.1b"><apply id="Thmproofx19.p1.9.m5.1.1.cmml" xref="Thmproofx19.p1.9.m5.1.1"><divide id="Thmproofx19.p1.9.m5.1.1.2.cmml" xref="Thmproofx19.p1.9.m5.1.1"></divide><apply id="Thmproofx19.p1.9.m5.1.1.1.cmml" xref="Thmproofx19.p1.9.m5.1.1.1"><minus id="Thmproofx19.p1.9.m5.1.1.1.2.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.2"></minus><apply id="Thmproofx19.p1.9.m5.1.1.1.3.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.9.m5.1.1.1.3.1.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.3">superscript</csymbol><ci id="Thmproofx19.p1.9.m5.1.1.1.3.2.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.3.2">𝑖</ci><ci id="Thmproofx19.p1.9.m5.1.1.1.3.3.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.3.3">𝑛</ci></apply><apply id="Thmproofx19.p1.9.m5.1.1.1.1.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.9.m5.1.1.1.1.2.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.1">superscript</csymbol><apply id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1"><minus id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.1"></minus><ci id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.2">𝑖</ci><cn id="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx19.p1.9.m5.1.1.1.1.1.1.1.3">1</cn></apply><ci id="Thmproofx19.p1.9.m5.1.1.1.1.3.cmml" xref="Thmproofx19.p1.9.m5.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmproofx19.p1.9.m5.1.1.3.cmml" xref="Thmproofx19.p1.9.m5.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.9.m5.1.1.3.1.cmml" xref="Thmproofx19.p1.9.m5.1.1.3">superscript</csymbol><ci id="Thmproofx19.p1.9.m5.1.1.3.2.cmml" xref="Thmproofx19.p1.9.m5.1.1.3.2">𝑚</ci><ci id="Thmproofx19.p1.9.m5.1.1.3.3.cmml" xref="Thmproofx19.p1.9.m5.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.9.m5.1c">\frac{i^{n}-(i-1)^{n}}{m^{n}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.9.m5.1d">divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> is strictly increasing in <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx19.p1.10.m6.1"><semantics id="Thmproofx19.p1.10.m6.1a"><mi id="Thmproofx19.p1.10.m6.1.1" xref="Thmproofx19.p1.10.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.10.m6.1b"><ci id="Thmproofx19.p1.10.m6.1.1.cmml" xref="Thmproofx19.p1.10.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.10.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.10.m6.1d">italic_i</annotation></semantics></math>. Therefore</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx51"> <tbody id="Pt0.A4.Ex85"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\alpha\sum_{i=1}^{m}k^{d}_{i}\frac{i^{n}-(i-1)^{n}}{m^{n}}\leq% \alpha k\frac{m^{n}-(m-1)^{n}}{m^{n}}=\frac{\mathcal{E}_{m,n}}{m}k\alpha" class="ltx_Math" display="inline" id="Pt0.A4.Ex85.m1.4"><semantics id="Pt0.A4.Ex85.m1.4a"><mrow id="Pt0.A4.Ex85.m1.4.5" xref="Pt0.A4.Ex85.m1.4.5.cmml"><mrow id="Pt0.A4.Ex85.m1.4.5.2" xref="Pt0.A4.Ex85.m1.4.5.2.cmml"><mi id="Pt0.A4.Ex85.m1.4.5.2.2" xref="Pt0.A4.Ex85.m1.4.5.2.2.cmml">α</mi><mo id="Pt0.A4.Ex85.m1.4.5.2.1" xref="Pt0.A4.Ex85.m1.4.5.2.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex85.m1.4.5.2.3" xref="Pt0.A4.Ex85.m1.4.5.2.3.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex85.m1.4.5.2.3.1" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.cmml"><munderover id="Pt0.A4.Ex85.m1.4.5.2.3.1a" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.cmml"><mo id="Pt0.A4.Ex85.m1.4.5.2.3.1.2.2" movablelimits="false" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3.cmml"><mi id="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3.2" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3.2.cmml">i</mi><mo id="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3.1" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3.1.cmml">=</mo><mn id="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3.3" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.2.3.3.cmml">1</mn></mrow><mi id="Pt0.A4.Ex85.m1.4.5.2.3.1.3" xref="Pt0.A4.Ex85.m1.4.5.2.3.1.3.cmml">m</mi></munderover></mstyle><mrow id="Pt0.A4.Ex85.m1.4.5.2.3.2" xref="Pt0.A4.Ex85.m1.4.5.2.3.2.cmml"><msubsup id="Pt0.A4.Ex85.m1.4.5.2.3.2.2" xref="Pt0.A4.Ex85.m1.4.5.2.3.2.2.cmml"><mi id="Pt0.A4.Ex85.m1.4.5.2.3.2.2.2.2" xref="Pt0.A4.Ex85.m1.4.5.2.3.2.2.2.2.cmml">k</mi><mi id="Pt0.A4.Ex85.m1.4.5.2.3.2.2.3" xref="Pt0.A4.Ex85.m1.4.5.2.3.2.2.3.cmml">i</mi><mi id="Pt0.A4.Ex85.m1.4.5.2.3.2.2.2.3" xref="Pt0.A4.Ex85.m1.4.5.2.3.2.2.2.3.cmml">d</mi></msubsup><mo id="Pt0.A4.Ex85.m1.4.5.2.3.2.1" xref="Pt0.A4.Ex85.m1.4.5.2.3.2.1.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A4.Ex85.m1.1.1" xref="Pt0.A4.Ex85.m1.1.1.cmml"><mfrac id="Pt0.A4.Ex85.m1.1.1a" xref="Pt0.A4.Ex85.m1.1.1.cmml"><mrow id="Pt0.A4.Ex85.m1.1.1.1" xref="Pt0.A4.Ex85.m1.1.1.1.cmml"><msup id="Pt0.A4.Ex85.m1.1.1.1.3" xref="Pt0.A4.Ex85.m1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex85.m1.1.1.1.3.2" xref="Pt0.A4.Ex85.m1.1.1.1.3.2.cmml">i</mi><mi id="Pt0.A4.Ex85.m1.1.1.1.3.3" xref="Pt0.A4.Ex85.m1.1.1.1.3.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex85.m1.1.1.1.2" xref="Pt0.A4.Ex85.m1.1.1.1.2.cmml">−</mo><msup id="Pt0.A4.Ex85.m1.1.1.1.1" xref="Pt0.A4.Ex85.m1.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex85.m1.1.1.1.1.1.1" xref="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex85.m1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.2.cmml">i</mi><mo id="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex85.m1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex85.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex85.m1.1.1.1.1.3" xref="Pt0.A4.Ex85.m1.1.1.1.1.3.cmml">n</mi></msup></mrow><msup id="Pt0.A4.Ex85.m1.1.1.3" xref="Pt0.A4.Ex85.m1.1.1.3.cmml"><mi id="Pt0.A4.Ex85.m1.1.1.3.2" xref="Pt0.A4.Ex85.m1.1.1.3.2.cmml">m</mi><mi id="Pt0.A4.Ex85.m1.1.1.3.3" xref="Pt0.A4.Ex85.m1.1.1.3.3.cmml">n</mi></msup></mfrac></mstyle></mrow></mrow></mrow><mo id="Pt0.A4.Ex85.m1.4.5.3" xref="Pt0.A4.Ex85.m1.4.5.3.cmml">≤</mo><mrow id="Pt0.A4.Ex85.m1.4.5.4" xref="Pt0.A4.Ex85.m1.4.5.4.cmml"><mi id="Pt0.A4.Ex85.m1.4.5.4.2" xref="Pt0.A4.Ex85.m1.4.5.4.2.cmml">α</mi><mo id="Pt0.A4.Ex85.m1.4.5.4.1" xref="Pt0.A4.Ex85.m1.4.5.4.1.cmml">⁢</mo><mi id="Pt0.A4.Ex85.m1.4.5.4.3" xref="Pt0.A4.Ex85.m1.4.5.4.3.cmml">k</mi><mo id="Pt0.A4.Ex85.m1.4.5.4.1a" xref="Pt0.A4.Ex85.m1.4.5.4.1.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A4.Ex85.m1.2.2" xref="Pt0.A4.Ex85.m1.2.2.cmml"><mfrac id="Pt0.A4.Ex85.m1.2.2a" xref="Pt0.A4.Ex85.m1.2.2.cmml"><mrow id="Pt0.A4.Ex85.m1.2.2.1" xref="Pt0.A4.Ex85.m1.2.2.1.cmml"><msup id="Pt0.A4.Ex85.m1.2.2.1.3" xref="Pt0.A4.Ex85.m1.2.2.1.3.cmml"><mi id="Pt0.A4.Ex85.m1.2.2.1.3.2" xref="Pt0.A4.Ex85.m1.2.2.1.3.2.cmml">m</mi><mi id="Pt0.A4.Ex85.m1.2.2.1.3.3" xref="Pt0.A4.Ex85.m1.2.2.1.3.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex85.m1.2.2.1.2" xref="Pt0.A4.Ex85.m1.2.2.1.2.cmml">−</mo><msup id="Pt0.A4.Ex85.m1.2.2.1.1" xref="Pt0.A4.Ex85.m1.2.2.1.1.cmml"><mrow id="Pt0.A4.Ex85.m1.2.2.1.1.1.1" xref="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex85.m1.2.2.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1" xref="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.2.cmml">m</mi><mo id="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.1.cmml">−</mo><mn id="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.3" xref="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex85.m1.2.2.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex85.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex85.m1.2.2.1.1.3" xref="Pt0.A4.Ex85.m1.2.2.1.1.3.cmml">n</mi></msup></mrow><msup id="Pt0.A4.Ex85.m1.2.2.3" xref="Pt0.A4.Ex85.m1.2.2.3.cmml"><mi id="Pt0.A4.Ex85.m1.2.2.3.2" xref="Pt0.A4.Ex85.m1.2.2.3.2.cmml">m</mi><mi id="Pt0.A4.Ex85.m1.2.2.3.3" xref="Pt0.A4.Ex85.m1.2.2.3.3.cmml">n</mi></msup></mfrac></mstyle></mrow><mo id="Pt0.A4.Ex85.m1.4.5.5" xref="Pt0.A4.Ex85.m1.4.5.5.cmml">=</mo><mrow id="Pt0.A4.Ex85.m1.4.5.6" xref="Pt0.A4.Ex85.m1.4.5.6.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex85.m1.4.4" xref="Pt0.A4.Ex85.m1.4.4.cmml"><mfrac 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id="Pt0.A4.Ex85.m1.4.5c.cmml" xref="Pt0.A4.Ex85.m1.4.5"><eq id="Pt0.A4.Ex85.m1.4.5.5.cmml" xref="Pt0.A4.Ex85.m1.4.5.5"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex85.m1.4.5.4.cmml" id="Pt0.A4.Ex85.m1.4.5d.cmml" xref="Pt0.A4.Ex85.m1.4.5"></share><apply id="Pt0.A4.Ex85.m1.4.5.6.cmml" xref="Pt0.A4.Ex85.m1.4.5.6"><times id="Pt0.A4.Ex85.m1.4.5.6.1.cmml" xref="Pt0.A4.Ex85.m1.4.5.6.1"></times><apply id="Pt0.A4.Ex85.m1.4.4.cmml" xref="Pt0.A4.Ex85.m1.4.4"><divide id="Pt0.A4.Ex85.m1.4.4.3.cmml" xref="Pt0.A4.Ex85.m1.4.4"></divide><apply id="Pt0.A4.Ex85.m1.4.4.2.cmml" xref="Pt0.A4.Ex85.m1.4.4.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex85.m1.4.4.2.3.cmml" xref="Pt0.A4.Ex85.m1.4.4.2">subscript</csymbol><ci id="Pt0.A4.Ex85.m1.4.4.2.4.cmml" xref="Pt0.A4.Ex85.m1.4.4.2.4">ℰ</ci><list id="Pt0.A4.Ex85.m1.4.4.2.2.2.3.cmml" xref="Pt0.A4.Ex85.m1.4.4.2.2.2.4"><ci id="Pt0.A4.Ex85.m1.3.3.1.1.1.1.cmml" xref="Pt0.A4.Ex85.m1.3.3.1.1.1.1">𝑚</ci><ci id="Pt0.A4.Ex85.m1.4.4.2.2.2.2.cmml" xref="Pt0.A4.Ex85.m1.4.4.2.2.2.2">𝑛</ci></list></apply><ci id="Pt0.A4.Ex85.m1.4.4.4.cmml" xref="Pt0.A4.Ex85.m1.4.4.4">𝑚</ci></apply><ci id="Pt0.A4.Ex85.m1.4.5.6.2.cmml" xref="Pt0.A4.Ex85.m1.4.5.6.2">𝑘</ci><ci id="Pt0.A4.Ex85.m1.4.5.6.3.cmml" xref="Pt0.A4.Ex85.m1.4.5.6.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex85.m1.4c">\displaystyle\alpha\sum_{i=1}^{m}k^{d}_{i}\frac{i^{n}-(i-1)^{n}}{m^{n}}\leq% \alpha k\frac{m^{n}-(m-1)^{n}}{m^{n}}=\frac{\mathcal{E}_{m,n}}{m}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex85.m1.4d">italic_α ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG ≤ italic_α italic_k divide start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_m - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG = divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx19.p1.31">This maximum with respect to <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="Thmproofx19.p1.11.m1.1"><semantics id="Thmproofx19.p1.11.m1.1a"><mrow id="Thmproofx19.p1.11.m1.1.1" xref="Thmproofx19.p1.11.m1.1.1.cmml"><mi id="Thmproofx19.p1.11.m1.1.1.2" xref="Thmproofx19.p1.11.m1.1.1.2.cmml">d</mi><mo id="Thmproofx19.p1.11.m1.1.1.1" xref="Thmproofx19.p1.11.m1.1.1.1.cmml">∈</mo><msup id="Thmproofx19.p1.11.m1.1.1.3" xref="Thmproofx19.p1.11.m1.1.1.3.cmml"><mi id="Thmproofx19.p1.11.m1.1.1.3.2" xref="Thmproofx19.p1.11.m1.1.1.3.2.cmml">E</mi><mi id="Thmproofx19.p1.11.m1.1.1.3.3" xref="Thmproofx19.p1.11.m1.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.11.m1.1b"><apply id="Thmproofx19.p1.11.m1.1.1.cmml" xref="Thmproofx19.p1.11.m1.1.1"><in id="Thmproofx19.p1.11.m1.1.1.1.cmml" xref="Thmproofx19.p1.11.m1.1.1.1"></in><ci id="Thmproofx19.p1.11.m1.1.1.2.cmml" xref="Thmproofx19.p1.11.m1.1.1.2">𝑑</ci><apply id="Thmproofx19.p1.11.m1.1.1.3.cmml" xref="Thmproofx19.p1.11.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.11.m1.1.1.3.1.cmml" xref="Thmproofx19.p1.11.m1.1.1.3">superscript</csymbol><ci id="Thmproofx19.p1.11.m1.1.1.3.2.cmml" xref="Thmproofx19.p1.11.m1.1.1.3.2">𝐸</ci><ci id="Thmproofx19.p1.11.m1.1.1.3.3.cmml" xref="Thmproofx19.p1.11.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.11.m1.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.11.m1.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> is obtained only if <math alttext="k^{d}_{i}=k\cdot 1_{i=m}" class="ltx_Math" display="inline" id="Thmproofx19.p1.12.m2.1"><semantics id="Thmproofx19.p1.12.m2.1a"><mrow id="Thmproofx19.p1.12.m2.1.1" xref="Thmproofx19.p1.12.m2.1.1.cmml"><msubsup id="Thmproofx19.p1.12.m2.1.1.2" xref="Thmproofx19.p1.12.m2.1.1.2.cmml"><mi id="Thmproofx19.p1.12.m2.1.1.2.2.2" xref="Thmproofx19.p1.12.m2.1.1.2.2.2.cmml">k</mi><mi id="Thmproofx19.p1.12.m2.1.1.2.3" xref="Thmproofx19.p1.12.m2.1.1.2.3.cmml">i</mi><mi id="Thmproofx19.p1.12.m2.1.1.2.2.3" xref="Thmproofx19.p1.12.m2.1.1.2.2.3.cmml">d</mi></msubsup><mo id="Thmproofx19.p1.12.m2.1.1.1" xref="Thmproofx19.p1.12.m2.1.1.1.cmml">=</mo><mrow id="Thmproofx19.p1.12.m2.1.1.3" xref="Thmproofx19.p1.12.m2.1.1.3.cmml"><mi id="Thmproofx19.p1.12.m2.1.1.3.2" xref="Thmproofx19.p1.12.m2.1.1.3.2.cmml">k</mi><mo id="Thmproofx19.p1.12.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="Thmproofx19.p1.12.m2.1.1.3.1.cmml">⋅</mo><msub id="Thmproofx19.p1.12.m2.1.1.3.3" xref="Thmproofx19.p1.12.m2.1.1.3.3.cmml"><mn id="Thmproofx19.p1.12.m2.1.1.3.3.2" xref="Thmproofx19.p1.12.m2.1.1.3.3.2.cmml">1</mn><mrow id="Thmproofx19.p1.12.m2.1.1.3.3.3" xref="Thmproofx19.p1.12.m2.1.1.3.3.3.cmml"><mi id="Thmproofx19.p1.12.m2.1.1.3.3.3.2" xref="Thmproofx19.p1.12.m2.1.1.3.3.3.2.cmml">i</mi><mo id="Thmproofx19.p1.12.m2.1.1.3.3.3.1" xref="Thmproofx19.p1.12.m2.1.1.3.3.3.1.cmml">=</mo><mi id="Thmproofx19.p1.12.m2.1.1.3.3.3.3" xref="Thmproofx19.p1.12.m2.1.1.3.3.3.3.cmml">m</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.12.m2.1b"><apply id="Thmproofx19.p1.12.m2.1.1.cmml" xref="Thmproofx19.p1.12.m2.1.1"><eq id="Thmproofx19.p1.12.m2.1.1.1.cmml" xref="Thmproofx19.p1.12.m2.1.1.1"></eq><apply id="Thmproofx19.p1.12.m2.1.1.2.cmml" xref="Thmproofx19.p1.12.m2.1.1.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.12.m2.1.1.2.1.cmml" xref="Thmproofx19.p1.12.m2.1.1.2">subscript</csymbol><apply id="Thmproofx19.p1.12.m2.1.1.2.2.cmml" xref="Thmproofx19.p1.12.m2.1.1.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.12.m2.1.1.2.2.1.cmml" xref="Thmproofx19.p1.12.m2.1.1.2">superscript</csymbol><ci id="Thmproofx19.p1.12.m2.1.1.2.2.2.cmml" xref="Thmproofx19.p1.12.m2.1.1.2.2.2">𝑘</ci><ci id="Thmproofx19.p1.12.m2.1.1.2.2.3.cmml" xref="Thmproofx19.p1.12.m2.1.1.2.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.12.m2.1.1.2.3.cmml" xref="Thmproofx19.p1.12.m2.1.1.2.3">𝑖</ci></apply><apply id="Thmproofx19.p1.12.m2.1.1.3.cmml" xref="Thmproofx19.p1.12.m2.1.1.3"><ci id="Thmproofx19.p1.12.m2.1.1.3.1.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.1">⋅</ci><ci id="Thmproofx19.p1.12.m2.1.1.3.2.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.2">𝑘</ci><apply id="Thmproofx19.p1.12.m2.1.1.3.3.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.12.m2.1.1.3.3.1.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.3">subscript</csymbol><cn id="Thmproofx19.p1.12.m2.1.1.3.3.2.cmml" type="integer" xref="Thmproofx19.p1.12.m2.1.1.3.3.2">1</cn><apply id="Thmproofx19.p1.12.m2.1.1.3.3.3.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.3.3"><eq id="Thmproofx19.p1.12.m2.1.1.3.3.3.1.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.3.3.1"></eq><ci id="Thmproofx19.p1.12.m2.1.1.3.3.3.2.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.3.3.2">𝑖</ci><ci id="Thmproofx19.p1.12.m2.1.1.3.3.3.3.cmml" xref="Thmproofx19.p1.12.m2.1.1.3.3.3.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.12.m2.1c">k^{d}_{i}=k\cdot 1_{i=m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.12.m2.1d">italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_k ⋅ 1 start_POSTSUBSCRIPT italic_i = italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. For <math alttext="d=(m,...,m)" class="ltx_Math" display="inline" id="Thmproofx19.p1.13.m3.3"><semantics id="Thmproofx19.p1.13.m3.3a"><mrow id="Thmproofx19.p1.13.m3.3.4" xref="Thmproofx19.p1.13.m3.3.4.cmml"><mi id="Thmproofx19.p1.13.m3.3.4.2" xref="Thmproofx19.p1.13.m3.3.4.2.cmml">d</mi><mo id="Thmproofx19.p1.13.m3.3.4.1" xref="Thmproofx19.p1.13.m3.3.4.1.cmml">=</mo><mrow id="Thmproofx19.p1.13.m3.3.4.3.2" xref="Thmproofx19.p1.13.m3.3.4.3.1.cmml"><mo id="Thmproofx19.p1.13.m3.3.4.3.2.1" stretchy="false" xref="Thmproofx19.p1.13.m3.3.4.3.1.cmml">(</mo><mi id="Thmproofx19.p1.13.m3.1.1" xref="Thmproofx19.p1.13.m3.1.1.cmml">m</mi><mo id="Thmproofx19.p1.13.m3.3.4.3.2.2" xref="Thmproofx19.p1.13.m3.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.13.m3.2.2" mathvariant="normal" xref="Thmproofx19.p1.13.m3.2.2.cmml">…</mi><mo id="Thmproofx19.p1.13.m3.3.4.3.2.3" xref="Thmproofx19.p1.13.m3.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.13.m3.3.3" xref="Thmproofx19.p1.13.m3.3.3.cmml">m</mi><mo id="Thmproofx19.p1.13.m3.3.4.3.2.4" stretchy="false" xref="Thmproofx19.p1.13.m3.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.13.m3.3b"><apply id="Thmproofx19.p1.13.m3.3.4.cmml" xref="Thmproofx19.p1.13.m3.3.4"><eq id="Thmproofx19.p1.13.m3.3.4.1.cmml" xref="Thmproofx19.p1.13.m3.3.4.1"></eq><ci id="Thmproofx19.p1.13.m3.3.4.2.cmml" xref="Thmproofx19.p1.13.m3.3.4.2">𝑑</ci><vector id="Thmproofx19.p1.13.m3.3.4.3.1.cmml" xref="Thmproofx19.p1.13.m3.3.4.3.2"><ci id="Thmproofx19.p1.13.m3.1.1.cmml" xref="Thmproofx19.p1.13.m3.1.1">𝑚</ci><ci id="Thmproofx19.p1.13.m3.2.2.cmml" xref="Thmproofx19.p1.13.m3.2.2">…</ci><ci id="Thmproofx19.p1.13.m3.3.3.cmml" xref="Thmproofx19.p1.13.m3.3.3">𝑚</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.13.m3.3c">d=(m,...,m)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.13.m3.3d">italic_d = ( italic_m , … , italic_m )</annotation></semantics></math>, we indeed get <math alttext="k^{d}_{i}=k\cdot 1_{i=m}" class="ltx_Math" display="inline" id="Thmproofx19.p1.14.m4.1"><semantics id="Thmproofx19.p1.14.m4.1a"><mrow id="Thmproofx19.p1.14.m4.1.1" xref="Thmproofx19.p1.14.m4.1.1.cmml"><msubsup id="Thmproofx19.p1.14.m4.1.1.2" xref="Thmproofx19.p1.14.m4.1.1.2.cmml"><mi id="Thmproofx19.p1.14.m4.1.1.2.2.2" xref="Thmproofx19.p1.14.m4.1.1.2.2.2.cmml">k</mi><mi id="Thmproofx19.p1.14.m4.1.1.2.3" xref="Thmproofx19.p1.14.m4.1.1.2.3.cmml">i</mi><mi id="Thmproofx19.p1.14.m4.1.1.2.2.3" xref="Thmproofx19.p1.14.m4.1.1.2.2.3.cmml">d</mi></msubsup><mo id="Thmproofx19.p1.14.m4.1.1.1" xref="Thmproofx19.p1.14.m4.1.1.1.cmml">=</mo><mrow id="Thmproofx19.p1.14.m4.1.1.3" xref="Thmproofx19.p1.14.m4.1.1.3.cmml"><mi id="Thmproofx19.p1.14.m4.1.1.3.2" xref="Thmproofx19.p1.14.m4.1.1.3.2.cmml">k</mi><mo id="Thmproofx19.p1.14.m4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="Thmproofx19.p1.14.m4.1.1.3.1.cmml">⋅</mo><msub id="Thmproofx19.p1.14.m4.1.1.3.3" xref="Thmproofx19.p1.14.m4.1.1.3.3.cmml"><mn id="Thmproofx19.p1.14.m4.1.1.3.3.2" xref="Thmproofx19.p1.14.m4.1.1.3.3.2.cmml">1</mn><mrow id="Thmproofx19.p1.14.m4.1.1.3.3.3" xref="Thmproofx19.p1.14.m4.1.1.3.3.3.cmml"><mi id="Thmproofx19.p1.14.m4.1.1.3.3.3.2" xref="Thmproofx19.p1.14.m4.1.1.3.3.3.2.cmml">i</mi><mo id="Thmproofx19.p1.14.m4.1.1.3.3.3.1" xref="Thmproofx19.p1.14.m4.1.1.3.3.3.1.cmml">=</mo><mi id="Thmproofx19.p1.14.m4.1.1.3.3.3.3" xref="Thmproofx19.p1.14.m4.1.1.3.3.3.3.cmml">m</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.14.m4.1b"><apply id="Thmproofx19.p1.14.m4.1.1.cmml" xref="Thmproofx19.p1.14.m4.1.1"><eq id="Thmproofx19.p1.14.m4.1.1.1.cmml" xref="Thmproofx19.p1.14.m4.1.1.1"></eq><apply id="Thmproofx19.p1.14.m4.1.1.2.cmml" xref="Thmproofx19.p1.14.m4.1.1.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.14.m4.1.1.2.1.cmml" xref="Thmproofx19.p1.14.m4.1.1.2">subscript</csymbol><apply id="Thmproofx19.p1.14.m4.1.1.2.2.cmml" xref="Thmproofx19.p1.14.m4.1.1.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.14.m4.1.1.2.2.1.cmml" xref="Thmproofx19.p1.14.m4.1.1.2">superscript</csymbol><ci id="Thmproofx19.p1.14.m4.1.1.2.2.2.cmml" xref="Thmproofx19.p1.14.m4.1.1.2.2.2">𝑘</ci><ci id="Thmproofx19.p1.14.m4.1.1.2.2.3.cmml" xref="Thmproofx19.p1.14.m4.1.1.2.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.14.m4.1.1.2.3.cmml" xref="Thmproofx19.p1.14.m4.1.1.2.3">𝑖</ci></apply><apply id="Thmproofx19.p1.14.m4.1.1.3.cmml" xref="Thmproofx19.p1.14.m4.1.1.3"><ci id="Thmproofx19.p1.14.m4.1.1.3.1.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.1">⋅</ci><ci id="Thmproofx19.p1.14.m4.1.1.3.2.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.2">𝑘</ci><apply id="Thmproofx19.p1.14.m4.1.1.3.3.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.14.m4.1.1.3.3.1.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.3">subscript</csymbol><cn id="Thmproofx19.p1.14.m4.1.1.3.3.2.cmml" type="integer" xref="Thmproofx19.p1.14.m4.1.1.3.3.2">1</cn><apply id="Thmproofx19.p1.14.m4.1.1.3.3.3.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.3.3"><eq id="Thmproofx19.p1.14.m4.1.1.3.3.3.1.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.3.3.1"></eq><ci id="Thmproofx19.p1.14.m4.1.1.3.3.3.2.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.3.3.2">𝑖</ci><ci id="Thmproofx19.p1.14.m4.1.1.3.3.3.3.cmml" xref="Thmproofx19.p1.14.m4.1.1.3.3.3.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.14.m4.1c">k^{d}_{i}=k\cdot 1_{i=m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.14.m4.1d">italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_k ⋅ 1 start_POSTSUBSCRIPT italic_i = italic_m end_POSTSUBSCRIPT</annotation></semantics></math>: because <math alttext="\alpha>\frac{|p_{m}|-|p_{1}|}{k}" class="ltx_Math" display="inline" id="Thmproofx19.p1.15.m5.2"><semantics id="Thmproofx19.p1.15.m5.2a"><mrow id="Thmproofx19.p1.15.m5.2.3" xref="Thmproofx19.p1.15.m5.2.3.cmml"><mi id="Thmproofx19.p1.15.m5.2.3.2" xref="Thmproofx19.p1.15.m5.2.3.2.cmml">α</mi><mo id="Thmproofx19.p1.15.m5.2.3.1" xref="Thmproofx19.p1.15.m5.2.3.1.cmml">></mo><mfrac id="Thmproofx19.p1.15.m5.2.2" xref="Thmproofx19.p1.15.m5.2.2.cmml"><mrow id="Thmproofx19.p1.15.m5.2.2.2" xref="Thmproofx19.p1.15.m5.2.2.2.cmml"><mrow id="Thmproofx19.p1.15.m5.1.1.1.1.1" xref="Thmproofx19.p1.15.m5.1.1.1.1.2.cmml"><mo id="Thmproofx19.p1.15.m5.1.1.1.1.1.2" stretchy="false" xref="Thmproofx19.p1.15.m5.1.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx19.p1.15.m5.1.1.1.1.1.1" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.1.cmml"><mi id="Thmproofx19.p1.15.m5.1.1.1.1.1.1.2" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx19.p1.15.m5.1.1.1.1.1.1.3" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="Thmproofx19.p1.15.m5.1.1.1.1.1.3" stretchy="false" xref="Thmproofx19.p1.15.m5.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx19.p1.15.m5.2.2.2.3" xref="Thmproofx19.p1.15.m5.2.2.2.3.cmml">−</mo><mrow id="Thmproofx19.p1.15.m5.2.2.2.2.1" xref="Thmproofx19.p1.15.m5.2.2.2.2.2.cmml"><mo id="Thmproofx19.p1.15.m5.2.2.2.2.1.2" stretchy="false" xref="Thmproofx19.p1.15.m5.2.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx19.p1.15.m5.2.2.2.2.1.1" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.1.cmml"><mi id="Thmproofx19.p1.15.m5.2.2.2.2.1.1.2" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.1.2.cmml">p</mi><mn id="Thmproofx19.p1.15.m5.2.2.2.2.1.1.3" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproofx19.p1.15.m5.2.2.2.2.1.3" stretchy="false" xref="Thmproofx19.p1.15.m5.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mi id="Thmproofx19.p1.15.m5.2.2.4" xref="Thmproofx19.p1.15.m5.2.2.4.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.15.m5.2b"><apply id="Thmproofx19.p1.15.m5.2.3.cmml" xref="Thmproofx19.p1.15.m5.2.3"><gt id="Thmproofx19.p1.15.m5.2.3.1.cmml" xref="Thmproofx19.p1.15.m5.2.3.1"></gt><ci id="Thmproofx19.p1.15.m5.2.3.2.cmml" xref="Thmproofx19.p1.15.m5.2.3.2">𝛼</ci><apply id="Thmproofx19.p1.15.m5.2.2.cmml" xref="Thmproofx19.p1.15.m5.2.2"><divide id="Thmproofx19.p1.15.m5.2.2.3.cmml" xref="Thmproofx19.p1.15.m5.2.2"></divide><apply id="Thmproofx19.p1.15.m5.2.2.2.cmml" xref="Thmproofx19.p1.15.m5.2.2.2"><minus id="Thmproofx19.p1.15.m5.2.2.2.3.cmml" xref="Thmproofx19.p1.15.m5.2.2.2.3"></minus><apply id="Thmproofx19.p1.15.m5.1.1.1.1.2.cmml" xref="Thmproofx19.p1.15.m5.1.1.1.1.1"><abs id="Thmproofx19.p1.15.m5.1.1.1.1.2.1.cmml" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.2"></abs><apply id="Thmproofx19.p1.15.m5.1.1.1.1.1.1.cmml" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.15.m5.1.1.1.1.1.1.1.cmml" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx19.p1.15.m5.1.1.1.1.1.1.2.cmml" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx19.p1.15.m5.1.1.1.1.1.1.3.cmml" xref="Thmproofx19.p1.15.m5.1.1.1.1.1.1.3">𝑚</ci></apply></apply><apply id="Thmproofx19.p1.15.m5.2.2.2.2.2.cmml" xref="Thmproofx19.p1.15.m5.2.2.2.2.1"><abs id="Thmproofx19.p1.15.m5.2.2.2.2.2.1.cmml" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.2"></abs><apply id="Thmproofx19.p1.15.m5.2.2.2.2.1.1.cmml" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.15.m5.2.2.2.2.1.1.1.cmml" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx19.p1.15.m5.2.2.2.2.1.1.2.cmml" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.1.2">𝑝</ci><cn id="Thmproofx19.p1.15.m5.2.2.2.2.1.1.3.cmml" type="integer" xref="Thmproofx19.p1.15.m5.2.2.2.2.1.1.3">1</cn></apply></apply></apply><ci id="Thmproofx19.p1.15.m5.2.2.4.cmml" xref="Thmproofx19.p1.15.m5.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.15.m5.2c">\alpha>\frac{|p_{m}|-|p_{1}|}{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.15.m5.2d">italic_α > divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>, so for that <math alttext="d" class="ltx_Math" display="inline" id="Thmproofx19.p1.16.m6.1"><semantics id="Thmproofx19.p1.16.m6.1a"><mi id="Thmproofx19.p1.16.m6.1.1" xref="Thmproofx19.p1.16.m6.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.16.m6.1b"><ci id="Thmproofx19.p1.16.m6.1.1.cmml" xref="Thmproofx19.p1.16.m6.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.16.m6.1c">d</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.16.m6.1d">italic_d</annotation></semantics></math> the unique longest path will be <math alttext="p^{d}_{m}" class="ltx_Math" display="inline" id="Thmproofx19.p1.17.m7.1"><semantics id="Thmproofx19.p1.17.m7.1a"><msubsup id="Thmproofx19.p1.17.m7.1.1" xref="Thmproofx19.p1.17.m7.1.1.cmml"><mi id="Thmproofx19.p1.17.m7.1.1.2.2" xref="Thmproofx19.p1.17.m7.1.1.2.2.cmml">p</mi><mi id="Thmproofx19.p1.17.m7.1.1.3" xref="Thmproofx19.p1.17.m7.1.1.3.cmml">m</mi><mi id="Thmproofx19.p1.17.m7.1.1.2.3" xref="Thmproofx19.p1.17.m7.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.17.m7.1b"><apply id="Thmproofx19.p1.17.m7.1.1.cmml" xref="Thmproofx19.p1.17.m7.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.17.m7.1.1.1.cmml" xref="Thmproofx19.p1.17.m7.1.1">subscript</csymbol><apply id="Thmproofx19.p1.17.m7.1.1.2.cmml" xref="Thmproofx19.p1.17.m7.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.17.m7.1.1.2.1.cmml" xref="Thmproofx19.p1.17.m7.1.1">superscript</csymbol><ci id="Thmproofx19.p1.17.m7.1.1.2.2.cmml" xref="Thmproofx19.p1.17.m7.1.1.2.2">𝑝</ci><ci id="Thmproofx19.p1.17.m7.1.1.2.3.cmml" xref="Thmproofx19.p1.17.m7.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.17.m7.1.1.3.cmml" xref="Thmproofx19.p1.17.m7.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.17.m7.1c">p^{d}_{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.17.m7.1d">italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> with length <math alttext="|p_{m}|+k\alpha" class="ltx_Math" display="inline" id="Thmproofx19.p1.18.m8.1"><semantics id="Thmproofx19.p1.18.m8.1a"><mrow id="Thmproofx19.p1.18.m8.1.1" xref="Thmproofx19.p1.18.m8.1.1.cmml"><mrow id="Thmproofx19.p1.18.m8.1.1.1.1" xref="Thmproofx19.p1.18.m8.1.1.1.2.cmml"><mo id="Thmproofx19.p1.18.m8.1.1.1.1.2" stretchy="false" xref="Thmproofx19.p1.18.m8.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx19.p1.18.m8.1.1.1.1.1" xref="Thmproofx19.p1.18.m8.1.1.1.1.1.cmml"><mi id="Thmproofx19.p1.18.m8.1.1.1.1.1.2" xref="Thmproofx19.p1.18.m8.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx19.p1.18.m8.1.1.1.1.1.3" xref="Thmproofx19.p1.18.m8.1.1.1.1.1.3.cmml">m</mi></msub><mo id="Thmproofx19.p1.18.m8.1.1.1.1.3" stretchy="false" xref="Thmproofx19.p1.18.m8.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx19.p1.18.m8.1.1.2" xref="Thmproofx19.p1.18.m8.1.1.2.cmml">+</mo><mrow id="Thmproofx19.p1.18.m8.1.1.3" xref="Thmproofx19.p1.18.m8.1.1.3.cmml"><mi id="Thmproofx19.p1.18.m8.1.1.3.2" xref="Thmproofx19.p1.18.m8.1.1.3.2.cmml">k</mi><mo id="Thmproofx19.p1.18.m8.1.1.3.1" xref="Thmproofx19.p1.18.m8.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx19.p1.18.m8.1.1.3.3" xref="Thmproofx19.p1.18.m8.1.1.3.3.cmml">α</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.18.m8.1b"><apply id="Thmproofx19.p1.18.m8.1.1.cmml" xref="Thmproofx19.p1.18.m8.1.1"><plus id="Thmproofx19.p1.18.m8.1.1.2.cmml" xref="Thmproofx19.p1.18.m8.1.1.2"></plus><apply id="Thmproofx19.p1.18.m8.1.1.1.2.cmml" xref="Thmproofx19.p1.18.m8.1.1.1.1"><abs id="Thmproofx19.p1.18.m8.1.1.1.2.1.cmml" xref="Thmproofx19.p1.18.m8.1.1.1.1.2"></abs><apply id="Thmproofx19.p1.18.m8.1.1.1.1.1.cmml" xref="Thmproofx19.p1.18.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.18.m8.1.1.1.1.1.1.cmml" xref="Thmproofx19.p1.18.m8.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx19.p1.18.m8.1.1.1.1.1.2.cmml" xref="Thmproofx19.p1.18.m8.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx19.p1.18.m8.1.1.1.1.1.3.cmml" xref="Thmproofx19.p1.18.m8.1.1.1.1.1.3">𝑚</ci></apply></apply><apply id="Thmproofx19.p1.18.m8.1.1.3.cmml" xref="Thmproofx19.p1.18.m8.1.1.3"><times id="Thmproofx19.p1.18.m8.1.1.3.1.cmml" xref="Thmproofx19.p1.18.m8.1.1.3.1"></times><ci id="Thmproofx19.p1.18.m8.1.1.3.2.cmml" xref="Thmproofx19.p1.18.m8.1.1.3.2">𝑘</ci><ci id="Thmproofx19.p1.18.m8.1.1.3.3.cmml" xref="Thmproofx19.p1.18.m8.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.18.m8.1c">|p_{m}|+k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.18.m8.1d">| italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | + italic_k italic_α</annotation></semantics></math>. On the other hand, if <math alttext="d\neq(m,...,m)" class="ltx_Math" display="inline" id="Thmproofx19.p1.19.m9.3"><semantics id="Thmproofx19.p1.19.m9.3a"><mrow id="Thmproofx19.p1.19.m9.3.4" xref="Thmproofx19.p1.19.m9.3.4.cmml"><mi id="Thmproofx19.p1.19.m9.3.4.2" xref="Thmproofx19.p1.19.m9.3.4.2.cmml">d</mi><mo id="Thmproofx19.p1.19.m9.3.4.1" xref="Thmproofx19.p1.19.m9.3.4.1.cmml">≠</mo><mrow id="Thmproofx19.p1.19.m9.3.4.3.2" xref="Thmproofx19.p1.19.m9.3.4.3.1.cmml"><mo id="Thmproofx19.p1.19.m9.3.4.3.2.1" stretchy="false" xref="Thmproofx19.p1.19.m9.3.4.3.1.cmml">(</mo><mi id="Thmproofx19.p1.19.m9.1.1" xref="Thmproofx19.p1.19.m9.1.1.cmml">m</mi><mo id="Thmproofx19.p1.19.m9.3.4.3.2.2" xref="Thmproofx19.p1.19.m9.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.19.m9.2.2" mathvariant="normal" xref="Thmproofx19.p1.19.m9.2.2.cmml">…</mi><mo id="Thmproofx19.p1.19.m9.3.4.3.2.3" xref="Thmproofx19.p1.19.m9.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.19.m9.3.3" xref="Thmproofx19.p1.19.m9.3.3.cmml">m</mi><mo id="Thmproofx19.p1.19.m9.3.4.3.2.4" stretchy="false" xref="Thmproofx19.p1.19.m9.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.19.m9.3b"><apply id="Thmproofx19.p1.19.m9.3.4.cmml" xref="Thmproofx19.p1.19.m9.3.4"><neq id="Thmproofx19.p1.19.m9.3.4.1.cmml" xref="Thmproofx19.p1.19.m9.3.4.1"></neq><ci id="Thmproofx19.p1.19.m9.3.4.2.cmml" xref="Thmproofx19.p1.19.m9.3.4.2">𝑑</ci><vector id="Thmproofx19.p1.19.m9.3.4.3.1.cmml" xref="Thmproofx19.p1.19.m9.3.4.3.2"><ci id="Thmproofx19.p1.19.m9.1.1.cmml" xref="Thmproofx19.p1.19.m9.1.1">𝑚</ci><ci id="Thmproofx19.p1.19.m9.2.2.cmml" xref="Thmproofx19.p1.19.m9.2.2">…</ci><ci id="Thmproofx19.p1.19.m9.3.3.cmml" xref="Thmproofx19.p1.19.m9.3.3">𝑚</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.19.m9.3c">d\neq(m,...,m)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.19.m9.3d">italic_d ≠ ( italic_m , … , italic_m )</annotation></semantics></math>, then <math alttext="k^{d}_{m}<k" class="ltx_Math" display="inline" id="Thmproofx19.p1.20.m10.1"><semantics id="Thmproofx19.p1.20.m10.1a"><mrow id="Thmproofx19.p1.20.m10.1.1" xref="Thmproofx19.p1.20.m10.1.1.cmml"><msubsup id="Thmproofx19.p1.20.m10.1.1.2" xref="Thmproofx19.p1.20.m10.1.1.2.cmml"><mi id="Thmproofx19.p1.20.m10.1.1.2.2.2" xref="Thmproofx19.p1.20.m10.1.1.2.2.2.cmml">k</mi><mi id="Thmproofx19.p1.20.m10.1.1.2.3" xref="Thmproofx19.p1.20.m10.1.1.2.3.cmml">m</mi><mi id="Thmproofx19.p1.20.m10.1.1.2.2.3" xref="Thmproofx19.p1.20.m10.1.1.2.2.3.cmml">d</mi></msubsup><mo id="Thmproofx19.p1.20.m10.1.1.1" xref="Thmproofx19.p1.20.m10.1.1.1.cmml"><</mo><mi id="Thmproofx19.p1.20.m10.1.1.3" xref="Thmproofx19.p1.20.m10.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.20.m10.1b"><apply id="Thmproofx19.p1.20.m10.1.1.cmml" xref="Thmproofx19.p1.20.m10.1.1"><lt id="Thmproofx19.p1.20.m10.1.1.1.cmml" xref="Thmproofx19.p1.20.m10.1.1.1"></lt><apply id="Thmproofx19.p1.20.m10.1.1.2.cmml" xref="Thmproofx19.p1.20.m10.1.1.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.20.m10.1.1.2.1.cmml" xref="Thmproofx19.p1.20.m10.1.1.2">subscript</csymbol><apply id="Thmproofx19.p1.20.m10.1.1.2.2.cmml" xref="Thmproofx19.p1.20.m10.1.1.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.20.m10.1.1.2.2.1.cmml" xref="Thmproofx19.p1.20.m10.1.1.2">superscript</csymbol><ci id="Thmproofx19.p1.20.m10.1.1.2.2.2.cmml" xref="Thmproofx19.p1.20.m10.1.1.2.2.2">𝑘</ci><ci id="Thmproofx19.p1.20.m10.1.1.2.2.3.cmml" xref="Thmproofx19.p1.20.m10.1.1.2.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.20.m10.1.1.2.3.cmml" xref="Thmproofx19.p1.20.m10.1.1.2.3">𝑚</ci></apply><ci id="Thmproofx19.p1.20.m10.1.1.3.cmml" xref="Thmproofx19.p1.20.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.20.m10.1c">k^{d}_{m}<k</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.20.m10.1d">italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT < italic_k</annotation></semantics></math> because the interceptors did not all pick edges from the same path. Therefore, the coefficient of <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx19.p1.21.m11.1"><semantics id="Thmproofx19.p1.21.m11.1a"><mi id="Thmproofx19.p1.21.m11.1.1" xref="Thmproofx19.p1.21.m11.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.21.m11.1b"><ci id="Thmproofx19.p1.21.m11.1.1.cmml" xref="Thmproofx19.p1.21.m11.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.21.m11.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.21.m11.1d">italic_α</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex83" title="Proof (Proof of Lemma 14) ‣ 0.D.4 Omitted proofs ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.D.4</span></a>) is strictly maximal iff <math alttext="d=(m,...,m)" class="ltx_Math" display="inline" id="Thmproofx19.p1.22.m12.3"><semantics id="Thmproofx19.p1.22.m12.3a"><mrow id="Thmproofx19.p1.22.m12.3.4" xref="Thmproofx19.p1.22.m12.3.4.cmml"><mi id="Thmproofx19.p1.22.m12.3.4.2" xref="Thmproofx19.p1.22.m12.3.4.2.cmml">d</mi><mo id="Thmproofx19.p1.22.m12.3.4.1" xref="Thmproofx19.p1.22.m12.3.4.1.cmml">=</mo><mrow id="Thmproofx19.p1.22.m12.3.4.3.2" xref="Thmproofx19.p1.22.m12.3.4.3.1.cmml"><mo id="Thmproofx19.p1.22.m12.3.4.3.2.1" stretchy="false" xref="Thmproofx19.p1.22.m12.3.4.3.1.cmml">(</mo><mi id="Thmproofx19.p1.22.m12.1.1" xref="Thmproofx19.p1.22.m12.1.1.cmml">m</mi><mo id="Thmproofx19.p1.22.m12.3.4.3.2.2" xref="Thmproofx19.p1.22.m12.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.22.m12.2.2" mathvariant="normal" xref="Thmproofx19.p1.22.m12.2.2.cmml">…</mi><mo id="Thmproofx19.p1.22.m12.3.4.3.2.3" xref="Thmproofx19.p1.22.m12.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p1.22.m12.3.3" xref="Thmproofx19.p1.22.m12.3.3.cmml">m</mi><mo id="Thmproofx19.p1.22.m12.3.4.3.2.4" stretchy="false" xref="Thmproofx19.p1.22.m12.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.22.m12.3b"><apply id="Thmproofx19.p1.22.m12.3.4.cmml" xref="Thmproofx19.p1.22.m12.3.4"><eq id="Thmproofx19.p1.22.m12.3.4.1.cmml" xref="Thmproofx19.p1.22.m12.3.4.1"></eq><ci id="Thmproofx19.p1.22.m12.3.4.2.cmml" xref="Thmproofx19.p1.22.m12.3.4.2">𝑑</ci><vector id="Thmproofx19.p1.22.m12.3.4.3.1.cmml" xref="Thmproofx19.p1.22.m12.3.4.3.2"><ci id="Thmproofx19.p1.22.m12.1.1.cmml" xref="Thmproofx19.p1.22.m12.1.1">𝑚</ci><ci id="Thmproofx19.p1.22.m12.2.2.cmml" xref="Thmproofx19.p1.22.m12.2.2">…</ci><ci id="Thmproofx19.p1.22.m12.3.3.cmml" xref="Thmproofx19.p1.22.m12.3.3">𝑚</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.22.m12.3c">d=(m,...,m)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.22.m12.3d">italic_d = ( italic_m , … , italic_m )</annotation></semantics></math>. As for <math alttext="\sum_{i=1}^{m}|p_{I_{d}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}" class="ltx_Math" display="inline" id="Thmproofx19.p1.23.m13.3"><semantics id="Thmproofx19.p1.23.m13.3a"><mrow id="Thmproofx19.p1.23.m13.3.3" xref="Thmproofx19.p1.23.m13.3.3.cmml"><msubsup id="Thmproofx19.p1.23.m13.3.3.2" xref="Thmproofx19.p1.23.m13.3.3.2.cmml"><mo id="Thmproofx19.p1.23.m13.3.3.2.2.2" xref="Thmproofx19.p1.23.m13.3.3.2.2.2.cmml">∑</mo><mrow id="Thmproofx19.p1.23.m13.3.3.2.2.3" xref="Thmproofx19.p1.23.m13.3.3.2.2.3.cmml"><mi id="Thmproofx19.p1.23.m13.3.3.2.2.3.2" xref="Thmproofx19.p1.23.m13.3.3.2.2.3.2.cmml">i</mi><mo id="Thmproofx19.p1.23.m13.3.3.2.2.3.1" xref="Thmproofx19.p1.23.m13.3.3.2.2.3.1.cmml">=</mo><mn id="Thmproofx19.p1.23.m13.3.3.2.2.3.3" xref="Thmproofx19.p1.23.m13.3.3.2.2.3.3.cmml">1</mn></mrow><mi id="Thmproofx19.p1.23.m13.3.3.2.3" xref="Thmproofx19.p1.23.m13.3.3.2.3.cmml">m</mi></msubsup><mrow id="Thmproofx19.p1.23.m13.3.3.1" xref="Thmproofx19.p1.23.m13.3.3.1.cmml"><mrow id="Thmproofx19.p1.23.m13.3.3.1.1.1" xref="Thmproofx19.p1.23.m13.3.3.1.1.2.cmml"><mo id="Thmproofx19.p1.23.m13.3.3.1.1.1.2" lspace="0em" stretchy="false" xref="Thmproofx19.p1.23.m13.3.3.1.1.2.1.cmml">|</mo><msub id="Thmproofx19.p1.23.m13.3.3.1.1.1.1" xref="Thmproofx19.p1.23.m13.3.3.1.1.1.1.cmml"><mi id="Thmproofx19.p1.23.m13.3.3.1.1.1.1.2" xref="Thmproofx19.p1.23.m13.3.3.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx19.p1.23.m13.1.1.1" xref="Thmproofx19.p1.23.m13.1.1.1.cmml"><msub id="Thmproofx19.p1.23.m13.1.1.1.3" xref="Thmproofx19.p1.23.m13.1.1.1.3.cmml"><mi id="Thmproofx19.p1.23.m13.1.1.1.3.2" xref="Thmproofx19.p1.23.m13.1.1.1.3.2.cmml">I</mi><mi id="Thmproofx19.p1.23.m13.1.1.1.3.3" xref="Thmproofx19.p1.23.m13.1.1.1.3.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.23.m13.1.1.1.2" xref="Thmproofx19.p1.23.m13.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx19.p1.23.m13.1.1.1.4.2" xref="Thmproofx19.p1.23.m13.1.1.1.cmml"><mo id="Thmproofx19.p1.23.m13.1.1.1.4.2.1" stretchy="false" xref="Thmproofx19.p1.23.m13.1.1.1.cmml">(</mo><mi id="Thmproofx19.p1.23.m13.1.1.1.1" xref="Thmproofx19.p1.23.m13.1.1.1.1.cmml">i</mi><mo id="Thmproofx19.p1.23.m13.1.1.1.4.2.2" stretchy="false" xref="Thmproofx19.p1.23.m13.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="Thmproofx19.p1.23.m13.3.3.1.1.1.3" stretchy="false" xref="Thmproofx19.p1.23.m13.3.3.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx19.p1.23.m13.3.3.1.2" xref="Thmproofx19.p1.23.m13.3.3.1.2.cmml">⁢</mo><mfrac id="Thmproofx19.p1.23.m13.2.2" xref="Thmproofx19.p1.23.m13.2.2.cmml"><mrow id="Thmproofx19.p1.23.m13.2.2.1" xref="Thmproofx19.p1.23.m13.2.2.1.cmml"><msup id="Thmproofx19.p1.23.m13.2.2.1.3" xref="Thmproofx19.p1.23.m13.2.2.1.3.cmml"><mi id="Thmproofx19.p1.23.m13.2.2.1.3.2" xref="Thmproofx19.p1.23.m13.2.2.1.3.2.cmml">i</mi><mi id="Thmproofx19.p1.23.m13.2.2.1.3.3" xref="Thmproofx19.p1.23.m13.2.2.1.3.3.cmml">n</mi></msup><mo id="Thmproofx19.p1.23.m13.2.2.1.2" xref="Thmproofx19.p1.23.m13.2.2.1.2.cmml">−</mo><msup id="Thmproofx19.p1.23.m13.2.2.1.1" xref="Thmproofx19.p1.23.m13.2.2.1.1.cmml"><mrow id="Thmproofx19.p1.23.m13.2.2.1.1.1.1" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.cmml"><mo id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.2" stretchy="false" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.cmml"><mi id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.2" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.2.cmml">i</mi><mo id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.1" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.1.cmml">−</mo><mn id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.3" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.3" stretchy="false" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx19.p1.23.m13.2.2.1.1.3" xref="Thmproofx19.p1.23.m13.2.2.1.1.3.cmml">n</mi></msup></mrow><msup id="Thmproofx19.p1.23.m13.2.2.3" xref="Thmproofx19.p1.23.m13.2.2.3.cmml"><mi id="Thmproofx19.p1.23.m13.2.2.3.2" xref="Thmproofx19.p1.23.m13.2.2.3.2.cmml">m</mi><mi id="Thmproofx19.p1.23.m13.2.2.3.3" xref="Thmproofx19.p1.23.m13.2.2.3.3.cmml">n</mi></msup></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.23.m13.3b"><apply id="Thmproofx19.p1.23.m13.3.3.cmml" xref="Thmproofx19.p1.23.m13.3.3"><apply id="Thmproofx19.p1.23.m13.3.3.2.cmml" xref="Thmproofx19.p1.23.m13.3.3.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.23.m13.3.3.2.1.cmml" xref="Thmproofx19.p1.23.m13.3.3.2">superscript</csymbol><apply id="Thmproofx19.p1.23.m13.3.3.2.2.cmml" xref="Thmproofx19.p1.23.m13.3.3.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.23.m13.3.3.2.2.1.cmml" xref="Thmproofx19.p1.23.m13.3.3.2">subscript</csymbol><sum id="Thmproofx19.p1.23.m13.3.3.2.2.2.cmml" xref="Thmproofx19.p1.23.m13.3.3.2.2.2"></sum><apply id="Thmproofx19.p1.23.m13.3.3.2.2.3.cmml" xref="Thmproofx19.p1.23.m13.3.3.2.2.3"><eq id="Thmproofx19.p1.23.m13.3.3.2.2.3.1.cmml" xref="Thmproofx19.p1.23.m13.3.3.2.2.3.1"></eq><ci id="Thmproofx19.p1.23.m13.3.3.2.2.3.2.cmml" xref="Thmproofx19.p1.23.m13.3.3.2.2.3.2">𝑖</ci><cn id="Thmproofx19.p1.23.m13.3.3.2.2.3.3.cmml" type="integer" xref="Thmproofx19.p1.23.m13.3.3.2.2.3.3">1</cn></apply></apply><ci id="Thmproofx19.p1.23.m13.3.3.2.3.cmml" xref="Thmproofx19.p1.23.m13.3.3.2.3">𝑚</ci></apply><apply id="Thmproofx19.p1.23.m13.3.3.1.cmml" xref="Thmproofx19.p1.23.m13.3.3.1"><times id="Thmproofx19.p1.23.m13.3.3.1.2.cmml" xref="Thmproofx19.p1.23.m13.3.3.1.2"></times><apply id="Thmproofx19.p1.23.m13.3.3.1.1.2.cmml" xref="Thmproofx19.p1.23.m13.3.3.1.1.1"><abs id="Thmproofx19.p1.23.m13.3.3.1.1.2.1.cmml" xref="Thmproofx19.p1.23.m13.3.3.1.1.1.2"></abs><apply id="Thmproofx19.p1.23.m13.3.3.1.1.1.1.cmml" xref="Thmproofx19.p1.23.m13.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.23.m13.3.3.1.1.1.1.1.cmml" xref="Thmproofx19.p1.23.m13.3.3.1.1.1.1">subscript</csymbol><ci id="Thmproofx19.p1.23.m13.3.3.1.1.1.1.2.cmml" xref="Thmproofx19.p1.23.m13.3.3.1.1.1.1.2">𝑝</ci><apply id="Thmproofx19.p1.23.m13.1.1.1.cmml" xref="Thmproofx19.p1.23.m13.1.1.1"><times id="Thmproofx19.p1.23.m13.1.1.1.2.cmml" xref="Thmproofx19.p1.23.m13.1.1.1.2"></times><apply id="Thmproofx19.p1.23.m13.1.1.1.3.cmml" xref="Thmproofx19.p1.23.m13.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.23.m13.1.1.1.3.1.cmml" xref="Thmproofx19.p1.23.m13.1.1.1.3">subscript</csymbol><ci id="Thmproofx19.p1.23.m13.1.1.1.3.2.cmml" xref="Thmproofx19.p1.23.m13.1.1.1.3.2">𝐼</ci><ci id="Thmproofx19.p1.23.m13.1.1.1.3.3.cmml" xref="Thmproofx19.p1.23.m13.1.1.1.3.3">𝑑</ci></apply><ci id="Thmproofx19.p1.23.m13.1.1.1.1.cmml" xref="Thmproofx19.p1.23.m13.1.1.1.1">𝑖</ci></apply></apply></apply><apply id="Thmproofx19.p1.23.m13.2.2.cmml" xref="Thmproofx19.p1.23.m13.2.2"><divide id="Thmproofx19.p1.23.m13.2.2.2.cmml" xref="Thmproofx19.p1.23.m13.2.2"></divide><apply id="Thmproofx19.p1.23.m13.2.2.1.cmml" xref="Thmproofx19.p1.23.m13.2.2.1"><minus id="Thmproofx19.p1.23.m13.2.2.1.2.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.2"></minus><apply id="Thmproofx19.p1.23.m13.2.2.1.3.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.23.m13.2.2.1.3.1.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.3">superscript</csymbol><ci id="Thmproofx19.p1.23.m13.2.2.1.3.2.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.3.2">𝑖</ci><ci id="Thmproofx19.p1.23.m13.2.2.1.3.3.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.3.3">𝑛</ci></apply><apply id="Thmproofx19.p1.23.m13.2.2.1.1.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.23.m13.2.2.1.1.2.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.1">superscript</csymbol><apply id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1"><minus id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.1.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.1"></minus><ci id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.2.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.2">𝑖</ci><cn id="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx19.p1.23.m13.2.2.1.1.1.1.1.3">1</cn></apply><ci id="Thmproofx19.p1.23.m13.2.2.1.1.3.cmml" xref="Thmproofx19.p1.23.m13.2.2.1.1.3">𝑛</ci></apply></apply><apply id="Thmproofx19.p1.23.m13.2.2.3.cmml" xref="Thmproofx19.p1.23.m13.2.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.23.m13.2.2.3.1.cmml" xref="Thmproofx19.p1.23.m13.2.2.3">superscript</csymbol><ci id="Thmproofx19.p1.23.m13.2.2.3.2.cmml" xref="Thmproofx19.p1.23.m13.2.2.3.2">𝑚</ci><ci id="Thmproofx19.p1.23.m13.2.2.3.3.cmml" xref="Thmproofx19.p1.23.m13.2.2.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.23.m13.3c">\sum_{i=1}^{m}|p_{I_{d}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.23.m13.3d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT | divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>, because <math alttext="|p_{1}|\leq...\leq|p_{m}|" class="ltx_Math" display="inline" id="Thmproofx19.p1.24.m14.2"><semantics id="Thmproofx19.p1.24.m14.2a"><mrow id="Thmproofx19.p1.24.m14.2.2" xref="Thmproofx19.p1.24.m14.2.2.cmml"><mrow id="Thmproofx19.p1.24.m14.1.1.1.1" xref="Thmproofx19.p1.24.m14.1.1.1.2.cmml"><mo id="Thmproofx19.p1.24.m14.1.1.1.1.2" stretchy="false" xref="Thmproofx19.p1.24.m14.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx19.p1.24.m14.1.1.1.1.1" xref="Thmproofx19.p1.24.m14.1.1.1.1.1.cmml"><mi id="Thmproofx19.p1.24.m14.1.1.1.1.1.2" xref="Thmproofx19.p1.24.m14.1.1.1.1.1.2.cmml">p</mi><mn id="Thmproofx19.p1.24.m14.1.1.1.1.1.3" xref="Thmproofx19.p1.24.m14.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx19.p1.24.m14.1.1.1.1.3" stretchy="false" xref="Thmproofx19.p1.24.m14.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx19.p1.24.m14.2.2.4" xref="Thmproofx19.p1.24.m14.2.2.4.cmml">≤</mo><mi id="Thmproofx19.p1.24.m14.2.2.5" mathvariant="normal" xref="Thmproofx19.p1.24.m14.2.2.5.cmml">…</mi><mo id="Thmproofx19.p1.24.m14.2.2.6" xref="Thmproofx19.p1.24.m14.2.2.6.cmml">≤</mo><mrow id="Thmproofx19.p1.24.m14.2.2.2.1" xref="Thmproofx19.p1.24.m14.2.2.2.2.cmml"><mo id="Thmproofx19.p1.24.m14.2.2.2.1.2" stretchy="false" xref="Thmproofx19.p1.24.m14.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx19.p1.24.m14.2.2.2.1.1" xref="Thmproofx19.p1.24.m14.2.2.2.1.1.cmml"><mi id="Thmproofx19.p1.24.m14.2.2.2.1.1.2" xref="Thmproofx19.p1.24.m14.2.2.2.1.1.2.cmml">p</mi><mi id="Thmproofx19.p1.24.m14.2.2.2.1.1.3" xref="Thmproofx19.p1.24.m14.2.2.2.1.1.3.cmml">m</mi></msub><mo id="Thmproofx19.p1.24.m14.2.2.2.1.3" stretchy="false" xref="Thmproofx19.p1.24.m14.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.24.m14.2b"><apply id="Thmproofx19.p1.24.m14.2.2.cmml" xref="Thmproofx19.p1.24.m14.2.2"><and id="Thmproofx19.p1.24.m14.2.2a.cmml" xref="Thmproofx19.p1.24.m14.2.2"></and><apply id="Thmproofx19.p1.24.m14.2.2b.cmml" xref="Thmproofx19.p1.24.m14.2.2"><leq id="Thmproofx19.p1.24.m14.2.2.4.cmml" xref="Thmproofx19.p1.24.m14.2.2.4"></leq><apply id="Thmproofx19.p1.24.m14.1.1.1.2.cmml" xref="Thmproofx19.p1.24.m14.1.1.1.1"><abs id="Thmproofx19.p1.24.m14.1.1.1.2.1.cmml" xref="Thmproofx19.p1.24.m14.1.1.1.1.2"></abs><apply id="Thmproofx19.p1.24.m14.1.1.1.1.1.cmml" xref="Thmproofx19.p1.24.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.24.m14.1.1.1.1.1.1.cmml" xref="Thmproofx19.p1.24.m14.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx19.p1.24.m14.1.1.1.1.1.2.cmml" xref="Thmproofx19.p1.24.m14.1.1.1.1.1.2">𝑝</ci><cn id="Thmproofx19.p1.24.m14.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx19.p1.24.m14.1.1.1.1.1.3">1</cn></apply></apply><ci id="Thmproofx19.p1.24.m14.2.2.5.cmml" xref="Thmproofx19.p1.24.m14.2.2.5">…</ci></apply><apply id="Thmproofx19.p1.24.m14.2.2c.cmml" xref="Thmproofx19.p1.24.m14.2.2"><leq id="Thmproofx19.p1.24.m14.2.2.6.cmml" xref="Thmproofx19.p1.24.m14.2.2.6"></leq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx19.p1.24.m14.2.2.5.cmml" id="Thmproofx19.p1.24.m14.2.2d.cmml" xref="Thmproofx19.p1.24.m14.2.2"></share><apply id="Thmproofx19.p1.24.m14.2.2.2.2.cmml" xref="Thmproofx19.p1.24.m14.2.2.2.1"><abs id="Thmproofx19.p1.24.m14.2.2.2.2.1.cmml" xref="Thmproofx19.p1.24.m14.2.2.2.1.2"></abs><apply id="Thmproofx19.p1.24.m14.2.2.2.1.1.cmml" xref="Thmproofx19.p1.24.m14.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.24.m14.2.2.2.1.1.1.cmml" xref="Thmproofx19.p1.24.m14.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx19.p1.24.m14.2.2.2.1.1.2.cmml" xref="Thmproofx19.p1.24.m14.2.2.2.1.1.2">𝑝</ci><ci id="Thmproofx19.p1.24.m14.2.2.2.1.1.3.cmml" xref="Thmproofx19.p1.24.m14.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.24.m14.2c">|p_{1}|\leq...\leq|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.24.m14.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ … ≤ | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math>, we argue that this term is maximal when <math alttext="I_{d}(i)=i" class="ltx_Math" display="inline" id="Thmproofx19.p1.25.m15.1"><semantics id="Thmproofx19.p1.25.m15.1a"><mrow id="Thmproofx19.p1.25.m15.1.2" xref="Thmproofx19.p1.25.m15.1.2.cmml"><mrow id="Thmproofx19.p1.25.m15.1.2.2" xref="Thmproofx19.p1.25.m15.1.2.2.cmml"><msub id="Thmproofx19.p1.25.m15.1.2.2.2" xref="Thmproofx19.p1.25.m15.1.2.2.2.cmml"><mi id="Thmproofx19.p1.25.m15.1.2.2.2.2" xref="Thmproofx19.p1.25.m15.1.2.2.2.2.cmml">I</mi><mi id="Thmproofx19.p1.25.m15.1.2.2.2.3" xref="Thmproofx19.p1.25.m15.1.2.2.2.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.25.m15.1.2.2.1" xref="Thmproofx19.p1.25.m15.1.2.2.1.cmml">⁢</mo><mrow id="Thmproofx19.p1.25.m15.1.2.2.3.2" xref="Thmproofx19.p1.25.m15.1.2.2.cmml"><mo id="Thmproofx19.p1.25.m15.1.2.2.3.2.1" stretchy="false" xref="Thmproofx19.p1.25.m15.1.2.2.cmml">(</mo><mi id="Thmproofx19.p1.25.m15.1.1" xref="Thmproofx19.p1.25.m15.1.1.cmml">i</mi><mo id="Thmproofx19.p1.25.m15.1.2.2.3.2.2" stretchy="false" xref="Thmproofx19.p1.25.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx19.p1.25.m15.1.2.1" xref="Thmproofx19.p1.25.m15.1.2.1.cmml">=</mo><mi id="Thmproofx19.p1.25.m15.1.2.3" xref="Thmproofx19.p1.25.m15.1.2.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.25.m15.1b"><apply id="Thmproofx19.p1.25.m15.1.2.cmml" xref="Thmproofx19.p1.25.m15.1.2"><eq id="Thmproofx19.p1.25.m15.1.2.1.cmml" xref="Thmproofx19.p1.25.m15.1.2.1"></eq><apply id="Thmproofx19.p1.25.m15.1.2.2.cmml" xref="Thmproofx19.p1.25.m15.1.2.2"><times id="Thmproofx19.p1.25.m15.1.2.2.1.cmml" xref="Thmproofx19.p1.25.m15.1.2.2.1"></times><apply id="Thmproofx19.p1.25.m15.1.2.2.2.cmml" xref="Thmproofx19.p1.25.m15.1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.25.m15.1.2.2.2.1.cmml" xref="Thmproofx19.p1.25.m15.1.2.2.2">subscript</csymbol><ci id="Thmproofx19.p1.25.m15.1.2.2.2.2.cmml" xref="Thmproofx19.p1.25.m15.1.2.2.2.2">𝐼</ci><ci id="Thmproofx19.p1.25.m15.1.2.2.2.3.cmml" xref="Thmproofx19.p1.25.m15.1.2.2.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.25.m15.1.1.cmml" xref="Thmproofx19.p1.25.m15.1.1">𝑖</ci></apply><ci id="Thmproofx19.p1.25.m15.1.2.3.cmml" xref="Thmproofx19.p1.25.m15.1.2.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.25.m15.1c">I_{d}(i)=i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.25.m15.1d">italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) = italic_i</annotation></semantics></math>, up to reordering of paths of identical lengths: Suppose that <math alttext="I_{d}(i)\neq i" class="ltx_Math" display="inline" id="Thmproofx19.p1.26.m16.1"><semantics id="Thmproofx19.p1.26.m16.1a"><mrow id="Thmproofx19.p1.26.m16.1.2" xref="Thmproofx19.p1.26.m16.1.2.cmml"><mrow id="Thmproofx19.p1.26.m16.1.2.2" xref="Thmproofx19.p1.26.m16.1.2.2.cmml"><msub id="Thmproofx19.p1.26.m16.1.2.2.2" xref="Thmproofx19.p1.26.m16.1.2.2.2.cmml"><mi id="Thmproofx19.p1.26.m16.1.2.2.2.2" xref="Thmproofx19.p1.26.m16.1.2.2.2.2.cmml">I</mi><mi id="Thmproofx19.p1.26.m16.1.2.2.2.3" xref="Thmproofx19.p1.26.m16.1.2.2.2.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.26.m16.1.2.2.1" xref="Thmproofx19.p1.26.m16.1.2.2.1.cmml">⁢</mo><mrow id="Thmproofx19.p1.26.m16.1.2.2.3.2" xref="Thmproofx19.p1.26.m16.1.2.2.cmml"><mo id="Thmproofx19.p1.26.m16.1.2.2.3.2.1" stretchy="false" xref="Thmproofx19.p1.26.m16.1.2.2.cmml">(</mo><mi id="Thmproofx19.p1.26.m16.1.1" xref="Thmproofx19.p1.26.m16.1.1.cmml">i</mi><mo id="Thmproofx19.p1.26.m16.1.2.2.3.2.2" stretchy="false" xref="Thmproofx19.p1.26.m16.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx19.p1.26.m16.1.2.1" xref="Thmproofx19.p1.26.m16.1.2.1.cmml">≠</mo><mi id="Thmproofx19.p1.26.m16.1.2.3" xref="Thmproofx19.p1.26.m16.1.2.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.26.m16.1b"><apply id="Thmproofx19.p1.26.m16.1.2.cmml" xref="Thmproofx19.p1.26.m16.1.2"><neq id="Thmproofx19.p1.26.m16.1.2.1.cmml" xref="Thmproofx19.p1.26.m16.1.2.1"></neq><apply id="Thmproofx19.p1.26.m16.1.2.2.cmml" xref="Thmproofx19.p1.26.m16.1.2.2"><times id="Thmproofx19.p1.26.m16.1.2.2.1.cmml" xref="Thmproofx19.p1.26.m16.1.2.2.1"></times><apply id="Thmproofx19.p1.26.m16.1.2.2.2.cmml" xref="Thmproofx19.p1.26.m16.1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.26.m16.1.2.2.2.1.cmml" xref="Thmproofx19.p1.26.m16.1.2.2.2">subscript</csymbol><ci id="Thmproofx19.p1.26.m16.1.2.2.2.2.cmml" xref="Thmproofx19.p1.26.m16.1.2.2.2.2">𝐼</ci><ci id="Thmproofx19.p1.26.m16.1.2.2.2.3.cmml" xref="Thmproofx19.p1.26.m16.1.2.2.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.26.m16.1.1.cmml" xref="Thmproofx19.p1.26.m16.1.1">𝑖</ci></apply><ci id="Thmproofx19.p1.26.m16.1.2.3.cmml" xref="Thmproofx19.p1.26.m16.1.2.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.26.m16.1c">I_{d}(i)\neq i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.26.m16.1d">italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) ≠ italic_i</annotation></semantics></math> for some <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx19.p1.27.m17.1"><semantics id="Thmproofx19.p1.27.m17.1a"><mi id="Thmproofx19.p1.27.m17.1.1" xref="Thmproofx19.p1.27.m17.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.27.m17.1b"><ci id="Thmproofx19.p1.27.m17.1.1.cmml" xref="Thmproofx19.p1.27.m17.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.27.m17.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.27.m17.1d">italic_i</annotation></semantics></math>. Then <math alttext="|p_{I_{d}(j)}|>|p_{I_{d}(j+1)}|" class="ltx_Math" display="inline" id="Thmproofx19.p1.28.m18.4"><semantics id="Thmproofx19.p1.28.m18.4a"><mrow id="Thmproofx19.p1.28.m18.4.4" xref="Thmproofx19.p1.28.m18.4.4.cmml"><mrow id="Thmproofx19.p1.28.m18.3.3.1.1" xref="Thmproofx19.p1.28.m18.3.3.1.2.cmml"><mo id="Thmproofx19.p1.28.m18.3.3.1.1.2" stretchy="false" xref="Thmproofx19.p1.28.m18.3.3.1.2.1.cmml">|</mo><msub id="Thmproofx19.p1.28.m18.3.3.1.1.1" xref="Thmproofx19.p1.28.m18.3.3.1.1.1.cmml"><mi id="Thmproofx19.p1.28.m18.3.3.1.1.1.2" xref="Thmproofx19.p1.28.m18.3.3.1.1.1.2.cmml">p</mi><mrow id="Thmproofx19.p1.28.m18.1.1.1" xref="Thmproofx19.p1.28.m18.1.1.1.cmml"><msub id="Thmproofx19.p1.28.m18.1.1.1.3" xref="Thmproofx19.p1.28.m18.1.1.1.3.cmml"><mi id="Thmproofx19.p1.28.m18.1.1.1.3.2" xref="Thmproofx19.p1.28.m18.1.1.1.3.2.cmml">I</mi><mi id="Thmproofx19.p1.28.m18.1.1.1.3.3" xref="Thmproofx19.p1.28.m18.1.1.1.3.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.28.m18.1.1.1.2" xref="Thmproofx19.p1.28.m18.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx19.p1.28.m18.1.1.1.4.2" xref="Thmproofx19.p1.28.m18.1.1.1.cmml"><mo id="Thmproofx19.p1.28.m18.1.1.1.4.2.1" stretchy="false" xref="Thmproofx19.p1.28.m18.1.1.1.cmml">(</mo><mi id="Thmproofx19.p1.28.m18.1.1.1.1" xref="Thmproofx19.p1.28.m18.1.1.1.1.cmml">j</mi><mo id="Thmproofx19.p1.28.m18.1.1.1.4.2.2" stretchy="false" xref="Thmproofx19.p1.28.m18.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="Thmproofx19.p1.28.m18.3.3.1.1.3" stretchy="false" xref="Thmproofx19.p1.28.m18.3.3.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx19.p1.28.m18.4.4.3" xref="Thmproofx19.p1.28.m18.4.4.3.cmml">></mo><mrow id="Thmproofx19.p1.28.m18.4.4.2.1" xref="Thmproofx19.p1.28.m18.4.4.2.2.cmml"><mo id="Thmproofx19.p1.28.m18.4.4.2.1.2" stretchy="false" xref="Thmproofx19.p1.28.m18.4.4.2.2.1.cmml">|</mo><msub id="Thmproofx19.p1.28.m18.4.4.2.1.1" xref="Thmproofx19.p1.28.m18.4.4.2.1.1.cmml"><mi id="Thmproofx19.p1.28.m18.4.4.2.1.1.2" xref="Thmproofx19.p1.28.m18.4.4.2.1.1.2.cmml">p</mi><mrow id="Thmproofx19.p1.28.m18.2.2.1" xref="Thmproofx19.p1.28.m18.2.2.1.cmml"><msub id="Thmproofx19.p1.28.m18.2.2.1.3" xref="Thmproofx19.p1.28.m18.2.2.1.3.cmml"><mi id="Thmproofx19.p1.28.m18.2.2.1.3.2" xref="Thmproofx19.p1.28.m18.2.2.1.3.2.cmml">I</mi><mi id="Thmproofx19.p1.28.m18.2.2.1.3.3" xref="Thmproofx19.p1.28.m18.2.2.1.3.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.28.m18.2.2.1.2" xref="Thmproofx19.p1.28.m18.2.2.1.2.cmml">⁢</mo><mrow id="Thmproofx19.p1.28.m18.2.2.1.1.1" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.cmml"><mo id="Thmproofx19.p1.28.m18.2.2.1.1.1.2" stretchy="false" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.cmml">(</mo><mrow id="Thmproofx19.p1.28.m18.2.2.1.1.1.1" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.cmml"><mi id="Thmproofx19.p1.28.m18.2.2.1.1.1.1.2" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.2.cmml">j</mi><mo id="Thmproofx19.p1.28.m18.2.2.1.1.1.1.1" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.1.cmml">+</mo><mn id="Thmproofx19.p1.28.m18.2.2.1.1.1.1.3" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="Thmproofx19.p1.28.m18.2.2.1.1.1.3" stretchy="false" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="Thmproofx19.p1.28.m18.4.4.2.1.3" stretchy="false" xref="Thmproofx19.p1.28.m18.4.4.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.28.m18.4b"><apply id="Thmproofx19.p1.28.m18.4.4.cmml" xref="Thmproofx19.p1.28.m18.4.4"><gt id="Thmproofx19.p1.28.m18.4.4.3.cmml" xref="Thmproofx19.p1.28.m18.4.4.3"></gt><apply id="Thmproofx19.p1.28.m18.3.3.1.2.cmml" xref="Thmproofx19.p1.28.m18.3.3.1.1"><abs id="Thmproofx19.p1.28.m18.3.3.1.2.1.cmml" xref="Thmproofx19.p1.28.m18.3.3.1.1.2"></abs><apply id="Thmproofx19.p1.28.m18.3.3.1.1.1.cmml" xref="Thmproofx19.p1.28.m18.3.3.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.28.m18.3.3.1.1.1.1.cmml" xref="Thmproofx19.p1.28.m18.3.3.1.1.1">subscript</csymbol><ci id="Thmproofx19.p1.28.m18.3.3.1.1.1.2.cmml" xref="Thmproofx19.p1.28.m18.3.3.1.1.1.2">𝑝</ci><apply id="Thmproofx19.p1.28.m18.1.1.1.cmml" xref="Thmproofx19.p1.28.m18.1.1.1"><times id="Thmproofx19.p1.28.m18.1.1.1.2.cmml" xref="Thmproofx19.p1.28.m18.1.1.1.2"></times><apply id="Thmproofx19.p1.28.m18.1.1.1.3.cmml" xref="Thmproofx19.p1.28.m18.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.28.m18.1.1.1.3.1.cmml" xref="Thmproofx19.p1.28.m18.1.1.1.3">subscript</csymbol><ci id="Thmproofx19.p1.28.m18.1.1.1.3.2.cmml" xref="Thmproofx19.p1.28.m18.1.1.1.3.2">𝐼</ci><ci id="Thmproofx19.p1.28.m18.1.1.1.3.3.cmml" xref="Thmproofx19.p1.28.m18.1.1.1.3.3">𝑑</ci></apply><ci id="Thmproofx19.p1.28.m18.1.1.1.1.cmml" xref="Thmproofx19.p1.28.m18.1.1.1.1">𝑗</ci></apply></apply></apply><apply id="Thmproofx19.p1.28.m18.4.4.2.2.cmml" xref="Thmproofx19.p1.28.m18.4.4.2.1"><abs id="Thmproofx19.p1.28.m18.4.4.2.2.1.cmml" xref="Thmproofx19.p1.28.m18.4.4.2.1.2"></abs><apply id="Thmproofx19.p1.28.m18.4.4.2.1.1.cmml" xref="Thmproofx19.p1.28.m18.4.4.2.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.28.m18.4.4.2.1.1.1.cmml" xref="Thmproofx19.p1.28.m18.4.4.2.1.1">subscript</csymbol><ci id="Thmproofx19.p1.28.m18.4.4.2.1.1.2.cmml" xref="Thmproofx19.p1.28.m18.4.4.2.1.1.2">𝑝</ci><apply id="Thmproofx19.p1.28.m18.2.2.1.cmml" xref="Thmproofx19.p1.28.m18.2.2.1"><times id="Thmproofx19.p1.28.m18.2.2.1.2.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.2"></times><apply id="Thmproofx19.p1.28.m18.2.2.1.3.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.28.m18.2.2.1.3.1.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.3">subscript</csymbol><ci id="Thmproofx19.p1.28.m18.2.2.1.3.2.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.3.2">𝐼</ci><ci id="Thmproofx19.p1.28.m18.2.2.1.3.3.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.3.3">𝑑</ci></apply><apply id="Thmproofx19.p1.28.m18.2.2.1.1.1.1.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.1.1"><plus id="Thmproofx19.p1.28.m18.2.2.1.1.1.1.1.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.1"></plus><ci id="Thmproofx19.p1.28.m18.2.2.1.1.1.1.2.cmml" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.2">𝑗</ci><cn id="Thmproofx19.p1.28.m18.2.2.1.1.1.1.3.cmml" type="integer" xref="Thmproofx19.p1.28.m18.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.28.m18.4c">|p_{I_{d}(j)}|>|p_{I_{d}(j+1)}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.28.m18.4d">| italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT | > | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j + 1 ) end_POSTSUBSCRIPT |</annotation></semantics></math> for some <math alttext="j" class="ltx_Math" display="inline" id="Thmproofx19.p1.29.m19.1"><semantics id="Thmproofx19.p1.29.m19.1a"><mi id="Thmproofx19.p1.29.m19.1.1" xref="Thmproofx19.p1.29.m19.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.29.m19.1b"><ci id="Thmproofx19.p1.29.m19.1.1.cmml" xref="Thmproofx19.p1.29.m19.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.29.m19.1c">j</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.29.m19.1d">italic_j</annotation></semantics></math> (after reordering paths of identical lengths). Then, defining <math alttext="I^{\prime}" class="ltx_Math" display="inline" id="Thmproofx19.p1.30.m20.1"><semantics id="Thmproofx19.p1.30.m20.1a"><msup id="Thmproofx19.p1.30.m20.1.1" xref="Thmproofx19.p1.30.m20.1.1.cmml"><mi id="Thmproofx19.p1.30.m20.1.1.2" xref="Thmproofx19.p1.30.m20.1.1.2.cmml">I</mi><mo id="Thmproofx19.p1.30.m20.1.1.3" xref="Thmproofx19.p1.30.m20.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.30.m20.1b"><apply id="Thmproofx19.p1.30.m20.1.1.cmml" xref="Thmproofx19.p1.30.m20.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.30.m20.1.1.1.cmml" xref="Thmproofx19.p1.30.m20.1.1">superscript</csymbol><ci id="Thmproofx19.p1.30.m20.1.1.2.cmml" xref="Thmproofx19.p1.30.m20.1.1.2">𝐼</ci><ci id="Thmproofx19.p1.30.m20.1.1.3.cmml" xref="Thmproofx19.p1.30.m20.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.30.m20.1c">I^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.30.m20.1d">italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> by swapping <math alttext="I_{d}(j),I_{d}(j+1)" class="ltx_Math" display="inline" id="Thmproofx19.p1.31.m21.3"><semantics id="Thmproofx19.p1.31.m21.3a"><mrow id="Thmproofx19.p1.31.m21.3.3.2" xref="Thmproofx19.p1.31.m21.3.3.3.cmml"><mrow id="Thmproofx19.p1.31.m21.2.2.1.1" xref="Thmproofx19.p1.31.m21.2.2.1.1.cmml"><msub id="Thmproofx19.p1.31.m21.2.2.1.1.2" xref="Thmproofx19.p1.31.m21.2.2.1.1.2.cmml"><mi id="Thmproofx19.p1.31.m21.2.2.1.1.2.2" xref="Thmproofx19.p1.31.m21.2.2.1.1.2.2.cmml">I</mi><mi id="Thmproofx19.p1.31.m21.2.2.1.1.2.3" xref="Thmproofx19.p1.31.m21.2.2.1.1.2.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.31.m21.2.2.1.1.1" xref="Thmproofx19.p1.31.m21.2.2.1.1.1.cmml">⁢</mo><mrow id="Thmproofx19.p1.31.m21.2.2.1.1.3.2" xref="Thmproofx19.p1.31.m21.2.2.1.1.cmml"><mo id="Thmproofx19.p1.31.m21.2.2.1.1.3.2.1" stretchy="false" xref="Thmproofx19.p1.31.m21.2.2.1.1.cmml">(</mo><mi id="Thmproofx19.p1.31.m21.1.1" xref="Thmproofx19.p1.31.m21.1.1.cmml">j</mi><mo id="Thmproofx19.p1.31.m21.2.2.1.1.3.2.2" stretchy="false" xref="Thmproofx19.p1.31.m21.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx19.p1.31.m21.3.3.2.3" xref="Thmproofx19.p1.31.m21.3.3.3.cmml">,</mo><mrow id="Thmproofx19.p1.31.m21.3.3.2.2" xref="Thmproofx19.p1.31.m21.3.3.2.2.cmml"><msub id="Thmproofx19.p1.31.m21.3.3.2.2.3" xref="Thmproofx19.p1.31.m21.3.3.2.2.3.cmml"><mi id="Thmproofx19.p1.31.m21.3.3.2.2.3.2" xref="Thmproofx19.p1.31.m21.3.3.2.2.3.2.cmml">I</mi><mi id="Thmproofx19.p1.31.m21.3.3.2.2.3.3" xref="Thmproofx19.p1.31.m21.3.3.2.2.3.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.31.m21.3.3.2.2.2" xref="Thmproofx19.p1.31.m21.3.3.2.2.2.cmml">⁢</mo><mrow id="Thmproofx19.p1.31.m21.3.3.2.2.1.1" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.cmml"><mo id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.2" stretchy="false" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.cmml"><mi id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.2" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.2.cmml">j</mi><mo id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.1" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.1.cmml">+</mo><mn id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.3" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.3" stretchy="false" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.31.m21.3b"><list id="Thmproofx19.p1.31.m21.3.3.3.cmml" xref="Thmproofx19.p1.31.m21.3.3.2"><apply id="Thmproofx19.p1.31.m21.2.2.1.1.cmml" xref="Thmproofx19.p1.31.m21.2.2.1.1"><times id="Thmproofx19.p1.31.m21.2.2.1.1.1.cmml" xref="Thmproofx19.p1.31.m21.2.2.1.1.1"></times><apply id="Thmproofx19.p1.31.m21.2.2.1.1.2.cmml" xref="Thmproofx19.p1.31.m21.2.2.1.1.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.31.m21.2.2.1.1.2.1.cmml" xref="Thmproofx19.p1.31.m21.2.2.1.1.2">subscript</csymbol><ci id="Thmproofx19.p1.31.m21.2.2.1.1.2.2.cmml" xref="Thmproofx19.p1.31.m21.2.2.1.1.2.2">𝐼</ci><ci id="Thmproofx19.p1.31.m21.2.2.1.1.2.3.cmml" xref="Thmproofx19.p1.31.m21.2.2.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.31.m21.1.1.cmml" xref="Thmproofx19.p1.31.m21.1.1">𝑗</ci></apply><apply id="Thmproofx19.p1.31.m21.3.3.2.2.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2"><times id="Thmproofx19.p1.31.m21.3.3.2.2.2.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.2"></times><apply id="Thmproofx19.p1.31.m21.3.3.2.2.3.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p1.31.m21.3.3.2.2.3.1.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.3">subscript</csymbol><ci id="Thmproofx19.p1.31.m21.3.3.2.2.3.2.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.3.2">𝐼</ci><ci id="Thmproofx19.p1.31.m21.3.3.2.2.3.3.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.3.3">𝑑</ci></apply><apply id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1"><plus id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.1.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.1"></plus><ci id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.2.cmml" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.2">𝑗</ci><cn id="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.3.cmml" type="integer" xref="Thmproofx19.p1.31.m21.3.3.2.2.1.1.1.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.31.m21.3c">I_{d}(j),I_{d}(j+1)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.31.m21.3d">italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j ) , italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j + 1 )</annotation></semantics></math> yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx52"> <tbody id="Pt0.A4.Ex86"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{i=1}^{m}|p_{I^{\prime}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}-% \sum_{i=1}^{m}|p_{I_{d}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}" class="ltx_Math" display="inline" id="Pt0.A4.Ex86.m1.6"><semantics id="Pt0.A4.Ex86.m1.6a"><mrow id="Pt0.A4.Ex86.m1.6.6" xref="Pt0.A4.Ex86.m1.6.6.cmml"><mrow id="Pt0.A4.Ex86.m1.5.5.1" xref="Pt0.A4.Ex86.m1.5.5.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex86.m1.5.5.1.2" xref="Pt0.A4.Ex86.m1.5.5.1.2.cmml"><munderover id="Pt0.A4.Ex86.m1.5.5.1.2a" xref="Pt0.A4.Ex86.m1.5.5.1.2.cmml"><mo id="Pt0.A4.Ex86.m1.5.5.1.2.2.2" movablelimits="false" xref="Pt0.A4.Ex86.m1.5.5.1.2.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex86.m1.5.5.1.2.2.3" xref="Pt0.A4.Ex86.m1.5.5.1.2.2.3.cmml"><mi id="Pt0.A4.Ex86.m1.5.5.1.2.2.3.2" xref="Pt0.A4.Ex86.m1.5.5.1.2.2.3.2.cmml">i</mi><mo id="Pt0.A4.Ex86.m1.5.5.1.2.2.3.1" xref="Pt0.A4.Ex86.m1.5.5.1.2.2.3.1.cmml">=</mo><mn id="Pt0.A4.Ex86.m1.5.5.1.2.2.3.3" xref="Pt0.A4.Ex86.m1.5.5.1.2.2.3.3.cmml">1</mn></mrow><mi id="Pt0.A4.Ex86.m1.5.5.1.2.3" xref="Pt0.A4.Ex86.m1.5.5.1.2.3.cmml">m</mi></munderover></mstyle><mrow id="Pt0.A4.Ex86.m1.5.5.1.1" xref="Pt0.A4.Ex86.m1.5.5.1.1.cmml"><mrow id="Pt0.A4.Ex86.m1.5.5.1.1.1.1" xref="Pt0.A4.Ex86.m1.5.5.1.1.1.2.cmml"><mo id="Pt0.A4.Ex86.m1.5.5.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex86.m1.5.5.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex86.m1.5.5.1.1.1.1.1" xref="Pt0.A4.Ex86.m1.5.5.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex86.m1.5.5.1.1.1.1.1.2" xref="Pt0.A4.Ex86.m1.5.5.1.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex86.m1.1.1.1" xref="Pt0.A4.Ex86.m1.1.1.1.cmml"><msup id="Pt0.A4.Ex86.m1.1.1.1.3" xref="Pt0.A4.Ex86.m1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex86.m1.1.1.1.3.2" xref="Pt0.A4.Ex86.m1.1.1.1.3.2.cmml">I</mi><mo id="Pt0.A4.Ex86.m1.1.1.1.3.3" xref="Pt0.A4.Ex86.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex86.m1.1.1.1.2" xref="Pt0.A4.Ex86.m1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex86.m1.1.1.1.4.2" xref="Pt0.A4.Ex86.m1.1.1.1.cmml"><mo id="Pt0.A4.Ex86.m1.1.1.1.4.2.1" stretchy="false" xref="Pt0.A4.Ex86.m1.1.1.1.cmml">(</mo><mi id="Pt0.A4.Ex86.m1.1.1.1.1" xref="Pt0.A4.Ex86.m1.1.1.1.1.cmml">i</mi><mo id="Pt0.A4.Ex86.m1.1.1.1.4.2.2" stretchy="false" xref="Pt0.A4.Ex86.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="Pt0.A4.Ex86.m1.5.5.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex86.m1.5.5.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex86.m1.5.5.1.1.2" xref="Pt0.A4.Ex86.m1.5.5.1.1.2.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A4.Ex86.m1.2.2" xref="Pt0.A4.Ex86.m1.2.2.cmml"><mfrac id="Pt0.A4.Ex86.m1.2.2a" xref="Pt0.A4.Ex86.m1.2.2.cmml"><mrow id="Pt0.A4.Ex86.m1.2.2.1" xref="Pt0.A4.Ex86.m1.2.2.1.cmml"><msup id="Pt0.A4.Ex86.m1.2.2.1.3" xref="Pt0.A4.Ex86.m1.2.2.1.3.cmml"><mi id="Pt0.A4.Ex86.m1.2.2.1.3.2" xref="Pt0.A4.Ex86.m1.2.2.1.3.2.cmml">i</mi><mi id="Pt0.A4.Ex86.m1.2.2.1.3.3" xref="Pt0.A4.Ex86.m1.2.2.1.3.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex86.m1.2.2.1.2" xref="Pt0.A4.Ex86.m1.2.2.1.2.cmml">−</mo><msup id="Pt0.A4.Ex86.m1.2.2.1.1" xref="Pt0.A4.Ex86.m1.2.2.1.1.cmml"><mrow id="Pt0.A4.Ex86.m1.2.2.1.1.1.1" xref="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex86.m1.2.2.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1" xref="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.2.cmml">i</mi><mo id="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.1.cmml">−</mo><mn id="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.3" xref="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex86.m1.2.2.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex86.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex86.m1.2.2.1.1.3" xref="Pt0.A4.Ex86.m1.2.2.1.1.3.cmml">n</mi></msup></mrow><msup id="Pt0.A4.Ex86.m1.2.2.3" xref="Pt0.A4.Ex86.m1.2.2.3.cmml"><mi id="Pt0.A4.Ex86.m1.2.2.3.2" xref="Pt0.A4.Ex86.m1.2.2.3.2.cmml">m</mi><mi id="Pt0.A4.Ex86.m1.2.2.3.3" xref="Pt0.A4.Ex86.m1.2.2.3.3.cmml">n</mi></msup></mfrac></mstyle></mrow></mrow><mo id="Pt0.A4.Ex86.m1.6.6.3" xref="Pt0.A4.Ex86.m1.6.6.3.cmml">−</mo><mrow id="Pt0.A4.Ex86.m1.6.6.2" xref="Pt0.A4.Ex86.m1.6.6.2.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex86.m1.6.6.2.2" xref="Pt0.A4.Ex86.m1.6.6.2.2.cmml"><munderover id="Pt0.A4.Ex86.m1.6.6.2.2a" xref="Pt0.A4.Ex86.m1.6.6.2.2.cmml"><mo id="Pt0.A4.Ex86.m1.6.6.2.2.2.2" movablelimits="false" xref="Pt0.A4.Ex86.m1.6.6.2.2.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex86.m1.6.6.2.2.2.3" xref="Pt0.A4.Ex86.m1.6.6.2.2.2.3.cmml"><mi id="Pt0.A4.Ex86.m1.6.6.2.2.2.3.2" xref="Pt0.A4.Ex86.m1.6.6.2.2.2.3.2.cmml">i</mi><mo id="Pt0.A4.Ex86.m1.6.6.2.2.2.3.1" xref="Pt0.A4.Ex86.m1.6.6.2.2.2.3.1.cmml">=</mo><mn id="Pt0.A4.Ex86.m1.6.6.2.2.2.3.3" xref="Pt0.A4.Ex86.m1.6.6.2.2.2.3.3.cmml">1</mn></mrow><mi id="Pt0.A4.Ex86.m1.6.6.2.2.3" xref="Pt0.A4.Ex86.m1.6.6.2.2.3.cmml">m</mi></munderover></mstyle><mrow id="Pt0.A4.Ex86.m1.6.6.2.1" xref="Pt0.A4.Ex86.m1.6.6.2.1.cmml"><mrow id="Pt0.A4.Ex86.m1.6.6.2.1.1.1" xref="Pt0.A4.Ex86.m1.6.6.2.1.1.2.cmml"><mo id="Pt0.A4.Ex86.m1.6.6.2.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex86.m1.6.6.2.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex86.m1.6.6.2.1.1.1.1" xref="Pt0.A4.Ex86.m1.6.6.2.1.1.1.1.cmml"><mi id="Pt0.A4.Ex86.m1.6.6.2.1.1.1.1.2" xref="Pt0.A4.Ex86.m1.6.6.2.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex86.m1.3.3.1" xref="Pt0.A4.Ex86.m1.3.3.1.cmml"><msub id="Pt0.A4.Ex86.m1.3.3.1.3" xref="Pt0.A4.Ex86.m1.3.3.1.3.cmml"><mi id="Pt0.A4.Ex86.m1.3.3.1.3.2" xref="Pt0.A4.Ex86.m1.3.3.1.3.2.cmml">I</mi><mi id="Pt0.A4.Ex86.m1.3.3.1.3.3" xref="Pt0.A4.Ex86.m1.3.3.1.3.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex86.m1.3.3.1.2" xref="Pt0.A4.Ex86.m1.3.3.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex86.m1.3.3.1.4.2" xref="Pt0.A4.Ex86.m1.3.3.1.cmml"><mo id="Pt0.A4.Ex86.m1.3.3.1.4.2.1" stretchy="false" xref="Pt0.A4.Ex86.m1.3.3.1.cmml">(</mo><mi id="Pt0.A4.Ex86.m1.3.3.1.1" xref="Pt0.A4.Ex86.m1.3.3.1.1.cmml">i</mi><mo id="Pt0.A4.Ex86.m1.3.3.1.4.2.2" stretchy="false" xref="Pt0.A4.Ex86.m1.3.3.1.cmml">)</mo></mrow></mrow></msub><mo id="Pt0.A4.Ex86.m1.6.6.2.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex86.m1.6.6.2.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex86.m1.6.6.2.1.2" xref="Pt0.A4.Ex86.m1.6.6.2.1.2.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A4.Ex86.m1.4.4" xref="Pt0.A4.Ex86.m1.4.4.cmml"><mfrac id="Pt0.A4.Ex86.m1.4.4a" xref="Pt0.A4.Ex86.m1.4.4.cmml"><mrow id="Pt0.A4.Ex86.m1.4.4.1" xref="Pt0.A4.Ex86.m1.4.4.1.cmml"><msup id="Pt0.A4.Ex86.m1.4.4.1.3" xref="Pt0.A4.Ex86.m1.4.4.1.3.cmml"><mi id="Pt0.A4.Ex86.m1.4.4.1.3.2" xref="Pt0.A4.Ex86.m1.4.4.1.3.2.cmml">i</mi><mi id="Pt0.A4.Ex86.m1.4.4.1.3.3" xref="Pt0.A4.Ex86.m1.4.4.1.3.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex86.m1.4.4.1.2" xref="Pt0.A4.Ex86.m1.4.4.1.2.cmml">−</mo><msup id="Pt0.A4.Ex86.m1.4.4.1.1" xref="Pt0.A4.Ex86.m1.4.4.1.1.cmml"><mrow 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xref="Pt0.A4.Ex86.m1.4.4.3.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex86.m1.6c">\displaystyle\sum_{i=1}^{m}|p_{I^{\prime}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}-% \sum_{i=1}^{m}|p_{I_{d}(i)}|\frac{i^{n}-(i-1)^{n}}{m^{n}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex86.m1.6d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_i ) end_POSTSUBSCRIPT | divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT | divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex87"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="Pt0.A4.Ex87.m1.1"><semantics id="Pt0.A4.Ex87.m1.1a"><mo id="Pt0.A4.Ex87.m1.1.1" xref="Pt0.A4.Ex87.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex87.m1.1b"><eq id="Pt0.A4.Ex87.m1.1.1.cmml" xref="Pt0.A4.Ex87.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex87.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex87.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(|p_{I^{\prime}(j)}|-|p_{I_{d}(j)}|)\frac{j^{n}-(j-1)^{n}}{m^{n}}% +(|p_{I^{\prime}(j+1)}|-|p_{I_{d}(j+1)}|)\frac{(j+1)^{n}-j^{n}}{m^{n}}" class="ltx_Math" display="inline" id="Pt0.A4.Ex87.m2.8"><semantics id="Pt0.A4.Ex87.m2.8a"><mrow id="Pt0.A4.Ex87.m2.8.8" xref="Pt0.A4.Ex87.m2.8.8.cmml"><mrow id="Pt0.A4.Ex87.m2.7.7.1" xref="Pt0.A4.Ex87.m2.7.7.1.cmml"><mrow id="Pt0.A4.Ex87.m2.7.7.1.1.1" xref="Pt0.A4.Ex87.m2.7.7.1.1.1.1.cmml"><mo id="Pt0.A4.Ex87.m2.7.7.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex87.m2.7.7.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex87.m2.7.7.1.1.1.1" xref="Pt0.A4.Ex87.m2.7.7.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex87.m2.7.7.1.1.1.1.1.1" xref="Pt0.A4.Ex87.m2.7.7.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex87.m2.7.7.1.1.1.1.1.1.2" stretchy="false" 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xref="Pt0.A4.Ex87.m2.3.3.1.3.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex87.m2.3.3.1.2" xref="Pt0.A4.Ex87.m2.3.3.1.2.cmml">−</mo><msup id="Pt0.A4.Ex87.m2.3.3.1.1" xref="Pt0.A4.Ex87.m2.3.3.1.1.cmml"><mrow id="Pt0.A4.Ex87.m2.3.3.1.1.1.1" xref="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex87.m2.3.3.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1" xref="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.2" xref="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.2.cmml">j</mi><mo id="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.1" xref="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.1.cmml">−</mo><mn id="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.3" xref="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex87.m2.3.3.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex87.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex87.m2.3.3.1.1.3" xref="Pt0.A4.Ex87.m2.3.3.1.1.3.cmml">n</mi></msup></mrow><msup id="Pt0.A4.Ex87.m2.3.3.3" xref="Pt0.A4.Ex87.m2.3.3.3.cmml"><mi id="Pt0.A4.Ex87.m2.3.3.3.2" xref="Pt0.A4.Ex87.m2.3.3.3.2.cmml">m</mi><mi id="Pt0.A4.Ex87.m2.3.3.3.3" xref="Pt0.A4.Ex87.m2.3.3.3.3.cmml">n</mi></msup></mfrac></mstyle></mrow><mo id="Pt0.A4.Ex87.m2.8.8.3" xref="Pt0.A4.Ex87.m2.8.8.3.cmml">+</mo><mrow id="Pt0.A4.Ex87.m2.8.8.2" xref="Pt0.A4.Ex87.m2.8.8.2.cmml"><mrow id="Pt0.A4.Ex87.m2.8.8.2.1.1" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.cmml"><mo id="Pt0.A4.Ex87.m2.8.8.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex87.m2.8.8.2.1.1.1" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.cmml"><mrow id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.1" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.1.1" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex87.m2.4.4.1" xref="Pt0.A4.Ex87.m2.4.4.1.cmml"><msup id="Pt0.A4.Ex87.m2.4.4.1.3" xref="Pt0.A4.Ex87.m2.4.4.1.3.cmml"><mi id="Pt0.A4.Ex87.m2.4.4.1.3.2" xref="Pt0.A4.Ex87.m2.4.4.1.3.2.cmml">I</mi><mo id="Pt0.A4.Ex87.m2.4.4.1.3.3" xref="Pt0.A4.Ex87.m2.4.4.1.3.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex87.m2.4.4.1.2" xref="Pt0.A4.Ex87.m2.4.4.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex87.m2.4.4.1.1.1" xref="Pt0.A4.Ex87.m2.4.4.1.1.1.1.cmml"><mo id="Pt0.A4.Ex87.m2.4.4.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex87.m2.4.4.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex87.m2.4.4.1.1.1.1" xref="Pt0.A4.Ex87.m2.4.4.1.1.1.1.cmml"><mi id="Pt0.A4.Ex87.m2.4.4.1.1.1.1.2" xref="Pt0.A4.Ex87.m2.4.4.1.1.1.1.2.cmml">j</mi><mo id="Pt0.A4.Ex87.m2.4.4.1.1.1.1.1" xref="Pt0.A4.Ex87.m2.4.4.1.1.1.1.1.cmml">+</mo><mn id="Pt0.A4.Ex87.m2.4.4.1.1.1.1.3" xref="Pt0.A4.Ex87.m2.4.4.1.1.1.1.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex87.m2.4.4.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex87.m2.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.3" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.1" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.2.cmml"><mo id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.1.2" stretchy="false" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.2.1.cmml">|</mo><msub id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.1.1" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.1.1.cmml"><mi id="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.1.1.2" xref="Pt0.A4.Ex87.m2.8.8.2.1.1.1.2.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex87.m2.5.5.1" xref="Pt0.A4.Ex87.m2.5.5.1.cmml"><msub id="Pt0.A4.Ex87.m2.5.5.1.3" xref="Pt0.A4.Ex87.m2.5.5.1.3.cmml"><mi id="Pt0.A4.Ex87.m2.5.5.1.3.2" xref="Pt0.A4.Ex87.m2.5.5.1.3.2.cmml">I</mi><mi id="Pt0.A4.Ex87.m2.5.5.1.3.3" xref="Pt0.A4.Ex87.m2.5.5.1.3.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex87.m2.5.5.1.2" xref="Pt0.A4.Ex87.m2.5.5.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex87.m2.5.5.1.1.1" 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id="Pt0.A4.Ex87.m2.6.6.2.cmml" xref="Pt0.A4.Ex87.m2.6.6"></divide><apply id="Pt0.A4.Ex87.m2.6.6.1.cmml" xref="Pt0.A4.Ex87.m2.6.6.1"><minus id="Pt0.A4.Ex87.m2.6.6.1.2.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.2"></minus><apply id="Pt0.A4.Ex87.m2.6.6.1.1.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex87.m2.6.6.1.1.2.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.1">superscript</csymbol><apply id="Pt0.A4.Ex87.m2.6.6.1.1.1.1.1.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.1.1.1"><plus id="Pt0.A4.Ex87.m2.6.6.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.1.1.1.1.1"></plus><ci id="Pt0.A4.Ex87.m2.6.6.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.1.1.1.1.2">𝑗</ci><cn id="Pt0.A4.Ex87.m2.6.6.1.1.1.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex87.m2.6.6.1.1.1.1.1.3">1</cn></apply><ci id="Pt0.A4.Ex87.m2.6.6.1.1.3.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.1.3">𝑛</ci></apply><apply id="Pt0.A4.Ex87.m2.6.6.1.3.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex87.m2.6.6.1.3.1.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.3">superscript</csymbol><ci id="Pt0.A4.Ex87.m2.6.6.1.3.2.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.3.2">𝑗</ci><ci id="Pt0.A4.Ex87.m2.6.6.1.3.3.cmml" xref="Pt0.A4.Ex87.m2.6.6.1.3.3">𝑛</ci></apply></apply><apply id="Pt0.A4.Ex87.m2.6.6.3.cmml" xref="Pt0.A4.Ex87.m2.6.6.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex87.m2.6.6.3.1.cmml" xref="Pt0.A4.Ex87.m2.6.6.3">superscript</csymbol><ci id="Pt0.A4.Ex87.m2.6.6.3.2.cmml" xref="Pt0.A4.Ex87.m2.6.6.3.2">𝑚</ci><ci id="Pt0.A4.Ex87.m2.6.6.3.3.cmml" xref="Pt0.A4.Ex87.m2.6.6.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex87.m2.8c">\displaystyle(|p_{I^{\prime}(j)}|-|p_{I_{d}(j)}|)\frac{j^{n}-(j-1)^{n}}{m^{n}}% +(|p_{I^{\prime}(j+1)}|-|p_{I_{d}(j+1)}|)\frac{(j+1)^{n}-j^{n}}{m^{n}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex87.m2.8d">( | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_j ) end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT | ) divide start_ARG italic_j start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_j - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG + ( | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_j + 1 ) end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j + 1 ) end_POSTSUBSCRIPT | ) divide start_ARG ( italic_j + 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_j start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex88"><tr class="ltx_equation 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xref="Pt0.A4.Ex88.m2.2.2.1.cmml">)</mo></mrow></mrow></msub><mo id="Pt0.A4.Ex88.m2.7.7.1.1.1.1.2.1.3" stretchy="false" xref="Pt0.A4.Ex88.m2.7.7.1.1.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Pt0.A4.Ex88.m2.7.7.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex88.m2.7.7.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex88.m2.7.7.1.2" xref="Pt0.A4.Ex88.m2.7.7.1.2.cmml">⁢</mo><mstyle displaystyle="true" id="Pt0.A4.Ex88.m2.3.3" xref="Pt0.A4.Ex88.m2.3.3.cmml"><mfrac id="Pt0.A4.Ex88.m2.3.3a" xref="Pt0.A4.Ex88.m2.3.3.cmml"><mrow id="Pt0.A4.Ex88.m2.3.3.1" xref="Pt0.A4.Ex88.m2.3.3.1.cmml"><msup id="Pt0.A4.Ex88.m2.3.3.1.3" xref="Pt0.A4.Ex88.m2.3.3.1.3.cmml"><mi id="Pt0.A4.Ex88.m2.3.3.1.3.2" xref="Pt0.A4.Ex88.m2.3.3.1.3.2.cmml">j</mi><mi id="Pt0.A4.Ex88.m2.3.3.1.3.3" xref="Pt0.A4.Ex88.m2.3.3.1.3.3.cmml">n</mi></msup><mo id="Pt0.A4.Ex88.m2.3.3.1.2" xref="Pt0.A4.Ex88.m2.3.3.1.2.cmml">−</mo><msup id="Pt0.A4.Ex88.m2.3.3.1.1" xref="Pt0.A4.Ex88.m2.3.3.1.1.cmml"><mrow id="Pt0.A4.Ex88.m2.3.3.1.1.1.1" 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xref="Pt0.A4.Ex88.m2.8.8.3.cmml">+</mo><mrow id="Pt0.A4.Ex88.m2.8.8.2" xref="Pt0.A4.Ex88.m2.8.8.2.cmml"><mrow id="Pt0.A4.Ex88.m2.8.8.2.1.1" xref="Pt0.A4.Ex88.m2.8.8.2.1.1.1.cmml"><mo id="Pt0.A4.Ex88.m2.8.8.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex88.m2.8.8.2.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex88.m2.8.8.2.1.1.1" xref="Pt0.A4.Ex88.m2.8.8.2.1.1.1.cmml"><mrow id="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.1" xref="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.1.1" xref="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex88.m2.8.8.2.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex88.m2.4.4.1" xref="Pt0.A4.Ex88.m2.4.4.1.cmml"><msub id="Pt0.A4.Ex88.m2.4.4.1.3" xref="Pt0.A4.Ex88.m2.4.4.1.3.cmml"><mi id="Pt0.A4.Ex88.m2.4.4.1.3.2" xref="Pt0.A4.Ex88.m2.4.4.1.3.2.cmml">I</mi><mi id="Pt0.A4.Ex88.m2.4.4.1.3.3" 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id="Pt0.A4.Ex88.m2.6.6.3.2.cmml" xref="Pt0.A4.Ex88.m2.6.6.3.2">𝑚</ci><ci id="Pt0.A4.Ex88.m2.6.6.3.3.cmml" xref="Pt0.A4.Ex88.m2.6.6.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex88.m2.8c">\displaystyle(|p_{I_{d}(j+1)}|-|p_{I_{d}(j)}|)\frac{j^{n}-(j-1)^{n}}{m^{n}}+(|% p_{I_{d}(j)}|-|p_{I_{d}(j+1)}|)\frac{(j+1)^{n}-j^{n}}{m^{n}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex88.m2.8d">( | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j + 1 ) end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT | ) divide start_ARG italic_j start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_j - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG + ( | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j + 1 ) end_POSTSUBSCRIPT | ) divide start_ARG ( italic_j + 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_j start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex89"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="Pt0.A4.Ex89.m1.1"><semantics id="Pt0.A4.Ex89.m1.1a"><mo id="Pt0.A4.Ex89.m1.1.1" xref="Pt0.A4.Ex89.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex89.m1.1b"><eq 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xref="Pt0.A4.Ex89.m2.4.4.3">superscript</csymbol><ci id="Pt0.A4.Ex89.m2.4.4.3.2.cmml" xref="Pt0.A4.Ex89.m2.4.4.3.2">𝑚</ci><ci id="Pt0.A4.Ex89.m2.4.4.3.3.cmml" xref="Pt0.A4.Ex89.m2.4.4.3.3">𝑛</ci></apply></apply></apply></apply><cn id="Pt0.A4.Ex89.m2.6.6.4.cmml" type="integer" xref="Pt0.A4.Ex89.m2.6.6.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex89.m2.6c">\displaystyle(|p_{I_{d}(j)}|-|p_{I_{d}(j+1)}|)\left(\frac{(j+1)^{n}-j^{n}}{m^{% n}}-\frac{j^{n}-(j-1)^{n}}{m^{n}}\right)>0</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex89.m2.6d">( | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_j + 1 ) end_POSTSUBSCRIPT | ) ( divide start_ARG ( italic_j + 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_j start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_j start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_j - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG ) > 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx19.p1.34">This proves the maximality of <math alttext="I_{d}" class="ltx_Math" display="inline" id="Thmproofx19.p1.32.m1.1"><semantics id="Thmproofx19.p1.32.m1.1a"><msub id="Thmproofx19.p1.32.m1.1.1" xref="Thmproofx19.p1.32.m1.1.1.cmml"><mi id="Thmproofx19.p1.32.m1.1.1.2" xref="Thmproofx19.p1.32.m1.1.1.2.cmml">I</mi><mi id="Thmproofx19.p1.32.m1.1.1.3" xref="Thmproofx19.p1.32.m1.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.32.m1.1b"><apply id="Thmproofx19.p1.32.m1.1.1.cmml" xref="Thmproofx19.p1.32.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p1.32.m1.1.1.1.cmml" xref="Thmproofx19.p1.32.m1.1.1">subscript</csymbol><ci id="Thmproofx19.p1.32.m1.1.1.2.cmml" xref="Thmproofx19.p1.32.m1.1.1.2">𝐼</ci><ci id="Thmproofx19.p1.32.m1.1.1.3.cmml" xref="Thmproofx19.p1.32.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.32.m1.1c">I_{d}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.32.m1.1d">italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> that satisfies <math alttext="I_{d}(i)=i" class="ltx_Math" display="inline" id="Thmproofx19.p1.33.m2.1"><semantics id="Thmproofx19.p1.33.m2.1a"><mrow id="Thmproofx19.p1.33.m2.1.2" xref="Thmproofx19.p1.33.m2.1.2.cmml"><mrow id="Thmproofx19.p1.33.m2.1.2.2" xref="Thmproofx19.p1.33.m2.1.2.2.cmml"><msub id="Thmproofx19.p1.33.m2.1.2.2.2" xref="Thmproofx19.p1.33.m2.1.2.2.2.cmml"><mi id="Thmproofx19.p1.33.m2.1.2.2.2.2" xref="Thmproofx19.p1.33.m2.1.2.2.2.2.cmml">I</mi><mi id="Thmproofx19.p1.33.m2.1.2.2.2.3" xref="Thmproofx19.p1.33.m2.1.2.2.2.3.cmml">d</mi></msub><mo id="Thmproofx19.p1.33.m2.1.2.2.1" xref="Thmproofx19.p1.33.m2.1.2.2.1.cmml">⁢</mo><mrow id="Thmproofx19.p1.33.m2.1.2.2.3.2" xref="Thmproofx19.p1.33.m2.1.2.2.cmml"><mo id="Thmproofx19.p1.33.m2.1.2.2.3.2.1" stretchy="false" xref="Thmproofx19.p1.33.m2.1.2.2.cmml">(</mo><mi id="Thmproofx19.p1.33.m2.1.1" xref="Thmproofx19.p1.33.m2.1.1.cmml">i</mi><mo id="Thmproofx19.p1.33.m2.1.2.2.3.2.2" stretchy="false" xref="Thmproofx19.p1.33.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx19.p1.33.m2.1.2.1" xref="Thmproofx19.p1.33.m2.1.2.1.cmml">=</mo><mi id="Thmproofx19.p1.33.m2.1.2.3" xref="Thmproofx19.p1.33.m2.1.2.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.33.m2.1b"><apply id="Thmproofx19.p1.33.m2.1.2.cmml" xref="Thmproofx19.p1.33.m2.1.2"><eq id="Thmproofx19.p1.33.m2.1.2.1.cmml" xref="Thmproofx19.p1.33.m2.1.2.1"></eq><apply id="Thmproofx19.p1.33.m2.1.2.2.cmml" xref="Thmproofx19.p1.33.m2.1.2.2"><times id="Thmproofx19.p1.33.m2.1.2.2.1.cmml" xref="Thmproofx19.p1.33.m2.1.2.2.1"></times><apply id="Thmproofx19.p1.33.m2.1.2.2.2.cmml" xref="Thmproofx19.p1.33.m2.1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx19.p1.33.m2.1.2.2.2.1.cmml" xref="Thmproofx19.p1.33.m2.1.2.2.2">subscript</csymbol><ci id="Thmproofx19.p1.33.m2.1.2.2.2.2.cmml" xref="Thmproofx19.p1.33.m2.1.2.2.2.2">𝐼</ci><ci id="Thmproofx19.p1.33.m2.1.2.2.2.3.cmml" xref="Thmproofx19.p1.33.m2.1.2.2.2.3">𝑑</ci></apply><ci id="Thmproofx19.p1.33.m2.1.1.cmml" xref="Thmproofx19.p1.33.m2.1.1">𝑖</ci></apply><ci id="Thmproofx19.p1.33.m2.1.2.3.cmml" xref="Thmproofx19.p1.33.m2.1.2.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.33.m2.1c">I_{d}(i)=i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.33.m2.1d">italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_i ) = italic_i</annotation></semantics></math> for all <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx19.p1.34.m3.1"><semantics id="Thmproofx19.p1.34.m3.1a"><mi id="Thmproofx19.p1.34.m3.1.1" xref="Thmproofx19.p1.34.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p1.34.m3.1b"><ci id="Thmproofx19.p1.34.m3.1.1.cmml" xref="Thmproofx19.p1.34.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p1.34.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p1.34.m3.1d">italic_i</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx19.p2"> <p class="ltx_p" id="Thmproofx19.p2.5">So, since <math alttext="I_{(m,...,m)}(i)\stackrel{{\scriptstyle\text{def}}}{{=}}i" class="ltx_Math" display="inline" id="Thmproofx19.p2.1.m1.4"><semantics id="Thmproofx19.p2.1.m1.4a"><mrow id="Thmproofx19.p2.1.m1.4.5" xref="Thmproofx19.p2.1.m1.4.5.cmml"><mrow id="Thmproofx19.p2.1.m1.4.5.2" xref="Thmproofx19.p2.1.m1.4.5.2.cmml"><msub id="Thmproofx19.p2.1.m1.4.5.2.2" xref="Thmproofx19.p2.1.m1.4.5.2.2.cmml"><mi id="Thmproofx19.p2.1.m1.4.5.2.2.2" xref="Thmproofx19.p2.1.m1.4.5.2.2.2.cmml">I</mi><mrow id="Thmproofx19.p2.1.m1.3.3.3.5" xref="Thmproofx19.p2.1.m1.3.3.3.4.cmml"><mo id="Thmproofx19.p2.1.m1.3.3.3.5.1" stretchy="false" xref="Thmproofx19.p2.1.m1.3.3.3.4.cmml">(</mo><mi id="Thmproofx19.p2.1.m1.1.1.1.1" xref="Thmproofx19.p2.1.m1.1.1.1.1.cmml">m</mi><mo id="Thmproofx19.p2.1.m1.3.3.3.5.2" xref="Thmproofx19.p2.1.m1.3.3.3.4.cmml">,</mo><mi id="Thmproofx19.p2.1.m1.2.2.2.2" mathvariant="normal" xref="Thmproofx19.p2.1.m1.2.2.2.2.cmml">…</mi><mo id="Thmproofx19.p2.1.m1.3.3.3.5.3" xref="Thmproofx19.p2.1.m1.3.3.3.4.cmml">,</mo><mi id="Thmproofx19.p2.1.m1.3.3.3.3" xref="Thmproofx19.p2.1.m1.3.3.3.3.cmml">m</mi><mo id="Thmproofx19.p2.1.m1.3.3.3.5.4" stretchy="false" xref="Thmproofx19.p2.1.m1.3.3.3.4.cmml">)</mo></mrow></msub><mo id="Thmproofx19.p2.1.m1.4.5.2.1" xref="Thmproofx19.p2.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="Thmproofx19.p2.1.m1.4.5.2.3.2" xref="Thmproofx19.p2.1.m1.4.5.2.cmml"><mo id="Thmproofx19.p2.1.m1.4.5.2.3.2.1" stretchy="false" xref="Thmproofx19.p2.1.m1.4.5.2.cmml">(</mo><mi id="Thmproofx19.p2.1.m1.4.4" xref="Thmproofx19.p2.1.m1.4.4.cmml">i</mi><mo id="Thmproofx19.p2.1.m1.4.5.2.3.2.2" stretchy="false" xref="Thmproofx19.p2.1.m1.4.5.2.cmml">)</mo></mrow></mrow><mover id="Thmproofx19.p2.1.m1.4.5.1" xref="Thmproofx19.p2.1.m1.4.5.1.cmml"><mo id="Thmproofx19.p2.1.m1.4.5.1.2" xref="Thmproofx19.p2.1.m1.4.5.1.2.cmml">=</mo><mtext id="Thmproofx19.p2.1.m1.4.5.1.3" xref="Thmproofx19.p2.1.m1.4.5.1.3a.cmml">def</mtext></mover><mi id="Thmproofx19.p2.1.m1.4.5.3" xref="Thmproofx19.p2.1.m1.4.5.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p2.1.m1.4b"><apply id="Thmproofx19.p2.1.m1.4.5.cmml" xref="Thmproofx19.p2.1.m1.4.5"><apply id="Thmproofx19.p2.1.m1.4.5.1.cmml" xref="Thmproofx19.p2.1.m1.4.5.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.1.m1.4.5.1.1.cmml" xref="Thmproofx19.p2.1.m1.4.5.1">superscript</csymbol><eq id="Thmproofx19.p2.1.m1.4.5.1.2.cmml" xref="Thmproofx19.p2.1.m1.4.5.1.2"></eq><ci id="Thmproofx19.p2.1.m1.4.5.1.3a.cmml" xref="Thmproofx19.p2.1.m1.4.5.1.3"><mtext id="Thmproofx19.p2.1.m1.4.5.1.3.cmml" mathsize="70%" xref="Thmproofx19.p2.1.m1.4.5.1.3">def</mtext></ci></apply><apply id="Thmproofx19.p2.1.m1.4.5.2.cmml" xref="Thmproofx19.p2.1.m1.4.5.2"><times id="Thmproofx19.p2.1.m1.4.5.2.1.cmml" xref="Thmproofx19.p2.1.m1.4.5.2.1"></times><apply id="Thmproofx19.p2.1.m1.4.5.2.2.cmml" xref="Thmproofx19.p2.1.m1.4.5.2.2"><csymbol cd="ambiguous" id="Thmproofx19.p2.1.m1.4.5.2.2.1.cmml" xref="Thmproofx19.p2.1.m1.4.5.2.2">subscript</csymbol><ci id="Thmproofx19.p2.1.m1.4.5.2.2.2.cmml" xref="Thmproofx19.p2.1.m1.4.5.2.2.2">𝐼</ci><vector id="Thmproofx19.p2.1.m1.3.3.3.4.cmml" xref="Thmproofx19.p2.1.m1.3.3.3.5"><ci id="Thmproofx19.p2.1.m1.1.1.1.1.cmml" xref="Thmproofx19.p2.1.m1.1.1.1.1">𝑚</ci><ci id="Thmproofx19.p2.1.m1.2.2.2.2.cmml" xref="Thmproofx19.p2.1.m1.2.2.2.2">…</ci><ci id="Thmproofx19.p2.1.m1.3.3.3.3.cmml" xref="Thmproofx19.p2.1.m1.3.3.3.3">𝑚</ci></vector></apply><ci id="Thmproofx19.p2.1.m1.4.4.cmml" xref="Thmproofx19.p2.1.m1.4.4">𝑖</ci></apply><ci id="Thmproofx19.p2.1.m1.4.5.3.cmml" xref="Thmproofx19.p2.1.m1.4.5.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p2.1.m1.4c">I_{(m,...,m)}(i)\stackrel{{\scriptstyle\text{def}}}{{=}}i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p2.1.m1.4d">italic_I start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_i ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP italic_i</annotation></semantics></math> for all <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx19.p2.2.m2.1"><semantics id="Thmproofx19.p2.2.m2.1a"><mi id="Thmproofx19.p2.2.m2.1.1" xref="Thmproofx19.p2.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx19.p2.2.m2.1b"><ci id="Thmproofx19.p2.2.m2.1.1.cmml" xref="Thmproofx19.p2.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p2.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p2.2.m2.1d">italic_i</annotation></semantics></math>, we have that (<a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex83" title="Proof (Proof of Lemma 14) ‣ 0.D.4 Omitted proofs ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">0.D.4</span></a>), i.e., <math alttext="f_{d}\left(U_{P_{st}}^{n},\alpha\right)" class="ltx_Math" display="inline" id="Thmproofx19.p2.3.m3.2"><semantics id="Thmproofx19.p2.3.m3.2a"><mrow id="Thmproofx19.p2.3.m3.2.2" xref="Thmproofx19.p2.3.m3.2.2.cmml"><msub id="Thmproofx19.p2.3.m3.2.2.3" xref="Thmproofx19.p2.3.m3.2.2.3.cmml"><mi id="Thmproofx19.p2.3.m3.2.2.3.2" xref="Thmproofx19.p2.3.m3.2.2.3.2.cmml">f</mi><mi id="Thmproofx19.p2.3.m3.2.2.3.3" xref="Thmproofx19.p2.3.m3.2.2.3.3.cmml">d</mi></msub><mo id="Thmproofx19.p2.3.m3.2.2.2" xref="Thmproofx19.p2.3.m3.2.2.2.cmml">⁢</mo><mrow id="Thmproofx19.p2.3.m3.2.2.1.1" xref="Thmproofx19.p2.3.m3.2.2.1.2.cmml"><mo id="Thmproofx19.p2.3.m3.2.2.1.1.2" xref="Thmproofx19.p2.3.m3.2.2.1.2.cmml">(</mo><msubsup id="Thmproofx19.p2.3.m3.2.2.1.1.1" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.cmml"><mi id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.2" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.cmml"><mi id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.2" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.cmml"><mi id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.2" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.1" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.3" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx19.p2.3.m3.2.2.1.1.1.3" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx19.p2.3.m3.2.2.1.1.3" xref="Thmproofx19.p2.3.m3.2.2.1.2.cmml">,</mo><mi id="Thmproofx19.p2.3.m3.1.1" xref="Thmproofx19.p2.3.m3.1.1.cmml">α</mi><mo id="Thmproofx19.p2.3.m3.2.2.1.1.4" xref="Thmproofx19.p2.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p2.3.m3.2b"><apply id="Thmproofx19.p2.3.m3.2.2.cmml" xref="Thmproofx19.p2.3.m3.2.2"><times id="Thmproofx19.p2.3.m3.2.2.2.cmml" xref="Thmproofx19.p2.3.m3.2.2.2"></times><apply id="Thmproofx19.p2.3.m3.2.2.3.cmml" xref="Thmproofx19.p2.3.m3.2.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p2.3.m3.2.2.3.1.cmml" xref="Thmproofx19.p2.3.m3.2.2.3">subscript</csymbol><ci id="Thmproofx19.p2.3.m3.2.2.3.2.cmml" xref="Thmproofx19.p2.3.m3.2.2.3.2">𝑓</ci><ci id="Thmproofx19.p2.3.m3.2.2.3.3.cmml" xref="Thmproofx19.p2.3.m3.2.2.3.3">𝑑</ci></apply><interval closure="open" id="Thmproofx19.p2.3.m3.2.2.1.2.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1"><apply id="Thmproofx19.p2.3.m3.2.2.1.1.1.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.3.m3.2.2.1.1.1.1.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1">superscript</csymbol><apply id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.1.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.2.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.2">𝑈</ci><apply id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.1.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.2.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.2">𝑃</ci><apply id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3"><times id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.1.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.1"></times><ci id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.2.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.3.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx19.p2.3.m3.2.2.1.1.1.3.cmml" xref="Thmproofx19.p2.3.m3.2.2.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx19.p2.3.m3.1.1.cmml" xref="Thmproofx19.p2.3.m3.1.1">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p2.3.m3.2c">f_{d}\left(U_{P_{st}}^{n},\alpha\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p2.3.m3.2d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math>, is strictly maximal for <math alttext="d=(m,...,m)" class="ltx_Math" display="inline" id="Thmproofx19.p2.4.m4.3"><semantics id="Thmproofx19.p2.4.m4.3a"><mrow id="Thmproofx19.p2.4.m4.3.4" xref="Thmproofx19.p2.4.m4.3.4.cmml"><mi id="Thmproofx19.p2.4.m4.3.4.2" xref="Thmproofx19.p2.4.m4.3.4.2.cmml">d</mi><mo id="Thmproofx19.p2.4.m4.3.4.1" xref="Thmproofx19.p2.4.m4.3.4.1.cmml">=</mo><mrow id="Thmproofx19.p2.4.m4.3.4.3.2" xref="Thmproofx19.p2.4.m4.3.4.3.1.cmml"><mo id="Thmproofx19.p2.4.m4.3.4.3.2.1" stretchy="false" xref="Thmproofx19.p2.4.m4.3.4.3.1.cmml">(</mo><mi id="Thmproofx19.p2.4.m4.1.1" xref="Thmproofx19.p2.4.m4.1.1.cmml">m</mi><mo id="Thmproofx19.p2.4.m4.3.4.3.2.2" xref="Thmproofx19.p2.4.m4.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p2.4.m4.2.2" mathvariant="normal" xref="Thmproofx19.p2.4.m4.2.2.cmml">…</mi><mo id="Thmproofx19.p2.4.m4.3.4.3.2.3" xref="Thmproofx19.p2.4.m4.3.4.3.1.cmml">,</mo><mi id="Thmproofx19.p2.4.m4.3.3" xref="Thmproofx19.p2.4.m4.3.3.cmml">m</mi><mo id="Thmproofx19.p2.4.m4.3.4.3.2.4" stretchy="false" xref="Thmproofx19.p2.4.m4.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p2.4.m4.3b"><apply id="Thmproofx19.p2.4.m4.3.4.cmml" xref="Thmproofx19.p2.4.m4.3.4"><eq id="Thmproofx19.p2.4.m4.3.4.1.cmml" xref="Thmproofx19.p2.4.m4.3.4.1"></eq><ci id="Thmproofx19.p2.4.m4.3.4.2.cmml" xref="Thmproofx19.p2.4.m4.3.4.2">𝑑</ci><vector id="Thmproofx19.p2.4.m4.3.4.3.1.cmml" xref="Thmproofx19.p2.4.m4.3.4.3.2"><ci id="Thmproofx19.p2.4.m4.1.1.cmml" xref="Thmproofx19.p2.4.m4.1.1">𝑚</ci><ci id="Thmproofx19.p2.4.m4.2.2.cmml" xref="Thmproofx19.p2.4.m4.2.2">…</ci><ci id="Thmproofx19.p2.4.m4.3.3.cmml" xref="Thmproofx19.p2.4.m4.3.3">𝑚</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p2.4.m4.3c">d=(m,...,m)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p2.4.m4.3d">italic_d = ( italic_m , … , italic_m )</annotation></semantics></math>, i.e., <math alttext="F_{n}\left(U_{P_{st}}^{n},\alpha\right)\stackrel{{\scriptstyle\text{def}}}{{=}% }f_{(m,...,m)}\left(U_{P_{st}}^{n},\alpha\right)>\max_{d\neq(m,...,m)}f_{d}% \left(U_{P_{st}}^{n},\alpha\right)" class="ltx_Math" display="inline" id="Thmproofx19.p2.5.m5.12"><semantics id="Thmproofx19.p2.5.m5.12a"><mrow id="Thmproofx19.p2.5.m5.12.12" xref="Thmproofx19.p2.5.m5.12.12.cmml"><mrow id="Thmproofx19.p2.5.m5.10.10.1" xref="Thmproofx19.p2.5.m5.10.10.1.cmml"><msub id="Thmproofx19.p2.5.m5.10.10.1.3" xref="Thmproofx19.p2.5.m5.10.10.1.3.cmml"><mi id="Thmproofx19.p2.5.m5.10.10.1.3.2" xref="Thmproofx19.p2.5.m5.10.10.1.3.2.cmml">F</mi><mi id="Thmproofx19.p2.5.m5.10.10.1.3.3" xref="Thmproofx19.p2.5.m5.10.10.1.3.3.cmml">n</mi></msub><mo id="Thmproofx19.p2.5.m5.10.10.1.2" xref="Thmproofx19.p2.5.m5.10.10.1.2.cmml">⁢</mo><mrow id="Thmproofx19.p2.5.m5.10.10.1.1.1" xref="Thmproofx19.p2.5.m5.10.10.1.1.2.cmml"><mo id="Thmproofx19.p2.5.m5.10.10.1.1.1.2" xref="Thmproofx19.p2.5.m5.10.10.1.1.2.cmml">(</mo><msubsup id="Thmproofx19.p2.5.m5.10.10.1.1.1.1" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.cmml"><mi id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.2" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.cmml"><mi id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.2" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.cmml"><mi id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.2" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.1" 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xref="Thmproofx19.p2.5.m5.11.11.2.3.cmml"><mi id="Thmproofx19.p2.5.m5.11.11.2.3.2" xref="Thmproofx19.p2.5.m5.11.11.2.3.2.cmml">f</mi><mrow id="Thmproofx19.p2.5.m5.3.3.3.5" xref="Thmproofx19.p2.5.m5.3.3.3.4.cmml"><mo id="Thmproofx19.p2.5.m5.3.3.3.5.1" stretchy="false" xref="Thmproofx19.p2.5.m5.3.3.3.4.cmml">(</mo><mi id="Thmproofx19.p2.5.m5.1.1.1.1" xref="Thmproofx19.p2.5.m5.1.1.1.1.cmml">m</mi><mo id="Thmproofx19.p2.5.m5.3.3.3.5.2" xref="Thmproofx19.p2.5.m5.3.3.3.4.cmml">,</mo><mi id="Thmproofx19.p2.5.m5.2.2.2.2" mathvariant="normal" xref="Thmproofx19.p2.5.m5.2.2.2.2.cmml">…</mi><mo id="Thmproofx19.p2.5.m5.3.3.3.5.3" xref="Thmproofx19.p2.5.m5.3.3.3.4.cmml">,</mo><mi id="Thmproofx19.p2.5.m5.3.3.3.3" xref="Thmproofx19.p2.5.m5.3.3.3.3.cmml">m</mi><mo id="Thmproofx19.p2.5.m5.3.3.3.5.4" stretchy="false" xref="Thmproofx19.p2.5.m5.3.3.3.4.cmml">)</mo></mrow></msub><mo id="Thmproofx19.p2.5.m5.11.11.2.2" xref="Thmproofx19.p2.5.m5.11.11.2.2.cmml">⁢</mo><mrow id="Thmproofx19.p2.5.m5.11.11.2.1.1" xref="Thmproofx19.p2.5.m5.11.11.2.1.2.cmml"><mo id="Thmproofx19.p2.5.m5.11.11.2.1.1.2" xref="Thmproofx19.p2.5.m5.11.11.2.1.2.cmml">(</mo><msubsup id="Thmproofx19.p2.5.m5.11.11.2.1.1.1" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.cmml"><mi id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.2" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.cmml"><mi id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.2" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.cmml"><mi id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.2" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.1" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.3" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.3" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx19.p2.5.m5.11.11.2.1.1.3" xref="Thmproofx19.p2.5.m5.11.11.2.1.2.cmml">,</mo><mi id="Thmproofx19.p2.5.m5.8.8" xref="Thmproofx19.p2.5.m5.8.8.cmml">α</mi><mo id="Thmproofx19.p2.5.m5.11.11.2.1.1.4" xref="Thmproofx19.p2.5.m5.11.11.2.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx19.p2.5.m5.12.12.6" xref="Thmproofx19.p2.5.m5.12.12.6.cmml">></mo><mrow id="Thmproofx19.p2.5.m5.12.12.3" xref="Thmproofx19.p2.5.m5.12.12.3.cmml"><mrow id="Thmproofx19.p2.5.m5.12.12.3.3" xref="Thmproofx19.p2.5.m5.12.12.3.3.cmml"><msub id="Thmproofx19.p2.5.m5.12.12.3.3.1" xref="Thmproofx19.p2.5.m5.12.12.3.3.1.cmml"><mi id="Thmproofx19.p2.5.m5.12.12.3.3.1.2" xref="Thmproofx19.p2.5.m5.12.12.3.3.1.2.cmml">max</mi><mrow id="Thmproofx19.p2.5.m5.6.6.3" xref="Thmproofx19.p2.5.m5.6.6.3.cmml"><mi id="Thmproofx19.p2.5.m5.6.6.3.5" xref="Thmproofx19.p2.5.m5.6.6.3.5.cmml">d</mi><mo id="Thmproofx19.p2.5.m5.6.6.3.4" xref="Thmproofx19.p2.5.m5.6.6.3.4.cmml">≠</mo><mrow id="Thmproofx19.p2.5.m5.6.6.3.6.2" xref="Thmproofx19.p2.5.m5.6.6.3.6.1.cmml"><mo id="Thmproofx19.p2.5.m5.6.6.3.6.2.1" stretchy="false" xref="Thmproofx19.p2.5.m5.6.6.3.6.1.cmml">(</mo><mi id="Thmproofx19.p2.5.m5.4.4.1.1" xref="Thmproofx19.p2.5.m5.4.4.1.1.cmml">m</mi><mo id="Thmproofx19.p2.5.m5.6.6.3.6.2.2" xref="Thmproofx19.p2.5.m5.6.6.3.6.1.cmml">,</mo><mi id="Thmproofx19.p2.5.m5.5.5.2.2" mathvariant="normal" xref="Thmproofx19.p2.5.m5.5.5.2.2.cmml">…</mi><mo id="Thmproofx19.p2.5.m5.6.6.3.6.2.3" xref="Thmproofx19.p2.5.m5.6.6.3.6.1.cmml">,</mo><mi id="Thmproofx19.p2.5.m5.6.6.3.3" xref="Thmproofx19.p2.5.m5.6.6.3.3.cmml">m</mi><mo id="Thmproofx19.p2.5.m5.6.6.3.6.2.4" stretchy="false" xref="Thmproofx19.p2.5.m5.6.6.3.6.1.cmml">)</mo></mrow></mrow></msub><mo id="Thmproofx19.p2.5.m5.12.12.3.3a" lspace="0.167em" xref="Thmproofx19.p2.5.m5.12.12.3.3.cmml">⁡</mo><msub id="Thmproofx19.p2.5.m5.12.12.3.3.2" xref="Thmproofx19.p2.5.m5.12.12.3.3.2.cmml"><mi id="Thmproofx19.p2.5.m5.12.12.3.3.2.2" xref="Thmproofx19.p2.5.m5.12.12.3.3.2.2.cmml">f</mi><mi id="Thmproofx19.p2.5.m5.12.12.3.3.2.3" xref="Thmproofx19.p2.5.m5.12.12.3.3.2.3.cmml">d</mi></msub></mrow><mo id="Thmproofx19.p2.5.m5.12.12.3.2" xref="Thmproofx19.p2.5.m5.12.12.3.2.cmml">⁢</mo><mrow id="Thmproofx19.p2.5.m5.12.12.3.1.1" xref="Thmproofx19.p2.5.m5.12.12.3.1.2.cmml"><mo id="Thmproofx19.p2.5.m5.12.12.3.1.1.2" xref="Thmproofx19.p2.5.m5.12.12.3.1.2.cmml">(</mo><msubsup id="Thmproofx19.p2.5.m5.12.12.3.1.1.1" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.cmml"><mi id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.2" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.cmml"><mi id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.2" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.cmml"><mi id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.2" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.1" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.3" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.3" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx19.p2.5.m5.12.12.3.1.1.3" xref="Thmproofx19.p2.5.m5.12.12.3.1.2.cmml">,</mo><mi id="Thmproofx19.p2.5.m5.9.9" xref="Thmproofx19.p2.5.m5.9.9.cmml">α</mi><mo id="Thmproofx19.p2.5.m5.12.12.3.1.1.4" xref="Thmproofx19.p2.5.m5.12.12.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx19.p2.5.m5.12b"><apply id="Thmproofx19.p2.5.m5.12.12.cmml" xref="Thmproofx19.p2.5.m5.12.12"><and id="Thmproofx19.p2.5.m5.12.12a.cmml" xref="Thmproofx19.p2.5.m5.12.12"></and><apply id="Thmproofx19.p2.5.m5.12.12b.cmml" xref="Thmproofx19.p2.5.m5.12.12"><apply id="Thmproofx19.p2.5.m5.12.12.5.cmml" xref="Thmproofx19.p2.5.m5.12.12.5"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.12.12.5.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.5">superscript</csymbol><eq id="Thmproofx19.p2.5.m5.12.12.5.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.5.2"></eq><ci id="Thmproofx19.p2.5.m5.12.12.5.3a.cmml" xref="Thmproofx19.p2.5.m5.12.12.5.3"><mtext id="Thmproofx19.p2.5.m5.12.12.5.3.cmml" mathsize="70%" xref="Thmproofx19.p2.5.m5.12.12.5.3">def</mtext></ci></apply><apply id="Thmproofx19.p2.5.m5.10.10.1.cmml" xref="Thmproofx19.p2.5.m5.10.10.1"><times id="Thmproofx19.p2.5.m5.10.10.1.2.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.2"></times><apply id="Thmproofx19.p2.5.m5.10.10.1.3.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.3"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.10.10.1.3.1.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.3">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.10.10.1.3.2.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.3.2">𝐹</ci><ci id="Thmproofx19.p2.5.m5.10.10.1.3.3.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.3.3">𝑛</ci></apply><interval closure="open" id="Thmproofx19.p2.5.m5.10.10.1.1.2.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1"><apply id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.1.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1">superscript</csymbol><apply id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.1.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.2.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.2">𝑈</ci><apply id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.1.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.2.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.2">𝑃</ci><apply id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3"><times id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.1.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.1"></times><ci id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.2.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.3.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx19.p2.5.m5.10.10.1.1.1.1.3.cmml" xref="Thmproofx19.p2.5.m5.10.10.1.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx19.p2.5.m5.7.7.cmml" xref="Thmproofx19.p2.5.m5.7.7">𝛼</ci></interval></apply><apply id="Thmproofx19.p2.5.m5.11.11.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2"><times id="Thmproofx19.p2.5.m5.11.11.2.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.2"></times><apply id="Thmproofx19.p2.5.m5.11.11.2.3.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.11.11.2.3.1.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.3">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.11.11.2.3.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.3.2">𝑓</ci><vector id="Thmproofx19.p2.5.m5.3.3.3.4.cmml" xref="Thmproofx19.p2.5.m5.3.3.3.5"><ci id="Thmproofx19.p2.5.m5.1.1.1.1.cmml" xref="Thmproofx19.p2.5.m5.1.1.1.1">𝑚</ci><ci id="Thmproofx19.p2.5.m5.2.2.2.2.cmml" xref="Thmproofx19.p2.5.m5.2.2.2.2">…</ci><ci id="Thmproofx19.p2.5.m5.3.3.3.3.cmml" xref="Thmproofx19.p2.5.m5.3.3.3.3">𝑚</ci></vector></apply><interval closure="open" id="Thmproofx19.p2.5.m5.11.11.2.1.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1"><apply id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.1.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1">superscript</csymbol><apply id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.1.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.2">𝑈</ci><apply id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.1.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.2">𝑃</ci><apply id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3"><times id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.1.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.1"></times><ci id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.2.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.3.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx19.p2.5.m5.11.11.2.1.1.1.3.cmml" xref="Thmproofx19.p2.5.m5.11.11.2.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx19.p2.5.m5.8.8.cmml" xref="Thmproofx19.p2.5.m5.8.8">𝛼</ci></interval></apply></apply><apply id="Thmproofx19.p2.5.m5.12.12c.cmml" xref="Thmproofx19.p2.5.m5.12.12"><gt id="Thmproofx19.p2.5.m5.12.12.6.cmml" xref="Thmproofx19.p2.5.m5.12.12.6"></gt><share href="https://arxiv.org/html/2308.08047v3#Thmproofx19.p2.5.m5.11.11.2.cmml" id="Thmproofx19.p2.5.m5.12.12d.cmml" xref="Thmproofx19.p2.5.m5.12.12"></share><apply id="Thmproofx19.p2.5.m5.12.12.3.cmml" xref="Thmproofx19.p2.5.m5.12.12.3"><times id="Thmproofx19.p2.5.m5.12.12.3.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.2"></times><apply id="Thmproofx19.p2.5.m5.12.12.3.3.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3"><apply id="Thmproofx19.p2.5.m5.12.12.3.3.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.12.12.3.3.1.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3.1">subscript</csymbol><max id="Thmproofx19.p2.5.m5.12.12.3.3.1.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3.1.2"></max><apply id="Thmproofx19.p2.5.m5.6.6.3.cmml" xref="Thmproofx19.p2.5.m5.6.6.3"><neq id="Thmproofx19.p2.5.m5.6.6.3.4.cmml" xref="Thmproofx19.p2.5.m5.6.6.3.4"></neq><ci id="Thmproofx19.p2.5.m5.6.6.3.5.cmml" xref="Thmproofx19.p2.5.m5.6.6.3.5">𝑑</ci><vector id="Thmproofx19.p2.5.m5.6.6.3.6.1.cmml" xref="Thmproofx19.p2.5.m5.6.6.3.6.2"><ci id="Thmproofx19.p2.5.m5.4.4.1.1.cmml" xref="Thmproofx19.p2.5.m5.4.4.1.1">𝑚</ci><ci id="Thmproofx19.p2.5.m5.5.5.2.2.cmml" xref="Thmproofx19.p2.5.m5.5.5.2.2">…</ci><ci id="Thmproofx19.p2.5.m5.6.6.3.3.cmml" xref="Thmproofx19.p2.5.m5.6.6.3.3">𝑚</ci></vector></apply></apply><apply id="Thmproofx19.p2.5.m5.12.12.3.3.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3.2"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.12.12.3.3.2.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3.2">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.12.12.3.3.2.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3.2.2">𝑓</ci><ci id="Thmproofx19.p2.5.m5.12.12.3.3.2.3.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.3.2.3">𝑑</ci></apply></apply><interval closure="open" id="Thmproofx19.p2.5.m5.12.12.3.1.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1"><apply id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1">superscript</csymbol><apply id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.2">𝑈</ci><apply id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.2">𝑃</ci><apply id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3"><times id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.1.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.1"></times><ci id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.2.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.3.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx19.p2.5.m5.12.12.3.1.1.1.3.cmml" xref="Thmproofx19.p2.5.m5.12.12.3.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx19.p2.5.m5.9.9.cmml" xref="Thmproofx19.p2.5.m5.9.9">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx19.p2.5.m5.12c">F_{n}\left(U_{P_{st}}^{n},\alpha\right)\stackrel{{\scriptstyle\text{def}}}{{=}% }f_{(m,...,m)}\left(U_{P_{st}}^{n},\alpha\right)>\max_{d\neq(m,...,m)}f_{d}% \left(U_{P_{st}}^{n},\alpha\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx19.p2.5.m5.12d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) > roman_max start_POSTSUBSCRIPT italic_d ≠ ( italic_m , … , italic_m ) end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx20"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma15" title="Lemma 15 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">15</span></a>)</h6> <div class="ltx_para" id="Thmproofx20.p1"> <p class="ltx_p" id="Thmproofx20.p1.5">Consider <math alttext="G_{i}" class="ltx_Math" display="inline" id="Thmproofx20.p1.1.m1.1"><semantics id="Thmproofx20.p1.1.m1.1a"><msub id="Thmproofx20.p1.1.m1.1.1" xref="Thmproofx20.p1.1.m1.1.1.cmml"><mi id="Thmproofx20.p1.1.m1.1.1.2" xref="Thmproofx20.p1.1.m1.1.1.2.cmml">G</mi><mi id="Thmproofx20.p1.1.m1.1.1.3" xref="Thmproofx20.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.1.m1.1b"><apply id="Thmproofx20.p1.1.m1.1.1.cmml" xref="Thmproofx20.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx20.p1.1.m1.1.1.1.cmml" xref="Thmproofx20.p1.1.m1.1.1">subscript</csymbol><ci id="Thmproofx20.p1.1.m1.1.1.2.cmml" xref="Thmproofx20.p1.1.m1.1.1.2">𝐺</ci><ci id="Thmproofx20.p1.1.m1.1.1.3.cmml" xref="Thmproofx20.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.1.m1.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.1.m1.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i\leq m_{1}" class="ltx_Math" display="inline" id="Thmproofx20.p1.2.m2.1"><semantics id="Thmproofx20.p1.2.m2.1a"><mrow id="Thmproofx20.p1.2.m2.1.1" xref="Thmproofx20.p1.2.m2.1.1.cmml"><mi id="Thmproofx20.p1.2.m2.1.1.2" xref="Thmproofx20.p1.2.m2.1.1.2.cmml">i</mi><mo id="Thmproofx20.p1.2.m2.1.1.1" xref="Thmproofx20.p1.2.m2.1.1.1.cmml">≤</mo><msub id="Thmproofx20.p1.2.m2.1.1.3" xref="Thmproofx20.p1.2.m2.1.1.3.cmml"><mi id="Thmproofx20.p1.2.m2.1.1.3.2" xref="Thmproofx20.p1.2.m2.1.1.3.2.cmml">m</mi><mn id="Thmproofx20.p1.2.m2.1.1.3.3" xref="Thmproofx20.p1.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.2.m2.1b"><apply id="Thmproofx20.p1.2.m2.1.1.cmml" xref="Thmproofx20.p1.2.m2.1.1"><leq id="Thmproofx20.p1.2.m2.1.1.1.cmml" xref="Thmproofx20.p1.2.m2.1.1.1"></leq><ci id="Thmproofx20.p1.2.m2.1.1.2.cmml" xref="Thmproofx20.p1.2.m2.1.1.2">𝑖</ci><apply id="Thmproofx20.p1.2.m2.1.1.3.cmml" xref="Thmproofx20.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx20.p1.2.m2.1.1.3.1.cmml" xref="Thmproofx20.p1.2.m2.1.1.3">subscript</csymbol><ci id="Thmproofx20.p1.2.m2.1.1.3.2.cmml" xref="Thmproofx20.p1.2.m2.1.1.3.2">𝑚</ci><cn id="Thmproofx20.p1.2.m2.1.1.3.3.cmml" type="integer" xref="Thmproofx20.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.2.m2.1c">i\leq m_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.2.m2.1d">italic_i ≤ italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Then to apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma14" title="Lemma 14 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">14</span></a> to it the <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx20.p1.3.m3.1"><semantics id="Thmproofx20.p1.3.m3.1a"><mi id="Thmproofx20.p1.3.m3.1.1" xref="Thmproofx20.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.3.m3.1b"><ci id="Thmproofx20.p1.3.m3.1.1.cmml" xref="Thmproofx20.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.3.m3.1d">italic_α</annotation></semantics></math> inequality constraint becomes <math alttext="\alpha>0" class="ltx_Math" display="inline" id="Thmproofx20.p1.4.m4.1"><semantics id="Thmproofx20.p1.4.m4.1a"><mrow id="Thmproofx20.p1.4.m4.1.1" xref="Thmproofx20.p1.4.m4.1.1.cmml"><mi id="Thmproofx20.p1.4.m4.1.1.2" xref="Thmproofx20.p1.4.m4.1.1.2.cmml">α</mi><mo id="Thmproofx20.p1.4.m4.1.1.1" xref="Thmproofx20.p1.4.m4.1.1.1.cmml">></mo><mn id="Thmproofx20.p1.4.m4.1.1.3" xref="Thmproofx20.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.4.m4.1b"><apply id="Thmproofx20.p1.4.m4.1.1.cmml" xref="Thmproofx20.p1.4.m4.1.1"><gt id="Thmproofx20.p1.4.m4.1.1.1.cmml" xref="Thmproofx20.p1.4.m4.1.1.1"></gt><ci id="Thmproofx20.p1.4.m4.1.1.2.cmml" xref="Thmproofx20.p1.4.m4.1.1.2">𝛼</ci><cn id="Thmproofx20.p1.4.m4.1.1.3.cmml" type="integer" xref="Thmproofx20.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.4.m4.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.4.m4.1d">italic_α > 0</annotation></semantics></math> which holds by definition of <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx20.p1.5.m5.1"><semantics id="Thmproofx20.p1.5.m5.1a"><mi id="Thmproofx20.p1.5.m5.1.1" xref="Thmproofx20.p1.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.5.m5.1b"><ci id="Thmproofx20.p1.5.m5.1.1.cmml" xref="Thmproofx20.p1.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.5.m5.1d">italic_α</annotation></semantics></math>. So</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx53"> <tbody id="Pt0.A4.Ex90"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle F_{n}(U_{P_{i}}^{n},\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex90.m1.2"><semantics id="Pt0.A4.Ex90.m1.2a"><mrow id="Pt0.A4.Ex90.m1.2.2" xref="Pt0.A4.Ex90.m1.2.2.cmml"><msub id="Pt0.A4.Ex90.m1.2.2.3" xref="Pt0.A4.Ex90.m1.2.2.3.cmml"><mi id="Pt0.A4.Ex90.m1.2.2.3.2" xref="Pt0.A4.Ex90.m1.2.2.3.2.cmml">F</mi><mi id="Pt0.A4.Ex90.m1.2.2.3.3" xref="Pt0.A4.Ex90.m1.2.2.3.3.cmml">n</mi></msub><mo id="Pt0.A4.Ex90.m1.2.2.2" xref="Pt0.A4.Ex90.m1.2.2.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex90.m1.2.2.1.1" xref="Pt0.A4.Ex90.m1.2.2.1.2.cmml"><mo id="Pt0.A4.Ex90.m1.2.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex90.m1.2.2.1.2.cmml">(</mo><msubsup id="Pt0.A4.Ex90.m1.2.2.1.1.1" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.cmml"><mi id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.2" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.2.cmml">U</mi><msub id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.cmml"><mi id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.2" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.2.cmml">P</mi><mi id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.3" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.3.cmml">i</mi></msub><mi id="Pt0.A4.Ex90.m1.2.2.1.1.1.3" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Pt0.A4.Ex90.m1.2.2.1.1.3" xref="Pt0.A4.Ex90.m1.2.2.1.2.cmml">,</mo><mi id="Pt0.A4.Ex90.m1.1.1" xref="Pt0.A4.Ex90.m1.1.1.cmml">α</mi><mo id="Pt0.A4.Ex90.m1.2.2.1.1.4" stretchy="false" xref="Pt0.A4.Ex90.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex90.m1.2b"><apply id="Pt0.A4.Ex90.m1.2.2.cmml" xref="Pt0.A4.Ex90.m1.2.2"><times id="Pt0.A4.Ex90.m1.2.2.2.cmml" xref="Pt0.A4.Ex90.m1.2.2.2"></times><apply id="Pt0.A4.Ex90.m1.2.2.3.cmml" xref="Pt0.A4.Ex90.m1.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m1.2.2.3.1.cmml" xref="Pt0.A4.Ex90.m1.2.2.3">subscript</csymbol><ci id="Pt0.A4.Ex90.m1.2.2.3.2.cmml" xref="Pt0.A4.Ex90.m1.2.2.3.2">𝐹</ci><ci id="Pt0.A4.Ex90.m1.2.2.3.3.cmml" xref="Pt0.A4.Ex90.m1.2.2.3.3">𝑛</ci></apply><interval closure="open" id="Pt0.A4.Ex90.m1.2.2.1.2.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1"><apply id="Pt0.A4.Ex90.m1.2.2.1.1.1.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m1.2.2.1.1.1.1.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.1.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.2.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.2">𝑈</ci><apply id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.1.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3">subscript</csymbol><ci id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.2.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.2">𝑃</ci><ci id="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.3.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Pt0.A4.Ex90.m1.2.2.1.1.1.3.cmml" xref="Pt0.A4.Ex90.m1.2.2.1.1.1.3">𝑛</ci></apply><ci id="Pt0.A4.Ex90.m1.1.1.cmml" xref="Pt0.A4.Ex90.m1.1.1">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex90.m1.2c">\displaystyle F_{n}(U_{P_{i}}^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex90.m1.2d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=f_{(i,...,i)}(U_{P_{i}}^{n},\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex90.m2.5"><semantics id="Pt0.A4.Ex90.m2.5a"><mrow id="Pt0.A4.Ex90.m2.5.5" xref="Pt0.A4.Ex90.m2.5.5.cmml"><mi id="Pt0.A4.Ex90.m2.5.5.3" xref="Pt0.A4.Ex90.m2.5.5.3.cmml"></mi><mo id="Pt0.A4.Ex90.m2.5.5.2" xref="Pt0.A4.Ex90.m2.5.5.2.cmml">=</mo><mrow id="Pt0.A4.Ex90.m2.5.5.1" xref="Pt0.A4.Ex90.m2.5.5.1.cmml"><msub id="Pt0.A4.Ex90.m2.5.5.1.3" xref="Pt0.A4.Ex90.m2.5.5.1.3.cmml"><mi id="Pt0.A4.Ex90.m2.5.5.1.3.2" xref="Pt0.A4.Ex90.m2.5.5.1.3.2.cmml">f</mi><mrow id="Pt0.A4.Ex90.m2.3.3.3.5" xref="Pt0.A4.Ex90.m2.3.3.3.4.cmml"><mo id="Pt0.A4.Ex90.m2.3.3.3.5.1" stretchy="false" xref="Pt0.A4.Ex90.m2.3.3.3.4.cmml">(</mo><mi id="Pt0.A4.Ex90.m2.1.1.1.1" xref="Pt0.A4.Ex90.m2.1.1.1.1.cmml">i</mi><mo id="Pt0.A4.Ex90.m2.3.3.3.5.2" xref="Pt0.A4.Ex90.m2.3.3.3.4.cmml">,</mo><mi id="Pt0.A4.Ex90.m2.2.2.2.2" mathvariant="normal" xref="Pt0.A4.Ex90.m2.2.2.2.2.cmml">…</mi><mo id="Pt0.A4.Ex90.m2.3.3.3.5.3" xref="Pt0.A4.Ex90.m2.3.3.3.4.cmml">,</mo><mi id="Pt0.A4.Ex90.m2.3.3.3.3" xref="Pt0.A4.Ex90.m2.3.3.3.3.cmml">i</mi><mo id="Pt0.A4.Ex90.m2.3.3.3.5.4" stretchy="false" xref="Pt0.A4.Ex90.m2.3.3.3.4.cmml">)</mo></mrow></msub><mo id="Pt0.A4.Ex90.m2.5.5.1.2" xref="Pt0.A4.Ex90.m2.5.5.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex90.m2.5.5.1.1.1" xref="Pt0.A4.Ex90.m2.5.5.1.1.2.cmml"><mo id="Pt0.A4.Ex90.m2.5.5.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex90.m2.5.5.1.1.2.cmml">(</mo><msubsup id="Pt0.A4.Ex90.m2.5.5.1.1.1.1" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.cmml"><mi id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.2" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.2.cmml">U</mi><msub id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.cmml"><mi id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.2" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.2.cmml">P</mi><mi id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.3" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.3.cmml">i</mi></msub><mi id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.3" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Pt0.A4.Ex90.m2.5.5.1.1.1.3" xref="Pt0.A4.Ex90.m2.5.5.1.1.2.cmml">,</mo><mi id="Pt0.A4.Ex90.m2.4.4" xref="Pt0.A4.Ex90.m2.4.4.cmml">α</mi><mo id="Pt0.A4.Ex90.m2.5.5.1.1.1.4" stretchy="false" xref="Pt0.A4.Ex90.m2.5.5.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex90.m2.5b"><apply id="Pt0.A4.Ex90.m2.5.5.cmml" xref="Pt0.A4.Ex90.m2.5.5"><eq id="Pt0.A4.Ex90.m2.5.5.2.cmml" xref="Pt0.A4.Ex90.m2.5.5.2"></eq><csymbol cd="latexml" id="Pt0.A4.Ex90.m2.5.5.3.cmml" xref="Pt0.A4.Ex90.m2.5.5.3">absent</csymbol><apply id="Pt0.A4.Ex90.m2.5.5.1.cmml" xref="Pt0.A4.Ex90.m2.5.5.1"><times id="Pt0.A4.Ex90.m2.5.5.1.2.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.2"></times><apply id="Pt0.A4.Ex90.m2.5.5.1.3.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m2.5.5.1.3.1.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.3">subscript</csymbol><ci id="Pt0.A4.Ex90.m2.5.5.1.3.2.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.3.2">𝑓</ci><vector id="Pt0.A4.Ex90.m2.3.3.3.4.cmml" xref="Pt0.A4.Ex90.m2.3.3.3.5"><ci id="Pt0.A4.Ex90.m2.1.1.1.1.cmml" xref="Pt0.A4.Ex90.m2.1.1.1.1">𝑖</ci><ci id="Pt0.A4.Ex90.m2.2.2.2.2.cmml" xref="Pt0.A4.Ex90.m2.2.2.2.2">…</ci><ci id="Pt0.A4.Ex90.m2.3.3.3.3.cmml" xref="Pt0.A4.Ex90.m2.3.3.3.3">𝑖</ci></vector></apply><interval closure="open" id="Pt0.A4.Ex90.m2.5.5.1.1.2.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1"><apply id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.1.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.1.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.2.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.2">𝑈</ci><apply id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.1.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3">subscript</csymbol><ci id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.2.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.2">𝑃</ci><ci id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.3.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Pt0.A4.Ex90.m2.5.5.1.1.1.1.3.cmml" xref="Pt0.A4.Ex90.m2.5.5.1.1.1.1.3">𝑛</ci></apply><ci id="Pt0.A4.Ex90.m2.4.4.cmml" xref="Pt0.A4.Ex90.m2.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex90.m2.5c">\displaystyle=f_{(i,...,i)}(U_{P_{i}}^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex90.m2.5d">= italic_f start_POSTSUBSCRIPT ( italic_i , … , italic_i ) end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> 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xref="Pt0.A4.Ex91.m1.5.5.1.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex91.m1.5.5.1.3.3" xref="Pt0.A4.Ex91.m1.5.5.1.3.3.cmml">k</mi><mo id="Pt0.A4.Ex91.m1.5.5.1.3.1a" xref="Pt0.A4.Ex91.m1.5.5.1.3.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex91.m1.5.5.1.3.4" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex91.m1.5.5.1.3.4.1" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.1.cmml"><munderover id="Pt0.A4.Ex91.m1.5.5.1.3.4.1a" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.1.cmml"><mo id="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.2" movablelimits="false" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.cmml"><msup id="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.2" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.2.cmml"><mi id="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.2.2" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.2.3" xref="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex91.m1.5.5.1.3.4.1.2.3.1" 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xref="Pt0.A4.Ex91.m1.4.4.4">superscript</csymbol><ci id="Pt0.A4.Ex91.m1.4.4.4.2.cmml" xref="Pt0.A4.Ex91.m1.4.4.4.2">𝑖</ci><ci id="Pt0.A4.Ex91.m1.4.4.4.3.cmml" xref="Pt0.A4.Ex91.m1.4.4.4.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex91.m1.5c">\displaystyle=\alpha k\sum_{i^{\prime}=1}^{i}1_{i^{\prime}=i}\frac{(i^{\prime}% )^{n}-(i^{\prime}-1)^{n}}{i^{n}}+\sum_{i^{\prime}=1}^{i}|p_{i^{\prime}}|\frac{% (i^{\prime})^{n}-(i^{\prime}-1)^{n}}{i^{n}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex91.m1.5d">= italic_α italic_k ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT 1 start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_i end_POSTSUBSCRIPT divide start_ARG ( italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG + ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT | italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | divide start_ARG ( italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex92"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=|p_{1}|+\frac{\mathcal{E}_{i,n}}{i}k\alpha" class="ltx_Math" display="inline" id="Pt0.A4.Ex92.m1.3"><semantics id="Pt0.A4.Ex92.m1.3a"><mrow id="Pt0.A4.Ex92.m1.3.3" xref="Pt0.A4.Ex92.m1.3.3.cmml"><mi id="Pt0.A4.Ex92.m1.3.3.3" xref="Pt0.A4.Ex92.m1.3.3.3.cmml"></mi><mo id="Pt0.A4.Ex92.m1.3.3.2" xref="Pt0.A4.Ex92.m1.3.3.2.cmml">=</mo><mrow id="Pt0.A4.Ex92.m1.3.3.1" xref="Pt0.A4.Ex92.m1.3.3.1.cmml"><mrow id="Pt0.A4.Ex92.m1.3.3.1.1.1" xref="Pt0.A4.Ex92.m1.3.3.1.1.2.cmml"><mo id="Pt0.A4.Ex92.m1.3.3.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex92.m1.3.3.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex92.m1.3.3.1.1.1.1" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.1.cmml"><mi id="Pt0.A4.Ex92.m1.3.3.1.1.1.1.2" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.1.2.cmml">p</mi><mn id="Pt0.A4.Ex92.m1.3.3.1.1.1.1.3" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.1.3.cmml">1</mn></msub><mo 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id="Pt0.A4.Ex92.m1.3.3.1.3.1" xref="Pt0.A4.Ex92.m1.3.3.1.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex92.m1.3.3.1.3.2" xref="Pt0.A4.Ex92.m1.3.3.1.3.2.cmml">k</mi><mo id="Pt0.A4.Ex92.m1.3.3.1.3.1a" xref="Pt0.A4.Ex92.m1.3.3.1.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex92.m1.3.3.1.3.3" xref="Pt0.A4.Ex92.m1.3.3.1.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex92.m1.3b"><apply id="Pt0.A4.Ex92.m1.3.3.cmml" xref="Pt0.A4.Ex92.m1.3.3"><eq id="Pt0.A4.Ex92.m1.3.3.2.cmml" xref="Pt0.A4.Ex92.m1.3.3.2"></eq><csymbol cd="latexml" id="Pt0.A4.Ex92.m1.3.3.3.cmml" xref="Pt0.A4.Ex92.m1.3.3.3">absent</csymbol><apply id="Pt0.A4.Ex92.m1.3.3.1.cmml" xref="Pt0.A4.Ex92.m1.3.3.1"><plus id="Pt0.A4.Ex92.m1.3.3.1.2.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.2"></plus><apply id="Pt0.A4.Ex92.m1.3.3.1.1.2.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.1.1"><abs id="Pt0.A4.Ex92.m1.3.3.1.1.2.1.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.2"></abs><apply id="Pt0.A4.Ex92.m1.3.3.1.1.1.1.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex92.m1.3.3.1.1.1.1.1.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex92.m1.3.3.1.1.1.1.2.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.1.2">𝑝</ci><cn id="Pt0.A4.Ex92.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex92.m1.3.3.1.1.1.1.3">1</cn></apply></apply><apply id="Pt0.A4.Ex92.m1.3.3.1.3.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.3"><times id="Pt0.A4.Ex92.m1.3.3.1.3.1.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.3.1"></times><apply id="Pt0.A4.Ex92.m1.2.2.cmml" xref="Pt0.A4.Ex92.m1.2.2"><divide id="Pt0.A4.Ex92.m1.2.2.3.cmml" xref="Pt0.A4.Ex92.m1.2.2"></divide><apply id="Pt0.A4.Ex92.m1.2.2.2.cmml" xref="Pt0.A4.Ex92.m1.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex92.m1.2.2.2.3.cmml" xref="Pt0.A4.Ex92.m1.2.2.2">subscript</csymbol><ci id="Pt0.A4.Ex92.m1.2.2.2.4.cmml" xref="Pt0.A4.Ex92.m1.2.2.2.4">ℰ</ci><list id="Pt0.A4.Ex92.m1.2.2.2.2.2.3.cmml" xref="Pt0.A4.Ex92.m1.2.2.2.2.2.4"><ci id="Pt0.A4.Ex92.m1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex92.m1.1.1.1.1.1.1">𝑖</ci><ci id="Pt0.A4.Ex92.m1.2.2.2.2.2.2.cmml" xref="Pt0.A4.Ex92.m1.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Pt0.A4.Ex92.m1.2.2.4.cmml" xref="Pt0.A4.Ex92.m1.2.2.4">𝑖</ci></apply><ci id="Pt0.A4.Ex92.m1.3.3.1.3.2.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.3.2">𝑘</ci><ci id="Pt0.A4.Ex92.m1.3.3.1.3.3.cmml" xref="Pt0.A4.Ex92.m1.3.3.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex92.m1.3c">\displaystyle=|p_{1}|+\frac{\mathcal{E}_{i,n}}{i}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex92.m1.3d">= | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx20.p1.8">The first equality is due to Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma14" title="Lemma 14 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">14</span></a>. The second is by Equation <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E6" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">6</span></a>, and the third is because the paths of <math alttext="G_{m_{1}}" class="ltx_Math" display="inline" id="Thmproofx20.p1.6.m1.1"><semantics id="Thmproofx20.p1.6.m1.1a"><msub id="Thmproofx20.p1.6.m1.1.1" xref="Thmproofx20.p1.6.m1.1.1.cmml"><mi id="Thmproofx20.p1.6.m1.1.1.2" xref="Thmproofx20.p1.6.m1.1.1.2.cmml">G</mi><msub id="Thmproofx20.p1.6.m1.1.1.3" xref="Thmproofx20.p1.6.m1.1.1.3.cmml"><mi id="Thmproofx20.p1.6.m1.1.1.3.2" xref="Thmproofx20.p1.6.m1.1.1.3.2.cmml">m</mi><mn id="Thmproofx20.p1.6.m1.1.1.3.3" xref="Thmproofx20.p1.6.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.6.m1.1b"><apply id="Thmproofx20.p1.6.m1.1.1.cmml" xref="Thmproofx20.p1.6.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx20.p1.6.m1.1.1.1.cmml" xref="Thmproofx20.p1.6.m1.1.1">subscript</csymbol><ci id="Thmproofx20.p1.6.m1.1.1.2.cmml" xref="Thmproofx20.p1.6.m1.1.1.2">𝐺</ci><apply id="Thmproofx20.p1.6.m1.1.1.3.cmml" xref="Thmproofx20.p1.6.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx20.p1.6.m1.1.1.3.1.cmml" xref="Thmproofx20.p1.6.m1.1.1.3">subscript</csymbol><ci id="Thmproofx20.p1.6.m1.1.1.3.2.cmml" xref="Thmproofx20.p1.6.m1.1.1.3.2">𝑚</ci><cn id="Thmproofx20.p1.6.m1.1.1.3.3.cmml" type="integer" xref="Thmproofx20.p1.6.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.6.m1.1c">G_{m_{1}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.6.m1.1d">italic_G start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> have identical lengths. The second part of the lemma follows by that <math alttext="\frac{\mathcal{E}_{i,n}}{i}" class="ltx_Math" display="inline" id="Thmproofx20.p1.7.m2.2"><semantics id="Thmproofx20.p1.7.m2.2a"><mfrac id="Thmproofx20.p1.7.m2.2.2" xref="Thmproofx20.p1.7.m2.2.2.cmml"><msub id="Thmproofx20.p1.7.m2.2.2.2" xref="Thmproofx20.p1.7.m2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx20.p1.7.m2.2.2.2.4" xref="Thmproofx20.p1.7.m2.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx20.p1.7.m2.2.2.2.2.2.4" xref="Thmproofx20.p1.7.m2.2.2.2.2.2.3.cmml"><mi id="Thmproofx20.p1.7.m2.1.1.1.1.1.1" xref="Thmproofx20.p1.7.m2.1.1.1.1.1.1.cmml">i</mi><mo id="Thmproofx20.p1.7.m2.2.2.2.2.2.4.1" xref="Thmproofx20.p1.7.m2.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx20.p1.7.m2.2.2.2.2.2.2" xref="Thmproofx20.p1.7.m2.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Thmproofx20.p1.7.m2.2.2.4" xref="Thmproofx20.p1.7.m2.2.2.4.cmml">i</mi></mfrac><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.7.m2.2b"><apply id="Thmproofx20.p1.7.m2.2.2.cmml" xref="Thmproofx20.p1.7.m2.2.2"><divide id="Thmproofx20.p1.7.m2.2.2.3.cmml" xref="Thmproofx20.p1.7.m2.2.2"></divide><apply id="Thmproofx20.p1.7.m2.2.2.2.cmml" xref="Thmproofx20.p1.7.m2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx20.p1.7.m2.2.2.2.3.cmml" xref="Thmproofx20.p1.7.m2.2.2.2">subscript</csymbol><ci id="Thmproofx20.p1.7.m2.2.2.2.4.cmml" xref="Thmproofx20.p1.7.m2.2.2.2.4">ℰ</ci><list id="Thmproofx20.p1.7.m2.2.2.2.2.2.3.cmml" xref="Thmproofx20.p1.7.m2.2.2.2.2.2.4"><ci id="Thmproofx20.p1.7.m2.1.1.1.1.1.1.cmml" xref="Thmproofx20.p1.7.m2.1.1.1.1.1.1">𝑖</ci><ci id="Thmproofx20.p1.7.m2.2.2.2.2.2.2.cmml" xref="Thmproofx20.p1.7.m2.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Thmproofx20.p1.7.m2.2.2.4.cmml" xref="Thmproofx20.p1.7.m2.2.2.4">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.7.m2.2c">\frac{\mathcal{E}_{i,n}}{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.7.m2.2d">divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_i end_ARG</annotation></semantics></math> is decreasing in <math alttext="i" class="ltx_Math" display="inline" id="Thmproofx20.p1.8.m3.1"><semantics id="Thmproofx20.p1.8.m3.1a"><mi id="Thmproofx20.p1.8.m3.1.1" xref="Thmproofx20.p1.8.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmproofx20.p1.8.m3.1b"><ci id="Thmproofx20.p1.8.m3.1.1.cmml" xref="Thmproofx20.p1.8.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx20.p1.8.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmproofx20.p1.8.m3.1d">italic_i</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx21"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma16" title="Lemma 16 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">16</span></a>)</h6> <div class="ltx_para" id="Thmproofx21.p1"> <p class="ltx_p" id="Thmproofx21.p1.6">Consider <math alttext="\sigma=U_{P_{m_{1}+1}}^{n}" class="ltx_Math" display="inline" id="Thmproofx21.p1.1.m1.1"><semantics id="Thmproofx21.p1.1.m1.1a"><mrow id="Thmproofx21.p1.1.m1.1.1" xref="Thmproofx21.p1.1.m1.1.1.cmml"><mi id="Thmproofx21.p1.1.m1.1.1.2" xref="Thmproofx21.p1.1.m1.1.1.2.cmml">σ</mi><mo id="Thmproofx21.p1.1.m1.1.1.1" xref="Thmproofx21.p1.1.m1.1.1.1.cmml">=</mo><msubsup id="Thmproofx21.p1.1.m1.1.1.3" xref="Thmproofx21.p1.1.m1.1.1.3.cmml"><mi id="Thmproofx21.p1.1.m1.1.1.3.2.2" xref="Thmproofx21.p1.1.m1.1.1.3.2.2.cmml">U</mi><msub id="Thmproofx21.p1.1.m1.1.1.3.2.3" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.cmml"><mi id="Thmproofx21.p1.1.m1.1.1.3.2.3.2" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.2.cmml">P</mi><mrow id="Thmproofx21.p1.1.m1.1.1.3.2.3.3" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.cmml"><msub id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.cmml"><mi id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.2" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.3" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.1" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.1.cmml">+</mo><mn id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.3" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.3.cmml">1</mn></mrow></msub><mi id="Thmproofx21.p1.1.m1.1.1.3.3" xref="Thmproofx21.p1.1.m1.1.1.3.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.1.m1.1b"><apply id="Thmproofx21.p1.1.m1.1.1.cmml" xref="Thmproofx21.p1.1.m1.1.1"><eq id="Thmproofx21.p1.1.m1.1.1.1.cmml" xref="Thmproofx21.p1.1.m1.1.1.1"></eq><ci id="Thmproofx21.p1.1.m1.1.1.2.cmml" xref="Thmproofx21.p1.1.m1.1.1.2">𝜎</ci><apply id="Thmproofx21.p1.1.m1.1.1.3.cmml" xref="Thmproofx21.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.1.m1.1.1.3.1.cmml" xref="Thmproofx21.p1.1.m1.1.1.3">superscript</csymbol><apply id="Thmproofx21.p1.1.m1.1.1.3.2.cmml" xref="Thmproofx21.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.1.m1.1.1.3.2.1.cmml" xref="Thmproofx21.p1.1.m1.1.1.3">subscript</csymbol><ci id="Thmproofx21.p1.1.m1.1.1.3.2.2.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.2">𝑈</ci><apply id="Thmproofx21.p1.1.m1.1.1.3.2.3.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.1.m1.1.1.3.2.3.1.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3">subscript</csymbol><ci id="Thmproofx21.p1.1.m1.1.1.3.2.3.2.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.2">𝑃</ci><apply id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3"><plus id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.1.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.1"></plus><apply id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.1.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2">subscript</csymbol><ci id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.2.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.1.m1.1.1.3.2.3.3.3.cmml" type="integer" xref="Thmproofx21.p1.1.m1.1.1.3.2.3.3.3">1</cn></apply></apply></apply><ci id="Thmproofx21.p1.1.m1.1.1.3.3.cmml" xref="Thmproofx21.p1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.1.m1.1c">\sigma=U_{P_{m_{1}+1}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.1.m1.1d">italic_σ = italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="G_{m_{1}+1}" class="ltx_Math" display="inline" id="Thmproofx21.p1.2.m2.1"><semantics id="Thmproofx21.p1.2.m2.1a"><msub id="Thmproofx21.p1.2.m2.1.1" xref="Thmproofx21.p1.2.m2.1.1.cmml"><mi id="Thmproofx21.p1.2.m2.1.1.2" xref="Thmproofx21.p1.2.m2.1.1.2.cmml">G</mi><mrow id="Thmproofx21.p1.2.m2.1.1.3" xref="Thmproofx21.p1.2.m2.1.1.3.cmml"><msub id="Thmproofx21.p1.2.m2.1.1.3.2" xref="Thmproofx21.p1.2.m2.1.1.3.2.cmml"><mi id="Thmproofx21.p1.2.m2.1.1.3.2.2" xref="Thmproofx21.p1.2.m2.1.1.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.2.m2.1.1.3.2.3" xref="Thmproofx21.p1.2.m2.1.1.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.2.m2.1.1.3.1" xref="Thmproofx21.p1.2.m2.1.1.3.1.cmml">+</mo><mn id="Thmproofx21.p1.2.m2.1.1.3.3" xref="Thmproofx21.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.2.m2.1b"><apply id="Thmproofx21.p1.2.m2.1.1.cmml" xref="Thmproofx21.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmproofx21.p1.2.m2.1.1.1.cmml" xref="Thmproofx21.p1.2.m2.1.1">subscript</csymbol><ci id="Thmproofx21.p1.2.m2.1.1.2.cmml" xref="Thmproofx21.p1.2.m2.1.1.2">𝐺</ci><apply id="Thmproofx21.p1.2.m2.1.1.3.cmml" xref="Thmproofx21.p1.2.m2.1.1.3"><plus id="Thmproofx21.p1.2.m2.1.1.3.1.cmml" xref="Thmproofx21.p1.2.m2.1.1.3.1"></plus><apply id="Thmproofx21.p1.2.m2.1.1.3.2.cmml" xref="Thmproofx21.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.2.m2.1.1.3.2.1.cmml" xref="Thmproofx21.p1.2.m2.1.1.3.2">subscript</csymbol><ci id="Thmproofx21.p1.2.m2.1.1.3.2.2.cmml" xref="Thmproofx21.p1.2.m2.1.1.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.2.m2.1.1.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.2.m2.1.1.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.2.m2.1.1.3.3.cmml" type="integer" xref="Thmproofx21.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.2.m2.1c">G_{m_{1}+1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.2.m2.1d">italic_G start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. We show that the bound on <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx21.p1.3.m3.1"><semantics id="Thmproofx21.p1.3.m3.1a"><mi id="Thmproofx21.p1.3.m3.1.1" xref="Thmproofx21.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.3.m3.1b"><ci id="Thmproofx21.p1.3.m3.1.1.cmml" xref="Thmproofx21.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.3.m3.1d">italic_α</annotation></semantics></math> in the lemma statement implies that <math alttext="\alpha>\frac{|p_{m_{1}+1}|-|p_{1}|}{k}" class="ltx_Math" display="inline" id="Thmproofx21.p1.4.m4.2"><semantics id="Thmproofx21.p1.4.m4.2a"><mrow id="Thmproofx21.p1.4.m4.2.3" xref="Thmproofx21.p1.4.m4.2.3.cmml"><mi id="Thmproofx21.p1.4.m4.2.3.2" xref="Thmproofx21.p1.4.m4.2.3.2.cmml">α</mi><mo id="Thmproofx21.p1.4.m4.2.3.1" xref="Thmproofx21.p1.4.m4.2.3.1.cmml">></mo><mfrac id="Thmproofx21.p1.4.m4.2.2" xref="Thmproofx21.p1.4.m4.2.2.cmml"><mrow id="Thmproofx21.p1.4.m4.2.2.2" xref="Thmproofx21.p1.4.m4.2.2.2.cmml"><mrow id="Thmproofx21.p1.4.m4.1.1.1.1.1" xref="Thmproofx21.p1.4.m4.1.1.1.1.2.cmml"><mo id="Thmproofx21.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="Thmproofx21.p1.4.m4.1.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx21.p1.4.m4.1.1.1.1.1.1" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.2" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.cmml"><msub id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.cmml"><mi id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.2" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.3" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.1" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.3" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmproofx21.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="Thmproofx21.p1.4.m4.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx21.p1.4.m4.2.2.2.3" xref="Thmproofx21.p1.4.m4.2.2.2.3.cmml">−</mo><mrow id="Thmproofx21.p1.4.m4.2.2.2.2.1" xref="Thmproofx21.p1.4.m4.2.2.2.2.2.cmml"><mo id="Thmproofx21.p1.4.m4.2.2.2.2.1.2" stretchy="false" xref="Thmproofx21.p1.4.m4.2.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx21.p1.4.m4.2.2.2.2.1.1" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.1.cmml"><mi id="Thmproofx21.p1.4.m4.2.2.2.2.1.1.2" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.1.2.cmml">p</mi><mn id="Thmproofx21.p1.4.m4.2.2.2.2.1.1.3" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.4.m4.2.2.2.2.1.3" stretchy="false" xref="Thmproofx21.p1.4.m4.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mi id="Thmproofx21.p1.4.m4.2.2.4" xref="Thmproofx21.p1.4.m4.2.2.4.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.4.m4.2b"><apply id="Thmproofx21.p1.4.m4.2.3.cmml" xref="Thmproofx21.p1.4.m4.2.3"><gt id="Thmproofx21.p1.4.m4.2.3.1.cmml" xref="Thmproofx21.p1.4.m4.2.3.1"></gt><ci id="Thmproofx21.p1.4.m4.2.3.2.cmml" xref="Thmproofx21.p1.4.m4.2.3.2">𝛼</ci><apply id="Thmproofx21.p1.4.m4.2.2.cmml" xref="Thmproofx21.p1.4.m4.2.2"><divide id="Thmproofx21.p1.4.m4.2.2.3.cmml" xref="Thmproofx21.p1.4.m4.2.2"></divide><apply id="Thmproofx21.p1.4.m4.2.2.2.cmml" xref="Thmproofx21.p1.4.m4.2.2.2"><minus id="Thmproofx21.p1.4.m4.2.2.2.3.cmml" xref="Thmproofx21.p1.4.m4.2.2.2.3"></minus><apply id="Thmproofx21.p1.4.m4.1.1.1.1.2.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1"><abs id="Thmproofx21.p1.4.m4.1.1.1.1.2.1.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.2"></abs><apply id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.2">𝑝</ci><apply id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3"><plus id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.1"></plus><apply id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.1.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.2.cmml" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmproofx21.p1.4.m4.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="Thmproofx21.p1.4.m4.2.2.2.2.2.cmml" xref="Thmproofx21.p1.4.m4.2.2.2.2.1"><abs id="Thmproofx21.p1.4.m4.2.2.2.2.2.1.cmml" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.2"></abs><apply id="Thmproofx21.p1.4.m4.2.2.2.2.1.1.cmml" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx21.p1.4.m4.2.2.2.2.1.1.1.cmml" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx21.p1.4.m4.2.2.2.2.1.1.2.cmml" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.1.2">𝑝</ci><cn id="Thmproofx21.p1.4.m4.2.2.2.2.1.1.3.cmml" type="integer" xref="Thmproofx21.p1.4.m4.2.2.2.2.1.1.3">1</cn></apply></apply></apply><ci id="Thmproofx21.p1.4.m4.2.2.4.cmml" xref="Thmproofx21.p1.4.m4.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.4.m4.2c">\alpha>\frac{|p_{m_{1}+1}|-|p_{1}|}{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.4.m4.2d">italic_α > divide start_ARG | italic_p start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>, namely that <math alttext="\frac{E^{\prime}_{m_{1}+1}}{k(E^{\prime}_{m_{1}}-E^{\prime}_{m_{1}+1})}\geq% \frac{1}{k}" class="ltx_Math" display="inline" id="Thmproofx21.p1.5.m5.1"><semantics id="Thmproofx21.p1.5.m5.1a"><mrow id="Thmproofx21.p1.5.m5.1.2" xref="Thmproofx21.p1.5.m5.1.2.cmml"><mfrac id="Thmproofx21.p1.5.m5.1.1" xref="Thmproofx21.p1.5.m5.1.1.cmml"><msubsup id="Thmproofx21.p1.5.m5.1.1.3" xref="Thmproofx21.p1.5.m5.1.1.3.cmml"><mi id="Thmproofx21.p1.5.m5.1.1.3.2.2" xref="Thmproofx21.p1.5.m5.1.1.3.2.2.cmml">E</mi><mrow id="Thmproofx21.p1.5.m5.1.1.3.3" xref="Thmproofx21.p1.5.m5.1.1.3.3.cmml"><msub id="Thmproofx21.p1.5.m5.1.1.3.3.2" xref="Thmproofx21.p1.5.m5.1.1.3.3.2.cmml"><mi id="Thmproofx21.p1.5.m5.1.1.3.3.2.2" xref="Thmproofx21.p1.5.m5.1.1.3.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.5.m5.1.1.3.3.2.3" xref="Thmproofx21.p1.5.m5.1.1.3.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.5.m5.1.1.3.3.1" xref="Thmproofx21.p1.5.m5.1.1.3.3.1.cmml">+</mo><mn id="Thmproofx21.p1.5.m5.1.1.3.3.3" xref="Thmproofx21.p1.5.m5.1.1.3.3.3.cmml">1</mn></mrow><mo id="Thmproofx21.p1.5.m5.1.1.3.2.3" xref="Thmproofx21.p1.5.m5.1.1.3.2.3.cmml">′</mo></msubsup><mrow id="Thmproofx21.p1.5.m5.1.1.1" xref="Thmproofx21.p1.5.m5.1.1.1.cmml"><mi id="Thmproofx21.p1.5.m5.1.1.1.3" xref="Thmproofx21.p1.5.m5.1.1.1.3.cmml">k</mi><mo id="Thmproofx21.p1.5.m5.1.1.1.2" xref="Thmproofx21.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx21.p1.5.m5.1.1.1.1.1" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="Thmproofx21.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx21.p1.5.m5.1.1.1.1.1.1" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.cmml"><msubsup id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.cmml"><mi id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.2" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.2.cmml">E</mi><msub id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.cmml"><mi id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.2" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.2.cmml">m</mi><mn id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.1" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.1.cmml">−</mo><msubsup id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.2" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.2.cmml">E</mi><mrow id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.cmml"><msub id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.cmml"><mi id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.2" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.1" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow><mo id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.3" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.3.cmml">′</mo></msubsup></mrow><mo id="Thmproofx21.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="Thmproofx21.p1.5.m5.1.2.1" xref="Thmproofx21.p1.5.m5.1.2.1.cmml">≥</mo><mfrac id="Thmproofx21.p1.5.m5.1.2.2" xref="Thmproofx21.p1.5.m5.1.2.2.cmml"><mn id="Thmproofx21.p1.5.m5.1.2.2.2" xref="Thmproofx21.p1.5.m5.1.2.2.2.cmml">1</mn><mi id="Thmproofx21.p1.5.m5.1.2.2.3" xref="Thmproofx21.p1.5.m5.1.2.2.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.5.m5.1b"><apply id="Thmproofx21.p1.5.m5.1.2.cmml" xref="Thmproofx21.p1.5.m5.1.2"><geq id="Thmproofx21.p1.5.m5.1.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.2.1"></geq><apply id="Thmproofx21.p1.5.m5.1.1.cmml" xref="Thmproofx21.p1.5.m5.1.1"><divide id="Thmproofx21.p1.5.m5.1.1.2.cmml" xref="Thmproofx21.p1.5.m5.1.1"></divide><apply id="Thmproofx21.p1.5.m5.1.1.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.3.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.3">subscript</csymbol><apply id="Thmproofx21.p1.5.m5.1.1.3.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.3.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.3">superscript</csymbol><ci id="Thmproofx21.p1.5.m5.1.1.3.2.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.3.2.2">𝐸</ci><ci id="Thmproofx21.p1.5.m5.1.1.3.2.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.3.2.3">′</ci></apply><apply id="Thmproofx21.p1.5.m5.1.1.3.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.3.3"><plus id="Thmproofx21.p1.5.m5.1.1.3.3.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.3.3.1"></plus><apply id="Thmproofx21.p1.5.m5.1.1.3.3.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.3.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.3.3.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.3.3.2">subscript</csymbol><ci id="Thmproofx21.p1.5.m5.1.1.3.3.2.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.3.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.5.m5.1.1.3.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.5.m5.1.1.3.3.3.cmml" type="integer" xref="Thmproofx21.p1.5.m5.1.1.3.3.3">1</cn></apply></apply><apply id="Thmproofx21.p1.5.m5.1.1.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1"><times id="Thmproofx21.p1.5.m5.1.1.1.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.2"></times><ci id="Thmproofx21.p1.5.m5.1.1.1.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.3">𝑘</ci><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1"><minus id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.1"></minus><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2">subscript</csymbol><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2">superscript</csymbol><ci id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.2">𝐸</ci><ci id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.2.3">′</ci></apply><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.2">𝑚</ci><cn id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3">subscript</csymbol><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.2">𝐸</ci><ci id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.2.3">′</ci></apply><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3"><plus id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.1"></plus><apply id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2">subscript</csymbol><ci id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.2.cmml" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="Thmproofx21.p1.5.m5.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply><apply id="Thmproofx21.p1.5.m5.1.2.2.cmml" xref="Thmproofx21.p1.5.m5.1.2.2"><divide id="Thmproofx21.p1.5.m5.1.2.2.1.cmml" xref="Thmproofx21.p1.5.m5.1.2.2"></divide><cn id="Thmproofx21.p1.5.m5.1.2.2.2.cmml" type="integer" xref="Thmproofx21.p1.5.m5.1.2.2.2">1</cn><ci id="Thmproofx21.p1.5.m5.1.2.2.3.cmml" xref="Thmproofx21.p1.5.m5.1.2.2.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.5.m5.1c">\frac{E^{\prime}_{m_{1}+1}}{k(E^{\prime}_{m_{1}}-E^{\prime}_{m_{1}+1})}\geq% \frac{1}{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.5.m5.1d">divide start_ARG italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_k ( italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT ) end_ARG ≥ divide start_ARG 1 end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>. Canceling out <math alttext="\frac{1}{k}" class="ltx_Math" display="inline" id="Thmproofx21.p1.6.m6.1"><semantics id="Thmproofx21.p1.6.m6.1a"><mfrac id="Thmproofx21.p1.6.m6.1.1" xref="Thmproofx21.p1.6.m6.1.1.cmml"><mn id="Thmproofx21.p1.6.m6.1.1.2" xref="Thmproofx21.p1.6.m6.1.1.2.cmml">1</mn><mi id="Thmproofx21.p1.6.m6.1.1.3" xref="Thmproofx21.p1.6.m6.1.1.3.cmml">k</mi></mfrac><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.6.m6.1b"><apply id="Thmproofx21.p1.6.m6.1.1.cmml" xref="Thmproofx21.p1.6.m6.1.1"><divide id="Thmproofx21.p1.6.m6.1.1.1.cmml" xref="Thmproofx21.p1.6.m6.1.1"></divide><cn id="Thmproofx21.p1.6.m6.1.1.2.cmml" type="integer" xref="Thmproofx21.p1.6.m6.1.1.2">1</cn><ci id="Thmproofx21.p1.6.m6.1.1.3.cmml" xref="Thmproofx21.p1.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.6.m6.1c">\frac{1}{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.6.m6.1d">divide start_ARG 1 end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> from both sides:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx54"> <tbody id="Pt0.A4.Ex93"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{E^{\prime}_{m_{1}+1}}{E^{\prime}_{m_{1}}-E^{\prime}_{m_{1}+% 1}}\geq 1" class="ltx_Math" display="inline" id="Pt0.A4.Ex93.m1.1"><semantics id="Pt0.A4.Ex93.m1.1a"><mrow id="Pt0.A4.Ex93.m1.1.1" xref="Pt0.A4.Ex93.m1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex93.m1.1.1.2" xref="Pt0.A4.Ex93.m1.1.1.2.cmml"><mfrac id="Pt0.A4.Ex93.m1.1.1.2a" xref="Pt0.A4.Ex93.m1.1.1.2.cmml"><msubsup id="Pt0.A4.Ex93.m1.1.1.2.2" xref="Pt0.A4.Ex93.m1.1.1.2.2.cmml"><mi id="Pt0.A4.Ex93.m1.1.1.2.2.2.2" xref="Pt0.A4.Ex93.m1.1.1.2.2.2.2.cmml">E</mi><mrow id="Pt0.A4.Ex93.m1.1.1.2.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.cmml"><msub id="Pt0.A4.Ex93.m1.1.1.2.2.3.2" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.2.cmml"><mi id="Pt0.A4.Ex93.m1.1.1.2.2.3.2.2" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.2.2.cmml">m</mi><mn id="Pt0.A4.Ex93.m1.1.1.2.2.3.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex93.m1.1.1.2.2.3.1" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.1.cmml">+</mo><mn id="Pt0.A4.Ex93.m1.1.1.2.2.3.3" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex93.m1.1.1.2.2.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.2.2.3.cmml">′</mo></msubsup><mrow id="Pt0.A4.Ex93.m1.1.1.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.cmml"><msubsup id="Pt0.A4.Ex93.m1.1.1.2.3.2" xref="Pt0.A4.Ex93.m1.1.1.2.3.2.cmml"><mi id="Pt0.A4.Ex93.m1.1.1.2.3.2.2.2" xref="Pt0.A4.Ex93.m1.1.1.2.3.2.2.2.cmml">E</mi><msub id="Pt0.A4.Ex93.m1.1.1.2.3.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.2.3.cmml"><mi id="Pt0.A4.Ex93.m1.1.1.2.3.2.3.2" xref="Pt0.A4.Ex93.m1.1.1.2.3.2.3.2.cmml">m</mi><mn id="Pt0.A4.Ex93.m1.1.1.2.3.2.3.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.2.3.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex93.m1.1.1.2.3.2.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.2.2.3.cmml">′</mo></msubsup><mo id="Pt0.A4.Ex93.m1.1.1.2.3.1" xref="Pt0.A4.Ex93.m1.1.1.2.3.1.cmml">−</mo><msubsup id="Pt0.A4.Ex93.m1.1.1.2.3.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex93.m1.1.1.2.3.3.2.2" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.2.2.cmml">E</mi><mrow id="Pt0.A4.Ex93.m1.1.1.2.3.3.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.cmml"><msub id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.cmml"><mi id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.2" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.2.cmml">m</mi><mn id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.1" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.1.cmml">+</mo><mn id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex93.m1.1.1.2.3.3.2.3" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.2.3.cmml">′</mo></msubsup></mrow></mfrac></mstyle><mo id="Pt0.A4.Ex93.m1.1.1.1" xref="Pt0.A4.Ex93.m1.1.1.1.cmml">≥</mo><mn id="Pt0.A4.Ex93.m1.1.1.3" xref="Pt0.A4.Ex93.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex93.m1.1b"><apply id="Pt0.A4.Ex93.m1.1.1.cmml" xref="Pt0.A4.Ex93.m1.1.1"><geq id="Pt0.A4.Ex93.m1.1.1.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.1"></geq><apply id="Pt0.A4.Ex93.m1.1.1.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2"><divide id="Pt0.A4.Ex93.m1.1.1.2.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2"></divide><apply id="Pt0.A4.Ex93.m1.1.1.2.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m1.1.1.2.2.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2">subscript</csymbol><apply id="Pt0.A4.Ex93.m1.1.1.2.2.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m1.1.1.2.2.2.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2">superscript</csymbol><ci id="Pt0.A4.Ex93.m1.1.1.2.2.2.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2.2.2">𝐸</ci><ci id="Pt0.A4.Ex93.m1.1.1.2.2.2.3.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2.2.3">′</ci></apply><apply id="Pt0.A4.Ex93.m1.1.1.2.2.3.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2.3"><plus id="Pt0.A4.Ex93.m1.1.1.2.2.3.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.1"></plus><apply id="Pt0.A4.Ex93.m1.1.1.2.2.3.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m1.1.1.2.2.3.2.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.2">subscript</csymbol><ci id="Pt0.A4.Ex93.m1.1.1.2.2.3.2.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.2.2">𝑚</ci><cn id="Pt0.A4.Ex93.m1.1.1.2.2.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.2.3">1</cn></apply><cn id="Pt0.A4.Ex93.m1.1.1.2.2.3.3.cmml" type="integer" xref="Pt0.A4.Ex93.m1.1.1.2.2.3.3">1</cn></apply></apply><apply id="Pt0.A4.Ex93.m1.1.1.2.3.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3"><minus id="Pt0.A4.Ex93.m1.1.1.2.3.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.1"></minus><apply id="Pt0.A4.Ex93.m1.1.1.2.3.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m1.1.1.2.3.2.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.2">subscript</csymbol><apply 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xref="Pt0.A4.Ex93.m1.1.1.2.3.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m1.1.1.2.3.3.2.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3">superscript</csymbol><ci id="Pt0.A4.Ex93.m1.1.1.2.3.3.2.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.2.2">𝐸</ci><ci id="Pt0.A4.Ex93.m1.1.1.2.3.3.2.3.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.2.3">′</ci></apply><apply id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3"><plus id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.1"></plus><apply id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.1.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2">subscript</csymbol><ci id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.2.cmml" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.2">𝑚</ci><cn id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.2.3">1</cn></apply><cn id="Pt0.A4.Ex93.m1.1.1.2.3.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex93.m1.1.1.2.3.3.3.3">1</cn></apply></apply></apply></apply><cn id="Pt0.A4.Ex93.m1.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex93.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex93.m1.1c">\displaystyle\frac{E^{\prime}_{m_{1}+1}}{E^{\prime}_{m_{1}}-E^{\prime}_{m_{1}+% 1}}\geq 1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex93.m1.1d">divide start_ARG italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT end_ARG ≥ 1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\iff 2E^{\prime}_{m_{1}+1}\geq E^{\prime}_{m_{1}}" class="ltx_Math" display="inline" id="Pt0.A4.Ex93.m2.1"><semantics id="Pt0.A4.Ex93.m2.1a"><mrow id="Pt0.A4.Ex93.m2.1.1" xref="Pt0.A4.Ex93.m2.1.1.cmml"><mi id="Pt0.A4.Ex93.m2.1.1.2" xref="Pt0.A4.Ex93.m2.1.1.2.cmml"></mi><mo id="Pt0.A4.Ex93.m2.1.1.1" stretchy="false" xref="Pt0.A4.Ex93.m2.1.1.1.cmml">⇔</mo><mrow id="Pt0.A4.Ex93.m2.1.1.3" xref="Pt0.A4.Ex93.m2.1.1.3.cmml"><mrow id="Pt0.A4.Ex93.m2.1.1.3.2" xref="Pt0.A4.Ex93.m2.1.1.3.2.cmml"><mn id="Pt0.A4.Ex93.m2.1.1.3.2.2" xref="Pt0.A4.Ex93.m2.1.1.3.2.2.cmml">2</mn><mo id="Pt0.A4.Ex93.m2.1.1.3.2.1" xref="Pt0.A4.Ex93.m2.1.1.3.2.1.cmml">⁢</mo><msubsup id="Pt0.A4.Ex93.m2.1.1.3.2.3" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.cmml"><mi id="Pt0.A4.Ex93.m2.1.1.3.2.3.2.2" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.2.2.cmml">E</mi><mrow id="Pt0.A4.Ex93.m2.1.1.3.2.3.3" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.3.cmml"><msub id="Pt0.A4.Ex93.m2.1.1.3.2.3.3.2" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.3.2.cmml"><mi id="Pt0.A4.Ex93.m2.1.1.3.2.3.3.2.2" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.3.2.2.cmml">m</mi><mn id="Pt0.A4.Ex93.m2.1.1.3.2.3.3.2.3" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.3.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex93.m2.1.1.3.2.3.3.1" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.3.1.cmml">+</mo><mn id="Pt0.A4.Ex93.m2.1.1.3.2.3.3.3" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.3.3.cmml">1</mn></mrow><mo id="Pt0.A4.Ex93.m2.1.1.3.2.3.2.3" xref="Pt0.A4.Ex93.m2.1.1.3.2.3.2.3.cmml">′</mo></msubsup></mrow><mo id="Pt0.A4.Ex93.m2.1.1.3.1" xref="Pt0.A4.Ex93.m2.1.1.3.1.cmml">≥</mo><msubsup id="Pt0.A4.Ex93.m2.1.1.3.3" xref="Pt0.A4.Ex93.m2.1.1.3.3.cmml"><mi id="Pt0.A4.Ex93.m2.1.1.3.3.2.2" xref="Pt0.A4.Ex93.m2.1.1.3.3.2.2.cmml">E</mi><msub id="Pt0.A4.Ex93.m2.1.1.3.3.3" xref="Pt0.A4.Ex93.m2.1.1.3.3.3.cmml"><mi id="Pt0.A4.Ex93.m2.1.1.3.3.3.2" xref="Pt0.A4.Ex93.m2.1.1.3.3.3.2.cmml">m</mi><mn id="Pt0.A4.Ex93.m2.1.1.3.3.3.3" xref="Pt0.A4.Ex93.m2.1.1.3.3.3.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex93.m2.1.1.3.3.2.3" xref="Pt0.A4.Ex93.m2.1.1.3.3.2.3.cmml">′</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex93.m2.1b"><apply id="Pt0.A4.Ex93.m2.1.1.cmml" xref="Pt0.A4.Ex93.m2.1.1"><csymbol cd="latexml" id="Pt0.A4.Ex93.m2.1.1.1.cmml" xref="Pt0.A4.Ex93.m2.1.1.1">iff</csymbol><csymbol cd="latexml" id="Pt0.A4.Ex93.m2.1.1.2.cmml" xref="Pt0.A4.Ex93.m2.1.1.2">absent</csymbol><apply id="Pt0.A4.Ex93.m2.1.1.3.cmml" xref="Pt0.A4.Ex93.m2.1.1.3"><geq id="Pt0.A4.Ex93.m2.1.1.3.1.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.1"></geq><apply id="Pt0.A4.Ex93.m2.1.1.3.2.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.2"><times id="Pt0.A4.Ex93.m2.1.1.3.2.1.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.2.1"></times><cn id="Pt0.A4.Ex93.m2.1.1.3.2.2.cmml" type="integer" xref="Pt0.A4.Ex93.m2.1.1.3.2.2">2</cn><apply id="Pt0.A4.Ex93.m2.1.1.3.2.3.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m2.1.1.3.2.3.1.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.2.3">subscript</csymbol><apply id="Pt0.A4.Ex93.m2.1.1.3.2.3.2.cmml" 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xref="Pt0.A4.Ex93.m2.1.1.3.2.3.3.3">1</cn></apply></apply></apply><apply id="Pt0.A4.Ex93.m2.1.1.3.3.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m2.1.1.3.3.1.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3">subscript</csymbol><apply id="Pt0.A4.Ex93.m2.1.1.3.3.2.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m2.1.1.3.3.2.1.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3">superscript</csymbol><ci id="Pt0.A4.Ex93.m2.1.1.3.3.2.2.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3.2.2">𝐸</ci><ci id="Pt0.A4.Ex93.m2.1.1.3.3.2.3.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3.2.3">′</ci></apply><apply id="Pt0.A4.Ex93.m2.1.1.3.3.3.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex93.m2.1.1.3.3.3.1.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3.3">subscript</csymbol><ci id="Pt0.A4.Ex93.m2.1.1.3.3.3.2.cmml" xref="Pt0.A4.Ex93.m2.1.1.3.3.3.2">𝑚</ci><cn id="Pt0.A4.Ex93.m2.1.1.3.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex93.m2.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex93.m2.1c">\displaystyle\iff 2E^{\prime}_{m_{1}+1}\geq E^{\prime}_{m_{1}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex93.m2.1d">⇔ 2 italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT ≥ italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex94"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\iff 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id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1"><minus id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.1"></minus><cn id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.2">1</cn><apply id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3"><divide id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3"></divide><cn id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.2">1</cn><apply id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3"><plus id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.1"></plus><apply id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2.1.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2">subscript</csymbol><ci id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2.2.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2.2">𝑚</ci><cn id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.2.3">1</cn></apply><cn id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="Pt0.A4.Ex94.m1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex94.m1.1.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Pt0.A4.Ex94.m1.2.2.2.2.2.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex94.m1.2.2.2.2.2.2.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2">superscript</csymbol><apply id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1"><minus id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.1.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.1"></minus><cn id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.2.cmml" type="integer" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.2">1</cn><apply id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3"><divide id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.1.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3"></divide><cn id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.2.cmml" type="integer" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.2">1</cn><apply id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3.1.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3">subscript</csymbol><ci id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3.2.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3.2">𝑚</ci><cn id="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="Pt0.A4.Ex94.m1.2.2.2.2.2.3.cmml" xref="Pt0.A4.Ex94.m1.2.2.2.2.2.3">𝑛</ci></apply></apply><cn id="Pt0.A4.Ex94.m1.2.2.2.4.cmml" type="integer" xref="Pt0.A4.Ex94.m1.2.2.2.4">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex94.m1.2c">\displaystyle\iff 2\left(1-\frac{1}{m_{1}+1}\right)^{n}-\left(1-\frac{1}{m_{1}% }\right)^{n}\leq 1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex94.m1.2d">⇔ 2 ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ≤ 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx21.p1.11">When <math alttext="n=2" class="ltx_Math" display="inline" id="Thmproofx21.p1.7.m1.1"><semantics id="Thmproofx21.p1.7.m1.1a"><mrow id="Thmproofx21.p1.7.m1.1.1" xref="Thmproofx21.p1.7.m1.1.1.cmml"><mi id="Thmproofx21.p1.7.m1.1.1.2" xref="Thmproofx21.p1.7.m1.1.1.2.cmml">n</mi><mo id="Thmproofx21.p1.7.m1.1.1.1" xref="Thmproofx21.p1.7.m1.1.1.1.cmml">=</mo><mn id="Thmproofx21.p1.7.m1.1.1.3" xref="Thmproofx21.p1.7.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.7.m1.1b"><apply id="Thmproofx21.p1.7.m1.1.1.cmml" xref="Thmproofx21.p1.7.m1.1.1"><eq id="Thmproofx21.p1.7.m1.1.1.1.cmml" xref="Thmproofx21.p1.7.m1.1.1.1"></eq><ci id="Thmproofx21.p1.7.m1.1.1.2.cmml" xref="Thmproofx21.p1.7.m1.1.1.2">𝑛</ci><cn id="Thmproofx21.p1.7.m1.1.1.3.cmml" type="integer" xref="Thmproofx21.p1.7.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.7.m1.1c">n=2</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.7.m1.1d">italic_n = 2</annotation></semantics></math>, the latter inequality simplifies to <math alttext="2\left(1-\frac{1}{m_{1}+1}\right)^{2}-\left(1-\frac{1}{m_{1}}\right)^{2}\leq% \frac{1}{2}<1" class="ltx_Math" display="inline" id="Thmproofx21.p1.8.m2.2"><semantics id="Thmproofx21.p1.8.m2.2a"><mrow id="Thmproofx21.p1.8.m2.2.2" xref="Thmproofx21.p1.8.m2.2.2.cmml"><mrow id="Thmproofx21.p1.8.m2.2.2.2" xref="Thmproofx21.p1.8.m2.2.2.2.cmml"><mrow id="Thmproofx21.p1.8.m2.1.1.1.1" xref="Thmproofx21.p1.8.m2.1.1.1.1.cmml"><mn id="Thmproofx21.p1.8.m2.1.1.1.1.3" xref="Thmproofx21.p1.8.m2.1.1.1.1.3.cmml">2</mn><mo id="Thmproofx21.p1.8.m2.1.1.1.1.2" xref="Thmproofx21.p1.8.m2.1.1.1.1.2.cmml">⁢</mo><msup id="Thmproofx21.p1.8.m2.1.1.1.1.1" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.cmml"><mrow id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.2" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.cmml"><mn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.2" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.1" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.cmml"><mn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.2" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.2.cmml">1</mn><mrow id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.cmml"><msub id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.cmml"><mi id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.2" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.3" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.1" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.3" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac></mrow><mo id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.3" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="Thmproofx21.p1.8.m2.1.1.1.1.1.3" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.3.cmml">2</mn></msup></mrow><mo id="Thmproofx21.p1.8.m2.2.2.2.3" xref="Thmproofx21.p1.8.m2.2.2.2.3.cmml">−</mo><msup id="Thmproofx21.p1.8.m2.2.2.2.2" xref="Thmproofx21.p1.8.m2.2.2.2.2.cmml"><mrow id="Thmproofx21.p1.8.m2.2.2.2.2.1.1" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.cmml"><mo id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.2" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.cmml"><mn id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.2" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.1" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.cmml"><mn id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.2" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.2.cmml">1</mn><msub id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.cmml"><mi id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.2" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.2.cmml">m</mi><mn id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.3" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.3.cmml">1</mn></msub></mfrac></mrow><mo id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.3" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.cmml">)</mo></mrow><mn id="Thmproofx21.p1.8.m2.2.2.2.2.3" xref="Thmproofx21.p1.8.m2.2.2.2.2.3.cmml">2</mn></msup></mrow><mo id="Thmproofx21.p1.8.m2.2.2.4" xref="Thmproofx21.p1.8.m2.2.2.4.cmml">≤</mo><mfrac id="Thmproofx21.p1.8.m2.2.2.5" xref="Thmproofx21.p1.8.m2.2.2.5.cmml"><mn id="Thmproofx21.p1.8.m2.2.2.5.2" xref="Thmproofx21.p1.8.m2.2.2.5.2.cmml">1</mn><mn id="Thmproofx21.p1.8.m2.2.2.5.3" xref="Thmproofx21.p1.8.m2.2.2.5.3.cmml">2</mn></mfrac><mo id="Thmproofx21.p1.8.m2.2.2.6" xref="Thmproofx21.p1.8.m2.2.2.6.cmml"><</mo><mn id="Thmproofx21.p1.8.m2.2.2.7" xref="Thmproofx21.p1.8.m2.2.2.7.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.8.m2.2b"><apply id="Thmproofx21.p1.8.m2.2.2.cmml" xref="Thmproofx21.p1.8.m2.2.2"><and id="Thmproofx21.p1.8.m2.2.2a.cmml" xref="Thmproofx21.p1.8.m2.2.2"></and><apply id="Thmproofx21.p1.8.m2.2.2b.cmml" xref="Thmproofx21.p1.8.m2.2.2"><leq id="Thmproofx21.p1.8.m2.2.2.4.cmml" xref="Thmproofx21.p1.8.m2.2.2.4"></leq><apply id="Thmproofx21.p1.8.m2.2.2.2.cmml" xref="Thmproofx21.p1.8.m2.2.2.2"><minus id="Thmproofx21.p1.8.m2.2.2.2.3.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.3"></minus><apply id="Thmproofx21.p1.8.m2.1.1.1.1.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1"><times id="Thmproofx21.p1.8.m2.1.1.1.1.2.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.2"></times><cn id="Thmproofx21.p1.8.m2.1.1.1.1.3.cmml" type="integer" xref="Thmproofx21.p1.8.m2.1.1.1.1.3">2</cn><apply id="Thmproofx21.p1.8.m2.1.1.1.1.1.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx21.p1.8.m2.1.1.1.1.1.2.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1">superscript</csymbol><apply id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1"><minus id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.1"></minus><cn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3"><divide id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3"></divide><cn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.2">1</cn><apply id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3"><plus id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.1"></plus><apply id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.1.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2">subscript</csymbol><ci id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.2.cmml" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply><cn id="Thmproofx21.p1.8.m2.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx21.p1.8.m2.1.1.1.1.1.3">2</cn></apply></apply><apply id="Thmproofx21.p1.8.m2.2.2.2.2.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.8.m2.2.2.2.2.2.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2">superscript</csymbol><apply id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1"><minus id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.1.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.1"></minus><cn id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.2.cmml" type="integer" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.2">1</cn><apply id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3"><divide id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3"></divide><cn id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.2.cmml" type="integer" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.2">1</cn><apply id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.1.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3">subscript</csymbol><ci id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.2.cmml" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.2">𝑚</ci><cn id="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.3.cmml" type="integer" xref="Thmproofx21.p1.8.m2.2.2.2.2.1.1.1.3.3.3">1</cn></apply></apply></apply><cn id="Thmproofx21.p1.8.m2.2.2.2.2.3.cmml" type="integer" xref="Thmproofx21.p1.8.m2.2.2.2.2.3">2</cn></apply></apply><apply id="Thmproofx21.p1.8.m2.2.2.5.cmml" xref="Thmproofx21.p1.8.m2.2.2.5"><divide id="Thmproofx21.p1.8.m2.2.2.5.1.cmml" xref="Thmproofx21.p1.8.m2.2.2.5"></divide><cn id="Thmproofx21.p1.8.m2.2.2.5.2.cmml" type="integer" xref="Thmproofx21.p1.8.m2.2.2.5.2">1</cn><cn id="Thmproofx21.p1.8.m2.2.2.5.3.cmml" type="integer" xref="Thmproofx21.p1.8.m2.2.2.5.3">2</cn></apply></apply><apply id="Thmproofx21.p1.8.m2.2.2c.cmml" xref="Thmproofx21.p1.8.m2.2.2"><lt id="Thmproofx21.p1.8.m2.2.2.6.cmml" xref="Thmproofx21.p1.8.m2.2.2.6"></lt><share href="https://arxiv.org/html/2308.08047v3#Thmproofx21.p1.8.m2.2.2.5.cmml" id="Thmproofx21.p1.8.m2.2.2d.cmml" xref="Thmproofx21.p1.8.m2.2.2"></share><cn id="Thmproofx21.p1.8.m2.2.2.7.cmml" type="integer" xref="Thmproofx21.p1.8.m2.2.2.7">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.8.m2.2c">2\left(1-\frac{1}{m_{1}+1}\right)^{2}-\left(1-\frac{1}{m_{1}}\right)^{2}\leq% \frac{1}{2}<1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.8.m2.2d">2 ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG < 1</annotation></semantics></math>. We show that <math alttext="2\left(1-\frac{1}{m_{1}+1}\right)^{n}-\left(1-\frac{1}{m_{1}}\right)^{n}" class="ltx_Math" display="inline" id="Thmproofx21.p1.9.m3.2"><semantics id="Thmproofx21.p1.9.m3.2a"><mrow id="Thmproofx21.p1.9.m3.2.2" xref="Thmproofx21.p1.9.m3.2.2.cmml"><mrow id="Thmproofx21.p1.9.m3.1.1.1" xref="Thmproofx21.p1.9.m3.1.1.1.cmml"><mn id="Thmproofx21.p1.9.m3.1.1.1.3" xref="Thmproofx21.p1.9.m3.1.1.1.3.cmml">2</mn><mo id="Thmproofx21.p1.9.m3.1.1.1.2" xref="Thmproofx21.p1.9.m3.1.1.1.2.cmml">⁢</mo><msup id="Thmproofx21.p1.9.m3.1.1.1.1" xref="Thmproofx21.p1.9.m3.1.1.1.1.cmml"><mrow id="Thmproofx21.p1.9.m3.1.1.1.1.1.1" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.2" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.cmml"><mn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.2" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.1" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.cmml"><mn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.2" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.2.cmml">1</mn><mrow id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.cmml"><msub id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.cmml"><mi id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.2" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.3" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.1" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.3" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac></mrow><mo id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.3" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx21.p1.9.m3.1.1.1.1.3" xref="Thmproofx21.p1.9.m3.1.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Thmproofx21.p1.9.m3.2.2.3" xref="Thmproofx21.p1.9.m3.2.2.3.cmml">−</mo><msup id="Thmproofx21.p1.9.m3.2.2.2" xref="Thmproofx21.p1.9.m3.2.2.2.cmml"><mrow id="Thmproofx21.p1.9.m3.2.2.2.1.1" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.cmml"><mo id="Thmproofx21.p1.9.m3.2.2.2.1.1.2" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx21.p1.9.m3.2.2.2.1.1.1" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.cmml"><mn id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.2" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.2.cmml">1</mn><mo id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.1" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.1.cmml">−</mo><mfrac id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.cmml"><mn id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.2" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.2.cmml">1</mn><msub id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.cmml"><mi id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.2" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.2.cmml">m</mi><mn id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.3" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.3.cmml">1</mn></msub></mfrac></mrow><mo id="Thmproofx21.p1.9.m3.2.2.2.1.1.3" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx21.p1.9.m3.2.2.2.3" xref="Thmproofx21.p1.9.m3.2.2.2.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.9.m3.2b"><apply id="Thmproofx21.p1.9.m3.2.2.cmml" xref="Thmproofx21.p1.9.m3.2.2"><minus id="Thmproofx21.p1.9.m3.2.2.3.cmml" xref="Thmproofx21.p1.9.m3.2.2.3"></minus><apply id="Thmproofx21.p1.9.m3.1.1.1.cmml" xref="Thmproofx21.p1.9.m3.1.1.1"><times id="Thmproofx21.p1.9.m3.1.1.1.2.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.2"></times><cn id="Thmproofx21.p1.9.m3.1.1.1.3.cmml" type="integer" xref="Thmproofx21.p1.9.m3.1.1.1.3">2</cn><apply id="Thmproofx21.p1.9.m3.1.1.1.1.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx21.p1.9.m3.1.1.1.1.2.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1">superscript</csymbol><apply id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1"><minus id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.1"></minus><cn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.2">1</cn><apply id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3"><divide id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3"></divide><cn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.2">1</cn><apply id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3"><plus id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.1.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.1"></plus><apply id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.1.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2">subscript</csymbol><ci id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.2.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="Thmproofx21.p1.9.m3.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="Thmproofx21.p1.9.m3.1.1.1.1.3.cmml" xref="Thmproofx21.p1.9.m3.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmproofx21.p1.9.m3.2.2.2.cmml" xref="Thmproofx21.p1.9.m3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.9.m3.2.2.2.2.cmml" xref="Thmproofx21.p1.9.m3.2.2.2">superscript</csymbol><apply id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.1.1"><minus id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.1.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.1"></minus><cn id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.2.cmml" type="integer" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.2">1</cn><apply id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3"><divide id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3"></divide><cn id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.2.cmml" type="integer" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.2">1</cn><apply id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.1.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3">subscript</csymbol><ci id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.2.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.2">𝑚</ci><cn id="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.3.cmml" type="integer" xref="Thmproofx21.p1.9.m3.2.2.2.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="Thmproofx21.p1.9.m3.2.2.2.3.cmml" xref="Thmproofx21.p1.9.m3.2.2.2.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.9.m3.2c">2\left(1-\frac{1}{m_{1}+1}\right)^{n}-\left(1-\frac{1}{m_{1}}\right)^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.9.m3.2d">2 ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> decreases as <math alttext="n" class="ltx_Math" display="inline" id="Thmproofx21.p1.10.m4.1"><semantics id="Thmproofx21.p1.10.m4.1a"><mi id="Thmproofx21.p1.10.m4.1.1" xref="Thmproofx21.p1.10.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.10.m4.1b"><ci id="Thmproofx21.p1.10.m4.1.1.cmml" xref="Thmproofx21.p1.10.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.10.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.10.m4.1d">italic_n</annotation></semantics></math> increases, to conclude that it is at most 1 for every <math alttext="n>1" class="ltx_Math" display="inline" id="Thmproofx21.p1.11.m5.1"><semantics id="Thmproofx21.p1.11.m5.1a"><mrow id="Thmproofx21.p1.11.m5.1.1" xref="Thmproofx21.p1.11.m5.1.1.cmml"><mi id="Thmproofx21.p1.11.m5.1.1.2" xref="Thmproofx21.p1.11.m5.1.1.2.cmml">n</mi><mo id="Thmproofx21.p1.11.m5.1.1.1" xref="Thmproofx21.p1.11.m5.1.1.1.cmml">></mo><mn id="Thmproofx21.p1.11.m5.1.1.3" xref="Thmproofx21.p1.11.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.11.m5.1b"><apply id="Thmproofx21.p1.11.m5.1.1.cmml" xref="Thmproofx21.p1.11.m5.1.1"><gt id="Thmproofx21.p1.11.m5.1.1.1.cmml" xref="Thmproofx21.p1.11.m5.1.1.1"></gt><ci id="Thmproofx21.p1.11.m5.1.1.2.cmml" xref="Thmproofx21.p1.11.m5.1.1.2">𝑛</ci><cn id="Thmproofx21.p1.11.m5.1.1.3.cmml" type="integer" xref="Thmproofx21.p1.11.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.11.m5.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.11.m5.1d">italic_n > 1</annotation></semantics></math>.</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx55"> <tbody id="Pt0.A4.Ex95"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 2\left(1-\frac{1}{m_{1}+1}\right)^{n+1}-\left(1-\frac{1}{m_{1}}% \right)^{n+1}-2\left(1-\frac{1}{m_{1}+1}\right)^{n}+\left(1-\frac{1}{m_{1}}% \right)^{n}" class="ltx_Math" display="inline" id="Pt0.A4.Ex95.m1.4"><semantics id="Pt0.A4.Ex95.m1.4a"><mrow id="Pt0.A4.Ex95.m1.4.4" xref="Pt0.A4.Ex95.m1.4.4.cmml"><mrow id="Pt0.A4.Ex95.m1.3.3.3" xref="Pt0.A4.Ex95.m1.3.3.3.cmml"><mrow id="Pt0.A4.Ex95.m1.1.1.1.1" xref="Pt0.A4.Ex95.m1.1.1.1.1.cmml"><mn id="Pt0.A4.Ex95.m1.1.1.1.1.3" xref="Pt0.A4.Ex95.m1.1.1.1.1.3.cmml">2</mn><mo id="Pt0.A4.Ex95.m1.1.1.1.1.2" xref="Pt0.A4.Ex95.m1.1.1.1.1.2.cmml">⁢</mo><msup id="Pt0.A4.Ex95.m1.1.1.1.1.1" xref="Pt0.A4.Ex95.m1.1.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex95.m1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex95.m1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex95.m1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex95.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex95.m1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex95.m1.1.1.1.1.1.1.1.1.cmml"><mn id="Pt0.A4.Ex95.m1.1.1.1.1.1.1.1.1.2" 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xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.1" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.1.cmml">−</mo><mstyle displaystyle="true" id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.cmml"><mfrac id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3a" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.cmml"><mn id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.2" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.2.cmml">1</mn><msub id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.3" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.3.cmml"><mi id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.3.2" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.3.2.cmml">m</mi><mn id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.3.3" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.3.3.3.cmml">1</mn></msub></mfrac></mstyle></mrow><mo id="Pt0.A4.Ex95.m1.2.2.2.2.1.1.3" xref="Pt0.A4.Ex95.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="Pt0.A4.Ex95.m1.2.2.2.2.3" xref="Pt0.A4.Ex95.m1.2.2.2.2.3.cmml"><mi id="Pt0.A4.Ex95.m1.2.2.2.2.3.2" xref="Pt0.A4.Ex95.m1.2.2.2.2.3.2.cmml">n</mi><mo 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xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.2">1</cn><apply id="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.cmml" xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3"><divide id="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.1.cmml" xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3"></divide><cn id="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.2.cmml" type="integer" xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.2">1</cn><apply id="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3.cmml" xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3.1.cmml" xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3">subscript</csymbol><ci id="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3.2.cmml" xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3.2">𝑚</ci><cn id="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex95.m1.4.4.4.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="Pt0.A4.Ex95.m1.4.4.4.3.cmml" xref="Pt0.A4.Ex95.m1.4.4.4.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex95.m1.4c">\displaystyle 2\left(1-\frac{1}{m_{1}+1}\right)^{n+1}-\left(1-\frac{1}{m_{1}}% \right)^{n+1}-2\left(1-\frac{1}{m_{1}+1}\right)^{n}+\left(1-\frac{1}{m_{1}}% \right)^{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex95.m1.4d">2 ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT - ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT - 2 ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody 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id="Pt0.A4.Ex96.m2.2.2.2.2.1.1.1.1.3.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex96.m2.2.2.2.2.1.1.1.1.3.3.2.3">1</cn></apply><cn id="Pt0.A4.Ex96.m2.2.2.2.2.1.1.1.1.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex96.m2.2.2.2.2.1.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="Pt0.A4.Ex96.m2.2.2.2.2.1.3.cmml" xref="Pt0.A4.Ex96.m2.2.2.2.2.1.3">𝑛</ci></apply></apply></apply><cn id="Pt0.A4.Ex96.m2.2.2.4.cmml" type="integer" xref="Pt0.A4.Ex96.m2.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex96.m2.2c">\displaystyle\frac{1}{m_{1}}\left(1-\frac{1}{m_{1}}\right)^{n}-\frac{2}{m_{1}+% 1}\left(1-\frac{1}{m_{1}+1}\right)^{n}\leq 0</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex96.m2.2d">divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - divide start_ARG 2 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ≤ 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex97"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\iff~{}" class="ltx_Math" display="inline" id="Pt0.A4.Ex97.m1.1"><semantics id="Pt0.A4.Ex97.m1.1a"><mo id="Pt0.A4.Ex97.m1.1.1" stretchy="false" xref="Pt0.A4.Ex97.m1.1.1.cmml">⇔</mo><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex97.m1.1b"><csymbol cd="latexml" id="Pt0.A4.Ex97.m1.1.1.cmml" xref="Pt0.A4.Ex97.m1.1.1">iff</csymbol></annotation-xml><annotation encoding="application/x-tex" 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id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.2.cmml" type="integer" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.2">1</cn><apply id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.cmml" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3"><plus id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.1.cmml" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.1"></plus><apply id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2.cmml" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2.1.cmml" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2">subscript</csymbol><ci id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2.2.cmml" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2.2">𝑚</ci><cn id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.2.3">1</cn></apply><cn id="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex97.m2.2.2.2.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="Pt0.A4.Ex97.m2.2.2.2.3.cmml" xref="Pt0.A4.Ex97.m2.2.2.2.3">𝑛</ci></apply></apply></apply><cn id="Pt0.A4.Ex97.m2.2.3.3.cmml" type="integer" xref="Pt0.A4.Ex97.m2.2.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex97.m2.2c">\displaystyle\frac{m_{1}+1}{m_{1}}\frac{\left(1-\frac{1}{m_{1}}\right)^{n}}{% \left(1-\frac{1}{m_{1}+1}\right)^{n}}\leq 2</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex97.m2.2d">divide start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG divide start_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG ≤ 2</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex98"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\iff~{}" class="ltx_Math" display="inline" id="Pt0.A4.Ex98.m1.1"><semantics id="Pt0.A4.Ex98.m1.1a"><mo id="Pt0.A4.Ex98.m1.1.1" stretchy="false" xref="Pt0.A4.Ex98.m1.1.1.cmml">⇔</mo><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex98.m1.1b"><csymbol cd="latexml" id="Pt0.A4.Ex98.m1.1.1.cmml" xref="Pt0.A4.Ex98.m1.1.1">iff</csymbol></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex98.m1.1c">\displaystyle\iff~{}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex98.m1.1d">⇔</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{m_{1}+1}{m_{1}}\left(\frac{m_{1}^{2}-1}{m_{1}^{2}}\right)^{% n}\leq 2" class="ltx_Math" display="inline" id="Pt0.A4.Ex98.m2.1"><semantics id="Pt0.A4.Ex98.m2.1a"><mrow id="Pt0.A4.Ex98.m2.1.2" xref="Pt0.A4.Ex98.m2.1.2.cmml"><mrow id="Pt0.A4.Ex98.m2.1.2.2" xref="Pt0.A4.Ex98.m2.1.2.2.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex98.m2.1.2.2.2" xref="Pt0.A4.Ex98.m2.1.2.2.2.cmml"><mfrac id="Pt0.A4.Ex98.m2.1.2.2.2a" xref="Pt0.A4.Ex98.m2.1.2.2.2.cmml"><mrow id="Pt0.A4.Ex98.m2.1.2.2.2.2" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.cmml"><msub id="Pt0.A4.Ex98.m2.1.2.2.2.2.2" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.2.cmml"><mi id="Pt0.A4.Ex98.m2.1.2.2.2.2.2.2" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.2.2.cmml">m</mi><mn id="Pt0.A4.Ex98.m2.1.2.2.2.2.2.3" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex98.m2.1.2.2.2.2.1" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.1.cmml">+</mo><mn id="Pt0.A4.Ex98.m2.1.2.2.2.2.3" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.3.cmml">1</mn></mrow><msub id="Pt0.A4.Ex98.m2.1.2.2.2.3" xref="Pt0.A4.Ex98.m2.1.2.2.2.3.cmml"><mi id="Pt0.A4.Ex98.m2.1.2.2.2.3.2" xref="Pt0.A4.Ex98.m2.1.2.2.2.3.2.cmml">m</mi><mn id="Pt0.A4.Ex98.m2.1.2.2.2.3.3" xref="Pt0.A4.Ex98.m2.1.2.2.2.3.3.cmml">1</mn></msub></mfrac></mstyle><mo id="Pt0.A4.Ex98.m2.1.2.2.1" xref="Pt0.A4.Ex98.m2.1.2.2.1.cmml">⁢</mo><msup id="Pt0.A4.Ex98.m2.1.2.2.3" xref="Pt0.A4.Ex98.m2.1.2.2.3.cmml"><mrow id="Pt0.A4.Ex98.m2.1.2.2.3.2.2" xref="Pt0.A4.Ex98.m2.1.1.cmml"><mo id="Pt0.A4.Ex98.m2.1.2.2.3.2.2.1" xref="Pt0.A4.Ex98.m2.1.1.cmml">(</mo><mstyle displaystyle="true" id="Pt0.A4.Ex98.m2.1.1" xref="Pt0.A4.Ex98.m2.1.1.cmml"><mfrac id="Pt0.A4.Ex98.m2.1.1a" xref="Pt0.A4.Ex98.m2.1.1.cmml"><mrow id="Pt0.A4.Ex98.m2.1.1.2" xref="Pt0.A4.Ex98.m2.1.1.2.cmml"><msubsup id="Pt0.A4.Ex98.m2.1.1.2.2" xref="Pt0.A4.Ex98.m2.1.1.2.2.cmml"><mi id="Pt0.A4.Ex98.m2.1.1.2.2.2.2" xref="Pt0.A4.Ex98.m2.1.1.2.2.2.2.cmml">m</mi><mn id="Pt0.A4.Ex98.m2.1.1.2.2.2.3" xref="Pt0.A4.Ex98.m2.1.1.2.2.2.3.cmml">1</mn><mn id="Pt0.A4.Ex98.m2.1.1.2.2.3" xref="Pt0.A4.Ex98.m2.1.1.2.2.3.cmml">2</mn></msubsup><mo id="Pt0.A4.Ex98.m2.1.1.2.1" xref="Pt0.A4.Ex98.m2.1.1.2.1.cmml">−</mo><mn id="Pt0.A4.Ex98.m2.1.1.2.3" xref="Pt0.A4.Ex98.m2.1.1.2.3.cmml">1</mn></mrow><msubsup id="Pt0.A4.Ex98.m2.1.1.3" xref="Pt0.A4.Ex98.m2.1.1.3.cmml"><mi id="Pt0.A4.Ex98.m2.1.1.3.2.2" xref="Pt0.A4.Ex98.m2.1.1.3.2.2.cmml">m</mi><mn id="Pt0.A4.Ex98.m2.1.1.3.2.3" xref="Pt0.A4.Ex98.m2.1.1.3.2.3.cmml">1</mn><mn id="Pt0.A4.Ex98.m2.1.1.3.3" xref="Pt0.A4.Ex98.m2.1.1.3.3.cmml">2</mn></msubsup></mfrac></mstyle><mo id="Pt0.A4.Ex98.m2.1.2.2.3.2.2.2" xref="Pt0.A4.Ex98.m2.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex98.m2.1.2.2.3.3" xref="Pt0.A4.Ex98.m2.1.2.2.3.3.cmml">n</mi></msup></mrow><mo id="Pt0.A4.Ex98.m2.1.2.1" xref="Pt0.A4.Ex98.m2.1.2.1.cmml">≤</mo><mn id="Pt0.A4.Ex98.m2.1.2.3" xref="Pt0.A4.Ex98.m2.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex98.m2.1b"><apply id="Pt0.A4.Ex98.m2.1.2.cmml" xref="Pt0.A4.Ex98.m2.1.2"><leq id="Pt0.A4.Ex98.m2.1.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.1"></leq><apply id="Pt0.A4.Ex98.m2.1.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.2.2"><times id="Pt0.A4.Ex98.m2.1.2.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.1"></times><apply id="Pt0.A4.Ex98.m2.1.2.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2"><divide id="Pt0.A4.Ex98.m2.1.2.2.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2"></divide><apply id="Pt0.A4.Ex98.m2.1.2.2.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.2"><plus id="Pt0.A4.Ex98.m2.1.2.2.2.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.1"></plus><apply id="Pt0.A4.Ex98.m2.1.2.2.2.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex98.m2.1.2.2.2.2.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.2">subscript</csymbol><ci id="Pt0.A4.Ex98.m2.1.2.2.2.2.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.2.2">𝑚</ci><cn id="Pt0.A4.Ex98.m2.1.2.2.2.2.2.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.2.3">1</cn></apply><cn id="Pt0.A4.Ex98.m2.1.2.2.2.2.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.2.2.2.2.3">1</cn></apply><apply id="Pt0.A4.Ex98.m2.1.2.2.2.3.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex98.m2.1.2.2.2.3.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.3">subscript</csymbol><ci id="Pt0.A4.Ex98.m2.1.2.2.2.3.2.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.2.3.2">𝑚</ci><cn id="Pt0.A4.Ex98.m2.1.2.2.2.3.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.2.2.2.3.3">1</cn></apply></apply><apply id="Pt0.A4.Ex98.m2.1.2.2.3.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex98.m2.1.2.2.3.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.3">superscript</csymbol><apply id="Pt0.A4.Ex98.m2.1.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.3.2.2"><divide id="Pt0.A4.Ex98.m2.1.1.1.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.3.2.2"></divide><apply id="Pt0.A4.Ex98.m2.1.1.2.cmml" xref="Pt0.A4.Ex98.m2.1.1.2"><minus id="Pt0.A4.Ex98.m2.1.1.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.1.2.1"></minus><apply id="Pt0.A4.Ex98.m2.1.1.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.1.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex98.m2.1.1.2.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.1.2.2">superscript</csymbol><apply id="Pt0.A4.Ex98.m2.1.1.2.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.1.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex98.m2.1.1.2.2.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.1.2.2">subscript</csymbol><ci id="Pt0.A4.Ex98.m2.1.1.2.2.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.1.2.2.2.2">𝑚</ci><cn id="Pt0.A4.Ex98.m2.1.1.2.2.2.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.1.2.2.2.3">1</cn></apply><cn id="Pt0.A4.Ex98.m2.1.1.2.2.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.1.2.2.3">2</cn></apply><cn id="Pt0.A4.Ex98.m2.1.1.2.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.1.2.3">1</cn></apply><apply id="Pt0.A4.Ex98.m2.1.1.3.cmml" xref="Pt0.A4.Ex98.m2.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex98.m2.1.1.3.1.cmml" xref="Pt0.A4.Ex98.m2.1.1.3">superscript</csymbol><apply id="Pt0.A4.Ex98.m2.1.1.3.2.cmml" xref="Pt0.A4.Ex98.m2.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex98.m2.1.1.3.2.1.cmml" xref="Pt0.A4.Ex98.m2.1.1.3">subscript</csymbol><ci id="Pt0.A4.Ex98.m2.1.1.3.2.2.cmml" xref="Pt0.A4.Ex98.m2.1.1.3.2.2">𝑚</ci><cn id="Pt0.A4.Ex98.m2.1.1.3.2.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.1.3.2.3">1</cn></apply><cn id="Pt0.A4.Ex98.m2.1.1.3.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.1.3.3">2</cn></apply></apply><ci id="Pt0.A4.Ex98.m2.1.2.2.3.3.cmml" xref="Pt0.A4.Ex98.m2.1.2.2.3.3">𝑛</ci></apply></apply><cn id="Pt0.A4.Ex98.m2.1.2.3.cmml" type="integer" xref="Pt0.A4.Ex98.m2.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex98.m2.1c">\displaystyle\frac{m_{1}+1}{m_{1}}\left(\frac{m_{1}^{2}-1}{m_{1}^{2}}\right)^{% n}\leq 2</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex98.m2.1d">divide start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( divide start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ≤ 2</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx21.p1.13">The latter holds because <math alttext="\frac{m_{1}+1}{m_{1}}\leq 2" class="ltx_Math" display="inline" id="Thmproofx21.p1.12.m1.1"><semantics id="Thmproofx21.p1.12.m1.1a"><mrow id="Thmproofx21.p1.12.m1.1.1" xref="Thmproofx21.p1.12.m1.1.1.cmml"><mfrac id="Thmproofx21.p1.12.m1.1.1.2" xref="Thmproofx21.p1.12.m1.1.1.2.cmml"><mrow id="Thmproofx21.p1.12.m1.1.1.2.2" xref="Thmproofx21.p1.12.m1.1.1.2.2.cmml"><msub id="Thmproofx21.p1.12.m1.1.1.2.2.2" xref="Thmproofx21.p1.12.m1.1.1.2.2.2.cmml"><mi id="Thmproofx21.p1.12.m1.1.1.2.2.2.2" xref="Thmproofx21.p1.12.m1.1.1.2.2.2.2.cmml">m</mi><mn id="Thmproofx21.p1.12.m1.1.1.2.2.2.3" xref="Thmproofx21.p1.12.m1.1.1.2.2.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p1.12.m1.1.1.2.2.1" xref="Thmproofx21.p1.12.m1.1.1.2.2.1.cmml">+</mo><mn id="Thmproofx21.p1.12.m1.1.1.2.2.3" xref="Thmproofx21.p1.12.m1.1.1.2.2.3.cmml">1</mn></mrow><msub id="Thmproofx21.p1.12.m1.1.1.2.3" xref="Thmproofx21.p1.12.m1.1.1.2.3.cmml"><mi id="Thmproofx21.p1.12.m1.1.1.2.3.2" xref="Thmproofx21.p1.12.m1.1.1.2.3.2.cmml">m</mi><mn id="Thmproofx21.p1.12.m1.1.1.2.3.3" xref="Thmproofx21.p1.12.m1.1.1.2.3.3.cmml">1</mn></msub></mfrac><mo id="Thmproofx21.p1.12.m1.1.1.1" xref="Thmproofx21.p1.12.m1.1.1.1.cmml">≤</mo><mn id="Thmproofx21.p1.12.m1.1.1.3" xref="Thmproofx21.p1.12.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.12.m1.1b"><apply id="Thmproofx21.p1.12.m1.1.1.cmml" xref="Thmproofx21.p1.12.m1.1.1"><leq id="Thmproofx21.p1.12.m1.1.1.1.cmml" xref="Thmproofx21.p1.12.m1.1.1.1"></leq><apply id="Thmproofx21.p1.12.m1.1.1.2.cmml" xref="Thmproofx21.p1.12.m1.1.1.2"><divide id="Thmproofx21.p1.12.m1.1.1.2.1.cmml" xref="Thmproofx21.p1.12.m1.1.1.2"></divide><apply id="Thmproofx21.p1.12.m1.1.1.2.2.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.2"><plus id="Thmproofx21.p1.12.m1.1.1.2.2.1.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.2.1"></plus><apply id="Thmproofx21.p1.12.m1.1.1.2.2.2.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.12.m1.1.1.2.2.2.1.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.2.2">subscript</csymbol><ci id="Thmproofx21.p1.12.m1.1.1.2.2.2.2.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.2.2.2">𝑚</ci><cn id="Thmproofx21.p1.12.m1.1.1.2.2.2.3.cmml" type="integer" xref="Thmproofx21.p1.12.m1.1.1.2.2.2.3">1</cn></apply><cn id="Thmproofx21.p1.12.m1.1.1.2.2.3.cmml" type="integer" xref="Thmproofx21.p1.12.m1.1.1.2.2.3">1</cn></apply><apply id="Thmproofx21.p1.12.m1.1.1.2.3.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.12.m1.1.1.2.3.1.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.3">subscript</csymbol><ci id="Thmproofx21.p1.12.m1.1.1.2.3.2.cmml" xref="Thmproofx21.p1.12.m1.1.1.2.3.2">𝑚</ci><cn id="Thmproofx21.p1.12.m1.1.1.2.3.3.cmml" type="integer" xref="Thmproofx21.p1.12.m1.1.1.2.3.3">1</cn></apply></apply><cn id="Thmproofx21.p1.12.m1.1.1.3.cmml" type="integer" xref="Thmproofx21.p1.12.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.12.m1.1c">\frac{m_{1}+1}{m_{1}}\leq 2</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.12.m1.1d">divide start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ≤ 2</annotation></semantics></math> and <math alttext="\left(\frac{m_{1}^{2}-1}{m_{1}^{2}}\right)^{n}\leq 1" class="ltx_Math" display="inline" id="Thmproofx21.p1.13.m2.1"><semantics id="Thmproofx21.p1.13.m2.1a"><mrow id="Thmproofx21.p1.13.m2.1.2" xref="Thmproofx21.p1.13.m2.1.2.cmml"><msup id="Thmproofx21.p1.13.m2.1.2.2" xref="Thmproofx21.p1.13.m2.1.2.2.cmml"><mrow id="Thmproofx21.p1.13.m2.1.2.2.2.2" xref="Thmproofx21.p1.13.m2.1.1.cmml"><mo id="Thmproofx21.p1.13.m2.1.2.2.2.2.1" xref="Thmproofx21.p1.13.m2.1.1.cmml">(</mo><mfrac id="Thmproofx21.p1.13.m2.1.1" xref="Thmproofx21.p1.13.m2.1.1.cmml"><mrow id="Thmproofx21.p1.13.m2.1.1.2" xref="Thmproofx21.p1.13.m2.1.1.2.cmml"><msubsup id="Thmproofx21.p1.13.m2.1.1.2.2" xref="Thmproofx21.p1.13.m2.1.1.2.2.cmml"><mi id="Thmproofx21.p1.13.m2.1.1.2.2.2.2" xref="Thmproofx21.p1.13.m2.1.1.2.2.2.2.cmml">m</mi><mn id="Thmproofx21.p1.13.m2.1.1.2.2.2.3" xref="Thmproofx21.p1.13.m2.1.1.2.2.2.3.cmml">1</mn><mn id="Thmproofx21.p1.13.m2.1.1.2.2.3" xref="Thmproofx21.p1.13.m2.1.1.2.2.3.cmml">2</mn></msubsup><mo id="Thmproofx21.p1.13.m2.1.1.2.1" xref="Thmproofx21.p1.13.m2.1.1.2.1.cmml">−</mo><mn id="Thmproofx21.p1.13.m2.1.1.2.3" xref="Thmproofx21.p1.13.m2.1.1.2.3.cmml">1</mn></mrow><msubsup id="Thmproofx21.p1.13.m2.1.1.3" xref="Thmproofx21.p1.13.m2.1.1.3.cmml"><mi id="Thmproofx21.p1.13.m2.1.1.3.2.2" xref="Thmproofx21.p1.13.m2.1.1.3.2.2.cmml">m</mi><mn id="Thmproofx21.p1.13.m2.1.1.3.2.3" xref="Thmproofx21.p1.13.m2.1.1.3.2.3.cmml">1</mn><mn id="Thmproofx21.p1.13.m2.1.1.3.3" xref="Thmproofx21.p1.13.m2.1.1.3.3.cmml">2</mn></msubsup></mfrac><mo id="Thmproofx21.p1.13.m2.1.2.2.2.2.2" xref="Thmproofx21.p1.13.m2.1.1.cmml">)</mo></mrow><mi id="Thmproofx21.p1.13.m2.1.2.2.3" xref="Thmproofx21.p1.13.m2.1.2.2.3.cmml">n</mi></msup><mo id="Thmproofx21.p1.13.m2.1.2.1" xref="Thmproofx21.p1.13.m2.1.2.1.cmml">≤</mo><mn id="Thmproofx21.p1.13.m2.1.2.3" xref="Thmproofx21.p1.13.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p1.13.m2.1b"><apply id="Thmproofx21.p1.13.m2.1.2.cmml" xref="Thmproofx21.p1.13.m2.1.2"><leq id="Thmproofx21.p1.13.m2.1.2.1.cmml" xref="Thmproofx21.p1.13.m2.1.2.1"></leq><apply id="Thmproofx21.p1.13.m2.1.2.2.cmml" xref="Thmproofx21.p1.13.m2.1.2.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.13.m2.1.2.2.1.cmml" xref="Thmproofx21.p1.13.m2.1.2.2">superscript</csymbol><apply id="Thmproofx21.p1.13.m2.1.1.cmml" xref="Thmproofx21.p1.13.m2.1.2.2.2.2"><divide id="Thmproofx21.p1.13.m2.1.1.1.cmml" xref="Thmproofx21.p1.13.m2.1.2.2.2.2"></divide><apply id="Thmproofx21.p1.13.m2.1.1.2.cmml" xref="Thmproofx21.p1.13.m2.1.1.2"><minus id="Thmproofx21.p1.13.m2.1.1.2.1.cmml" xref="Thmproofx21.p1.13.m2.1.1.2.1"></minus><apply id="Thmproofx21.p1.13.m2.1.1.2.2.cmml" xref="Thmproofx21.p1.13.m2.1.1.2.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.13.m2.1.1.2.2.1.cmml" xref="Thmproofx21.p1.13.m2.1.1.2.2">superscript</csymbol><apply id="Thmproofx21.p1.13.m2.1.1.2.2.2.cmml" xref="Thmproofx21.p1.13.m2.1.1.2.2"><csymbol cd="ambiguous" id="Thmproofx21.p1.13.m2.1.1.2.2.2.1.cmml" xref="Thmproofx21.p1.13.m2.1.1.2.2">subscript</csymbol><ci id="Thmproofx21.p1.13.m2.1.1.2.2.2.2.cmml" xref="Thmproofx21.p1.13.m2.1.1.2.2.2.2">𝑚</ci><cn id="Thmproofx21.p1.13.m2.1.1.2.2.2.3.cmml" type="integer" xref="Thmproofx21.p1.13.m2.1.1.2.2.2.3">1</cn></apply><cn id="Thmproofx21.p1.13.m2.1.1.2.2.3.cmml" type="integer" xref="Thmproofx21.p1.13.m2.1.1.2.2.3">2</cn></apply><cn id="Thmproofx21.p1.13.m2.1.1.2.3.cmml" type="integer" xref="Thmproofx21.p1.13.m2.1.1.2.3">1</cn></apply><apply id="Thmproofx21.p1.13.m2.1.1.3.cmml" xref="Thmproofx21.p1.13.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.13.m2.1.1.3.1.cmml" xref="Thmproofx21.p1.13.m2.1.1.3">superscript</csymbol><apply id="Thmproofx21.p1.13.m2.1.1.3.2.cmml" xref="Thmproofx21.p1.13.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p1.13.m2.1.1.3.2.1.cmml" xref="Thmproofx21.p1.13.m2.1.1.3">subscript</csymbol><ci id="Thmproofx21.p1.13.m2.1.1.3.2.2.cmml" xref="Thmproofx21.p1.13.m2.1.1.3.2.2">𝑚</ci><cn id="Thmproofx21.p1.13.m2.1.1.3.2.3.cmml" type="integer" xref="Thmproofx21.p1.13.m2.1.1.3.2.3">1</cn></apply><cn id="Thmproofx21.p1.13.m2.1.1.3.3.cmml" type="integer" xref="Thmproofx21.p1.13.m2.1.1.3.3">2</cn></apply></apply><ci id="Thmproofx21.p1.13.m2.1.2.2.3.cmml" xref="Thmproofx21.p1.13.m2.1.2.2.3">𝑛</ci></apply><cn id="Thmproofx21.p1.13.m2.1.2.3.cmml" type="integer" xref="Thmproofx21.p1.13.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p1.13.m2.1c">\left(\frac{m_{1}^{2}-1}{m_{1}^{2}}\right)^{n}\leq 1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p1.13.m2.1d">( divide start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ≤ 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx21.p2"> <p class="ltx_p" id="Thmproofx21.p2.4">Therefore, we can use Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma14" title="Lemma 14 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">14</span></a> to obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx56"> <tbody id="Pt0.A4.Ex99"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle F_{n}(U_{P_{m_{1}+1}}^{n},\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex99.m1.2"><semantics id="Pt0.A4.Ex99.m1.2a"><mrow id="Pt0.A4.Ex99.m1.2.2" xref="Pt0.A4.Ex99.m1.2.2.cmml"><msub id="Pt0.A4.Ex99.m1.2.2.3" xref="Pt0.A4.Ex99.m1.2.2.3.cmml"><mi id="Pt0.A4.Ex99.m1.2.2.3.2" xref="Pt0.A4.Ex99.m1.2.2.3.2.cmml">F</mi><mi id="Pt0.A4.Ex99.m1.2.2.3.3" xref="Pt0.A4.Ex99.m1.2.2.3.3.cmml">n</mi></msub><mo id="Pt0.A4.Ex99.m1.2.2.2" xref="Pt0.A4.Ex99.m1.2.2.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex99.m1.2.2.1.1" xref="Pt0.A4.Ex99.m1.2.2.1.2.cmml"><mo id="Pt0.A4.Ex99.m1.2.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex99.m1.2.2.1.2.cmml">(</mo><msubsup id="Pt0.A4.Ex99.m1.2.2.1.1.1" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.cmml"><mi id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.2" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.2.cmml">U</mi><msub id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.cmml"><mi id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.2" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.2.cmml">P</mi><mrow id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.cmml"><msub id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2.cmml"><mi id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2.2" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2.2.cmml">m</mi><mn id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2.3" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.1" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.1.cmml">+</mo><mn id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.3" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.3.cmml">1</mn></mrow></msub><mi id="Pt0.A4.Ex99.m1.2.2.1.1.1.3" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Pt0.A4.Ex99.m1.2.2.1.1.3" xref="Pt0.A4.Ex99.m1.2.2.1.2.cmml">,</mo><mi id="Pt0.A4.Ex99.m1.1.1" xref="Pt0.A4.Ex99.m1.1.1.cmml">α</mi><mo id="Pt0.A4.Ex99.m1.2.2.1.1.4" stretchy="false" xref="Pt0.A4.Ex99.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex99.m1.2b"><apply id="Pt0.A4.Ex99.m1.2.2.cmml" xref="Pt0.A4.Ex99.m1.2.2"><times id="Pt0.A4.Ex99.m1.2.2.2.cmml" xref="Pt0.A4.Ex99.m1.2.2.2"></times><apply id="Pt0.A4.Ex99.m1.2.2.3.cmml" xref="Pt0.A4.Ex99.m1.2.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex99.m1.2.2.3.1.cmml" xref="Pt0.A4.Ex99.m1.2.2.3">subscript</csymbol><ci id="Pt0.A4.Ex99.m1.2.2.3.2.cmml" xref="Pt0.A4.Ex99.m1.2.2.3.2">𝐹</ci><ci id="Pt0.A4.Ex99.m1.2.2.3.3.cmml" xref="Pt0.A4.Ex99.m1.2.2.3.3">𝑛</ci></apply><interval closure="open" id="Pt0.A4.Ex99.m1.2.2.1.2.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1"><apply id="Pt0.A4.Ex99.m1.2.2.1.1.1.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex99.m1.2.2.1.1.1.1.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.1.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.2.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.2">𝑈</ci><apply id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.1.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3">subscript</csymbol><ci id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.2.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.2">𝑃</ci><apply id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3"><plus id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.1.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.1"></plus><apply id="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2.cmml" xref="Pt0.A4.Ex99.m1.2.2.1.1.1.2.3.3.2"><csymbol cd="ambiguous" 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id="Pt0.A4.Ex99.m2.5c">\displaystyle=\alpha\sum_{i=1}^{m_{1}+1}k\cdot 1_{i=m_{1}+1}\frac{i^{n}-(i-1)^% {n}}{(m_{1}+1)^{n}}+\sum_{i=1}^{m_{1}+1}|p_{i}|\frac{i^{n}-(i-1)^{n}}{(m_{1}+1% )^{n}}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex99.m2.5d">= italic_α ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUPERSCRIPT italic_k ⋅ 1 start_POSTSUBSCRIPT italic_i = italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUPERSCRIPT | italic_p 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encoding="application/x-llamapun" id="Pt0.A4.Ex100.m1.2d">= | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + ( | italic_p start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ) italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT + italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex101"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle<|p_{1}|+E^{\prime}_{m_{1}}k\alpha" class="ltx_Math" display="inline" id="Pt0.A4.Ex101.m1.1"><semantics id="Pt0.A4.Ex101.m1.1a"><mrow id="Pt0.A4.Ex101.m1.1.1" xref="Pt0.A4.Ex101.m1.1.1.cmml"><mi id="Pt0.A4.Ex101.m1.1.1.3" xref="Pt0.A4.Ex101.m1.1.1.3.cmml"></mi><mo id="Pt0.A4.Ex101.m1.1.1.2" xref="Pt0.A4.Ex101.m1.1.1.2.cmml"><</mo><mrow id="Pt0.A4.Ex101.m1.1.1.1" xref="Pt0.A4.Ex101.m1.1.1.1.cmml"><mrow id="Pt0.A4.Ex101.m1.1.1.1.1.1" xref="Pt0.A4.Ex101.m1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex101.m1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex101.m1.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex101.m1.1.1.1.1.1.1" xref="Pt0.A4.Ex101.m1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex101.m1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex101.m1.1.1.1.1.1.1.2.cmml">p</mi><mn id="Pt0.A4.Ex101.m1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex101.m1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex101.m1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex101.m1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex101.m1.1.1.1.2" xref="Pt0.A4.Ex101.m1.1.1.1.2.cmml">+</mo><mrow id="Pt0.A4.Ex101.m1.1.1.1.3" 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xref="Pt0.A4.Ex101.m1.1.1.1.3.2.3.3">1</cn></apply></apply><ci id="Pt0.A4.Ex101.m1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex101.m1.1.1.1.3.3">𝑘</ci><ci id="Pt0.A4.Ex101.m1.1.1.1.3.4.cmml" xref="Pt0.A4.Ex101.m1.1.1.1.3.4">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex101.m1.1c">\displaystyle<|p_{1}|+E^{\prime}_{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex101.m1.1d">< | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx21.p2.3">The second equality uses <math alttext="\frac{(m_{1}+1)^{n}-m_{1}^{n}}{(m_{1}+1)^{n}}=E^{\prime}_{m_{1}+1}" class="ltx_Math" display="inline" 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id="Thmproofx21.p2.1.m1.2.3.1.cmml" xref="Thmproofx21.p2.1.m1.2.3.1"></eq><apply id="Thmproofx21.p2.1.m1.2.2.cmml" xref="Thmproofx21.p2.1.m1.2.2"><divide id="Thmproofx21.p2.1.m1.2.2.3.cmml" xref="Thmproofx21.p2.1.m1.2.2"></divide><apply id="Thmproofx21.p2.1.m1.1.1.1.cmml" xref="Thmproofx21.p2.1.m1.1.1.1"><minus id="Thmproofx21.p2.1.m1.1.1.1.2.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.2"></minus><apply id="Thmproofx21.p2.1.m1.1.1.1.1.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.1.1.1.1.2.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1">superscript</csymbol><apply id="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1.1.1"><plus id="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.1"></plus><apply id="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2.2.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2.2">𝑚</ci><cn id="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><cn id="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx21.p2.1.m1.1.1.1.1.1.1.1.3">1</cn></apply><ci id="Thmproofx21.p2.1.m1.1.1.1.1.3.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.1.3">𝑛</ci></apply><apply id="Thmproofx21.p2.1.m1.1.1.1.3.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.1.1.1.3.1.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.3">superscript</csymbol><apply id="Thmproofx21.p2.1.m1.1.1.1.3.2.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.1.1.1.3.2.1.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.3">subscript</csymbol><ci id="Thmproofx21.p2.1.m1.1.1.1.3.2.2.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.3.2.2">𝑚</ci><cn id="Thmproofx21.p2.1.m1.1.1.1.3.2.3.cmml" type="integer" xref="Thmproofx21.p2.1.m1.1.1.1.3.2.3">1</cn></apply><ci id="Thmproofx21.p2.1.m1.1.1.1.3.3.cmml" xref="Thmproofx21.p2.1.m1.1.1.1.3.3">𝑛</ci></apply></apply><apply id="Thmproofx21.p2.1.m1.2.2.2.cmml" xref="Thmproofx21.p2.1.m1.2.2.2"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.2.2.2.2.cmml" xref="Thmproofx21.p2.1.m1.2.2.2">superscript</csymbol><apply id="Thmproofx21.p2.1.m1.2.2.2.1.1.1.cmml" xref="Thmproofx21.p2.1.m1.2.2.2.1.1"><plus id="Thmproofx21.p2.1.m1.2.2.2.1.1.1.1.cmml" xref="Thmproofx21.p2.1.m1.2.2.2.1.1.1.1"></plus><apply id="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2.cmml" xref="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2.1.cmml" xref="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2">subscript</csymbol><ci id="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2.2.cmml" xref="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2.2">𝑚</ci><cn id="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2.3.cmml" type="integer" xref="Thmproofx21.p2.1.m1.2.2.2.1.1.1.2.3">1</cn></apply><cn id="Thmproofx21.p2.1.m1.2.2.2.1.1.1.3.cmml" type="integer" xref="Thmproofx21.p2.1.m1.2.2.2.1.1.1.3">1</cn></apply><ci id="Thmproofx21.p2.1.m1.2.2.2.3.cmml" xref="Thmproofx21.p2.1.m1.2.2.2.3">𝑛</ci></apply></apply><apply id="Thmproofx21.p2.1.m1.2.3.2.cmml" xref="Thmproofx21.p2.1.m1.2.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.2.3.2.1.cmml" xref="Thmproofx21.p2.1.m1.2.3.2">subscript</csymbol><apply id="Thmproofx21.p2.1.m1.2.3.2.2.cmml" xref="Thmproofx21.p2.1.m1.2.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.2.3.2.2.1.cmml" xref="Thmproofx21.p2.1.m1.2.3.2">superscript</csymbol><ci id="Thmproofx21.p2.1.m1.2.3.2.2.2.cmml" xref="Thmproofx21.p2.1.m1.2.3.2.2.2">𝐸</ci><ci id="Thmproofx21.p2.1.m1.2.3.2.2.3.cmml" xref="Thmproofx21.p2.1.m1.2.3.2.2.3">′</ci></apply><apply id="Thmproofx21.p2.1.m1.2.3.2.3.cmml" xref="Thmproofx21.p2.1.m1.2.3.2.3"><plus id="Thmproofx21.p2.1.m1.2.3.2.3.1.cmml" xref="Thmproofx21.p2.1.m1.2.3.2.3.1"></plus><apply id="Thmproofx21.p2.1.m1.2.3.2.3.2.cmml" xref="Thmproofx21.p2.1.m1.2.3.2.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p2.1.m1.2.3.2.3.2.1.cmml" xref="Thmproofx21.p2.1.m1.2.3.2.3.2">subscript</csymbol><ci id="Thmproofx21.p2.1.m1.2.3.2.3.2.2.cmml" xref="Thmproofx21.p2.1.m1.2.3.2.3.2.2">𝑚</ci><cn id="Thmproofx21.p2.1.m1.2.3.2.3.2.3.cmml" type="integer" xref="Thmproofx21.p2.1.m1.2.3.2.3.2.3">1</cn></apply><cn id="Thmproofx21.p2.1.m1.2.3.2.3.3.cmml" type="integer" xref="Thmproofx21.p2.1.m1.2.3.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p2.1.m1.2c">\frac{(m_{1}+1)^{n}-m_{1}^{n}}{(m_{1}+1)^{n}}=E^{\prime}_{m_{1}+1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p2.1.m1.2d">divide start_ARG ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG = italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sum_{i=1}^{m_{1}+1}\frac{i^{n}-(i-1)^{n}}{(m_{1}+1)^{n}}=1" class="ltx_Math" display="inline" id="Thmproofx21.p2.2.m2.2"><semantics id="Thmproofx21.p2.2.m2.2a"><mrow id="Thmproofx21.p2.2.m2.2.3" xref="Thmproofx21.p2.2.m2.2.3.cmml"><mrow id="Thmproofx21.p2.2.m2.2.3.2" xref="Thmproofx21.p2.2.m2.2.3.2.cmml"><msubsup id="Thmproofx21.p2.2.m2.2.3.2.1" xref="Thmproofx21.p2.2.m2.2.3.2.1.cmml"><mo id="Thmproofx21.p2.2.m2.2.3.2.1.2.2" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.2.cmml">∑</mo><mrow id="Thmproofx21.p2.2.m2.2.3.2.1.2.3" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3.cmml"><mi id="Thmproofx21.p2.2.m2.2.3.2.1.2.3.2" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3.2.cmml">i</mi><mo id="Thmproofx21.p2.2.m2.2.3.2.1.2.3.1" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3.1.cmml">=</mo><mn id="Thmproofx21.p2.2.m2.2.3.2.1.2.3.3" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3.3.cmml">1</mn></mrow><mrow id="Thmproofx21.p2.2.m2.2.3.2.1.3" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.cmml"><msub id="Thmproofx21.p2.2.m2.2.3.2.1.3.2" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.2.cmml"><mi id="Thmproofx21.p2.2.m2.2.3.2.1.3.2.2" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.2.2.cmml">m</mi><mn id="Thmproofx21.p2.2.m2.2.3.2.1.3.2.3" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p2.2.m2.2.3.2.1.3.1" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.1.cmml">+</mo><mn id="Thmproofx21.p2.2.m2.2.3.2.1.3.3" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.3.cmml">1</mn></mrow></msubsup><mfrac id="Thmproofx21.p2.2.m2.2.2" xref="Thmproofx21.p2.2.m2.2.2.cmml"><mrow id="Thmproofx21.p2.2.m2.1.1.1" xref="Thmproofx21.p2.2.m2.1.1.1.cmml"><msup id="Thmproofx21.p2.2.m2.1.1.1.3" xref="Thmproofx21.p2.2.m2.1.1.1.3.cmml"><mi id="Thmproofx21.p2.2.m2.1.1.1.3.2" xref="Thmproofx21.p2.2.m2.1.1.1.3.2.cmml">i</mi><mi id="Thmproofx21.p2.2.m2.1.1.1.3.3" xref="Thmproofx21.p2.2.m2.1.1.1.3.3.cmml">n</mi></msup><mo id="Thmproofx21.p2.2.m2.1.1.1.2" xref="Thmproofx21.p2.2.m2.1.1.1.2.cmml">−</mo><msup id="Thmproofx21.p2.2.m2.1.1.1.1" xref="Thmproofx21.p2.2.m2.1.1.1.1.cmml"><mrow id="Thmproofx21.p2.2.m2.1.1.1.1.1.1" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.2" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.2.cmml">i</mi><mo id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.1" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.3" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx21.p2.2.m2.1.1.1.1.3" xref="Thmproofx21.p2.2.m2.1.1.1.1.3.cmml">n</mi></msup></mrow><msup id="Thmproofx21.p2.2.m2.2.2.2" xref="Thmproofx21.p2.2.m2.2.2.2.cmml"><mrow id="Thmproofx21.p2.2.m2.2.2.2.1.1" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.cmml"><mo id="Thmproofx21.p2.2.m2.2.2.2.1.1.2" stretchy="false" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmproofx21.p2.2.m2.2.2.2.1.1.1" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.cmml"><msub id="Thmproofx21.p2.2.m2.2.2.2.1.1.1.2" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.2.cmml"><mi id="Thmproofx21.p2.2.m2.2.2.2.1.1.1.2.2" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.2.2.cmml">m</mi><mn id="Thmproofx21.p2.2.m2.2.2.2.1.1.1.2.3" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmproofx21.p2.2.m2.2.2.2.1.1.1.1" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.1.cmml">+</mo><mn id="Thmproofx21.p2.2.m2.2.2.2.1.1.1.3" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="Thmproofx21.p2.2.m2.2.2.2.1.1.3" stretchy="false" xref="Thmproofx21.p2.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx21.p2.2.m2.2.2.2.3" xref="Thmproofx21.p2.2.m2.2.2.2.3.cmml">n</mi></msup></mfrac></mrow><mo id="Thmproofx21.p2.2.m2.2.3.1" xref="Thmproofx21.p2.2.m2.2.3.1.cmml">=</mo><mn id="Thmproofx21.p2.2.m2.2.3.3" xref="Thmproofx21.p2.2.m2.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx21.p2.2.m2.2b"><apply id="Thmproofx21.p2.2.m2.2.3.cmml" xref="Thmproofx21.p2.2.m2.2.3"><eq id="Thmproofx21.p2.2.m2.2.3.1.cmml" xref="Thmproofx21.p2.2.m2.2.3.1"></eq><apply id="Thmproofx21.p2.2.m2.2.3.2.cmml" xref="Thmproofx21.p2.2.m2.2.3.2"><apply id="Thmproofx21.p2.2.m2.2.3.2.1.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1"><csymbol cd="ambiguous" id="Thmproofx21.p2.2.m2.2.3.2.1.1.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1">superscript</csymbol><apply id="Thmproofx21.p2.2.m2.2.3.2.1.2.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1"><csymbol cd="ambiguous" id="Thmproofx21.p2.2.m2.2.3.2.1.2.1.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1">subscript</csymbol><sum id="Thmproofx21.p2.2.m2.2.3.2.1.2.2.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.2"></sum><apply id="Thmproofx21.p2.2.m2.2.3.2.1.2.3.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3"><eq id="Thmproofx21.p2.2.m2.2.3.2.1.2.3.1.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3.1"></eq><ci id="Thmproofx21.p2.2.m2.2.3.2.1.2.3.2.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3.2">𝑖</ci><cn id="Thmproofx21.p2.2.m2.2.3.2.1.2.3.3.cmml" type="integer" xref="Thmproofx21.p2.2.m2.2.3.2.1.2.3.3">1</cn></apply></apply><apply id="Thmproofx21.p2.2.m2.2.3.2.1.3.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.3"><plus id="Thmproofx21.p2.2.m2.2.3.2.1.3.1.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.1"></plus><apply id="Thmproofx21.p2.2.m2.2.3.2.1.3.2.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.2"><csymbol cd="ambiguous" id="Thmproofx21.p2.2.m2.2.3.2.1.3.2.1.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.2">subscript</csymbol><ci id="Thmproofx21.p2.2.m2.2.3.2.1.3.2.2.cmml" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.2.2">𝑚</ci><cn id="Thmproofx21.p2.2.m2.2.3.2.1.3.2.3.cmml" type="integer" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.2.3">1</cn></apply><cn id="Thmproofx21.p2.2.m2.2.3.2.1.3.3.cmml" type="integer" xref="Thmproofx21.p2.2.m2.2.3.2.1.3.3">1</cn></apply></apply><apply id="Thmproofx21.p2.2.m2.2.2.cmml" xref="Thmproofx21.p2.2.m2.2.2"><divide id="Thmproofx21.p2.2.m2.2.2.3.cmml" xref="Thmproofx21.p2.2.m2.2.2"></divide><apply id="Thmproofx21.p2.2.m2.1.1.1.cmml" xref="Thmproofx21.p2.2.m2.1.1.1"><minus id="Thmproofx21.p2.2.m2.1.1.1.2.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.2"></minus><apply id="Thmproofx21.p2.2.m2.1.1.1.3.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx21.p2.2.m2.1.1.1.3.1.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.3">superscript</csymbol><ci id="Thmproofx21.p2.2.m2.1.1.1.3.2.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.3.2">𝑖</ci><ci id="Thmproofx21.p2.2.m2.1.1.1.3.3.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.3.3">𝑛</ci></apply><apply id="Thmproofx21.p2.2.m2.1.1.1.1.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx21.p2.2.m2.1.1.1.1.2.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.1">superscript</csymbol><apply id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1"><minus id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.1"></minus><ci id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.2">𝑖</ci><cn id="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx21.p2.2.m2.1.1.1.1.1.1.1.3">1</cn></apply><ci id="Thmproofx21.p2.2.m2.1.1.1.1.3.cmml" xref="Thmproofx21.p2.2.m2.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Thmproofx21.p2.2.m2.2.2.2.cmml" 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xref="Thmproofx21.p2.2.m2.2.2.2.3">𝑛</ci></apply></apply></apply><cn id="Thmproofx21.p2.2.m2.2.3.3.cmml" type="integer" xref="Thmproofx21.p2.2.m2.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p2.2.m2.2c">\sum_{i=1}^{m_{1}+1}\frac{i^{n}-(i-1)^{n}}{(m_{1}+1)^{n}}=1</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p2.2.m2.2d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUPERSCRIPT divide start_ARG italic_i start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_i - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG = 1</annotation></semantics></math>, and the inequality uses the assumption on <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmproofx21.p2.3.m3.1"><semantics id="Thmproofx21.p2.3.m3.1a"><mi id="Thmproofx21.p2.3.m3.1.1" xref="Thmproofx21.p2.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmproofx21.p2.3.m3.1b"><ci id="Thmproofx21.p2.3.m3.1.1.cmml" xref="Thmproofx21.p2.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx21.p2.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx21.p2.3.m3.1d">italic_α</annotation></semantics></math>. The lemma now follows by definition. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx22"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem2" title="Theorem 5.2 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.2</span></a>)</h6> <div class="ltx_para" id="Thmproofx22.p1"> <p class="ltx_p" id="Thmproofx22.p1.9">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma14" title="Lemma 14 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">14</span></a>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx57"> <tbody 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xref="Pt0.A4.Ex102.m1.12.12.3.1.1.1.3">𝑛</ci></apply><ci id="Pt0.A4.Ex102.m1.9.9.cmml" xref="Pt0.A4.Ex102.m1.9.9">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex102.m1.12c">\displaystyle F_{n}(U_{P_{st}}^{n},\alpha)=f_{(m,...,m)}(U_{P_{st}}^{n},\alpha% )>\max_{d\neq(m,...,m)}f_{d}(U_{P_{st}}^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex102.m1.12d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) > roman_max start_POSTSUBSCRIPT italic_d ≠ ( italic_m , … , italic_m ) end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx22.p1.8">For a small <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="Thmproofx22.p1.1.m1.1"><semantics id="Thmproofx22.p1.1.m1.1a"><mrow id="Thmproofx22.p1.1.m1.1.1" xref="Thmproofx22.p1.1.m1.1.1.cmml"><mi id="Thmproofx22.p1.1.m1.1.1.2" xref="Thmproofx22.p1.1.m1.1.1.2.cmml">ε</mi><mo id="Thmproofx22.p1.1.m1.1.1.1" xref="Thmproofx22.p1.1.m1.1.1.1.cmml">></mo><mn id="Thmproofx22.p1.1.m1.1.1.3" xref="Thmproofx22.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.1.m1.1b"><apply id="Thmproofx22.p1.1.m1.1.1.cmml" xref="Thmproofx22.p1.1.m1.1.1"><gt id="Thmproofx22.p1.1.m1.1.1.1.cmml" xref="Thmproofx22.p1.1.m1.1.1.1"></gt><ci id="Thmproofx22.p1.1.m1.1.1.2.cmml" xref="Thmproofx22.p1.1.m1.1.1.2">𝜀</ci><cn id="Thmproofx22.p1.1.m1.1.1.3.cmml" type="integer" xref="Thmproofx22.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.1.m1.1d">italic_ε > 0</annotation></semantics></math>, let <math alttext="\nu" class="ltx_Math" display="inline" id="Thmproofx22.p1.2.m2.1"><semantics id="Thmproofx22.p1.2.m2.1a"><mi id="Thmproofx22.p1.2.m2.1.1" xref="Thmproofx22.p1.2.m2.1.1.cmml">ν</mi><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.2.m2.1b"><ci id="Thmproofx22.p1.2.m2.1.1.cmml" xref="Thmproofx22.p1.2.m2.1.1">𝜈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.2.m2.1c">\nu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.2.m2.1d">italic_ν</annotation></semantics></math> such that <math alttext="\nu(p_{1})=\frac{1}{m}+\varepsilon,\nu(p_{2})=...=\nu(p_{m-1})=\frac{1}{m},\nu% (p_{m})=\frac{1}{m}-\varepsilon" class="ltx_Math" display="inline" id="Thmproofx22.p1.3.m3.2"><semantics id="Thmproofx22.p1.3.m3.2a"><mrow id="Thmproofx22.p1.3.m3.2.2.2" xref="Thmproofx22.p1.3.m3.2.2.3.cmml"><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.3.cmml"><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.cmml"><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.3.cmml">ν</mi><mo id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><mn id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.2.cmml">=</mo><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.cmml"><mfrac id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.cmml"><mn id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.2.cmml">1</mn><mi id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.3.cmml">m</mi></mfrac><mo id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.3.cmml">ε</mi></mrow></mrow><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.3a.cmml">,</mo><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.2.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.cmml"><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.cmml"><mi id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.3.cmml">ν</mi><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.cmml"><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.2" stretchy="false" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.cmml">(</mo><msub id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.cmml"><mi id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.2.cmml">p</mi><mn id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.3.cmml">2</mn></msub><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.3" stretchy="false" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.4" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.4.cmml">=</mo><mi id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.5" mathvariant="normal" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.5.cmml">…</mi><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.6" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.6.cmml">=</mo><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.cmml"><mi id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.3.cmml">ν</mi><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.cmml"><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.2" stretchy="false" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.cmml"><mi id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.2.cmml">p</mi><mrow id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.cmml"><mi id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.2.cmml">m</mi><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.1" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.1.cmml">−</mo><mn id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.3" stretchy="false" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.7" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.7.cmml">=</mo><mfrac id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.cmml"><mn id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.2" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.2.cmml">1</mn><mi id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.3" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.3.cmml">m</mi></mfrac></mrow></mrow><mo id="Thmproofx22.p1.3.m3.2.2.2.3" xref="Thmproofx22.p1.3.m3.2.2.3a.cmml">,</mo><mrow id="Thmproofx22.p1.3.m3.2.2.2.2" xref="Thmproofx22.p1.3.m3.2.2.2.2.cmml"><mrow id="Thmproofx22.p1.3.m3.2.2.2.2.1" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.cmml"><mi id="Thmproofx22.p1.3.m3.2.2.2.2.1.3" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.3.cmml">ν</mi><mo id="Thmproofx22.p1.3.m3.2.2.2.2.1.2" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.cmml"><mo id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.2" stretchy="false" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.cmml">(</mo><msub id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.cmml"><mi id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.2" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.3" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.3.cmml">m</mi></msub><mo id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.3" stretchy="false" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p1.3.m3.2.2.2.2.2" xref="Thmproofx22.p1.3.m3.2.2.2.2.2.cmml">=</mo><mrow id="Thmproofx22.p1.3.m3.2.2.2.2.3" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.cmml"><mfrac id="Thmproofx22.p1.3.m3.2.2.2.2.3.2" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.2.cmml"><mn id="Thmproofx22.p1.3.m3.2.2.2.2.3.2.2" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.2.2.cmml">1</mn><mi id="Thmproofx22.p1.3.m3.2.2.2.2.3.2.3" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.2.3.cmml">m</mi></mfrac><mo id="Thmproofx22.p1.3.m3.2.2.2.2.3.1" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.1.cmml">−</mo><mi id="Thmproofx22.p1.3.m3.2.2.2.2.3.3" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.3.cmml">ε</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.3.m3.2b"><apply id="Thmproofx22.p1.3.m3.2.2.3.cmml" xref="Thmproofx22.p1.3.m3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx22.p1.3.m3.2.2.3a.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.3">formulae-sequence</csymbol><apply id="Thmproofx22.p1.3.m3.1.1.1.1.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx22.p1.3.m3.1.1.1.1.3a.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1"><eq id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.2"></eq><apply id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1"><times id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.2"></times><ci id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.3">𝜈</ci><apply id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><cn id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3"><plus id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.1"></plus><apply id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2"><divide id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2"></divide><cn id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.2">1</cn><ci id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.2.3">𝑚</ci></apply><ci id="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.1.1.3.3">𝜀</ci></apply></apply><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2"><and id="Thmproofx22.p1.3.m3.1.1.1.1.2.2a.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2"></and><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2b.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2"><eq id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.4.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.4"></eq><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1"><times id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.2"></times><ci id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.3">𝜈</ci><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.2">𝑝</ci><cn id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.3.cmml" type="integer" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.1.1.1.1.3">2</cn></apply></apply><ci id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.5.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.5">…</ci></apply><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2c.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2"><eq id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.6.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.6"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx22.p1.3.m3.1.1.1.1.2.2.5.cmml" id="Thmproofx22.p1.3.m3.1.1.1.1.2.2d.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2"></share><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2"><times id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.2"></times><ci id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.3">𝜈</ci><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.2">𝑝</ci><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3"><minus id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.1"></minus><ci id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.2.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.2">𝑚</ci><cn id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.3.cmml" type="integer" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2e.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2"><eq id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.7.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.7"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx22.p1.3.m3.1.1.1.1.2.2.2.cmml" id="Thmproofx22.p1.3.m3.1.1.1.1.2.2f.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2"></share><apply id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8"><divide id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.1.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8"></divide><cn id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.2.cmml" type="integer" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.2">1</cn><ci id="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.3.cmml" xref="Thmproofx22.p1.3.m3.1.1.1.1.2.2.8.3">𝑚</ci></apply></apply></apply></apply><apply id="Thmproofx22.p1.3.m3.2.2.2.2.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2"><eq id="Thmproofx22.p1.3.m3.2.2.2.2.2.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.2"></eq><apply id="Thmproofx22.p1.3.m3.2.2.2.2.1.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.1"><times id="Thmproofx22.p1.3.m3.2.2.2.2.1.2.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.2"></times><ci id="Thmproofx22.p1.3.m3.2.2.2.2.1.3.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.3">𝜈</ci><apply id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.1.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1">subscript</csymbol><ci id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.2.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.2">𝑝</ci><ci id="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.3.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.1.1.1.1.3">𝑚</ci></apply></apply><apply id="Thmproofx22.p1.3.m3.2.2.2.2.3.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.3"><minus id="Thmproofx22.p1.3.m3.2.2.2.2.3.1.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.1"></minus><apply id="Thmproofx22.p1.3.m3.2.2.2.2.3.2.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.2"><divide id="Thmproofx22.p1.3.m3.2.2.2.2.3.2.1.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.2"></divide><cn id="Thmproofx22.p1.3.m3.2.2.2.2.3.2.2.cmml" type="integer" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.2.2">1</cn><ci id="Thmproofx22.p1.3.m3.2.2.2.2.3.2.3.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.2.3">𝑚</ci></apply><ci id="Thmproofx22.p1.3.m3.2.2.2.2.3.3.cmml" xref="Thmproofx22.p1.3.m3.2.2.2.2.3.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.3.m3.2c">\nu(p_{1})=\frac{1}{m}+\varepsilon,\nu(p_{2})=...=\nu(p_{m-1})=\frac{1}{m},\nu% (p_{m})=\frac{1}{m}-\varepsilon</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.3.m3.2d">italic_ν ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_m end_ARG + italic_ε , italic_ν ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = … = italic_ν ( italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_m end_ARG , italic_ν ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_m end_ARG - italic_ε</annotation></semantics></math>. Observe that for every <math alttext="d" class="ltx_Math" display="inline" id="Thmproofx22.p1.4.m4.1"><semantics id="Thmproofx22.p1.4.m4.1a"><mi id="Thmproofx22.p1.4.m4.1.1" xref="Thmproofx22.p1.4.m4.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.4.m4.1b"><ci id="Thmproofx22.p1.4.m4.1.1.cmml" xref="Thmproofx22.p1.4.m4.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.4.m4.1c">d</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.4.m4.1d">italic_d</annotation></semantics></math>, <math alttext="f_{d}" class="ltx_Math" display="inline" id="Thmproofx22.p1.5.m5.1"><semantics id="Thmproofx22.p1.5.m5.1a"><msub id="Thmproofx22.p1.5.m5.1.1" xref="Thmproofx22.p1.5.m5.1.1.cmml"><mi id="Thmproofx22.p1.5.m5.1.1.2" xref="Thmproofx22.p1.5.m5.1.1.2.cmml">f</mi><mi id="Thmproofx22.p1.5.m5.1.1.3" xref="Thmproofx22.p1.5.m5.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.5.m5.1b"><apply id="Thmproofx22.p1.5.m5.1.1.cmml" xref="Thmproofx22.p1.5.m5.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.5.m5.1.1.1.cmml" xref="Thmproofx22.p1.5.m5.1.1">subscript</csymbol><ci id="Thmproofx22.p1.5.m5.1.1.2.cmml" xref="Thmproofx22.p1.5.m5.1.1.2">𝑓</ci><ci id="Thmproofx22.p1.5.m5.1.1.3.cmml" xref="Thmproofx22.p1.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.5.m5.1c">f_{d}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.5.m5.1d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> is continuous, and <math alttext="|E^{k}|<\infty" class="ltx_Math" display="inline" id="Thmproofx22.p1.6.m6.1"><semantics id="Thmproofx22.p1.6.m6.1a"><mrow id="Thmproofx22.p1.6.m6.1.1" xref="Thmproofx22.p1.6.m6.1.1.cmml"><mrow id="Thmproofx22.p1.6.m6.1.1.1.1" xref="Thmproofx22.p1.6.m6.1.1.1.2.cmml"><mo id="Thmproofx22.p1.6.m6.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p1.6.m6.1.1.1.2.1.cmml">|</mo><msup id="Thmproofx22.p1.6.m6.1.1.1.1.1" xref="Thmproofx22.p1.6.m6.1.1.1.1.1.cmml"><mi id="Thmproofx22.p1.6.m6.1.1.1.1.1.2" xref="Thmproofx22.p1.6.m6.1.1.1.1.1.2.cmml">E</mi><mi id="Thmproofx22.p1.6.m6.1.1.1.1.1.3" xref="Thmproofx22.p1.6.m6.1.1.1.1.1.3.cmml">k</mi></msup><mo id="Thmproofx22.p1.6.m6.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p1.6.m6.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx22.p1.6.m6.1.1.2" xref="Thmproofx22.p1.6.m6.1.1.2.cmml"><</mo><mi id="Thmproofx22.p1.6.m6.1.1.3" mathvariant="normal" xref="Thmproofx22.p1.6.m6.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.6.m6.1b"><apply id="Thmproofx22.p1.6.m6.1.1.cmml" xref="Thmproofx22.p1.6.m6.1.1"><lt id="Thmproofx22.p1.6.m6.1.1.2.cmml" xref="Thmproofx22.p1.6.m6.1.1.2"></lt><apply id="Thmproofx22.p1.6.m6.1.1.1.2.cmml" xref="Thmproofx22.p1.6.m6.1.1.1.1"><abs id="Thmproofx22.p1.6.m6.1.1.1.2.1.cmml" xref="Thmproofx22.p1.6.m6.1.1.1.1.2"></abs><apply id="Thmproofx22.p1.6.m6.1.1.1.1.1.cmml" xref="Thmproofx22.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.6.m6.1.1.1.1.1.1.cmml" xref="Thmproofx22.p1.6.m6.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx22.p1.6.m6.1.1.1.1.1.2.cmml" xref="Thmproofx22.p1.6.m6.1.1.1.1.1.2">𝐸</ci><ci id="Thmproofx22.p1.6.m6.1.1.1.1.1.3.cmml" xref="Thmproofx22.p1.6.m6.1.1.1.1.1.3">𝑘</ci></apply></apply><infinity id="Thmproofx22.p1.6.m6.1.1.3.cmml" xref="Thmproofx22.p1.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.6.m6.1c">|E^{k}|<\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.6.m6.1d">| italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT | < ∞</annotation></semantics></math>. Hence, when <math alttext="\varepsilon" class="ltx_Math" display="inline" id="Thmproofx22.p1.7.m7.1"><semantics id="Thmproofx22.p1.7.m7.1a"><mi id="Thmproofx22.p1.7.m7.1.1" xref="Thmproofx22.p1.7.m7.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.7.m7.1b"><ci id="Thmproofx22.p1.7.m7.1.1.cmml" xref="Thmproofx22.p1.7.m7.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.7.m7.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.7.m7.1d">italic_ε</annotation></semantics></math> is sufficiently small, we still have <math alttext="F_{n}(\nu^{n},\alpha)=f_{(m,...,m)}\left(\nu^{n},\alpha\right)>\max_{d\neq(m,.% ..,m)}f_{d}\left(\nu^{n},\alpha\right)" class="ltx_Math" display="inline" id="Thmproofx22.p1.8.m8.12"><semantics id="Thmproofx22.p1.8.m8.12a"><mrow id="Thmproofx22.p1.8.m8.12.12" xref="Thmproofx22.p1.8.m8.12.12.cmml"><mrow id="Thmproofx22.p1.8.m8.10.10.1" xref="Thmproofx22.p1.8.m8.10.10.1.cmml"><msub id="Thmproofx22.p1.8.m8.10.10.1.3" xref="Thmproofx22.p1.8.m8.10.10.1.3.cmml"><mi id="Thmproofx22.p1.8.m8.10.10.1.3.2" xref="Thmproofx22.p1.8.m8.10.10.1.3.2.cmml">F</mi><mi id="Thmproofx22.p1.8.m8.10.10.1.3.3" xref="Thmproofx22.p1.8.m8.10.10.1.3.3.cmml">n</mi></msub><mo id="Thmproofx22.p1.8.m8.10.10.1.2" xref="Thmproofx22.p1.8.m8.10.10.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p1.8.m8.10.10.1.1.1" xref="Thmproofx22.p1.8.m8.10.10.1.1.2.cmml"><mo id="Thmproofx22.p1.8.m8.10.10.1.1.1.2" stretchy="false" xref="Thmproofx22.p1.8.m8.10.10.1.1.2.cmml">(</mo><msup id="Thmproofx22.p1.8.m8.10.10.1.1.1.1" xref="Thmproofx22.p1.8.m8.10.10.1.1.1.1.cmml"><mi id="Thmproofx22.p1.8.m8.10.10.1.1.1.1.2" xref="Thmproofx22.p1.8.m8.10.10.1.1.1.1.2.cmml">ν</mi><mi id="Thmproofx22.p1.8.m8.10.10.1.1.1.1.3" xref="Thmproofx22.p1.8.m8.10.10.1.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx22.p1.8.m8.10.10.1.1.1.3" xref="Thmproofx22.p1.8.m8.10.10.1.1.2.cmml">,</mo><mi id="Thmproofx22.p1.8.m8.7.7" xref="Thmproofx22.p1.8.m8.7.7.cmml">α</mi><mo id="Thmproofx22.p1.8.m8.10.10.1.1.1.4" stretchy="false" xref="Thmproofx22.p1.8.m8.10.10.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p1.8.m8.12.12.5" xref="Thmproofx22.p1.8.m8.12.12.5.cmml">=</mo><mrow id="Thmproofx22.p1.8.m8.11.11.2" xref="Thmproofx22.p1.8.m8.11.11.2.cmml"><msub id="Thmproofx22.p1.8.m8.11.11.2.3" xref="Thmproofx22.p1.8.m8.11.11.2.3.cmml"><mi id="Thmproofx22.p1.8.m8.11.11.2.3.2" xref="Thmproofx22.p1.8.m8.11.11.2.3.2.cmml">f</mi><mrow id="Thmproofx22.p1.8.m8.3.3.3.5" xref="Thmproofx22.p1.8.m8.3.3.3.4.cmml"><mo id="Thmproofx22.p1.8.m8.3.3.3.5.1" stretchy="false" xref="Thmproofx22.p1.8.m8.3.3.3.4.cmml">(</mo><mi id="Thmproofx22.p1.8.m8.1.1.1.1" xref="Thmproofx22.p1.8.m8.1.1.1.1.cmml">m</mi><mo id="Thmproofx22.p1.8.m8.3.3.3.5.2" xref="Thmproofx22.p1.8.m8.3.3.3.4.cmml">,</mo><mi id="Thmproofx22.p1.8.m8.2.2.2.2" mathvariant="normal" xref="Thmproofx22.p1.8.m8.2.2.2.2.cmml">…</mi><mo id="Thmproofx22.p1.8.m8.3.3.3.5.3" xref="Thmproofx22.p1.8.m8.3.3.3.4.cmml">,</mo><mi id="Thmproofx22.p1.8.m8.3.3.3.3" xref="Thmproofx22.p1.8.m8.3.3.3.3.cmml">m</mi><mo id="Thmproofx22.p1.8.m8.3.3.3.5.4" stretchy="false" xref="Thmproofx22.p1.8.m8.3.3.3.4.cmml">)</mo></mrow></msub><mo id="Thmproofx22.p1.8.m8.11.11.2.2" xref="Thmproofx22.p1.8.m8.11.11.2.2.cmml">⁢</mo><mrow id="Thmproofx22.p1.8.m8.11.11.2.1.1" xref="Thmproofx22.p1.8.m8.11.11.2.1.2.cmml"><mo id="Thmproofx22.p1.8.m8.11.11.2.1.1.2" xref="Thmproofx22.p1.8.m8.11.11.2.1.2.cmml">(</mo><msup id="Thmproofx22.p1.8.m8.11.11.2.1.1.1" xref="Thmproofx22.p1.8.m8.11.11.2.1.1.1.cmml"><mi id="Thmproofx22.p1.8.m8.11.11.2.1.1.1.2" xref="Thmproofx22.p1.8.m8.11.11.2.1.1.1.2.cmml">ν</mi><mi id="Thmproofx22.p1.8.m8.11.11.2.1.1.1.3" xref="Thmproofx22.p1.8.m8.11.11.2.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx22.p1.8.m8.11.11.2.1.1.3" xref="Thmproofx22.p1.8.m8.11.11.2.1.2.cmml">,</mo><mi id="Thmproofx22.p1.8.m8.8.8" xref="Thmproofx22.p1.8.m8.8.8.cmml">α</mi><mo id="Thmproofx22.p1.8.m8.11.11.2.1.1.4" xref="Thmproofx22.p1.8.m8.11.11.2.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p1.8.m8.12.12.6" xref="Thmproofx22.p1.8.m8.12.12.6.cmml">></mo><mrow id="Thmproofx22.p1.8.m8.12.12.3" xref="Thmproofx22.p1.8.m8.12.12.3.cmml"><mrow id="Thmproofx22.p1.8.m8.12.12.3.3" xref="Thmproofx22.p1.8.m8.12.12.3.3.cmml"><msub id="Thmproofx22.p1.8.m8.12.12.3.3.1" xref="Thmproofx22.p1.8.m8.12.12.3.3.1.cmml"><mi id="Thmproofx22.p1.8.m8.12.12.3.3.1.2" xref="Thmproofx22.p1.8.m8.12.12.3.3.1.2.cmml">max</mi><mrow id="Thmproofx22.p1.8.m8.6.6.3" xref="Thmproofx22.p1.8.m8.6.6.3.cmml"><mi id="Thmproofx22.p1.8.m8.6.6.3.5" xref="Thmproofx22.p1.8.m8.6.6.3.5.cmml">d</mi><mo id="Thmproofx22.p1.8.m8.6.6.3.4" xref="Thmproofx22.p1.8.m8.6.6.3.4.cmml">≠</mo><mrow id="Thmproofx22.p1.8.m8.6.6.3.6.2" xref="Thmproofx22.p1.8.m8.6.6.3.6.1.cmml"><mo id="Thmproofx22.p1.8.m8.6.6.3.6.2.1" stretchy="false" xref="Thmproofx22.p1.8.m8.6.6.3.6.1.cmml">(</mo><mi id="Thmproofx22.p1.8.m8.4.4.1.1" xref="Thmproofx22.p1.8.m8.4.4.1.1.cmml">m</mi><mo id="Thmproofx22.p1.8.m8.6.6.3.6.2.2" xref="Thmproofx22.p1.8.m8.6.6.3.6.1.cmml">,</mo><mi id="Thmproofx22.p1.8.m8.5.5.2.2" mathvariant="normal" xref="Thmproofx22.p1.8.m8.5.5.2.2.cmml">…</mi><mo id="Thmproofx22.p1.8.m8.6.6.3.6.2.3" xref="Thmproofx22.p1.8.m8.6.6.3.6.1.cmml">,</mo><mi id="Thmproofx22.p1.8.m8.6.6.3.3" xref="Thmproofx22.p1.8.m8.6.6.3.3.cmml">m</mi><mo id="Thmproofx22.p1.8.m8.6.6.3.6.2.4" stretchy="false" xref="Thmproofx22.p1.8.m8.6.6.3.6.1.cmml">)</mo></mrow></mrow></msub><mo id="Thmproofx22.p1.8.m8.12.12.3.3a" lspace="0.167em" xref="Thmproofx22.p1.8.m8.12.12.3.3.cmml">⁡</mo><msub id="Thmproofx22.p1.8.m8.12.12.3.3.2" xref="Thmproofx22.p1.8.m8.12.12.3.3.2.cmml"><mi id="Thmproofx22.p1.8.m8.12.12.3.3.2.2" xref="Thmproofx22.p1.8.m8.12.12.3.3.2.2.cmml">f</mi><mi id="Thmproofx22.p1.8.m8.12.12.3.3.2.3" xref="Thmproofx22.p1.8.m8.12.12.3.3.2.3.cmml">d</mi></msub></mrow><mo id="Thmproofx22.p1.8.m8.12.12.3.2" xref="Thmproofx22.p1.8.m8.12.12.3.2.cmml">⁢</mo><mrow id="Thmproofx22.p1.8.m8.12.12.3.1.1" xref="Thmproofx22.p1.8.m8.12.12.3.1.2.cmml"><mo id="Thmproofx22.p1.8.m8.12.12.3.1.1.2" xref="Thmproofx22.p1.8.m8.12.12.3.1.2.cmml">(</mo><msup id="Thmproofx22.p1.8.m8.12.12.3.1.1.1" xref="Thmproofx22.p1.8.m8.12.12.3.1.1.1.cmml"><mi id="Thmproofx22.p1.8.m8.12.12.3.1.1.1.2" xref="Thmproofx22.p1.8.m8.12.12.3.1.1.1.2.cmml">ν</mi><mi id="Thmproofx22.p1.8.m8.12.12.3.1.1.1.3" xref="Thmproofx22.p1.8.m8.12.12.3.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx22.p1.8.m8.12.12.3.1.1.3" xref="Thmproofx22.p1.8.m8.12.12.3.1.2.cmml">,</mo><mi id="Thmproofx22.p1.8.m8.9.9" xref="Thmproofx22.p1.8.m8.9.9.cmml">α</mi><mo id="Thmproofx22.p1.8.m8.12.12.3.1.1.4" xref="Thmproofx22.p1.8.m8.12.12.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p1.8.m8.12b"><apply id="Thmproofx22.p1.8.m8.12.12.cmml" xref="Thmproofx22.p1.8.m8.12.12"><and id="Thmproofx22.p1.8.m8.12.12a.cmml" xref="Thmproofx22.p1.8.m8.12.12"></and><apply id="Thmproofx22.p1.8.m8.12.12b.cmml" xref="Thmproofx22.p1.8.m8.12.12"><eq id="Thmproofx22.p1.8.m8.12.12.5.cmml" xref="Thmproofx22.p1.8.m8.12.12.5"></eq><apply id="Thmproofx22.p1.8.m8.10.10.1.cmml" xref="Thmproofx22.p1.8.m8.10.10.1"><times id="Thmproofx22.p1.8.m8.10.10.1.2.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.2"></times><apply id="Thmproofx22.p1.8.m8.10.10.1.3.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.3"><csymbol cd="ambiguous" id="Thmproofx22.p1.8.m8.10.10.1.3.1.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.3">subscript</csymbol><ci id="Thmproofx22.p1.8.m8.10.10.1.3.2.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.3.2">𝐹</ci><ci id="Thmproofx22.p1.8.m8.10.10.1.3.3.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.3.3">𝑛</ci></apply><interval closure="open" id="Thmproofx22.p1.8.m8.10.10.1.1.2.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.1.1"><apply id="Thmproofx22.p1.8.m8.10.10.1.1.1.1.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.8.m8.10.10.1.1.1.1.1.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.1.1.1">superscript</csymbol><ci id="Thmproofx22.p1.8.m8.10.10.1.1.1.1.2.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.1.1.1.2">𝜈</ci><ci id="Thmproofx22.p1.8.m8.10.10.1.1.1.1.3.cmml" xref="Thmproofx22.p1.8.m8.10.10.1.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx22.p1.8.m8.7.7.cmml" xref="Thmproofx22.p1.8.m8.7.7">𝛼</ci></interval></apply><apply id="Thmproofx22.p1.8.m8.11.11.2.cmml" xref="Thmproofx22.p1.8.m8.11.11.2"><times id="Thmproofx22.p1.8.m8.11.11.2.2.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.2"></times><apply id="Thmproofx22.p1.8.m8.11.11.2.3.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p1.8.m8.11.11.2.3.1.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.3">subscript</csymbol><ci id="Thmproofx22.p1.8.m8.11.11.2.3.2.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.3.2">𝑓</ci><vector id="Thmproofx22.p1.8.m8.3.3.3.4.cmml" xref="Thmproofx22.p1.8.m8.3.3.3.5"><ci id="Thmproofx22.p1.8.m8.1.1.1.1.cmml" xref="Thmproofx22.p1.8.m8.1.1.1.1">𝑚</ci><ci id="Thmproofx22.p1.8.m8.2.2.2.2.cmml" xref="Thmproofx22.p1.8.m8.2.2.2.2">…</ci><ci id="Thmproofx22.p1.8.m8.3.3.3.3.cmml" xref="Thmproofx22.p1.8.m8.3.3.3.3">𝑚</ci></vector></apply><interval closure="open" id="Thmproofx22.p1.8.m8.11.11.2.1.2.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.1.1"><apply id="Thmproofx22.p1.8.m8.11.11.2.1.1.1.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.8.m8.11.11.2.1.1.1.1.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.1.1.1">superscript</csymbol><ci id="Thmproofx22.p1.8.m8.11.11.2.1.1.1.2.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.1.1.1.2">𝜈</ci><ci id="Thmproofx22.p1.8.m8.11.11.2.1.1.1.3.cmml" xref="Thmproofx22.p1.8.m8.11.11.2.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx22.p1.8.m8.8.8.cmml" xref="Thmproofx22.p1.8.m8.8.8">𝛼</ci></interval></apply></apply><apply id="Thmproofx22.p1.8.m8.12.12c.cmml" xref="Thmproofx22.p1.8.m8.12.12"><gt id="Thmproofx22.p1.8.m8.12.12.6.cmml" xref="Thmproofx22.p1.8.m8.12.12.6"></gt><share href="https://arxiv.org/html/2308.08047v3#Thmproofx22.p1.8.m8.11.11.2.cmml" id="Thmproofx22.p1.8.m8.12.12d.cmml" xref="Thmproofx22.p1.8.m8.12.12"></share><apply id="Thmproofx22.p1.8.m8.12.12.3.cmml" xref="Thmproofx22.p1.8.m8.12.12.3"><times id="Thmproofx22.p1.8.m8.12.12.3.2.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.2"></times><apply id="Thmproofx22.p1.8.m8.12.12.3.3.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3"><apply id="Thmproofx22.p1.8.m8.12.12.3.3.1.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.8.m8.12.12.3.3.1.1.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3.1">subscript</csymbol><max id="Thmproofx22.p1.8.m8.12.12.3.3.1.2.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3.1.2"></max><apply id="Thmproofx22.p1.8.m8.6.6.3.cmml" xref="Thmproofx22.p1.8.m8.6.6.3"><neq id="Thmproofx22.p1.8.m8.6.6.3.4.cmml" xref="Thmproofx22.p1.8.m8.6.6.3.4"></neq><ci id="Thmproofx22.p1.8.m8.6.6.3.5.cmml" xref="Thmproofx22.p1.8.m8.6.6.3.5">𝑑</ci><vector id="Thmproofx22.p1.8.m8.6.6.3.6.1.cmml" xref="Thmproofx22.p1.8.m8.6.6.3.6.2"><ci id="Thmproofx22.p1.8.m8.4.4.1.1.cmml" xref="Thmproofx22.p1.8.m8.4.4.1.1">𝑚</ci><ci id="Thmproofx22.p1.8.m8.5.5.2.2.cmml" xref="Thmproofx22.p1.8.m8.5.5.2.2">…</ci><ci id="Thmproofx22.p1.8.m8.6.6.3.3.cmml" xref="Thmproofx22.p1.8.m8.6.6.3.3">𝑚</ci></vector></apply></apply><apply id="Thmproofx22.p1.8.m8.12.12.3.3.2.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3.2"><csymbol cd="ambiguous" id="Thmproofx22.p1.8.m8.12.12.3.3.2.1.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3.2">subscript</csymbol><ci id="Thmproofx22.p1.8.m8.12.12.3.3.2.2.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3.2.2">𝑓</ci><ci id="Thmproofx22.p1.8.m8.12.12.3.3.2.3.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.3.2.3">𝑑</ci></apply></apply><interval closure="open" id="Thmproofx22.p1.8.m8.12.12.3.1.2.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.1.1"><apply id="Thmproofx22.p1.8.m8.12.12.3.1.1.1.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p1.8.m8.12.12.3.1.1.1.1.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.1.1.1">superscript</csymbol><ci id="Thmproofx22.p1.8.m8.12.12.3.1.1.1.2.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.1.1.1.2">𝜈</ci><ci id="Thmproofx22.p1.8.m8.12.12.3.1.1.1.3.cmml" xref="Thmproofx22.p1.8.m8.12.12.3.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx22.p1.8.m8.9.9.cmml" xref="Thmproofx22.p1.8.m8.9.9">𝛼</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p1.8.m8.12c">F_{n}(\nu^{n},\alpha)=f_{(m,...,m)}\left(\nu^{n},\alpha\right)>\max_{d\neq(m,.% ..,m)}f_{d}\left(\nu^{n},\alpha\right)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p1.8.m8.12d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_ν start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_ν start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) > roman_max start_POSTSUBSCRIPT italic_d ≠ ( italic_m , … , italic_m ) end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ν start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx22.p2"> <p class="ltx_p" id="Thmproofx22.p2.14"><math alttext="f_{(m,...,m)}" class="ltx_Math" display="inline" id="Thmproofx22.p2.1.m1.3"><semantics id="Thmproofx22.p2.1.m1.3a"><msub id="Thmproofx22.p2.1.m1.3.4" xref="Thmproofx22.p2.1.m1.3.4.cmml"><mi id="Thmproofx22.p2.1.m1.3.4.2" xref="Thmproofx22.p2.1.m1.3.4.2.cmml">f</mi><mrow id="Thmproofx22.p2.1.m1.3.3.3.5" xref="Thmproofx22.p2.1.m1.3.3.3.4.cmml"><mo id="Thmproofx22.p2.1.m1.3.3.3.5.1" stretchy="false" xref="Thmproofx22.p2.1.m1.3.3.3.4.cmml">(</mo><mi id="Thmproofx22.p2.1.m1.1.1.1.1" xref="Thmproofx22.p2.1.m1.1.1.1.1.cmml">m</mi><mo id="Thmproofx22.p2.1.m1.3.3.3.5.2" xref="Thmproofx22.p2.1.m1.3.3.3.4.cmml">,</mo><mi id="Thmproofx22.p2.1.m1.2.2.2.2" mathvariant="normal" xref="Thmproofx22.p2.1.m1.2.2.2.2.cmml">…</mi><mo id="Thmproofx22.p2.1.m1.3.3.3.5.3" xref="Thmproofx22.p2.1.m1.3.3.3.4.cmml">,</mo><mi id="Thmproofx22.p2.1.m1.3.3.3.3" xref="Thmproofx22.p2.1.m1.3.3.3.3.cmml">m</mi><mo id="Thmproofx22.p2.1.m1.3.3.3.5.4" stretchy="false" xref="Thmproofx22.p2.1.m1.3.3.3.4.cmml">)</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.1.m1.3b"><apply id="Thmproofx22.p2.1.m1.3.4.cmml" xref="Thmproofx22.p2.1.m1.3.4"><csymbol cd="ambiguous" id="Thmproofx22.p2.1.m1.3.4.1.cmml" xref="Thmproofx22.p2.1.m1.3.4">subscript</csymbol><ci id="Thmproofx22.p2.1.m1.3.4.2.cmml" xref="Thmproofx22.p2.1.m1.3.4.2">𝑓</ci><vector id="Thmproofx22.p2.1.m1.3.3.3.4.cmml" xref="Thmproofx22.p2.1.m1.3.3.3.5"><ci id="Thmproofx22.p2.1.m1.1.1.1.1.cmml" xref="Thmproofx22.p2.1.m1.1.1.1.1">𝑚</ci><ci id="Thmproofx22.p2.1.m1.2.2.2.2.cmml" xref="Thmproofx22.p2.1.m1.2.2.2.2">…</ci><ci id="Thmproofx22.p2.1.m1.3.3.3.3.cmml" xref="Thmproofx22.p2.1.m1.3.3.3.3">𝑚</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.1.m1.3c">f_{(m,...,m)}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.1.m1.3d">italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT</annotation></semantics></math> is the cost to the agents when the interceptors pick <math alttext="k" class="ltx_Math" display="inline" id="Thmproofx22.p2.2.m2.1"><semantics id="Thmproofx22.p2.2.m2.1a"><mi id="Thmproofx22.p2.2.m2.1.1" xref="Thmproofx22.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.2.m2.1b"><ci id="Thmproofx22.p2.2.m2.1.1.cmml" xref="Thmproofx22.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.2.m2.1d">italic_k</annotation></semantics></math> edges of <math alttext="p_{m}" class="ltx_Math" display="inline" id="Thmproofx22.p2.3.m3.1"><semantics id="Thmproofx22.p2.3.m3.1a"><msub id="Thmproofx22.p2.3.m3.1.1" xref="Thmproofx22.p2.3.m3.1.1.cmml"><mi id="Thmproofx22.p2.3.m3.1.1.2" xref="Thmproofx22.p2.3.m3.1.1.2.cmml">p</mi><mi id="Thmproofx22.p2.3.m3.1.1.3" xref="Thmproofx22.p2.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.3.m3.1b"><apply id="Thmproofx22.p2.3.m3.1.1.cmml" xref="Thmproofx22.p2.3.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.3.m3.1.1.1.cmml" xref="Thmproofx22.p2.3.m3.1.1">subscript</csymbol><ci id="Thmproofx22.p2.3.m3.1.1.2.cmml" xref="Thmproofx22.p2.3.m3.1.1.2">𝑝</ci><ci id="Thmproofx22.p2.3.m3.1.1.3.cmml" xref="Thmproofx22.p2.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.3.m3.1c">p_{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.3.m3.1d">italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="\nu" class="ltx_Math" display="inline" id="Thmproofx22.p2.4.m4.1"><semantics id="Thmproofx22.p2.4.m4.1a"><mi id="Thmproofx22.p2.4.m4.1.1" xref="Thmproofx22.p2.4.m4.1.1.cmml">ν</mi><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.4.m4.1b"><ci id="Thmproofx22.p2.4.m4.1.1.cmml" xref="Thmproofx22.p2.4.m4.1.1">𝜈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.4.m4.1c">\nu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.4.m4.1d">italic_ν</annotation></semantics></math> differs from <math alttext="U_{P_{st}}" class="ltx_Math" display="inline" id="Thmproofx22.p2.5.m5.1"><semantics id="Thmproofx22.p2.5.m5.1a"><msub id="Thmproofx22.p2.5.m5.1.1" xref="Thmproofx22.p2.5.m5.1.1.cmml"><mi id="Thmproofx22.p2.5.m5.1.1.2" xref="Thmproofx22.p2.5.m5.1.1.2.cmml">U</mi><msub id="Thmproofx22.p2.5.m5.1.1.3" xref="Thmproofx22.p2.5.m5.1.1.3.cmml"><mi id="Thmproofx22.p2.5.m5.1.1.3.2" xref="Thmproofx22.p2.5.m5.1.1.3.2.cmml">P</mi><mrow id="Thmproofx22.p2.5.m5.1.1.3.3" xref="Thmproofx22.p2.5.m5.1.1.3.3.cmml"><mi id="Thmproofx22.p2.5.m5.1.1.3.3.2" xref="Thmproofx22.p2.5.m5.1.1.3.3.2.cmml">s</mi><mo id="Thmproofx22.p2.5.m5.1.1.3.3.1" xref="Thmproofx22.p2.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="Thmproofx22.p2.5.m5.1.1.3.3.3" xref="Thmproofx22.p2.5.m5.1.1.3.3.3.cmml">t</mi></mrow></msub></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.5.m5.1b"><apply id="Thmproofx22.p2.5.m5.1.1.cmml" xref="Thmproofx22.p2.5.m5.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.5.m5.1.1.1.cmml" xref="Thmproofx22.p2.5.m5.1.1">subscript</csymbol><ci id="Thmproofx22.p2.5.m5.1.1.2.cmml" xref="Thmproofx22.p2.5.m5.1.1.2">𝑈</ci><apply id="Thmproofx22.p2.5.m5.1.1.3.cmml" xref="Thmproofx22.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.5.m5.1.1.3.1.cmml" xref="Thmproofx22.p2.5.m5.1.1.3">subscript</csymbol><ci id="Thmproofx22.p2.5.m5.1.1.3.2.cmml" xref="Thmproofx22.p2.5.m5.1.1.3.2">𝑃</ci><apply id="Thmproofx22.p2.5.m5.1.1.3.3.cmml" xref="Thmproofx22.p2.5.m5.1.1.3.3"><times id="Thmproofx22.p2.5.m5.1.1.3.3.1.cmml" xref="Thmproofx22.p2.5.m5.1.1.3.3.1"></times><ci id="Thmproofx22.p2.5.m5.1.1.3.3.2.cmml" xref="Thmproofx22.p2.5.m5.1.1.3.3.2">𝑠</ci><ci id="Thmproofx22.p2.5.m5.1.1.3.3.3.cmml" xref="Thmproofx22.p2.5.m5.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.5.m5.1c">U_{P_{st}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.5.m5.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> by shifting <math alttext="\varepsilon" class="ltx_Math" display="inline" id="Thmproofx22.p2.6.m6.1"><semantics id="Thmproofx22.p2.6.m6.1a"><mi id="Thmproofx22.p2.6.m6.1.1" xref="Thmproofx22.p2.6.m6.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.6.m6.1b"><ci id="Thmproofx22.p2.6.m6.1.1.cmml" xref="Thmproofx22.p2.6.m6.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.6.m6.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.6.m6.1d">italic_ε</annotation></semantics></math> weight from picking <math alttext="p_{m}" class="ltx_Math" display="inline" id="Thmproofx22.p2.7.m7.1"><semantics id="Thmproofx22.p2.7.m7.1a"><msub id="Thmproofx22.p2.7.m7.1.1" xref="Thmproofx22.p2.7.m7.1.1.cmml"><mi id="Thmproofx22.p2.7.m7.1.1.2" xref="Thmproofx22.p2.7.m7.1.1.2.cmml">p</mi><mi id="Thmproofx22.p2.7.m7.1.1.3" xref="Thmproofx22.p2.7.m7.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.7.m7.1b"><apply id="Thmproofx22.p2.7.m7.1.1.cmml" xref="Thmproofx22.p2.7.m7.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.7.m7.1.1.1.cmml" xref="Thmproofx22.p2.7.m7.1.1">subscript</csymbol><ci id="Thmproofx22.p2.7.m7.1.1.2.cmml" xref="Thmproofx22.p2.7.m7.1.1.2">𝑝</ci><ci id="Thmproofx22.p2.7.m7.1.1.3.cmml" xref="Thmproofx22.p2.7.m7.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.7.m7.1c">p_{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.7.m7.1d">italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> to picking <math alttext="p_{1}" class="ltx_Math" display="inline" id="Thmproofx22.p2.8.m8.1"><semantics id="Thmproofx22.p2.8.m8.1a"><msub id="Thmproofx22.p2.8.m8.1.1" xref="Thmproofx22.p2.8.m8.1.1.cmml"><mi id="Thmproofx22.p2.8.m8.1.1.2" xref="Thmproofx22.p2.8.m8.1.1.2.cmml">p</mi><mn id="Thmproofx22.p2.8.m8.1.1.3" xref="Thmproofx22.p2.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.8.m8.1b"><apply id="Thmproofx22.p2.8.m8.1.1.cmml" xref="Thmproofx22.p2.8.m8.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.8.m8.1.1.1.cmml" xref="Thmproofx22.p2.8.m8.1.1">subscript</csymbol><ci id="Thmproofx22.p2.8.m8.1.1.2.cmml" xref="Thmproofx22.p2.8.m8.1.1.2">𝑝</ci><cn id="Thmproofx22.p2.8.m8.1.1.3.cmml" type="integer" xref="Thmproofx22.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.8.m8.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.8.m8.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="|p_{1}|<|p_{m}|" class="ltx_Math" display="inline" id="Thmproofx22.p2.9.m9.2"><semantics id="Thmproofx22.p2.9.m9.2a"><mrow id="Thmproofx22.p2.9.m9.2.2" xref="Thmproofx22.p2.9.m9.2.2.cmml"><mrow id="Thmproofx22.p2.9.m9.1.1.1.1" xref="Thmproofx22.p2.9.m9.1.1.1.2.cmml"><mo id="Thmproofx22.p2.9.m9.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p2.9.m9.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx22.p2.9.m9.1.1.1.1.1" xref="Thmproofx22.p2.9.m9.1.1.1.1.1.cmml"><mi id="Thmproofx22.p2.9.m9.1.1.1.1.1.2" xref="Thmproofx22.p2.9.m9.1.1.1.1.1.2.cmml">p</mi><mn id="Thmproofx22.p2.9.m9.1.1.1.1.1.3" xref="Thmproofx22.p2.9.m9.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx22.p2.9.m9.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p2.9.m9.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx22.p2.9.m9.2.2.3" xref="Thmproofx22.p2.9.m9.2.2.3.cmml"><</mo><mrow id="Thmproofx22.p2.9.m9.2.2.2.1" xref="Thmproofx22.p2.9.m9.2.2.2.2.cmml"><mo id="Thmproofx22.p2.9.m9.2.2.2.1.2" stretchy="false" xref="Thmproofx22.p2.9.m9.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx22.p2.9.m9.2.2.2.1.1" xref="Thmproofx22.p2.9.m9.2.2.2.1.1.cmml"><mi id="Thmproofx22.p2.9.m9.2.2.2.1.1.2" xref="Thmproofx22.p2.9.m9.2.2.2.1.1.2.cmml">p</mi><mi id="Thmproofx22.p2.9.m9.2.2.2.1.1.3" xref="Thmproofx22.p2.9.m9.2.2.2.1.1.3.cmml">m</mi></msub><mo id="Thmproofx22.p2.9.m9.2.2.2.1.3" stretchy="false" xref="Thmproofx22.p2.9.m9.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.9.m9.2b"><apply id="Thmproofx22.p2.9.m9.2.2.cmml" xref="Thmproofx22.p2.9.m9.2.2"><lt id="Thmproofx22.p2.9.m9.2.2.3.cmml" xref="Thmproofx22.p2.9.m9.2.2.3"></lt><apply id="Thmproofx22.p2.9.m9.1.1.1.2.cmml" xref="Thmproofx22.p2.9.m9.1.1.1.1"><abs id="Thmproofx22.p2.9.m9.1.1.1.2.1.cmml" xref="Thmproofx22.p2.9.m9.1.1.1.1.2"></abs><apply id="Thmproofx22.p2.9.m9.1.1.1.1.1.cmml" xref="Thmproofx22.p2.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.9.m9.1.1.1.1.1.1.cmml" xref="Thmproofx22.p2.9.m9.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p2.9.m9.1.1.1.1.1.2.cmml" xref="Thmproofx22.p2.9.m9.1.1.1.1.1.2">𝑝</ci><cn id="Thmproofx22.p2.9.m9.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx22.p2.9.m9.1.1.1.1.1.3">1</cn></apply></apply><apply id="Thmproofx22.p2.9.m9.2.2.2.2.cmml" xref="Thmproofx22.p2.9.m9.2.2.2.1"><abs id="Thmproofx22.p2.9.m9.2.2.2.2.1.cmml" xref="Thmproofx22.p2.9.m9.2.2.2.1.2"></abs><apply id="Thmproofx22.p2.9.m9.2.2.2.1.1.cmml" xref="Thmproofx22.p2.9.m9.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.9.m9.2.2.2.1.1.1.cmml" xref="Thmproofx22.p2.9.m9.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx22.p2.9.m9.2.2.2.1.1.2.cmml" xref="Thmproofx22.p2.9.m9.2.2.2.1.1.2">𝑝</ci><ci id="Thmproofx22.p2.9.m9.2.2.2.1.1.3.cmml" xref="Thmproofx22.p2.9.m9.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.9.m9.2c">|p_{1}|<|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.9.m9.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math> by our assumption. So, when we go from <math alttext="U_{P_{st}}^{n}" class="ltx_Math" display="inline" id="Thmproofx22.p2.10.m10.1"><semantics id="Thmproofx22.p2.10.m10.1a"><msubsup id="Thmproofx22.p2.10.m10.1.1" xref="Thmproofx22.p2.10.m10.1.1.cmml"><mi id="Thmproofx22.p2.10.m10.1.1.2.2" xref="Thmproofx22.p2.10.m10.1.1.2.2.cmml">U</mi><msub id="Thmproofx22.p2.10.m10.1.1.2.3" xref="Thmproofx22.p2.10.m10.1.1.2.3.cmml"><mi id="Thmproofx22.p2.10.m10.1.1.2.3.2" xref="Thmproofx22.p2.10.m10.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx22.p2.10.m10.1.1.2.3.3" xref="Thmproofx22.p2.10.m10.1.1.2.3.3.cmml"><mi id="Thmproofx22.p2.10.m10.1.1.2.3.3.2" xref="Thmproofx22.p2.10.m10.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx22.p2.10.m10.1.1.2.3.3.1" xref="Thmproofx22.p2.10.m10.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx22.p2.10.m10.1.1.2.3.3.3" xref="Thmproofx22.p2.10.m10.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx22.p2.10.m10.1.1.3" xref="Thmproofx22.p2.10.m10.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.10.m10.1b"><apply id="Thmproofx22.p2.10.m10.1.1.cmml" xref="Thmproofx22.p2.10.m10.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.10.m10.1.1.1.cmml" xref="Thmproofx22.p2.10.m10.1.1">superscript</csymbol><apply id="Thmproofx22.p2.10.m10.1.1.2.cmml" xref="Thmproofx22.p2.10.m10.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.10.m10.1.1.2.1.cmml" xref="Thmproofx22.p2.10.m10.1.1">subscript</csymbol><ci id="Thmproofx22.p2.10.m10.1.1.2.2.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.2">𝑈</ci><apply id="Thmproofx22.p2.10.m10.1.1.2.3.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.10.m10.1.1.2.3.1.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.3">subscript</csymbol><ci id="Thmproofx22.p2.10.m10.1.1.2.3.2.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.3.2">𝑃</ci><apply id="Thmproofx22.p2.10.m10.1.1.2.3.3.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.3.3"><times id="Thmproofx22.p2.10.m10.1.1.2.3.3.1.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.3.3.1"></times><ci id="Thmproofx22.p2.10.m10.1.1.2.3.3.2.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx22.p2.10.m10.1.1.2.3.3.3.cmml" xref="Thmproofx22.p2.10.m10.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx22.p2.10.m10.1.1.3.cmml" xref="Thmproofx22.p2.10.m10.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.10.m10.1c">U_{P_{st}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.10.m10.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\nu^{n}" class="ltx_Math" display="inline" id="Thmproofx22.p2.11.m11.1"><semantics id="Thmproofx22.p2.11.m11.1a"><msup id="Thmproofx22.p2.11.m11.1.1" xref="Thmproofx22.p2.11.m11.1.1.cmml"><mi id="Thmproofx22.p2.11.m11.1.1.2" xref="Thmproofx22.p2.11.m11.1.1.2.cmml">ν</mi><mi id="Thmproofx22.p2.11.m11.1.1.3" xref="Thmproofx22.p2.11.m11.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.11.m11.1b"><apply id="Thmproofx22.p2.11.m11.1.1.cmml" xref="Thmproofx22.p2.11.m11.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.11.m11.1.1.1.cmml" xref="Thmproofx22.p2.11.m11.1.1">superscript</csymbol><ci id="Thmproofx22.p2.11.m11.1.1.2.cmml" xref="Thmproofx22.p2.11.m11.1.1.2">𝜈</ci><ci id="Thmproofx22.p2.11.m11.1.1.3.cmml" xref="Thmproofx22.p2.11.m11.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.11.m11.1c">\nu^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.11.m11.1d">italic_ν start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, we get <math alttext="f_{(m,...,m)}(U_{P_{st}}^{n},\alpha)>f_{(m,...,m)}(\nu^{n},\alpha)" class="ltx_Math" display="inline" id="Thmproofx22.p2.12.m12.10"><semantics id="Thmproofx22.p2.12.m12.10a"><mrow id="Thmproofx22.p2.12.m12.10.10" xref="Thmproofx22.p2.12.m12.10.10.cmml"><mrow id="Thmproofx22.p2.12.m12.9.9.1" xref="Thmproofx22.p2.12.m12.9.9.1.cmml"><msub id="Thmproofx22.p2.12.m12.9.9.1.3" xref="Thmproofx22.p2.12.m12.9.9.1.3.cmml"><mi id="Thmproofx22.p2.12.m12.9.9.1.3.2" xref="Thmproofx22.p2.12.m12.9.9.1.3.2.cmml">f</mi><mrow id="Thmproofx22.p2.12.m12.3.3.3.5" xref="Thmproofx22.p2.12.m12.3.3.3.4.cmml"><mo id="Thmproofx22.p2.12.m12.3.3.3.5.1" stretchy="false" xref="Thmproofx22.p2.12.m12.3.3.3.4.cmml">(</mo><mi id="Thmproofx22.p2.12.m12.1.1.1.1" xref="Thmproofx22.p2.12.m12.1.1.1.1.cmml">m</mi><mo id="Thmproofx22.p2.12.m12.3.3.3.5.2" xref="Thmproofx22.p2.12.m12.3.3.3.4.cmml">,</mo><mi id="Thmproofx22.p2.12.m12.2.2.2.2" mathvariant="normal" xref="Thmproofx22.p2.12.m12.2.2.2.2.cmml">…</mi><mo id="Thmproofx22.p2.12.m12.3.3.3.5.3" xref="Thmproofx22.p2.12.m12.3.3.3.4.cmml">,</mo><mi id="Thmproofx22.p2.12.m12.3.3.3.3" xref="Thmproofx22.p2.12.m12.3.3.3.3.cmml">m</mi><mo id="Thmproofx22.p2.12.m12.3.3.3.5.4" stretchy="false" xref="Thmproofx22.p2.12.m12.3.3.3.4.cmml">)</mo></mrow></msub><mo id="Thmproofx22.p2.12.m12.9.9.1.2" xref="Thmproofx22.p2.12.m12.9.9.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p2.12.m12.9.9.1.1.1" xref="Thmproofx22.p2.12.m12.9.9.1.1.2.cmml"><mo id="Thmproofx22.p2.12.m12.9.9.1.1.1.2" stretchy="false" xref="Thmproofx22.p2.12.m12.9.9.1.1.2.cmml">(</mo><msubsup id="Thmproofx22.p2.12.m12.9.9.1.1.1.1" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.cmml"><mi id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.2" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.cmml"><mi id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.2" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.cmml"><mi id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.2" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.1" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.3" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.3" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx22.p2.12.m12.9.9.1.1.1.3" xref="Thmproofx22.p2.12.m12.9.9.1.1.2.cmml">,</mo><mi id="Thmproofx22.p2.12.m12.7.7" xref="Thmproofx22.p2.12.m12.7.7.cmml">α</mi><mo id="Thmproofx22.p2.12.m12.9.9.1.1.1.4" stretchy="false" xref="Thmproofx22.p2.12.m12.9.9.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p2.12.m12.10.10.3" xref="Thmproofx22.p2.12.m12.10.10.3.cmml">></mo><mrow id="Thmproofx22.p2.12.m12.10.10.2" xref="Thmproofx22.p2.12.m12.10.10.2.cmml"><msub id="Thmproofx22.p2.12.m12.10.10.2.3" 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xref="Thmproofx22.p2.12.m12.10.10"><gt id="Thmproofx22.p2.12.m12.10.10.3.cmml" xref="Thmproofx22.p2.12.m12.10.10.3"></gt><apply id="Thmproofx22.p2.12.m12.9.9.1.cmml" xref="Thmproofx22.p2.12.m12.9.9.1"><times id="Thmproofx22.p2.12.m12.9.9.1.2.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.2"></times><apply id="Thmproofx22.p2.12.m12.9.9.1.3.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.12.m12.9.9.1.3.1.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.3">subscript</csymbol><ci id="Thmproofx22.p2.12.m12.9.9.1.3.2.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.3.2">𝑓</ci><vector id="Thmproofx22.p2.12.m12.3.3.3.4.cmml" xref="Thmproofx22.p2.12.m12.3.3.3.5"><ci id="Thmproofx22.p2.12.m12.1.1.1.1.cmml" xref="Thmproofx22.p2.12.m12.1.1.1.1">𝑚</ci><ci id="Thmproofx22.p2.12.m12.2.2.2.2.cmml" xref="Thmproofx22.p2.12.m12.2.2.2.2">…</ci><ci id="Thmproofx22.p2.12.m12.3.3.3.3.cmml" xref="Thmproofx22.p2.12.m12.3.3.3.3">𝑚</ci></vector></apply><interval closure="open" id="Thmproofx22.p2.12.m12.9.9.1.1.2.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1"><apply id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.1.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1">superscript</csymbol><apply id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.1.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.2.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.2">𝑈</ci><apply id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.1.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.2.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.2">𝑃</ci><apply id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3"><times id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.1.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.1"></times><ci id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.2.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.3.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx22.p2.12.m12.9.9.1.1.1.1.3.cmml" xref="Thmproofx22.p2.12.m12.9.9.1.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx22.p2.12.m12.7.7.cmml" xref="Thmproofx22.p2.12.m12.7.7">𝛼</ci></interval></apply><apply id="Thmproofx22.p2.12.m12.10.10.2.cmml" xref="Thmproofx22.p2.12.m12.10.10.2"><times id="Thmproofx22.p2.12.m12.10.10.2.2.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.2"></times><apply id="Thmproofx22.p2.12.m12.10.10.2.3.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.12.m12.10.10.2.3.1.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.3">subscript</csymbol><ci id="Thmproofx22.p2.12.m12.10.10.2.3.2.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.3.2">𝑓</ci><vector id="Thmproofx22.p2.12.m12.6.6.3.4.cmml" xref="Thmproofx22.p2.12.m12.6.6.3.5"><ci id="Thmproofx22.p2.12.m12.4.4.1.1.cmml" xref="Thmproofx22.p2.12.m12.4.4.1.1">𝑚</ci><ci id="Thmproofx22.p2.12.m12.5.5.2.2.cmml" xref="Thmproofx22.p2.12.m12.5.5.2.2">…</ci><ci id="Thmproofx22.p2.12.m12.6.6.3.3.cmml" xref="Thmproofx22.p2.12.m12.6.6.3.3">𝑚</ci></vector></apply><interval closure="open" id="Thmproofx22.p2.12.m12.10.10.2.1.2.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.1.1"><apply id="Thmproofx22.p2.12.m12.10.10.2.1.1.1.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.12.m12.10.10.2.1.1.1.1.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.1.1.1">superscript</csymbol><ci id="Thmproofx22.p2.12.m12.10.10.2.1.1.1.2.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.1.1.1.2">𝜈</ci><ci id="Thmproofx22.p2.12.m12.10.10.2.1.1.1.3.cmml" xref="Thmproofx22.p2.12.m12.10.10.2.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx22.p2.12.m12.8.8.cmml" xref="Thmproofx22.p2.12.m12.8.8">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.12.m12.10c">f_{(m,...,m)}(U_{P_{st}}^{n},\alpha)>f_{(m,...,m)}(\nu^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.12.m12.10d">italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) > italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_ν start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math>, hence <math alttext="F_{n}(U_{P_{st}}^{n},\alpha)>F_{n}(\nu^{n},\alpha)" class="ltx_Math" 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xref="Thmproofx22.p2.13.m13.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p2.13.m13.4.4.3" xref="Thmproofx22.p2.13.m13.4.4.3.cmml">></mo><mrow id="Thmproofx22.p2.13.m13.4.4.2" xref="Thmproofx22.p2.13.m13.4.4.2.cmml"><msub id="Thmproofx22.p2.13.m13.4.4.2.3" xref="Thmproofx22.p2.13.m13.4.4.2.3.cmml"><mi id="Thmproofx22.p2.13.m13.4.4.2.3.2" xref="Thmproofx22.p2.13.m13.4.4.2.3.2.cmml">F</mi><mi id="Thmproofx22.p2.13.m13.4.4.2.3.3" xref="Thmproofx22.p2.13.m13.4.4.2.3.3.cmml">n</mi></msub><mo id="Thmproofx22.p2.13.m13.4.4.2.2" xref="Thmproofx22.p2.13.m13.4.4.2.2.cmml">⁢</mo><mrow id="Thmproofx22.p2.13.m13.4.4.2.1.1" xref="Thmproofx22.p2.13.m13.4.4.2.1.2.cmml"><mo id="Thmproofx22.p2.13.m13.4.4.2.1.1.2" stretchy="false" xref="Thmproofx22.p2.13.m13.4.4.2.1.2.cmml">(</mo><msup id="Thmproofx22.p2.13.m13.4.4.2.1.1.1" xref="Thmproofx22.p2.13.m13.4.4.2.1.1.1.cmml"><mi id="Thmproofx22.p2.13.m13.4.4.2.1.1.1.2" xref="Thmproofx22.p2.13.m13.4.4.2.1.1.1.2.cmml">ν</mi><mi id="Thmproofx22.p2.13.m13.4.4.2.1.1.1.3" xref="Thmproofx22.p2.13.m13.4.4.2.1.1.1.3.cmml">n</mi></msup><mo id="Thmproofx22.p2.13.m13.4.4.2.1.1.3" xref="Thmproofx22.p2.13.m13.4.4.2.1.2.cmml">,</mo><mi id="Thmproofx22.p2.13.m13.2.2" xref="Thmproofx22.p2.13.m13.2.2.cmml">α</mi><mo id="Thmproofx22.p2.13.m13.4.4.2.1.1.4" stretchy="false" xref="Thmproofx22.p2.13.m13.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.13.m13.4b"><apply id="Thmproofx22.p2.13.m13.4.4.cmml" xref="Thmproofx22.p2.13.m13.4.4"><gt id="Thmproofx22.p2.13.m13.4.4.3.cmml" xref="Thmproofx22.p2.13.m13.4.4.3"></gt><apply id="Thmproofx22.p2.13.m13.3.3.1.cmml" xref="Thmproofx22.p2.13.m13.3.3.1"><times id="Thmproofx22.p2.13.m13.3.3.1.2.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.2"></times><apply id="Thmproofx22.p2.13.m13.3.3.1.3.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.13.m13.3.3.1.3.1.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.3">subscript</csymbol><ci id="Thmproofx22.p2.13.m13.3.3.1.3.2.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.3.2">𝐹</ci><ci id="Thmproofx22.p2.13.m13.3.3.1.3.3.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.3.3">𝑛</ci></apply><interval closure="open" id="Thmproofx22.p2.13.m13.3.3.1.1.2.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1"><apply id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.1.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1">superscript</csymbol><apply id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.1.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.2.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.2">𝑈</ci><apply id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.1.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3">subscript</csymbol><ci id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.2.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.2">𝑃</ci><apply id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3"><times id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3.1.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3.1"></times><ci id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3.2.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3.3.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx22.p2.13.m13.3.3.1.1.1.1.3.cmml" xref="Thmproofx22.p2.13.m13.3.3.1.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx22.p2.13.m13.1.1.cmml" xref="Thmproofx22.p2.13.m13.1.1">𝛼</ci></interval></apply><apply id="Thmproofx22.p2.13.m13.4.4.2.cmml" xref="Thmproofx22.p2.13.m13.4.4.2"><times id="Thmproofx22.p2.13.m13.4.4.2.2.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.2"></times><apply id="Thmproofx22.p2.13.m13.4.4.2.3.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.13.m13.4.4.2.3.1.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.3">subscript</csymbol><ci id="Thmproofx22.p2.13.m13.4.4.2.3.2.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.3.2">𝐹</ci><ci id="Thmproofx22.p2.13.m13.4.4.2.3.3.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.3.3">𝑛</ci></apply><interval closure="open" id="Thmproofx22.p2.13.m13.4.4.2.1.2.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.1.1"><apply id="Thmproofx22.p2.13.m13.4.4.2.1.1.1.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.13.m13.4.4.2.1.1.1.1.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.1.1.1">superscript</csymbol><ci id="Thmproofx22.p2.13.m13.4.4.2.1.1.1.2.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.1.1.1.2">𝜈</ci><ci id="Thmproofx22.p2.13.m13.4.4.2.1.1.1.3.cmml" xref="Thmproofx22.p2.13.m13.4.4.2.1.1.1.3">𝑛</ci></apply><ci id="Thmproofx22.p2.13.m13.2.2.cmml" xref="Thmproofx22.p2.13.m13.2.2">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.13.m13.4c">F_{n}(U_{P_{st}}^{n},\alpha)>F_{n}(\nu^{n},\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.13.m13.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) > italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_ν start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α )</annotation></semantics></math>, namely <math alttext="U_{P_{st}}^{n}" class="ltx_Math" display="inline" id="Thmproofx22.p2.14.m14.1"><semantics id="Thmproofx22.p2.14.m14.1a"><msubsup id="Thmproofx22.p2.14.m14.1.1" xref="Thmproofx22.p2.14.m14.1.1.cmml"><mi id="Thmproofx22.p2.14.m14.1.1.2.2" xref="Thmproofx22.p2.14.m14.1.1.2.2.cmml">U</mi><msub id="Thmproofx22.p2.14.m14.1.1.2.3" xref="Thmproofx22.p2.14.m14.1.1.2.3.cmml"><mi id="Thmproofx22.p2.14.m14.1.1.2.3.2" xref="Thmproofx22.p2.14.m14.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx22.p2.14.m14.1.1.2.3.3" xref="Thmproofx22.p2.14.m14.1.1.2.3.3.cmml"><mi id="Thmproofx22.p2.14.m14.1.1.2.3.3.2" xref="Thmproofx22.p2.14.m14.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx22.p2.14.m14.1.1.2.3.3.1" xref="Thmproofx22.p2.14.m14.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx22.p2.14.m14.1.1.2.3.3.3" xref="Thmproofx22.p2.14.m14.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx22.p2.14.m14.1.1.3" xref="Thmproofx22.p2.14.m14.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx22.p2.14.m14.1b"><apply id="Thmproofx22.p2.14.m14.1.1.cmml" xref="Thmproofx22.p2.14.m14.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.14.m14.1.1.1.cmml" xref="Thmproofx22.p2.14.m14.1.1">superscript</csymbol><apply id="Thmproofx22.p2.14.m14.1.1.2.cmml" xref="Thmproofx22.p2.14.m14.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p2.14.m14.1.1.2.1.cmml" xref="Thmproofx22.p2.14.m14.1.1">subscript</csymbol><ci id="Thmproofx22.p2.14.m14.1.1.2.2.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.2">𝑈</ci><apply id="Thmproofx22.p2.14.m14.1.1.2.3.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p2.14.m14.1.1.2.3.1.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.3">subscript</csymbol><ci id="Thmproofx22.p2.14.m14.1.1.2.3.2.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.3.2">𝑃</ci><apply id="Thmproofx22.p2.14.m14.1.1.2.3.3.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.3.3"><times id="Thmproofx22.p2.14.m14.1.1.2.3.3.1.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.3.3.1"></times><ci id="Thmproofx22.p2.14.m14.1.1.2.3.3.2.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.3.3.2">𝑠</ci><ci id="Thmproofx22.p2.14.m14.1.1.2.3.3.3.cmml" xref="Thmproofx22.p2.14.m14.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx22.p2.14.m14.1.1.3.cmml" xref="Thmproofx22.p2.14.m14.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p2.14.m14.1c">U_{P_{st}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p2.14.m14.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is not optimal for the agents.</p> </div> <div class="ltx_para" id="Thmproofx22.p3"> <p class="ltx_p" id="Thmproofx22.p3.5">Let <math alttext="m_{1}<i<m" class="ltx_Math" display="inline" id="Thmproofx22.p3.1.m1.1"><semantics id="Thmproofx22.p3.1.m1.1a"><mrow id="Thmproofx22.p3.1.m1.1.1" xref="Thmproofx22.p3.1.m1.1.1.cmml"><msub id="Thmproofx22.p3.1.m1.1.1.2" xref="Thmproofx22.p3.1.m1.1.1.2.cmml"><mi id="Thmproofx22.p3.1.m1.1.1.2.2" xref="Thmproofx22.p3.1.m1.1.1.2.2.cmml">m</mi><mn id="Thmproofx22.p3.1.m1.1.1.2.3" xref="Thmproofx22.p3.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="Thmproofx22.p3.1.m1.1.1.3" xref="Thmproofx22.p3.1.m1.1.1.3.cmml"><</mo><mi id="Thmproofx22.p3.1.m1.1.1.4" xref="Thmproofx22.p3.1.m1.1.1.4.cmml">i</mi><mo id="Thmproofx22.p3.1.m1.1.1.5" xref="Thmproofx22.p3.1.m1.1.1.5.cmml"><</mo><mi id="Thmproofx22.p3.1.m1.1.1.6" xref="Thmproofx22.p3.1.m1.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p3.1.m1.1b"><apply id="Thmproofx22.p3.1.m1.1.1.cmml" xref="Thmproofx22.p3.1.m1.1.1"><and id="Thmproofx22.p3.1.m1.1.1a.cmml" xref="Thmproofx22.p3.1.m1.1.1"></and><apply id="Thmproofx22.p3.1.m1.1.1b.cmml" xref="Thmproofx22.p3.1.m1.1.1"><lt id="Thmproofx22.p3.1.m1.1.1.3.cmml" xref="Thmproofx22.p3.1.m1.1.1.3"></lt><apply id="Thmproofx22.p3.1.m1.1.1.2.cmml" xref="Thmproofx22.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx22.p3.1.m1.1.1.2.1.cmml" xref="Thmproofx22.p3.1.m1.1.1.2">subscript</csymbol><ci id="Thmproofx22.p3.1.m1.1.1.2.2.cmml" xref="Thmproofx22.p3.1.m1.1.1.2.2">𝑚</ci><cn id="Thmproofx22.p3.1.m1.1.1.2.3.cmml" type="integer" xref="Thmproofx22.p3.1.m1.1.1.2.3">1</cn></apply><ci id="Thmproofx22.p3.1.m1.1.1.4.cmml" xref="Thmproofx22.p3.1.m1.1.1.4">𝑖</ci></apply><apply id="Thmproofx22.p3.1.m1.1.1c.cmml" xref="Thmproofx22.p3.1.m1.1.1"><lt id="Thmproofx22.p3.1.m1.1.1.5.cmml" xref="Thmproofx22.p3.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2308.08047v3#Thmproofx22.p3.1.m1.1.1.4.cmml" id="Thmproofx22.p3.1.m1.1.1d.cmml" xref="Thmproofx22.p3.1.m1.1.1"></share><ci id="Thmproofx22.p3.1.m1.1.1.6.cmml" xref="Thmproofx22.p3.1.m1.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.1.m1.1c">m_{1}<i<m</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.1.m1.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_i < italic_m</annotation></semantics></math>, and consider <math alttext="G_{i},U_{P_{i}}^{n}" class="ltx_Math" display="inline" id="Thmproofx22.p3.2.m2.2"><semantics id="Thmproofx22.p3.2.m2.2a"><mrow id="Thmproofx22.p3.2.m2.2.2.2" xref="Thmproofx22.p3.2.m2.2.2.3.cmml"><msub id="Thmproofx22.p3.2.m2.1.1.1.1" xref="Thmproofx22.p3.2.m2.1.1.1.1.cmml"><mi id="Thmproofx22.p3.2.m2.1.1.1.1.2" xref="Thmproofx22.p3.2.m2.1.1.1.1.2.cmml">G</mi><mi id="Thmproofx22.p3.2.m2.1.1.1.1.3" xref="Thmproofx22.p3.2.m2.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx22.p3.2.m2.2.2.2.3" xref="Thmproofx22.p3.2.m2.2.2.3.cmml">,</mo><msubsup id="Thmproofx22.p3.2.m2.2.2.2.2" xref="Thmproofx22.p3.2.m2.2.2.2.2.cmml"><mi id="Thmproofx22.p3.2.m2.2.2.2.2.2.2" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.2.cmml">U</mi><msub id="Thmproofx22.p3.2.m2.2.2.2.2.2.3" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.3.cmml"><mi id="Thmproofx22.p3.2.m2.2.2.2.2.2.3.2" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.3.2.cmml">P</mi><mi id="Thmproofx22.p3.2.m2.2.2.2.2.2.3.3" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.3.3.cmml">i</mi></msub><mi id="Thmproofx22.p3.2.m2.2.2.2.2.3" xref="Thmproofx22.p3.2.m2.2.2.2.2.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p3.2.m2.2b"><list id="Thmproofx22.p3.2.m2.2.2.3.cmml" xref="Thmproofx22.p3.2.m2.2.2.2"><apply id="Thmproofx22.p3.2.m2.1.1.1.1.cmml" xref="Thmproofx22.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.2.m2.1.1.1.1.1.cmml" xref="Thmproofx22.p3.2.m2.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p3.2.m2.1.1.1.1.2.cmml" xref="Thmproofx22.p3.2.m2.1.1.1.1.2">𝐺</ci><ci id="Thmproofx22.p3.2.m2.1.1.1.1.3.cmml" xref="Thmproofx22.p3.2.m2.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx22.p3.2.m2.2.2.2.2.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx22.p3.2.m2.2.2.2.2.1.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2">superscript</csymbol><apply id="Thmproofx22.p3.2.m2.2.2.2.2.2.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx22.p3.2.m2.2.2.2.2.2.1.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2">subscript</csymbol><ci id="Thmproofx22.p3.2.m2.2.2.2.2.2.2.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.2">𝑈</ci><apply id="Thmproofx22.p3.2.m2.2.2.2.2.2.3.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p3.2.m2.2.2.2.2.2.3.1.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.3">subscript</csymbol><ci id="Thmproofx22.p3.2.m2.2.2.2.2.2.3.2.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.3.2">𝑃</ci><ci id="Thmproofx22.p3.2.m2.2.2.2.2.2.3.3.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2.2.3.3">𝑖</ci></apply></apply><ci id="Thmproofx22.p3.2.m2.2.2.2.2.3.cmml" xref="Thmproofx22.p3.2.m2.2.2.2.2.3">𝑛</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.2.m2.2c">G_{i},U_{P_{i}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.2.m2.2d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. <math alttext="|p_{i}|\leq|p_{m}|" class="ltx_Math" display="inline" id="Thmproofx22.p3.3.m3.2"><semantics id="Thmproofx22.p3.3.m3.2a"><mrow id="Thmproofx22.p3.3.m3.2.2" xref="Thmproofx22.p3.3.m3.2.2.cmml"><mrow id="Thmproofx22.p3.3.m3.1.1.1.1" xref="Thmproofx22.p3.3.m3.1.1.1.2.cmml"><mo id="Thmproofx22.p3.3.m3.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p3.3.m3.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx22.p3.3.m3.1.1.1.1.1" xref="Thmproofx22.p3.3.m3.1.1.1.1.1.cmml"><mi id="Thmproofx22.p3.3.m3.1.1.1.1.1.2" xref="Thmproofx22.p3.3.m3.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx22.p3.3.m3.1.1.1.1.1.3" xref="Thmproofx22.p3.3.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx22.p3.3.m3.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p3.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx22.p3.3.m3.2.2.3" xref="Thmproofx22.p3.3.m3.2.2.3.cmml">≤</mo><mrow id="Thmproofx22.p3.3.m3.2.2.2.1" xref="Thmproofx22.p3.3.m3.2.2.2.2.cmml"><mo id="Thmproofx22.p3.3.m3.2.2.2.1.2" stretchy="false" xref="Thmproofx22.p3.3.m3.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx22.p3.3.m3.2.2.2.1.1" xref="Thmproofx22.p3.3.m3.2.2.2.1.1.cmml"><mi id="Thmproofx22.p3.3.m3.2.2.2.1.1.2" xref="Thmproofx22.p3.3.m3.2.2.2.1.1.2.cmml">p</mi><mi id="Thmproofx22.p3.3.m3.2.2.2.1.1.3" xref="Thmproofx22.p3.3.m3.2.2.2.1.1.3.cmml">m</mi></msub><mo id="Thmproofx22.p3.3.m3.2.2.2.1.3" stretchy="false" xref="Thmproofx22.p3.3.m3.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p3.3.m3.2b"><apply id="Thmproofx22.p3.3.m3.2.2.cmml" xref="Thmproofx22.p3.3.m3.2.2"><leq id="Thmproofx22.p3.3.m3.2.2.3.cmml" xref="Thmproofx22.p3.3.m3.2.2.3"></leq><apply id="Thmproofx22.p3.3.m3.1.1.1.2.cmml" xref="Thmproofx22.p3.3.m3.1.1.1.1"><abs id="Thmproofx22.p3.3.m3.1.1.1.2.1.cmml" xref="Thmproofx22.p3.3.m3.1.1.1.1.2"></abs><apply id="Thmproofx22.p3.3.m3.1.1.1.1.1.cmml" xref="Thmproofx22.p3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.3.m3.1.1.1.1.1.1.cmml" xref="Thmproofx22.p3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p3.3.m3.1.1.1.1.1.2.cmml" xref="Thmproofx22.p3.3.m3.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx22.p3.3.m3.1.1.1.1.1.3.cmml" xref="Thmproofx22.p3.3.m3.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx22.p3.3.m3.2.2.2.2.cmml" xref="Thmproofx22.p3.3.m3.2.2.2.1"><abs id="Thmproofx22.p3.3.m3.2.2.2.2.1.cmml" xref="Thmproofx22.p3.3.m3.2.2.2.1.2"></abs><apply id="Thmproofx22.p3.3.m3.2.2.2.1.1.cmml" xref="Thmproofx22.p3.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.3.m3.2.2.2.1.1.1.cmml" xref="Thmproofx22.p3.3.m3.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx22.p3.3.m3.2.2.2.1.1.2.cmml" xref="Thmproofx22.p3.3.m3.2.2.2.1.1.2">𝑝</ci><ci id="Thmproofx22.p3.3.m3.2.2.2.1.1.3.cmml" xref="Thmproofx22.p3.3.m3.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.3.m3.2c">|p_{i}|\leq|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.3.m3.2d">| italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ≤ | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math>, so <math alttext="\alpha>\frac{|p_{i}|-|p_{1}|}{k}" class="ltx_Math" display="inline" id="Thmproofx22.p3.4.m4.2"><semantics id="Thmproofx22.p3.4.m4.2a"><mrow id="Thmproofx22.p3.4.m4.2.3" xref="Thmproofx22.p3.4.m4.2.3.cmml"><mi id="Thmproofx22.p3.4.m4.2.3.2" xref="Thmproofx22.p3.4.m4.2.3.2.cmml">α</mi><mo id="Thmproofx22.p3.4.m4.2.3.1" xref="Thmproofx22.p3.4.m4.2.3.1.cmml">></mo><mfrac id="Thmproofx22.p3.4.m4.2.2" xref="Thmproofx22.p3.4.m4.2.2.cmml"><mrow id="Thmproofx22.p3.4.m4.2.2.2" xref="Thmproofx22.p3.4.m4.2.2.2.cmml"><mrow id="Thmproofx22.p3.4.m4.1.1.1.1.1" xref="Thmproofx22.p3.4.m4.1.1.1.1.2.cmml"><mo id="Thmproofx22.p3.4.m4.1.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p3.4.m4.1.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx22.p3.4.m4.1.1.1.1.1.1" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmproofx22.p3.4.m4.1.1.1.1.1.1.2" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx22.p3.4.m4.1.1.1.1.1.1.3" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx22.p3.4.m4.1.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p3.4.m4.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx22.p3.4.m4.2.2.2.3" xref="Thmproofx22.p3.4.m4.2.2.2.3.cmml">−</mo><mrow id="Thmproofx22.p3.4.m4.2.2.2.2.1" xref="Thmproofx22.p3.4.m4.2.2.2.2.2.cmml"><mo id="Thmproofx22.p3.4.m4.2.2.2.2.1.2" stretchy="false" xref="Thmproofx22.p3.4.m4.2.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx22.p3.4.m4.2.2.2.2.1.1" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.1.cmml"><mi id="Thmproofx22.p3.4.m4.2.2.2.2.1.1.2" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.1.2.cmml">p</mi><mn id="Thmproofx22.p3.4.m4.2.2.2.2.1.1.3" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproofx22.p3.4.m4.2.2.2.2.1.3" stretchy="false" xref="Thmproofx22.p3.4.m4.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mi id="Thmproofx22.p3.4.m4.2.2.4" xref="Thmproofx22.p3.4.m4.2.2.4.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p3.4.m4.2b"><apply id="Thmproofx22.p3.4.m4.2.3.cmml" xref="Thmproofx22.p3.4.m4.2.3"><gt id="Thmproofx22.p3.4.m4.2.3.1.cmml" xref="Thmproofx22.p3.4.m4.2.3.1"></gt><ci id="Thmproofx22.p3.4.m4.2.3.2.cmml" xref="Thmproofx22.p3.4.m4.2.3.2">𝛼</ci><apply id="Thmproofx22.p3.4.m4.2.2.cmml" xref="Thmproofx22.p3.4.m4.2.2"><divide id="Thmproofx22.p3.4.m4.2.2.3.cmml" xref="Thmproofx22.p3.4.m4.2.2"></divide><apply id="Thmproofx22.p3.4.m4.2.2.2.cmml" xref="Thmproofx22.p3.4.m4.2.2.2"><minus id="Thmproofx22.p3.4.m4.2.2.2.3.cmml" xref="Thmproofx22.p3.4.m4.2.2.2.3"></minus><apply id="Thmproofx22.p3.4.m4.1.1.1.1.2.cmml" xref="Thmproofx22.p3.4.m4.1.1.1.1.1"><abs id="Thmproofx22.p3.4.m4.1.1.1.1.2.1.cmml" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.2"></abs><apply id="Thmproofx22.p3.4.m4.1.1.1.1.1.1.cmml" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx22.p3.4.m4.1.1.1.1.1.1.3.cmml" xref="Thmproofx22.p3.4.m4.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx22.p3.4.m4.2.2.2.2.2.cmml" xref="Thmproofx22.p3.4.m4.2.2.2.2.1"><abs id="Thmproofx22.p3.4.m4.2.2.2.2.2.1.cmml" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.2"></abs><apply id="Thmproofx22.p3.4.m4.2.2.2.2.1.1.cmml" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.4.m4.2.2.2.2.1.1.1.cmml" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx22.p3.4.m4.2.2.2.2.1.1.2.cmml" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.1.2">𝑝</ci><cn id="Thmproofx22.p3.4.m4.2.2.2.2.1.1.3.cmml" type="integer" xref="Thmproofx22.p3.4.m4.2.2.2.2.1.1.3">1</cn></apply></apply></apply><ci id="Thmproofx22.p3.4.m4.2.2.4.cmml" xref="Thmproofx22.p3.4.m4.2.2.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.4.m4.2c">\alpha>\frac{|p_{i}|-|p_{1}|}{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.4.m4.2d">italic_α > divide start_ARG | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>. Now there are 2 cases. In the first case, <math alttext="\min_{\sigma\in(\Delta(P_{i}))^{n}}F_{n}(\sigma,\alpha)=F_{n}(U_{P_{m_{1}}}^{n% },\alpha)=|p_{1}|+\frac{\mathcal{E}_{m_{1},n}}{m_{1}}k\alpha" class="ltx_Math" display="inline" id="Thmproofx22.p3.5.m5.8"><semantics id="Thmproofx22.p3.5.m5.8a"><mrow id="Thmproofx22.p3.5.m5.8.8" xref="Thmproofx22.p3.5.m5.8.8.cmml"><mrow id="Thmproofx22.p3.5.m5.8.8.4" xref="Thmproofx22.p3.5.m5.8.8.4.cmml"><mrow id="Thmproofx22.p3.5.m5.8.8.4.2" xref="Thmproofx22.p3.5.m5.8.8.4.2.cmml"><msub id="Thmproofx22.p3.5.m5.8.8.4.2.1" xref="Thmproofx22.p3.5.m5.8.8.4.2.1.cmml"><mi id="Thmproofx22.p3.5.m5.8.8.4.2.1.2" xref="Thmproofx22.p3.5.m5.8.8.4.2.1.2.cmml">min</mi><mrow id="Thmproofx22.p3.5.m5.1.1.1" xref="Thmproofx22.p3.5.m5.1.1.1.cmml"><mi id="Thmproofx22.p3.5.m5.1.1.1.3" xref="Thmproofx22.p3.5.m5.1.1.1.3.cmml">σ</mi><mo id="Thmproofx22.p3.5.m5.1.1.1.2" xref="Thmproofx22.p3.5.m5.1.1.1.2.cmml">∈</mo><msup id="Thmproofx22.p3.5.m5.1.1.1.1" xref="Thmproofx22.p3.5.m5.1.1.1.1.cmml"><mrow id="Thmproofx22.p3.5.m5.1.1.1.1.1.1" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.3" mathvariant="normal" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.2" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.2" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mi id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.3" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p3.5.m5.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p3.5.m5.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx22.p3.5.m5.1.1.1.1.3" xref="Thmproofx22.p3.5.m5.1.1.1.1.3.cmml">n</mi></msup></mrow></msub><mo id="Thmproofx22.p3.5.m5.8.8.4.2a" lspace="0.167em" xref="Thmproofx22.p3.5.m5.8.8.4.2.cmml">⁡</mo><msub id="Thmproofx22.p3.5.m5.8.8.4.2.2" xref="Thmproofx22.p3.5.m5.8.8.4.2.2.cmml"><mi id="Thmproofx22.p3.5.m5.8.8.4.2.2.2" xref="Thmproofx22.p3.5.m5.8.8.4.2.2.2.cmml">F</mi><mi id="Thmproofx22.p3.5.m5.8.8.4.2.2.3" xref="Thmproofx22.p3.5.m5.8.8.4.2.2.3.cmml">n</mi></msub></mrow><mo id="Thmproofx22.p3.5.m5.8.8.4.1" xref="Thmproofx22.p3.5.m5.8.8.4.1.cmml">⁢</mo><mrow id="Thmproofx22.p3.5.m5.8.8.4.3.2" xref="Thmproofx22.p3.5.m5.8.8.4.3.1.cmml"><mo id="Thmproofx22.p3.5.m5.8.8.4.3.2.1" stretchy="false" xref="Thmproofx22.p3.5.m5.8.8.4.3.1.cmml">(</mo><mi id="Thmproofx22.p3.5.m5.4.4" xref="Thmproofx22.p3.5.m5.4.4.cmml">σ</mi><mo id="Thmproofx22.p3.5.m5.8.8.4.3.2.2" xref="Thmproofx22.p3.5.m5.8.8.4.3.1.cmml">,</mo><mi id="Thmproofx22.p3.5.m5.5.5" xref="Thmproofx22.p3.5.m5.5.5.cmml">α</mi><mo id="Thmproofx22.p3.5.m5.8.8.4.3.2.3" stretchy="false" xref="Thmproofx22.p3.5.m5.8.8.4.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p3.5.m5.8.8.5" xref="Thmproofx22.p3.5.m5.8.8.5.cmml">=</mo><mrow id="Thmproofx22.p3.5.m5.7.7.1" xref="Thmproofx22.p3.5.m5.7.7.1.cmml"><msub id="Thmproofx22.p3.5.m5.7.7.1.3" xref="Thmproofx22.p3.5.m5.7.7.1.3.cmml"><mi id="Thmproofx22.p3.5.m5.7.7.1.3.2" xref="Thmproofx22.p3.5.m5.7.7.1.3.2.cmml">F</mi><mi id="Thmproofx22.p3.5.m5.7.7.1.3.3" xref="Thmproofx22.p3.5.m5.7.7.1.3.3.cmml">n</mi></msub><mo id="Thmproofx22.p3.5.m5.7.7.1.2" xref="Thmproofx22.p3.5.m5.7.7.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p3.5.m5.7.7.1.1.1" xref="Thmproofx22.p3.5.m5.7.7.1.1.2.cmml"><mo id="Thmproofx22.p3.5.m5.7.7.1.1.1.2" stretchy="false" xref="Thmproofx22.p3.5.m5.7.7.1.1.2.cmml">(</mo><msubsup id="Thmproofx22.p3.5.m5.7.7.1.1.1.1" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.cmml"><mi id="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.2" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.cmml"><mi id="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.2" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.2.cmml">P</mi><msub id="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.3" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.3.cmml"><mi id="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.3.2" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.3.2.cmml">m</mi><mn id="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.3.3" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.2.3.3.3.cmml">1</mn></msub></msub><mi id="Thmproofx22.p3.5.m5.7.7.1.1.1.1.3" xref="Thmproofx22.p3.5.m5.7.7.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx22.p3.5.m5.7.7.1.1.1.3" xref="Thmproofx22.p3.5.m5.7.7.1.1.2.cmml">,</mo><mi id="Thmproofx22.p3.5.m5.6.6" xref="Thmproofx22.p3.5.m5.6.6.cmml">α</mi><mo id="Thmproofx22.p3.5.m5.7.7.1.1.1.4" stretchy="false" xref="Thmproofx22.p3.5.m5.7.7.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p3.5.m5.8.8.6" xref="Thmproofx22.p3.5.m5.8.8.6.cmml">=</mo><mrow id="Thmproofx22.p3.5.m5.8.8.2" xref="Thmproofx22.p3.5.m5.8.8.2.cmml"><mrow id="Thmproofx22.p3.5.m5.8.8.2.1.1" xref="Thmproofx22.p3.5.m5.8.8.2.1.2.cmml"><mo id="Thmproofx22.p3.5.m5.8.8.2.1.1.2" stretchy="false" xref="Thmproofx22.p3.5.m5.8.8.2.1.2.1.cmml">|</mo><msub id="Thmproofx22.p3.5.m5.8.8.2.1.1.1" xref="Thmproofx22.p3.5.m5.8.8.2.1.1.1.cmml"><mi id="Thmproofx22.p3.5.m5.8.8.2.1.1.1.2" xref="Thmproofx22.p3.5.m5.8.8.2.1.1.1.2.cmml">p</mi><mn id="Thmproofx22.p3.5.m5.8.8.2.1.1.1.3" xref="Thmproofx22.p3.5.m5.8.8.2.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx22.p3.5.m5.8.8.2.1.1.3" stretchy="false" xref="Thmproofx22.p3.5.m5.8.8.2.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx22.p3.5.m5.8.8.2.2" xref="Thmproofx22.p3.5.m5.8.8.2.2.cmml">+</mo><mrow id="Thmproofx22.p3.5.m5.8.8.2.3" xref="Thmproofx22.p3.5.m5.8.8.2.3.cmml"><mfrac id="Thmproofx22.p3.5.m5.3.3" xref="Thmproofx22.p3.5.m5.3.3.cmml"><msub id="Thmproofx22.p3.5.m5.3.3.2" xref="Thmproofx22.p3.5.m5.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx22.p3.5.m5.3.3.2.4" xref="Thmproofx22.p3.5.m5.3.3.2.4.cmml">ℰ</mi><mrow id="Thmproofx22.p3.5.m5.3.3.2.2.2.2" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.3.cmml"><msub id="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.cmml"><mi id="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.2" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.2.cmml">m</mi><mn id="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.3" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.3.cmml">1</mn></msub><mo id="Thmproofx22.p3.5.m5.3.3.2.2.2.2.2" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.3.cmml">,</mo><mi id="Thmproofx22.p3.5.m5.2.2.1.1.1.1" xref="Thmproofx22.p3.5.m5.2.2.1.1.1.1.cmml">n</mi></mrow></msub><msub id="Thmproofx22.p3.5.m5.3.3.4" xref="Thmproofx22.p3.5.m5.3.3.4.cmml"><mi id="Thmproofx22.p3.5.m5.3.3.4.2" xref="Thmproofx22.p3.5.m5.3.3.4.2.cmml">m</mi><mn id="Thmproofx22.p3.5.m5.3.3.4.3" xref="Thmproofx22.p3.5.m5.3.3.4.3.cmml">1</mn></msub></mfrac><mo id="Thmproofx22.p3.5.m5.8.8.2.3.1" xref="Thmproofx22.p3.5.m5.8.8.2.3.1.cmml">⁢</mo><mi id="Thmproofx22.p3.5.m5.8.8.2.3.2" xref="Thmproofx22.p3.5.m5.8.8.2.3.2.cmml">k</mi><mo id="Thmproofx22.p3.5.m5.8.8.2.3.1a" xref="Thmproofx22.p3.5.m5.8.8.2.3.1.cmml">⁢</mo><mi id="Thmproofx22.p3.5.m5.8.8.2.3.3" xref="Thmproofx22.p3.5.m5.8.8.2.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p3.5.m5.8b"><apply id="Thmproofx22.p3.5.m5.8.8.cmml" xref="Thmproofx22.p3.5.m5.8.8"><and id="Thmproofx22.p3.5.m5.8.8a.cmml" 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id="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.1.cmml" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1">subscript</csymbol><ci id="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.2.cmml" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.2">𝑚</ci><cn id="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.3.cmml" type="integer" xref="Thmproofx22.p3.5.m5.3.3.2.2.2.2.1.3">1</cn></apply><ci id="Thmproofx22.p3.5.m5.2.2.1.1.1.1.cmml" xref="Thmproofx22.p3.5.m5.2.2.1.1.1.1">𝑛</ci></list></apply><apply id="Thmproofx22.p3.5.m5.3.3.4.cmml" xref="Thmproofx22.p3.5.m5.3.3.4"><csymbol cd="ambiguous" id="Thmproofx22.p3.5.m5.3.3.4.1.cmml" xref="Thmproofx22.p3.5.m5.3.3.4">subscript</csymbol><ci id="Thmproofx22.p3.5.m5.3.3.4.2.cmml" xref="Thmproofx22.p3.5.m5.3.3.4.2">𝑚</ci><cn id="Thmproofx22.p3.5.m5.3.3.4.3.cmml" type="integer" xref="Thmproofx22.p3.5.m5.3.3.4.3">1</cn></apply></apply><ci id="Thmproofx22.p3.5.m5.8.8.2.3.2.cmml" xref="Thmproofx22.p3.5.m5.8.8.2.3.2">𝑘</ci><ci id="Thmproofx22.p3.5.m5.8.8.2.3.3.cmml" xref="Thmproofx22.p3.5.m5.8.8.2.3.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.5.m5.8c">\min_{\sigma\in(\Delta(P_{i}))^{n}}F_{n}(\sigma,\alpha)=F_{n}(U_{P_{m_{1}}}^{n% },\alpha)=|p_{1}|+\frac{\mathcal{E}_{m_{1},n}}{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.5.m5.8d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math>. Then, in particular</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx58"> <tbody id="Pt0.A4.Ex103"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min_{\sigma\in\mathcal{U}_{n}\cap(\Delta(P_{i}))^{n}}F_{n}(% \sigma,\alpha)\geq|p_{1}|+\frac{\mathcal{E}_{m_{1},n}}{m_{1}}k\alpha>\min_{% \sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex103.m1.9"><semantics id="Pt0.A4.Ex103.m1.9a"><mrow id="Pt0.A4.Ex103.m1.9.9" xref="Pt0.A4.Ex103.m1.9.9.cmml"><mrow id="Pt0.A4.Ex103.m1.9.9.3" xref="Pt0.A4.Ex103.m1.9.9.3.cmml"><mrow id="Pt0.A4.Ex103.m1.9.9.3.2" xref="Pt0.A4.Ex103.m1.9.9.3.2.cmml"><munder id="Pt0.A4.Ex103.m1.9.9.3.2.1" xref="Pt0.A4.Ex103.m1.9.9.3.2.1.cmml"><mi id="Pt0.A4.Ex103.m1.9.9.3.2.1.2" xref="Pt0.A4.Ex103.m1.9.9.3.2.1.2.cmml">min</mi><mrow id="Pt0.A4.Ex103.m1.1.1.1" 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end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) ≥ | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α > roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx22.p3.9">by the theorem statement. 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xref="Thmproofx22.p3.6.m1.6.6"><lt id="Thmproofx22.p3.6.m1.6.6.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.2"></lt><apply id="Thmproofx22.p3.6.m1.6.6.3.cmml" xref="Thmproofx22.p3.6.m1.6.6.3"><times id="Thmproofx22.p3.6.m1.6.6.3.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.1"></times><apply id="Thmproofx22.p3.6.m1.6.6.3.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2"><apply id="Thmproofx22.p3.6.m1.6.6.3.2.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.6.6.3.2.1.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2.1">subscript</csymbol><min id="Thmproofx22.p3.6.m1.6.6.3.2.1.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2.1.2"></min><apply id="Thmproofx22.p3.6.m1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.1.1.1"><in id="Thmproofx22.p3.6.m1.1.1.1.2.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.2"></in><ci id="Thmproofx22.p3.6.m1.1.1.1.3.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.3">𝜎</ci><apply id="Thmproofx22.p3.6.m1.1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.1.1.1.1.2.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1">superscript</csymbol><apply id="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.1.1"><times id="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.2"></times><ci id="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.3">Δ</ci><apply id="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><ci id="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><ci id="Thmproofx22.p3.6.m1.1.1.1.1.3.cmml" xref="Thmproofx22.p3.6.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmproofx22.p3.6.m1.6.6.3.2.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2.2"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.6.6.3.2.2.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2.2">subscript</csymbol><ci id="Thmproofx22.p3.6.m1.6.6.3.2.2.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2.2.2">𝐹</ci><ci id="Thmproofx22.p3.6.m1.6.6.3.2.2.3.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmproofx22.p3.6.m1.6.6.3.3.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.3.3.2"><ci id="Thmproofx22.p3.6.m1.4.4.cmml" xref="Thmproofx22.p3.6.m1.4.4">𝜎</ci><ci id="Thmproofx22.p3.6.m1.5.5.cmml" xref="Thmproofx22.p3.6.m1.5.5">𝛼</ci></interval></apply><apply id="Thmproofx22.p3.6.m1.6.6.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.1"><plus id="Thmproofx22.p3.6.m1.6.6.1.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.2"></plus><apply id="Thmproofx22.p3.6.m1.6.6.1.1.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.1.1"><abs id="Thmproofx22.p3.6.m1.6.6.1.1.2.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.1.1.2"></abs><apply id="Thmproofx22.p3.6.m1.6.6.1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.6.6.1.1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p3.6.m1.6.6.1.1.1.1.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.1.1.1.2">𝑝</ci><cn id="Thmproofx22.p3.6.m1.6.6.1.1.1.1.3.cmml" type="integer" xref="Thmproofx22.p3.6.m1.6.6.1.1.1.1.3">1</cn></apply></apply><apply id="Thmproofx22.p3.6.m1.6.6.1.3.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.3"><times id="Thmproofx22.p3.6.m1.6.6.1.3.1.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.3.1"></times><apply id="Thmproofx22.p3.6.m1.3.3.cmml" xref="Thmproofx22.p3.6.m1.3.3"><divide id="Thmproofx22.p3.6.m1.3.3.3.cmml" xref="Thmproofx22.p3.6.m1.3.3"></divide><apply id="Thmproofx22.p3.6.m1.3.3.2.cmml" xref="Thmproofx22.p3.6.m1.3.3.2"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.3.3.2.3.cmml" xref="Thmproofx22.p3.6.m1.3.3.2">subscript</csymbol><ci id="Thmproofx22.p3.6.m1.3.3.2.4.cmml" xref="Thmproofx22.p3.6.m1.3.3.2.4">ℰ</ci><list id="Thmproofx22.p3.6.m1.3.3.2.2.2.3.cmml" xref="Thmproofx22.p3.6.m1.3.3.2.2.2.2"><apply id="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1.cmml" xref="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1.1.cmml" xref="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1">subscript</csymbol><ci id="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1.2.cmml" xref="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1.2">𝑚</ci><cn id="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1.3.cmml" type="integer" xref="Thmproofx22.p3.6.m1.3.3.2.2.2.2.1.3">1</cn></apply><ci id="Thmproofx22.p3.6.m1.2.2.1.1.1.1.cmml" xref="Thmproofx22.p3.6.m1.2.2.1.1.1.1">𝑛</ci></list></apply><apply id="Thmproofx22.p3.6.m1.3.3.4.cmml" xref="Thmproofx22.p3.6.m1.3.3.4"><csymbol cd="ambiguous" id="Thmproofx22.p3.6.m1.3.3.4.1.cmml" xref="Thmproofx22.p3.6.m1.3.3.4">subscript</csymbol><ci id="Thmproofx22.p3.6.m1.3.3.4.2.cmml" xref="Thmproofx22.p3.6.m1.3.3.4.2">𝑚</ci><cn id="Thmproofx22.p3.6.m1.3.3.4.3.cmml" type="integer" xref="Thmproofx22.p3.6.m1.3.3.4.3">1</cn></apply></apply><ci id="Thmproofx22.p3.6.m1.6.6.1.3.2.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.3.2">𝑘</ci><ci id="Thmproofx22.p3.6.m1.6.6.1.3.3.cmml" xref="Thmproofx22.p3.6.m1.6.6.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.6.m1.6c">\min_{\sigma\in(\Delta(P_{i}))^{n}}F_{n}(\sigma,\alpha)<|p_{1}|+\frac{\mathcal% {E}_{m_{1},n}}{m_{1}}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.6.m1.6d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_k italic_α</annotation></semantics></math> holds. Then, we apply the argument above to the graph <math alttext="G_{i}" class="ltx_Math" display="inline" id="Thmproofx22.p3.7.m2.1"><semantics id="Thmproofx22.p3.7.m2.1a"><msub id="Thmproofx22.p3.7.m2.1.1" xref="Thmproofx22.p3.7.m2.1.1.cmml"><mi id="Thmproofx22.p3.7.m2.1.1.2" xref="Thmproofx22.p3.7.m2.1.1.2.cmml">G</mi><mi id="Thmproofx22.p3.7.m2.1.1.3" xref="Thmproofx22.p3.7.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p3.7.m2.1b"><apply id="Thmproofx22.p3.7.m2.1.1.cmml" xref="Thmproofx22.p3.7.m2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.7.m2.1.1.1.cmml" xref="Thmproofx22.p3.7.m2.1.1">subscript</csymbol><ci id="Thmproofx22.p3.7.m2.1.1.2.cmml" xref="Thmproofx22.p3.7.m2.1.1.2">𝐺</ci><ci id="Thmproofx22.p3.7.m2.1.1.3.cmml" xref="Thmproofx22.p3.7.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.7.m2.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.7.m2.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to get <math alttext="F_{n}(U_{P_{i}}^{n},\alpha)>\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Thmproofx22.p3.8.m3.5"><semantics id="Thmproofx22.p3.8.m3.5a"><mrow id="Thmproofx22.p3.8.m3.5.5" xref="Thmproofx22.p3.8.m3.5.5.cmml"><mrow id="Thmproofx22.p3.8.m3.5.5.1" xref="Thmproofx22.p3.8.m3.5.5.1.cmml"><msub id="Thmproofx22.p3.8.m3.5.5.1.3" xref="Thmproofx22.p3.8.m3.5.5.1.3.cmml"><mi id="Thmproofx22.p3.8.m3.5.5.1.3.2" xref="Thmproofx22.p3.8.m3.5.5.1.3.2.cmml">F</mi><mi id="Thmproofx22.p3.8.m3.5.5.1.3.3" xref="Thmproofx22.p3.8.m3.5.5.1.3.3.cmml">n</mi></msub><mo id="Thmproofx22.p3.8.m3.5.5.1.2" xref="Thmproofx22.p3.8.m3.5.5.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p3.8.m3.5.5.1.1.1" xref="Thmproofx22.p3.8.m3.5.5.1.1.2.cmml"><mo id="Thmproofx22.p3.8.m3.5.5.1.1.1.2" stretchy="false" xref="Thmproofx22.p3.8.m3.5.5.1.1.2.cmml">(</mo><msubsup id="Thmproofx22.p3.8.m3.5.5.1.1.1.1" xref="Thmproofx22.p3.8.m3.5.5.1.1.1.1.cmml"><mi id="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.2" xref="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.3" xref="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.3.cmml"><mi id="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.3.2" xref="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.3.2.cmml">P</mi><mi id="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.3.3" xref="Thmproofx22.p3.8.m3.5.5.1.1.1.1.2.3.3.cmml">i</mi></msub><mi id="Thmproofx22.p3.8.m3.5.5.1.1.1.1.3" xref="Thmproofx22.p3.8.m3.5.5.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx22.p3.8.m3.5.5.1.1.1.3" xref="Thmproofx22.p3.8.m3.5.5.1.1.2.cmml">,</mo><mi id="Thmproofx22.p3.8.m3.2.2" xref="Thmproofx22.p3.8.m3.2.2.cmml">α</mi><mo id="Thmproofx22.p3.8.m3.5.5.1.1.1.4" stretchy="false" xref="Thmproofx22.p3.8.m3.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p3.8.m3.5.5.2" xref="Thmproofx22.p3.8.m3.5.5.2.cmml">></mo><mrow id="Thmproofx22.p3.8.m3.5.5.3" xref="Thmproofx22.p3.8.m3.5.5.3.cmml"><mrow id="Thmproofx22.p3.8.m3.5.5.3.2" xref="Thmproofx22.p3.8.m3.5.5.3.2.cmml"><msub id="Thmproofx22.p3.8.m3.5.5.3.2.1" xref="Thmproofx22.p3.8.m3.5.5.3.2.1.cmml"><mi id="Thmproofx22.p3.8.m3.5.5.3.2.1.2" xref="Thmproofx22.p3.8.m3.5.5.3.2.1.2.cmml">min</mi><mrow id="Thmproofx22.p3.8.m3.1.1.1" xref="Thmproofx22.p3.8.m3.1.1.1.cmml"><mi id="Thmproofx22.p3.8.m3.1.1.1.3" xref="Thmproofx22.p3.8.m3.1.1.1.3.cmml">σ</mi><mo id="Thmproofx22.p3.8.m3.1.1.1.2" xref="Thmproofx22.p3.8.m3.1.1.1.2.cmml">∈</mo><msup id="Thmproofx22.p3.8.m3.1.1.1.1" xref="Thmproofx22.p3.8.m3.1.1.1.1.cmml"><mrow id="Thmproofx22.p3.8.m3.1.1.1.1.1.1" xref="Thmproofx22.p3.8.m3.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx22.p3.8.m3.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p3.8.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx22.p3.8.m3.1.1.1.1.1.1.1" xref="Thmproofx22.p3.8.m3.1.1.1.1.1.1.1.cmml"><mi 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xref="Thmproofx22.p3.8.m3.5.5.3.2.2">subscript</csymbol><ci id="Thmproofx22.p3.8.m3.5.5.3.2.2.2.cmml" xref="Thmproofx22.p3.8.m3.5.5.3.2.2.2">𝐹</ci><ci id="Thmproofx22.p3.8.m3.5.5.3.2.2.3.cmml" xref="Thmproofx22.p3.8.m3.5.5.3.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmproofx22.p3.8.m3.5.5.3.3.1.cmml" xref="Thmproofx22.p3.8.m3.5.5.3.3.2"><ci id="Thmproofx22.p3.8.m3.3.3.cmml" xref="Thmproofx22.p3.8.m3.3.3">𝜎</ci><ci id="Thmproofx22.p3.8.m3.4.4.cmml" xref="Thmproofx22.p3.8.m3.4.4">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.8.m3.5c">F_{n}(U_{P_{i}}^{n},\alpha)>\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.8.m3.5d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) > roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math>. In both cases, we conclude that <math alttext="U_{P_{i}}^{n}" class="ltx_Math" display="inline" id="Thmproofx22.p3.9.m4.1"><semantics id="Thmproofx22.p3.9.m4.1a"><msubsup id="Thmproofx22.p3.9.m4.1.1" xref="Thmproofx22.p3.9.m4.1.1.cmml"><mi id="Thmproofx22.p3.9.m4.1.1.2.2" xref="Thmproofx22.p3.9.m4.1.1.2.2.cmml">U</mi><msub id="Thmproofx22.p3.9.m4.1.1.2.3" xref="Thmproofx22.p3.9.m4.1.1.2.3.cmml"><mi id="Thmproofx22.p3.9.m4.1.1.2.3.2" xref="Thmproofx22.p3.9.m4.1.1.2.3.2.cmml">P</mi><mi id="Thmproofx22.p3.9.m4.1.1.2.3.3" xref="Thmproofx22.p3.9.m4.1.1.2.3.3.cmml">i</mi></msub><mi id="Thmproofx22.p3.9.m4.1.1.3" xref="Thmproofx22.p3.9.m4.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx22.p3.9.m4.1b"><apply id="Thmproofx22.p3.9.m4.1.1.cmml" xref="Thmproofx22.p3.9.m4.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.9.m4.1.1.1.cmml" xref="Thmproofx22.p3.9.m4.1.1">superscript</csymbol><apply id="Thmproofx22.p3.9.m4.1.1.2.cmml" xref="Thmproofx22.p3.9.m4.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p3.9.m4.1.1.2.1.cmml" xref="Thmproofx22.p3.9.m4.1.1">subscript</csymbol><ci id="Thmproofx22.p3.9.m4.1.1.2.2.cmml" xref="Thmproofx22.p3.9.m4.1.1.2.2">𝑈</ci><apply id="Thmproofx22.p3.9.m4.1.1.2.3.cmml" xref="Thmproofx22.p3.9.m4.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p3.9.m4.1.1.2.3.1.cmml" xref="Thmproofx22.p3.9.m4.1.1.2.3">subscript</csymbol><ci id="Thmproofx22.p3.9.m4.1.1.2.3.2.cmml" xref="Thmproofx22.p3.9.m4.1.1.2.3.2">𝑃</ci><ci id="Thmproofx22.p3.9.m4.1.1.2.3.3.cmml" xref="Thmproofx22.p3.9.m4.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Thmproofx22.p3.9.m4.1.1.3.cmml" xref="Thmproofx22.p3.9.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p3.9.m4.1c">U_{P_{i}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p3.9.m4.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is not optimal for the agents.</p> </div> <div class="ltx_para" id="Thmproofx22.p4"> <p class="ltx_p" id="Thmproofx22.p4.3">Now consider <math alttext="G_{m_{1}}" class="ltx_Math" display="inline" id="Thmproofx22.p4.1.m1.1"><semantics id="Thmproofx22.p4.1.m1.1a"><msub id="Thmproofx22.p4.1.m1.1.1" xref="Thmproofx22.p4.1.m1.1.1.cmml"><mi id="Thmproofx22.p4.1.m1.1.1.2" xref="Thmproofx22.p4.1.m1.1.1.2.cmml">G</mi><msub id="Thmproofx22.p4.1.m1.1.1.3" xref="Thmproofx22.p4.1.m1.1.1.3.cmml"><mi id="Thmproofx22.p4.1.m1.1.1.3.2" xref="Thmproofx22.p4.1.m1.1.1.3.2.cmml">m</mi><mn id="Thmproofx22.p4.1.m1.1.1.3.3" xref="Thmproofx22.p4.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p4.1.m1.1b"><apply id="Thmproofx22.p4.1.m1.1.1.cmml" xref="Thmproofx22.p4.1.m1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p4.1.m1.1.1.1.cmml" xref="Thmproofx22.p4.1.m1.1.1">subscript</csymbol><ci id="Thmproofx22.p4.1.m1.1.1.2.cmml" xref="Thmproofx22.p4.1.m1.1.1.2">𝐺</ci><apply id="Thmproofx22.p4.1.m1.1.1.3.cmml" xref="Thmproofx22.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx22.p4.1.m1.1.1.3.1.cmml" xref="Thmproofx22.p4.1.m1.1.1.3">subscript</csymbol><ci id="Thmproofx22.p4.1.m1.1.1.3.2.cmml" xref="Thmproofx22.p4.1.m1.1.1.3.2">𝑚</ci><cn id="Thmproofx22.p4.1.m1.1.1.3.3.cmml" type="integer" xref="Thmproofx22.p4.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p4.1.m1.1c">G_{m_{1}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p4.1.m1.1d">italic_G start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Then by Lemmas <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma15" title="Lemma 15 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">15</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma16" title="Lemma 16 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">16</span></a>, we again get that <math alttext="U_{P_{i}}^{n}" class="ltx_Math" display="inline" id="Thmproofx22.p4.2.m2.1"><semantics id="Thmproofx22.p4.2.m2.1a"><msubsup id="Thmproofx22.p4.2.m2.1.1" xref="Thmproofx22.p4.2.m2.1.1.cmml"><mi id="Thmproofx22.p4.2.m2.1.1.2.2" xref="Thmproofx22.p4.2.m2.1.1.2.2.cmml">U</mi><msub id="Thmproofx22.p4.2.m2.1.1.2.3" xref="Thmproofx22.p4.2.m2.1.1.2.3.cmml"><mi id="Thmproofx22.p4.2.m2.1.1.2.3.2" xref="Thmproofx22.p4.2.m2.1.1.2.3.2.cmml">P</mi><mi id="Thmproofx22.p4.2.m2.1.1.2.3.3" xref="Thmproofx22.p4.2.m2.1.1.2.3.3.cmml">i</mi></msub><mi id="Thmproofx22.p4.2.m2.1.1.3" xref="Thmproofx22.p4.2.m2.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmproofx22.p4.2.m2.1b"><apply id="Thmproofx22.p4.2.m2.1.1.cmml" xref="Thmproofx22.p4.2.m2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p4.2.m2.1.1.1.cmml" xref="Thmproofx22.p4.2.m2.1.1">superscript</csymbol><apply id="Thmproofx22.p4.2.m2.1.1.2.cmml" xref="Thmproofx22.p4.2.m2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p4.2.m2.1.1.2.1.cmml" xref="Thmproofx22.p4.2.m2.1.1">subscript</csymbol><ci id="Thmproofx22.p4.2.m2.1.1.2.2.cmml" xref="Thmproofx22.p4.2.m2.1.1.2.2">𝑈</ci><apply id="Thmproofx22.p4.2.m2.1.1.2.3.cmml" xref="Thmproofx22.p4.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="Thmproofx22.p4.2.m2.1.1.2.3.1.cmml" xref="Thmproofx22.p4.2.m2.1.1.2.3">subscript</csymbol><ci id="Thmproofx22.p4.2.m2.1.1.2.3.2.cmml" xref="Thmproofx22.p4.2.m2.1.1.2.3.2">𝑃</ci><ci id="Thmproofx22.p4.2.m2.1.1.2.3.3.cmml" xref="Thmproofx22.p4.2.m2.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Thmproofx22.p4.2.m2.1.1.3.cmml" xref="Thmproofx22.p4.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p4.2.m2.1c">U_{P_{i}}^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p4.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is not optimal for the agents, now for all <math alttext="i\leq m_{1}" class="ltx_Math" display="inline" id="Thmproofx22.p4.3.m3.1"><semantics id="Thmproofx22.p4.3.m3.1a"><mrow id="Thmproofx22.p4.3.m3.1.1" xref="Thmproofx22.p4.3.m3.1.1.cmml"><mi id="Thmproofx22.p4.3.m3.1.1.2" xref="Thmproofx22.p4.3.m3.1.1.2.cmml">i</mi><mo id="Thmproofx22.p4.3.m3.1.1.1" xref="Thmproofx22.p4.3.m3.1.1.1.cmml">≤</mo><msub id="Thmproofx22.p4.3.m3.1.1.3" xref="Thmproofx22.p4.3.m3.1.1.3.cmml"><mi id="Thmproofx22.p4.3.m3.1.1.3.2" xref="Thmproofx22.p4.3.m3.1.1.3.2.cmml">m</mi><mn id="Thmproofx22.p4.3.m3.1.1.3.3" xref="Thmproofx22.p4.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx22.p4.3.m3.1b"><apply id="Thmproofx22.p4.3.m3.1.1.cmml" xref="Thmproofx22.p4.3.m3.1.1"><leq id="Thmproofx22.p4.3.m3.1.1.1.cmml" xref="Thmproofx22.p4.3.m3.1.1.1"></leq><ci id="Thmproofx22.p4.3.m3.1.1.2.cmml" xref="Thmproofx22.p4.3.m3.1.1.2">𝑖</ci><apply id="Thmproofx22.p4.3.m3.1.1.3.cmml" xref="Thmproofx22.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="Thmproofx22.p4.3.m3.1.1.3.1.cmml" xref="Thmproofx22.p4.3.m3.1.1.3">subscript</csymbol><ci id="Thmproofx22.p4.3.m3.1.1.3.2.cmml" xref="Thmproofx22.p4.3.m3.1.1.3.2">𝑚</ci><cn id="Thmproofx22.p4.3.m3.1.1.3.3.cmml" type="integer" xref="Thmproofx22.p4.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p4.3.m3.1c">i\leq m_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p4.3.m3.1d">italic_i ≤ italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmproofx22.p5"> <p class="ltx_p" id="Thmproofx22.p5.2">In total, <math alttext="\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<\min_{\sigma\in% \mathcal{U}_{n}}F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Thmproofx22.p5.1.m1.5"><semantics id="Thmproofx22.p5.1.m1.5a"><mrow id="Thmproofx22.p5.1.m1.5.6" xref="Thmproofx22.p5.1.m1.5.6.cmml"><mrow id="Thmproofx22.p5.1.m1.5.6.2" xref="Thmproofx22.p5.1.m1.5.6.2.cmml"><mrow id="Thmproofx22.p5.1.m1.5.6.2.2" xref="Thmproofx22.p5.1.m1.5.6.2.2.cmml"><msub id="Thmproofx22.p5.1.m1.5.6.2.2.1" xref="Thmproofx22.p5.1.m1.5.6.2.2.1.cmml"><mi id="Thmproofx22.p5.1.m1.5.6.2.2.1.2" xref="Thmproofx22.p5.1.m1.5.6.2.2.1.2.cmml">min</mi><mrow id="Thmproofx22.p5.1.m1.1.1.1" xref="Thmproofx22.p5.1.m1.1.1.1.cmml"><mi id="Thmproofx22.p5.1.m1.1.1.1.3" xref="Thmproofx22.p5.1.m1.1.1.1.3.cmml">σ</mi><mo id="Thmproofx22.p5.1.m1.1.1.1.2" xref="Thmproofx22.p5.1.m1.1.1.1.2.cmml">∈</mo><msup id="Thmproofx22.p5.1.m1.1.1.1.1" xref="Thmproofx22.p5.1.m1.1.1.1.1.cmml"><mrow id="Thmproofx22.p5.1.m1.1.1.1.1.1.1" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.3.cmml">Δ</mi><mo id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.2" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.2" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx22.p5.1.m1.1.1.1.1.3" xref="Thmproofx22.p5.1.m1.1.1.1.1.3.cmml">n</mi></msup></mrow></msub><mo id="Thmproofx22.p5.1.m1.5.6.2.2a" lspace="0.167em" xref="Thmproofx22.p5.1.m1.5.6.2.2.cmml">⁡</mo><msub id="Thmproofx22.p5.1.m1.5.6.2.2.2" xref="Thmproofx22.p5.1.m1.5.6.2.2.2.cmml"><mi id="Thmproofx22.p5.1.m1.5.6.2.2.2.2" xref="Thmproofx22.p5.1.m1.5.6.2.2.2.2.cmml">F</mi><mi id="Thmproofx22.p5.1.m1.5.6.2.2.2.3" xref="Thmproofx22.p5.1.m1.5.6.2.2.2.3.cmml">n</mi></msub></mrow><mo id="Thmproofx22.p5.1.m1.5.6.2.1" xref="Thmproofx22.p5.1.m1.5.6.2.1.cmml">⁢</mo><mrow id="Thmproofx22.p5.1.m1.5.6.2.3.2" xref="Thmproofx22.p5.1.m1.5.6.2.3.1.cmml"><mo id="Thmproofx22.p5.1.m1.5.6.2.3.2.1" stretchy="false" xref="Thmproofx22.p5.1.m1.5.6.2.3.1.cmml">(</mo><mi id="Thmproofx22.p5.1.m1.2.2" xref="Thmproofx22.p5.1.m1.2.2.cmml">σ</mi><mo id="Thmproofx22.p5.1.m1.5.6.2.3.2.2" xref="Thmproofx22.p5.1.m1.5.6.2.3.1.cmml">,</mo><mi id="Thmproofx22.p5.1.m1.3.3" xref="Thmproofx22.p5.1.m1.3.3.cmml">α</mi><mo id="Thmproofx22.p5.1.m1.5.6.2.3.2.3" stretchy="false" xref="Thmproofx22.p5.1.m1.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx22.p5.1.m1.5.6.1" xref="Thmproofx22.p5.1.m1.5.6.1.cmml"><</mo><mrow id="Thmproofx22.p5.1.m1.5.6.3" xref="Thmproofx22.p5.1.m1.5.6.3.cmml"><mrow id="Thmproofx22.p5.1.m1.5.6.3.2" xref="Thmproofx22.p5.1.m1.5.6.3.2.cmml"><msub id="Thmproofx22.p5.1.m1.5.6.3.2.1" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.cmml"><mi id="Thmproofx22.p5.1.m1.5.6.3.2.1.2" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.2.cmml">min</mi><mrow id="Thmproofx22.p5.1.m1.5.6.3.2.1.3" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.cmml"><mi id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.2" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.2.cmml">σ</mi><mo id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.1" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.1.cmml">∈</mo><msub id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3" 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xref="Thmproofx22.p5.1.m1.5.6.2.2.1">subscript</csymbol><min id="Thmproofx22.p5.1.m1.5.6.2.2.1.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.2.2.1.2"></min><apply id="Thmproofx22.p5.1.m1.1.1.1.cmml" xref="Thmproofx22.p5.1.m1.1.1.1"><in id="Thmproofx22.p5.1.m1.1.1.1.2.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.2"></in><ci id="Thmproofx22.p5.1.m1.1.1.1.3.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.3">𝜎</ci><apply id="Thmproofx22.p5.1.m1.1.1.1.1.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p5.1.m1.1.1.1.1.2.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1">superscript</csymbol><apply id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1"><times id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.2"></times><ci id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.3">Δ</ci><apply id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3"><times id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.1"></times><ci id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><ci id="Thmproofx22.p5.1.m1.1.1.1.1.3.cmml" xref="Thmproofx22.p5.1.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmproofx22.p5.1.m1.5.6.2.2.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.2.2.2"><csymbol cd="ambiguous" id="Thmproofx22.p5.1.m1.5.6.2.2.2.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.2.2.2">subscript</csymbol><ci id="Thmproofx22.p5.1.m1.5.6.2.2.2.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.2.2.2.2">𝐹</ci><ci id="Thmproofx22.p5.1.m1.5.6.2.2.2.3.cmml" xref="Thmproofx22.p5.1.m1.5.6.2.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmproofx22.p5.1.m1.5.6.2.3.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.2.3.2"><ci id="Thmproofx22.p5.1.m1.2.2.cmml" xref="Thmproofx22.p5.1.m1.2.2">𝜎</ci><ci id="Thmproofx22.p5.1.m1.3.3.cmml" xref="Thmproofx22.p5.1.m1.3.3">𝛼</ci></interval></apply><apply id="Thmproofx22.p5.1.m1.5.6.3.cmml" xref="Thmproofx22.p5.1.m1.5.6.3"><times id="Thmproofx22.p5.1.m1.5.6.3.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.1"></times><apply id="Thmproofx22.p5.1.m1.5.6.3.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2"><apply id="Thmproofx22.p5.1.m1.5.6.3.2.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1"><csymbol cd="ambiguous" id="Thmproofx22.p5.1.m1.5.6.3.2.1.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1">subscript</csymbol><min id="Thmproofx22.p5.1.m1.5.6.3.2.1.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.2"></min><apply id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3"><in id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.1"></in><ci id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.2">𝜎</ci><apply id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3"><csymbol cd="ambiguous" id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3">subscript</csymbol><ci id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3.2">𝒰</ci><ci id="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3.3.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.1.3.3.3">𝑛</ci></apply></apply></apply><apply id="Thmproofx22.p5.1.m1.5.6.3.2.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.2"><csymbol cd="ambiguous" id="Thmproofx22.p5.1.m1.5.6.3.2.2.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.2">subscript</csymbol><ci id="Thmproofx22.p5.1.m1.5.6.3.2.2.2.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.2.2">𝐹</ci><ci id="Thmproofx22.p5.1.m1.5.6.3.2.2.3.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.2.2.3">𝑛</ci></apply></apply><interval closure="open" id="Thmproofx22.p5.1.m1.5.6.3.3.1.cmml" xref="Thmproofx22.p5.1.m1.5.6.3.3.2"><ci id="Thmproofx22.p5.1.m1.4.4.cmml" xref="Thmproofx22.p5.1.m1.4.4">𝜎</ci><ci id="Thmproofx22.p5.1.m1.5.5.cmml" xref="Thmproofx22.p5.1.m1.5.5">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p5.1.m1.5c">\min_{\sigma\in(\Delta(P_{st}))^{n}}F_{n}(\sigma,\alpha)<\min_{\sigma\in% \mathcal{U}_{n}}F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p5.1.m1.5d">roman_min start_POSTSUBSCRIPT italic_σ ∈ ( roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) < roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math> by the definition of <math alttext="\mathcal{U}_{n}" class="ltx_Math" display="inline" id="Thmproofx22.p5.2.m2.1"><semantics id="Thmproofx22.p5.2.m2.1a"><msub id="Thmproofx22.p5.2.m2.1.1" xref="Thmproofx22.p5.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx22.p5.2.m2.1.1.2" xref="Thmproofx22.p5.2.m2.1.1.2.cmml">𝒰</mi><mi id="Thmproofx22.p5.2.m2.1.1.3" xref="Thmproofx22.p5.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx22.p5.2.m2.1b"><apply id="Thmproofx22.p5.2.m2.1.1.cmml" xref="Thmproofx22.p5.2.m2.1.1"><csymbol cd="ambiguous" id="Thmproofx22.p5.2.m2.1.1.1.cmml" xref="Thmproofx22.p5.2.m2.1.1">subscript</csymbol><ci id="Thmproofx22.p5.2.m2.1.1.2.cmml" xref="Thmproofx22.p5.2.m2.1.1.2">𝒰</ci><ci id="Thmproofx22.p5.2.m2.1.1.3.cmml" xref="Thmproofx22.p5.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx22.p5.2.m2.1c">\mathcal{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx22.p5.2.m2.1d">caligraphic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx23"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#S5.Thmtheorem3" title="Theorem 5.3 ‣ The general 𝑘 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">5.3</span></a>)</h6> <div class="ltx_para" id="Thmproofx23.p1"> <p class="ltx_p" id="Thmproofx23.p1.2">Using Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma13" title="Lemma 13 ‣ 5.1 General 𝑠-𝑡 graphs ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">13</span></a>, we can wlog consider only <math alttext="\sigma=X^{n}" class="ltx_Math" display="inline" id="Thmproofx23.p1.1.m1.1"><semantics id="Thmproofx23.p1.1.m1.1a"><mrow id="Thmproofx23.p1.1.m1.1.1" xref="Thmproofx23.p1.1.m1.1.1.cmml"><mi id="Thmproofx23.p1.1.m1.1.1.2" xref="Thmproofx23.p1.1.m1.1.1.2.cmml">σ</mi><mo id="Thmproofx23.p1.1.m1.1.1.1" xref="Thmproofx23.p1.1.m1.1.1.1.cmml">=</mo><msup id="Thmproofx23.p1.1.m1.1.1.3" xref="Thmproofx23.p1.1.m1.1.1.3.cmml"><mi id="Thmproofx23.p1.1.m1.1.1.3.2" xref="Thmproofx23.p1.1.m1.1.1.3.2.cmml">X</mi><mi id="Thmproofx23.p1.1.m1.1.1.3.3" xref="Thmproofx23.p1.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.1.m1.1b"><apply id="Thmproofx23.p1.1.m1.1.1.cmml" xref="Thmproofx23.p1.1.m1.1.1"><eq id="Thmproofx23.p1.1.m1.1.1.1.cmml" xref="Thmproofx23.p1.1.m1.1.1.1"></eq><ci id="Thmproofx23.p1.1.m1.1.1.2.cmml" xref="Thmproofx23.p1.1.m1.1.1.2">𝜎</ci><apply id="Thmproofx23.p1.1.m1.1.1.3.cmml" xref="Thmproofx23.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.1.m1.1.1.3.1.cmml" xref="Thmproofx23.p1.1.m1.1.1.3">superscript</csymbol><ci id="Thmproofx23.p1.1.m1.1.1.3.2.cmml" xref="Thmproofx23.p1.1.m1.1.1.3.2">𝑋</ci><ci id="Thmproofx23.p1.1.m1.1.1.3.3.cmml" xref="Thmproofx23.p1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.1.m1.1c">\sigma=X^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.1.m1.1d">italic_σ = italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. For <math alttext="d\in E^{k}" class="ltx_Math" display="inline" id="Thmproofx23.p1.2.m2.1"><semantics id="Thmproofx23.p1.2.m2.1a"><mrow id="Thmproofx23.p1.2.m2.1.1" xref="Thmproofx23.p1.2.m2.1.1.cmml"><mi id="Thmproofx23.p1.2.m2.1.1.2" xref="Thmproofx23.p1.2.m2.1.1.2.cmml">d</mi><mo id="Thmproofx23.p1.2.m2.1.1.1" xref="Thmproofx23.p1.2.m2.1.1.1.cmml">∈</mo><msup id="Thmproofx23.p1.2.m2.1.1.3" xref="Thmproofx23.p1.2.m2.1.1.3.cmml"><mi id="Thmproofx23.p1.2.m2.1.1.3.2" xref="Thmproofx23.p1.2.m2.1.1.3.2.cmml">E</mi><mi id="Thmproofx23.p1.2.m2.1.1.3.3" xref="Thmproofx23.p1.2.m2.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.2.m2.1b"><apply id="Thmproofx23.p1.2.m2.1.1.cmml" xref="Thmproofx23.p1.2.m2.1.1"><in id="Thmproofx23.p1.2.m2.1.1.1.cmml" xref="Thmproofx23.p1.2.m2.1.1.1"></in><ci id="Thmproofx23.p1.2.m2.1.1.2.cmml" xref="Thmproofx23.p1.2.m2.1.1.2">𝑑</ci><apply id="Thmproofx23.p1.2.m2.1.1.3.cmml" xref="Thmproofx23.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.2.m2.1.1.3.1.cmml" xref="Thmproofx23.p1.2.m2.1.1.3">superscript</csymbol><ci id="Thmproofx23.p1.2.m2.1.1.3.2.cmml" xref="Thmproofx23.p1.2.m2.1.1.3.2">𝐸</ci><ci id="Thmproofx23.p1.2.m2.1.1.3.3.cmml" xref="Thmproofx23.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.2.m2.1c">d\in E^{k}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.2.m2.1d">italic_d ∈ italic_E start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>, by Equation <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E6" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">6</span></a>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx59"> <tbody id="Pt0.A4.Ex104"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{d}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.Ex104.m1.2"><semantics id="Pt0.A4.Ex104.m1.2a"><mrow id="Pt0.A4.Ex104.m1.2.3" xref="Pt0.A4.Ex104.m1.2.3.cmml"><msub id="Pt0.A4.Ex104.m1.2.3.2" xref="Pt0.A4.Ex104.m1.2.3.2.cmml"><mi id="Pt0.A4.Ex104.m1.2.3.2.2" xref="Pt0.A4.Ex104.m1.2.3.2.2.cmml">f</mi><mi id="Pt0.A4.Ex104.m1.2.3.2.3" xref="Pt0.A4.Ex104.m1.2.3.2.3.cmml">d</mi></msub><mo id="Pt0.A4.Ex104.m1.2.3.1" xref="Pt0.A4.Ex104.m1.2.3.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex104.m1.2.3.3.2" xref="Pt0.A4.Ex104.m1.2.3.3.1.cmml"><mo id="Pt0.A4.Ex104.m1.2.3.3.2.1" stretchy="false" xref="Pt0.A4.Ex104.m1.2.3.3.1.cmml">(</mo><mi id="Pt0.A4.Ex104.m1.1.1" xref="Pt0.A4.Ex104.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.Ex104.m1.2.3.3.2.2" xref="Pt0.A4.Ex104.m1.2.3.3.1.cmml">,</mo><mi id="Pt0.A4.Ex104.m1.2.2" xref="Pt0.A4.Ex104.m1.2.2.cmml">α</mi><mo id="Pt0.A4.Ex104.m1.2.3.3.2.3" stretchy="false" xref="Pt0.A4.Ex104.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex104.m1.2b"><apply id="Pt0.A4.Ex104.m1.2.3.cmml" xref="Pt0.A4.Ex104.m1.2.3"><times id="Pt0.A4.Ex104.m1.2.3.1.cmml" xref="Pt0.A4.Ex104.m1.2.3.1"></times><apply id="Pt0.A4.Ex104.m1.2.3.2.cmml" xref="Pt0.A4.Ex104.m1.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex104.m1.2.3.2.1.cmml" xref="Pt0.A4.Ex104.m1.2.3.2">subscript</csymbol><ci id="Pt0.A4.Ex104.m1.2.3.2.2.cmml" xref="Pt0.A4.Ex104.m1.2.3.2.2">𝑓</ci><ci id="Pt0.A4.Ex104.m1.2.3.2.3.cmml" xref="Pt0.A4.Ex104.m1.2.3.2.3">𝑑</ci></apply><interval closure="open" id="Pt0.A4.Ex104.m1.2.3.3.1.cmml" xref="Pt0.A4.Ex104.m1.2.3.3.2"><ci id="Pt0.A4.Ex104.m1.1.1.cmml" xref="Pt0.A4.Ex104.m1.1.1">𝜎</ci><ci id="Pt0.A4.Ex104.m1.2.2.cmml" xref="Pt0.A4.Ex104.m1.2.2">𝛼</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex104.m1.2c">\displaystyle f_{d}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex104.m1.2d">italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i\in[m]}c_{d}(p^{d}_{i})\left(\left(\sum_{i^{\prime}\leq i% }X(p^{d}_{i^{\prime}})\right)^{n}-\left(\sum_{i^{\prime}\leq i-1}X(p^{d}_{i^{% \prime}})\right)^{n}\right)=(*)" class="ltx_Math" display="inline" id="Pt0.A4.Ex104.m2.4"><semantics id="Pt0.A4.Ex104.m2.4a"><mrow id="Pt0.A4.Ex104.m2.4.4" xref="Pt0.A4.Ex104.m2.4.4.cmml"><mi id="Pt0.A4.Ex104.m2.4.4.4" xref="Pt0.A4.Ex104.m2.4.4.4.cmml"></mi><mo id="Pt0.A4.Ex104.m2.4.4.5" xref="Pt0.A4.Ex104.m2.4.4.5.cmml">=</mo><mrow id="Pt0.A4.Ex104.m2.4.4.2" xref="Pt0.A4.Ex104.m2.4.4.2.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex104.m2.4.4.2.3" xref="Pt0.A4.Ex104.m2.4.4.2.3.cmml"><munder 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id="Pt0.A4.Ex104.m2.4.4d.cmml" xref="Pt0.A4.Ex104.m2.4.4"></share><times id="Pt0.A4.Ex104.m2.2.2.cmml" xref="Pt0.A4.Ex104.m2.2.2"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex104.m2.4c">\displaystyle=\sum_{i\in[m]}c_{d}(p^{d}_{i})\left(\left(\sum_{i^{\prime}\leq i% }X(p^{d}_{i^{\prime}})\right)^{n}-\left(\sum_{i^{\prime}\leq i-1}X(p^{d}_{i^{% \prime}})\right)^{n}\right)=(*)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex104.m2.4d">= ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) = ( ∗ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx23.p1.5">Let <math alttext="y_{i}=\sum_{i^{\prime}\leq i}X(p^{d}_{i^{\prime}})" class="ltx_Math" display="inline" id="Thmproofx23.p1.3.m1.1"><semantics id="Thmproofx23.p1.3.m1.1a"><mrow id="Thmproofx23.p1.3.m1.1.1" xref="Thmproofx23.p1.3.m1.1.1.cmml"><msub id="Thmproofx23.p1.3.m1.1.1.3" xref="Thmproofx23.p1.3.m1.1.1.3.cmml"><mi id="Thmproofx23.p1.3.m1.1.1.3.2" xref="Thmproofx23.p1.3.m1.1.1.3.2.cmml">y</mi><mi id="Thmproofx23.p1.3.m1.1.1.3.3" xref="Thmproofx23.p1.3.m1.1.1.3.3.cmml">i</mi></msub><mo id="Thmproofx23.p1.3.m1.1.1.2" rspace="0.111em" xref="Thmproofx23.p1.3.m1.1.1.2.cmml">=</mo><mrow id="Thmproofx23.p1.3.m1.1.1.1" xref="Thmproofx23.p1.3.m1.1.1.1.cmml"><msub id="Thmproofx23.p1.3.m1.1.1.1.2" xref="Thmproofx23.p1.3.m1.1.1.1.2.cmml"><mo id="Thmproofx23.p1.3.m1.1.1.1.2.2" xref="Thmproofx23.p1.3.m1.1.1.1.2.2.cmml">∑</mo><mrow id="Thmproofx23.p1.3.m1.1.1.1.2.3" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.cmml"><msup id="Thmproofx23.p1.3.m1.1.1.1.2.3.2" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.2.cmml"><mi id="Thmproofx23.p1.3.m1.1.1.1.2.3.2.2" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Thmproofx23.p1.3.m1.1.1.1.2.3.2.3" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Thmproofx23.p1.3.m1.1.1.1.2.3.1" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.1.cmml">≤</mo><mi id="Thmproofx23.p1.3.m1.1.1.1.2.3.3" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.3.cmml">i</mi></mrow></msub><mrow id="Thmproofx23.p1.3.m1.1.1.1.1" xref="Thmproofx23.p1.3.m1.1.1.1.1.cmml"><mi id="Thmproofx23.p1.3.m1.1.1.1.1.3" xref="Thmproofx23.p1.3.m1.1.1.1.1.3.cmml">X</mi><mo id="Thmproofx23.p1.3.m1.1.1.1.1.2" xref="Thmproofx23.p1.3.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx23.p1.3.m1.1.1.1.1.1.1" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.2" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.2.cmml">p</mi><msup id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.2" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.3" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup><mi id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.3" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.3.m1.1b"><apply id="Thmproofx23.p1.3.m1.1.1.cmml" xref="Thmproofx23.p1.3.m1.1.1"><eq id="Thmproofx23.p1.3.m1.1.1.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.2"></eq><apply id="Thmproofx23.p1.3.m1.1.1.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.3.m1.1.1.3.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.3">subscript</csymbol><ci id="Thmproofx23.p1.3.m1.1.1.3.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.3.2">𝑦</ci><ci id="Thmproofx23.p1.3.m1.1.1.3.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.3.3">𝑖</ci></apply><apply id="Thmproofx23.p1.3.m1.1.1.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1"><apply id="Thmproofx23.p1.3.m1.1.1.1.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.3.m1.1.1.1.2.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2">subscript</csymbol><sum id="Thmproofx23.p1.3.m1.1.1.1.2.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.2"></sum><apply id="Thmproofx23.p1.3.m1.1.1.1.2.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.3"><leq id="Thmproofx23.p1.3.m1.1.1.1.2.3.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.1"></leq><apply id="Thmproofx23.p1.3.m1.1.1.1.2.3.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.3.m1.1.1.1.2.3.2.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.2">superscript</csymbol><ci id="Thmproofx23.p1.3.m1.1.1.1.2.3.2.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.2.2">𝑖</ci><ci id="Thmproofx23.p1.3.m1.1.1.1.2.3.2.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.2.3">′</ci></apply><ci id="Thmproofx23.p1.3.m1.1.1.1.2.3.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.2.3.3">𝑖</ci></apply></apply><apply id="Thmproofx23.p1.3.m1.1.1.1.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1"><times id="Thmproofx23.p1.3.m1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.2"></times><ci id="Thmproofx23.p1.3.m1.1.1.1.1.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.3">𝑋</ci><apply id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1">subscript</csymbol><apply id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.2">𝑝</ci><ci id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.2.3">𝑑</ci></apply><apply id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx23.p1.3.m1.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.3.m1.1c">y_{i}=\sum_{i^{\prime}\leq i}X(p^{d}_{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.3.m1.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_i end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>, let <math alttext="\delta(X)=\max_{i\in[m]}(y_{i}^{n}-y_{i-1}^{n})" class="ltx_Math" display="inline" id="Thmproofx23.p1.4.m2.4"><semantics id="Thmproofx23.p1.4.m2.4a"><mrow id="Thmproofx23.p1.4.m2.4.4" xref="Thmproofx23.p1.4.m2.4.4.cmml"><mrow id="Thmproofx23.p1.4.m2.4.4.4" xref="Thmproofx23.p1.4.m2.4.4.4.cmml"><mi id="Thmproofx23.p1.4.m2.4.4.4.2" xref="Thmproofx23.p1.4.m2.4.4.4.2.cmml">δ</mi><mo id="Thmproofx23.p1.4.m2.4.4.4.1" xref="Thmproofx23.p1.4.m2.4.4.4.1.cmml">⁢</mo><mrow id="Thmproofx23.p1.4.m2.4.4.4.3.2" xref="Thmproofx23.p1.4.m2.4.4.4.cmml"><mo id="Thmproofx23.p1.4.m2.4.4.4.3.2.1" stretchy="false" xref="Thmproofx23.p1.4.m2.4.4.4.cmml">(</mo><mi id="Thmproofx23.p1.4.m2.2.2" xref="Thmproofx23.p1.4.m2.2.2.cmml">X</mi><mo id="Thmproofx23.p1.4.m2.4.4.4.3.2.2" stretchy="false" xref="Thmproofx23.p1.4.m2.4.4.4.cmml">)</mo></mrow></mrow><mo id="Thmproofx23.p1.4.m2.4.4.3" xref="Thmproofx23.p1.4.m2.4.4.3.cmml">=</mo><mrow id="Thmproofx23.p1.4.m2.4.4.2.2" xref="Thmproofx23.p1.4.m2.4.4.2.3.cmml"><msub id="Thmproofx23.p1.4.m2.3.3.1.1.1" xref="Thmproofx23.p1.4.m2.3.3.1.1.1.cmml"><mi id="Thmproofx23.p1.4.m2.3.3.1.1.1.2" xref="Thmproofx23.p1.4.m2.3.3.1.1.1.2.cmml">max</mi><mrow id="Thmproofx23.p1.4.m2.1.1.1" xref="Thmproofx23.p1.4.m2.1.1.1.cmml"><mi id="Thmproofx23.p1.4.m2.1.1.1.3" xref="Thmproofx23.p1.4.m2.1.1.1.3.cmml">i</mi><mo id="Thmproofx23.p1.4.m2.1.1.1.2" xref="Thmproofx23.p1.4.m2.1.1.1.2.cmml">∈</mo><mrow id="Thmproofx23.p1.4.m2.1.1.1.4.2" xref="Thmproofx23.p1.4.m2.1.1.1.4.1.cmml"><mo id="Thmproofx23.p1.4.m2.1.1.1.4.2.1" stretchy="false" xref="Thmproofx23.p1.4.m2.1.1.1.4.1.1.cmml">[</mo><mi id="Thmproofx23.p1.4.m2.1.1.1.1" xref="Thmproofx23.p1.4.m2.1.1.1.1.cmml">m</mi><mo id="Thmproofx23.p1.4.m2.1.1.1.4.2.2" stretchy="false" xref="Thmproofx23.p1.4.m2.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mo id="Thmproofx23.p1.4.m2.4.4.2.2a" xref="Thmproofx23.p1.4.m2.4.4.2.3.cmml">⁡</mo><mrow id="Thmproofx23.p1.4.m2.4.4.2.2.2" xref="Thmproofx23.p1.4.m2.4.4.2.3.cmml"><mo id="Thmproofx23.p1.4.m2.4.4.2.2.2.2" stretchy="false" xref="Thmproofx23.p1.4.m2.4.4.2.3.cmml">(</mo><mrow id="Thmproofx23.p1.4.m2.4.4.2.2.2.1" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.cmml"><msubsup id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.cmml"><mi id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.2" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.2.cmml">y</mi><mi id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.3" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.3.cmml">i</mi><mi id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.3" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.3.cmml">n</mi></msubsup><mo id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.1" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.1.cmml">−</mo><msubsup id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.cmml"><mi id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.2" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.2.cmml">y</mi><mrow id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.cmml"><mi id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.2" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.2.cmml">i</mi><mo id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.1" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.1.cmml">−</mo><mn id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.3" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.3.cmml">1</mn></mrow><mi id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.3" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.3.cmml">n</mi></msubsup></mrow><mo id="Thmproofx23.p1.4.m2.4.4.2.2.2.3" stretchy="false" xref="Thmproofx23.p1.4.m2.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.4.m2.4b"><apply id="Thmproofx23.p1.4.m2.4.4.cmml" xref="Thmproofx23.p1.4.m2.4.4"><eq id="Thmproofx23.p1.4.m2.4.4.3.cmml" xref="Thmproofx23.p1.4.m2.4.4.3"></eq><apply id="Thmproofx23.p1.4.m2.4.4.4.cmml" xref="Thmproofx23.p1.4.m2.4.4.4"><times id="Thmproofx23.p1.4.m2.4.4.4.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.4.1"></times><ci id="Thmproofx23.p1.4.m2.4.4.4.2.cmml" xref="Thmproofx23.p1.4.m2.4.4.4.2">𝛿</ci><ci id="Thmproofx23.p1.4.m2.2.2.cmml" xref="Thmproofx23.p1.4.m2.2.2">𝑋</ci></apply><apply id="Thmproofx23.p1.4.m2.4.4.2.3.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2"><apply id="Thmproofx23.p1.4.m2.3.3.1.1.1.cmml" xref="Thmproofx23.p1.4.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.4.m2.3.3.1.1.1.1.cmml" xref="Thmproofx23.p1.4.m2.3.3.1.1.1">subscript</csymbol><max id="Thmproofx23.p1.4.m2.3.3.1.1.1.2.cmml" xref="Thmproofx23.p1.4.m2.3.3.1.1.1.2"></max><apply id="Thmproofx23.p1.4.m2.1.1.1.cmml" xref="Thmproofx23.p1.4.m2.1.1.1"><in id="Thmproofx23.p1.4.m2.1.1.1.2.cmml" xref="Thmproofx23.p1.4.m2.1.1.1.2"></in><ci id="Thmproofx23.p1.4.m2.1.1.1.3.cmml" xref="Thmproofx23.p1.4.m2.1.1.1.3">𝑖</ci><apply id="Thmproofx23.p1.4.m2.1.1.1.4.1.cmml" xref="Thmproofx23.p1.4.m2.1.1.1.4.2"><csymbol cd="latexml" id="Thmproofx23.p1.4.m2.1.1.1.4.1.1.cmml" xref="Thmproofx23.p1.4.m2.1.1.1.4.2.1">delimited-[]</csymbol><ci id="Thmproofx23.p1.4.m2.1.1.1.1.cmml" xref="Thmproofx23.p1.4.m2.1.1.1.1">𝑚</ci></apply></apply></apply><apply id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1"><minus id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.1"></minus><apply id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2">superscript</csymbol><apply id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2">subscript</csymbol><ci id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.2.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.2">𝑦</ci><ci id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.3.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.2.3">𝑖</ci></apply><ci id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.3.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.2.3">𝑛</ci></apply><apply id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3">superscript</csymbol><apply id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3">subscript</csymbol><ci id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.2.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.2">𝑦</ci><apply id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3"><minus id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.1.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.1"></minus><ci id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.2.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.2">𝑖</ci><cn id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.3.cmml" type="integer" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.2.3.3">1</cn></apply></apply><ci id="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.3.cmml" xref="Thmproofx23.p1.4.m2.4.4.2.2.2.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.4.m2.4c">\delta(X)=\max_{i\in[m]}(y_{i}^{n}-y_{i-1}^{n})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.4.m2.4d">italic_δ ( italic_X ) = roman_max start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math> and let <math alttext="i^{*}\in\operatorname*{arg\,max}_{i}(y_{i}^{n}-y_{i-1}^{n})" class="ltx_Math" display="inline" id="Thmproofx23.p1.5.m3.2"><semantics id="Thmproofx23.p1.5.m3.2a"><mrow id="Thmproofx23.p1.5.m3.2.2" xref="Thmproofx23.p1.5.m3.2.2.cmml"><msup id="Thmproofx23.p1.5.m3.2.2.4" xref="Thmproofx23.p1.5.m3.2.2.4.cmml"><mi id="Thmproofx23.p1.5.m3.2.2.4.2" xref="Thmproofx23.p1.5.m3.2.2.4.2.cmml">i</mi><mo id="Thmproofx23.p1.5.m3.2.2.4.3" xref="Thmproofx23.p1.5.m3.2.2.4.3.cmml">∗</mo></msup><mo id="Thmproofx23.p1.5.m3.2.2.3" xref="Thmproofx23.p1.5.m3.2.2.3.cmml">∈</mo><mrow id="Thmproofx23.p1.5.m3.2.2.2.2" xref="Thmproofx23.p1.5.m3.2.2.2.3.cmml"><msub id="Thmproofx23.p1.5.m3.1.1.1.1.1" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.cmml"><mrow id="Thmproofx23.p1.5.m3.1.1.1.1.1.2" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2.cmml"><mi id="Thmproofx23.p1.5.m3.1.1.1.1.1.2.2" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2.2.cmml">arg</mi><mo id="Thmproofx23.p1.5.m3.1.1.1.1.1.2.1" lspace="0.170em" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="Thmproofx23.p1.5.m3.1.1.1.1.1.2.3" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2.3.cmml">max</mi></mrow><mi id="Thmproofx23.p1.5.m3.1.1.1.1.1.3" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx23.p1.5.m3.2.2.2.2a" xref="Thmproofx23.p1.5.m3.2.2.2.3.cmml">⁡</mo><mrow id="Thmproofx23.p1.5.m3.2.2.2.2.2" xref="Thmproofx23.p1.5.m3.2.2.2.3.cmml"><mo id="Thmproofx23.p1.5.m3.2.2.2.2.2.2" stretchy="false" xref="Thmproofx23.p1.5.m3.2.2.2.3.cmml">(</mo><mrow id="Thmproofx23.p1.5.m3.2.2.2.2.2.1" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.cmml"><msubsup id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.cmml"><mi id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.2" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.2.cmml">y</mi><mi id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.3" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.3.cmml">i</mi><mi id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.3" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.3.cmml">n</mi></msubsup><mo id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.1" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.1.cmml">−</mo><msubsup id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.cmml"><mi id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.2" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.2.cmml">y</mi><mrow id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.cmml"><mi id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.2" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.2.cmml">i</mi><mo id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.1" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.1.cmml">−</mo><mn id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.3" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.3.cmml">1</mn></mrow><mi id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.3" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.3.cmml">n</mi></msubsup></mrow><mo id="Thmproofx23.p1.5.m3.2.2.2.2.2.3" stretchy="false" xref="Thmproofx23.p1.5.m3.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.5.m3.2b"><apply id="Thmproofx23.p1.5.m3.2.2.cmml" xref="Thmproofx23.p1.5.m3.2.2"><in id="Thmproofx23.p1.5.m3.2.2.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.3"></in><apply id="Thmproofx23.p1.5.m3.2.2.4.cmml" xref="Thmproofx23.p1.5.m3.2.2.4"><csymbol cd="ambiguous" id="Thmproofx23.p1.5.m3.2.2.4.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.4">superscript</csymbol><ci id="Thmproofx23.p1.5.m3.2.2.4.2.cmml" xref="Thmproofx23.p1.5.m3.2.2.4.2">𝑖</ci><times id="Thmproofx23.p1.5.m3.2.2.4.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.4.3"></times></apply><apply id="Thmproofx23.p1.5.m3.2.2.2.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2"><apply id="Thmproofx23.p1.5.m3.1.1.1.1.1.cmml" xref="Thmproofx23.p1.5.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.5.m3.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.5.m3.1.1.1.1.1">subscript</csymbol><apply id="Thmproofx23.p1.5.m3.1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2"><times id="Thmproofx23.p1.5.m3.1.1.1.1.1.2.1.cmml" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2.1"></times><ci id="Thmproofx23.p1.5.m3.1.1.1.1.1.2.2.cmml" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2.2">arg</ci><ci id="Thmproofx23.p1.5.m3.1.1.1.1.1.2.3.cmml" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.2.3">max</ci></apply><ci id="Thmproofx23.p1.5.m3.1.1.1.1.1.3.cmml" xref="Thmproofx23.p1.5.m3.1.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1"><minus id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.1"></minus><apply id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2">superscript</csymbol><apply id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2">subscript</csymbol><ci id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.2.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.2">𝑦</ci><ci id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.2.3">𝑖</ci></apply><ci id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.2.3">𝑛</ci></apply><apply id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3">superscript</csymbol><apply id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3">subscript</csymbol><ci id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.2.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.2">𝑦</ci><apply id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3"><minus id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.1.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.1"></minus><ci id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.2.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.2">𝑖</ci><cn id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.3.cmml" type="integer" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.2.3.3">1</cn></apply></apply><ci id="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.3.cmml" xref="Thmproofx23.p1.5.m3.2.2.2.2.2.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.5.m3.2c">i^{*}\in\operatorname*{arg\,max}_{i}(y_{i}^{n}-y_{i-1}^{n})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.5.m3.2d">italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx60"> <tbody id="Pt0.A4.Ex105"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(*)" class="ltx_Math" display="inline" id="Pt0.A4.Ex105.m1.1"><semantics id="Pt0.A4.Ex105.m1.1a"><mrow id="Pt0.A4.Ex105.m1.1.2.2"><mo id="Pt0.A4.Ex105.m1.1.2.2.1" stretchy="false">(</mo><mo id="Pt0.A4.Ex105.m1.1.1" lspace="0em" rspace="0em" xref="Pt0.A4.Ex105.m1.1.1.cmml">∗</mo><mo id="Pt0.A4.Ex105.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex105.m1.1b"><times id="Pt0.A4.Ex105.m1.1.1.cmml" xref="Pt0.A4.Ex105.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex105.m1.1c">\displaystyle(*)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex105.m1.1d">( ∗ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\alpha\sum_{i\in[m]}k^{d}_{i}\left(y_{i}^{n}-y_{i-1}^{n}\right)+% \sum_{i\in[m]}|p_{I_{d}(i)}|\left(y_{i}^{n}-y_{i-1}^{n}\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex105.m2.6"><semantics id="Pt0.A4.Ex105.m2.6a"><mrow id="Pt0.A4.Ex105.m2.6.6" xref="Pt0.A4.Ex105.m2.6.6.cmml"><mi id="Pt0.A4.Ex105.m2.6.6.5" xref="Pt0.A4.Ex105.m2.6.6.5.cmml"></mi><mo id="Pt0.A4.Ex105.m2.6.6.4" xref="Pt0.A4.Ex105.m2.6.6.4.cmml">=</mo><mrow id="Pt0.A4.Ex105.m2.6.6.3" xref="Pt0.A4.Ex105.m2.6.6.3.cmml"><mrow id="Pt0.A4.Ex105.m2.4.4.1.1" xref="Pt0.A4.Ex105.m2.4.4.1.1.cmml"><mi id="Pt0.A4.Ex105.m2.4.4.1.1.3" xref="Pt0.A4.Ex105.m2.4.4.1.1.3.cmml">α</mi><mo id="Pt0.A4.Ex105.m2.4.4.1.1.2" xref="Pt0.A4.Ex105.m2.4.4.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex105.m2.4.4.1.1.1" xref="Pt0.A4.Ex105.m2.4.4.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex105.m2.4.4.1.1.1.2" 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xref="Pt0.A4.Ex105.m2.6.6.3.3.2.2.1.1.3.2.3.3">1</cn></apply></apply><ci id="Pt0.A4.Ex105.m2.6.6.3.3.2.2.1.1.3.3.cmml" xref="Pt0.A4.Ex105.m2.6.6.3.3.2.2.1.1.3.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex105.m2.6c">\displaystyle=\alpha\sum_{i\in[m]}k^{d}_{i}\left(y_{i}^{n}-y_{i-1}^{n}\right)+% \sum_{i\in[m]}|p_{I_{d}(i)}|\left(y_{i}^{n}-y_{i-1}^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex105.m2.6d">= italic_α ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) + ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT | 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id="Pt0.A4.Ex106.m1.5d">= italic_α ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) + ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_m ] end_POSTSUBSCRIPT | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx23.p1.7">as <math alttext="|p_{1}|=...=|p_{m}|" class="ltx_Math" display="inline" 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id="Thmproofx23.p1.6.m1.1.1.1.2.1.cmml" xref="Thmproofx23.p1.6.m1.1.1.1.1.2"></abs><apply id="Thmproofx23.p1.6.m1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.6.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.6.m1.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.6.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx23.p1.6.m1.1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.6.m1.1.1.1.1.1.2">𝑝</ci><cn id="Thmproofx23.p1.6.m1.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx23.p1.6.m1.1.1.1.1.1.3">1</cn></apply></apply><ci id="Thmproofx23.p1.6.m1.2.2.5.cmml" xref="Thmproofx23.p1.6.m1.2.2.5">…</ci></apply><apply id="Thmproofx23.p1.6.m1.2.2c.cmml" xref="Thmproofx23.p1.6.m1.2.2"><eq id="Thmproofx23.p1.6.m1.2.2.6.cmml" xref="Thmproofx23.p1.6.m1.2.2.6"></eq><share href="https://arxiv.org/html/2308.08047v3#Thmproofx23.p1.6.m1.2.2.5.cmml" id="Thmproofx23.p1.6.m1.2.2d.cmml" xref="Thmproofx23.p1.6.m1.2.2"></share><apply id="Thmproofx23.p1.6.m1.2.2.2.2.cmml" xref="Thmproofx23.p1.6.m1.2.2.2.1"><abs id="Thmproofx23.p1.6.m1.2.2.2.2.1.cmml" xref="Thmproofx23.p1.6.m1.2.2.2.1.2"></abs><apply id="Thmproofx23.p1.6.m1.2.2.2.1.1.cmml" xref="Thmproofx23.p1.6.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.6.m1.2.2.2.1.1.1.cmml" xref="Thmproofx23.p1.6.m1.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx23.p1.6.m1.2.2.2.1.1.2.cmml" xref="Thmproofx23.p1.6.m1.2.2.2.1.1.2">𝑝</ci><ci id="Thmproofx23.p1.6.m1.2.2.2.1.1.3.cmml" xref="Thmproofx23.p1.6.m1.2.2.2.1.1.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.6.m1.2c">|p_{1}|=...=|p_{m}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.6.m1.2d">| italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = … = | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |</annotation></semantics></math>. So the above is maximal when <math alttext="k^{d}_{i}=k\cdot 1_{i=i^{*}}" class="ltx_Math" display="inline" id="Thmproofx23.p1.7.m2.1"><semantics id="Thmproofx23.p1.7.m2.1a"><mrow id="Thmproofx23.p1.7.m2.1.1" xref="Thmproofx23.p1.7.m2.1.1.cmml"><msubsup id="Thmproofx23.p1.7.m2.1.1.2" xref="Thmproofx23.p1.7.m2.1.1.2.cmml"><mi id="Thmproofx23.p1.7.m2.1.1.2.2.2" xref="Thmproofx23.p1.7.m2.1.1.2.2.2.cmml">k</mi><mi id="Thmproofx23.p1.7.m2.1.1.2.3" xref="Thmproofx23.p1.7.m2.1.1.2.3.cmml">i</mi><mi id="Thmproofx23.p1.7.m2.1.1.2.2.3" xref="Thmproofx23.p1.7.m2.1.1.2.2.3.cmml">d</mi></msubsup><mo id="Thmproofx23.p1.7.m2.1.1.1" xref="Thmproofx23.p1.7.m2.1.1.1.cmml">=</mo><mrow id="Thmproofx23.p1.7.m2.1.1.3" xref="Thmproofx23.p1.7.m2.1.1.3.cmml"><mi id="Thmproofx23.p1.7.m2.1.1.3.2" xref="Thmproofx23.p1.7.m2.1.1.3.2.cmml">k</mi><mo id="Thmproofx23.p1.7.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="Thmproofx23.p1.7.m2.1.1.3.1.cmml">⋅</mo><msub id="Thmproofx23.p1.7.m2.1.1.3.3" xref="Thmproofx23.p1.7.m2.1.1.3.3.cmml"><mn id="Thmproofx23.p1.7.m2.1.1.3.3.2" xref="Thmproofx23.p1.7.m2.1.1.3.3.2.cmml">1</mn><mrow id="Thmproofx23.p1.7.m2.1.1.3.3.3" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.cmml"><mi id="Thmproofx23.p1.7.m2.1.1.3.3.3.2" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.2.cmml">i</mi><mo id="Thmproofx23.p1.7.m2.1.1.3.3.3.1" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.1.cmml">=</mo><msup id="Thmproofx23.p1.7.m2.1.1.3.3.3.3" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.3.cmml"><mi id="Thmproofx23.p1.7.m2.1.1.3.3.3.3.2" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.3.2.cmml">i</mi><mo id="Thmproofx23.p1.7.m2.1.1.3.3.3.3.3" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.3.3.cmml">∗</mo></msup></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.7.m2.1b"><apply id="Thmproofx23.p1.7.m2.1.1.cmml" xref="Thmproofx23.p1.7.m2.1.1"><eq id="Thmproofx23.p1.7.m2.1.1.1.cmml" xref="Thmproofx23.p1.7.m2.1.1.1"></eq><apply id="Thmproofx23.p1.7.m2.1.1.2.cmml" xref="Thmproofx23.p1.7.m2.1.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.7.m2.1.1.2.1.cmml" xref="Thmproofx23.p1.7.m2.1.1.2">subscript</csymbol><apply id="Thmproofx23.p1.7.m2.1.1.2.2.cmml" xref="Thmproofx23.p1.7.m2.1.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.7.m2.1.1.2.2.1.cmml" xref="Thmproofx23.p1.7.m2.1.1.2">superscript</csymbol><ci id="Thmproofx23.p1.7.m2.1.1.2.2.2.cmml" xref="Thmproofx23.p1.7.m2.1.1.2.2.2">𝑘</ci><ci id="Thmproofx23.p1.7.m2.1.1.2.2.3.cmml" xref="Thmproofx23.p1.7.m2.1.1.2.2.3">𝑑</ci></apply><ci id="Thmproofx23.p1.7.m2.1.1.2.3.cmml" xref="Thmproofx23.p1.7.m2.1.1.2.3">𝑖</ci></apply><apply id="Thmproofx23.p1.7.m2.1.1.3.cmml" xref="Thmproofx23.p1.7.m2.1.1.3"><ci id="Thmproofx23.p1.7.m2.1.1.3.1.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.1">⋅</ci><ci id="Thmproofx23.p1.7.m2.1.1.3.2.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.2">𝑘</ci><apply id="Thmproofx23.p1.7.m2.1.1.3.3.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.7.m2.1.1.3.3.1.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3">subscript</csymbol><cn id="Thmproofx23.p1.7.m2.1.1.3.3.2.cmml" type="integer" xref="Thmproofx23.p1.7.m2.1.1.3.3.2">1</cn><apply id="Thmproofx23.p1.7.m2.1.1.3.3.3.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3.3"><eq id="Thmproofx23.p1.7.m2.1.1.3.3.3.1.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.1"></eq><ci id="Thmproofx23.p1.7.m2.1.1.3.3.3.2.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.2">𝑖</ci><apply id="Thmproofx23.p1.7.m2.1.1.3.3.3.3.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.7.m2.1.1.3.3.3.3.1.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.3">superscript</csymbol><ci id="Thmproofx23.p1.7.m2.1.1.3.3.3.3.2.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.3.2">𝑖</ci><times id="Thmproofx23.p1.7.m2.1.1.3.3.3.3.3.cmml" xref="Thmproofx23.p1.7.m2.1.1.3.3.3.3.3"></times></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.7.m2.1c">k^{d}_{i}=k\cdot 1_{i=i^{*}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.7.m2.1d">italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_k ⋅ 1 start_POSTSUBSCRIPT italic_i = italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and the cost to the agents is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx61"> <tbody id="Pt0.A4.Ex107"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle F_{n}(\sigma,\alpha)=f_{(i^{*},...,i^{*})}(\sigma,\alpha)=|p_{1}% |+\delta(X)k\alpha" class="ltx_Math" display="inline" id="Pt0.A4.Ex107.m1.9"><semantics id="Pt0.A4.Ex107.m1.9a"><mrow id="Pt0.A4.Ex107.m1.9.9" xref="Pt0.A4.Ex107.m1.9.9.cmml"><mrow id="Pt0.A4.Ex107.m1.9.9.3" xref="Pt0.A4.Ex107.m1.9.9.3.cmml"><msub id="Pt0.A4.Ex107.m1.9.9.3.2" xref="Pt0.A4.Ex107.m1.9.9.3.2.cmml"><mi id="Pt0.A4.Ex107.m1.9.9.3.2.2" xref="Pt0.A4.Ex107.m1.9.9.3.2.2.cmml">F</mi><mi id="Pt0.A4.Ex107.m1.9.9.3.2.3" xref="Pt0.A4.Ex107.m1.9.9.3.2.3.cmml">n</mi></msub><mo id="Pt0.A4.Ex107.m1.9.9.3.1" xref="Pt0.A4.Ex107.m1.9.9.3.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex107.m1.9.9.3.3.2" xref="Pt0.A4.Ex107.m1.9.9.3.3.1.cmml"><mo id="Pt0.A4.Ex107.m1.9.9.3.3.2.1" stretchy="false" xref="Pt0.A4.Ex107.m1.9.9.3.3.1.cmml">(</mo><mi id="Pt0.A4.Ex107.m1.4.4" xref="Pt0.A4.Ex107.m1.4.4.cmml">σ</mi><mo id="Pt0.A4.Ex107.m1.9.9.3.3.2.2" xref="Pt0.A4.Ex107.m1.9.9.3.3.1.cmml">,</mo><mi id="Pt0.A4.Ex107.m1.5.5" xref="Pt0.A4.Ex107.m1.5.5.cmml">α</mi><mo id="Pt0.A4.Ex107.m1.9.9.3.3.2.3" stretchy="false" xref="Pt0.A4.Ex107.m1.9.9.3.3.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex107.m1.9.9.4" xref="Pt0.A4.Ex107.m1.9.9.4.cmml">=</mo><mrow id="Pt0.A4.Ex107.m1.9.9.5" xref="Pt0.A4.Ex107.m1.9.9.5.cmml"><msub id="Pt0.A4.Ex107.m1.9.9.5.2" xref="Pt0.A4.Ex107.m1.9.9.5.2.cmml"><mi id="Pt0.A4.Ex107.m1.9.9.5.2.2" xref="Pt0.A4.Ex107.m1.9.9.5.2.2.cmml">f</mi><mrow id="Pt0.A4.Ex107.m1.3.3.3.3" xref="Pt0.A4.Ex107.m1.3.3.3.4.cmml"><mo id="Pt0.A4.Ex107.m1.3.3.3.3.3" stretchy="false" xref="Pt0.A4.Ex107.m1.3.3.3.4.cmml">(</mo><msup id="Pt0.A4.Ex107.m1.2.2.2.2.1" xref="Pt0.A4.Ex107.m1.2.2.2.2.1.cmml"><mi id="Pt0.A4.Ex107.m1.2.2.2.2.1.2" xref="Pt0.A4.Ex107.m1.2.2.2.2.1.2.cmml">i</mi><mo id="Pt0.A4.Ex107.m1.2.2.2.2.1.3" xref="Pt0.A4.Ex107.m1.2.2.2.2.1.3.cmml">∗</mo></msup><mo id="Pt0.A4.Ex107.m1.3.3.3.3.4" xref="Pt0.A4.Ex107.m1.3.3.3.4.cmml">,</mo><mi id="Pt0.A4.Ex107.m1.1.1.1.1" mathvariant="normal" xref="Pt0.A4.Ex107.m1.1.1.1.1.cmml">…</mi><mo id="Pt0.A4.Ex107.m1.3.3.3.3.5" xref="Pt0.A4.Ex107.m1.3.3.3.4.cmml">,</mo><msup id="Pt0.A4.Ex107.m1.3.3.3.3.2" xref="Pt0.A4.Ex107.m1.3.3.3.3.2.cmml"><mi id="Pt0.A4.Ex107.m1.3.3.3.3.2.2" xref="Pt0.A4.Ex107.m1.3.3.3.3.2.2.cmml">i</mi><mo 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id="Pt0.A4.Ex107.m1.9d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = italic_f start_POSTSUBSCRIPT ( italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , … , italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT ( italic_σ , italic_α ) = | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + italic_δ ( italic_X ) italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx23.p1.10">If <math alttext="X=U_{P_{st}}" class="ltx_Math" display="inline" id="Thmproofx23.p1.8.m1.1"><semantics id="Thmproofx23.p1.8.m1.1a"><mrow id="Thmproofx23.p1.8.m1.1.1" xref="Thmproofx23.p1.8.m1.1.1.cmml"><mi id="Thmproofx23.p1.8.m1.1.1.2" xref="Thmproofx23.p1.8.m1.1.1.2.cmml">X</mi><mo id="Thmproofx23.p1.8.m1.1.1.1" xref="Thmproofx23.p1.8.m1.1.1.1.cmml">=</mo><msub id="Thmproofx23.p1.8.m1.1.1.3" xref="Thmproofx23.p1.8.m1.1.1.3.cmml"><mi 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id="Thmproofx23.p1.9.m2.2.2.2.2.2.3.cmml" xref="Thmproofx23.p1.9.m2.2.2.2.2.2.4"><ci id="Thmproofx23.p1.9.m2.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.9.m2.1.1.1.1.1.1">𝑚</ci><ci id="Thmproofx23.p1.9.m2.2.2.2.2.2.2.cmml" xref="Thmproofx23.p1.9.m2.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Thmproofx23.p1.9.m2.2.2.4.cmml" xref="Thmproofx23.p1.9.m2.2.2.4">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.9.m2.3c">\delta(X)=\frac{\mathcal{E}_{m,n}}{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.9.m2.3d">italic_δ ( italic_X ) = divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG</annotation></semantics></math>, <math alttext="i^{*}=m" class="ltx_Math" display="inline" id="Thmproofx23.p1.10.m3.1"><semantics id="Thmproofx23.p1.10.m3.1a"><mrow id="Thmproofx23.p1.10.m3.1.1" xref="Thmproofx23.p1.10.m3.1.1.cmml"><msup id="Thmproofx23.p1.10.m3.1.1.2" xref="Thmproofx23.p1.10.m3.1.1.2.cmml"><mi id="Thmproofx23.p1.10.m3.1.1.2.2" xref="Thmproofx23.p1.10.m3.1.1.2.2.cmml">i</mi><mo id="Thmproofx23.p1.10.m3.1.1.2.3" xref="Thmproofx23.p1.10.m3.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproofx23.p1.10.m3.1.1.1" xref="Thmproofx23.p1.10.m3.1.1.1.cmml">=</mo><mi id="Thmproofx23.p1.10.m3.1.1.3" xref="Thmproofx23.p1.10.m3.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.10.m3.1b"><apply id="Thmproofx23.p1.10.m3.1.1.cmml" xref="Thmproofx23.p1.10.m3.1.1"><eq id="Thmproofx23.p1.10.m3.1.1.1.cmml" xref="Thmproofx23.p1.10.m3.1.1.1"></eq><apply id="Thmproofx23.p1.10.m3.1.1.2.cmml" xref="Thmproofx23.p1.10.m3.1.1.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.10.m3.1.1.2.1.cmml" xref="Thmproofx23.p1.10.m3.1.1.2">superscript</csymbol><ci id="Thmproofx23.p1.10.m3.1.1.2.2.cmml" xref="Thmproofx23.p1.10.m3.1.1.2.2">𝑖</ci><times id="Thmproofx23.p1.10.m3.1.1.2.3.cmml" xref="Thmproofx23.p1.10.m3.1.1.2.3"></times></apply><ci id="Thmproofx23.p1.10.m3.1.1.3.cmml" xref="Thmproofx23.p1.10.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.10.m3.1c">i^{*}=m</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.10.m3.1d">italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_m</annotation></semantics></math> and we get</p> <table class="ltx_equation ltx_eqn_table" id="Pt0.A4.Ex108"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F_{n}(U_{P_{st}}^{n},\alpha)=f_{(m,...,m)}(U_{P_{st}}^{n},\alpha)=|p_{1}|+% \frac{\mathcal{E}_{m,n}}{m}k\alpha" class="ltx_Math" display="block" id="Pt0.A4.Ex108.m1.10"><semantics id="Pt0.A4.Ex108.m1.10a"><mrow id="Pt0.A4.Ex108.m1.10.10" xref="Pt0.A4.Ex108.m1.10.10.cmml"><mrow id="Pt0.A4.Ex108.m1.8.8.1" xref="Pt0.A4.Ex108.m1.8.8.1.cmml"><msub id="Pt0.A4.Ex108.m1.8.8.1.3" xref="Pt0.A4.Ex108.m1.8.8.1.3.cmml"><mi id="Pt0.A4.Ex108.m1.8.8.1.3.2" xref="Pt0.A4.Ex108.m1.8.8.1.3.2.cmml">F</mi><mi id="Pt0.A4.Ex108.m1.8.8.1.3.3" xref="Pt0.A4.Ex108.m1.8.8.1.3.3.cmml">n</mi></msub><mo id="Pt0.A4.Ex108.m1.8.8.1.2" xref="Pt0.A4.Ex108.m1.8.8.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex108.m1.8.8.1.1.1" xref="Pt0.A4.Ex108.m1.8.8.1.1.2.cmml"><mo id="Pt0.A4.Ex108.m1.8.8.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex108.m1.8.8.1.1.2.cmml">(</mo><msubsup id="Pt0.A4.Ex108.m1.8.8.1.1.1.1" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.cmml"><mi id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.2" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.2.cmml">U</mi><msub id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.cmml"><mi id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.2" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.2.cmml">P</mi><mrow id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3.2" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3.2.cmml">s</mi><mo id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3.1" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3.3" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Pt0.A4.Ex108.m1.8.8.1.1.1.1.3" xref="Pt0.A4.Ex108.m1.8.8.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Pt0.A4.Ex108.m1.8.8.1.1.1.3" xref="Pt0.A4.Ex108.m1.8.8.1.1.2.cmml">,</mo><mi id="Pt0.A4.Ex108.m1.6.6" xref="Pt0.A4.Ex108.m1.6.6.cmml">α</mi><mo id="Pt0.A4.Ex108.m1.8.8.1.1.1.4" stretchy="false" xref="Pt0.A4.Ex108.m1.8.8.1.1.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex108.m1.10.10.5" xref="Pt0.A4.Ex108.m1.10.10.5.cmml">=</mo><mrow id="Pt0.A4.Ex108.m1.9.9.2" xref="Pt0.A4.Ex108.m1.9.9.2.cmml"><msub id="Pt0.A4.Ex108.m1.9.9.2.3" xref="Pt0.A4.Ex108.m1.9.9.2.3.cmml"><mi id="Pt0.A4.Ex108.m1.9.9.2.3.2" xref="Pt0.A4.Ex108.m1.9.9.2.3.2.cmml">f</mi><mrow id="Pt0.A4.Ex108.m1.3.3.3.5" xref="Pt0.A4.Ex108.m1.3.3.3.4.cmml"><mo id="Pt0.A4.Ex108.m1.3.3.3.5.1" stretchy="false" xref="Pt0.A4.Ex108.m1.3.3.3.4.cmml">(</mo><mi id="Pt0.A4.Ex108.m1.1.1.1.1" xref="Pt0.A4.Ex108.m1.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.Ex108.m1.3.3.3.5.2" xref="Pt0.A4.Ex108.m1.3.3.3.4.cmml">,</mo><mi id="Pt0.A4.Ex108.m1.2.2.2.2" mathvariant="normal" xref="Pt0.A4.Ex108.m1.2.2.2.2.cmml">…</mi><mo id="Pt0.A4.Ex108.m1.3.3.3.5.3" xref="Pt0.A4.Ex108.m1.3.3.3.4.cmml">,</mo><mi id="Pt0.A4.Ex108.m1.3.3.3.3" xref="Pt0.A4.Ex108.m1.3.3.3.3.cmml">m</mi><mo id="Pt0.A4.Ex108.m1.3.3.3.5.4" stretchy="false" xref="Pt0.A4.Ex108.m1.3.3.3.4.cmml">)</mo></mrow></msub><mo id="Pt0.A4.Ex108.m1.9.9.2.2" xref="Pt0.A4.Ex108.m1.9.9.2.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex108.m1.9.9.2.1.1" xref="Pt0.A4.Ex108.m1.9.9.2.1.2.cmml"><mo id="Pt0.A4.Ex108.m1.9.9.2.1.1.2" stretchy="false" xref="Pt0.A4.Ex108.m1.9.9.2.1.2.cmml">(</mo><msubsup id="Pt0.A4.Ex108.m1.9.9.2.1.1.1" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.cmml"><mi id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.2" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.2.cmml">U</mi><msub id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.cmml"><mi id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.2" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.2.cmml">P</mi><mrow id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3.2" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3.2.cmml">s</mi><mo id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3.1" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3.3" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Pt0.A4.Ex108.m1.9.9.2.1.1.1.3" xref="Pt0.A4.Ex108.m1.9.9.2.1.1.1.3.cmml">n</mi></msubsup><mo id="Pt0.A4.Ex108.m1.9.9.2.1.1.3" xref="Pt0.A4.Ex108.m1.9.9.2.1.2.cmml">,</mo><mi id="Pt0.A4.Ex108.m1.7.7" xref="Pt0.A4.Ex108.m1.7.7.cmml">α</mi><mo id="Pt0.A4.Ex108.m1.9.9.2.1.1.4" stretchy="false" xref="Pt0.A4.Ex108.m1.9.9.2.1.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex108.m1.10.10.6" xref="Pt0.A4.Ex108.m1.10.10.6.cmml">=</mo><mrow id="Pt0.A4.Ex108.m1.10.10.3" xref="Pt0.A4.Ex108.m1.10.10.3.cmml"><mrow id="Pt0.A4.Ex108.m1.10.10.3.1.1" xref="Pt0.A4.Ex108.m1.10.10.3.1.2.cmml"><mo id="Pt0.A4.Ex108.m1.10.10.3.1.1.2" stretchy="false" xref="Pt0.A4.Ex108.m1.10.10.3.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex108.m1.10.10.3.1.1.1" xref="Pt0.A4.Ex108.m1.10.10.3.1.1.1.cmml"><mi id="Pt0.A4.Ex108.m1.10.10.3.1.1.1.2" xref="Pt0.A4.Ex108.m1.10.10.3.1.1.1.2.cmml">p</mi><mn id="Pt0.A4.Ex108.m1.10.10.3.1.1.1.3" xref="Pt0.A4.Ex108.m1.10.10.3.1.1.1.3.cmml">1</mn></msub><mo id="Pt0.A4.Ex108.m1.10.10.3.1.1.3" stretchy="false" xref="Pt0.A4.Ex108.m1.10.10.3.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex108.m1.10.10.3.2" xref="Pt0.A4.Ex108.m1.10.10.3.2.cmml">+</mo><mrow id="Pt0.A4.Ex108.m1.10.10.3.3" xref="Pt0.A4.Ex108.m1.10.10.3.3.cmml"><mfrac id="Pt0.A4.Ex108.m1.5.5" xref="Pt0.A4.Ex108.m1.5.5.cmml"><msub id="Pt0.A4.Ex108.m1.5.5.2" xref="Pt0.A4.Ex108.m1.5.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Pt0.A4.Ex108.m1.5.5.2.4" xref="Pt0.A4.Ex108.m1.5.5.2.4.cmml">ℰ</mi><mrow id="Pt0.A4.Ex108.m1.5.5.2.2.2.4" xref="Pt0.A4.Ex108.m1.5.5.2.2.2.3.cmml"><mi id="Pt0.A4.Ex108.m1.4.4.1.1.1.1" xref="Pt0.A4.Ex108.m1.4.4.1.1.1.1.cmml">m</mi><mo id="Pt0.A4.Ex108.m1.5.5.2.2.2.4.1" xref="Pt0.A4.Ex108.m1.5.5.2.2.2.3.cmml">,</mo><mi id="Pt0.A4.Ex108.m1.5.5.2.2.2.2" xref="Pt0.A4.Ex108.m1.5.5.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Pt0.A4.Ex108.m1.5.5.4" xref="Pt0.A4.Ex108.m1.5.5.4.cmml">m</mi></mfrac><mo id="Pt0.A4.Ex108.m1.10.10.3.3.1" xref="Pt0.A4.Ex108.m1.10.10.3.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex108.m1.10.10.3.3.2" xref="Pt0.A4.Ex108.m1.10.10.3.3.2.cmml">k</mi><mo id="Pt0.A4.Ex108.m1.10.10.3.3.1a" xref="Pt0.A4.Ex108.m1.10.10.3.3.1.cmml">⁢</mo><mi id="Pt0.A4.Ex108.m1.10.10.3.3.3" xref="Pt0.A4.Ex108.m1.10.10.3.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex108.m1.10b"><apply id="Pt0.A4.Ex108.m1.10.10.cmml" xref="Pt0.A4.Ex108.m1.10.10"><and 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id="Pt0.A4.Ex108.m1.10d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = italic_f start_POSTSUBSCRIPT ( italic_m , … , italic_m ) end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG italic_k italic_α</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmproofx23.p1.14">Now suppose otherwise. The paths have identical lengths, so wlog we reorder them such that <math alttext="X(p^{d}_{m})>\frac{1}{m}" class="ltx_Math" display="inline" id="Thmproofx23.p1.11.m1.1"><semantics id="Thmproofx23.p1.11.m1.1a"><mrow id="Thmproofx23.p1.11.m1.1.1" xref="Thmproofx23.p1.11.m1.1.1.cmml"><mrow id="Thmproofx23.p1.11.m1.1.1.1" xref="Thmproofx23.p1.11.m1.1.1.1.cmml"><mi id="Thmproofx23.p1.11.m1.1.1.1.3" xref="Thmproofx23.p1.11.m1.1.1.1.3.cmml">X</mi><mo id="Thmproofx23.p1.11.m1.1.1.1.2" xref="Thmproofx23.p1.11.m1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx23.p1.11.m1.1.1.1.1.1" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.cmml"><mo id="Thmproofx23.p1.11.m1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Thmproofx23.p1.11.m1.1.1.1.1.1.1" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.cmml"><mi id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.2" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.2.cmml">p</mi><mi id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.3" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.3.cmml">m</mi><mi id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.3" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Thmproofx23.p1.11.m1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx23.p1.11.m1.1.1.2" xref="Thmproofx23.p1.11.m1.1.1.2.cmml">></mo><mfrac id="Thmproofx23.p1.11.m1.1.1.3" xref="Thmproofx23.p1.11.m1.1.1.3.cmml"><mn id="Thmproofx23.p1.11.m1.1.1.3.2" xref="Thmproofx23.p1.11.m1.1.1.3.2.cmml">1</mn><mi id="Thmproofx23.p1.11.m1.1.1.3.3" xref="Thmproofx23.p1.11.m1.1.1.3.3.cmml">m</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.11.m1.1b"><apply id="Thmproofx23.p1.11.m1.1.1.cmml" xref="Thmproofx23.p1.11.m1.1.1"><gt id="Thmproofx23.p1.11.m1.1.1.2.cmml" xref="Thmproofx23.p1.11.m1.1.1.2"></gt><apply id="Thmproofx23.p1.11.m1.1.1.1.cmml" xref="Thmproofx23.p1.11.m1.1.1.1"><times id="Thmproofx23.p1.11.m1.1.1.1.2.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.2"></times><ci id="Thmproofx23.p1.11.m1.1.1.1.3.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.3">𝑋</ci><apply id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.1.1">subscript</csymbol><apply id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.1.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.2.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.2">𝑝</ci><ci id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.3.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx23.p1.11.m1.1.1.1.1.1.1.3.cmml" xref="Thmproofx23.p1.11.m1.1.1.1.1.1.1.3">𝑚</ci></apply></apply><apply id="Thmproofx23.p1.11.m1.1.1.3.cmml" xref="Thmproofx23.p1.11.m1.1.1.3"><divide id="Thmproofx23.p1.11.m1.1.1.3.1.cmml" xref="Thmproofx23.p1.11.m1.1.1.3"></divide><cn id="Thmproofx23.p1.11.m1.1.1.3.2.cmml" type="integer" xref="Thmproofx23.p1.11.m1.1.1.3.2">1</cn><ci id="Thmproofx23.p1.11.m1.1.1.3.3.cmml" xref="Thmproofx23.p1.11.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.11.m1.1c">X(p^{d}_{m})>\frac{1}{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.11.m1.1d">italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) > divide start_ARG 1 end_ARG start_ARG italic_m end_ARG</annotation></semantics></math>. Then <math alttext="\delta(X)\geq y_{m}^{n}-y_{m-1}^{n}\stackrel{{\scriptstyle\text{def}}}{{=}}1-(% 1-X(p^{d}_{m}))^{n}>\frac{\mathcal{E}_{m,n}}{m}" class="ltx_Math" display="inline" id="Thmproofx23.p1.12.m2.4"><semantics id="Thmproofx23.p1.12.m2.4a"><mrow id="Thmproofx23.p1.12.m2.4.4" xref="Thmproofx23.p1.12.m2.4.4.cmml"><mrow id="Thmproofx23.p1.12.m2.4.4.3" xref="Thmproofx23.p1.12.m2.4.4.3.cmml"><mi id="Thmproofx23.p1.12.m2.4.4.3.2" xref="Thmproofx23.p1.12.m2.4.4.3.2.cmml">δ</mi><mo id="Thmproofx23.p1.12.m2.4.4.3.1" xref="Thmproofx23.p1.12.m2.4.4.3.1.cmml">⁢</mo><mrow id="Thmproofx23.p1.12.m2.4.4.3.3.2" xref="Thmproofx23.p1.12.m2.4.4.3.cmml"><mo id="Thmproofx23.p1.12.m2.4.4.3.3.2.1" stretchy="false" xref="Thmproofx23.p1.12.m2.4.4.3.cmml">(</mo><mi id="Thmproofx23.p1.12.m2.3.3" xref="Thmproofx23.p1.12.m2.3.3.cmml">X</mi><mo id="Thmproofx23.p1.12.m2.4.4.3.3.2.2" stretchy="false" xref="Thmproofx23.p1.12.m2.4.4.3.cmml">)</mo></mrow></mrow><mo id="Thmproofx23.p1.12.m2.4.4.4" xref="Thmproofx23.p1.12.m2.4.4.4.cmml">≥</mo><mrow id="Thmproofx23.p1.12.m2.4.4.5" xref="Thmproofx23.p1.12.m2.4.4.5.cmml"><msubsup id="Thmproofx23.p1.12.m2.4.4.5.2" xref="Thmproofx23.p1.12.m2.4.4.5.2.cmml"><mi id="Thmproofx23.p1.12.m2.4.4.5.2.2.2" xref="Thmproofx23.p1.12.m2.4.4.5.2.2.2.cmml">y</mi><mi id="Thmproofx23.p1.12.m2.4.4.5.2.2.3" xref="Thmproofx23.p1.12.m2.4.4.5.2.2.3.cmml">m</mi><mi id="Thmproofx23.p1.12.m2.4.4.5.2.3" xref="Thmproofx23.p1.12.m2.4.4.5.2.3.cmml">n</mi></msubsup><mo id="Thmproofx23.p1.12.m2.4.4.5.1" xref="Thmproofx23.p1.12.m2.4.4.5.1.cmml">−</mo><msubsup id="Thmproofx23.p1.12.m2.4.4.5.3" xref="Thmproofx23.p1.12.m2.4.4.5.3.cmml"><mi id="Thmproofx23.p1.12.m2.4.4.5.3.2.2" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.2.cmml">y</mi><mrow id="Thmproofx23.p1.12.m2.4.4.5.3.2.3" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3.cmml"><mi id="Thmproofx23.p1.12.m2.4.4.5.3.2.3.2" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3.2.cmml">m</mi><mo id="Thmproofx23.p1.12.m2.4.4.5.3.2.3.1" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3.1.cmml">−</mo><mn id="Thmproofx23.p1.12.m2.4.4.5.3.2.3.3" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3.3.cmml">1</mn></mrow><mi id="Thmproofx23.p1.12.m2.4.4.5.3.3" xref="Thmproofx23.p1.12.m2.4.4.5.3.3.cmml">n</mi></msubsup></mrow><mover id="Thmproofx23.p1.12.m2.4.4.6" xref="Thmproofx23.p1.12.m2.4.4.6.cmml"><mo id="Thmproofx23.p1.12.m2.4.4.6.2" xref="Thmproofx23.p1.12.m2.4.4.6.2.cmml">=</mo><mtext id="Thmproofx23.p1.12.m2.4.4.6.3" xref="Thmproofx23.p1.12.m2.4.4.6.3a.cmml">def</mtext></mover><mrow id="Thmproofx23.p1.12.m2.4.4.1" xref="Thmproofx23.p1.12.m2.4.4.1.cmml"><mn id="Thmproofx23.p1.12.m2.4.4.1.3" xref="Thmproofx23.p1.12.m2.4.4.1.3.cmml">1</mn><mo id="Thmproofx23.p1.12.m2.4.4.1.2" xref="Thmproofx23.p1.12.m2.4.4.1.2.cmml">−</mo><msup id="Thmproofx23.p1.12.m2.4.4.1.1" xref="Thmproofx23.p1.12.m2.4.4.1.1.cmml"><mrow id="Thmproofx23.p1.12.m2.4.4.1.1.1.1" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.cmml"><mo id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.2" stretchy="false" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.cmml">(</mo><mrow id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.cmml"><mn id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.3" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.3.cmml">1</mn><mo id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.2" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.2.cmml">−</mo><mrow id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.cmml"><mi id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.3" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.3.cmml">X</mi><mo id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.2" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.2" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.2.cmml">p</mi><mi id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.3" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.3.cmml">m</mi><mi id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.3" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.3" stretchy="false" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx23.p1.12.m2.4.4.1.1.3" xref="Thmproofx23.p1.12.m2.4.4.1.1.3.cmml">n</mi></msup></mrow><mo id="Thmproofx23.p1.12.m2.4.4.7" xref="Thmproofx23.p1.12.m2.4.4.7.cmml">></mo><mfrac id="Thmproofx23.p1.12.m2.2.2" xref="Thmproofx23.p1.12.m2.2.2.cmml"><msub id="Thmproofx23.p1.12.m2.2.2.2" xref="Thmproofx23.p1.12.m2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx23.p1.12.m2.2.2.2.4" xref="Thmproofx23.p1.12.m2.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx23.p1.12.m2.2.2.2.2.2.4" xref="Thmproofx23.p1.12.m2.2.2.2.2.2.3.cmml"><mi id="Thmproofx23.p1.12.m2.1.1.1.1.1.1" xref="Thmproofx23.p1.12.m2.1.1.1.1.1.1.cmml">m</mi><mo id="Thmproofx23.p1.12.m2.2.2.2.2.2.4.1" xref="Thmproofx23.p1.12.m2.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx23.p1.12.m2.2.2.2.2.2.2" xref="Thmproofx23.p1.12.m2.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Thmproofx23.p1.12.m2.2.2.4" xref="Thmproofx23.p1.12.m2.2.2.4.cmml">m</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.12.m2.4b"><apply id="Thmproofx23.p1.12.m2.4.4.cmml" xref="Thmproofx23.p1.12.m2.4.4"><and id="Thmproofx23.p1.12.m2.4.4a.cmml" xref="Thmproofx23.p1.12.m2.4.4"></and><apply id="Thmproofx23.p1.12.m2.4.4b.cmml" xref="Thmproofx23.p1.12.m2.4.4"><geq id="Thmproofx23.p1.12.m2.4.4.4.cmml" xref="Thmproofx23.p1.12.m2.4.4.4"></geq><apply id="Thmproofx23.p1.12.m2.4.4.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.3"><times id="Thmproofx23.p1.12.m2.4.4.3.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.3.1"></times><ci id="Thmproofx23.p1.12.m2.4.4.3.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.3.2">𝛿</ci><ci id="Thmproofx23.p1.12.m2.3.3.cmml" xref="Thmproofx23.p1.12.m2.3.3">𝑋</ci></apply><apply id="Thmproofx23.p1.12.m2.4.4.5.cmml" xref="Thmproofx23.p1.12.m2.4.4.5"><minus id="Thmproofx23.p1.12.m2.4.4.5.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.1"></minus><apply id="Thmproofx23.p1.12.m2.4.4.5.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.5.2.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.2">superscript</csymbol><apply id="Thmproofx23.p1.12.m2.4.4.5.2.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.5.2.2.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.2">subscript</csymbol><ci id="Thmproofx23.p1.12.m2.4.4.5.2.2.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.2.2.2">𝑦</ci><ci id="Thmproofx23.p1.12.m2.4.4.5.2.2.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.2.2.3">𝑚</ci></apply><ci id="Thmproofx23.p1.12.m2.4.4.5.2.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.2.3">𝑛</ci></apply><apply id="Thmproofx23.p1.12.m2.4.4.5.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.5.3.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3">superscript</csymbol><apply id="Thmproofx23.p1.12.m2.4.4.5.3.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.5.3.2.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3">subscript</csymbol><ci id="Thmproofx23.p1.12.m2.4.4.5.3.2.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.2">𝑦</ci><apply id="Thmproofx23.p1.12.m2.4.4.5.3.2.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3"><minus id="Thmproofx23.p1.12.m2.4.4.5.3.2.3.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3.1"></minus><ci id="Thmproofx23.p1.12.m2.4.4.5.3.2.3.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3.2">𝑚</ci><cn id="Thmproofx23.p1.12.m2.4.4.5.3.2.3.3.cmml" type="integer" xref="Thmproofx23.p1.12.m2.4.4.5.3.2.3.3">1</cn></apply></apply><ci id="Thmproofx23.p1.12.m2.4.4.5.3.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.5.3.3">𝑛</ci></apply></apply></apply><apply id="Thmproofx23.p1.12.m2.4.4c.cmml" xref="Thmproofx23.p1.12.m2.4.4"><apply id="Thmproofx23.p1.12.m2.4.4.6.cmml" xref="Thmproofx23.p1.12.m2.4.4.6"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.6.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.6">superscript</csymbol><eq id="Thmproofx23.p1.12.m2.4.4.6.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.6.2"></eq><ci id="Thmproofx23.p1.12.m2.4.4.6.3a.cmml" xref="Thmproofx23.p1.12.m2.4.4.6.3"><mtext id="Thmproofx23.p1.12.m2.4.4.6.3.cmml" mathsize="70%" xref="Thmproofx23.p1.12.m2.4.4.6.3">def</mtext></ci></apply><share href="https://arxiv.org/html/2308.08047v3#Thmproofx23.p1.12.m2.4.4.5.cmml" id="Thmproofx23.p1.12.m2.4.4d.cmml" xref="Thmproofx23.p1.12.m2.4.4"></share><apply id="Thmproofx23.p1.12.m2.4.4.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.1"><minus id="Thmproofx23.p1.12.m2.4.4.1.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.2"></minus><cn id="Thmproofx23.p1.12.m2.4.4.1.3.cmml" type="integer" xref="Thmproofx23.p1.12.m2.4.4.1.3">1</cn><apply id="Thmproofx23.p1.12.m2.4.4.1.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.1.1.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1">superscript</csymbol><apply id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1"><minus id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.2"></minus><cn id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.3.cmml" type="integer" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.3">1</cn><apply id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1"><times id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.2"></times><ci id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.3">𝑋</ci><apply id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1">subscript</csymbol><apply id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.2">𝑝</ci><ci id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.2.3">𝑑</ci></apply><ci id="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.1.1.1.1.1.1.1.3">𝑚</ci></apply></apply></apply><ci id="Thmproofx23.p1.12.m2.4.4.1.1.3.cmml" xref="Thmproofx23.p1.12.m2.4.4.1.1.3">𝑛</ci></apply></apply></apply><apply id="Thmproofx23.p1.12.m2.4.4e.cmml" xref="Thmproofx23.p1.12.m2.4.4"><gt id="Thmproofx23.p1.12.m2.4.4.7.cmml" xref="Thmproofx23.p1.12.m2.4.4.7"></gt><share href="https://arxiv.org/html/2308.08047v3#Thmproofx23.p1.12.m2.4.4.1.cmml" id="Thmproofx23.p1.12.m2.4.4f.cmml" xref="Thmproofx23.p1.12.m2.4.4"></share><apply id="Thmproofx23.p1.12.m2.2.2.cmml" xref="Thmproofx23.p1.12.m2.2.2"><divide id="Thmproofx23.p1.12.m2.2.2.3.cmml" xref="Thmproofx23.p1.12.m2.2.2"></divide><apply id="Thmproofx23.p1.12.m2.2.2.2.cmml" xref="Thmproofx23.p1.12.m2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.12.m2.2.2.2.3.cmml" xref="Thmproofx23.p1.12.m2.2.2.2">subscript</csymbol><ci id="Thmproofx23.p1.12.m2.2.2.2.4.cmml" xref="Thmproofx23.p1.12.m2.2.2.2.4">ℰ</ci><list id="Thmproofx23.p1.12.m2.2.2.2.2.2.3.cmml" xref="Thmproofx23.p1.12.m2.2.2.2.2.2.4"><ci id="Thmproofx23.p1.12.m2.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.12.m2.1.1.1.1.1.1">𝑚</ci><ci id="Thmproofx23.p1.12.m2.2.2.2.2.2.2.cmml" xref="Thmproofx23.p1.12.m2.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Thmproofx23.p1.12.m2.2.2.4.cmml" xref="Thmproofx23.p1.12.m2.2.2.4">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.12.m2.4c">\delta(X)\geq y_{m}^{n}-y_{m-1}^{n}\stackrel{{\scriptstyle\text{def}}}{{=}}1-(% 1-X(p^{d}_{m}))^{n}>\frac{\mathcal{E}_{m,n}}{m}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.12.m2.4d">italic_δ ( italic_X ) ≥ italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP 1 - ( 1 - italic_X ( italic_p start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT > divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG</annotation></semantics></math>. Therefore <math alttext="F_{n}(\sigma,\alpha)>|p_{1}|+\frac{\mathcal{E}_{m,n}}{m}k\alpha" class="ltx_Math" display="inline" id="Thmproofx23.p1.13.m3.5"><semantics id="Thmproofx23.p1.13.m3.5a"><mrow id="Thmproofx23.p1.13.m3.5.5" xref="Thmproofx23.p1.13.m3.5.5.cmml"><mrow id="Thmproofx23.p1.13.m3.5.5.3" xref="Thmproofx23.p1.13.m3.5.5.3.cmml"><msub id="Thmproofx23.p1.13.m3.5.5.3.2" xref="Thmproofx23.p1.13.m3.5.5.3.2.cmml"><mi id="Thmproofx23.p1.13.m3.5.5.3.2.2" xref="Thmproofx23.p1.13.m3.5.5.3.2.2.cmml">F</mi><mi id="Thmproofx23.p1.13.m3.5.5.3.2.3" xref="Thmproofx23.p1.13.m3.5.5.3.2.3.cmml">n</mi></msub><mo id="Thmproofx23.p1.13.m3.5.5.3.1" xref="Thmproofx23.p1.13.m3.5.5.3.1.cmml">⁢</mo><mrow id="Thmproofx23.p1.13.m3.5.5.3.3.2" xref="Thmproofx23.p1.13.m3.5.5.3.3.1.cmml"><mo id="Thmproofx23.p1.13.m3.5.5.3.3.2.1" stretchy="false" xref="Thmproofx23.p1.13.m3.5.5.3.3.1.cmml">(</mo><mi id="Thmproofx23.p1.13.m3.3.3" xref="Thmproofx23.p1.13.m3.3.3.cmml">σ</mi><mo id="Thmproofx23.p1.13.m3.5.5.3.3.2.2" xref="Thmproofx23.p1.13.m3.5.5.3.3.1.cmml">,</mo><mi id="Thmproofx23.p1.13.m3.4.4" xref="Thmproofx23.p1.13.m3.4.4.cmml">α</mi><mo id="Thmproofx23.p1.13.m3.5.5.3.3.2.3" stretchy="false" xref="Thmproofx23.p1.13.m3.5.5.3.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx23.p1.13.m3.5.5.2" xref="Thmproofx23.p1.13.m3.5.5.2.cmml">></mo><mrow id="Thmproofx23.p1.13.m3.5.5.1" xref="Thmproofx23.p1.13.m3.5.5.1.cmml"><mrow id="Thmproofx23.p1.13.m3.5.5.1.1.1" xref="Thmproofx23.p1.13.m3.5.5.1.1.2.cmml"><mo id="Thmproofx23.p1.13.m3.5.5.1.1.1.2" stretchy="false" xref="Thmproofx23.p1.13.m3.5.5.1.1.2.1.cmml">|</mo><msub id="Thmproofx23.p1.13.m3.5.5.1.1.1.1" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.1.cmml"><mi id="Thmproofx23.p1.13.m3.5.5.1.1.1.1.2" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.1.2.cmml">p</mi><mn id="Thmproofx23.p1.13.m3.5.5.1.1.1.1.3" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmproofx23.p1.13.m3.5.5.1.1.1.3" stretchy="false" xref="Thmproofx23.p1.13.m3.5.5.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx23.p1.13.m3.5.5.1.2" xref="Thmproofx23.p1.13.m3.5.5.1.2.cmml">+</mo><mrow id="Thmproofx23.p1.13.m3.5.5.1.3" xref="Thmproofx23.p1.13.m3.5.5.1.3.cmml"><mfrac id="Thmproofx23.p1.13.m3.2.2" xref="Thmproofx23.p1.13.m3.2.2.cmml"><msub id="Thmproofx23.p1.13.m3.2.2.2" xref="Thmproofx23.p1.13.m3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproofx23.p1.13.m3.2.2.2.4" xref="Thmproofx23.p1.13.m3.2.2.2.4.cmml">ℰ</mi><mrow id="Thmproofx23.p1.13.m3.2.2.2.2.2.4" xref="Thmproofx23.p1.13.m3.2.2.2.2.2.3.cmml"><mi id="Thmproofx23.p1.13.m3.1.1.1.1.1.1" xref="Thmproofx23.p1.13.m3.1.1.1.1.1.1.cmml">m</mi><mo id="Thmproofx23.p1.13.m3.2.2.2.2.2.4.1" xref="Thmproofx23.p1.13.m3.2.2.2.2.2.3.cmml">,</mo><mi id="Thmproofx23.p1.13.m3.2.2.2.2.2.2" xref="Thmproofx23.p1.13.m3.2.2.2.2.2.2.cmml">n</mi></mrow></msub><mi id="Thmproofx23.p1.13.m3.2.2.4" xref="Thmproofx23.p1.13.m3.2.2.4.cmml">m</mi></mfrac><mo id="Thmproofx23.p1.13.m3.5.5.1.3.1" xref="Thmproofx23.p1.13.m3.5.5.1.3.1.cmml">⁢</mo><mi id="Thmproofx23.p1.13.m3.5.5.1.3.2" xref="Thmproofx23.p1.13.m3.5.5.1.3.2.cmml">k</mi><mo id="Thmproofx23.p1.13.m3.5.5.1.3.1a" xref="Thmproofx23.p1.13.m3.5.5.1.3.1.cmml">⁢</mo><mi id="Thmproofx23.p1.13.m3.5.5.1.3.3" xref="Thmproofx23.p1.13.m3.5.5.1.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx23.p1.13.m3.5b"><apply id="Thmproofx23.p1.13.m3.5.5.cmml" xref="Thmproofx23.p1.13.m3.5.5"><gt id="Thmproofx23.p1.13.m3.5.5.2.cmml" xref="Thmproofx23.p1.13.m3.5.5.2"></gt><apply id="Thmproofx23.p1.13.m3.5.5.3.cmml" xref="Thmproofx23.p1.13.m3.5.5.3"><times id="Thmproofx23.p1.13.m3.5.5.3.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.3.1"></times><apply id="Thmproofx23.p1.13.m3.5.5.3.2.cmml" xref="Thmproofx23.p1.13.m3.5.5.3.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.13.m3.5.5.3.2.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.3.2">subscript</csymbol><ci id="Thmproofx23.p1.13.m3.5.5.3.2.2.cmml" xref="Thmproofx23.p1.13.m3.5.5.3.2.2">𝐹</ci><ci id="Thmproofx23.p1.13.m3.5.5.3.2.3.cmml" xref="Thmproofx23.p1.13.m3.5.5.3.2.3">𝑛</ci></apply><interval closure="open" id="Thmproofx23.p1.13.m3.5.5.3.3.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.3.3.2"><ci id="Thmproofx23.p1.13.m3.3.3.cmml" xref="Thmproofx23.p1.13.m3.3.3">𝜎</ci><ci id="Thmproofx23.p1.13.m3.4.4.cmml" xref="Thmproofx23.p1.13.m3.4.4">𝛼</ci></interval></apply><apply id="Thmproofx23.p1.13.m3.5.5.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.1"><plus id="Thmproofx23.p1.13.m3.5.5.1.2.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.2"></plus><apply id="Thmproofx23.p1.13.m3.5.5.1.1.2.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.1.1"><abs id="Thmproofx23.p1.13.m3.5.5.1.1.2.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.2"></abs><apply id="Thmproofx23.p1.13.m3.5.5.1.1.1.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.13.m3.5.5.1.1.1.1.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.1">subscript</csymbol><ci id="Thmproofx23.p1.13.m3.5.5.1.1.1.1.2.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.1.2">𝑝</ci><cn id="Thmproofx23.p1.13.m3.5.5.1.1.1.1.3.cmml" type="integer" xref="Thmproofx23.p1.13.m3.5.5.1.1.1.1.3">1</cn></apply></apply><apply id="Thmproofx23.p1.13.m3.5.5.1.3.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.3"><times id="Thmproofx23.p1.13.m3.5.5.1.3.1.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.3.1"></times><apply id="Thmproofx23.p1.13.m3.2.2.cmml" xref="Thmproofx23.p1.13.m3.2.2"><divide id="Thmproofx23.p1.13.m3.2.2.3.cmml" xref="Thmproofx23.p1.13.m3.2.2"></divide><apply id="Thmproofx23.p1.13.m3.2.2.2.cmml" xref="Thmproofx23.p1.13.m3.2.2.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.13.m3.2.2.2.3.cmml" xref="Thmproofx23.p1.13.m3.2.2.2">subscript</csymbol><ci id="Thmproofx23.p1.13.m3.2.2.2.4.cmml" xref="Thmproofx23.p1.13.m3.2.2.2.4">ℰ</ci><list id="Thmproofx23.p1.13.m3.2.2.2.2.2.3.cmml" xref="Thmproofx23.p1.13.m3.2.2.2.2.2.4"><ci id="Thmproofx23.p1.13.m3.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.13.m3.1.1.1.1.1.1">𝑚</ci><ci id="Thmproofx23.p1.13.m3.2.2.2.2.2.2.cmml" xref="Thmproofx23.p1.13.m3.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Thmproofx23.p1.13.m3.2.2.4.cmml" xref="Thmproofx23.p1.13.m3.2.2.4">𝑚</ci></apply><ci id="Thmproofx23.p1.13.m3.5.5.1.3.2.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.3.2">𝑘</ci><ci id="Thmproofx23.p1.13.m3.5.5.1.3.3.cmml" xref="Thmproofx23.p1.13.m3.5.5.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.13.m3.5c">F_{n}(\sigma,\alpha)>|p_{1}|+\frac{\mathcal{E}_{m,n}}{m}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.13.m3.5d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) > | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG italic_k italic_α</annotation></semantics></math>. Hence <math alttext="\min_{\sigma}F_{n}(\sigma,\alpha)=F_{n}(U_{P_{st}}^{n},\alpha)=|p_{1}|+\frac{% \mathcal{E}_{m,n}}{m}k\alpha" class="ltx_Math" display="inline" id="Thmproofx23.p1.14.m4.7"><semantics id="Thmproofx23.p1.14.m4.7a"><mrow id="Thmproofx23.p1.14.m4.7.7" xref="Thmproofx23.p1.14.m4.7.7.cmml"><mrow id="Thmproofx23.p1.14.m4.7.7.4" xref="Thmproofx23.p1.14.m4.7.7.4.cmml"><mrow id="Thmproofx23.p1.14.m4.7.7.4.2" xref="Thmproofx23.p1.14.m4.7.7.4.2.cmml"><msub id="Thmproofx23.p1.14.m4.7.7.4.2.1" xref="Thmproofx23.p1.14.m4.7.7.4.2.1.cmml"><mi id="Thmproofx23.p1.14.m4.7.7.4.2.1.2" xref="Thmproofx23.p1.14.m4.7.7.4.2.1.2.cmml">min</mi><mi id="Thmproofx23.p1.14.m4.7.7.4.2.1.3" xref="Thmproofx23.p1.14.m4.7.7.4.2.1.3.cmml">σ</mi></msub><mo id="Thmproofx23.p1.14.m4.7.7.4.2a" lspace="0.167em" xref="Thmproofx23.p1.14.m4.7.7.4.2.cmml">⁡</mo><msub id="Thmproofx23.p1.14.m4.7.7.4.2.2" xref="Thmproofx23.p1.14.m4.7.7.4.2.2.cmml"><mi id="Thmproofx23.p1.14.m4.7.7.4.2.2.2" xref="Thmproofx23.p1.14.m4.7.7.4.2.2.2.cmml">F</mi><mi id="Thmproofx23.p1.14.m4.7.7.4.2.2.3" xref="Thmproofx23.p1.14.m4.7.7.4.2.2.3.cmml">n</mi></msub></mrow><mo id="Thmproofx23.p1.14.m4.7.7.4.1" xref="Thmproofx23.p1.14.m4.7.7.4.1.cmml">⁢</mo><mrow id="Thmproofx23.p1.14.m4.7.7.4.3.2" xref="Thmproofx23.p1.14.m4.7.7.4.3.1.cmml"><mo id="Thmproofx23.p1.14.m4.7.7.4.3.2.1" stretchy="false" xref="Thmproofx23.p1.14.m4.7.7.4.3.1.cmml">(</mo><mi id="Thmproofx23.p1.14.m4.3.3" xref="Thmproofx23.p1.14.m4.3.3.cmml">σ</mi><mo id="Thmproofx23.p1.14.m4.7.7.4.3.2.2" xref="Thmproofx23.p1.14.m4.7.7.4.3.1.cmml">,</mo><mi id="Thmproofx23.p1.14.m4.4.4" xref="Thmproofx23.p1.14.m4.4.4.cmml">α</mi><mo id="Thmproofx23.p1.14.m4.7.7.4.3.2.3" stretchy="false" xref="Thmproofx23.p1.14.m4.7.7.4.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx23.p1.14.m4.7.7.5" xref="Thmproofx23.p1.14.m4.7.7.5.cmml">=</mo><mrow id="Thmproofx23.p1.14.m4.6.6.1" xref="Thmproofx23.p1.14.m4.6.6.1.cmml"><msub id="Thmproofx23.p1.14.m4.6.6.1.3" xref="Thmproofx23.p1.14.m4.6.6.1.3.cmml"><mi id="Thmproofx23.p1.14.m4.6.6.1.3.2" xref="Thmproofx23.p1.14.m4.6.6.1.3.2.cmml">F</mi><mi id="Thmproofx23.p1.14.m4.6.6.1.3.3" xref="Thmproofx23.p1.14.m4.6.6.1.3.3.cmml">n</mi></msub><mo id="Thmproofx23.p1.14.m4.6.6.1.2" xref="Thmproofx23.p1.14.m4.6.6.1.2.cmml">⁢</mo><mrow id="Thmproofx23.p1.14.m4.6.6.1.1.1" xref="Thmproofx23.p1.14.m4.6.6.1.1.2.cmml"><mo id="Thmproofx23.p1.14.m4.6.6.1.1.1.2" stretchy="false" xref="Thmproofx23.p1.14.m4.6.6.1.1.2.cmml">(</mo><msubsup id="Thmproofx23.p1.14.m4.6.6.1.1.1.1" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.cmml"><mi id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.2" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.2.cmml">U</mi><msub id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.cmml"><mi id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.2" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.2.cmml">P</mi><mrow id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3.cmml"><mi id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3.2" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3.2.cmml">s</mi><mo id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3.1" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3.3" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></msub><mi id="Thmproofx23.p1.14.m4.6.6.1.1.1.1.3" xref="Thmproofx23.p1.14.m4.6.6.1.1.1.1.3.cmml">n</mi></msubsup><mo id="Thmproofx23.p1.14.m4.6.6.1.1.1.3" xref="Thmproofx23.p1.14.m4.6.6.1.1.2.cmml">,</mo><mi id="Thmproofx23.p1.14.m4.5.5" xref="Thmproofx23.p1.14.m4.5.5.cmml">α</mi><mo id="Thmproofx23.p1.14.m4.6.6.1.1.1.4" stretchy="false" xref="Thmproofx23.p1.14.m4.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmproofx23.p1.14.m4.7.7.6" xref="Thmproofx23.p1.14.m4.7.7.6.cmml">=</mo><mrow id="Thmproofx23.p1.14.m4.7.7.2" xref="Thmproofx23.p1.14.m4.7.7.2.cmml"><mrow id="Thmproofx23.p1.14.m4.7.7.2.1.1" 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href="https://arxiv.org/html/2308.08047v3#Thmproofx23.p1.14.m4.6.6.1.cmml" id="Thmproofx23.p1.14.m4.7.7d.cmml" xref="Thmproofx23.p1.14.m4.7.7"></share><apply id="Thmproofx23.p1.14.m4.7.7.2.cmml" xref="Thmproofx23.p1.14.m4.7.7.2"><plus id="Thmproofx23.p1.14.m4.7.7.2.2.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.2"></plus><apply id="Thmproofx23.p1.14.m4.7.7.2.1.2.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.1.1"><abs id="Thmproofx23.p1.14.m4.7.7.2.1.2.1.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.1.1.2"></abs><apply id="Thmproofx23.p1.14.m4.7.7.2.1.1.1.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.1.1.1"><csymbol cd="ambiguous" id="Thmproofx23.p1.14.m4.7.7.2.1.1.1.1.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.1.1.1">subscript</csymbol><ci id="Thmproofx23.p1.14.m4.7.7.2.1.1.1.2.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.1.1.1.2">𝑝</ci><cn id="Thmproofx23.p1.14.m4.7.7.2.1.1.1.3.cmml" type="integer" xref="Thmproofx23.p1.14.m4.7.7.2.1.1.1.3">1</cn></apply></apply><apply id="Thmproofx23.p1.14.m4.7.7.2.3.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.3"><times id="Thmproofx23.p1.14.m4.7.7.2.3.1.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.3.1"></times><apply id="Thmproofx23.p1.14.m4.2.2.cmml" xref="Thmproofx23.p1.14.m4.2.2"><divide id="Thmproofx23.p1.14.m4.2.2.3.cmml" xref="Thmproofx23.p1.14.m4.2.2"></divide><apply id="Thmproofx23.p1.14.m4.2.2.2.cmml" xref="Thmproofx23.p1.14.m4.2.2.2"><csymbol cd="ambiguous" id="Thmproofx23.p1.14.m4.2.2.2.3.cmml" xref="Thmproofx23.p1.14.m4.2.2.2">subscript</csymbol><ci id="Thmproofx23.p1.14.m4.2.2.2.4.cmml" xref="Thmproofx23.p1.14.m4.2.2.2.4">ℰ</ci><list id="Thmproofx23.p1.14.m4.2.2.2.2.2.3.cmml" xref="Thmproofx23.p1.14.m4.2.2.2.2.2.4"><ci id="Thmproofx23.p1.14.m4.1.1.1.1.1.1.cmml" xref="Thmproofx23.p1.14.m4.1.1.1.1.1.1">𝑚</ci><ci id="Thmproofx23.p1.14.m4.2.2.2.2.2.2.cmml" xref="Thmproofx23.p1.14.m4.2.2.2.2.2.2">𝑛</ci></list></apply><ci id="Thmproofx23.p1.14.m4.2.2.4.cmml" xref="Thmproofx23.p1.14.m4.2.2.4">𝑚</ci></apply><ci id="Thmproofx23.p1.14.m4.7.7.2.3.2.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.3.2">𝑘</ci><ci id="Thmproofx23.p1.14.m4.7.7.2.3.3.cmml" xref="Thmproofx23.p1.14.m4.7.7.2.3.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx23.p1.14.m4.7c">\min_{\sigma}F_{n}(\sigma,\alpha)=F_{n}(U_{P_{st}}^{n},\alpha)=|p_{1}|+\frac{% \mathcal{E}_{m,n}}{m}k\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmproofx23.p1.14.m4.7d">roman_min start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α ) = italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_α ) = | italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | + divide start_ARG caligraphic_E start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG italic_k italic_α</annotation></semantics></math>, completing the proof. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx24"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Proof</span> (Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Thmlemma17" title="Lemma 17 ‣ The 𝑘=1 case ‣ 5.2 Disjoint paths ‣ 5 MAX Cost ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">17</span></a>)</h6> <div class="ltx_para" id="Thmproofx24.p1"> <p class="ltx_p" id="Thmproofx24.p1.13">Let <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmproofx24.p1.1.m1.1"><semantics id="Thmproofx24.p1.1.m1.1a"><mi id="Thmproofx24.p1.1.m1.1.1" xref="Thmproofx24.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.1.m1.1b"><ci id="Thmproofx24.p1.1.m1.1.1.cmml" xref="Thmproofx24.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.1.m1.1d">italic_σ</annotation></semantics></math>, and wlog it’s iid, namely <math alttext="\sigma=X^{n}" class="ltx_Math" display="inline" id="Thmproofx24.p1.2.m2.1"><semantics id="Thmproofx24.p1.2.m2.1a"><mrow id="Thmproofx24.p1.2.m2.1.1" xref="Thmproofx24.p1.2.m2.1.1.cmml"><mi id="Thmproofx24.p1.2.m2.1.1.2" xref="Thmproofx24.p1.2.m2.1.1.2.cmml">σ</mi><mo id="Thmproofx24.p1.2.m2.1.1.1" xref="Thmproofx24.p1.2.m2.1.1.1.cmml">=</mo><msup id="Thmproofx24.p1.2.m2.1.1.3" xref="Thmproofx24.p1.2.m2.1.1.3.cmml"><mi id="Thmproofx24.p1.2.m2.1.1.3.2" xref="Thmproofx24.p1.2.m2.1.1.3.2.cmml">X</mi><mi id="Thmproofx24.p1.2.m2.1.1.3.3" xref="Thmproofx24.p1.2.m2.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.2.m2.1b"><apply id="Thmproofx24.p1.2.m2.1.1.cmml" xref="Thmproofx24.p1.2.m2.1.1"><eq id="Thmproofx24.p1.2.m2.1.1.1.cmml" xref="Thmproofx24.p1.2.m2.1.1.1"></eq><ci id="Thmproofx24.p1.2.m2.1.1.2.cmml" xref="Thmproofx24.p1.2.m2.1.1.2">𝜎</ci><apply id="Thmproofx24.p1.2.m2.1.1.3.cmml" xref="Thmproofx24.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproofx24.p1.2.m2.1.1.3.1.cmml" xref="Thmproofx24.p1.2.m2.1.1.3">superscript</csymbol><ci id="Thmproofx24.p1.2.m2.1.1.3.2.cmml" xref="Thmproofx24.p1.2.m2.1.1.3.2">𝑋</ci><ci id="Thmproofx24.p1.2.m2.1.1.3.3.cmml" xref="Thmproofx24.p1.2.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.2.m2.1c">\sigma=X^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.2.m2.1d">italic_σ = italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="X\in\Delta(P_{st})" class="ltx_Math" display="inline" id="Thmproofx24.p1.3.m3.1"><semantics id="Thmproofx24.p1.3.m3.1a"><mrow id="Thmproofx24.p1.3.m3.1.1" xref="Thmproofx24.p1.3.m3.1.1.cmml"><mi id="Thmproofx24.p1.3.m3.1.1.3" xref="Thmproofx24.p1.3.m3.1.1.3.cmml">X</mi><mo id="Thmproofx24.p1.3.m3.1.1.2" xref="Thmproofx24.p1.3.m3.1.1.2.cmml">∈</mo><mrow id="Thmproofx24.p1.3.m3.1.1.1" xref="Thmproofx24.p1.3.m3.1.1.1.cmml"><mi id="Thmproofx24.p1.3.m3.1.1.1.3" mathvariant="normal" xref="Thmproofx24.p1.3.m3.1.1.1.3.cmml">Δ</mi><mo id="Thmproofx24.p1.3.m3.1.1.1.2" xref="Thmproofx24.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx24.p1.3.m3.1.1.1.1.1" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="Thmproofx24.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx24.p1.3.m3.1.1.1.1.1.1" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.2" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.2.cmml">P</mi><mrow id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.cmml"><mi id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.2" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.1" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.3" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo id="Thmproofx24.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.3.m3.1b"><apply id="Thmproofx24.p1.3.m3.1.1.cmml" xref="Thmproofx24.p1.3.m3.1.1"><in id="Thmproofx24.p1.3.m3.1.1.2.cmml" xref="Thmproofx24.p1.3.m3.1.1.2"></in><ci id="Thmproofx24.p1.3.m3.1.1.3.cmml" xref="Thmproofx24.p1.3.m3.1.1.3">𝑋</ci><apply id="Thmproofx24.p1.3.m3.1.1.1.cmml" xref="Thmproofx24.p1.3.m3.1.1.1"><times id="Thmproofx24.p1.3.m3.1.1.1.2.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.2"></times><ci id="Thmproofx24.p1.3.m3.1.1.1.3.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.3">Δ</ci><apply id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3"><times id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.1.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.1"></times><ci id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.2.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.2">𝑠</ci><ci id="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.3.cmml" xref="Thmproofx24.p1.3.m3.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.3.m3.1c">X\in\Delta(P_{st})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.3.m3.1d">italic_X ∈ roman_Δ ( italic_P start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT )</annotation></semantics></math>. Let <math alttext="i<m" class="ltx_Math" display="inline" id="Thmproofx24.p1.4.m4.1"><semantics id="Thmproofx24.p1.4.m4.1a"><mrow id="Thmproofx24.p1.4.m4.1.1" xref="Thmproofx24.p1.4.m4.1.1.cmml"><mi id="Thmproofx24.p1.4.m4.1.1.2" xref="Thmproofx24.p1.4.m4.1.1.2.cmml">i</mi><mo id="Thmproofx24.p1.4.m4.1.1.1" xref="Thmproofx24.p1.4.m4.1.1.1.cmml"><</mo><mi id="Thmproofx24.p1.4.m4.1.1.3" xref="Thmproofx24.p1.4.m4.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.4.m4.1b"><apply id="Thmproofx24.p1.4.m4.1.1.cmml" xref="Thmproofx24.p1.4.m4.1.1"><lt id="Thmproofx24.p1.4.m4.1.1.1.cmml" xref="Thmproofx24.p1.4.m4.1.1.1"></lt><ci id="Thmproofx24.p1.4.m4.1.1.2.cmml" xref="Thmproofx24.p1.4.m4.1.1.2">𝑖</ci><ci id="Thmproofx24.p1.4.m4.1.1.3.cmml" xref="Thmproofx24.p1.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.4.m4.1c">i<m</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.4.m4.1d">italic_i < italic_m</annotation></semantics></math> such that <math alttext="|p_{i}|<|p_{i+1}|" class="ltx_Math" display="inline" id="Thmproofx24.p1.5.m5.2"><semantics id="Thmproofx24.p1.5.m5.2a"><mrow id="Thmproofx24.p1.5.m5.2.2" xref="Thmproofx24.p1.5.m5.2.2.cmml"><mrow id="Thmproofx24.p1.5.m5.1.1.1.1" xref="Thmproofx24.p1.5.m5.1.1.1.2.cmml"><mo id="Thmproofx24.p1.5.m5.1.1.1.1.2" stretchy="false" xref="Thmproofx24.p1.5.m5.1.1.1.2.1.cmml">|</mo><msub id="Thmproofx24.p1.5.m5.1.1.1.1.1" xref="Thmproofx24.p1.5.m5.1.1.1.1.1.cmml"><mi id="Thmproofx24.p1.5.m5.1.1.1.1.1.2" xref="Thmproofx24.p1.5.m5.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx24.p1.5.m5.1.1.1.1.1.3" xref="Thmproofx24.p1.5.m5.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx24.p1.5.m5.1.1.1.1.3" stretchy="false" xref="Thmproofx24.p1.5.m5.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmproofx24.p1.5.m5.2.2.3" xref="Thmproofx24.p1.5.m5.2.2.3.cmml"><</mo><mrow id="Thmproofx24.p1.5.m5.2.2.2.1" xref="Thmproofx24.p1.5.m5.2.2.2.2.cmml"><mo id="Thmproofx24.p1.5.m5.2.2.2.1.2" stretchy="false" xref="Thmproofx24.p1.5.m5.2.2.2.2.1.cmml">|</mo><msub id="Thmproofx24.p1.5.m5.2.2.2.1.1" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.cmml"><mi id="Thmproofx24.p1.5.m5.2.2.2.1.1.2" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.2.cmml">p</mi><mrow id="Thmproofx24.p1.5.m5.2.2.2.1.1.3" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3.cmml"><mi id="Thmproofx24.p1.5.m5.2.2.2.1.1.3.2" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3.2.cmml">i</mi><mo id="Thmproofx24.p1.5.m5.2.2.2.1.1.3.1" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3.1.cmml">+</mo><mn id="Thmproofx24.p1.5.m5.2.2.2.1.1.3.3" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmproofx24.p1.5.m5.2.2.2.1.3" stretchy="false" xref="Thmproofx24.p1.5.m5.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.5.m5.2b"><apply id="Thmproofx24.p1.5.m5.2.2.cmml" xref="Thmproofx24.p1.5.m5.2.2"><lt id="Thmproofx24.p1.5.m5.2.2.3.cmml" xref="Thmproofx24.p1.5.m5.2.2.3"></lt><apply id="Thmproofx24.p1.5.m5.1.1.1.2.cmml" xref="Thmproofx24.p1.5.m5.1.1.1.1"><abs id="Thmproofx24.p1.5.m5.1.1.1.2.1.cmml" xref="Thmproofx24.p1.5.m5.1.1.1.1.2"></abs><apply id="Thmproofx24.p1.5.m5.1.1.1.1.1.cmml" xref="Thmproofx24.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.5.m5.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx24.p1.5.m5.1.1.1.1.1.2.cmml" xref="Thmproofx24.p1.5.m5.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx24.p1.5.m5.1.1.1.1.1.3.cmml" xref="Thmproofx24.p1.5.m5.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx24.p1.5.m5.2.2.2.2.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1"><abs id="Thmproofx24.p1.5.m5.2.2.2.2.1.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1.2"></abs><apply id="Thmproofx24.p1.5.m5.2.2.2.1.1.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.5.m5.2.2.2.1.1.1.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx24.p1.5.m5.2.2.2.1.1.2.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.2">𝑝</ci><apply id="Thmproofx24.p1.5.m5.2.2.2.1.1.3.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3"><plus id="Thmproofx24.p1.5.m5.2.2.2.1.1.3.1.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3.1"></plus><ci id="Thmproofx24.p1.5.m5.2.2.2.1.1.3.2.cmml" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3.2">𝑖</ci><cn id="Thmproofx24.p1.5.m5.2.2.2.1.1.3.3.cmml" type="integer" xref="Thmproofx24.p1.5.m5.2.2.2.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.5.m5.2c">|p_{i}|<|p_{i+1}|</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.5.m5.2d">| italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT |</annotation></semantics></math> and <math alttext="\sigma(p_{i})<\sigma(p_{i+1})" class="ltx_Math" display="inline" id="Thmproofx24.p1.6.m6.2"><semantics id="Thmproofx24.p1.6.m6.2a"><mrow id="Thmproofx24.p1.6.m6.2.2" xref="Thmproofx24.p1.6.m6.2.2.cmml"><mrow id="Thmproofx24.p1.6.m6.1.1.1" xref="Thmproofx24.p1.6.m6.1.1.1.cmml"><mi id="Thmproofx24.p1.6.m6.1.1.1.3" xref="Thmproofx24.p1.6.m6.1.1.1.3.cmml">σ</mi><mo id="Thmproofx24.p1.6.m6.1.1.1.2" xref="Thmproofx24.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx24.p1.6.m6.1.1.1.1.1" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmproofx24.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx24.p1.6.m6.1.1.1.1.1.1" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="Thmproofx24.p1.6.m6.1.1.1.1.1.1.2" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx24.p1.6.m6.1.1.1.1.1.1.3" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx24.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx24.p1.6.m6.2.2.3" xref="Thmproofx24.p1.6.m6.2.2.3.cmml"><</mo><mrow id="Thmproofx24.p1.6.m6.2.2.2" xref="Thmproofx24.p1.6.m6.2.2.2.cmml"><mi id="Thmproofx24.p1.6.m6.2.2.2.3" xref="Thmproofx24.p1.6.m6.2.2.2.3.cmml">σ</mi><mo id="Thmproofx24.p1.6.m6.2.2.2.2" xref="Thmproofx24.p1.6.m6.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx24.p1.6.m6.2.2.2.1.1" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.cmml"><mo id="Thmproofx24.p1.6.m6.2.2.2.1.1.2" stretchy="false" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx24.p1.6.m6.2.2.2.1.1.1" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.cmml"><mi id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.2" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.2.cmml">p</mi><mrow id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.cmml"><mi id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.2" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.1" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.1.cmml">+</mo><mn id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.3" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmproofx24.p1.6.m6.2.2.2.1.1.3" stretchy="false" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.6.m6.2b"><apply id="Thmproofx24.p1.6.m6.2.2.cmml" xref="Thmproofx24.p1.6.m6.2.2"><lt id="Thmproofx24.p1.6.m6.2.2.3.cmml" xref="Thmproofx24.p1.6.m6.2.2.3"></lt><apply id="Thmproofx24.p1.6.m6.1.1.1.cmml" xref="Thmproofx24.p1.6.m6.1.1.1"><times id="Thmproofx24.p1.6.m6.1.1.1.2.cmml" xref="Thmproofx24.p1.6.m6.1.1.1.2"></times><ci id="Thmproofx24.p1.6.m6.1.1.1.3.cmml" xref="Thmproofx24.p1.6.m6.1.1.1.3">𝜎</ci><apply id="Thmproofx24.p1.6.m6.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx24.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx24.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="Thmproofx24.p1.6.m6.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx24.p1.6.m6.2.2.2.cmml" xref="Thmproofx24.p1.6.m6.2.2.2"><times id="Thmproofx24.p1.6.m6.2.2.2.2.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.2"></times><ci id="Thmproofx24.p1.6.m6.2.2.2.3.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.3">𝜎</ci><apply id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.1.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.2.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.2">𝑝</ci><apply id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3"><plus id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.1"></plus><ci id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.2.cmml" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.2">𝑖</ci><cn id="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.3.cmml" type="integer" xref="Thmproofx24.p1.6.m6.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.6.m6.2c">\sigma(p_{i})<\sigma(p_{i+1})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.6.m6.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) < italic_σ ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Let <math alttext="X^{\prime}" class="ltx_Math" display="inline" id="Thmproofx24.p1.7.m7.1"><semantics id="Thmproofx24.p1.7.m7.1a"><msup id="Thmproofx24.p1.7.m7.1.1" xref="Thmproofx24.p1.7.m7.1.1.cmml"><mi id="Thmproofx24.p1.7.m7.1.1.2" xref="Thmproofx24.p1.7.m7.1.1.2.cmml">X</mi><mo id="Thmproofx24.p1.7.m7.1.1.3" xref="Thmproofx24.p1.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.7.m7.1b"><apply id="Thmproofx24.p1.7.m7.1.1.cmml" xref="Thmproofx24.p1.7.m7.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.7.m7.1.1.1.cmml" xref="Thmproofx24.p1.7.m7.1.1">superscript</csymbol><ci id="Thmproofx24.p1.7.m7.1.1.2.cmml" xref="Thmproofx24.p1.7.m7.1.1.2">𝑋</ci><ci id="Thmproofx24.p1.7.m7.1.1.3.cmml" xref="Thmproofx24.p1.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.7.m7.1c">X^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.7.m7.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> result from <math alttext="X" class="ltx_Math" display="inline" id="Thmproofx24.p1.8.m8.1"><semantics id="Thmproofx24.p1.8.m8.1a"><mi id="Thmproofx24.p1.8.m8.1.1" xref="Thmproofx24.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.8.m8.1b"><ci id="Thmproofx24.p1.8.m8.1.1.cmml" xref="Thmproofx24.p1.8.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.8.m8.1d">italic_X</annotation></semantics></math> by swapping <math alttext="X_{i},X_{i+1}" class="ltx_Math" display="inline" id="Thmproofx24.p1.9.m9.2"><semantics id="Thmproofx24.p1.9.m9.2a"><mrow id="Thmproofx24.p1.9.m9.2.2.2" xref="Thmproofx24.p1.9.m9.2.2.3.cmml"><msub id="Thmproofx24.p1.9.m9.1.1.1.1" xref="Thmproofx24.p1.9.m9.1.1.1.1.cmml"><mi id="Thmproofx24.p1.9.m9.1.1.1.1.2" xref="Thmproofx24.p1.9.m9.1.1.1.1.2.cmml">X</mi><mi id="Thmproofx24.p1.9.m9.1.1.1.1.3" xref="Thmproofx24.p1.9.m9.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx24.p1.9.m9.2.2.2.3" xref="Thmproofx24.p1.9.m9.2.2.3.cmml">,</mo><msub id="Thmproofx24.p1.9.m9.2.2.2.2" xref="Thmproofx24.p1.9.m9.2.2.2.2.cmml"><mi id="Thmproofx24.p1.9.m9.2.2.2.2.2" xref="Thmproofx24.p1.9.m9.2.2.2.2.2.cmml">X</mi><mrow id="Thmproofx24.p1.9.m9.2.2.2.2.3" xref="Thmproofx24.p1.9.m9.2.2.2.2.3.cmml"><mi id="Thmproofx24.p1.9.m9.2.2.2.2.3.2" xref="Thmproofx24.p1.9.m9.2.2.2.2.3.2.cmml">i</mi><mo id="Thmproofx24.p1.9.m9.2.2.2.2.3.1" xref="Thmproofx24.p1.9.m9.2.2.2.2.3.1.cmml">+</mo><mn id="Thmproofx24.p1.9.m9.2.2.2.2.3.3" xref="Thmproofx24.p1.9.m9.2.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.9.m9.2b"><list id="Thmproofx24.p1.9.m9.2.2.3.cmml" xref="Thmproofx24.p1.9.m9.2.2.2"><apply id="Thmproofx24.p1.9.m9.1.1.1.1.cmml" xref="Thmproofx24.p1.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.9.m9.1.1.1.1.1.cmml" xref="Thmproofx24.p1.9.m9.1.1.1.1">subscript</csymbol><ci id="Thmproofx24.p1.9.m9.1.1.1.1.2.cmml" xref="Thmproofx24.p1.9.m9.1.1.1.1.2">𝑋</ci><ci id="Thmproofx24.p1.9.m9.1.1.1.1.3.cmml" xref="Thmproofx24.p1.9.m9.1.1.1.1.3">𝑖</ci></apply><apply id="Thmproofx24.p1.9.m9.2.2.2.2.cmml" xref="Thmproofx24.p1.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="Thmproofx24.p1.9.m9.2.2.2.2.1.cmml" xref="Thmproofx24.p1.9.m9.2.2.2.2">subscript</csymbol><ci id="Thmproofx24.p1.9.m9.2.2.2.2.2.cmml" xref="Thmproofx24.p1.9.m9.2.2.2.2.2">𝑋</ci><apply id="Thmproofx24.p1.9.m9.2.2.2.2.3.cmml" xref="Thmproofx24.p1.9.m9.2.2.2.2.3"><plus id="Thmproofx24.p1.9.m9.2.2.2.2.3.1.cmml" xref="Thmproofx24.p1.9.m9.2.2.2.2.3.1"></plus><ci id="Thmproofx24.p1.9.m9.2.2.2.2.3.2.cmml" xref="Thmproofx24.p1.9.m9.2.2.2.2.3.2">𝑖</ci><cn id="Thmproofx24.p1.9.m9.2.2.2.2.3.3.cmml" type="integer" xref="Thmproofx24.p1.9.m9.2.2.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.9.m9.2c">X_{i},X_{i+1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.9.m9.2d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and let <math alttext="\sigma^{\prime}=(X^{\prime})^{n}" class="ltx_Math" display="inline" id="Thmproofx24.p1.10.m10.1"><semantics id="Thmproofx24.p1.10.m10.1a"><mrow id="Thmproofx24.p1.10.m10.1.1" xref="Thmproofx24.p1.10.m10.1.1.cmml"><msup id="Thmproofx24.p1.10.m10.1.1.3" xref="Thmproofx24.p1.10.m10.1.1.3.cmml"><mi id="Thmproofx24.p1.10.m10.1.1.3.2" xref="Thmproofx24.p1.10.m10.1.1.3.2.cmml">σ</mi><mo id="Thmproofx24.p1.10.m10.1.1.3.3" xref="Thmproofx24.p1.10.m10.1.1.3.3.cmml">′</mo></msup><mo id="Thmproofx24.p1.10.m10.1.1.2" xref="Thmproofx24.p1.10.m10.1.1.2.cmml">=</mo><msup id="Thmproofx24.p1.10.m10.1.1.1" xref="Thmproofx24.p1.10.m10.1.1.1.cmml"><mrow id="Thmproofx24.p1.10.m10.1.1.1.1.1" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.cmml"><mo id="Thmproofx24.p1.10.m10.1.1.1.1.1.2" stretchy="false" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.cmml">(</mo><msup id="Thmproofx24.p1.10.m10.1.1.1.1.1.1" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.cmml"><mi id="Thmproofx24.p1.10.m10.1.1.1.1.1.1.2" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.2.cmml">X</mi><mo id="Thmproofx24.p1.10.m10.1.1.1.1.1.1.3" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Thmproofx24.p1.10.m10.1.1.1.1.1.3" stretchy="false" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Thmproofx24.p1.10.m10.1.1.1.3" xref="Thmproofx24.p1.10.m10.1.1.1.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.10.m10.1b"><apply id="Thmproofx24.p1.10.m10.1.1.cmml" xref="Thmproofx24.p1.10.m10.1.1"><eq id="Thmproofx24.p1.10.m10.1.1.2.cmml" xref="Thmproofx24.p1.10.m10.1.1.2"></eq><apply id="Thmproofx24.p1.10.m10.1.1.3.cmml" xref="Thmproofx24.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="Thmproofx24.p1.10.m10.1.1.3.1.cmml" xref="Thmproofx24.p1.10.m10.1.1.3">superscript</csymbol><ci id="Thmproofx24.p1.10.m10.1.1.3.2.cmml" xref="Thmproofx24.p1.10.m10.1.1.3.2">𝜎</ci><ci id="Thmproofx24.p1.10.m10.1.1.3.3.cmml" xref="Thmproofx24.p1.10.m10.1.1.3.3">′</ci></apply><apply id="Thmproofx24.p1.10.m10.1.1.1.cmml" xref="Thmproofx24.p1.10.m10.1.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.10.m10.1.1.1.2.cmml" xref="Thmproofx24.p1.10.m10.1.1.1">superscript</csymbol><apply id="Thmproofx24.p1.10.m10.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.10.m10.1.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.10.m10.1.1.1.1.1">superscript</csymbol><ci id="Thmproofx24.p1.10.m10.1.1.1.1.1.1.2.cmml" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.2">𝑋</ci><ci id="Thmproofx24.p1.10.m10.1.1.1.1.1.1.3.cmml" xref="Thmproofx24.p1.10.m10.1.1.1.1.1.1.3">′</ci></apply><ci id="Thmproofx24.p1.10.m10.1.1.1.3.cmml" xref="Thmproofx24.p1.10.m10.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.10.m10.1c">\sigma^{\prime}=(X^{\prime})^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.10.m10.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Then <math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="Thmproofx24.p1.11.m11.1"><semantics id="Thmproofx24.p1.11.m11.1a"><msup id="Thmproofx24.p1.11.m11.1.1" xref="Thmproofx24.p1.11.m11.1.1.cmml"><mi id="Thmproofx24.p1.11.m11.1.1.2" xref="Thmproofx24.p1.11.m11.1.1.2.cmml">σ</mi><mo id="Thmproofx24.p1.11.m11.1.1.3" xref="Thmproofx24.p1.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.11.m11.1b"><apply id="Thmproofx24.p1.11.m11.1.1.cmml" xref="Thmproofx24.p1.11.m11.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.11.m11.1.1.1.cmml" xref="Thmproofx24.p1.11.m11.1.1">superscript</csymbol><ci id="Thmproofx24.p1.11.m11.1.1.2.cmml" xref="Thmproofx24.p1.11.m11.1.1.2">𝜎</ci><ci id="Thmproofx24.p1.11.m11.1.1.3.cmml" xref="Thmproofx24.p1.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.11.m11.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.11.m11.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> also results from <math alttext="\sigma" class="ltx_Math" display="inline" id="Thmproofx24.p1.12.m12.1"><semantics id="Thmproofx24.p1.12.m12.1a"><mi id="Thmproofx24.p1.12.m12.1.1" xref="Thmproofx24.p1.12.m12.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.12.m12.1b"><ci id="Thmproofx24.p1.12.m12.1.1.cmml" xref="Thmproofx24.p1.12.m12.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.12.m12.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.12.m12.1d">italic_σ</annotation></semantics></math> by swapping <math alttext="\sigma(p_{i}),\sigma(p_{i+1})" class="ltx_Math" display="inline" id="Thmproofx24.p1.13.m13.2"><semantics id="Thmproofx24.p1.13.m13.2a"><mrow id="Thmproofx24.p1.13.m13.2.2.2" xref="Thmproofx24.p1.13.m13.2.2.3.cmml"><mrow id="Thmproofx24.p1.13.m13.1.1.1.1" xref="Thmproofx24.p1.13.m13.1.1.1.1.cmml"><mi id="Thmproofx24.p1.13.m13.1.1.1.1.3" xref="Thmproofx24.p1.13.m13.1.1.1.1.3.cmml">σ</mi><mo id="Thmproofx24.p1.13.m13.1.1.1.1.2" xref="Thmproofx24.p1.13.m13.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmproofx24.p1.13.m13.1.1.1.1.1.1" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.cmml"><mo id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.2" stretchy="false" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.cmml"><mi id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.2" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.2.cmml">p</mi><mi id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.3" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.3" stretchy="false" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmproofx24.p1.13.m13.2.2.2.3" xref="Thmproofx24.p1.13.m13.2.2.3.cmml">,</mo><mrow id="Thmproofx24.p1.13.m13.2.2.2.2" xref="Thmproofx24.p1.13.m13.2.2.2.2.cmml"><mi id="Thmproofx24.p1.13.m13.2.2.2.2.3" xref="Thmproofx24.p1.13.m13.2.2.2.2.3.cmml">σ</mi><mo id="Thmproofx24.p1.13.m13.2.2.2.2.2" xref="Thmproofx24.p1.13.m13.2.2.2.2.2.cmml">⁢</mo><mrow id="Thmproofx24.p1.13.m13.2.2.2.2.1.1" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.cmml"><mo id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.2" stretchy="false" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.cmml">(</mo><msub id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.cmml"><mi id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.2" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.2.cmml">p</mi><mrow id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.cmml"><mi id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.2" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.1" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.1.cmml">+</mo><mn id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.3" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.3" stretchy="false" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx24.p1.13.m13.2b"><list id="Thmproofx24.p1.13.m13.2.2.3.cmml" xref="Thmproofx24.p1.13.m13.2.2.2"><apply id="Thmproofx24.p1.13.m13.1.1.1.1.cmml" xref="Thmproofx24.p1.13.m13.1.1.1.1"><times id="Thmproofx24.p1.13.m13.1.1.1.1.2.cmml" xref="Thmproofx24.p1.13.m13.1.1.1.1.2"></times><ci id="Thmproofx24.p1.13.m13.1.1.1.1.3.cmml" xref="Thmproofx24.p1.13.m13.1.1.1.1.3">𝜎</ci><apply id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.1.cmml" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1">subscript</csymbol><ci id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.2.cmml" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.2">𝑝</ci><ci id="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.3.cmml" xref="Thmproofx24.p1.13.m13.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Thmproofx24.p1.13.m13.2.2.2.2.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2"><times id="Thmproofx24.p1.13.m13.2.2.2.2.2.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.2"></times><ci id="Thmproofx24.p1.13.m13.2.2.2.2.3.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.3">𝜎</ci><apply id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.1.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1">subscript</csymbol><ci id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.2.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.2">𝑝</ci><apply id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3"><plus id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.1.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.1"></plus><ci id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.2.cmml" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.2">𝑖</ci><cn id="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.3.cmml" type="integer" xref="Thmproofx24.p1.13.m13.2.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx24.p1.13.m13.2c">\sigma(p_{i}),\sigma(p_{i+1})</annotation><annotation encoding="application/x-llamapun" id="Thmproofx24.p1.13.m13.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_σ ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. We split into two cases.</p> </div> </div> <section class="ltx_paragraph" id="Pt0.A4.SS4.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Case <math alttext="i<m-1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.1.m1.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.1.m1.1b"><mrow id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.1.cmml"><</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.1.m1.1c"><apply id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1"><lt id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.1"></lt><ci id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.2">𝑖</ci><apply id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3"><minus id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.1"></minus><ci id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.2">𝑚</ci><cn id="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.1.m1.1d">i<m-1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.1.m1.1e">italic_i < italic_m - 1</annotation></semantics></math>:</h5> <div class="ltx_para" id="Pt0.A4.SS4.SSS0.Px1.p1"> <p class="ltx_p" id="Pt0.A4.SS4.SSS0.Px1.p1.13">Let <math alttext="i^{\prime}\in[m]" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.cmml"><msup id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.1.1.cmml">[</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.1.cmml">m</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2"><in id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.1"></in><apply id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.2.3">′</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.2"><csymbol cd="latexml" id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1c">i^{\prime}\in[m]</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.1.m1.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ [ italic_m ]</annotation></semantics></math>. First, suppose <math alttext="i^{\prime}\neq i" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.cmml"><msup id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.1.cmml">≠</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1"><neq id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.1"></neq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.2.3">′</ci></apply><ci id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1c">i^{\prime}\neq i</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.2.m2.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ italic_i</annotation></semantics></math>. If <math alttext="i^{\prime}\neq i+1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.cmml"><msup id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.1.cmml">≠</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1"><neq id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.1"></neq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.2.3">′</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1c">i^{\prime}\neq i+1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.3.m3.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ italic_i + 1</annotation></semantics></math> too then <math alttext="f_{i^{\prime}}(\sigma^{\prime})<f_{i^{\prime}}(\sigma)" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.2.cmml">f</mi><msup id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.cmml">(</mo><msup id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.2.cmml"><</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.2.cmml">f</mi><msup id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.1.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.3.2.1" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.cmml">(</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.1.1.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.3.2.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2"><lt id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.2"></lt><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1"><times id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.2"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.2">𝑓</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.3.3.3">′</ci></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2">𝜎</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3"><times id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.1"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.2">𝑓</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2.2.3.2.3.3">′</ci></apply></apply><ci id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.1.1">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2c">f_{i^{\prime}}(\sigma^{\prime})<f_{i^{\prime}}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.4.m4.2d">italic_f start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) < italic_f start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_σ )</annotation></semantics></math>. This is because <math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1a"><msup id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.2.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.2">𝜎</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.5.m5.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> results from <math alttext="\sigma" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1a"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1b"><ci id="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.6.m6.1d">italic_σ</annotation></semantics></math> by swapping the probabilities of <math alttext="p_{i},p_{i+1}" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.3.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.3.cmml">,</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.2.cmml">p</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2b"><list id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.2">𝑝</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.1.1.1.1.3">𝑖</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2.2.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2c">p_{i},p_{i+1}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.7.m7.2d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math> being picked, and <math alttext="\sigma(p_{i})<\sigma(p_{i+1})" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.3.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.3.cmml"><</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.3.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2"><lt id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.3"></lt><apply id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1"><times id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.2"></times><ci id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.3">𝜎</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.2">𝑝</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2"><times id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.2"></times><ci id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.3">𝜎</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2c">\sigma(p_{i})<\sigma(p_{i+1})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.8.m8.2d">italic_σ ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) < italic_σ ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="c_{i^{\prime}}(p_{i})=|p_{i}|<|p_{i+1}|=c_{i^{\prime}}(p_{i+1})" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4" xref="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.1.1.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.1.1.1.3.cmml"><mi 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id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4.4.4.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4c">c_{i^{\prime}}(p_{i})=|p_{i}|<|p_{i+1}|=c_{i^{\prime}}(p_{i+1})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.9.m9.4d">italic_c start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | = italic_c start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. If <math alttext="i^{\prime}=i+1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.cmml"><msup id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.1.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1"><eq id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.1"></eq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.2.3">′</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1c">i^{\prime}=i+1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.10.m10.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_i + 1</annotation></semantics></math> then this inequality holds as well, because <math alttext="c_{i+1}(p_{i})=|p_{i}|<|p_{i+1}|+\alpha=c_{i+1}(p_{i+1})" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.2.cmml">c</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.6" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.6.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.2.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.2.1.cmml">|</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.2.2.2.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.7" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.7.cmml"><</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.2.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.2.1.cmml">|</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.2.cmml">+</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.3.3.3.3.cmml">α</mi></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.8" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.8.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.2.cmml">c</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4b"><apply 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id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.2"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.2">𝑐</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.3.3.3">1</cn></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4.4.4.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4c">c_{i+1}(p_{i})=|p_{i}|<|p_{i+1}|+\alpha=c_{i+1}(p_{i+1})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.11.m11.4d">italic_c start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | < | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | + italic_α = italic_c start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, so the argument above holds here too. Therefore <math alttext="i^{\prime}=i" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.cmml"><msup id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.1.cmml">=</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1"><eq id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.1"></eq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.2.3">′</ci></apply><ci id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1c">i^{\prime}=i</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.12.m12.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_i</annotation></semantics></math> is left. We show that <math alttext="f_{i}(\sigma^{\prime})\leq f_{i+1}(\sigma)" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.2.cmml">f</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.cmml">(</mo><msup id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.2.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.2.cmml">≤</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.2.cmml">f</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.1.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.3.2.1" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.cmml">(</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.1.1.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.3.2.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2"><leq id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.2"></leq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1"><times id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.2"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.2">𝑓</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.3.3">𝑖</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.2">𝜎</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3"><times id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.1"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.2">𝑓</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2.2.3.2.3.3">1</cn></apply></apply><ci id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.1.1">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2c">f_{i}(\sigma^{\prime})\leq f_{i+1}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.13.m13.2d">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_f start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_σ )</annotation></semantics></math>. Using Equation <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E7" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">7</span></a>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx62"> <tbody id="Pt0.A4.Ex109"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle f_{i+1}(\sigma)-f_{i}(\sigma^{\prime})=(|p_{i}|-|p_{i-1}|)\left(% 1-\left(\sum_{i^{\prime}<i}X(p_{i^{\prime}})\right)^{n}\right)-(|p_{i+1}|-|p_{% i-1}|)\left(1-\left(\sum_{i^{\prime}<i}X(p_{i^{\prime}})\right)^{n}\right)~{}+" class="ltx_Math" display="inline" id="Pt0.A4.Ex109.m1.6"><semantics id="Pt0.A4.Ex109.m1.6a"><mrow id="Pt0.A4.Ex109.m1.6.6" xref="Pt0.A4.Ex109.m1.6.6.cmml"><mrow id="Pt0.A4.Ex109.m1.2.2.1" xref="Pt0.A4.Ex109.m1.2.2.1.cmml"><mrow id="Pt0.A4.Ex109.m1.2.2.1.3" xref="Pt0.A4.Ex109.m1.2.2.1.3.cmml"><msub id="Pt0.A4.Ex109.m1.2.2.1.3.2" xref="Pt0.A4.Ex109.m1.2.2.1.3.2.cmml"><mi id="Pt0.A4.Ex109.m1.2.2.1.3.2.2" xref="Pt0.A4.Ex109.m1.2.2.1.3.2.2.cmml">f</mi><mrow id="Pt0.A4.Ex109.m1.2.2.1.3.2.3" xref="Pt0.A4.Ex109.m1.2.2.1.3.2.3.cmml"><mi id="Pt0.A4.Ex109.m1.2.2.1.3.2.3.2" xref="Pt0.A4.Ex109.m1.2.2.1.3.2.3.2.cmml">i</mi><mo id="Pt0.A4.Ex109.m1.2.2.1.3.2.3.1" xref="Pt0.A4.Ex109.m1.2.2.1.3.2.3.1.cmml">+</mo><mn id="Pt0.A4.Ex109.m1.2.2.1.3.2.3.3" xref="Pt0.A4.Ex109.m1.2.2.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.Ex109.m1.2.2.1.3.1" xref="Pt0.A4.Ex109.m1.2.2.1.3.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex109.m1.2.2.1.3.3.2" xref="Pt0.A4.Ex109.m1.2.2.1.3.cmml"><mo id="Pt0.A4.Ex109.m1.2.2.1.3.3.2.1" stretchy="false" xref="Pt0.A4.Ex109.m1.2.2.1.3.cmml">(</mo><mi id="Pt0.A4.Ex109.m1.1.1" xref="Pt0.A4.Ex109.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.Ex109.m1.2.2.1.3.3.2.2" stretchy="false" xref="Pt0.A4.Ex109.m1.2.2.1.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex109.m1.2.2.1.2" xref="Pt0.A4.Ex109.m1.2.2.1.2.cmml">−</mo><mrow id="Pt0.A4.Ex109.m1.2.2.1.1" xref="Pt0.A4.Ex109.m1.2.2.1.1.cmml"><msub id="Pt0.A4.Ex109.m1.2.2.1.1.3" xref="Pt0.A4.Ex109.m1.2.2.1.1.3.cmml"><mi id="Pt0.A4.Ex109.m1.2.2.1.1.3.2" xref="Pt0.A4.Ex109.m1.2.2.1.1.3.2.cmml">f</mi><mi id="Pt0.A4.Ex109.m1.2.2.1.1.3.3" xref="Pt0.A4.Ex109.m1.2.2.1.1.3.3.cmml">i</mi></msub><mo id="Pt0.A4.Ex109.m1.2.2.1.1.2" xref="Pt0.A4.Ex109.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex109.m1.2.2.1.1.1.1" xref="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex109.m1.2.2.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.cmml">(</mo><msup id="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.2.cmml">σ</mi><mo id="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.3" xref="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex109.m1.2.2.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex109.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex109.m1.6.6.6" xref="Pt0.A4.Ex109.m1.6.6.6.cmml">=</mo><mrow id="Pt0.A4.Ex109.m1.6.6.5" xref="Pt0.A4.Ex109.m1.6.6.5.cmml"><mrow id="Pt0.A4.Ex109.m1.4.4.3.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.cmml"><mrow id="Pt0.A4.Ex109.m1.3.3.2.1.1.1" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.cmml"><mo id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.1.1.1.cmml"><mi 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xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.2.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.2.1.1.3.1" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.2.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.2.1.1.3.3" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.2.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.2.1.3" stretchy="false" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Pt0.A4.Ex109.m1.3.3.2.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex109.m1.3.3.2.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex109.m1.4.4.3.2.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex109.m1.4.4.3.2.2.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.cmml"><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.cmml"><mn id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.3.cmml">1</mn><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.2.cmml">−</mo><msup id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.cmml"><mrow id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.cmml"><munder id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2a" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.2" movablelimits="false" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.2.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.2.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.1.cmml"><</mo><mi id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.2.3.3.cmml">i</mi></mrow></munder></mstyle><mrow id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><msup id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Pt0.A4.Ex109.m1.4.4.3.2.2.1.3" xref="Pt0.A4.Ex109.m1.4.4.3.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex109.m1.6.6.5.5" xref="Pt0.A4.Ex109.m1.6.6.5.5.cmml">−</mo><mrow id="Pt0.A4.Ex109.m1.6.6.5.4" xref="Pt0.A4.Ex109.m1.6.6.5.4.cmml"><mrow id="Pt0.A4.Ex109.m1.6.6.5.4.2" xref="Pt0.A4.Ex109.m1.6.6.5.4.2.cmml"><mrow id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.cmml"><mo id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3.1" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.3" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.2.1" xref="Pt0.A4.Ex109.m1.5.5.4.3.1.1.1.1.2.2.cmml"><mo 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xref="Pt0.A4.Ex109.m1.6.6.5.4.4"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex109.m1.6c">\displaystyle f_{i+1}(\sigma)-f_{i}(\sigma^{\prime})=(|p_{i}|-|p_{i-1}|)\left(% 1-\left(\sum_{i^{\prime}<i}X(p_{i^{\prime}})\right)^{n}\right)-(|p_{i+1}|-|p_{% i-1}|)\left(1-\left(\sum_{i^{\prime}<i}X(p_{i^{\prime}})\right)^{n}\right)~{}+</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex109.m1.6d">italic_f start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_σ ) - italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ( | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT | ) ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_i end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - ( | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT | ) ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_i end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) +</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex110"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(|p_{i+2}|-|p_{i}|)\left(1-\left(\sum_{i^{\prime}<i+1}X(p_{i^{% \prime}})\right)^{n}\right)-(|p_{i+2}|-|p_{i+1}|)\left(1-\left(\sum_{i^{\prime% }<i+1}X(p_{i^{\prime}})\right)^{n}\right)~{}+" 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xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.3">𝑋</ci><apply id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply><ci id="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.3.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.2.2.1.1.1.3">𝑛</ci></apply></apply></apply><plus id="Pt0.A4.Ex110.m1.4.4.4.4.cmml" xref="Pt0.A4.Ex110.m1.4.4.4.4"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex110.m1.4c">\displaystyle(|p_{i+2}|-|p_{i}|)\left(1-\left(\sum_{i^{\prime}<i+1}X(p_{i^{% \prime}})\right)^{n}\right)-(|p_{i+2}|-|p_{i+1}|)\left(1-\left(\sum_{i^{\prime% }<i+1}X(p_{i^{\prime}})\right)^{n}\right)~{}+</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex110.m1.4d">( | italic_p start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_i + 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - ( | italic_p start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | ) ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_i + 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) +</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex111"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(|p_{i+1}|+\alpha-|p_{m}|)\left(1-\left(1-X(p_{i+1})\right)^{n}% \right)-(|p_{i}|+\alpha-|p_{m}|)\left(1-\left(1-X(p_{i+1})\right)^{n}\right)" class="ltx_Math" 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xref="Pt0.A4.Ex111.m1.3.3.3.1.1.1.2.1.1.3">𝑚</ci></apply></apply></apply><apply id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1"><minus id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.2.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.2"></minus><cn id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.3">1</cn><apply id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.2.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1"><minus id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.2"></minus><cn id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.3">1</cn><apply id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1"><times id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.2"></times><ci id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.3">𝑋</ci><apply id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3"><plus id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.1"></plus><ci id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.2">𝑖</ci><cn id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply><ci id="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.3.cmml" xref="Pt0.A4.Ex111.m1.4.4.4.2.1.1.1.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex111.m1.4c">\displaystyle(|p_{i+1}|+\alpha-|p_{m}|)\left(1-\left(1-X(p_{i+1})\right)^{n}% \right)-(|p_{i}|+\alpha-|p_{m}|)\left(1-\left(1-X(p_{i+1})\right)^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex111.m1.4d">( | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | + italic_α - | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ) ( 1 - ( 1 - italic_X ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - ( | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | + italic_α - | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ) ( 1 - ( 1 - italic_X ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A4.SS4.SSS0.Px1.p1.14">Let <math alttext="x=\sum_{i^{\prime}<i}X(p_{i^{\prime}}),y=X(p_{i}),z=X(p_{i+1})" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.3.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.3.cmml">x</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.2" rspace="0.111em" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.2.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.1.cmml"><</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.3.cmml">i</mi></mrow></msub><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><msup id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.3a.cmml">,</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.3.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.3.cmml">y</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.2.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.3a.cmml">,</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.3.cmml">z</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.2.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.3.cmml">X</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.3a.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.3">formulae-sequence</csymbol><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1"><eq id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.2"></eq><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.3">𝑥</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2">subscript</csymbol><sum id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.2"></sum><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3"><lt id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.1"></lt><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.2.3.2.2.cmml" 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xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.1.1.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.3a.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.3">formulae-sequence</csymbol><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1"><eq id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.2"></eq><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.3">𝑦</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1"><times id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.2"></times><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.3">𝑋</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.2">𝑝</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2"><eq id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.2"></eq><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.3">𝑧</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1"><times id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.2"></times><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.3">𝑋</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2.2.2.2.2.2.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2c">x=\sum_{i^{\prime}<i}X(p_{i^{\prime}}),y=X(p_{i}),z=X(p_{i+1})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.14.m1.2d">italic_x = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_i end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) , italic_y = italic_X ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_z = italic_X ( italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Then we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx63"> <tbody id="Pt0.A4.Ex112"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{i+1}(\sigma)-f_{i}(\sigma^{\prime})=" class="ltx_Math" display="inline" id="Pt0.A4.Ex112.m1.2"><semantics id="Pt0.A4.Ex112.m1.2a"><mrow id="Pt0.A4.Ex112.m1.2.2" xref="Pt0.A4.Ex112.m1.2.2.cmml"><mrow id="Pt0.A4.Ex112.m1.2.2.1" xref="Pt0.A4.Ex112.m1.2.2.1.cmml"><mrow id="Pt0.A4.Ex112.m1.2.2.1.3" xref="Pt0.A4.Ex112.m1.2.2.1.3.cmml"><msub id="Pt0.A4.Ex112.m1.2.2.1.3.2" xref="Pt0.A4.Ex112.m1.2.2.1.3.2.cmml"><mi id="Pt0.A4.Ex112.m1.2.2.1.3.2.2" xref="Pt0.A4.Ex112.m1.2.2.1.3.2.2.cmml">f</mi><mrow id="Pt0.A4.Ex112.m1.2.2.1.3.2.3" xref="Pt0.A4.Ex112.m1.2.2.1.3.2.3.cmml"><mi id="Pt0.A4.Ex112.m1.2.2.1.3.2.3.2" xref="Pt0.A4.Ex112.m1.2.2.1.3.2.3.2.cmml">i</mi><mo id="Pt0.A4.Ex112.m1.2.2.1.3.2.3.1" xref="Pt0.A4.Ex112.m1.2.2.1.3.2.3.1.cmml">+</mo><mn id="Pt0.A4.Ex112.m1.2.2.1.3.2.3.3" xref="Pt0.A4.Ex112.m1.2.2.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.Ex112.m1.2.2.1.3.1" xref="Pt0.A4.Ex112.m1.2.2.1.3.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex112.m1.2.2.1.3.3.2" xref="Pt0.A4.Ex112.m1.2.2.1.3.cmml"><mo id="Pt0.A4.Ex112.m1.2.2.1.3.3.2.1" stretchy="false" xref="Pt0.A4.Ex112.m1.2.2.1.3.cmml">(</mo><mi id="Pt0.A4.Ex112.m1.1.1" xref="Pt0.A4.Ex112.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.Ex112.m1.2.2.1.3.3.2.2" stretchy="false" xref="Pt0.A4.Ex112.m1.2.2.1.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex112.m1.2.2.1.2" xref="Pt0.A4.Ex112.m1.2.2.1.2.cmml">−</mo><mrow id="Pt0.A4.Ex112.m1.2.2.1.1" xref="Pt0.A4.Ex112.m1.2.2.1.1.cmml"><msub id="Pt0.A4.Ex112.m1.2.2.1.1.3" xref="Pt0.A4.Ex112.m1.2.2.1.1.3.cmml"><mi id="Pt0.A4.Ex112.m1.2.2.1.1.3.2" xref="Pt0.A4.Ex112.m1.2.2.1.1.3.2.cmml">f</mi><mi id="Pt0.A4.Ex112.m1.2.2.1.1.3.3" 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xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1"><minus id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.2.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.2"></minus><cn id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.3">1</cn><apply id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.2.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1"><plus id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1.1.1"></plus><ci id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1.1.2">𝑥</ci><ci id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.1.1.1.3">𝑦</ci></apply><ci id="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.3.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.2.2.1.1.1.3">𝑛</ci></apply></apply></apply><plus id="Pt0.A4.Ex113.m1.4.4.4.4.cmml" xref="Pt0.A4.Ex113.m1.4.4.4.4"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex113.m1.4c">\displaystyle(|p_{i+2}|-|p_{i}|)\left(1-\left(x+y\right)^{n}\right)-(|p_{i+2}|% -|p_{i+1}|)\left(1-\left(x+y\right)^{n}\right)~{}+</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex113.m1.4d">( | italic_p start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) ( 1 - ( italic_x + italic_y ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - ( | italic_p start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | ) ( 1 - ( italic_x + italic_y ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) +</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> 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cd="ambiguous" id="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1"><minus id="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1.1.1"></minus><cn id="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1.1.2">1</cn><ci id="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.1.1.1.3">𝑧</ci></apply><ci id="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.1.1.3">𝑛</ci></apply><apply id="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.2.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2">superscript</csymbol><apply id="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1.1.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1"><plus id="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1.1.1.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1.1.1"></plus><ci id="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1.1.2.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1.1.2">𝑥</ci><ci id="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1.1.3.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.1.1.1.3">𝑦</ci></apply><ci id="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.3.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.2.2.3">𝑛</ci></apply></apply><apply id="Pt0.A4.Ex115.m2.2.2.2.1.1.4.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.4"><csymbol cd="ambiguous" id="Pt0.A4.Ex115.m2.2.2.2.1.1.4.1.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.4">superscript</csymbol><ci id="Pt0.A4.Ex115.m2.2.2.2.1.1.4.2.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.4.2">𝑥</ci><ci id="Pt0.A4.Ex115.m2.2.2.2.1.1.4.3.cmml" xref="Pt0.A4.Ex115.m2.2.2.2.1.1.4.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex115.m2.2c">\displaystyle(|p_{i+1}|-|p_{i}|)\left(1-(1-z)^{n}-(x+y)^{n}+x^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex115.m2.2d">( | italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) ( 1 - ( 1 - italic_z ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_x + italic_y ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A4.SS4.SSS0.Px1.p1.19"><math alttext="|p_{i+1}|>|p_{i}|" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.2.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.3.cmml">></mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.2.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.2.1.cmml">|</mo><msub id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2"><gt id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.3"></gt><apply id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1"><abs id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.2"></abs><apply id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.1.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1"><abs id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.2"></abs><apply id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.2">𝑝</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2c">|p_{i+1}|>|p_{i}|</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.15.m1.2d">| italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT | > | italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math>. Also, there holds <math alttext="y<z" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.2.cmml">y</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.1.cmml"><</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1"><lt id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.1"></lt><ci id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.2">𝑦</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1c">y<z</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.16.m2.1d">italic_y < italic_z</annotation></semantics></math> and <math alttext="x+y+z\leq 1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.2.cmml">x</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.1.cmml">+</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.3.cmml">y</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.1a" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.1.cmml">+</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.4" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.4.cmml">z</mi></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.1.cmml">≤</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1"><leq id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.1"></leq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.2">𝑥</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.3">𝑦</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.4.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.2.4">𝑧</ci></apply><cn id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1c">x+y+z\leq 1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.17.m3.1d">italic_x + italic_y + italic_z ≤ 1</annotation></semantics></math>, so <math alttext="x+y\leq 1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.2.cmml">x</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.1.cmml">+</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.3.cmml">y</mi></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.1.cmml">≤</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1"><leq id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.1"></leq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.2">𝑥</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.2.3">𝑦</ci></apply><cn id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1c">x+y\leq 1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.18.m4.1d">italic_x + italic_y ≤ 1</annotation></semantics></math> and <math alttext="x\leq 1-z" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.2.cmml">x</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.1.cmml">≤</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.cmml"><mn id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.2.cmml">1</mn><mo id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.1.cmml">−</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.3.cmml">z</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1"><leq id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.1"></leq><ci id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.2">𝑥</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3"><minus id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.1"></minus><cn id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.2.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.2">1</cn><ci id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1.1.3.3">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1c">x\leq 1-z</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.19.m5.1d">italic_x ≤ 1 - italic_z</annotation></semantics></math>. Then</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx64"> <tbody id="Pt0.A4.Ex116"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 1-(1-z)^{n}=z\sum_{i=0}^{n}(1-z)^{i}\geq y\sum_{i=0}^{n}x^{i}(x+% y)^{n-i}=(x+y)^{n}-x^{n}" class="ltx_Math" display="inline" id="Pt0.A4.Ex116.m1.4"><semantics id="Pt0.A4.Ex116.m1.4a"><mrow id="Pt0.A4.Ex116.m1.4.4" xref="Pt0.A4.Ex116.m1.4.4.cmml"><mrow id="Pt0.A4.Ex116.m1.1.1.1" xref="Pt0.A4.Ex116.m1.1.1.1.cmml"><mn id="Pt0.A4.Ex116.m1.1.1.1.3" xref="Pt0.A4.Ex116.m1.1.1.1.3.cmml">1</mn><mo id="Pt0.A4.Ex116.m1.1.1.1.2" xref="Pt0.A4.Ex116.m1.1.1.1.2.cmml">−</mo><msup id="Pt0.A4.Ex116.m1.1.1.1.1" xref="Pt0.A4.Ex116.m1.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex116.m1.1.1.1.1.1.1" xref="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex116.m1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.cmml"><mn id="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="Pt0.A4.Ex116.m1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex116.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex116.m1.1.1.1.1.3" xref="Pt0.A4.Ex116.m1.1.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Pt0.A4.Ex116.m1.4.4.6" xref="Pt0.A4.Ex116.m1.4.4.6.cmml">=</mo><mrow id="Pt0.A4.Ex116.m1.2.2.2" xref="Pt0.A4.Ex116.m1.2.2.2.cmml"><mi id="Pt0.A4.Ex116.m1.2.2.2.3" xref="Pt0.A4.Ex116.m1.2.2.2.3.cmml">z</mi><mo id="Pt0.A4.Ex116.m1.2.2.2.2" xref="Pt0.A4.Ex116.m1.2.2.2.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex116.m1.2.2.2.1" xref="Pt0.A4.Ex116.m1.2.2.2.1.cmml"><mstyle 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id="Pt0.A4.Ex116.m1.3.3.3.1.1.1.3.2.cmml" xref="Pt0.A4.Ex116.m1.3.3.3.1.1.1.3.2">𝑛</ci><ci id="Pt0.A4.Ex116.m1.3.3.3.1.1.1.3.3.cmml" xref="Pt0.A4.Ex116.m1.3.3.3.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply><apply id="Pt0.A4.Ex116.m1.4.4e.cmml" xref="Pt0.A4.Ex116.m1.4.4"><eq id="Pt0.A4.Ex116.m1.4.4.8.cmml" xref="Pt0.A4.Ex116.m1.4.4.8"></eq><share href="https://arxiv.org/html/2308.08047v3#Pt0.A4.Ex116.m1.3.3.3.cmml" id="Pt0.A4.Ex116.m1.4.4f.cmml" xref="Pt0.A4.Ex116.m1.4.4"></share><apply id="Pt0.A4.Ex116.m1.4.4.4.cmml" xref="Pt0.A4.Ex116.m1.4.4.4"><minus id="Pt0.A4.Ex116.m1.4.4.4.2.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.2"></minus><apply id="Pt0.A4.Ex116.m1.4.4.4.1.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex116.m1.4.4.4.1.2.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.1">superscript</csymbol><apply id="Pt0.A4.Ex116.m1.4.4.4.1.1.1.1.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.1.1.1"><plus id="Pt0.A4.Ex116.m1.4.4.4.1.1.1.1.1.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.1.1.1.1.1"></plus><ci id="Pt0.A4.Ex116.m1.4.4.4.1.1.1.1.2.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.1.1.1.1.2">𝑥</ci><ci id="Pt0.A4.Ex116.m1.4.4.4.1.1.1.1.3.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.1.1.1.1.3">𝑦</ci></apply><ci id="Pt0.A4.Ex116.m1.4.4.4.1.3.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.1.3">𝑛</ci></apply><apply id="Pt0.A4.Ex116.m1.4.4.4.3.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex116.m1.4.4.4.3.1.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.3">superscript</csymbol><ci id="Pt0.A4.Ex116.m1.4.4.4.3.2.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.3.2">𝑥</ci><ci id="Pt0.A4.Ex116.m1.4.4.4.3.3.cmml" xref="Pt0.A4.Ex116.m1.4.4.4.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex116.m1.4c">\displaystyle 1-(1-z)^{n}=z\sum_{i=0}^{n}(1-z)^{i}\geq y\sum_{i=0}^{n}x^{i}(x+% y)^{n-i}=(x+y)^{n}-x^{n}</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex116.m1.4d">1 - ( 1 - italic_z ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_z ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 1 - italic_z ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ≥ italic_y ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_x + italic_y ) start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT = ( italic_x + italic_y ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A4.SS4.SSS0.Px1.p1.21">Hence <math alttext="f_{i+1}(\sigma)-f_{i}(\sigma^{\prime})\geq 0" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.2.cmml">f</mi><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.1.cmml">+</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.1.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.3.2.1" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.cmml">(</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.3.2.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.2.cmml">−</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.2.cmml">f</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.3.cmml">i</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.cmml">(</mo><msup id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.2.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.2.cmml">≥</mo><mn id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2"><geq id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.2"></geq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1"><minus id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.2"></minus><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3"><times id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.1"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.2">𝑓</ci><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3"><plus id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.2">𝑖</ci><cn id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.3.2.3.3">1</cn></apply></apply><ci id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.1.1">𝜎</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1"><times id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.2"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.2">𝑓</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.3.3">𝑖</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.2">𝜎</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.1.1.1.1.1.3">′</ci></apply></apply></apply><cn id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2c">f_{i+1}(\sigma)-f_{i}(\sigma^{\prime})\geq 0</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.20.m1.2d">italic_f start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_σ ) - italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≥ 0</annotation></semantics></math>. Therefore by definition <math alttext="F_{n}(\sigma^{\prime},\alpha)\leq F_{n}(\sigma,\alpha)" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4"><semantics id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4a"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.2.cmml">F</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.3.cmml">n</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.2.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.2.cmml">(</mo><msup id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.2.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.2.cmml">,</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.1.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.1.1.cmml">α</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.4" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.2.cmml">≤</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.cmml"><msub id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.2.cmml">F</mi><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.3.cmml">n</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.1" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.1.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.2.1" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.1.cmml">(</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.2.2.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.2.2" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.1.cmml">,</mo><mi id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.3.3" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.3.3.cmml">α</mi><mo id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.2.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4b"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4"><leq id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.2"></leq><apply id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1"><times id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.2"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.2">𝐹</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.3.3">𝑛</ci></apply><interval closure="open" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1"><apply id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.2">𝜎</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.1.1.1.1.3">′</ci></apply><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.1.1">𝛼</ci></interval></apply><apply id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3"><times id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.1"></times><apply id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.2">𝐹</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.2.3">𝑛</ci></apply><interval closure="open" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4.4.3.3.2"><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.2.2">𝜎</ci><ci id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.3.3">𝛼</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4c">F_{n}(\sigma^{\prime},\alpha)\leq F_{n}(\sigma,\alpha)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px1.p1.21.m2.4d">italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_α ) ≤ italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ , italic_α )</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="Pt0.A4.SS4.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Case <math alttext="i=m-1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px2.1.m1.1"><semantics id="Pt0.A4.SS4.SSS0.Px2.1.m1.1b"><mrow id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.1.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px2.1.m1.1c"><apply id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1"><eq id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.1"></eq><ci id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.2">𝑖</ci><apply id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3"><minus id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.1"></minus><ci id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.2">𝑚</ci><cn id="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px2.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px2.1.m1.1d">i=m-1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px2.1.m1.1e">italic_i = italic_m - 1</annotation></semantics></math>:</h5> <div class="ltx_para" id="Pt0.A4.SS4.SSS0.Px2.p1"> <p class="ltx_p" id="Pt0.A4.SS4.SSS0.Px2.p1.3">Note that the same proof as in the other case proves that <math alttext="f_{i^{\prime}}(\sigma^{\prime})<f_{i^{\prime}}(\sigma)" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2"><semantics id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2a"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.2.cmml">f</mi><msup id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><msup id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.2.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.2.cmml"><</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.cmml"><msub id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.2.cmml">f</mi><msup id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.1.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.cmml"><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.3.2.1" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.cmml">(</mo><mi id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.3.2.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2b"><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2"><lt id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.2"></lt><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1"><times id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.2"></times><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.2">𝑓</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.3.3.3">′</ci></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.2">𝜎</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3"><times id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.1"></times><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.2">𝑓</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2.2.3.2.3.3">′</ci></apply></apply><ci id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.1.1">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2c">f_{i^{\prime}}(\sigma^{\prime})<f_{i^{\prime}}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px2.p1.1.m1.2d">italic_f start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) < italic_f start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_σ )</annotation></semantics></math> for <math alttext="i^{\prime}\neq m-1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1"><semantics id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1a"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.cmml"><msup id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.1.cmml">≠</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1b"><apply id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1"><neq id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.1"></neq><apply id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.2.3">′</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3"><minus id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.1"></minus><ci id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.2">𝑚</ci><cn id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1c">i^{\prime}\neq m-1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px2.p1.2.m2.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ italic_m - 1</annotation></semantics></math>. Case <math alttext="i^{\prime}=m-1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1"><semantics id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1a"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.cmml"><msup id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.1.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1b"><apply id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1"><eq id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.1"></eq><apply id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.2.3">′</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3"><minus id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.1"></minus><ci id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.2">𝑚</ci><cn id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1c">i^{\prime}=m-1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px2.p1.3.m3.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_m - 1</annotation></semantics></math> is left. In the same way as in the previous case, using Equations <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E7" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">7</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2308.08047v3#Pt0.A4.E8" title="In 0.D.3 Expressions for MAX cost ‣ Appendix 0.D Omitted Details in Section 5 ‣ Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games"><span class="ltx_text ltx_ref_tag">8</span></a>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx65"> <tbody id="Pt0.A4.Ex117"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{m}(\sigma)" class="ltx_Math" display="inline" id="Pt0.A4.Ex117.m1.1"><semantics id="Pt0.A4.Ex117.m1.1a"><mrow id="Pt0.A4.Ex117.m1.1.2" xref="Pt0.A4.Ex117.m1.1.2.cmml"><msub id="Pt0.A4.Ex117.m1.1.2.2" xref="Pt0.A4.Ex117.m1.1.2.2.cmml"><mi id="Pt0.A4.Ex117.m1.1.2.2.2" xref="Pt0.A4.Ex117.m1.1.2.2.2.cmml">f</mi><mi id="Pt0.A4.Ex117.m1.1.2.2.3" xref="Pt0.A4.Ex117.m1.1.2.2.3.cmml">m</mi></msub><mo id="Pt0.A4.Ex117.m1.1.2.1" xref="Pt0.A4.Ex117.m1.1.2.1.cmml">⁢</mo><mrow id="Pt0.A4.Ex117.m1.1.2.3.2" xref="Pt0.A4.Ex117.m1.1.2.cmml"><mo id="Pt0.A4.Ex117.m1.1.2.3.2.1" stretchy="false" xref="Pt0.A4.Ex117.m1.1.2.cmml">(</mo><mi id="Pt0.A4.Ex117.m1.1.1" xref="Pt0.A4.Ex117.m1.1.1.cmml">σ</mi><mo id="Pt0.A4.Ex117.m1.1.2.3.2.2" stretchy="false" xref="Pt0.A4.Ex117.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex117.m1.1b"><apply id="Pt0.A4.Ex117.m1.1.2.cmml" xref="Pt0.A4.Ex117.m1.1.2"><times id="Pt0.A4.Ex117.m1.1.2.1.cmml" xref="Pt0.A4.Ex117.m1.1.2.1"></times><apply id="Pt0.A4.Ex117.m1.1.2.2.cmml" xref="Pt0.A4.Ex117.m1.1.2.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex117.m1.1.2.2.1.cmml" xref="Pt0.A4.Ex117.m1.1.2.2">subscript</csymbol><ci id="Pt0.A4.Ex117.m1.1.2.2.2.cmml" xref="Pt0.A4.Ex117.m1.1.2.2.2">𝑓</ci><ci id="Pt0.A4.Ex117.m1.1.2.2.3.cmml" xref="Pt0.A4.Ex117.m1.1.2.2.3">𝑚</ci></apply><ci id="Pt0.A4.Ex117.m1.1.1.cmml" xref="Pt0.A4.Ex117.m1.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex117.m1.1c">\displaystyle f_{m}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex117.m1.1d">italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_σ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-f_{m-1}(\sigma^{\prime})" class="ltx_Math" display="inline" id="Pt0.A4.Ex117.m2.1"><semantics id="Pt0.A4.Ex117.m2.1a"><mrow id="Pt0.A4.Ex117.m2.1.1" xref="Pt0.A4.Ex117.m2.1.1.cmml"><mo id="Pt0.A4.Ex117.m2.1.1a" xref="Pt0.A4.Ex117.m2.1.1.cmml">−</mo><mrow id="Pt0.A4.Ex117.m2.1.1.1" xref="Pt0.A4.Ex117.m2.1.1.1.cmml"><msub id="Pt0.A4.Ex117.m2.1.1.1.3" xref="Pt0.A4.Ex117.m2.1.1.1.3.cmml"><mi id="Pt0.A4.Ex117.m2.1.1.1.3.2" xref="Pt0.A4.Ex117.m2.1.1.1.3.2.cmml">f</mi><mrow id="Pt0.A4.Ex117.m2.1.1.1.3.3" xref="Pt0.A4.Ex117.m2.1.1.1.3.3.cmml"><mi id="Pt0.A4.Ex117.m2.1.1.1.3.3.2" xref="Pt0.A4.Ex117.m2.1.1.1.3.3.2.cmml">m</mi><mo id="Pt0.A4.Ex117.m2.1.1.1.3.3.1" xref="Pt0.A4.Ex117.m2.1.1.1.3.3.1.cmml">−</mo><mn id="Pt0.A4.Ex117.m2.1.1.1.3.3.3" xref="Pt0.A4.Ex117.m2.1.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.Ex117.m2.1.1.1.2" xref="Pt0.A4.Ex117.m2.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex117.m2.1.1.1.1.1" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex117.m2.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.cmml">(</mo><msup id="Pt0.A4.Ex117.m2.1.1.1.1.1.1" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex117.m2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.2.cmml">σ</mi><mo id="Pt0.A4.Ex117.m2.1.1.1.1.1.1.3" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex117.m2.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.Ex117.m2.1b"><apply id="Pt0.A4.Ex117.m2.1.1.cmml" xref="Pt0.A4.Ex117.m2.1.1"><minus id="Pt0.A4.Ex117.m2.1.1.2.cmml" xref="Pt0.A4.Ex117.m2.1.1"></minus><apply id="Pt0.A4.Ex117.m2.1.1.1.cmml" xref="Pt0.A4.Ex117.m2.1.1.1"><times id="Pt0.A4.Ex117.m2.1.1.1.2.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.2"></times><apply id="Pt0.A4.Ex117.m2.1.1.1.3.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex117.m2.1.1.1.3.1.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.3">subscript</csymbol><ci id="Pt0.A4.Ex117.m2.1.1.1.3.2.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.3.2">𝑓</ci><apply id="Pt0.A4.Ex117.m2.1.1.1.3.3.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.3.3"><minus id="Pt0.A4.Ex117.m2.1.1.1.3.3.1.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.3.3.1"></minus><ci id="Pt0.A4.Ex117.m2.1.1.1.3.3.2.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.3.3.2">𝑚</ci><cn id="Pt0.A4.Ex117.m2.1.1.1.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex117.m2.1.1.1.3.3.3">1</cn></apply></apply><apply id="Pt0.A4.Ex117.m2.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex117.m2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.1.1">superscript</csymbol><ci id="Pt0.A4.Ex117.m2.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.2">𝜎</ci><ci id="Pt0.A4.Ex117.m2.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex117.m2.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex117.m2.1c">\displaystyle-f_{m-1}(\sigma^{\prime})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex117.m2.1d">- italic_f start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex118"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=(|p_{m-1}|-|p_{m-2}|)\left(1-\left(\sum_{i^{\prime}<m-1}X(p_{i^{% \prime}})\right)^{n}\right)-(|p_{m}|-|p_{m-2}|)\left(1-\left(\sum_{i^{\prime}<% m-1}X(p_{i^{\prime}})\right)^{n}\right)~{}+" class="ltx_Math" display="inline" id="Pt0.A4.Ex118.m1.4"><semantics id="Pt0.A4.Ex118.m1.4a"><mrow id="Pt0.A4.Ex118.m1.4.4" xref="Pt0.A4.Ex118.m1.4.4.cmml"><mi id="Pt0.A4.Ex118.m1.4.4.6" xref="Pt0.A4.Ex118.m1.4.4.6.cmml"></mi><mo id="Pt0.A4.Ex118.m1.4.4.5" xref="Pt0.A4.Ex118.m1.4.4.5.cmml">=</mo><mrow id="Pt0.A4.Ex118.m1.4.4.4" xref="Pt0.A4.Ex118.m1.4.4.4.cmml"><mrow id="Pt0.A4.Ex118.m1.2.2.2.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.cmml"><mrow id="Pt0.A4.Ex118.m1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.2.cmml"><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.2" stretchy="false" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.2.1.cmml">|</mo><msub id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.cmml"><mi id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.2" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3.cmml"><mi id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3.2" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3.1" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3.3" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.1.3.3.cmml">2</mn></mrow></msub><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.1.3" stretchy="false" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Pt0.A4.Ex118.m1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex118.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex118.m1.2.2.2.2.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.cmml"><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.cmml"><mn id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.3.cmml">1</mn><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.2.cmml">−</mo><msup id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.cmml"><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.cmml"><munder id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2a" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.2" movablelimits="false" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.2.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.2.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.1.cmml"><</mo><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3.2.cmml">m</mi><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3.1.cmml">−</mo><mn id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><msup id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Pt0.A4.Ex118.m1.2.2.2.2.2.1.3" xref="Pt0.A4.Ex118.m1.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="Pt0.A4.Ex118.m1.4.4.4.5" xref="Pt0.A4.Ex118.m1.4.4.4.5.cmml">−</mo><mrow id="Pt0.A4.Ex118.m1.4.4.4.4" xref="Pt0.A4.Ex118.m1.4.4.4.4.cmml"><mrow id="Pt0.A4.Ex118.m1.4.4.4.4.2" xref="Pt0.A4.Ex118.m1.4.4.4.4.2.cmml"><mrow id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.cmml"><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.1" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.3" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.2.cmml"><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.2" stretchy="false" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.2.1.cmml">|</mo><msub id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.cmml"><mi id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.2" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3.cmml"><mi id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3.2" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3.1" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3.3" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.1.3.3.cmml">2</mn></mrow></msub><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.1.3" stretchy="false" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex118.m1.3.3.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex118.m1.4.4.4.4.2.3" 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xref="Pt0.A4.Ex118.m1.4.4.4.4.4"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex118.m1.4c">\displaystyle=(|p_{m-1}|-|p_{m-2}|)\left(1-\left(\sum_{i^{\prime}<m-1}X(p_{i^{% \prime}})\right)^{n}\right)-(|p_{m}|-|p_{m-2}|)\left(1-\left(\sum_{i^{\prime}<% m-1}X(p_{i^{\prime}})\right)^{n}\right)~{}+</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex118.m1.4d">= ( | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT | ) ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_m - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - ( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT | ) ( 1 - ( ∑ 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id="Pt0.A4.Ex119.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex119.m1.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3">𝑚</ci></apply></apply></apply><ci id="Pt0.A4.Ex119.m1.4.4.4.2.1.1.1.3.cmml" xref="Pt0.A4.Ex119.m1.4.4.4.2.1.1.1.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex119.m1.4c">\displaystyle(|p_{m}|+\alpha-|p_{m-1}|)\left(1-(1-X(p_{m}))^{n}\right)-(|p_{m-% 1}|+\alpha-|p_{m}|)\left(1-(1-X(p_{m}))^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex119.m1.4d">( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | + italic_α - | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | ) ( 1 - ( 1 - italic_X ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - ( | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | + italic_α - | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ) ( 1 - ( 1 - italic_X ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex120"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-(|p_{m}|-|p_{m-1}|)\left(1-\left(\sum_{i^{\prime}<m-1}X(p_{i^{% \prime}})\right)^{n}\right)+2(|p_{m}|-|p_{m-1}|)\left(1-(1-X(p_{m}))^{n}\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex120.m1.4"><semantics id="Pt0.A4.Ex120.m1.4a"><mrow id="Pt0.A4.Ex120.m1.4.4" xref="Pt0.A4.Ex120.m1.4.4.cmml"><mi id="Pt0.A4.Ex120.m1.4.4.6" xref="Pt0.A4.Ex120.m1.4.4.6.cmml"></mi><mo id="Pt0.A4.Ex120.m1.4.4.5" xref="Pt0.A4.Ex120.m1.4.4.5.cmml">=</mo><mrow id="Pt0.A4.Ex120.m1.4.4.4" xref="Pt0.A4.Ex120.m1.4.4.4.cmml"><mrow 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xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.2.cmml"><mo id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.2" stretchy="false" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.2.1.cmml">|</mo><msub id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.cmml"><mi id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.2" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.2.cmml">p</mi><mrow id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3.cmml"><mi id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3.2" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3.1" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3.3" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.1.3" stretchy="false" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.2.2.1.cmml">|</mo></mrow></mrow><mo id="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex120.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.cmml"><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.cmml"><mn id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.3.cmml">1</mn><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.2.cmml">−</mo><msup id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.cmml"><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.cmml"><munder id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2a" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.2" movablelimits="false" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.2.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.2.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.1.cmml"><</mo><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3.2.cmml">m</mi><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3.1.cmml">−</mo><mn id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></mrow></munder></mstyle><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><msup id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.1.3.cmml">n</mi></msup></mrow><mo id="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.3" xref="Pt0.A4.Ex120.m1.2.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.Ex120.m1.4.4.4.5" xref="Pt0.A4.Ex120.m1.4.4.4.5.cmml">+</mo><mrow id="Pt0.A4.Ex120.m1.4.4.4.4" xref="Pt0.A4.Ex120.m1.4.4.4.4.cmml"><mn id="Pt0.A4.Ex120.m1.4.4.4.4.4" xref="Pt0.A4.Ex120.m1.4.4.4.4.4.cmml">2</mn><mo id="Pt0.A4.Ex120.m1.4.4.4.4.3" xref="Pt0.A4.Ex120.m1.4.4.4.4.3.cmml">⁢</mo><mrow id="Pt0.A4.Ex120.m1.3.3.3.3.1.1" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.cmml"><mo id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.2" stretchy="false" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.cmml"><mrow id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.1" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.1.2" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.1.3" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex120.m1.3.3.3.3.1.1.1.3" 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xref="Pt0.A4.Ex120.m1.4.4.4.4.2.1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><ci id="Pt0.A4.Ex120.m1.4.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex120.m1.4.4.4.4.2.1.1.1.1.1.1.1.1.1.1.3">𝑚</ci></apply></apply></apply><ci id="Pt0.A4.Ex120.m1.4.4.4.4.2.1.1.1.3.cmml" xref="Pt0.A4.Ex120.m1.4.4.4.4.2.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex120.m1.4c">\displaystyle=-(|p_{m}|-|p_{m-1}|)\left(1-\left(\sum_{i^{\prime}<m-1}X(p_{i^{% \prime}})\right)^{n}\right)+2(|p_{m}|-|p_{m-1}|)\left(1-(1-X(p_{m}))^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex120.m1.4d">= - ( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | ) ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_m - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) + 2 ( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | ) ( 1 - ( 1 - italic_X ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="Pt0.A4.Ex121"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=(|p_{m}|-|p_{m-1}|)\left(2\left(1-(1-X(p_{m}))^{n}\right)-\left(% 1-\left(\sum_{i^{\prime}<m-1}X(p_{i^{\prime}})\right)^{n}\right)\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex121.m1.2"><semantics id="Pt0.A4.Ex121.m1.2a"><mrow id="Pt0.A4.Ex121.m1.2.2" 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xref="Pt0.A4.Ex121.m1.2.2.2.2.1.1.2.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex121.m1.2c">\displaystyle=(|p_{m}|-|p_{m-1}|)\left(2\left(1-(1-X(p_{m}))^{n}\right)-\left(% 1-\left(\sum_{i^{\prime}<m-1}X(p_{i^{\prime}})\right)^{n}\right)\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex121.m1.2d">= ( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | ) ( 2 ( 1 - ( 1 - italic_X ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) - ( 1 - ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_m - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) 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cd="ambiguous" id="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1">subscript</csymbol><ci id="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.2">𝑝</ci><apply id="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3.cmml" xref="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3"><minus id="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3.1.cmml" xref="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3.1"></minus><ci id="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3.2.cmml" xref="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3.2">𝑚</ci><cn id="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3.3.cmml" type="integer" xref="Pt0.A4.Ex122.m1.1.1.1.1.1.1.2.1.1.3.3">1</cn></apply></apply></apply></apply><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1"><minus id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.3"></minus><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1"><plus id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.2"></plus><cn id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.3">1</cn><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1">superscript</csymbol><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1"><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2">subscript</csymbol><sum id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.2"></sum><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3"><lt id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.1"></lt><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2">superscript</csymbol><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2.2">𝑖</ci><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.2.3">′</ci></apply><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3"><minus id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3.1"></minus><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3.2">𝑚</ci><cn id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3.3.cmml" type="integer" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.2.3.3.3">1</cn></apply></apply></apply><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1"><times id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.2"></times><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.3">𝑋</ci><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.1.1.3">𝑛</ci></apply></apply><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2"><times id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.2"></times><cn id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.3.cmml" type="integer" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.3">2</cn><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1">superscript</csymbol><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1"><minus id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.2"></minus><cn id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.3.cmml" type="integer" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.3">1</cn><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1"><times id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.2"></times><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.3">𝑋</ci><apply id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1.1.2">𝑝</ci><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.1.1.1.1.1.1.1.3">𝑚</ci></apply></apply></apply><ci id="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.3.cmml" xref="Pt0.A4.Ex122.m1.2.2.2.2.1.1.2.1.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.Ex122.m1.2c">\displaystyle=(|p_{m}|-|p_{m-1}|)\left(1+\left(\sum_{i^{\prime}<m-1}X(p_{i^{% \prime}})\right)^{n}-2(1-X(p_{m}))^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex122.m1.2d">= ( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | ) ( 1 + ( ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_m - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - 2 ( 1 - italic_X ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A4.SS4.SSS0.Px2.p1.5">Let <math alttext="x=\sum_{i^{\prime}<m-1}X(p_{i^{\prime}}),y=X(p_{m-1}),z=X(p_{m})" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2"><semantics id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2a"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.3.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.3.cmml">x</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.2" rspace="0.111em" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.2.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.cmml"><msub id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.cmml"><msup id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.1.cmml"><</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.2.cmml">m</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.1.cmml">−</mo><mn id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></mrow></msub><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><msup id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.3a.cmml">,</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.3.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.3.cmml">y</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.2.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.3.cmml">X</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.2.cmml">p</mi><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.3a.cmml">,</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3.cmml">z</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2.cmml">=</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.3.cmml">X</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.2.cmml">⁢</mo><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.cmml"><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.2" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.cmml">(</mo><msub id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.3.cmml">m</mi></msub><mo id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.3" stretchy="false" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2b"><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.3a.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.3">formulae-sequence</csymbol><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1"><eq id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.2"></eq><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.3">𝑥</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1"><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2">subscript</csymbol><sum id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.2"></sum><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3"><lt id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.1"></lt><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.2.3">′</ci></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3"><minus id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.1"></minus><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.2">𝑚</ci><cn id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.2.3.3.3">1</cn></apply></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1"><times id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2"></times><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3">𝑋</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.3a.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.3">formulae-sequence</csymbol><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1"><eq id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.2"></eq><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.3">𝑦</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1"><times id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.2"></times><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.3">𝑋</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.2">𝑝</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3"><minus id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.1"></minus><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.2">𝑚</ci><cn id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2"><eq id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2"></eq><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3">𝑧</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1"><times id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.2"></times><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.3">𝑋</ci><apply id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1">subscript</csymbol><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.2">𝑝</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.1.1.1.3">𝑚</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2c">x=\sum_{i^{\prime}<m-1}X(p_{i^{\prime}}),y=X(p_{m-1}),z=X(p_{m})</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px2.p1.4.m1.2d">italic_x = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_m - 1 end_POSTSUBSCRIPT italic_X ( italic_p start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) , italic_y = italic_X ( italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ) , italic_z = italic_X ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math>. Because <math alttext="x+y+z=1" class="ltx_Math" display="inline" id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1"><semantics id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1a"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.cmml"><mrow id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.cmml"><mi id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.2" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.2.cmml">x</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.1.cmml">+</mo><mi id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.3.cmml">y</mi><mo id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.1a" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.1.cmml">+</mo><mi id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.4" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.4.cmml">z</mi></mrow><mo id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.1" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.1.cmml">=</mo><mn id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.3" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1b"><apply id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1"><eq id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.1"></eq><apply id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2"><plus id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.1.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.1"></plus><ci id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.2.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.2">𝑥</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.3.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.3">𝑦</ci><ci id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.4.cmml" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.2.4">𝑧</ci></apply><cn id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.3.cmml" type="integer" xref="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1c">x+y+z=1</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.SS4.SSS0.Px2.p1.5.m2.1d">italic_x + italic_y + italic_z = 1</annotation></semantics></math>, this becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Pt0.A4.EGx66"> <tbody id="Pt0.A4.Ex123"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(|p_{m}|-|p_{m-1}|)\left(1-(1-z)^{n}-(x+y)^{n}+x^{n}\right)" class="ltx_Math" display="inline" id="Pt0.A4.Ex123.m1.2"><semantics id="Pt0.A4.Ex123.m1.2a"><mrow id="Pt0.A4.Ex123.m1.2.2" xref="Pt0.A4.Ex123.m1.2.2.cmml"><mrow id="Pt0.A4.Ex123.m1.1.1.1.1" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.cmml"><mo id="Pt0.A4.Ex123.m1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.cmml">(</mo><mrow id="Pt0.A4.Ex123.m1.1.1.1.1.1" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.cmml"><mrow id="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.1.2.cmml"><mo id="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.1.2.1.cmml">|</mo><msub id="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.1" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.1.cmml"><mi id="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.1.2" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.1.2.cmml">p</mi><mi id="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.1.3" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="Pt0.A4.Ex123.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Pt0.A4.Ex123.m1.1.1.1.1.1.3" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.3.cmml">−</mo><mrow id="Pt0.A4.Ex123.m1.1.1.1.1.1.2.1" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.2.2.cmml"><mo id="Pt0.A4.Ex123.m1.1.1.1.1.1.2.1.2" stretchy="false" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.2.2.1.cmml">|</mo><msub id="Pt0.A4.Ex123.m1.1.1.1.1.1.2.1.1" xref="Pt0.A4.Ex123.m1.1.1.1.1.1.2.1.1.cmml"><mi id="Pt0.A4.Ex123.m1.1.1.1.1.1.2.1.1.2" 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id="Pt0.A4.Ex123.m1.2c">\displaystyle(|p_{m}|-|p_{m-1}|)\left(1-(1-z)^{n}-(x+y)^{n}+x^{n}\right)</annotation><annotation encoding="application/x-llamapun" id="Pt0.A4.Ex123.m1.2d">( | italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | - | italic_p start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT | ) ( 1 - ( 1 - italic_z ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - ( italic_x + italic_y ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Pt0.A4.SS4.SSS0.Px2.p1.6">and the proof of the previous case completes the proof of this case, establishing the lemma. ∎</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </section> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue May 14 16:09:31 2024 by <a 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