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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives</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> <base href="/html/2308.09443v2/"/></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/2308.09443v2#S1" title="In Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><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_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S1.SS0.SSS0.Px1" title="In 1 Introduction ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Our contribution.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S1.SS0.SSS0.Px2" title="In 1 Introduction ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Related work.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S1.SS0.SSS0.Px3" title="In 1 Introduction ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Structure of the paper.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2" title="In Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries and Studied Problem</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/2308.09443v2#S2.SS1" title="In 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Graph Games</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/2308.09443v2#S2.SS1.SSS0.Px1" title="In 2.1 Graph Games ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Game arenas.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS1.SSS0.Px2" title="In 2.1 Graph Games ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Plays and histories.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS1.SSS0.Px3" title="In 2.1 Graph Games ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Strategies.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS1.SSS0.Px4" title="In 2.1 Graph Games ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Reachability costs.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS2" title="In 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Stackelberg-Pareto Synthesis Problem</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/2308.09443v2#S2.SS2.SSS0.Px1" title="In 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Stackelberg-Pareto games.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS2.SSS0.Px2" title="In 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Stackelberg-Pareto synthesis problem.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS2.SSS0.Px3" title="In 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Example.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS3" title="In 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Tools</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/2308.09443v2#S2.SS3.SSS0.Px1" title="In 2.3 Tools ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Witnesses.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.SS3.SSS0.Px2" title="In 2.3 Tools ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title">Reduction to binary arenas.</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3" title="In Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Improving a Solution</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/2308.09443v2#S3.SS1" title="In 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Order on Strategies and Subgames</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.SS2" title="In 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Improving a Solution in a Subgame</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.SS3" title="In 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Eliminating a Cycle in a Witness</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4" title="In Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Bounding Pareto-Optimal Payoffs</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/2308.09443v2#S4.SS1" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Dimension One</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.SS2" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Bounding the Pareto-Optimal Costs by Induction</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.SS3" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Bounding the Minimum Component of Pareto-Optimal Costs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.SS4" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.4 </span>Proof of Theorem <span class="ltx_text ltx_ref_tag">4.1</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5" title="In Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Complexity of the SPS Problem</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/2308.09443v2#S5.SS1" title="In 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Finite-Memory Solutions</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.SS2" title="In 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Punishing Strategies</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.SS3" title="In 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Lasso Witnesses</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.SS4" title="In 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.4 </span><span class="ltx_text ltx_font_sansserif">NEXPTIME</span>-Completeness</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S6" title="In Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusion and Future Work</span></a></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"> <p class="ltx_p" id="p1.1">Quantitative Reachability Stackelberg-Pareto Synthesis <span class="ltx_ERROR undefined" id="p1.1.1">\NewDocumentCommand</span><span class="ltx_ERROR undefined" id="p1.1.2">\Last</span>m<span class="ltx_text ltx_font_sansserif" id="p1.1.3">last</span>(#1) </p> </div> <h1 class="ltx_title ltx_title_document">Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Thomas Brihaye <br class="ltx_break"/>Université de Mons (UMONS) </span><span class="ltx_author_notes">Partly supported by the F.R.S.- FNRS under grant n°T.0027.21.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Belgium </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Véronique Bruyère <br class="ltx_break"/>Université de Mons (UMONS) </span><span class="ltx_author_notes">Partly supported by the F.R.S.- FNRS under grant n°T.0023.22.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Belgium </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Gaspard Reghem <br class="ltx_break"/>ENS Paris-Saclay </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Université Paris-Saclay </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> France </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id7.7">In this paper, we deepen the study of two-player Stackelberg games played on graphs in which Player <math alttext="0" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mn id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><cn id="id1.1.m1.1.1.cmml" type="integer" xref="id1.1.m1.1.1">0</cn></annotation-xml></semantics></math> announces a strategy and Player <math alttext="1" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mn id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><cn id="id2.2.m2.1.1.cmml" type="integer" xref="id2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">1</annotation></semantics></math>, having several objectives, responds rationally by following plays providing him Pareto-optimal payoffs given the strategy of Player <math alttext="0" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mn id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><cn id="id3.3.m3.1.1.cmml" type="integer" xref="id3.3.m3.1.1">0</cn></annotation-xml></semantics></math>. The Stackelberg-Pareto synthesis problem, asking whether Player <math alttext="0" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mn id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><cn id="id4.4.m4.1.1.cmml" type="integer" xref="id4.4.m4.1.1">0</cn></annotation-xml></semantics></math> can announce a strategy which satisfies his objective, whatever the rational response of Player <math alttext="1" class="ltx_Math" display="inline" id="id5.5.m5.1"><semantics id="id5.5.m5.1a"><mn id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="id5.5.m5.1b"><cn id="id5.5.m5.1.1.cmml" type="integer" xref="id5.5.m5.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.1c">1</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.1d">1</annotation></semantics></math>, has been recently investigated for <math alttext="\omega" class="ltx_Math" display="inline" id="id6.6.m6.1"><semantics id="id6.6.m6.1a"><mi id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="id6.6.m6.1b"><ci id="id6.6.m6.1.1.cmml" xref="id6.6.m6.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m6.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="id6.6.m6.1d">italic_ω</annotation></semantics></math>-regular objectives. We solve this problem for weighted graph games and quantitative reachability objectives such that Player <math alttext="0" class="ltx_Math" display="inline" id="id7.7.m7.1"><semantics id="id7.7.m7.1a"><mn id="id7.7.m7.1.1" xref="id7.7.m7.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="id7.7.m7.1b"><cn id="id7.7.m7.1.1.cmml" type="integer" xref="id7.7.m7.1.1">0</cn></annotation-xml></semantics></math> wants to reach his target set with a total cost less than some given upper bound. We show that it is <span class="ltx_text ltx_font_sansserif" id="id7.7.1">NEXPTIME</span>-complete, as for Boolean reachability objectives.</p> </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"><em class="ltx_emph ltx_font_italic" id="S1.p1.1.1">Formal verification</em>, and more specifically <em class="ltx_emph ltx_font_italic" id="S1.p1.1.2">model-checking</em>, is a branch of computer science which offers techniques to check automatically whether a system is correct <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib20" title="">20</a>]</cite>. This is essential for systems responsible for critical tasks like air traffic management or control of nuclear power plants. Much progress has been made in model-checking both theoretically and in tool development, and the technique is now widely used in industry.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Nowadays, it is common to face more complex systems, called <em class="ltx_emph ltx_font_italic" id="S1.p2.1.1">multi-agent systems</em>, that are composed of heterogeneous components, ranging from traditional pieces of reactive code, to wholly autonomous robots or human users. Modelling and verifying such systems is a challenging problem that is far from being solved. One possible approach is to rely on <em class="ltx_emph ltx_font_italic" id="S1.p2.1.2">game theory</em>, a branch of mathematics that studies mathematical models of interaction between agents and the understanding of their decisions assuming that they are <em class="ltx_emph ltx_font_italic" id="S1.p2.1.3">rational</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib40" title="">40</a>]</cite>. Typically, each agent (i.e. player) composing the system has his own objectives or preferences, and the way he manages to achieve them is influenced by the behavior of the other agents.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Rationality can be formalized in several ways. A famous model of agents’ rational behavior is the concept of <em class="ltx_emph ltx_font_italic" id="S1.p3.1.1">Nash equilibrium</em> (NE) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib33" title="">33</a>]</cite> in a multiplayer non-zero sum game graph that represents the possible interactions between the players <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib38" title="">38</a>]</cite>. Another model is the one of <em class="ltx_emph ltx_font_italic" id="S1.p3.1.2">Stackelberg games</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib41" title="">41</a>]</cite>, in which one designated player – the leader, announces a strategy to achieve his goal, and the other players – the followers, respond rationally with an optimal response depending on their goals (e.g. with an NE). This framework is well-suited for the verification of correctness of a controller intending to enforce a given property, while interacting with an environment composed of several agents each having his own objective. In practical applications, a strategy for interacting with the environment is committed before the interaction actually happens.</p> </div> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Our contribution.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px1.p1.1">In this paper, we investigate the recent concept of two-player Stackelberg games, where the environment is composed of <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p1.1.1">one player</em> aiming at satisfying <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p1.1.2">several objectives</em>, and its related <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p1.1.3">Stackelberg-Pareto synthesis</em> (SPS) problem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite>. In this framework, for Boolean objectives, given the strategy announced by the leader, the follower responses rationally with a strategy that ensures him a vector of Boolean payoffs that is <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p1.1.4">Pareto-optimal</em>, that is, with a maximal number of satisfied objectives. This setting encompasses scenarios where, for instance, several components of the environment can collaborate and agree on trade-offs. The SPS problem is to decide whether the leader can announce a strategy that guarantees him to satisfy his own objective, whatever the rational response of the follower.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="S1.SS0.SSS0.Px1.p2.1">The SPS problem has been solved in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite> for <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p2.1.1"><math alttext="\omega" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px1.p2.1.1.m1.1"><semantics id="S1.SS0.SSS0.Px1.p2.1.1.m1.1a"><mi id="S1.SS0.SSS0.Px1.p2.1.1.m1.1.1" xref="S1.SS0.SSS0.Px1.p2.1.1.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px1.p2.1.1.m1.1b"><ci id="S1.SS0.SSS0.Px1.p2.1.1.m1.1.1.cmml" xref="S1.SS0.SSS0.Px1.p2.1.1.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px1.p2.1.1.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px1.p2.1.1.m1.1d">italic_ω</annotation></semantics></math>-regular</em> objectives. We here solve this problem for weighted game graphs and <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p2.1.2">quantitative reachability</em> objectives for both players. Given a target of vertices, the goal is to reach this target with a cost as small as possible. In this quantitative context, the follower responds to the strategy of the leader with a strategy that ensures him a Pareto-optimal cost vector given his series of targets. The aim of the leader is to announce a strategy in a way to reach his target with a total cost less than some given upper bound, whatever the rational response of the follower. We show that the SPS problem is <span class="ltx_text ltx_font_sansserif" id="S1.SS0.SSS0.Px1.p2.1.3">NEXPTIME</span>-complete (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem1" title="Theorem 2.1 ‣ Stackelberg-Pareto synthesis problem. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.1</span></a>), as for Boolean reachability objectives.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px1.p3"> <p class="ltx_p" id="S1.SS0.SSS0.Px1.p3.1">It is well-known that moving from Boolean objectives to quantitative ones allows to model <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p3.1.1">richer properties</em>. This paper is a first step in this direction for the SPS problem for two-player Stackelberg games with multiple objectives for the follower. Our proof follows the same pattern as for Boolean reachability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite>: if there is a solution to the SPS problem, then there is one that is finite-memory whose memory size is at most exponential. The non-deterministic algorithm thus guesses such a strategy and checks whether it is a solution. However, a crucial intermediate step is to prove that if there exists a solution, then there exists one whose Pareto-optimal costs for the follower are <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px1.p3.1.2">exponential</em> in the size of the instance (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>). The proof of this non trivial step (which is meaningless in the Boolean case) is the main contribution of the paper. Given a solution, we first present some hypotheses and techniques that allow to locally modify it into a solution with smaller Pareto-optimal cost vectors. We then conduct a proof by induction on the number of follower’s targets, to globally decrease the cost vectors and to get an exponential number of Pareto-optimal cost vectors. The <span class="ltx_text ltx_font_sansserif" id="S1.SS0.SSS0.Px1.p3.1.3">NEXPTIME</span>-hardness of the SPS problem is trivially obtained by reduction from this problem for Boolean reachability. Indeed, the Boolean version is equivalent to the quantitative one with all weights put to zero and with the given upper bound equal to zero. Notice that the two versions differ: we exhibit an example of game that has a solution to the SPS problem for quantitative reachability, but none for Boolean reachability.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px1.p4"> <p class="ltx_p" id="S1.SS0.SSS0.Px1.p4.1">Our result first appeared in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib11" title="">11</a>]</cite>. In this long version, we provide all the proofs.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Related work.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px2.p1.1">During the last decade, multiplayer non-zero sum games and their applications to reactive synthesis have raised a growing attention, see for instance the surveys <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib26" title="">26</a>]</cite>. When several players (like the followers) play with the aim to satisfy their objectives, several <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px2.p1.1.1">solution concepts</em> exist such as NE, subgame perfect equilibrium (SPE) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib35" title="">35</a>]</cite>, secure equilibria <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib19" title="">19</a>]</cite>, or admissibility <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib5" title="">5</a>]</cite>. Several results have been obtained, for Boolean and quantitative objectives, about the constrained existence problem which consists in deciding whether there exists a solution concept such that the payoff obtained by each player is larger than some given threshold. Let us mention <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib39" title="">39</a>]</cite> for results on the constrained existence for NEs and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib37" title="">37</a>]</cite> for SPEs. Some of them rely on a recent elegant characterization of SPE outcomes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib24" title="">24</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px2.p2"> <p class="ltx_p" id="S1.SS0.SSS0.Px2.p2.1">Stackelberg games with <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px2.p2.1.1">several followers</em> have been recently studied in the context of <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px2.p2.1.2">rational synthesis</em>: in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib23" title="">23</a>]</cite> in a setting where the followers are cooperative with the leader, and later in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib31" title="">31</a>]</cite> where they are adversarial. Rational responses of the followers are, for instance, to play an NE or an SPE. The rational synthesis problem and the SPS problem are incomparable, as illustrated in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib36" title="">36</a>, Section 4.3.2]</cite>: in rational synthesis, each component of the environment acts selfishly, whereas in SPS, the components cooperate in a way to obtain a Pareto-optimal cost. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib32" title="">32</a>]</cite>, the authors solve the rational synthesis problem that consists in deciding whether the leader can announce a strategy satisfying his objective, when the objectives of the players are specified by LTL formulas. Complexity classes for various <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px2.p2.1.m1.1"><semantics id="S1.SS0.SSS0.Px2.p2.1.m1.1a"><mi id="S1.SS0.SSS0.Px2.p2.1.m1.1.1" xref="S1.SS0.SSS0.Px2.p2.1.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px2.p2.1.m1.1b"><ci id="S1.SS0.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S1.SS0.SSS0.Px2.p2.1.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px2.p2.1.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px2.p2.1.m1.1d">italic_ω</annotation></semantics></math>-regular objectives are established in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib21" title="">21</a>]</cite> for both cooperative and adversarial settings. Extension to quantitative payoffs, like mean-payoff or discounted sum, is studied in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib28" title="">28</a>]</cite> in the cooperative setting and in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib22" title="">22</a>]</cite> in the adversarial setting.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px2.p3"> <p class="ltx_p" id="S1.SS0.SSS0.Px2.p3.1">The concept of <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px2.p3.1.1">rational verification</em> has been introduced in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib29" title="">29</a>]</cite>, where instead of deciding the existence of a strategy for the leader, one verifies that some given leader’s strategy satisfies his objective, whatever the NE responses of the followers. An algorithm and its implementation in the EVE system are presented in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib29" title="">29</a>]</cite> for objectives specified by LTL formulas. This verification problem is studied in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib30" title="">30</a>]</cite> for mean-payoff objectives for the followers and an omega-regular objective for the leader, and it is solved in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib9" title="">9</a>]</cite> for both NE and SPE responses of the followers and for a variety of objectives including quantitative objectives. The Stackelberg-Pareto verification problem is solved in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib17" title="">17</a>]</cite> for some <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px2.p3.1.m1.1"><semantics id="S1.SS0.SSS0.Px2.p3.1.m1.1a"><mi id="S1.SS0.SSS0.Px2.p3.1.m1.1.1" xref="S1.SS0.SSS0.Px2.p3.1.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px2.p3.1.m1.1b"><ci id="S1.SS0.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S1.SS0.SSS0.Px2.p3.1.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px2.p3.1.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px2.p3.1.m1.1d">italic_ω</annotation></semantics></math>-regular or LTL objectives.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px2.p4"> <p class="ltx_p" id="S1.SS0.SSS0.Px2.p4.1">Since our paper <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib11" title="">11</a>]</cite>, the rational synthesis and verification problems have been further studied for quantitative reachability objectives in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib14" title="">14</a>]</cite>, for both cases of rational responses: NE responses and Pareto-optimal responses.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Structure of the paper.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px3.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p1.1">In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2" title="2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2</span></a>, we introduce the concept of Stackelberg-Pareto games with quantitative reachability costs. We also recall several useful related notions. In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4" title="4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>, we show that if there exists a solution to the SPS problem, then there exists one whose Pareto-optimal costs are exponential in the size of the instance. In Section <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5" title="5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5</span></a>, we prove that the SPS problem is <span class="ltx_text ltx_font_sansserif" id="S1.SS0.SSS0.Px3.p1.1.1">NEXPTIME</span>-complete by using the result of the previous section. Finally, we give a conclusion and some future work.</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>Preliminaries and Studied Problem</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">We introduce the concept of Stackelberg-Pareto games with quantitative reachability costs. We present the related Stackelberg-Pareto synthesis problem and state our main result.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Graph Games</h3> <section class="ltx_paragraph" id="S2.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Game arenas.</h4> <div class="ltx_para" id="S2.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p1.14">A <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px1.p1.14.1">game arena</em> is a tuple <math alttext="A=(V,V_{0},V_{1},E,v_{0},w)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.1.m1.6"><semantics id="S2.SS1.SSS0.Px1.p1.1.m1.6a"><mrow id="S2.SS1.SSS0.Px1.p1.1.m1.6.6" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.cmml"><mi id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.5" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.5.cmml">A</mi><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.4" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.4.cmml">=</mo><mrow id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml"><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.4" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1.cmml">V</mi><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.5" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml">,</mo><msub id="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1" xref="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.2.cmml">V</mi><mn id="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.6" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml">,</mo><msub id="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.2.cmml">V</mi><mn id="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.7" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml">,</mo><mi id="S2.SS1.SSS0.Px1.p1.1.m1.2.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.2.cmml">E</mi><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.8" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml">,</mo><msub id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.2.cmml">v</mi><mn id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.9" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml">,</mo><mi id="S2.SS1.SSS0.Px1.p1.1.m1.3.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.3.3.cmml">w</mi><mo id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.10" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.1.m1.6b"><apply id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6"><eq id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.4.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.4"></eq><ci id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.5.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.5">𝐴</ci><vector id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.4.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3"><ci id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1">𝑉</ci><apply id="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.2">𝑉</ci><cn id="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.3">0</cn></apply><apply id="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.2">𝑉</ci><cn id="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.3">1</cn></apply><ci id="S2.SS1.SSS0.Px1.p1.1.m1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.2">𝐸</ci><apply id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.2">𝑣</ci><cn id="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.1.m1.6.6.3.3.3.3">0</cn></apply><ci id="S2.SS1.SSS0.Px1.p1.1.m1.3.3.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.3.3">𝑤</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.1.m1.6c">A=(V,V_{0},V_{1},E,v_{0},w)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.1.m1.6d">italic_A = ( italic_V , italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_E , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_w )</annotation></semantics></math> where: <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px1.p1.14.2">(1)</em> <math alttext="(V,E)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.2.m2.2"><semantics id="S2.SS1.SSS0.Px1.p1.2.m2.2a"><mrow id="S2.SS1.SSS0.Px1.p1.2.m2.2.3.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.3.1.cmml"><mo id="S2.SS1.SSS0.Px1.p1.2.m2.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.3.1.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1.cmml">V</mi><mo id="S2.SS1.SSS0.Px1.p1.2.m2.2.3.2.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.3.1.cmml">,</mo><mi id="S2.SS1.SSS0.Px1.p1.2.m2.2.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.2.cmml">E</mi><mo id="S2.SS1.SSS0.Px1.p1.2.m2.2.3.2.3" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.2.m2.2b"><interval closure="open" id="S2.SS1.SSS0.Px1.p1.2.m2.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.3.2"><ci id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1">𝑉</ci><ci id="S2.SS1.SSS0.Px1.p1.2.m2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.2">𝐸</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.2.m2.2c">(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.2.m2.2d">( italic_V , italic_E )</annotation></semantics></math> is a finite directed graph with <math alttext="V" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.3.m3.1"><semantics id="S2.SS1.SSS0.Px1.p1.3.m3.1a"><mi id="S2.SS1.SSS0.Px1.p1.3.m3.1.1" xref="S2.SS1.SSS0.Px1.p1.3.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.3.m3.1b"><ci id="S2.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.3.m3.1d">italic_V</annotation></semantics></math> as set of vertices and <math alttext="E" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.4.m4.1"><semantics id="S2.SS1.SSS0.Px1.p1.4.m4.1a"><mi id="S2.SS1.SSS0.Px1.p1.4.m4.1.1" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.4.m4.1b"><ci id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.4.m4.1d">italic_E</annotation></semantics></math> as set of edges (it is supposed that every vertex has a successor), <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px1.p1.14.3">(2)</em> <math alttext="V" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.5.m5.1"><semantics id="S2.SS1.SSS0.Px1.p1.5.m5.1a"><mi id="S2.SS1.SSS0.Px1.p1.5.m5.1.1" xref="S2.SS1.SSS0.Px1.p1.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.5.m5.1b"><ci id="S2.SS1.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.5.m5.1d">italic_V</annotation></semantics></math> is partitioned as <math alttext="V_{0}\cup V_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.6.m6.1"><semantics id="S2.SS1.SSS0.Px1.p1.6.m6.1a"><mrow id="S2.SS1.SSS0.Px1.p1.6.m6.1.1" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.cmml"><msub id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.2" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.2.cmml">V</mi><mn id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.3" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.cmml">∪</mo><msub id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.2" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.2.cmml">V</mi><mn id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.3" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.6.m6.1b"><apply id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1"><union id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1"></union><apply id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.2">𝑉</ci><cn id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.2.3">0</cn></apply><apply id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.2">𝑉</ci><cn id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.6.m6.1c">V_{0}\cup V_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.6.m6.1d">italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="V_{0}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.7.m7.1"><semantics id="S2.SS1.SSS0.Px1.p1.7.m7.1a"><msub id="S2.SS1.SSS0.Px1.p1.7.m7.1.1" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml">V</mi><mn id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.7.m7.1b"><apply id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2">𝑉</ci><cn id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.7.m7.1c">V_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.7.m7.1d">italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> (resp. <math alttext="V_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.8.m8.1"><semantics id="S2.SS1.SSS0.Px1.p1.8.m8.1a"><msub id="S2.SS1.SSS0.Px1.p1.8.m8.1.1" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml">V</mi><mn id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.8.m8.1b"><apply id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2">𝑉</ci><cn id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.8.m8.1c">V_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.8.m8.1d">italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>) represents the vertices controlled by Player 0 (resp. Player 1), <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px1.p1.14.4">(3)</em> <math alttext="v_{0}\in V" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.9.m9.1"><semantics id="S2.SS1.SSS0.Px1.p1.9.m9.1a"><mrow id="S2.SS1.SSS0.Px1.p1.9.m9.1.1" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.cmml"><msub id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.2" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.2.cmml">v</mi><mn id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.3" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.1" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.1.cmml">∈</mo><mi id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.3" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.9.m9.1b"><apply id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1"><in id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.1"></in><apply id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.2">𝑣</ci><cn id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.2.3">0</cn></apply><ci id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.9.m9.1c">v_{0}\in V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.9.m9.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V</annotation></semantics></math> is the initial vertex, and <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px1.p1.14.5">(4)</em> <math alttext="w\colon E\rightarrow{\rm Nature}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.10.m10.1"><semantics id="S2.SS1.SSS0.Px1.p1.10.m10.1a"><mrow id="S2.SS1.SSS0.Px1.p1.10.m10.1.1" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.2" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.2.cmml">w</mi><mo id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.1.cmml">:</mo><mrow id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.2" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.2.cmml">E</mi><mo id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.1.cmml">→</mo><mi id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.3" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.3.cmml">Nature</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.10.m10.1b"><apply id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1"><ci id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.1">:</ci><ci id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.2">𝑤</ci><apply id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3"><ci id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.1">→</ci><ci id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.2">𝐸</ci><ci id="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m10.1.1.3.3">Nature</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.10.m10.1c">w\colon E\rightarrow{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.10.m10.1d">italic_w : italic_E → roman_Nature</annotation></semantics></math> is a weight function that assigns a non-negative integer<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>Notice that null weights are allowed.</span></span></span> to each edge, such that <math alttext="W=\max_{e\in E}w(e)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.11.m11.1"><semantics id="S2.SS1.SSS0.Px1.p1.11.m11.1a"><mrow id="S2.SS1.SSS0.Px1.p1.11.m11.1.2" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.2" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.2.cmml">W</mi><mo id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.1" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.cmml"><msub id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.2" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.2.cmml">max</mi><mrow id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.2" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.2.cmml">e</mi><mo id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.1" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.1.cmml">∈</mo><mi id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.3" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.3.cmml">E</mi></mrow></msub><mo id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2a" lspace="0.167em" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.cmml">⁡</mo><mi id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.2" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.2.cmml">w</mi></mrow><mo id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.1" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.3.2" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.cmml"><mo id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.11.m11.1.1" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.1.cmml">e</mi><mo id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.11.m11.1b"><apply id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2"><eq id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.1"></eq><ci id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.2">𝑊</ci><apply id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3"><times id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.1"></times><apply id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2"><apply id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1">subscript</csymbol><max id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.2"></max><apply id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3"><in id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.1"></in><ci id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.2">𝑒</ci><ci id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.3.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.1.3.3">𝐸</ci></apply></apply><ci id="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.2.3.2.2">𝑤</ci></apply><ci id="S2.SS1.SSS0.Px1.p1.11.m11.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m11.1.1">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.11.m11.1c">W=\max_{e\in E}w(e)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.11.m11.1d">italic_W = roman_max start_POSTSUBSCRIPT italic_e ∈ italic_E end_POSTSUBSCRIPT italic_w ( italic_e )</annotation></semantics></math> denotes the maximum weight. An arena <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.12.m12.1"><semantics id="S2.SS1.SSS0.Px1.p1.12.m12.1a"><mi id="S2.SS1.SSS0.Px1.p1.12.m12.1.1" xref="S2.SS1.SSS0.Px1.p1.12.m12.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.12.m12.1b"><ci id="S2.SS1.SSS0.Px1.p1.12.m12.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.12.m12.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.12.m12.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.12.m12.1d">italic_A</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px1.p1.14.6">binary</em> if <math alttext="w(e)\in\{0,1\}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.13.m13.3"><semantics id="S2.SS1.SSS0.Px1.p1.13.m13.3a"><mrow id="S2.SS1.SSS0.Px1.p1.13.m13.3.4" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.2" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.2.cmml">w</mi><mo id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.1" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.3.2" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.cmml"><mo id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.13.m13.1.1" xref="S2.SS1.SSS0.Px1.p1.13.m13.1.1.cmml">e</mi><mo id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.1" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.1.cmml">∈</mo><mrow id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.2" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.1.cmml"><mo id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.1.cmml">{</mo><mn id="S2.SS1.SSS0.Px1.p1.13.m13.2.2" xref="S2.SS1.SSS0.Px1.p1.13.m13.2.2.cmml">0</mn><mo id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.2.2" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.1.cmml">,</mo><mn id="S2.SS1.SSS0.Px1.p1.13.m13.3.3" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.3.cmml">1</mn><mo id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.2.3" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.13.m13.3b"><apply id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4"><in id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.1.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.1"></in><apply id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2"><times id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.1"></times><ci id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.2.2">𝑤</ci><ci id="S2.SS1.SSS0.Px1.p1.13.m13.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m13.1.1">𝑒</ci></apply><set id="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.4.3.2"><cn id="S2.SS1.SSS0.Px1.p1.13.m13.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.13.m13.2.2">0</cn><cn id="S2.SS1.SSS0.Px1.p1.13.m13.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.13.m13.3.3">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.13.m13.3c">w(e)\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.13.m13.3d">italic_w ( italic_e ) ∈ { 0 , 1 }</annotation></semantics></math> for all <math alttext="e\in E" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.14.m14.1"><semantics id="S2.SS1.SSS0.Px1.p1.14.m14.1a"><mrow id="S2.SS1.SSS0.Px1.p1.14.m14.1.1" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.14.m14.1.1.2" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1.2.cmml">e</mi><mo id="S2.SS1.SSS0.Px1.p1.14.m14.1.1.1" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1.1.cmml">∈</mo><mi id="S2.SS1.SSS0.Px1.p1.14.m14.1.1.3" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.14.m14.1b"><apply id="S2.SS1.SSS0.Px1.p1.14.m14.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1"><in id="S2.SS1.SSS0.Px1.p1.14.m14.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1.1"></in><ci id="S2.SS1.SSS0.Px1.p1.14.m14.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1.2">𝑒</ci><ci id="S2.SS1.SSS0.Px1.p1.14.m14.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.14.m14.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.14.m14.1c">e\in E</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.14.m14.1d">italic_e ∈ italic_E</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Plays and histories.</h4> <div class="ltx_para" id="S2.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS1.SSS0.Px2.p1.32">A <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px2.p1.32.1">play</em> in an arena <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S2.SS1.SSS0.Px2.p1.1.m1.1a"><mi id="S2.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.1.m1.1b"><ci id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.1.m1.1d">italic_A</annotation></semantics></math> is an infinite sequence of vertices <math alttext="\rho=\rho_{0}\rho_{1}\ldots\in V^{\omega}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S2.SS1.SSS0.Px2.p1.2.m2.1a"><mrow id="S2.SS1.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml">ρ</mi><mo id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml">=</mo><mrow id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.cmml"><msub id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.2" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.3" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.1" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.2" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.3" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.1a" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.4" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.4.cmml">…</mi></mrow><mo id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.5" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.5.cmml">∈</mo><msup id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.cmml"><mi id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.2" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.2.cmml">V</mi><mi id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.3" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.3.cmml">ω</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.2.m2.1b"><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1"><and id="S2.SS1.SSS0.Px2.p1.2.m2.1.1a.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1"></and><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1b.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1"><eq id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3"></eq><ci id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2">𝜌</ci><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4"><times id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.1.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.1"></times><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.2">𝜌</ci><cn id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.2.3">0</cn></apply><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.2">𝜌</ci><cn id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.3.3">1</cn></apply><ci id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.4.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.4">…</ci></apply></apply><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1c.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1"><in id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.5.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.5"></in><share href="https://arxiv.org/html/2308.09443v2#S2.SS1.SSS0.Px2.p1.2.m2.1.1.4.cmml" id="S2.SS1.SSS0.Px2.p1.2.m2.1.1d.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1"></share><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.1.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6">superscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.2.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.2">𝑉</ci><ci id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.3.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.6.3">𝜔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.2.m2.1c">\rho=\rho_{0}\rho_{1}\ldots\in V^{\omega}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.2.m2.1d">italic_ρ = italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … ∈ italic_V start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\rho_{0}=v_{0}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="S2.SS1.SSS0.Px2.p1.3.m3.1a"><mrow id="S2.SS1.SSS0.Px2.p1.3.m3.1.1" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.cmml"><msub id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.1" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.1.cmml">=</mo><msub id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.2.cmml">v</mi><mn id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.3.m3.1b"><apply id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1"><eq id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.1"></eq><apply id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.2">𝜌</ci><cn id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.3">0</cn></apply><apply id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.2">𝑣</ci><cn id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.3.m3.1c">\rho_{0}=v_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.3.m3.1d">italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="(\rho_{k},\rho_{k+1})\in E" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.4.m4.2"><semantics id="S2.SS1.SSS0.Px2.p1.4.m4.2a"><mrow id="S2.SS1.SSS0.Px2.p1.4.m4.2.2" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.cmml"><mrow id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.3.cmml">(</mo><msub id="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1" xref="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.4" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.3.cmml">,</mo><msub id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.2" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.2" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.2.cmml">k</mi><mo id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.1" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.1.cmml">+</mo><mn id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.3" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.5" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.3.cmml">)</mo></mrow><mo id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.3" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.3.cmml">∈</mo><mi id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.4" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.4.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.4.m4.2b"><apply id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2"><in id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.3"></in><interval closure="open" id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2"><apply id="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.2">𝜌</ci><ci id="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.1.1.1.1.1.3">𝑘</ci></apply><apply id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.2">𝜌</ci><apply id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3"><plus id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.1"></plus><ci id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.2">𝑘</ci><cn id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.2.2.2.3.3">1</cn></apply></apply></interval><ci id="S2.SS1.SSS0.Px2.p1.4.m4.2.2.4.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m4.2.2.4">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.4.m4.2c">(\rho_{k},\rho_{k+1})\in E</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.4.m4.2d">( italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_ρ start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) ∈ italic_E</annotation></semantics></math> for all <math alttext="k\in{\rm Nature}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.5.m5.1"><semantics id="S2.SS1.SSS0.Px2.p1.5.m5.1a"><mrow id="S2.SS1.SSS0.Px2.p1.5.m5.1.1" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.5.m5.1.1.2" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1.2.cmml">k</mi><mo id="S2.SS1.SSS0.Px2.p1.5.m5.1.1.1" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S2.SS1.SSS0.Px2.p1.5.m5.1.1.3" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.5.m5.1b"><apply id="S2.SS1.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1"><in id="S2.SS1.SSS0.Px2.p1.5.m5.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1.1"></in><ci id="S2.SS1.SSS0.Px2.p1.5.m5.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1.2">𝑘</ci><ci id="S2.SS1.SSS0.Px2.p1.5.m5.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.5.m5.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.5.m5.1c">k\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.5.m5.1d">italic_k ∈ roman_Nature</annotation></semantics></math>. <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px2.p1.32.2">Histories</em> are finite sequences <math alttext="h=h_{0}\ldots h_{k}\in V^{+}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.6.m6.1"><semantics id="S2.SS1.SSS0.Px2.p1.6.m6.1a"><mrow id="S2.SS1.SSS0.Px2.p1.6.m6.1.1" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.2" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.2.cmml">h</mi><mo id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.3" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.3.cmml">=</mo><mrow id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.cmml"><msub id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.2" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.2.cmml">h</mi><mn id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.3" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.1" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.3" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.3.cmml">…</mi><mo id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.1a" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.cmml"><mi id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.2" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.2.cmml">h</mi><mi id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.3" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.3.cmml">k</mi></msub></mrow><mo id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.5" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.5.cmml">∈</mo><msup id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.cmml"><mi id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.2" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.2.cmml">V</mi><mo id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.3" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.6.m6.1b"><apply id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1"><and id="S2.SS1.SSS0.Px2.p1.6.m6.1.1a.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1"></and><apply id="S2.SS1.SSS0.Px2.p1.6.m6.1.1b.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1"><eq id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.3"></eq><ci id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.2">ℎ</ci><apply id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4"><times id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.1"></times><apply id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.2">ℎ</ci><cn id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.2.3">0</cn></apply><ci id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.3.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.3">…</ci><apply id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.2">ℎ</ci><ci id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.3.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.4.3">𝑘</ci></apply></apply></apply><apply id="S2.SS1.SSS0.Px2.p1.6.m6.1.1c.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1"><in id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.5.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.5"></in><share href="https://arxiv.org/html/2308.09443v2#S2.SS1.SSS0.Px2.p1.6.m6.1.1.4.cmml" id="S2.SS1.SSS0.Px2.p1.6.m6.1.1d.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1"></share><apply id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6">superscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.2">𝑉</ci><plus id="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.3.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m6.1.1.6.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.6.m6.1c">h=h_{0}\ldots h_{k}\in V^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.6.m6.1d">italic_h = italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT … italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> defined similarly. We denote by <math alttext="\Last{h}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.7.m7.1"><semantics id="S2.SS1.SSS0.Px2.p1.7.m7.1a"><mrow id="S2.SS1.SSS0.Px2.p1.7.m7.1.1" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2b.cmml"><mtext id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2a" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2b.cmml">\Last</mtext></merror><mo id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.1" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.3" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.7.m7.1b"><apply id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1"><times id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.1"></times><ci id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2b.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2"><mtext id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2a.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.2">\Last</mtext></merror></ci><ci id="S2.SS1.SSS0.Px2.p1.7.m7.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m7.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.7.m7.1c">\Last{h}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.7.m7.1d">italic_h</annotation></semantics></math> the last vertex <math alttext="h_{k}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.8.m8.1"><semantics id="S2.SS1.SSS0.Px2.p1.8.m8.1a"><msub id="S2.SS1.SSS0.Px2.p1.8.m8.1.1" xref="S2.SS1.SSS0.Px2.p1.8.m8.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.8.m8.1.1.2" xref="S2.SS1.SSS0.Px2.p1.8.m8.1.1.2.cmml">h</mi><mi id="S2.SS1.SSS0.Px2.p1.8.m8.1.1.3" xref="S2.SS1.SSS0.Px2.p1.8.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.8.m8.1b"><apply id="S2.SS1.SSS0.Px2.p1.8.m8.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.8.m8.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.8.m8.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m8.1.1.2">ℎ</ci><ci id="S2.SS1.SSS0.Px2.p1.8.m8.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.8.m8.1c">h_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.8.m8.1d">italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> of the history <math alttext="h" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.9.m9.1"><semantics id="S2.SS1.SSS0.Px2.p1.9.m9.1a"><mi id="S2.SS1.SSS0.Px2.p1.9.m9.1.1" xref="S2.SS1.SSS0.Px2.p1.9.m9.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.9.m9.1b"><ci id="S2.SS1.SSS0.Px2.p1.9.m9.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.9.m9.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.9.m9.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.9.m9.1d">italic_h</annotation></semantics></math> and by <math alttext="|h|" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.10.m10.1"><semantics id="S2.SS1.SSS0.Px2.p1.10.m10.1a"><mrow id="S2.SS1.SSS0.Px2.p1.10.m10.1.2.2" xref="S2.SS1.SSS0.Px2.p1.10.m10.1.2.1.cmml"><mo id="S2.SS1.SSS0.Px2.p1.10.m10.1.2.2.1" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.10.m10.1.2.1.1.cmml">|</mo><mi id="S2.SS1.SSS0.Px2.p1.10.m10.1.1" xref="S2.SS1.SSS0.Px2.p1.10.m10.1.1.cmml">h</mi><mo id="S2.SS1.SSS0.Px2.p1.10.m10.1.2.2.2" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.10.m10.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.10.m10.1b"><apply id="S2.SS1.SSS0.Px2.p1.10.m10.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m10.1.2.2"><abs id="S2.SS1.SSS0.Px2.p1.10.m10.1.2.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m10.1.2.2.1"></abs><ci id="S2.SS1.SSS0.Px2.p1.10.m10.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m10.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.10.m10.1c">|h|</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.10.m10.1d">| italic_h |</annotation></semantics></math> its length (equal to <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.11.m11.1"><semantics id="S2.SS1.SSS0.Px2.p1.11.m11.1a"><mi id="S2.SS1.SSS0.Px2.p1.11.m11.1.1" xref="S2.SS1.SSS0.Px2.p1.11.m11.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.11.m11.1b"><ci id="S2.SS1.SSS0.Px2.p1.11.m11.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.11.m11.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.11.m11.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.11.m11.1d">italic_k</annotation></semantics></math>). Let <math alttext="\textsf{Play}_{A}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.12.m12.1"><semantics id="S2.SS1.SSS0.Px2.p1.12.m12.1a"><msub id="S2.SS1.SSS0.Px2.p1.12.m12.1.1" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px2.p1.12.m12.1.1.2" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1.2a.cmml">Play</mtext><mi id="S2.SS1.SSS0.Px2.p1.12.m12.1.1.3" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.12.m12.1b"><apply id="S2.SS1.SSS0.Px2.p1.12.m12.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.12.m12.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.12.m12.1.1.2a.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px2.p1.12.m12.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1.2">Play</mtext></ci><ci id="S2.SS1.SSS0.Px2.p1.12.m12.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m12.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.12.m12.1c">\textsf{Play}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.12.m12.1d">Play start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> denote the set of all plays in <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.13.m13.1"><semantics id="S2.SS1.SSS0.Px2.p1.13.m13.1a"><mi id="S2.SS1.SSS0.Px2.p1.13.m13.1.1" xref="S2.SS1.SSS0.Px2.p1.13.m13.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.13.m13.1b"><ci id="S2.SS1.SSS0.Px2.p1.13.m13.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.13.m13.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.13.m13.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.13.m13.1d">italic_A</annotation></semantics></math>, <math alttext="\textsf{Hist}_{A}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.14.m14.1"><semantics id="S2.SS1.SSS0.Px2.p1.14.m14.1a"><msub id="S2.SS1.SSS0.Px2.p1.14.m14.1.1" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px2.p1.14.m14.1.1.2" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1.2a.cmml">Hist</mtext><mi id="S2.SS1.SSS0.Px2.p1.14.m14.1.1.3" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.14.m14.1b"><apply id="S2.SS1.SSS0.Px2.p1.14.m14.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.14.m14.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.14.m14.1.1.2a.cmml" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px2.p1.14.m14.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1.2">Hist</mtext></ci><ci id="S2.SS1.SSS0.Px2.p1.14.m14.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.14.m14.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.14.m14.1c">\textsf{Hist}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.14.m14.1d">Hist start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> the set of all histories in <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.15.m15.1"><semantics id="S2.SS1.SSS0.Px2.p1.15.m15.1a"><mi id="S2.SS1.SSS0.Px2.p1.15.m15.1.1" xref="S2.SS1.SSS0.Px2.p1.15.m15.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.15.m15.1b"><ci id="S2.SS1.SSS0.Px2.p1.15.m15.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.15.m15.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.15.m15.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.15.m15.1d">italic_A</annotation></semantics></math>, and <math alttext="\textsf{Hist}^{i}_{A}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.16.m16.1"><semantics id="S2.SS1.SSS0.Px2.p1.16.m16.1a"><msubsup id="S2.SS1.SSS0.Px2.p1.16.m16.1.1" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.2a.cmml">Hist</mtext><mi id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.3" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.3.cmml">A</mi><mi id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.16.m16.1b"><apply id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1">subscript</csymbol><apply id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1">superscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.2a.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.2">Hist</mtext></ci><ci id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.SSS0.Px2.p1.16.m16.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.16.m16.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.16.m16.1c">\textsf{Hist}^{i}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.16.m16.1d">Hist start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> the set of all histories in <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.17.m17.1"><semantics id="S2.SS1.SSS0.Px2.p1.17.m17.1a"><mi id="S2.SS1.SSS0.Px2.p1.17.m17.1.1" xref="S2.SS1.SSS0.Px2.p1.17.m17.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.17.m17.1b"><ci id="S2.SS1.SSS0.Px2.p1.17.m17.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.17.m17.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.17.m17.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.17.m17.1d">italic_A</annotation></semantics></math> ending on a vertex in <math alttext="V_{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.18.m18.1"><semantics id="S2.SS1.SSS0.Px2.p1.18.m18.1a"><msub id="S2.SS1.SSS0.Px2.p1.18.m18.1.1" xref="S2.SS1.SSS0.Px2.p1.18.m18.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.18.m18.1.1.2" xref="S2.SS1.SSS0.Px2.p1.18.m18.1.1.2.cmml">V</mi><mi id="S2.SS1.SSS0.Px2.p1.18.m18.1.1.3" xref="S2.SS1.SSS0.Px2.p1.18.m18.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.18.m18.1b"><apply id="S2.SS1.SSS0.Px2.p1.18.m18.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.18.m18.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.18.m18.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.18.m18.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.18.m18.1.1.2">𝑉</ci><ci id="S2.SS1.SSS0.Px2.p1.18.m18.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.18.m18.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.18.m18.1c">V_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.18.m18.1d">italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="i=0,1" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.19.m19.2"><semantics id="S2.SS1.SSS0.Px2.p1.19.m19.2a"><mrow id="S2.SS1.SSS0.Px2.p1.19.m19.2.3" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.2" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.2.cmml">i</mi><mo id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.1" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.3.2" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.3.1.cmml"><mn id="S2.SS1.SSS0.Px2.p1.19.m19.1.1" xref="S2.SS1.SSS0.Px2.p1.19.m19.1.1.cmml">0</mn><mo id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.3.2.1" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.3.1.cmml">,</mo><mn id="S2.SS1.SSS0.Px2.p1.19.m19.2.2" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.19.m19.2b"><apply id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3"><eq id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.1"></eq><ci id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.2">𝑖</ci><list id="S2.SS1.SSS0.Px2.p1.19.m19.2.3.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.3.3.2"><cn id="S2.SS1.SSS0.Px2.p1.19.m19.1.1.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.19.m19.1.1">0</cn><cn id="S2.SS1.SSS0.Px2.p1.19.m19.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.19.m19.2.2">1</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.19.m19.2c">i=0,1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.19.m19.2d">italic_i = 0 , 1</annotation></semantics></math>. The mention of the arena will be omitted when it is clear from the context. If a history <math alttext="h" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.20.m20.1"><semantics id="S2.SS1.SSS0.Px2.p1.20.m20.1a"><mi id="S2.SS1.SSS0.Px2.p1.20.m20.1.1" xref="S2.SS1.SSS0.Px2.p1.20.m20.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.20.m20.1b"><ci id="S2.SS1.SSS0.Px2.p1.20.m20.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.20.m20.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.20.m20.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.20.m20.1d">italic_h</annotation></semantics></math> is prefix of a play <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.21.m21.1"><semantics id="S2.SS1.SSS0.Px2.p1.21.m21.1a"><mi id="S2.SS1.SSS0.Px2.p1.21.m21.1.1" xref="S2.SS1.SSS0.Px2.p1.21.m21.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.21.m21.1b"><ci id="S2.SS1.SSS0.Px2.p1.21.m21.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.21.m21.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.21.m21.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.21.m21.1d">italic_ρ</annotation></semantics></math>, we denote it by <math alttext="h\rho" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.22.m22.1"><semantics id="S2.SS1.SSS0.Px2.p1.22.m22.1a"><mrow id="S2.SS1.SSS0.Px2.p1.22.m22.1.1" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.22.m22.1.1.2" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1.2.cmml">h</mi><mo id="S2.SS1.SSS0.Px2.p1.22.m22.1.1.1" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.22.m22.1.1.3" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.22.m22.1b"><apply id="S2.SS1.SSS0.Px2.p1.22.m22.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1"><times id="S2.SS1.SSS0.Px2.p1.22.m22.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1.1"></times><ci id="S2.SS1.SSS0.Px2.p1.22.m22.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1.2">ℎ</ci><ci id="S2.SS1.SSS0.Px2.p1.22.m22.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.22.m22.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.22.m22.1c">h\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.22.m22.1d">italic_h italic_ρ</annotation></semantics></math>. Given a play <math alttext="\rho=\rho_{0}\rho_{1}\ldots" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.23.m23.1"><semantics id="S2.SS1.SSS0.Px2.p1.23.m23.1a"><mrow id="S2.SS1.SSS0.Px2.p1.23.m23.1.1" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.2" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.2.cmml">ρ</mi><mo id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.1" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.cmml"><msub id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.2" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.3" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.1" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.2" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.3" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.1a" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.4" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.4.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.23.m23.1b"><apply id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1"><eq id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.1"></eq><ci id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.2">𝜌</ci><apply id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3"><times id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.1"></times><apply id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.2">𝜌</ci><cn id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.2.3">0</cn></apply><apply id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.2">𝜌</ci><cn id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.3.3">1</cn></apply><ci id="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.4.cmml" xref="S2.SS1.SSS0.Px2.p1.23.m23.1.1.3.4">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.23.m23.1c">\rho=\rho_{0}\rho_{1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.23.m23.1d">italic_ρ = italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT …</annotation></semantics></math>, we denote by <math alttext="\rho_{\leq k}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.24.m24.1"><semantics id="S2.SS1.SSS0.Px2.p1.24.m24.1a"><msub id="S2.SS1.SSS0.Px2.p1.24.m24.1.1" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.2" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.2.cmml"></mi><mo id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.1" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.1.cmml">≤</mo><mi id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.3.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.24.m24.1b"><apply id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.2">𝜌</ci><apply id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3"><leq id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.1"></leq><csymbol cd="latexml" id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.2">absent</csymbol><ci id="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px2.p1.24.m24.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.24.m24.1c">\rho_{\leq k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.24.m24.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_k end_POSTSUBSCRIPT</annotation></semantics></math> the prefix <math alttext="\rho_{0}\ldots\rho_{k}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.25.m25.1"><semantics id="S2.SS1.SSS0.Px2.p1.25.m25.1a"><mrow id="S2.SS1.SSS0.Px2.p1.25.m25.1.1" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.cmml"><msub id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.1" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.3" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.3.cmml">…</mi><mo id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.1a" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.cmml"><mi id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.2" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.3" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.25.m25.1b"><apply id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1"><times id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.1"></times><apply id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.2">𝜌</ci><cn id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.2.3">0</cn></apply><ci id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.3">…</ci><apply id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.1.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.2.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.2">𝜌</ci><ci id="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.3.cmml" xref="S2.SS1.SSS0.Px2.p1.25.m25.1.1.4.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.25.m25.1c">\rho_{0}\ldots\rho_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.25.m25.1d">italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT … italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.26.m26.1"><semantics id="S2.SS1.SSS0.Px2.p1.26.m26.1a"><mi id="S2.SS1.SSS0.Px2.p1.26.m26.1.1" xref="S2.SS1.SSS0.Px2.p1.26.m26.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.26.m26.1b"><ci id="S2.SS1.SSS0.Px2.p1.26.m26.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.26.m26.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.26.m26.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.26.m26.1d">italic_ρ</annotation></semantics></math>, and by <math alttext="\rho_{\geq k}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.27.m27.1"><semantics id="S2.SS1.SSS0.Px2.p1.27.m27.1a"><msub id="S2.SS1.SSS0.Px2.p1.27.m27.1.1" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.2" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.2.cmml"></mi><mo id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.1" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.1.cmml">≥</mo><mi id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.3.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.27.m27.1b"><apply id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.2">𝜌</ci><apply id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3"><geq id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.1"></geq><csymbol cd="latexml" id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.2">absent</csymbol><ci id="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px2.p1.27.m27.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.27.m27.1c">\rho_{\geq k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.27.m27.1d">italic_ρ start_POSTSUBSCRIPT ≥ italic_k end_POSTSUBSCRIPT</annotation></semantics></math> its suffix <math alttext="\rho_{k}\rho_{k+1}\ldots" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.28.m28.1"><semantics id="S2.SS1.SSS0.Px2.p1.28.m28.1a"><mrow id="S2.SS1.SSS0.Px2.p1.28.m28.1.1" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.cmml"><msub id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.1" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.2" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.2.cmml">k</mi><mo id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.1" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.1.cmml">+</mo><mn id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.3" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.1a" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.4" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.4.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.28.m28.1b"><apply id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1"><times id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.1"></times><apply id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.2">𝜌</ci><ci id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.2.3">𝑘</ci></apply><apply id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.2">𝜌</ci><apply id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3"><plus id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.1"></plus><ci id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.2">𝑘</ci><cn id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.3.3.3">1</cn></apply></apply><ci id="S2.SS1.SSS0.Px2.p1.28.m28.1.1.4.cmml" xref="S2.SS1.SSS0.Px2.p1.28.m28.1.1.4">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.28.m28.1c">\rho_{k}\rho_{k+1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.28.m28.1d">italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT …</annotation></semantics></math>. We also write <math alttext="\rho_{[k,\ell]}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.29.m29.2"><semantics id="S2.SS1.SSS0.Px2.p1.29.m29.2a"><msub id="S2.SS1.SSS0.Px2.p1.29.m29.2.3" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.29.m29.2.3.2" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.3.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.4" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.4.1" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.3.cmml">[</mo><mi id="S2.SS1.SSS0.Px2.p1.29.m29.1.1.1.1" xref="S2.SS1.SSS0.Px2.p1.29.m29.1.1.1.1.cmml">k</mi><mo id="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.4.2" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.3.cmml">,</mo><mi id="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.2" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.2.cmml">ℓ</mi><mo id="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.4.3" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.29.m29.2b"><apply id="S2.SS1.SSS0.Px2.p1.29.m29.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.29.m29.2.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.29.m29.2.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.3.2">𝜌</ci><interval closure="closed" id="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.4"><ci id="S2.SS1.SSS0.Px2.p1.29.m29.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.29.m29.1.1.1.1">𝑘</ci><ci id="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.29.m29.2.2.2.2">ℓ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.29.m29.2c">\rho_{[k,\ell]}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.29.m29.2d">italic_ρ start_POSTSUBSCRIPT [ italic_k , roman_ℓ ] end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="\rho_{k}\ldots\rho_{\ell}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.30.m30.1"><semantics id="S2.SS1.SSS0.Px2.p1.30.m30.1a"><mrow id="S2.SS1.SSS0.Px2.p1.30.m30.1.1" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.cmml"><msub id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.1" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.3" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.3.cmml">…</mi><mo id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.1a" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.cmml"><mi id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.2" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.3" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.30.m30.1b"><apply id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1"><times id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.1"></times><apply id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.2">𝜌</ci><ci id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.2.3">𝑘</ci></apply><ci id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.3">…</ci><apply id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.1.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.2.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.2">𝜌</ci><ci id="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.3.cmml" xref="S2.SS1.SSS0.Px2.p1.30.m30.1.1.4.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.30.m30.1c">\rho_{k}\ldots\rho_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.30.m30.1d">italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT … italic_ρ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px2.p1.32.3">weight</em> of <math alttext="\rho_{[k,\ell]}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.31.m31.2"><semantics id="S2.SS1.SSS0.Px2.p1.31.m31.2a"><msub id="S2.SS1.SSS0.Px2.p1.31.m31.2.3" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.31.m31.2.3.2" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.3.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.4" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.4.1" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.3.cmml">[</mo><mi id="S2.SS1.SSS0.Px2.p1.31.m31.1.1.1.1" xref="S2.SS1.SSS0.Px2.p1.31.m31.1.1.1.1.cmml">k</mi><mo id="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.4.2" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.3.cmml">,</mo><mi id="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.2" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.2.cmml">ℓ</mi><mo id="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.4.3" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.31.m31.2b"><apply id="S2.SS1.SSS0.Px2.p1.31.m31.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.31.m31.2.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.31.m31.2.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.3.2">𝜌</ci><interval closure="closed" id="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.4"><ci id="S2.SS1.SSS0.Px2.p1.31.m31.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.31.m31.1.1.1.1">𝑘</ci><ci id="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.31.m31.2.2.2.2">ℓ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.31.m31.2c">\rho_{[k,\ell]}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.31.m31.2d">italic_ρ start_POSTSUBSCRIPT [ italic_k , roman_ℓ ] end_POSTSUBSCRIPT</annotation></semantics></math> is equal to <math alttext="w(\rho_{[k,\ell]})=\Sigma_{j=k}^{\ell-1}w(\rho_{j},\rho_{j+1})" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.32.m32.5"><semantics id="S2.SS1.SSS0.Px2.p1.32.m32.5a"><mrow id="S2.SS1.SSS0.Px2.p1.32.m32.5.5" xref="S2.SS1.SSS0.Px2.p1.32.m32.5.5.cmml"><mrow id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.3" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.3.cmml">w</mi><mo id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.2" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.1.cmml">(</mo><msub id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.1" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.1.2" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.1.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.4" xref="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.4.1" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.3.cmml">[</mo><mi id="S2.SS1.SSS0.Px2.p1.32.m32.1.1.1.1" xref="S2.SS1.SSS0.Px2.p1.32.m32.1.1.1.1.cmml">k</mi><mo id="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.4.2" xref="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.3.cmml">,</mo><mi id="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.2" mathvariant="normal" xref="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.2.cmml">ℓ</mi><mo id="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.4.3" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.32.m32.2.2.2.3.cmml">]</mo></mrow></msub><mo id="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.32.m32.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px2.p1.32.m32.5.5.4" xref="S2.SS1.SSS0.Px2.p1.32.m32.5.5.4.cmml">=</mo><mrow id="S2.SS1.SSS0.Px2.p1.32.m32.5.5.3" 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id="S2.SS1.SSS0.Px2.p1.32.m32.5.5.3.2.2.2.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.32.m32.5.5.3.2.2.2.3.1"></plus><ci id="S2.SS1.SSS0.Px2.p1.32.m32.5.5.3.2.2.2.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.32.m32.5.5.3.2.2.2.3.2">𝑗</ci><cn id="S2.SS1.SSS0.Px2.p1.32.m32.5.5.3.2.2.2.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.32.m32.5.5.3.2.2.2.3.3">1</cn></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.32.m32.5c">w(\rho_{[k,\ell]})=\Sigma_{j=k}^{\ell-1}w(\rho_{j},\rho_{j+1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.32.m32.5d">italic_w ( italic_ρ start_POSTSUBSCRIPT [ italic_k , roman_ℓ ] end_POSTSUBSCRIPT ) = roman_Σ start_POSTSUBSCRIPT italic_j = italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ - 1 end_POSTSUPERSCRIPT italic_w ( italic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_ρ start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS1.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Strategies.</h4> <div class="ltx_para" id="S2.SS1.SSS0.Px3.p1"> <p class="ltx_p" id="S2.SS1.SSS0.Px3.p1.12">Let <math alttext="i\in\{0,1\}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.1.m1.2"><semantics id="S2.SS1.SSS0.Px3.p1.1.m1.2a"><mrow id="S2.SS1.SSS0.Px3.p1.1.m1.2.3" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.cmml"><mi id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.2" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.2.cmml">i</mi><mo id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.1" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.2" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.1.cmml"><mo id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.1.cmml">{</mo><mn id="S2.SS1.SSS0.Px3.p1.1.m1.1.1" xref="S2.SS1.SSS0.Px3.p1.1.m1.1.1.cmml">0</mn><mo id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.2.2" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.1.cmml">,</mo><mn id="S2.SS1.SSS0.Px3.p1.1.m1.2.2" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.2.cmml">1</mn><mo id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.1.m1.2b"><apply id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3"><in id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.1"></in><ci id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.2">𝑖</ci><set id="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.3.3.2"><cn id="S2.SS1.SSS0.Px3.p1.1.m1.1.1.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p1.1.m1.1.1">0</cn><cn id="S2.SS1.SSS0.Px3.p1.1.m1.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p1.1.m1.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.1.m1.2c">i\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.1.m1.2d">italic_i ∈ { 0 , 1 }</annotation></semantics></math>, a <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px3.p1.12.1">strategy</em> for Player <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.2.m2.1"><semantics id="S2.SS1.SSS0.Px3.p1.2.m2.1a"><mi id="S2.SS1.SSS0.Px3.p1.2.m2.1.1" xref="S2.SS1.SSS0.Px3.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.2.m2.1b"><ci id="S2.SS1.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.2.m2.1d">italic_i</annotation></semantics></math> is a function <math alttext="\sigma_{i}\colon\textsf{Hist}^{i}\rightarrow V" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.3.m3.1"><semantics id="S2.SS1.SSS0.Px3.p1.3.m3.1a"><mrow id="S2.SS1.SSS0.Px3.p1.3.m3.1.1" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.cmml"><msub id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.2" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.3" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.cmml"><msup id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.2" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.2a.cmml">Hist</mtext><mi id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.3" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.3.cmml">i</mi></msup><mo id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.1" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.1.cmml">→</mo><mi id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.3" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.3.m3.1b"><apply id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1"><ci id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.1">:</ci><apply id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3"><ci id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.1">→</ci><apply id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.1.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.2a.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.2">Hist</mtext></ci><ci id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.3">𝑖</ci></apply><ci id="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p1.3.m3.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.3.m3.1c">\sigma_{i}\colon\textsf{Hist}^{i}\rightarrow V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : Hist start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → italic_V</annotation></semantics></math> assigning to each history <math alttext="h\in\textsf{Hist}^{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.4.m4.1"><semantics id="S2.SS1.SSS0.Px3.p1.4.m4.1a"><mrow id="S2.SS1.SSS0.Px3.p1.4.m4.1.1" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.2" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.2.cmml">h</mi><mo id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.1" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.2" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.2a.cmml">Hist</mtext><mi id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.3" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.4.m4.1b"><apply id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1"><in id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.1"></in><ci id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.2">ℎ</ci><apply id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3">superscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.2a.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.2">Hist</mtext></ci><ci id="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p1.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.4.m4.1c">h\in\textsf{Hist}^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.4.m4.1d">italic_h ∈ Hist start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> a vertex <math alttext="v=\sigma_{i}(h)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.5.m5.1"><semantics id="S2.SS1.SSS0.Px3.p1.5.m5.1a"><mrow id="S2.SS1.SSS0.Px3.p1.5.m5.1.2" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.cmml"><mi id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.2" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.2.cmml">v</mi><mo id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.1" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.cmml"><msub id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.cmml"><mi id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.2" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.3" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.1" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.3.2" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.cmml"><mo id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.cmml">(</mo><mi id="S2.SS1.SSS0.Px3.p1.5.m5.1.1" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.1.cmml">h</mi><mo id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.5.m5.1b"><apply id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2"><eq id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.1.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.1"></eq><ci id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.2">𝑣</ci><apply id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3"><times id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.1"></times><apply id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.1.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.2.3.2.3">𝑖</ci></apply><ci id="S2.SS1.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.5.m5.1.1">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.5.m5.1c">v=\sigma_{i}(h)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.5.m5.1d">italic_v = italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_h )</annotation></semantics></math> such that <math alttext="(\Last{h},v)\in E" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.6.m6.2"><semantics id="S2.SS1.SSS0.Px3.p1.6.m6.2a"><mrow id="S2.SS1.SSS0.Px3.p1.6.m6.2.2" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.cmml"><mrow id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.2.cmml"><mo id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.2.cmml">(</mo><mrow id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2b.cmml"><mtext id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2a" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2b.cmml">\Last</mtext></merror><mo id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.1" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.3" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.3.cmml">h</mi></mrow><mo id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.3" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.2.cmml">,</mo><mi id="S2.SS1.SSS0.Px3.p1.6.m6.1.1" xref="S2.SS1.SSS0.Px3.p1.6.m6.1.1.cmml">v</mi><mo id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.4" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.2.cmml">)</mo></mrow><mo id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.2" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.2.cmml">∈</mo><mi id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.3" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.6.m6.2b"><apply id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2"><in id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.2"></in><interval closure="open" id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1"><apply id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1"><times id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.1"></times><ci id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2b.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2"><mtext id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2a.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.2">\Last</mtext></merror></ci><ci id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.1.1.1.3">ℎ</ci></apply><ci id="S2.SS1.SSS0.Px3.p1.6.m6.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.1.1">𝑣</ci></interval><ci id="S2.SS1.SSS0.Px3.p1.6.m6.2.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.6.m6.2.2.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.6.m6.2c">(\Last{h},v)\in E</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.6.m6.2d">( italic_h , italic_v ) ∈ italic_E</annotation></semantics></math>. We denote by <math alttext="\Sigma_{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.7.m7.1"><semantics id="S2.SS1.SSS0.Px3.p1.7.m7.1a"><msub id="S2.SS1.SSS0.Px3.p1.7.m7.1.1" xref="S2.SS1.SSS0.Px3.p1.7.m7.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p1.7.m7.1.1.2" mathvariant="normal" xref="S2.SS1.SSS0.Px3.p1.7.m7.1.1.2.cmml">Σ</mi><mi id="S2.SS1.SSS0.Px3.p1.7.m7.1.1.3" xref="S2.SS1.SSS0.Px3.p1.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.7.m7.1b"><apply id="S2.SS1.SSS0.Px3.p1.7.m7.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.7.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.7.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.7.m7.1.1.2">Σ</ci><ci id="S2.SS1.SSS0.Px3.p1.7.m7.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.7.m7.1c">\Sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.7.m7.1d">roman_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> the set of all strategies for Player <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.8.m8.1"><semantics id="S2.SS1.SSS0.Px3.p1.8.m8.1a"><mi id="S2.SS1.SSS0.Px3.p1.8.m8.1.1" xref="S2.SS1.SSS0.Px3.p1.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.8.m8.1b"><ci id="S2.SS1.SSS0.Px3.p1.8.m8.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.8.m8.1d">italic_i</annotation></semantics></math>. We say that a strategy <math alttext="\sigma_{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.9.m9.1"><semantics id="S2.SS1.SSS0.Px3.p1.9.m9.1a"><msub id="S2.SS1.SSS0.Px3.p1.9.m9.1.1" xref="S2.SS1.SSS0.Px3.p1.9.m9.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p1.9.m9.1.1.2" xref="S2.SS1.SSS0.Px3.p1.9.m9.1.1.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p1.9.m9.1.1.3" xref="S2.SS1.SSS0.Px3.p1.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.9.m9.1b"><apply id="S2.SS1.SSS0.Px3.p1.9.m9.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.9.m9.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.9.m9.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.9.m9.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.9.m9.1.1.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p1.9.m9.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.9.m9.1c">\sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.9.m9.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px3.p1.12.2">memoryless</em> if for all <math alttext="h,h^{\prime}\in\textsf{Hist}^{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.10.m10.2"><semantics id="S2.SS1.SSS0.Px3.p1.10.m10.2a"><mrow id="S2.SS1.SSS0.Px3.p1.10.m10.2.2" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.cmml"><mrow id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.2.cmml"><mi id="S2.SS1.SSS0.Px3.p1.10.m10.1.1" xref="S2.SS1.SSS0.Px3.p1.10.m10.1.1.cmml">h</mi><mo id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.2" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.2.cmml">,</mo><msup id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.2" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.2.cmml">h</mi><mo id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.3" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.2" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.2.cmml">∈</mo><msup id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.2" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.2a.cmml">Hist</mtext><mi id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.3" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.10.m10.2b"><apply id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2"><in id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.2"></in><list id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1"><ci id="S2.SS1.SSS0.Px3.p1.10.m10.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.1.1">ℎ</ci><apply id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1">superscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.2">ℎ</ci><ci id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.1.1.1.3">′</ci></apply></list><apply id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3">superscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.2a.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.2">Hist</mtext></ci><ci id="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.3.cmml" xref="S2.SS1.SSS0.Px3.p1.10.m10.2.2.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.10.m10.2c">h,h^{\prime}\in\textsf{Hist}^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.10.m10.2d">italic_h , italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Hist start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, if <math alttext="\Last{h}=\Last{h^{\prime}}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.11.m11.1"><semantics id="S2.SS1.SSS0.Px3.p1.11.m11.1a"><mrow id="S2.SS1.SSS0.Px3.p1.11.m11.1.1" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.cmml"><mrow id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2b.cmml"><mtext id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2a" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2b.cmml">\Last</mtext></merror><mo id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.1" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.3" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.3.cmml">h</mi></mrow><mo id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.1" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.cmml"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2b.cmml"><mtext id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2a" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2b.cmml">\Last</mtext></merror><mo id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.1" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.1.cmml">⁢</mo><msup id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.cmml"><mi id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.2" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.2.cmml">h</mi><mo id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.3" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.11.m11.1b"><apply id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1"><eq id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.1"></eq><apply id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2"><times id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.1"></times><ci id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2b.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2"><mtext id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2a.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.2">\Last</mtext></merror></ci><ci id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.2.3">ℎ</ci></apply><apply id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3"><times id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.1"></times><ci id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2b.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2"><merror class="ltx_ERROR undefined undefined" id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2"><mtext id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2a.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.2">\Last</mtext></merror></ci><apply id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3">superscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.2">ℎ</ci><ci id="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.3.cmml" xref="S2.SS1.SSS0.Px3.p1.11.m11.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.11.m11.1c">\Last{h}=\Last{h^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.11.m11.1d">italic_h = italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="\sigma_{i}(h)=\sigma_{i}(h^{\prime})" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p1.12.m12.2"><semantics id="S2.SS1.SSS0.Px3.p1.12.m12.2a"><mrow id="S2.SS1.SSS0.Px3.p1.12.m12.2.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.cmml"><mrow id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.cmml"><msub id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.cmml"><mi id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.3" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.1" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.3.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.cmml">(</mo><mi id="S2.SS1.SSS0.Px3.p1.12.m12.1.1" xref="S2.SS1.SSS0.Px3.p1.12.m12.1.1.cmml">h</mi><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.2.cmml">=</mo><mrow id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.cmml"><msub id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.cmml"><mi id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.3" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.3.cmml">i</mi></msub><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.cmml">(</mo><msup id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.2" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.2.cmml">h</mi><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.3" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p1.12.m12.2b"><apply id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2"><eq id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.2"></eq><apply id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3"><times id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.1"></times><apply id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.1.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.2.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.3.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.3.2.3">𝑖</ci></apply><ci id="S2.SS1.SSS0.Px3.p1.12.m12.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.1.1">ℎ</ci></apply><apply id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1"><times id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.2"></times><apply id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.3.3">𝑖</ci></apply><apply id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1">superscript</csymbol><ci id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.2">ℎ</ci><ci id="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p1.12.m12.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p1.12.m12.2c">\sigma_{i}(h)=\sigma_{i}(h^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p1.12.m12.2d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_h ) = italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. A strategy is considered <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px3.p1.12.3">finite-memory</em> if it can be encoded by a Mealy machine and its <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px3.p1.12.4">memory size</em> is the number of states of the machine <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib25" title="">25</a>]</cite>.<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>We assume that the reader is familiar with the concept of finite-memory strategy and memoryless strategy.</span></span></span></p> </div> <div class="ltx_para" id="S2.SS1.SSS0.Px3.p2"> <p class="ltx_p" id="S2.SS1.SSS0.Px3.p2.11">A play <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.1.m1.1"><semantics id="S2.SS1.SSS0.Px3.p2.1.m1.1a"><mi id="S2.SS1.SSS0.Px3.p2.1.m1.1.1" xref="S2.SS1.SSS0.Px3.p2.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.1.m1.1b"><ci id="S2.SS1.SSS0.Px3.p2.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.1.m1.1d">italic_ρ</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px3.p2.11.1">consistent</em> with a strategy <math alttext="\sigma_{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.2.m2.1"><semantics id="S2.SS1.SSS0.Px3.p2.2.m2.1a"><msub id="S2.SS1.SSS0.Px3.p2.2.m2.1.1" xref="S2.SS1.SSS0.Px3.p2.2.m2.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p2.2.m2.1.1.2" xref="S2.SS1.SSS0.Px3.p2.2.m2.1.1.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p2.2.m2.1.1.3" xref="S2.SS1.SSS0.Px3.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.2.m2.1b"><apply id="S2.SS1.SSS0.Px3.p2.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.2.m2.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.2.m2.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.2.m2.1.1.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p2.2.m2.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.2.m2.1c">\sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.2.m2.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> if for all <math alttext="k\in{\rm Nature}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.3.m3.1"><semantics id="S2.SS1.SSS0.Px3.p2.3.m3.1a"><mrow id="S2.SS1.SSS0.Px3.p2.3.m3.1.1" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p2.3.m3.1.1.2" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1.2.cmml">k</mi><mo id="S2.SS1.SSS0.Px3.p2.3.m3.1.1.1" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1.1.cmml">∈</mo><mi id="S2.SS1.SSS0.Px3.p2.3.m3.1.1.3" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.3.m3.1b"><apply id="S2.SS1.SSS0.Px3.p2.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1"><in id="S2.SS1.SSS0.Px3.p2.3.m3.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1.1"></in><ci id="S2.SS1.SSS0.Px3.p2.3.m3.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1.2">𝑘</ci><ci id="S2.SS1.SSS0.Px3.p2.3.m3.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.3.m3.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.3.m3.1c">k\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.3.m3.1d">italic_k ∈ roman_Nature</annotation></semantics></math>, <math alttext="\rho_{k}\in V_{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.4.m4.1"><semantics id="S2.SS1.SSS0.Px3.p2.4.m4.1a"><mrow id="S2.SS1.SSS0.Px3.p2.4.m4.1.1" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.cmml"><msub id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.2" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.3" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.1" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.1.cmml">∈</mo><msub id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.2" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.2.cmml">V</mi><mi id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.3" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.4.m4.1b"><apply id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1"><in id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.1"></in><apply id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.2">𝜌</ci><ci id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.2.3">𝑘</ci></apply><apply id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.2">𝑉</ci><ci id="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p2.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.4.m4.1c">\rho_{k}\in V_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.4.m4.1d">italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> implies that <math alttext="\rho_{k+1}=\sigma_{i}(\rho_{\leq k})" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.5.m5.1"><semantics id="S2.SS1.SSS0.Px3.p2.5.m5.1a"><mrow id="S2.SS1.SSS0.Px3.p2.5.m5.1.1" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.cmml"><msub id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.2" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.cmml"><mi id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.2" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.2.cmml">k</mi><mo id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.1" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.1.cmml">+</mo><mn id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.3" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.2" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.2.cmml">=</mo><mrow id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.cmml"><msub id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.2" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.3" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.3.cmml">i</mi></msub><mo id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.2" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.2" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.2.cmml"></mi><mo id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.1" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.1.cmml">≤</mo><mi id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.3" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.3.cmml">k</mi></mrow></msub><mo id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.5.m5.1b"><apply id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1"><eq id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.2"></eq><apply id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.2">𝜌</ci><apply id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3"><plus id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.1"></plus><ci id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.2.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.2">𝑘</ci><cn id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.3.3.3">1</cn></apply></apply><apply id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1"><times id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.2"></times><apply id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.3.3">𝑖</ci></apply><apply id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.2">𝜌</ci><apply id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3"><leq id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.1"></leq><csymbol cd="latexml" id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.2">absent</csymbol><ci id="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p2.5.m5.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.5.m5.1c">\rho_{k+1}=\sigma_{i}(\rho_{\leq k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.5.m5.1d">italic_ρ start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT ≤ italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. Consistency is extended to histories as expected. We denote <math alttext="\textsf{Play}_{\sigma_{i}}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.6.m6.1"><semantics id="S2.SS1.SSS0.Px3.p2.6.m6.1a"><msub id="S2.SS1.SSS0.Px3.p2.6.m6.1.1" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.2" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.2a.cmml">Play</mtext><msub id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.2" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.3" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.3.cmml">i</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.6.m6.1b"><apply id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.2a.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.2">Play</mtext></ci><apply id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p2.6.m6.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.6.m6.1c">\textsf{Play}_{\sigma_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.6.m6.1d">Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> (resp. <math alttext="\textsf{Hist}_{\sigma_{i}}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.7.m7.1"><semantics id="S2.SS1.SSS0.Px3.p2.7.m7.1a"><msub id="S2.SS1.SSS0.Px3.p2.7.m7.1.1" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.2" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.2a.cmml">Hist</mtext><msub id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.2" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.3" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.3.cmml">i</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.7.m7.1b"><apply id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.2a.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.2">Hist</mtext></ci><apply id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px3.p2.7.m7.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.7.m7.1c">\textsf{Hist}_{\sigma_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.7.m7.1d">Hist start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>) the set of all plays (resp. histories) consistent with <math alttext="\sigma_{i}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.8.m8.1"><semantics id="S2.SS1.SSS0.Px3.p2.8.m8.1a"><msub id="S2.SS1.SSS0.Px3.p2.8.m8.1.1" xref="S2.SS1.SSS0.Px3.p2.8.m8.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p2.8.m8.1.1.2" xref="S2.SS1.SSS0.Px3.p2.8.m8.1.1.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px3.p2.8.m8.1.1.3" xref="S2.SS1.SSS0.Px3.p2.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.8.m8.1b"><apply id="S2.SS1.SSS0.Px3.p2.8.m8.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.8.m8.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.8.m8.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.8.m8.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.8.m8.1.1.2">𝜎</ci><ci id="S2.SS1.SSS0.Px3.p2.8.m8.1.1.3.cmml" xref="S2.SS1.SSS0.Px3.p2.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.8.m8.1c">\sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.8.m8.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Given a couple of strategies <math alttext="(\sigma_{0},\sigma_{1})" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.9.m9.2"><semantics id="S2.SS1.SSS0.Px3.p2.9.m9.2a"><mrow id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.3" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.3.cmml">(</mo><msub id="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1" xref="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.2" xref="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.2.cmml">σ</mi><mn id="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.3" xref="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.4" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.3.cmml">,</mo><msub id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.cmml"><mi id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.2" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.2.cmml">σ</mi><mn id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.3" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.5" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.9.m9.2b"><interval closure="open" id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.3.cmml" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2"><apply id="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.2">𝜎</ci><cn id="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p2.9.m9.1.1.1.1.3">0</cn></apply><apply id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.2">𝜎</ci><cn id="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p2.9.m9.2.2.2.2.3">1</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.9.m9.2c">(\sigma_{0},\sigma_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.9.m9.2d">( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> for Players 0 and 1, there exists a single play that is consistent with both of them, that we denote by <math alttext="\textsf{out}(\sigma_{0},\sigma_{1})" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.10.m10.2"><semantics id="S2.SS1.SSS0.Px3.p2.10.m10.2a"><mrow id="S2.SS1.SSS0.Px3.p2.10.m10.2.2" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.4" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.4a.cmml">out</mtext><mo id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.3" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.3.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.3" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.3.cmml">(</mo><msub id="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1" xref="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.2.cmml">σ</mi><mn id="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.4" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.3.cmml">,</mo><msub id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.cmml"><mi id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.2" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.2.cmml">σ</mi><mn id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.3" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.5" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.10.m10.2b"><apply id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2"><times id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.3.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.3"></times><ci id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.4a.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.4"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.4.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.4">out</mtext></ci><interval closure="open" id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2"><apply id="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.2">𝜎</ci><cn id="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p2.10.m10.1.1.1.1.1.3">0</cn></apply><apply id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.2">𝜎</ci><cn id="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p2.10.m10.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.10.m10.2c">\textsf{out}(\sigma_{0},\sigma_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.10.m10.2d">out ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and call the <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px3.p2.11.2">outcome</em> of <math alttext="(\sigma_{0},\sigma_{1})" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px3.p2.11.m11.2"><semantics id="S2.SS1.SSS0.Px3.p2.11.m11.2a"><mrow id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.3" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.3.cmml">(</mo><msub id="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1" xref="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.2" xref="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.2.cmml">σ</mi><mn id="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.3" xref="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.4" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.3.cmml">,</mo><msub id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.cmml"><mi id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.2" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.2.cmml">σ</mi><mn id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.3" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.5" stretchy="false" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px3.p2.11.m11.2b"><interval closure="open" id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.3.cmml" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2"><apply id="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.2">𝜎</ci><cn id="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p2.11.m11.1.1.1.1.3">0</cn></apply><apply id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.2">𝜎</ci><cn id="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px3.p2.11.m11.2.2.2.2.3">1</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px3.p2.11.m11.2c">(\sigma_{0},\sigma_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px3.p2.11.m11.2d">( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS1.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Reachability costs.</h4> <div class="ltx_para" id="S2.SS1.SSS0.Px4.p1"> <p class="ltx_p" id="S2.SS1.SSS0.Px4.p1.26">Given an arena <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.1.m1.1"><semantics id="S2.SS1.SSS0.Px4.p1.1.m1.1a"><mi id="S2.SS1.SSS0.Px4.p1.1.m1.1.1" xref="S2.SS1.SSS0.Px4.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.1.m1.1b"><ci id="S2.SS1.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.1.m1.1d">italic_A</annotation></semantics></math>, let us consider a non-empty subset <math alttext="T\subseteq V" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.2.m2.1"><semantics id="S2.SS1.SSS0.Px4.p1.2.m2.1a"><mrow id="S2.SS1.SSS0.Px4.p1.2.m2.1.1" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.2.m2.1.1.2" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1.2.cmml">T</mi><mo id="S2.SS1.SSS0.Px4.p1.2.m2.1.1.1" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S2.SS1.SSS0.Px4.p1.2.m2.1.1.3" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.2.m2.1b"><apply id="S2.SS1.SSS0.Px4.p1.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1"><subset id="S2.SS1.SSS0.Px4.p1.2.m2.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1.1"></subset><ci id="S2.SS1.SSS0.Px4.p1.2.m2.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1.2">𝑇</ci><ci id="S2.SS1.SSS0.Px4.p1.2.m2.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.2.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.2.m2.1c">T\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.2.m2.1d">italic_T ⊆ italic_V</annotation></semantics></math> of vertices called <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px4.p1.26.1">target</em>. We say that a play <math alttext="\rho=\rho_{0}\rho_{1}\ldots" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.3.m3.1"><semantics id="S2.SS1.SSS0.Px4.p1.3.m3.1a"><mrow id="S2.SS1.SSS0.Px4.p1.3.m3.1.1" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.2" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.2.cmml">ρ</mi><mo id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.1" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.cmml"><msub id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.cmml"><mi id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.2" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.3" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.1" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.cmml"><mi id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.2" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.2.cmml">ρ</mi><mn id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.3" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.1a" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.4" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.4.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.3.m3.1b"><apply id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1"><eq id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.1"></eq><ci id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.2">𝜌</ci><apply id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3"><times id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.1"></times><apply id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.2">𝜌</ci><cn id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.3">0</cn></apply><apply id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.2">𝜌</ci><cn id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.3">1</cn></apply><ci id="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.4.cmml" xref="S2.SS1.SSS0.Px4.p1.3.m3.1.1.3.4">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.3.m3.1c">\rho=\rho_{0}\rho_{1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.3.m3.1d">italic_ρ = italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT …</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px4.p1.26.2">visits</em> the target <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.4.m4.1"><semantics id="S2.SS1.SSS0.Px4.p1.4.m4.1a"><mi id="S2.SS1.SSS0.Px4.p1.4.m4.1.1" xref="S2.SS1.SSS0.Px4.p1.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.4.m4.1b"><ci id="S2.SS1.SSS0.Px4.p1.4.m4.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.4.m4.1d">italic_T</annotation></semantics></math>, if <math alttext="\rho_{k}\in T" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.5.m5.1"><semantics id="S2.SS1.SSS0.Px4.p1.5.m5.1a"><mrow id="S2.SS1.SSS0.Px4.p1.5.m5.1.1" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.cmml"><msub id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.2" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.3" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.1" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.3" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.5.m5.1b"><apply id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1"><in id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.1"></in><apply id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.2">𝜌</ci><ci id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.2.3">𝑘</ci></apply><ci id="S2.SS1.SSS0.Px4.p1.5.m5.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.5.m5.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.5.m5.1c">\rho_{k}\in T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.5.m5.1d">italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_T</annotation></semantics></math> for some <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.6.m6.1"><semantics id="S2.SS1.SSS0.Px4.p1.6.m6.1a"><mi id="S2.SS1.SSS0.Px4.p1.6.m6.1.1" xref="S2.SS1.SSS0.Px4.p1.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.6.m6.1b"><ci id="S2.SS1.SSS0.Px4.p1.6.m6.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.6.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.6.m6.1d">italic_k</annotation></semantics></math>. We define a <em class="ltx_emph ltx_font_italic" id="S2.SS1.SSS0.Px4.p1.26.3">cost function</em> <math alttext="\textsf{cost}_{T}\colon\textsf{Play}\rightarrow\overline{{\rm Nature}}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.7.m7.1"><semantics id="S2.SS1.SSS0.Px4.p1.7.m7.1a"><mrow id="S2.SS1.SSS0.Px4.p1.7.m7.1.1" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.cmml"><msub id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.2" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.2a.cmml">cost</mtext><mi id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.3" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.3.cmml">T</mi></msub><mo id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.1.cmml">:</mo><mrow id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.2" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.2a.cmml">Play</mtext><mo id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.1.cmml">→</mo><mover accent="true" id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.cmml"><mi id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.2" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.2.cmml">Nature</mi><mo id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.1" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.7.m7.1b"><apply id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1"><ci id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.1">:</ci><apply id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.2a.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.2">cost</mtext></ci><ci id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.2.3">𝑇</ci></apply><apply id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3"><ci id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.1">→</ci><ci id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.2a.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.2">Play</mtext></ci><apply id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3"><ci id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.1">¯</ci><ci id="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.7.m7.1.1.3.3.2">Nature</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.7.m7.1c">\textsf{cost}_{T}\colon\textsf{Play}\rightarrow\overline{{\rm Nature}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.7.m7.1d">cost start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT : Play → over¯ start_ARG roman_Nature end_ARG</annotation></semantics></math>, where <math alttext="\overline{{\rm Nature}}={\rm Nature}\cup\{\infty\}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.8.m8.1"><semantics id="S2.SS1.SSS0.Px4.p1.8.m8.1a"><mrow id="S2.SS1.SSS0.Px4.p1.8.m8.1.2" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.cmml"><mover accent="true" id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.cmml"><mi id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.2" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.2.cmml">Nature</mi><mo id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.1" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.1.cmml">¯</mo></mover><mo id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.1" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.cmml"><mi id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.2" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.2.cmml">Nature</mi><mo id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.1" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.1.cmml">∪</mo><mrow id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.2" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.1.cmml"><mo id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.1.cmml">{</mo><mi id="S2.SS1.SSS0.Px4.p1.8.m8.1.1" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.1.cmml">∞</mi><mo id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.8.m8.1b"><apply id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2"><eq id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.1"></eq><apply id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2"><ci id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.1">¯</ci><ci id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.2.2">Nature</ci></apply><apply id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3"><union id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.1"></union><ci id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.2">Nature</ci><set id="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.2.3.3.2"><infinity id="S2.SS1.SSS0.Px4.p1.8.m8.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.8.m8.1.1"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.8.m8.1c">\overline{{\rm Nature}}={\rm Nature}\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.8.m8.1d">over¯ start_ARG roman_Nature end_ARG = roman_Nature ∪ { ∞ }</annotation></semantics></math>, that assigns to every play <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.9.m9.1"><semantics id="S2.SS1.SSS0.Px4.p1.9.m9.1a"><mi id="S2.SS1.SSS0.Px4.p1.9.m9.1.1" xref="S2.SS1.SSS0.Px4.p1.9.m9.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.9.m9.1b"><ci id="S2.SS1.SSS0.Px4.p1.9.m9.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.9.m9.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.9.m9.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.9.m9.1d">italic_ρ</annotation></semantics></math> the quantity <math alttext="\textsf{cost}_{T}(\rho)=\min\{w(\rho_{\leq k})\mid\rho_{k}\in T\}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.10.m10.3"><semantics id="S2.SS1.SSS0.Px4.p1.10.m10.3a"><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.cmml"><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.cmml"><msub id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.2a.cmml">cost</mtext><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.3.cmml">T</mi></msub><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.3.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.cmml"><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.cmml">(</mo><mi id="S2.SS1.SSS0.Px4.p1.10.m10.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.1.1.cmml">ρ</mi><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.2.cmml">=</mo><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.2.cmml"><mi id="S2.SS1.SSS0.Px4.p1.10.m10.2.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.2.2.cmml">min</mi><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1a" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.2.cmml">⁡</mo><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.2.cmml"><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.2.cmml">{</mo><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.cmml"><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.cmml"><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.3.cmml">w</mi><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.2.cmml">ρ</mi><mrow id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.2.cmml"></mi><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.1" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.1.cmml">≤</mo><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.3.cmml">k</mi></mrow></msub><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.2.cmml">∣</mo><msub id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.2.cmml">ρ</mi><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.2" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.2.cmml">∈</mo><mi id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.3" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.3.cmml">T</mi></mrow><mo id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.10.m10.3b"><apply id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3"><eq id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.2"></eq><apply id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3"><times id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.1"></times><apply id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.3.2.2a.cmml" 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xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.2">conditional</csymbol><apply id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1"><times id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.2"></times><ci id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.3">𝑤</ci><apply id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.2">𝜌</ci><apply id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3"><leq id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.1"></leq><csymbol cd="latexml" id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.2">absent</csymbol><ci id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply><apply id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.2">𝜌</ci><ci id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.1.3.3">𝑘</ci></apply></apply><ci id="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.10.m10.3.3.1.1.1.1.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.10.m10.3c">\textsf{cost}_{T}(\rho)=\min\{w(\rho_{\leq k})\mid\rho_{k}\in T\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.10.m10.3d">cost start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_ρ ) = roman_min { italic_w ( italic_ρ start_POSTSUBSCRIPT ≤ italic_k end_POSTSUBSCRIPT ) ∣ italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_T }</annotation></semantics></math>, that is, the weight to the first visit of <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.11.m11.1"><semantics id="S2.SS1.SSS0.Px4.p1.11.m11.1a"><mi id="S2.SS1.SSS0.Px4.p1.11.m11.1.1" xref="S2.SS1.SSS0.Px4.p1.11.m11.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.11.m11.1b"><ci id="S2.SS1.SSS0.Px4.p1.11.m11.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.11.m11.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.11.m11.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.11.m11.1d">italic_T</annotation></semantics></math> if <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.12.m12.1"><semantics id="S2.SS1.SSS0.Px4.p1.12.m12.1a"><mi id="S2.SS1.SSS0.Px4.p1.12.m12.1.1" xref="S2.SS1.SSS0.Px4.p1.12.m12.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.12.m12.1b"><ci id="S2.SS1.SSS0.Px4.p1.12.m12.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.12.m12.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.12.m12.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.12.m12.1d">italic_ρ</annotation></semantics></math> visits <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.13.m13.1"><semantics id="S2.SS1.SSS0.Px4.p1.13.m13.1a"><mi id="S2.SS1.SSS0.Px4.p1.13.m13.1.1" xref="S2.SS1.SSS0.Px4.p1.13.m13.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.13.m13.1b"><ci id="S2.SS1.SSS0.Px4.p1.13.m13.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.13.m13.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.13.m13.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.13.m13.1d">italic_T</annotation></semantics></math>, and <math alttext="\infty" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.14.m14.1"><semantics id="S2.SS1.SSS0.Px4.p1.14.m14.1a"><mi id="S2.SS1.SSS0.Px4.p1.14.m14.1.1" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.14.m14.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.14.m14.1b"><infinity id="S2.SS1.SSS0.Px4.p1.14.m14.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.14.m14.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.14.m14.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.14.m14.1d">∞</annotation></semantics></math> otherwise. The cost function is extended to histories in the expected way. Note that for two histories <math alttext="g,h" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.15.m15.2"><semantics id="S2.SS1.SSS0.Px4.p1.15.m15.2a"><mrow id="S2.SS1.SSS0.Px4.p1.15.m15.2.3.2" xref="S2.SS1.SSS0.Px4.p1.15.m15.2.3.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.15.m15.1.1" xref="S2.SS1.SSS0.Px4.p1.15.m15.1.1.cmml">g</mi><mo id="S2.SS1.SSS0.Px4.p1.15.m15.2.3.2.1" xref="S2.SS1.SSS0.Px4.p1.15.m15.2.3.1.cmml">,</mo><mi id="S2.SS1.SSS0.Px4.p1.15.m15.2.2" xref="S2.SS1.SSS0.Px4.p1.15.m15.2.2.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.15.m15.2b"><list id="S2.SS1.SSS0.Px4.p1.15.m15.2.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.15.m15.2.3.2"><ci id="S2.SS1.SSS0.Px4.p1.15.m15.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.15.m15.1.1">𝑔</ci><ci id="S2.SS1.SSS0.Px4.p1.15.m15.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.15.m15.2.2">ℎ</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.15.m15.2c">g,h</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.15.m15.2d">italic_g , italic_h</annotation></semantics></math> such that <math alttext="gh" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.16.m16.1"><semantics id="S2.SS1.SSS0.Px4.p1.16.m16.1a"><mrow id="S2.SS1.SSS0.Px4.p1.16.m16.1.1" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.16.m16.1.1.2" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1.2.cmml">g</mi><mo id="S2.SS1.SSS0.Px4.p1.16.m16.1.1.1" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px4.p1.16.m16.1.1.3" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.16.m16.1b"><apply id="S2.SS1.SSS0.Px4.p1.16.m16.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1"><times id="S2.SS1.SSS0.Px4.p1.16.m16.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1.1"></times><ci id="S2.SS1.SSS0.Px4.p1.16.m16.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1.2">𝑔</ci><ci id="S2.SS1.SSS0.Px4.p1.16.m16.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.16.m16.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.16.m16.1c">gh</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.16.m16.1d">italic_g italic_h</annotation></semantics></math> (i.e. <math alttext="g=h_{0}\ldots h_{k}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.17.m17.1"><semantics id="S2.SS1.SSS0.Px4.p1.17.m17.1a"><mrow id="S2.SS1.SSS0.Px4.p1.17.m17.1.1" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.2" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.2.cmml">g</mi><mo id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.1" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.cmml"><msub id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.cmml"><mi id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.2" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.2.cmml">h</mi><mn id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.3" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.1" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.3" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.3.cmml">…</mi><mo id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.1a" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.cmml"><mi id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.2" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.2.cmml">h</mi><mi id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.3" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.3.cmml">k</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.17.m17.1b"><apply id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1"><eq id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.1"></eq><ci id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.2">𝑔</ci><apply id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3"><times id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.1"></times><apply id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.2">ℎ</ci><cn id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.2.3">0</cn></apply><ci id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.3">…</ci><apply id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.1.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.2.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.2">ℎ</ci><ci id="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.3.cmml" xref="S2.SS1.SSS0.Px4.p1.17.m17.1.1.3.4.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.17.m17.1c">g=h_{0}\ldots h_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.17.m17.1d">italic_g = italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT … italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="h=h_{0}\ldots h_{\ell}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.18.m18.1"><semantics id="S2.SS1.SSS0.Px4.p1.18.m18.1a"><mrow id="S2.SS1.SSS0.Px4.p1.18.m18.1.1" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.2" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.2.cmml">h</mi><mo id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.1" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.cmml"><msub id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.cmml"><mi id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.2" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.2.cmml">h</mi><mn id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.3" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.1" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.3" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.3.cmml">…</mi><mo id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.1a" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.cmml"><mi id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.2" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.2.cmml">h</mi><mi id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.3" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.3.cmml">ℓ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.18.m18.1b"><apply id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1"><eq id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.1"></eq><ci id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.2">ℎ</ci><apply id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3"><times id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.1"></times><apply id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.2">ℎ</ci><cn id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.2.3">0</cn></apply><ci id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.3">…</ci><apply id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.1.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.2.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.2">ℎ</ci><ci id="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.3.cmml" xref="S2.SS1.SSS0.Px4.p1.18.m18.1.1.3.4.3">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.18.m18.1c">h=h_{0}\ldots h_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.18.m18.1d">italic_h = italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT … italic_h start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>, with <math alttext="k\leq\ell" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.19.m19.1"><semantics id="S2.SS1.SSS0.Px4.p1.19.m19.1a"><mrow id="S2.SS1.SSS0.Px4.p1.19.m19.1.1" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.19.m19.1.1.2" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1.2.cmml">k</mi><mo id="S2.SS1.SSS0.Px4.p1.19.m19.1.1.1" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1.1.cmml">≤</mo><mi id="S2.SS1.SSS0.Px4.p1.19.m19.1.1.3" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1.3.cmml">ℓ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.19.m19.1b"><apply id="S2.SS1.SSS0.Px4.p1.19.m19.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1"><leq id="S2.SS1.SSS0.Px4.p1.19.m19.1.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1.1"></leq><ci id="S2.SS1.SSS0.Px4.p1.19.m19.1.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1.2">𝑘</ci><ci id="S2.SS1.SSS0.Px4.p1.19.m19.1.1.3.cmml" xref="S2.SS1.SSS0.Px4.p1.19.m19.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.19.m19.1c">k\leq\ell</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.19.m19.1d">italic_k ≤ roman_ℓ</annotation></semantics></math>), it may happen that <math alttext="\textsf{cost}_{T}(g)=\infty" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.20.m20.1"><semantics id="S2.SS1.SSS0.Px4.p1.20.m20.1a"><mrow id="S2.SS1.SSS0.Px4.p1.20.m20.1.2" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.cmml"><mrow id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.cmml"><msub id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.2" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.2a.cmml">cost</mtext><mi id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.3" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.3.cmml">T</mi></msub><mo id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.1" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.3.2" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.cmml"><mo id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px4.p1.20.m20.1.1" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.1.cmml">g</mi><mo id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.1" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.1.cmml">=</mo><mi id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.3" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.20.m20.1b"><apply id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2"><eq id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.1"></eq><apply id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2"><times id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.1"></times><apply id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.2a.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.2">cost</mtext></ci><ci id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.2.2.3">𝑇</ci></apply><ci id="S2.SS1.SSS0.Px4.p1.20.m20.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.1">𝑔</ci></apply><infinity id="S2.SS1.SSS0.Px4.p1.20.m20.1.2.3.cmml" xref="S2.SS1.SSS0.Px4.p1.20.m20.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.20.m20.1c">\textsf{cost}_{T}(g)=\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.20.m20.1d">cost start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_g ) = ∞</annotation></semantics></math> and <math alttext="\textsf{cost}_{T}(h)<\infty" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.21.m21.1"><semantics id="S2.SS1.SSS0.Px4.p1.21.m21.1a"><mrow id="S2.SS1.SSS0.Px4.p1.21.m21.1.2" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.cmml"><mrow id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.cmml"><msub id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.2" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.2a.cmml">cost</mtext><mi id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.3" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.3.cmml">T</mi></msub><mo id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.1" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.3.2" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.cmml"><mo id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px4.p1.21.m21.1.1" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.1.cmml">h</mi><mo id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.1" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.1.cmml"><</mo><mi id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.3" mathvariant="normal" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.21.m21.1b"><apply id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2"><lt id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.1"></lt><apply id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2"><times id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.1"></times><apply id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.2a.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.2">cost</mtext></ci><ci id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.2.2.3">𝑇</ci></apply><ci id="S2.SS1.SSS0.Px4.p1.21.m21.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.1">ℎ</ci></apply><infinity id="S2.SS1.SSS0.Px4.p1.21.m21.1.2.3.cmml" xref="S2.SS1.SSS0.Px4.p1.21.m21.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.21.m21.1c">\textsf{cost}_{T}(h)<\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.21.m21.1d">cost start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_h ) < ∞</annotation></semantics></math> because <math alttext="h" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.22.m22.1"><semantics id="S2.SS1.SSS0.Px4.p1.22.m22.1a"><mi id="S2.SS1.SSS0.Px4.p1.22.m22.1.1" xref="S2.SS1.SSS0.Px4.p1.22.m22.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.22.m22.1b"><ci id="S2.SS1.SSS0.Px4.p1.22.m22.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.22.m22.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.22.m22.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.22.m22.1d">italic_h</annotation></semantics></math> visits <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.23.m23.1"><semantics id="S2.SS1.SSS0.Px4.p1.23.m23.1a"><mi id="S2.SS1.SSS0.Px4.p1.23.m23.1.1" xref="S2.SS1.SSS0.Px4.p1.23.m23.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.23.m23.1b"><ci id="S2.SS1.SSS0.Px4.p1.23.m23.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.23.m23.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.23.m23.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.23.m23.1d">italic_T</annotation></semantics></math> after <math alttext="g" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.24.m24.1"><semantics id="S2.SS1.SSS0.Px4.p1.24.m24.1a"><mi id="S2.SS1.SSS0.Px4.p1.24.m24.1.1" xref="S2.SS1.SSS0.Px4.p1.24.m24.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.24.m24.1b"><ci id="S2.SS1.SSS0.Px4.p1.24.m24.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.24.m24.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.24.m24.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.24.m24.1d">italic_g</annotation></semantics></math>. However, <math alttext="w(g)\leq w(h)=w(g)+w(h_{[k,\ell]})" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.25.m25.6"><semantics id="S2.SS1.SSS0.Px4.p1.25.m25.6a"><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.cmml"><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.cmml"><mi id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.2.cmml">w</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.3.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.cmml"><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.cmml">(</mo><mi id="S2.SS1.SSS0.Px4.p1.25.m25.3.3" xref="S2.SS1.SSS0.Px4.p1.25.m25.3.3.cmml">g</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.3.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.4" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.4.cmml">≤</mo><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.cmml"><mi id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.2.cmml">w</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.3.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.cmml"><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.cmml">(</mo><mi id="S2.SS1.SSS0.Px4.p1.25.m25.4.4" xref="S2.SS1.SSS0.Px4.p1.25.m25.4.4.cmml">h</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.5.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.6" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.6.cmml">=</mo><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.cmml"><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.cmml"><mi id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.2.cmml">w</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.3.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.cmml"><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.cmml">(</mo><mi id="S2.SS1.SSS0.Px4.p1.25.m25.5.5" xref="S2.SS1.SSS0.Px4.p1.25.m25.5.5.cmml">g</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.3.cmml">)</mo></mrow></mrow><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.2.cmml">+</mo><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.3" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.3.cmml">w</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1.1.cmml"><mi id="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.6.6.1.1.1.1.1.2.cmml">h</mi><mrow id="S2.SS1.SSS0.Px4.p1.25.m25.2.2.2.4" xref="S2.SS1.SSS0.Px4.p1.25.m25.2.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px4.p1.25.m25.2.2.2.4.1" stretchy="false" xref="S2.SS1.SSS0.Px4.p1.25.m25.2.2.2.3.cmml">[</mo><mi id="S2.SS1.SSS0.Px4.p1.25.m25.1.1.1.1" xref="S2.SS1.SSS0.Px4.p1.25.m25.1.1.1.1.cmml">k</mi><mo id="S2.SS1.SSS0.Px4.p1.25.m25.2.2.2.4.2" xref="S2.SS1.SSS0.Px4.p1.25.m25.2.2.2.3.cmml">,</mo><mi id="S2.SS1.SSS0.Px4.p1.25.m25.2.2.2.2" mathvariant="normal" 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not to <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px4.p1.26.m26.1"><semantics id="S2.SS1.SSS0.Px4.p1.26.m26.1a"><mi id="S2.SS1.SSS0.Px4.p1.26.m26.1.1" xref="S2.SS1.SSS0.Px4.p1.26.m26.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px4.p1.26.m26.1b"><ci id="S2.SS1.SSS0.Px4.p1.26.m26.1.1.cmml" xref="S2.SS1.SSS0.Px4.p1.26.m26.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px4.p1.26.m26.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px4.p1.26.m26.1d">italic_T</annotation></semantics></math>. In the proofs of this paper, weight and cost functions should not be confused.</p> </div> </section> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Stackelberg-Pareto Synthesis Problem</h3> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Stackelberg-Pareto games.</h4> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p1.19">Let <math alttext="t\in{\rm Nature}\setminus\{0\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p1.1.m1.1a"><mrow id="S2.SS2.SSS0.Px1.p1.1.m1.1.2" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p1.1.m1.1.2.2" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.2.2.cmml">t</mi><mo id="S2.SS2.SSS0.Px1.p1.1.m1.1.2.1" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S2.SS2.SSS0.Px1.p1.1.m1.1.2.3" 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id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.9" stretchy="false" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.2.m2.5b"><apply id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5"><eq id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.4.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.4"></eq><ci id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.5.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.5">𝐺</ci><vector id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.4.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3"><ci id="S2.SS2.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.1.1">𝐴</ci><apply id="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.2.m2.3.3.1.1.1.3">0</cn></apply><apply id="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2.2">𝑇</ci><cn id="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.2.m2.4.4.2.2.2.3">1</cn></apply><ci id="S2.SS2.SSS0.Px1.p1.2.m2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.2.2">…</ci><apply id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3.2.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3.2">𝑇</ci><ci id="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3.3.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m2.5.5.3.3.3.3">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.2.m2.5c">G=(A,T_{0},T_{1},\ldots,T_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.2.m2.5d">italic_G = ( italic_A , italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> where <math alttext="A" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.3.m3.1"><semantics id="S2.SS2.SSS0.Px1.p1.3.m3.1a"><mi id="S2.SS2.SSS0.Px1.p1.3.m3.1.1" xref="S2.SS2.SSS0.Px1.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.3.m3.1b"><ci id="S2.SS2.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.3.m3.1d">italic_A</annotation></semantics></math> is a game arena and <math alttext="T_{i}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.4.m4.1"><semantics id="S2.SS2.SSS0.Px1.p1.4.m4.1a"><msub id="S2.SS2.SSS0.Px1.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px1.p1.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p1.4.m4.1.1.2" xref="S2.SS2.SSS0.Px1.p1.4.m4.1.1.2.cmml">T</mi><mi id="S2.SS2.SSS0.Px1.p1.4.m4.1.1.3" xref="S2.SS2.SSS0.Px1.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.4.m4.1b"><apply id="S2.SS2.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.4.m4.1.1.2">𝑇</ci><ci id="S2.SS2.SSS0.Px1.p1.4.m4.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.4.m4.1c">T_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are targets for all <math alttext="i\in\{0,\ldots,t\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.5.m5.3"><semantics id="S2.SS2.SSS0.Px1.p1.5.m5.3a"><mrow id="S2.SS2.SSS0.Px1.p1.5.m5.3.4" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.cmml"><mi id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.2" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.2.cmml">i</mi><mo id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.1" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.1.cmml">∈</mo><mrow id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.2" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.1.cmml">{</mo><mn id="S2.SS2.SSS0.Px1.p1.5.m5.1.1" xref="S2.SS2.SSS0.Px1.p1.5.m5.1.1.cmml">0</mn><mo id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.2.2" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px1.p1.5.m5.2.2" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p1.5.m5.2.2.cmml">…</mi><mo id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.2.3" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px1.p1.5.m5.3.3" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.3.cmml">t</mi><mo id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.2.4" stretchy="false" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.5.m5.3b"><apply id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4"><in id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.1.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.1"></in><ci id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.2.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.2">𝑖</ci><set id="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.1.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.4.3.2"><cn id="S2.SS2.SSS0.Px1.p1.5.m5.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.5.m5.1.1">0</cn><ci id="S2.SS2.SSS0.Px1.p1.5.m5.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m5.2.2">…</ci><ci id="S2.SS2.SSS0.Px1.p1.5.m5.3.3.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m5.3.3">𝑡</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.5.m5.3c">i\in\{0,\ldots,t\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.5.m5.3d">italic_i ∈ { 0 , … , italic_t }</annotation></semantics></math>, such that <math alttext="T_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.6.m6.1"><semantics id="S2.SS2.SSS0.Px1.p1.6.m6.1a"><msub id="S2.SS2.SSS0.Px1.p1.6.m6.1.1" xref="S2.SS2.SSS0.Px1.p1.6.m6.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p1.6.m6.1.1.2" xref="S2.SS2.SSS0.Px1.p1.6.m6.1.1.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px1.p1.6.m6.1.1.3" xref="S2.SS2.SSS0.Px1.p1.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.6.m6.1b"><apply id="S2.SS2.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.6.m6.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px1.p1.6.m6.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.6.m6.1c">T_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.6.m6.1d">italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is Player 0’s target and <math alttext="T_{1},\ldots,T_{t}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.7.m7.3"><semantics id="S2.SS2.SSS0.Px1.p1.7.m7.3a"><mrow id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml"><msub id="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1" xref="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2" xref="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3" xref="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.3" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml">,</mo><mi id="S2.SS2.SSS0.Px1.p1.7.m7.1.1" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p1.7.m7.1.1.cmml">…</mi><mo id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.4" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml">,</mo><msub id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.cmml"><mi id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2.cmml">T</mi><mi id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.7.m7.3b"><list id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2"><apply id="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3">1</cn></apply><ci id="S2.SS2.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.1.1">…</ci><apply id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2">𝑇</ci><ci id="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.7.m7.3c">T_{1},\ldots,T_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.7.m7.3d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> are the <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.8.m8.1"><semantics id="S2.SS2.SSS0.Px1.p1.8.m8.1a"><mi id="S2.SS2.SSS0.Px1.p1.8.m8.1.1" xref="S2.SS2.SSS0.Px1.p1.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.8.m8.1b"><ci id="S2.SS2.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.8.m8.1d">italic_t</annotation></semantics></math> targets of Player 1. When <math alttext="A" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.9.m9.1"><semantics id="S2.SS2.SSS0.Px1.p1.9.m9.1a"><mi id="S2.SS2.SSS0.Px1.p1.9.m9.1.1" xref="S2.SS2.SSS0.Px1.p1.9.m9.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.9.m9.1b"><ci id="S2.SS2.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m9.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.9.m9.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.9.m9.1d">italic_A</annotation></semantics></math> is binary, we say that <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.10.m10.1"><semantics id="S2.SS2.SSS0.Px1.p1.10.m10.1a"><mi id="S2.SS2.SSS0.Px1.p1.10.m10.1.1" xref="S2.SS2.SSS0.Px1.p1.10.m10.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.10.m10.1b"><ci id="S2.SS2.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.10.m10.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.10.m10.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.10.m10.1d">italic_G</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px1.p1.19.2">binary</em>. The <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px1.p1.19.3">dimension</em> <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.11.m11.1"><semantics id="S2.SS2.SSS0.Px1.p1.11.m11.1a"><mi id="S2.SS2.SSS0.Px1.p1.11.m11.1.1" xref="S2.SS2.SSS0.Px1.p1.11.m11.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.11.m11.1b"><ci id="S2.SS2.SSS0.Px1.p1.11.m11.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.11.m11.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.11.m11.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.11.m11.1d">italic_t</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.12.m12.1"><semantics id="S2.SS2.SSS0.Px1.p1.12.m12.1a"><mi id="S2.SS2.SSS0.Px1.p1.12.m12.1.1" xref="S2.SS2.SSS0.Px1.p1.12.m12.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.12.m12.1b"><ci id="S2.SS2.SSS0.Px1.p1.12.m12.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m12.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.12.m12.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.12.m12.1d">italic_G</annotation></semantics></math> is the number of Player 1’s targets, and we denote by <math alttext="\textsf{Games}_{t}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.13.m13.1"><semantics id="S2.SS2.SSS0.Px1.p1.13.m13.1a"><msub id="S2.SS2.SSS0.Px1.p1.13.m13.1.1" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.13.m13.1.1.2" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1.2a.cmml">Games</mtext><mi id="S2.SS2.SSS0.Px1.p1.13.m13.1.1.3" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.13.m13.1b"><apply id="S2.SS2.SSS0.Px1.p1.13.m13.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.13.m13.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.13.m13.1.1.2a.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.13.m13.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1.2">Games</mtext></ci><ci id="S2.SS2.SSS0.Px1.p1.13.m13.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m13.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.13.m13.1c">\textsf{Games}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.13.m13.1d">Games start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> (resp. <math alttext="\textsf{BinGames}_{t}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.14.m14.1"><semantics id="S2.SS2.SSS0.Px1.p1.14.m14.1a"><msub id="S2.SS2.SSS0.Px1.p1.14.m14.1.1" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.14.m14.1.1.2" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1.2a.cmml">BinGames</mtext><mi id="S2.SS2.SSS0.Px1.p1.14.m14.1.1.3" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.14.m14.1b"><apply id="S2.SS2.SSS0.Px1.p1.14.m14.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.14.m14.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.14.m14.1.1.2a.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.14.m14.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1.2">BinGames</mtext></ci><ci id="S2.SS2.SSS0.Px1.p1.14.m14.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m14.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.14.m14.1c">\textsf{BinGames}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.14.m14.1d">BinGames start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>) the set of all (resp. binary) SP games with dimension <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.15.m15.1"><semantics id="S2.SS2.SSS0.Px1.p1.15.m15.1a"><mi id="S2.SS2.SSS0.Px1.p1.15.m15.1.1" xref="S2.SS2.SSS0.Px1.p1.15.m15.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.15.m15.1b"><ci id="S2.SS2.SSS0.Px1.p1.15.m15.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.15.m15.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.15.m15.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.15.m15.1d">italic_t</annotation></semantics></math>. The notations <math alttext="\textsf{Play}_{G}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.16.m16.1"><semantics id="S2.SS2.SSS0.Px1.p1.16.m16.1a"><msub id="S2.SS2.SSS0.Px1.p1.16.m16.1.1" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.16.m16.1.1.2" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1.2a.cmml">Play</mtext><mi id="S2.SS2.SSS0.Px1.p1.16.m16.1.1.3" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.16.m16.1b"><apply id="S2.SS2.SSS0.Px1.p1.16.m16.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.16.m16.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.16.m16.1.1.2a.cmml" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.16.m16.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1.2">Play</mtext></ci><ci id="S2.SS2.SSS0.Px1.p1.16.m16.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.16.m16.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.16.m16.1c">\textsf{Play}_{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.16.m16.1d">Play start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\textsf{Hist}_{G}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.17.m17.1"><semantics id="S2.SS2.SSS0.Px1.p1.17.m17.1a"><msub id="S2.SS2.SSS0.Px1.p1.17.m17.1.1" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.17.m17.1.1.2" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1.2a.cmml">Hist</mtext><mi id="S2.SS2.SSS0.Px1.p1.17.m17.1.1.3" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.17.m17.1b"><apply id="S2.SS2.SSS0.Px1.p1.17.m17.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.17.m17.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.17.m17.1.1.2a.cmml" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.17.m17.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1.2">Hist</mtext></ci><ci id="S2.SS2.SSS0.Px1.p1.17.m17.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.17.m17.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.17.m17.1c">\textsf{Hist}_{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.17.m17.1d">Hist start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> may be used instead of <math alttext="\textsf{Play}_{A}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.18.m18.1"><semantics id="S2.SS2.SSS0.Px1.p1.18.m18.1a"><msub id="S2.SS2.SSS0.Px1.p1.18.m18.1.1" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.18.m18.1.1.2" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1.2a.cmml">Play</mtext><mi id="S2.SS2.SSS0.Px1.p1.18.m18.1.1.3" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.18.m18.1b"><apply id="S2.SS2.SSS0.Px1.p1.18.m18.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.18.m18.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.18.m18.1.1.2a.cmml" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.18.m18.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1.2">Play</mtext></ci><ci id="S2.SS2.SSS0.Px1.p1.18.m18.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.18.m18.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.18.m18.1c">\textsf{Play}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.18.m18.1d">Play start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\textsf{Hist}_{A}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.19.m19.1"><semantics id="S2.SS2.SSS0.Px1.p1.19.m19.1a"><msub id="S2.SS2.SSS0.Px1.p1.19.m19.1.1" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.19.m19.1.1.2" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1.2a.cmml">Hist</mtext><mi id="S2.SS2.SSS0.Px1.p1.19.m19.1.1.3" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.19.m19.1b"><apply id="S2.SS2.SSS0.Px1.p1.19.m19.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.19.m19.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.19.m19.1.1.2a.cmml" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p1.19.m19.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1.2">Hist</mtext></ci><ci id="S2.SS2.SSS0.Px1.p1.19.m19.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.19.m19.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.19.m19.1c">\textsf{Hist}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.19.m19.1d">Hist start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p2.13">To distinguish the two players with respect to their targets, we introduce the following terminology. The <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px1.p2.13.1">cost</em> of a play <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p2.1.m1.1a"><mi id="S2.SS2.SSS0.Px1.p2.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.1.m1.1b"><ci id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.1.m1.1d">italic_ρ</annotation></semantics></math> is the tuple <math alttext="\textsf{cost}({\rho})\in\overline{{\rm Nature}}^{t}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.2.m2.1"><semantics id="S2.SS2.SSS0.Px1.p2.2.m2.1a"><mrow id="S2.SS2.SSS0.Px1.p2.2.m2.1.2" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.cmml"><mrow id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.2" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.2a.cmml">cost</mtext><mo id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.1" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.3.2" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.cmml"><mo id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p2.2.m2.1.1" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.1.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.1" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.1.cmml">∈</mo><msup id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.cmml"><mover accent="true" id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.cmml"><mi id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.2" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.2.cmml">Nature</mi><mo id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.1" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.1.cmml">¯</mo></mover><mi id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.3" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.2.m2.1b"><apply id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2"><in id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.1"></in><apply id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2"><times id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.1"></times><ci id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.2a.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.2.2">cost</mtext></ci><ci id="S2.SS2.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.1">𝜌</ci></apply><apply id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3">superscript</csymbol><apply id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2"><ci id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.1">¯</ci><ci id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.2.2">Nature</ci></apply><ci id="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.3.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.2.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.2.m2.1c">\textsf{cost}({\rho})\in\overline{{\rm Nature}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.2.m2.1d">cost ( italic_ρ ) ∈ over¯ start_ARG roman_Nature end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\textsf{cost}({\rho})=(\textsf{cost}_{T_{1}}(\rho),\ldots,\textsf{cost}_{T_{t}% }(\rho))" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.3.m3.6"><semantics id="S2.SS2.SSS0.Px1.p2.3.m3.6a"><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.6.6" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.cmml"><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.2a.cmml">cost</mtext><mo id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.1" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.3.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.cmml"><mo id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p2.3.m3.1.1" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.4.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.3" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.3.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.3.cmml"><mo id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.3.cmml">(</mo><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.cmml"><msub id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.2a.cmml">cost</mtext><msub id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.3" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.3.cmml"><mi id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.3.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.3.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.3.3" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.2.3.3.cmml">1</mn></msub></msub><mo id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.1" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.3.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.cmml"><mo id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p2.3.m3.2.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.2.2.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.3.m3.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.2.4" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.3.cmml">,</mo><mi id="S2.SS2.SSS0.Px1.p2.3.m3.4.4" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p2.3.m3.4.4.cmml">…</mi><mo id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.2.5" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.3.cmml">,</mo><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.2.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.6.6.2.2.2.cmml"><msub 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start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) , … , cost start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) )</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px1.p2.13.2">value</em> of a play <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.4.m4.1"><semantics id="S2.SS2.SSS0.Px1.p2.4.m4.1a"><mi id="S2.SS2.SSS0.Px1.p2.4.m4.1.1" xref="S2.SS2.SSS0.Px1.p2.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.4.m4.1b"><ci id="S2.SS2.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.4.m4.1d">italic_ρ</annotation></semantics></math> is a non-negative integer or <math alttext="\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.5.m5.1"><semantics id="S2.SS2.SSS0.Px1.p2.5.m5.1a"><mi id="S2.SS2.SSS0.Px1.p2.5.m5.1.1" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p2.5.m5.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.5.m5.1b"><infinity id="S2.SS2.SSS0.Px1.p2.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.5.m5.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.5.m5.1d">∞</annotation></semantics></math> defined by <math alttext="\textsf{val}(\rho)=\textsf{cost}_{T_{0}}(\rho)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.6.m6.2"><semantics id="S2.SS2.SSS0.Px1.p2.6.m6.2a"><mrow id="S2.SS2.SSS0.Px1.p2.6.m6.2.3" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.cmml"><mrow id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.2a.cmml">val</mtext><mo id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.1" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.3.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.cmml"><mo id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p2.6.m6.1.1" xref="S2.SS2.SSS0.Px1.p2.6.m6.1.1.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.1" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.1.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.cmml"><msub id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.2a.cmml">cost</mtext><msub id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.cmml"><mi id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.3" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.3.cmml">0</mn></msub></msub><mo id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.1" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.3.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.cmml"><mo id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p2.6.m6.2.2" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.2.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.6.m6.2b"><apply id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3"><eq id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.1"></eq><apply id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2"><times id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.1"></times><ci id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.2a.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.2.2">val</mtext></ci><ci id="S2.SS2.SSS0.Px1.p2.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.1.1">𝜌</ci></apply><apply id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3"><times id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.1"></times><apply id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.2a.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.2">cost</mtext></ci><apply id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.2">𝑇</ci><cn id="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.3.3.2.3.3">0</cn></apply></apply><ci id="S2.SS2.SSS0.Px1.p2.6.m6.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.2.2">𝜌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.6.m6.2c">\textsf{val}(\rho)=\textsf{cost}_{T_{0}}(\rho)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.6.m6.2d">val ( italic_ρ ) = cost start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ )</annotation></semantics></math>. The value can be viewed as the score of Player 0 and the cost as the score of Player 1. Both functions are extended to histories in the expected way. In the sequel, given a cost <math alttext="c\in\overline{{\rm Nature}}^{t}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.7.m7.1"><semantics id="S2.SS2.SSS0.Px1.p2.7.m7.1a"><mrow id="S2.SS2.SSS0.Px1.p2.7.m7.1.1" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.2" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.1" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.1.cmml">∈</mo><msup id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.cmml"><mover accent="true" id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.cmml"><mi id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.2" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.2.cmml">Nature</mi><mo id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.1" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.1.cmml">¯</mo></mover><mi id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.3" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.7.m7.1b"><apply id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1"><in id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.1"></in><ci id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3">superscript</csymbol><apply id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2"><ci id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.1">¯</ci><ci id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.2.2">Nature</ci></apply><ci id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.7.m7.1c">c\in\overline{{\rm Nature}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.7.m7.1d">italic_c ∈ over¯ start_ARG roman_Nature end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, we denote by <math alttext="c_{i}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.8.m8.1"><semantics id="S2.SS2.SSS0.Px1.p2.8.m8.1a"><msub id="S2.SS2.SSS0.Px1.p2.8.m8.1.1" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p2.8.m8.1.1.2" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1.2.cmml">c</mi><mi id="S2.SS2.SSS0.Px1.p2.8.m8.1.1.3" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.8.m8.1b"><apply id="S2.SS2.SSS0.Px1.p2.8.m8.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.8.m8.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.8.m8.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p2.8.m8.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.8.m8.1c">c_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.8.m8.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> the <math alttext="i" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.9.m9.1"><semantics id="S2.SS2.SSS0.Px1.p2.9.m9.1a"><mi id="S2.SS2.SSS0.Px1.p2.9.m9.1.1" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.9.m9.1b"><ci id="S2.SS2.SSS0.Px1.p2.9.m9.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.9.m9.1d">italic_i</annotation></semantics></math>-th component of <math alttext="c" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.10.m10.1"><semantics id="S2.SS2.SSS0.Px1.p2.10.m10.1a"><mi id="S2.SS2.SSS0.Px1.p2.10.m10.1.1" xref="S2.SS2.SSS0.Px1.p2.10.m10.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.10.m10.1b"><ci id="S2.SS2.SSS0.Px1.p2.10.m10.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.10.m10.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.10.m10.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.10.m10.1d">italic_c</annotation></semantics></math> and by <math alttext="c_{min}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.11.m11.1"><semantics id="S2.SS2.SSS0.Px1.p2.11.m11.1a"><msub id="S2.SS2.SSS0.Px1.p2.11.m11.1.1" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.2" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.2.cmml">c</mi><mrow id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.2" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.2.cmml">m</mi><mo id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.1" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.3" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.3.cmml">i</mi><mo id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.1a" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.4" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.4.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.11.m11.1b"><apply id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3"><times id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.1"></times><ci id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.2">𝑚</ci><ci id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.3">𝑖</ci><ci id="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.4.cmml" xref="S2.SS2.SSS0.Px1.p2.11.m11.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.11.m11.1c">c_{min}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.11.m11.1d">italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math> the component of <math alttext="c" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.12.m12.1"><semantics id="S2.SS2.SSS0.Px1.p2.12.m12.1a"><mi id="S2.SS2.SSS0.Px1.p2.12.m12.1.1" xref="S2.SS2.SSS0.Px1.p2.12.m12.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.12.m12.1b"><ci id="S2.SS2.SSS0.Px1.p2.12.m12.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.12.m12.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.12.m12.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.12.m12.1d">italic_c</annotation></semantics></math> that is minimum, i.e. <math alttext="c_{min}=\min\{c_{i}\mid i\in\{1,\ldots,t\}\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.13.m13.5"><semantics id="S2.SS2.SSS0.Px1.p2.13.m13.5a"><mrow id="S2.SS2.SSS0.Px1.p2.13.m13.5.5" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.cmml"><msub id="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3.cmml"><mi id="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3.2" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3.2.cmml">c</mi><mrow id="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3.3" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3.3.cmml"><mi id="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3.3.2" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.3.3.2.cmml">m</mi><mo 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id="S2.SS2.SSS0.Px1.p2.13.m13.5.5.1.1.1.1.2.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.1.1.1.1.2.2.3">𝑖</ci></apply><ci id="S2.SS2.SSS0.Px1.p2.13.m13.5.5.1.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.1.1.1.1.2.3">𝑖</ci></apply><set id="S2.SS2.SSS0.Px1.p2.13.m13.5.5.1.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.13.m13.5.5.1.1.1.1.3.2"><cn id="S2.SS2.SSS0.Px1.p2.13.m13.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p2.13.m13.1.1">1</cn><ci id="S2.SS2.SSS0.Px1.p2.13.m13.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.13.m13.2.2">…</ci><ci id="S2.SS2.SSS0.Px1.p2.13.m13.3.3.cmml" xref="S2.SS2.SSS0.Px1.p2.13.m13.3.3">𝑡</ci></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.13.m13.5c">c_{min}=\min\{c_{i}\mid i\in\{1,\ldots,t\}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.13.m13.5d">italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT = roman_min { italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_i ∈ { 1 , … , italic_t } }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p3"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p3.18">In an SP game, Player 0 wishes to minimize the value of a play with respect to the usual order <math alttext="<" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p3.1.m1.1a"><mo id="S2.SS2.SSS0.Px1.p3.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p3.1.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.1.m1.1b"><lt id="S2.SS2.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.1.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.1.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.1.m1.1d"><</annotation></semantics></math> on <math alttext="{\rm Nature}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.2.m2.1"><semantics id="S2.SS2.SSS0.Px1.p3.2.m2.1a"><mi id="S2.SS2.SSS0.Px1.p3.2.m2.1.1" xref="S2.SS2.SSS0.Px1.p3.2.m2.1.1.cmml">Nature</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.2.m2.1b"><ci id="S2.SS2.SSS0.Px1.p3.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.2.m2.1.1">Nature</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.2.m2.1c">{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.2.m2.1d">roman_Nature</annotation></semantics></math> extended to <math alttext="\overline{{\rm Nature}}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.3.m3.1"><semantics id="S2.SS2.SSS0.Px1.p3.3.m3.1a"><mover accent="true" id="S2.SS2.SSS0.Px1.p3.3.m3.1.1" xref="S2.SS2.SSS0.Px1.p3.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.3.m3.1.1.2" xref="S2.SS2.SSS0.Px1.p3.3.m3.1.1.2.cmml">Nature</mi><mo id="S2.SS2.SSS0.Px1.p3.3.m3.1.1.1" xref="S2.SS2.SSS0.Px1.p3.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.3.m3.1b"><apply id="S2.SS2.SSS0.Px1.p3.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.1.1"><ci id="S2.SS2.SSS0.Px1.p3.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.1.1.1">¯</ci><ci id="S2.SS2.SSS0.Px1.p3.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.1.1.2">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.3.m3.1c">\overline{{\rm Nature}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.3.m3.1d">over¯ start_ARG roman_Nature end_ARG</annotation></semantics></math> such that <math alttext="n<\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.4.m4.1"><semantics id="S2.SS2.SSS0.Px1.p3.4.m4.1a"><mrow id="S2.SS2.SSS0.Px1.p3.4.m4.1.1" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.2" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.2.cmml">n</mi><mo id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.1" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.1.cmml"><</mo><mi id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.3" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.4.m4.1b"><apply id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1"><lt id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.1"></lt><ci id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.2">𝑛</ci><infinity id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.4.m4.1c">n<\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.4.m4.1d">italic_n < ∞</annotation></semantics></math> for all <math alttext="n\in{\rm Nature}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.5.m5.1"><semantics id="S2.SS2.SSS0.Px1.p3.5.m5.1a"><mrow id="S2.SS2.SSS0.Px1.p3.5.m5.1.1" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.2" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.2.cmml">n</mi><mo id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.1" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.1.cmml">∈</mo><mi id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.3" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.5.m5.1b"><apply id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1"><in id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.1"></in><ci id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.2">𝑛</ci><ci id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.5.m5.1c">n\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.5.m5.1d">italic_n ∈ roman_Nature</annotation></semantics></math>. To compare the costs of Player <math alttext="1" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.6.m6.1"><semantics id="S2.SS2.SSS0.Px1.p3.6.m6.1a"><mn id="S2.SS2.SSS0.Px1.p3.6.m6.1.1" xref="S2.SS2.SSS0.Px1.p3.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.6.m6.1b"><cn id="S2.SS2.SSS0.Px1.p3.6.m6.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p3.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.6.m6.1d">1</annotation></semantics></math>, the following component-wise order is introduced. Let <math alttext="c,c^{\prime}\in\overline{{\rm Nature}}^{t}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.7.m7.2"><semantics id="S2.SS2.SSS0.Px1.p3.7.m7.2a"><mrow id="S2.SS2.SSS0.Px1.p3.7.m7.2.2" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.cmml"><mrow id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p3.7.m7.1.1" xref="S2.SS2.SSS0.Px1.p3.7.m7.1.1.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.2" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.2.cmml">,</mo><msup id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.2" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.3" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.2" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.2.cmml">∈</mo><msup id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.cmml"><mover accent="true" id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.cmml"><mi id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.2" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.2.cmml">Nature</mi><mo id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.1" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.1.cmml">¯</mo></mover><mi id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.3" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.7.m7.2b"><apply id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2"><in id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.2"></in><list id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1"><ci id="S2.SS2.SSS0.Px1.p3.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.1.1">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.1.1.1.3">′</ci></apply></list><apply id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3">superscript</csymbol><apply id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2"><ci id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.1">¯</ci><ci id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.2.2">Nature</ci></apply><ci id="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.7.m7.2.2.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.7.m7.2c">c,c^{\prime}\in\overline{{\rm Nature}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.7.m7.2d">italic_c , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ over¯ start_ARG roman_Nature end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> be two costs, we say that <math alttext="c\leq c^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.8.m8.1"><semantics id="S2.SS2.SSS0.Px1.p3.8.m8.1a"><mrow id="S2.SS2.SSS0.Px1.p3.8.m8.1.1" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.2" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.1" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.1.cmml">≤</mo><msup id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.2" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.3" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.8.m8.1b"><apply id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1"><leq id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.1"></leq><ci id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.8.m8.1c">c\leq c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.8.m8.1d">italic_c ≤ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> if <math alttext="c_{i}\leq c_{i}^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.9.m9.1"><semantics id="S2.SS2.SSS0.Px1.p3.9.m9.1a"><mrow id="S2.SS2.SSS0.Px1.p3.9.m9.1.1" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.cmml"><msub id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.2" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.2.cmml">c</mi><mi id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.3" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.1" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.1.cmml">≤</mo><msubsup id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.2" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.2.cmml">c</mi><mi id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.3" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.3.cmml">i</mi><mo id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.3" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.9.m9.1b"><apply id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1"><leq id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.1"></leq><apply id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.2.3">𝑖</ci></apply><apply id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3">superscript</csymbol><apply id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.3.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.2.3">𝑖</ci></apply><ci id="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.9.m9.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.9.m9.1c">c_{i}\leq c_{i}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.9.m9.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for all <math alttext="i\in\{1,\ldots,t\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.10.m10.3"><semantics id="S2.SS2.SSS0.Px1.p3.10.m10.3a"><mrow id="S2.SS2.SSS0.Px1.p3.10.m10.3.4" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.cmml"><mi id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.2" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.2.cmml">i</mi><mo id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.1" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.1.cmml">∈</mo><mrow id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.2" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.1.cmml">{</mo><mn id="S2.SS2.SSS0.Px1.p3.10.m10.1.1" xref="S2.SS2.SSS0.Px1.p3.10.m10.1.1.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.2.2" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px1.p3.10.m10.2.2" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p3.10.m10.2.2.cmml">…</mi><mo id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.2.3" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px1.p3.10.m10.3.3" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.3.cmml">t</mi><mo id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.2.4" stretchy="false" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.10.m10.3b"><apply id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.cmml" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4"><in id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.1.cmml" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.1"></in><ci id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.2.cmml" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.2">𝑖</ci><set id="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.4.3.2"><cn id="S2.SS2.SSS0.Px1.p3.10.m10.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p3.10.m10.1.1">1</cn><ci id="S2.SS2.SSS0.Px1.p3.10.m10.2.2.cmml" xref="S2.SS2.SSS0.Px1.p3.10.m10.2.2">…</ci><ci id="S2.SS2.SSS0.Px1.p3.10.m10.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.10.m10.3.3">𝑡</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.10.m10.3c">i\in\{1,\ldots,t\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.10.m10.3d">italic_i ∈ { 1 , … , italic_t }</annotation></semantics></math>. Moreover, we write <math alttext="c<c^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.11.m11.1"><semantics id="S2.SS2.SSS0.Px1.p3.11.m11.1a"><mrow id="S2.SS2.SSS0.Px1.p3.11.m11.1.1" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.2" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.1" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.1.cmml"><</mo><msup id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.2" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.3" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.11.m11.1b"><apply id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1"><lt id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.1"></lt><ci id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.11.m11.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.11.m11.1c">c<c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.11.m11.1d">italic_c < italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> if <math alttext="c\leq c^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.12.m12.1"><semantics id="S2.SS2.SSS0.Px1.p3.12.m12.1a"><mrow id="S2.SS2.SSS0.Px1.p3.12.m12.1.1" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.2" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.1" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.1.cmml">≤</mo><msup id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.2" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.3" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.12.m12.1b"><apply id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1"><leq id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.1"></leq><ci id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.12.m12.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.12.m12.1c">c\leq c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.12.m12.1d">italic_c ≤ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="c\neq c^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.13.m13.1"><semantics id="S2.SS2.SSS0.Px1.p3.13.m13.1a"><mrow id="S2.SS2.SSS0.Px1.p3.13.m13.1.1" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.2" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.1" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.1.cmml">≠</mo><msup id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.2" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.3" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.13.m13.1b"><apply id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1"><neq id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.1"></neq><ci id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.13.m13.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.13.m13.1c">c\neq c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.13.m13.1d">italic_c ≠ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Notice that the order defined on costs is not total. Given two plays with respective costs <math alttext="c" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.14.m14.1"><semantics id="S2.SS2.SSS0.Px1.p3.14.m14.1a"><mi id="S2.SS2.SSS0.Px1.p3.14.m14.1.1" xref="S2.SS2.SSS0.Px1.p3.14.m14.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.14.m14.1b"><ci id="S2.SS2.SSS0.Px1.p3.14.m14.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.14.m14.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.14.m14.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.14.m14.1d">italic_c</annotation></semantics></math> and <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.15.m15.1"><semantics id="S2.SS2.SSS0.Px1.p3.15.m15.1a"><msup id="S2.SS2.SSS0.Px1.p3.15.m15.1.1" xref="S2.SS2.SSS0.Px1.p3.15.m15.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.15.m15.1.1.2" xref="S2.SS2.SSS0.Px1.p3.15.m15.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.15.m15.1.1.3" xref="S2.SS2.SSS0.Px1.p3.15.m15.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.15.m15.1b"><apply id="S2.SS2.SSS0.Px1.p3.15.m15.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.15.m15.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.15.m15.1.1">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.15.m15.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.15.m15.1.1.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.15.m15.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.15.m15.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.15.m15.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.15.m15.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, if <math alttext="c<c^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.16.m16.1"><semantics id="S2.SS2.SSS0.Px1.p3.16.m16.1a"><mrow id="S2.SS2.SSS0.Px1.p3.16.m16.1.1" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.2" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.1" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.1.cmml"><</mo><msup id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.2" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.3" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.16.m16.1b"><apply id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1"><lt id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.1"></lt><ci id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.p3.16.m16.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.16.m16.1c">c<c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.16.m16.1d">italic_c < italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then Player <math alttext="1" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.17.m17.1"><semantics id="S2.SS2.SSS0.Px1.p3.17.m17.1a"><mn id="S2.SS2.SSS0.Px1.p3.17.m17.1.1" xref="S2.SS2.SSS0.Px1.p3.17.m17.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.17.m17.1b"><cn id="S2.SS2.SSS0.Px1.p3.17.m17.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p3.17.m17.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.17.m17.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.17.m17.1d">1</annotation></semantics></math> prefers the play with lower cost <math alttext="c" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.18.m18.1"><semantics id="S2.SS2.SSS0.Px1.p3.18.m18.1a"><mi id="S2.SS2.SSS0.Px1.p3.18.m18.1.1" xref="S2.SS2.SSS0.Px1.p3.18.m18.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.18.m18.1b"><ci id="S2.SS2.SSS0.Px1.p3.18.m18.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.18.m18.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.18.m18.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.18.m18.1d">italic_c</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Stackelberg-Pareto synthesis problem.</h4> <div class="ltx_para" id="S2.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p1.5">Given an SP game and a strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p1.1.m1.1a"><msub id="S2.SS2.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.2" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.3" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.1.m1.1b"><apply id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> for Player 0, we consider the set <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.2.m2.1"><semantics id="S2.SS2.SSS0.Px2.p1.2.m2.1a"><msub id="S2.SS2.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.2" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.2.cmml">C</mi><msub id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.2" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.3" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.2.m2.1b"><apply id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.2">𝐶</ci><apply id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.2.m2.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.2.m2.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> of costs of plays consistent with <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.3.m3.1"><semantics id="S2.SS2.SSS0.Px2.p1.3.m3.1a"><msub id="S2.SS2.SSS0.Px2.p1.3.m3.1.1" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.2" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.3" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.3.m3.1b"><apply id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.3.m3.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> that are <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px2.p1.5.1">Pareto-optimal</em> for Player <math alttext="1" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.4.m4.1"><semantics id="S2.SS2.SSS0.Px2.p1.4.m4.1a"><mn id="S2.SS2.SSS0.Px2.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.4.m4.1b"><cn id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.4.m4.1d">1</annotation></semantics></math>, i.e., minimal with respect to the order <math alttext="\leq" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.5.m5.1"><semantics id="S2.SS2.SSS0.Px2.p1.5.m5.1a"><mo id="S2.SS2.SSS0.Px2.p1.5.m5.1.1" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.5.m5.1b"><leq id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.5.m5.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.5.m5.1d">≤</annotation></semantics></math> on costs. Hence,</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="C_{\sigma_{0}}=\min\{\textsf{cost}({\rho})\mid\rho\in\textsf{Play}_{\sigma_{0}% }\}." class="ltx_Math" display="block" id="S2.Ex1.m1.3"><semantics id="S2.Ex1.m1.3a"><mrow id="S2.Ex1.m1.3.3.1" xref="S2.Ex1.m1.3.3.1.1.cmml"><mrow id="S2.Ex1.m1.3.3.1.1" xref="S2.Ex1.m1.3.3.1.1.cmml"><msub id="S2.Ex1.m1.3.3.1.1.3" xref="S2.Ex1.m1.3.3.1.1.3.cmml"><mi id="S2.Ex1.m1.3.3.1.1.3.2" xref="S2.Ex1.m1.3.3.1.1.3.2.cmml">C</mi><msub id="S2.Ex1.m1.3.3.1.1.3.3" xref="S2.Ex1.m1.3.3.1.1.3.3.cmml"><mi id="S2.Ex1.m1.3.3.1.1.3.3.2" xref="S2.Ex1.m1.3.3.1.1.3.3.2.cmml">σ</mi><mn id="S2.Ex1.m1.3.3.1.1.3.3.3" xref="S2.Ex1.m1.3.3.1.1.3.3.3.cmml">0</mn></msub></msub><mo id="S2.Ex1.m1.3.3.1.1.2" xref="S2.Ex1.m1.3.3.1.1.2.cmml">=</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1" xref="S2.Ex1.m1.3.3.1.1.1.2.cmml"><mi id="S2.Ex1.m1.2.2" xref="S2.Ex1.m1.2.2.cmml">min</mi><mo id="S2.Ex1.m1.3.3.1.1.1.1a" xref="S2.Ex1.m1.3.3.1.1.1.2.cmml">⁡</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.1" xref="S2.Ex1.m1.3.3.1.1.1.2.cmml"><mo id="S2.Ex1.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.2.cmml">{</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.1.1" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.cmml"><mrow id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.cmml"><mrow id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.2" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.2a.cmml">cost</mtext><mo id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.1" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.3.2" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.cmml"><mo id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.cmml">(</mo><mi id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">ρ</mi><mo id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.3.2.2" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.1" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.1.cmml">∣</mo><mi id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.3" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.3.cmml">ρ</mi></mrow><mo id="S2.Ex1.m1.3.3.1.1.1.1.1.1.1" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.1.cmml">∈</mo><msub id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.2" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.2a.cmml">Play</mtext><msub id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.cmml"><mi id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.2" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.2.cmml">σ</mi><mn id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.3" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><mo id="S2.Ex1.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><mo id="S2.Ex1.m1.3.3.1.2" lspace="0em" xref="S2.Ex1.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.3b"><apply id="S2.Ex1.m1.3.3.1.1.cmml" xref="S2.Ex1.m1.3.3.1"><eq id="S2.Ex1.m1.3.3.1.1.2.cmml" xref="S2.Ex1.m1.3.3.1.1.2"></eq><apply id="S2.Ex1.m1.3.3.1.1.3.cmml" xref="S2.Ex1.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.3">subscript</csymbol><ci id="S2.Ex1.m1.3.3.1.1.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.3.2">𝐶</ci><apply id="S2.Ex1.m1.3.3.1.1.3.3.cmml" xref="S2.Ex1.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.3.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.3.3">subscript</csymbol><ci id="S2.Ex1.m1.3.3.1.1.3.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.3.3.2">𝜎</ci><cn id="S2.Ex1.m1.3.3.1.1.3.3.3.cmml" type="integer" xref="S2.Ex1.m1.3.3.1.1.3.3.3">0</cn></apply></apply><apply id="S2.Ex1.m1.3.3.1.1.1.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1"><min id="S2.Ex1.m1.2.2.cmml" xref="S2.Ex1.m1.2.2"></min><apply id="S2.Ex1.m1.3.3.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1"><in id="S2.Ex1.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.1"></in><apply id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2"><csymbol cd="latexml" id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.1">conditional</csymbol><apply id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2"><times id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.1"></times><ci id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.2a.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.2.2">cost</mtext></ci><ci id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1">𝜌</ci></apply><ci id="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.2.3">𝜌</ci></apply><apply id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.2a.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.2">Play</mtext></ci><apply id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.2">𝜎</ci><cn id="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.Ex1.m1.3.3.1.1.1.1.1.1.3.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.3c">C_{\sigma_{0}}=\min\{\textsf{cost}({\rho})\mid\rho\in\textsf{Play}_{\sigma_{0}% }\}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.3d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_min { cost ( italic_ρ ) ∣ italic_ρ ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p1.12">Notice that <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.6.m1.1"><semantics id="S2.SS2.SSS0.Px2.p1.6.m1.1a"><msub id="S2.SS2.SSS0.Px2.p1.6.m1.1.1" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.2" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.2.cmml">C</mi><msub id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.2" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.3" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.6.m1.1b"><apply id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.2">𝐶</ci><apply id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.6.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.6.m1.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.6.m1.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is an antichain. A cost <math alttext="c" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.7.m2.1"><semantics id="S2.SS2.SSS0.Px2.p1.7.m2.1a"><mi id="S2.SS2.SSS0.Px2.p1.7.m2.1.1" xref="S2.SS2.SSS0.Px2.p1.7.m2.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.7.m2.1b"><ci id="S2.SS2.SSS0.Px2.p1.7.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.7.m2.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.7.m2.1d">italic_c</annotation></semantics></math> is said to be <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.8.m3.1"><semantics id="S2.SS2.SSS0.Px2.p1.8.m3.1a"><msub id="S2.SS2.SSS0.Px2.p1.8.m3.1.1" xref="S2.SS2.SSS0.Px2.p1.8.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.8.m3.1.1.2" xref="S2.SS2.SSS0.Px2.p1.8.m3.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.8.m3.1.1.3" xref="S2.SS2.SSS0.Px2.p1.8.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.8.m3.1b"><apply id="S2.SS2.SSS0.Px2.p1.8.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.8.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.8.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.8.m3.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.8.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.8.m3.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.8.m3.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.8.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.8.m3.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.8.m3.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px2.p1.12.1">-fixed Pareto-optimal</em> if <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.9.m4.1"><semantics id="S2.SS2.SSS0.Px2.p1.9.m4.1a"><mrow id="S2.SS2.SSS0.Px2.p1.9.m4.1.1" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.2" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.2.cmml">c</mi><mo id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.1" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.1.cmml">∈</mo><msub id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.2" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.2.cmml">C</mi><msub id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.cmml"><mi id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.2" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.3" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.9.m4.1b"><apply id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1"><in id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.1"></in><ci id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.2">𝑐</ci><apply id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.2">𝐶</ci><apply id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.1.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.2.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.9.m4.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.9.m4.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.9.m4.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, a play is said to be <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.10.m5.1"><semantics id="S2.SS2.SSS0.Px2.p1.10.m5.1a"><msub id="S2.SS2.SSS0.Px2.p1.10.m5.1.1" xref="S2.SS2.SSS0.Px2.p1.10.m5.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.10.m5.1.1.2" xref="S2.SS2.SSS0.Px2.p1.10.m5.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.10.m5.1.1.3" xref="S2.SS2.SSS0.Px2.p1.10.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.10.m5.1b"><apply id="S2.SS2.SSS0.Px2.p1.10.m5.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.10.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.10.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.10.m5.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.10.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.10.m5.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.10.m5.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.10.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.10.m5.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.10.m5.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>-fixed Pareto-optimal if its cost is <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.11.m6.1"><semantics id="S2.SS2.SSS0.Px2.p1.11.m6.1a"><msub id="S2.SS2.SSS0.Px2.p1.11.m6.1.1" xref="S2.SS2.SSS0.Px2.p1.11.m6.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.11.m6.1.1.2" xref="S2.SS2.SSS0.Px2.p1.11.m6.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.11.m6.1.1.3" xref="S2.SS2.SSS0.Px2.p1.11.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.11.m6.1b"><apply id="S2.SS2.SSS0.Px2.p1.11.m6.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.11.m6.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m6.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.11.m6.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m6.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.11.m6.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.11.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.11.m6.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.11.m6.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>-fixed Pareto-optimal. We will omit the mention of <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.12.m7.1"><semantics id="S2.SS2.SSS0.Px2.p1.12.m7.1a"><msub id="S2.SS2.SSS0.Px2.p1.12.m7.1.1" xref="S2.SS2.SSS0.Px2.p1.12.m7.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.12.m7.1.1.2" xref="S2.SS2.SSS0.Px2.p1.12.m7.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p1.12.m7.1.1.3" xref="S2.SS2.SSS0.Px2.p1.12.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.12.m7.1b"><apply id="S2.SS2.SSS0.Px2.p1.12.m7.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.12.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.12.m7.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.12.m7.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.12.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.12.m7.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p1.12.m7.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.12.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.12.m7.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.12.m7.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> when it is clear from context.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px2.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p2.11">The problem we study is the following one: given an SP game <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p2.1.m1.1a"><mi id="S2.SS2.SSS0.Px2.p2.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.1.m1.1b"><ci id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.1.m1.1d">italic_G</annotation></semantics></math> and a bound <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.2.m2.1"><semantics id="S2.SS2.SSS0.Px2.p2.2.m2.1a"><mrow id="S2.SS2.SSS0.Px2.p2.2.m2.1.1" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.cmml">B</mi><mo id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1.cmml">∈</mo><mi id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.2.m2.1b"><apply id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1"><in id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1"></in><ci id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2">𝐵</ci><ci id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, is there a strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.3.m3.1"><semantics id="S2.SS2.SSS0.Px2.p2.3.m3.1a"><msub id="S2.SS2.SSS0.Px2.p2.3.m3.1.1" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.3.m3.1b"><apply id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.3.m3.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.3.m3.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> for Player 0 such that, for all strategies <math alttext="\sigma_{1}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.4.m4.1"><semantics id="S2.SS2.SSS0.Px2.p2.4.m4.1a"><msub id="S2.SS2.SSS0.Px2.p2.4.m4.1.1" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.4.m4.1b"><apply id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.4.m4.1c">\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.4.m4.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> for Player 1, if the outcome <math alttext="\textsf{out}(\sigma_{0},\sigma_{1})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.5.m5.2"><semantics id="S2.SS2.SSS0.Px2.p2.5.m5.2a"><mrow id="S2.SS2.SSS0.Px2.p2.5.m5.2.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.4" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.4a.cmml">out</mtext><mo id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.3" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.3.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.3.cmml"><mo id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.3" stretchy="false" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.3.cmml">(</mo><msub id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.4" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.3.cmml">,</mo><msub id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.cmml"><mi id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.3" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.5" stretchy="false" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.5.m5.2b"><apply id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2"><times id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.3.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.3"></times><ci id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.4a.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.4"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.4.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.4">out</mtext></ci><interval closure="open" id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.3.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2"><apply id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.1.1.3">0</cn></apply><apply id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.1.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.5.m5.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.5.m5.2c">\textsf{out}(\sigma_{0},\sigma_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.5.m5.2d">out ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> is Pareto-optimal, then the value of the outcome is below <math alttext="B" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.6.m6.1"><semantics id="S2.SS2.SSS0.Px2.p2.6.m6.1a"><mi id="S2.SS2.SSS0.Px2.p2.6.m6.1.1" xref="S2.SS2.SSS0.Px2.p2.6.m6.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.6.m6.1b"><ci id="S2.SS2.SSS0.Px2.p2.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.6.m6.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.6.m6.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.6.m6.1d">italic_B</annotation></semantics></math>. This is equivalent to say that for all <math alttext="\rho\in\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.7.m7.1"><semantics id="S2.SS2.SSS0.Px2.p2.7.m7.1a"><mrow id="S2.SS2.SSS0.Px2.p2.7.m7.1.1" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.2" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.2.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.1" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.1.cmml">∈</mo><msub id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.2" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.2a.cmml">Play</mtext><msub id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.cmml"><mi id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.2" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.3" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.7.m7.1b"><apply id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1"><in id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.1"></in><ci id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.2">𝜌</ci><apply id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.2a.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.2">Play</mtext></ci><apply id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.1.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.2.cmml" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.7.m7.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.7.m7.1c">\rho\in\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.7.m7.1d">italic_ρ ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, if <math alttext="\textsf{cost}({\rho})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.8.m8.1"><semantics id="S2.SS2.SSS0.Px2.p2.8.m8.1a"><mrow id="S2.SS2.SSS0.Px2.p2.8.m8.1.2" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.2" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.2a.cmml">cost</mtext><mo id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.1" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.3.2" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.cmml"><mo id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px2.p2.8.m8.1.1" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.1.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.8.m8.1b"><apply id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2"><times id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.1.cmml" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.1"></times><ci id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.2a.cmml" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.8.m8.1.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.2.2">cost</mtext></ci><ci id="S2.SS2.SSS0.Px2.p2.8.m8.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.8.m8.1.1">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.8.m8.1c">\textsf{cost}({\rho})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.8.m8.1d">cost ( italic_ρ )</annotation></semantics></math> is <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.9.m9.1"><semantics id="S2.SS2.SSS0.Px2.p2.9.m9.1a"><msub id="S2.SS2.SSS0.Px2.p2.9.m9.1.1" xref="S2.SS2.SSS0.Px2.p2.9.m9.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p2.9.m9.1.1.2" xref="S2.SS2.SSS0.Px2.p2.9.m9.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p2.9.m9.1.1.3" xref="S2.SS2.SSS0.Px2.p2.9.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.9.m9.1b"><apply id="S2.SS2.SSS0.Px2.p2.9.m9.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.9.m9.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.9.m9.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.9.m9.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.9.m9.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p2.9.m9.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.9.m9.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.9.m9.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>-fixed Pareto-optimal, then <math alttext="\textsf{val}(\rho)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.10.m10.1"><semantics id="S2.SS2.SSS0.Px2.p2.10.m10.1a"><mrow id="S2.SS2.SSS0.Px2.p2.10.m10.1.2" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.2" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.2a.cmml">val</mtext><mo id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.1" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.3.2" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.cmml"><mo id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px2.p2.10.m10.1.1" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.1.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.10.m10.1b"><apply id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2"><times id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.1.cmml" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.1"></times><ci id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.2a.cmml" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS2.SSS0.Px2.p2.10.m10.1.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.2.2">val</mtext></ci><ci id="S2.SS2.SSS0.Px2.p2.10.m10.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.10.m10.1.1">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.10.m10.1c">\textsf{val}(\rho)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.10.m10.1d">val ( italic_ρ )</annotation></semantics></math> is below <math alttext="B" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.11.m11.1"><semantics id="S2.SS2.SSS0.Px2.p2.11.m11.1a"><mi id="S2.SS2.SSS0.Px2.p2.11.m11.1.1" xref="S2.SS2.SSS0.Px2.p2.11.m11.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.11.m11.1b"><ci id="S2.SS2.SSS0.Px2.p2.11.m11.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.11.m11.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.11.m11.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.11.m11.1d">italic_B</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S2.SS2.SSS0.Px2.p3"> <br class="ltx_break"/> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p3.2"><span class="ltx_text ltx_font_bold" id="S2.SS2.SSS0.Px2.p3.2.1">Problem. </span> The <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px2.p3.2.2">Stackelberg-Pareto Synthesis problem</em> (SPS problem) is to decide, given an SP game <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p3.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p3.1.m1.1a"><mi id="S2.SS2.SSS0.Px2.p3.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p3.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p3.1.m1.1b"><ci id="S2.SS2.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p3.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p3.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p3.1.m1.1d">italic_G</annotation></semantics></math> and a bound <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p3.2.m2.1"><semantics id="S2.SS2.SSS0.Px2.p3.2.m2.1a"><mrow id="S2.SS2.SSS0.Px2.p3.2.m2.1.1" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p3.2.m2.1.1.2" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1.2.cmml">B</mi><mo id="S2.SS2.SSS0.Px2.p3.2.m2.1.1.1" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1.1.cmml">∈</mo><mi id="S2.SS2.SSS0.Px2.p3.2.m2.1.1.3" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p3.2.m2.1b"><apply id="S2.SS2.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1"><in id="S2.SS2.SSS0.Px2.p3.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1.1"></in><ci id="S2.SS2.SSS0.Px2.p3.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1.2">𝐵</ci><ci id="S2.SS2.SSS0.Px2.p3.2.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p3.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p3.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p3.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, whether</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx1"> <tbody id="S2.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\exists\sigma_{0}\in\Sigma_{0},\forall\sigma_{1}\in\Sigma_{1},% \textsf{cost}({\textsf{out}(\sigma_{0},\sigma_{1})})\in C_{\sigma_{0}}% \Rightarrow\textsf{val}(\textsf{out}(\sigma_{0},\sigma_{1}))\leq B." class="ltx_Math" display="inline" id="S2.E1.m1.1"><semantics id="S2.E1.m1.1a"><mrow id="S2.E1.m1.1.1.1"><mrow id="S2.E1.m1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.3.cmml"><mrow id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml"><mrow id="S2.E1.m1.1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.1.2.cmml"><mo id="S2.E1.m1.1.1.1.1.1.1.2.1" rspace="0.167em" xref="S2.E1.m1.1.1.1.1.1.1.2.1.cmml">∃</mo><msub id="S2.E1.m1.1.1.1.1.1.1.2.2" xref="S2.E1.m1.1.1.1.1.1.1.2.2.cmml"><mi id="S2.E1.m1.1.1.1.1.1.1.2.2.2" xref="S2.E1.m1.1.1.1.1.1.1.2.2.2.cmml">σ</mi><mn id="S2.E1.m1.1.1.1.1.1.1.2.2.3" xref="S2.E1.m1.1.1.1.1.1.1.2.2.3.cmml">0</mn></msub></mrow><mo id="S2.E1.m1.1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml">∈</mo><msub id="S2.E1.m1.1.1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.1.1.3.cmml"><mi id="S2.E1.m1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.E1.m1.1.1.1.1.1.1.3.2.cmml">Σ</mi><mn id="S2.E1.m1.1.1.1.1.1.1.3.3" xref="S2.E1.m1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.E1.m1.1.1.1.1.2.3" xref="S2.E1.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.3.cmml"><mrow id="S2.E1.m1.1.1.1.1.2.2.1.1" xref="S2.E1.m1.1.1.1.1.2.2.1.1.cmml"><mrow id="S2.E1.m1.1.1.1.1.2.2.1.1.2" xref="S2.E1.m1.1.1.1.1.2.2.1.1.2.cmml"><mo id="S2.E1.m1.1.1.1.1.2.2.1.1.2.1" rspace="0.167em" xref="S2.E1.m1.1.1.1.1.2.2.1.1.2.1.cmml">∀</mo><msub id="S2.E1.m1.1.1.1.1.2.2.1.1.2.2" xref="S2.E1.m1.1.1.1.1.2.2.1.1.2.2.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2.1.1.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.1.1.2.2.2.cmml">σ</mi><mn id="S2.E1.m1.1.1.1.1.2.2.1.1.2.2.3" xref="S2.E1.m1.1.1.1.1.2.2.1.1.2.2.3.cmml">1</mn></msub></mrow><mo id="S2.E1.m1.1.1.1.1.2.2.1.1.1" xref="S2.E1.m1.1.1.1.1.2.2.1.1.1.cmml">∈</mo><msub id="S2.E1.m1.1.1.1.1.2.2.1.1.3" xref="S2.E1.m1.1.1.1.1.2.2.1.1.3.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2.1.1.3.2" mathvariant="normal" xref="S2.E1.m1.1.1.1.1.2.2.1.1.3.2.cmml">Σ</mi><mn id="S2.E1.m1.1.1.1.1.2.2.1.1.3.3" xref="S2.E1.m1.1.1.1.1.2.2.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S2.E1.m1.1.1.1.1.2.2.2.3" xref="S2.E1.m1.1.1.1.1.2.2.3a.cmml">,</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.cmml"><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2.1" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.E1.m1.1.1.1.1.2.2.2.2.1.3" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.3a.cmml">cost</mtext><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.1.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.cmml"><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.2" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.4" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.4a.cmml">out</mtext><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.3.cmml">⁢</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.3.cmml"><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.3" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.3.cmml">(</mo><msub id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.1.1.1.2.cmml">σ</mi><mn id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.4" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.3.cmml">,</mo><msub id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.2.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.2.2.cmml">σ</mi><mn id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.2.3" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.2.3.cmml">1</mn></msub><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.2.5" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.3" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.4" xref="S2.E1.m1.1.1.1.1.2.2.2.2.4.cmml">∈</mo><msub id="S2.E1.m1.1.1.1.1.2.2.2.2.5" xref="S2.E1.m1.1.1.1.1.2.2.2.2.5.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2.2.2.5.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.5.2.cmml">C</mi><msub id="S2.E1.m1.1.1.1.1.2.2.2.2.5.3" xref="S2.E1.m1.1.1.1.1.2.2.2.2.5.3.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2.2.2.5.3.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.5.3.2.cmml">σ</mi><mn id="S2.E1.m1.1.1.1.1.2.2.2.2.5.3.3" xref="S2.E1.m1.1.1.1.1.2.2.2.2.5.3.3.cmml">0</mn></msub></msub><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.6" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.2.2.2.6.cmml">⇒</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.E1.m1.1.1.1.1.2.2.2.2.2.3" xref="S2.E1.m1.1.1.1.1.2.2.2.2.2.3a.cmml">val</mtext><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2.2.1.1" xref="S2.E1.m1.1.1.1.1.2.2.2.2.2.1.1.1.cmml"><mo id="S2.E1.m1.1.1.1.1.2.2.2.2.2.1.1.2" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.1.1.1.1.2.2.2.2.2.1.1.1" 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id="S2.E1.m1.1.1.1.1.2.2.2.2f.cmml" xref="S2.E1.m1.1.1.1.1.2.2.2.2"></share><ci id="S2.E1.m1.1.1.1.1.2.2.2.2.8.cmml" xref="S2.E1.m1.1.1.1.1.2.2.2.2.8">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.1c">\displaystyle\exists\sigma_{0}\in\Sigma_{0},\forall\sigma_{1}\in\Sigma_{1},% \textsf{cost}({\textsf{out}(\sigma_{0},\sigma_{1})})\in C_{\sigma_{0}}% \Rightarrow\textsf{val}(\textsf{out}(\sigma_{0},\sigma_{1}))\leq B.</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.1d">∃ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , ∀ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , cost ( out ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⇒ val ( out ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) ≤ italic_B .</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 class="ltx_para" id="S2.SS2.SSS0.Px2.p4"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p4.2">Any strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p4.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p4.1.m1.1a"><msub id="S2.SS2.SSS0.Px2.p4.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p4.1.m1.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p4.1.m1.1.1.2" xref="S2.SS2.SSS0.Px2.p4.1.m1.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p4.1.m1.1.1.3" xref="S2.SS2.SSS0.Px2.p4.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p4.1.m1.1b"><apply id="S2.SS2.SSS0.Px2.p4.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p4.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p4.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p4.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p4.1.m1.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p4.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p4.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p4.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> satisfying (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.E1" title="In Stackelberg-Pareto synthesis problem. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">1</span></a>) is called a <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS0.Px2.p4.2.1">solution</em> and we denote it by <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p4.2.m2.2"><semantics id="S2.SS2.SSS0.Px2.p4.2.m2.2a"><mrow id="S2.SS2.SSS0.Px2.p4.2.m2.2.3" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.cmml"><msub id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.2" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.2.cmml"><mi id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.2.2" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.2.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.2.3" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.1" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.1.cmml">∈</mo><mrow id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.3" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.3.cmml"><mtext id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.3.2" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.3.2a.cmml">SPS</mtext><mo id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.3.1" xref="S2.SS2.SSS0.Px2.p4.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px2.p4.2.m2.2.3.3.3.2" 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id="S2.SS2.SSS0.Px2.p4.2.m2.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p4.2.m2.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math>. Our main result is the following theorem.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.1.1.1">Theorem 2.1</span></span></h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p1.1.1">The SPS problem is <span class="ltx_text ltx_font_sansserif" id="S2.Thmtheorem1.p1.1.1.1">NEXPTIME</span>-complete.</span></p> </div> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px2.p5"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p5.4">The non-deterministic algorithm is exponential in the number of targets <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p5.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p5.1.m1.1a"><mi id="S2.SS2.SSS0.Px2.p5.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p5.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p5.1.m1.1b"><ci id="S2.SS2.SSS0.Px2.p5.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p5.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p5.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p5.1.m1.1d">italic_t</annotation></semantics></math> and in the size of the binary encoding of the maximum weight <math alttext="W" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p5.2.m2.1"><semantics id="S2.SS2.SSS0.Px2.p5.2.m2.1a"><mi id="S2.SS2.SSS0.Px2.p5.2.m2.1.1" xref="S2.SS2.SSS0.Px2.p5.2.m2.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p5.2.m2.1b"><ci id="S2.SS2.SSS0.Px2.p5.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p5.2.m2.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p5.2.m2.1c">W</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p5.2.m2.1d">italic_W</annotation></semantics></math> and the bound <math alttext="B" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p5.3.m3.1"><semantics id="S2.SS2.SSS0.Px2.p5.3.m3.1a"><mi id="S2.SS2.SSS0.Px2.p5.3.m3.1.1" xref="S2.SS2.SSS0.Px2.p5.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p5.3.m3.1b"><ci id="S2.SS2.SSS0.Px2.p5.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p5.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p5.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p5.3.m3.1d">italic_B</annotation></semantics></math>. The general approach to obtain this <span class="ltx_text ltx_font_sansserif" id="S2.SS2.SSS0.Px2.p5.4.1">NEXPTIME</span>-membership is to show that when there is a solution <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p5.4.m4.2"><semantics id="S2.SS2.SSS0.Px2.p5.4.m4.2a"><mrow id="S2.SS2.SSS0.Px2.p5.4.m4.2.3" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.cmml"><msub id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.cmml"><mi id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.2" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.3" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.1" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.1.cmml">∈</mo><mrow id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.cmml"><mtext id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.2" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.1" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.2" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.1.cmml"><mo id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px2.p5.4.m4.1.1" xref="S2.SS2.SSS0.Px2.p5.4.m4.1.1.cmml">G</mi><mo id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.2.2" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px2.p5.4.m4.2.2" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.2.cmml">B</mi><mo id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p5.4.m4.2b"><apply id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3"><in id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.1.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.1"></in><apply id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.1.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.2">𝜎</ci><cn id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.2.3">0</cn></apply><apply id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3"><times id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.1.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.1"></times><ci id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.2a.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.2"><mtext id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.2.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.1.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.3.3.3.2"><ci id="S2.SS2.SSS0.Px2.p5.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.1.1">𝐺</ci><ci id="S2.SS2.SSS0.Px2.p5.4.m4.2.2.cmml" xref="S2.SS2.SSS0.Px2.p5.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p5.4.m4.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p5.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math>, then there exists one that is finite-memory and whose memory size is exponential. An important part of this paper is devoted to this proof. Then we show that such a strategy can be guessed and checked to be a solution in exponential time.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Example.</h4> <div class="ltx_para" id="S2.SS2.SSS0.Px3.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px3.p1.13">To provide a better understanding of the SPS problem, let us solve it on a specific example. The arena <math alttext="A" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.1.m1.1"><semantics id="S2.SS2.SSS0.Px3.p1.1.m1.1a"><mi id="S2.SS2.SSS0.Px3.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.1.m1.1b"><ci id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.1.m1.1d">italic_A</annotation></semantics></math> is displayed on Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.F1" title="Figure 1 ‣ Example. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">1</span></a> (left part) where the vertices controlled by Player 0 (resp. Player <math alttext="1" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.2.m2.1"><semantics id="S2.SS2.SSS0.Px3.p1.2.m2.1a"><mn id="S2.SS2.SSS0.Px3.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.2.m2.1b"><cn id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.2.m2.1d">1</annotation></semantics></math>) are represented as circles (resp. squares). The weights are indicated only if they are different from 1 (e.g., the edge <math alttext="(v_{0},v_{6})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.3.m3.2"><semantics id="S2.SS2.SSS0.Px3.p1.3.m3.2a"><mrow id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.3.cmml"><mo id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.3" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.3.cmml">(</mo><msub id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.2" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.3" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.4" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.3.cmml">,</mo><msub id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.cmml"><mi id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.3" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.3.cmml">6</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.5" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.3.m3.2b"><interval closure="open" id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2"><apply id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.1.3">0</cn></apply><apply id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.1.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.3.m3.2.2.2.2.3">6</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.3.m3.2c">(v_{0},v_{6})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.3.m3.2d">( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT )</annotation></semantics></math> has a weight of 1). The initial vertex is <math alttext="v_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.4.m4.1"><semantics id="S2.SS2.SSS0.Px3.p1.4.m4.1a"><msub id="S2.SS2.SSS0.Px3.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.4.m4.1b"><apply id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.4.m4.1c">v_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.4.m4.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. The target of Player 0 is <math alttext="T_{0}=\{v_{3},v_{9}\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.5.m5.2"><semantics id="S2.SS2.SSS0.Px3.p1.5.m5.2a"><mrow id="S2.SS2.SSS0.Px3.p1.5.m5.2.2" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.cmml"><msub id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.cmml"><mi id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.2" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.3" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.3" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.3.cmml">=</mo><mrow id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.3.cmml"><mo id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.3" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.3.cmml">{</mo><msub id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.3.cmml">3</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.4" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.3.cmml">,</mo><msub id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.cmml"><mi id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.3" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.3.cmml">9</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.5" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.5.m5.2b"><apply id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2"><eq id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.3"></eq><apply id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.1.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.2.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.4.3">0</cn></apply><set id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2"><apply id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.1.1.3">3</cn></apply><apply id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.1.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.5.m5.2.2.2.2.2.3">9</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.5.m5.2c">T_{0}=\{v_{3},v_{9}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.5.m5.2d">italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT }</annotation></semantics></math> and is represented by doubled vertices. Player <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.F1" title="Figure 1 ‣ Example. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">1</span></a> has three targets: <math alttext="T_{1}=\{v_{1},v_{8}\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.6.m6.2"><semantics id="S2.SS2.SSS0.Px3.p1.6.m6.2a"><mrow id="S2.SS2.SSS0.Px3.p1.6.m6.2.2" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.cmml"><msub id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.cmml"><mi id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.2" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.3" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.3.cmml">1</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.3" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.3.cmml">=</mo><mrow id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.3.cmml"><mo id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.3" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.3.cmml">{</mo><msub id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.4" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.3.cmml">,</mo><msub id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.cmml"><mi id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.3" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.3.cmml">8</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.5" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.6.m6.2b"><apply id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2"><eq id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.3"></eq><apply id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.1.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.2.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.4.3">1</cn></apply><set id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2"><apply id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.1.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.6.m6.2.2.2.2.2.3">8</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.6.m6.2c">T_{1}=\{v_{1},v_{8}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.6.m6.2d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT }</annotation></semantics></math>, <math alttext="T_{2}=\{v_{9}\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.7.m7.1"><semantics id="S2.SS2.SSS0.Px3.p1.7.m7.1a"><mrow id="S2.SS2.SSS0.Px3.p1.7.m7.1.1" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.2" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.3" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.3.cmml">2</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.2" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.2.cmml">=</mo><mrow id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.2.cmml"><mo id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.2.cmml">{</mo><msub id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.3.cmml">9</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.7.m7.1b"><apply id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1"><eq id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.2"></eq><apply id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.3">2</cn></apply><set id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1"><apply id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.3">9</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.7.m7.1c">T_{2}=\{v_{9}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.7.m7.1d">italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = { italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT }</annotation></semantics></math> and <math alttext="T_{3}=\{v_{2},v_{4}\}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.8.m8.2"><semantics id="S2.SS2.SSS0.Px3.p1.8.m8.2a"><mrow id="S2.SS2.SSS0.Px3.p1.8.m8.2.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.cmml"><msub id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.cmml"><mi id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.3" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.3.cmml">3</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.3" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.3.cmml">=</mo><mrow id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.3.cmml"><mo id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.3" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.3.cmml">{</mo><msub id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.4" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.3.cmml">,</mo><msub id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.cmml"><mi id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.3" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.3.cmml">4</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.5" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.8.m8.2b"><apply id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2"><eq id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.3"></eq><apply id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.1.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.2.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.4.3">3</cn></apply><set id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2"><apply id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.1.1.1.3">2</cn></apply><apply id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.1.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.8.m8.2.2.2.2.2.3">4</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.8.m8.2c">T_{3}=\{v_{2},v_{4}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.8.m8.2d">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = { italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT }</annotation></semantics></math>, that are represented using colors (green for <math alttext="T_{1}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.9.m9.1"><semantics id="S2.SS2.SSS0.Px3.p1.9.m9.1a"><msub id="S2.SS2.SSS0.Px3.p1.9.m9.1.1" xref="S2.SS2.SSS0.Px3.p1.9.m9.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.9.m9.1.1.2" xref="S2.SS2.SSS0.Px3.p1.9.m9.1.1.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p1.9.m9.1.1.3" xref="S2.SS2.SSS0.Px3.p1.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.9.m9.1b"><apply id="S2.SS2.SSS0.Px3.p1.9.m9.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.9.m9.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.9.m9.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.9.m9.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.9.m9.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p1.9.m9.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.9.m9.1c">T_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.9.m9.1d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, red for <math alttext="T_{2}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.10.m10.1"><semantics id="S2.SS2.SSS0.Px3.p1.10.m10.1a"><msub id="S2.SS2.SSS0.Px3.p1.10.m10.1.1" xref="S2.SS2.SSS0.Px3.p1.10.m10.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.10.m10.1.1.2" xref="S2.SS2.SSS0.Px3.p1.10.m10.1.1.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p1.10.m10.1.1.3" xref="S2.SS2.SSS0.Px3.p1.10.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.10.m10.1b"><apply id="S2.SS2.SSS0.Px3.p1.10.m10.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.10.m10.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.10.m10.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.10.m10.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.10.m10.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p1.10.m10.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.10.m10.1c">T_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.10.m10.1d">italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, blue for <math alttext="T_{3}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.11.m11.1"><semantics id="S2.SS2.SSS0.Px3.p1.11.m11.1a"><msub id="S2.SS2.SSS0.Px3.p1.11.m11.1.1" xref="S2.SS2.SSS0.Px3.p1.11.m11.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.11.m11.1.1.2" xref="S2.SS2.SSS0.Px3.p1.11.m11.1.1.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p1.11.m11.1.1.3" xref="S2.SS2.SSS0.Px3.p1.11.m11.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.11.m11.1b"><apply id="S2.SS2.SSS0.Px3.p1.11.m11.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.11.m11.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.11.m11.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.11.m11.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.11.m11.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p1.11.m11.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.11.m11.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.11.m11.1c">T_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.11.m11.1d">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>). Let us exhibit a solution <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.12.m12.1"><semantics id="S2.SS2.SSS0.Px3.p1.12.m12.1a"><msub id="S2.SS2.SSS0.Px3.p1.12.m12.1.1" xref="S2.SS2.SSS0.Px3.p1.12.m12.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.12.m12.1.1.2" xref="S2.SS2.SSS0.Px3.p1.12.m12.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px3.p1.12.m12.1.1.3" xref="S2.SS2.SSS0.Px3.p1.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.12.m12.1b"><apply id="S2.SS2.SSS0.Px3.p1.12.m12.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.12.m12.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.12.m12.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.12.m12.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.12.m12.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px3.p1.12.m12.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.12.m12.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.12.m12.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mbox{SPS}({G},{5})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.13.m13.2"><semantics id="S2.SS2.SSS0.Px3.p1.13.m13.2a"><mrow id="S2.SS2.SSS0.Px3.p1.13.m13.2.3" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.cmml"><mtext id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.2" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.2a.cmml">SPS</mtext><mo id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.1" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.2" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.1.cmml"><mo id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px3.p1.13.m13.1.1" xref="S2.SS2.SSS0.Px3.p1.13.m13.1.1.cmml">G</mi><mo id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.2.2" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.1.cmml">,</mo><mn id="S2.SS2.SSS0.Px3.p1.13.m13.2.2" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.2.cmml">5</mn><mo id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.13.m13.2b"><apply id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.cmml" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3"><times id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.1.cmml" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.1"></times><ci id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.2a.cmml" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.2"><mtext id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.2.cmml" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.2">SPS</mtext></ci><interval closure="open" id="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.1.cmml" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.3.3.2"><ci id="S2.SS2.SSS0.Px3.p1.13.m13.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.13.m13.1.1">𝐺</ci><cn id="S2.SS2.SSS0.Px3.p1.13.m13.2.2.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.13.m13.2.2">5</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.13.m13.2c">\mbox{SPS}({G},{5})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.13.m13.2d">SPS ( italic_G , 5 )</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S2.F1"><svg class="ltx_picture ltx_centering" height="244.09" id="S2.F1.1.pic1" overflow="visible" version="1.1" width="522.75"><g fill="#000000" stroke="#000000" transform="translate(0,244.09) matrix(1 0 0 -1 0 0) translate(40.16,0) translate(0,117.14)"><g stroke-width="0.8pt"><path d="M -9.84 -9.84 h 19.69 v 19.69 h -19.69 Z" style="fill:none"></path></g><g fill="#000000" stroke="#000000" stroke-width="0.8pt" transform="matrix(1.0 0.0 0.0 1.0 -5.54 -1.73)"><foreignobject height="8.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="11.08"><math alttext="v_{0}" class="ltx_Math" display="inline" 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start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 388.16 -60.79)"><foreignobject height="8.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="11.08"><math alttext="v_{9}" class="ltx_Math" display="inline" id="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">v</mi><mn id="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1.1.3" 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xref="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1.1.3">9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1c">v_{9}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.1.pic1.22.22.22.22.22.22.22.22.22.22.22.22.12.1.1.1.1.1.1.1.1.1.m1.1d">italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 388.51 -109.76)"><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="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1.1" mathvariant="normal" xref="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">⋮</mi><annotation-xml encoding="MathML-Content" id="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1.1">⋮</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1c">\vdots</annotation><annotation encoding="application/x-llamapun" id="S2.F1.1.pic1.23.23.23.23.23.23.23.23.23.23.23.23.13.1.1.1.1.1.1.1.1.1.m1.1d">⋮</annotation></semantics></math></foreignobject></g></g><g stroke-width="1.1381pt"><path d="M 443.2 109 L 456.56 96.97" style="fill:none"></path><g transform="matrix(0.74333 -0.66891 0.66891 0.74333 456.56 96.97)"><path d="M 7.74 0 C 5.45 0.43 1.72 1.72 -0.86 3.22 L -0.86 -3.22 C 1.72 -1.72 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style="stroke:none"></path></g></g><g stroke-width="1.1381pt"><path d="M 393.7 38.13 L 393.7 28.67" style="fill:none"></path><g transform="matrix(0.0 -1.0 1.0 0.0 393.7 28.67)"><path d="M 7.74 0 C 5.45 0.43 1.72 1.72 -0.86 3.22 L -0.86 -3.22 C 1.72 -1.72 5.45 -0.43 7.74 0" style="stroke:none"></path></g></g><g stroke-width="1.1381pt"><path d="M 393.7 2.7 L 393.7 -6.77" style="fill:none"></path><g transform="matrix(0.0 -1.0 1.0 0.0 393.7 -6.77)"><path d="M 7.74 0 C 5.45 0.43 1.72 1.72 -0.86 3.22 L -0.86 -3.22 C 1.72 -1.72 5.45 -0.43 7.74 0" style="stroke:none"></path></g></g><g stroke-width="1.1381pt"><path d="M 393.7 -32.74 L 393.7 -42.2" style="fill:none"></path><g transform="matrix(0.0 -1.0 1.0 0.0 393.7 -42.2)"><path d="M 7.74 0 C 5.45 0.43 1.72 1.72 -0.86 3.22 L -0.86 -3.22 C 1.72 -1.72 5.45 -0.43 7.74 0" style="stroke:none"></path></g></g><g stroke-width="1.1381pt"><path d="M 393.7 -68.17 L 393.7 -87.45" style="fill:none"></path><g transform="matrix(0.0 -1.0 1.0 0.0 393.7 -87.45)"><path d="M 7.74 0 C 5.45 0.43 1.72 1.72 -0.86 3.22 L -0.86 -3.22 C 1.72 -1.72 5.45 -0.43 7.74 0" style="stroke:none"></path></g></g></g></svg> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Arena <math alttext="A" class="ltx_Math" display="inline" id="S2.F1.3.m1.1"><semantics id="S2.F1.3.m1.1b"><mi id="S2.F1.3.m1.1.1" xref="S2.F1.3.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.F1.3.m1.1c"><ci id="S2.F1.3.m1.1.1.cmml" xref="S2.F1.3.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.3.m1.1d">A</annotation><annotation encoding="application/x-llamapun" id="S2.F1.3.m1.1e">italic_A</annotation></semantics></math> (on the left) – Witness tree (on the right)</figcaption> </figure> <div class="ltx_para" id="S2.SS2.SSS0.Px3.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px3.p2.18">We define <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.1.m1.1"><semantics id="S2.SS2.SSS0.Px3.p2.1.m1.1a"><msub id="S2.SS2.SSS0.Px3.p2.1.m1.1.1" xref="S2.SS2.SSS0.Px3.p2.1.m1.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.1.m1.1.1.2" xref="S2.SS2.SSS0.Px3.p2.1.m1.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px3.p2.1.m1.1.1.3" xref="S2.SS2.SSS0.Px3.p2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.1.m1.1b"><apply id="S2.SS2.SSS0.Px3.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.1.m1.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px3.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> as the strategy that always moves from <math alttext="v_{3}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.2.m2.1"><semantics id="S2.SS2.SSS0.Px3.p2.2.m2.1a"><msub id="S2.SS2.SSS0.Px3.p2.2.m2.1.1" xref="S2.SS2.SSS0.Px3.p2.2.m2.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.2.m2.1.1.2" xref="S2.SS2.SSS0.Px3.p2.2.m2.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.2.m2.1.1.3" xref="S2.SS2.SSS0.Px3.p2.2.m2.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.2.m2.1b"><apply id="S2.SS2.SSS0.Px3.p2.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.2.m2.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.2.m2.1c">v_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.2.m2.1d">italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="v_{4}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.3.m3.1"><semantics id="S2.SS2.SSS0.Px3.p2.3.m3.1a"><msub id="S2.SS2.SSS0.Px3.p2.3.m3.1.1" xref="S2.SS2.SSS0.Px3.p2.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.3.m3.1.1.2" xref="S2.SS2.SSS0.Px3.p2.3.m3.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.3.m3.1.1.3" xref="S2.SS2.SSS0.Px3.p2.3.m3.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.3.m3.1b"><apply id="S2.SS2.SSS0.Px3.p2.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.3.m3.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.3.m3.1c">v_{4}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.3.m3.1d">italic_v start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>, and that loops once on <math alttext="v_{6}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.4.m4.1"><semantics id="S2.SS2.SSS0.Px3.p2.4.m4.1a"><msub id="S2.SS2.SSS0.Px3.p2.4.m4.1.1" xref="S2.SS2.SSS0.Px3.p2.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.4.m4.1.1.2" xref="S2.SS2.SSS0.Px3.p2.4.m4.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.4.m4.1.1.3" xref="S2.SS2.SSS0.Px3.p2.4.m4.1.1.3.cmml">6</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.4.m4.1b"><apply id="S2.SS2.SSS0.Px3.p2.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.4.m4.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.4.m4.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.4.m4.1c">v_{6}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.4.m4.1d">italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT</annotation></semantics></math> and then moves to <math alttext="v_{7}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.5.m5.1"><semantics id="S2.SS2.SSS0.Px3.p2.5.m5.1a"><msub id="S2.SS2.SSS0.Px3.p2.5.m5.1.1" xref="S2.SS2.SSS0.Px3.p2.5.m5.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.5.m5.1.1.2" xref="S2.SS2.SSS0.Px3.p2.5.m5.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.5.m5.1.1.3" xref="S2.SS2.SSS0.Px3.p2.5.m5.1.1.3.cmml">7</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.5.m5.1b"><apply id="S2.SS2.SSS0.Px3.p2.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.5.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.5.m5.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.5.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.5.m5.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.5.m5.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.5.m5.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.5.m5.1c">v_{7}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.5.m5.1d">italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT</annotation></semantics></math>. The plays consistent with <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.6.m6.1"><semantics id="S2.SS2.SSS0.Px3.p2.6.m6.1a"><msub id="S2.SS2.SSS0.Px3.p2.6.m6.1.1" xref="S2.SS2.SSS0.Px3.p2.6.m6.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.6.m6.1.1.2" xref="S2.SS2.SSS0.Px3.p2.6.m6.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px3.p2.6.m6.1.1.3" xref="S2.SS2.SSS0.Px3.p2.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.6.m6.1b"><apply id="S2.SS2.SSS0.Px3.p2.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.6.m6.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.6.m6.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.6.m6.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.6.m6.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px3.p2.6.m6.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.6.m6.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.6.m6.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> are <math alttext="v_{0}v_{1}v_{2}^{\omega}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.7.m7.1"><semantics id="S2.SS2.SSS0.Px3.p2.7.m7.1a"><mrow id="S2.SS2.SSS0.Px3.p2.7.m7.1.1" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.2" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.3" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.1" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.1.cmml">⁢</mo><msub id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.2" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.3" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.1a" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.1.cmml">⁢</mo><msubsup id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.cmml"><mi id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.2.2" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.2.3" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.2.3.cmml">2</mn><mi id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.3" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.3.cmml">ω</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.7.m7.1b"><apply id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1"><times id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.1"></times><apply id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.2.3">0</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.3.3">1</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.1.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4">superscript</csymbol><apply id="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4.2.cmml" xref="S2.SS2.SSS0.Px3.p2.7.m7.1.1.4"><csymbol cd="ambiguous" 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id="S2.SS2.SSS0.Px3.p2.9.m9.1.1.6.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.9.m9.1.1.6.2.3">8</cn></apply><ci id="S2.SS2.SSS0.Px3.p2.9.m9.1.1.6.3.cmml" xref="S2.SS2.SSS0.Px3.p2.9.m9.1.1.6.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.9.m9.1c">v_{0}v_{6}v_{6}v_{7}v_{8}^{\omega}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.9.m9.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="v_{0}v_{6}v_{6}v_{7}v_{9}^{\omega}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.10.m10.1"><semantics id="S2.SS2.SSS0.Px3.p2.10.m10.1a"><mrow id="S2.SS2.SSS0.Px3.p2.10.m10.1.1" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.2" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.3" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.2" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.3" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.3.cmml">6</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1a" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.cmml"><mi id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.2" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.3" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.3.cmml">6</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1b" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.cmml"><mi id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.2" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.3" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.3.cmml">7</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1c" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1.cmml">⁢</mo><msubsup id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.cmml"><mi id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.2" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.3" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.3.cmml">9</mn><mi id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.3" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.3.cmml">ω</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.10.m10.1b"><apply id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1"><times id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.1"></times><apply id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.2.3">0</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.3.3">6</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.2.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.4.3">6</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.2.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.5.3">7</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6">superscript</csymbol><apply id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.1.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.2.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.2.3">9</cn></apply><ci id="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.3.cmml" xref="S2.SS2.SSS0.Px3.p2.10.m10.1.1.6.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.10.m10.1c">v_{0}v_{6}v_{6}v_{7}v_{9}^{\omega}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.10.m10.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math>. The Pareto-optimal plays are <math alttext="v_{0}v_{1}(v_{3}v_{4})^{\omega}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.11.m11.1"><semantics id="S2.SS2.SSS0.Px3.p2.11.m11.1a"><mrow id="S2.SS2.SSS0.Px3.p2.11.m11.1.1" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.3" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.3.2" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.3.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.3.3" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.3.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.2" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.2.cmml">⁢</mo><msub id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.4" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.4.cmml"><mi id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.4.2" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.4.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.4.3" xref="S2.SS2.SSS0.Px3.p2.11.m11.1.1.4.3.cmml">1</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.11.m11.1.1.2a" 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6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math> with respective costs <math alttext="(4,\infty,7)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.13.m13.3"><semantics id="S2.SS2.SSS0.Px3.p2.13.m13.3a"><mrow id="S2.SS2.SSS0.Px3.p2.13.m13.3.4.2" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.4.1.cmml"><mo id="S2.SS2.SSS0.Px3.p2.13.m13.3.4.2.1" stretchy="false" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.4.1.cmml">(</mo><mn id="S2.SS2.SSS0.Px3.p2.13.m13.1.1" xref="S2.SS2.SSS0.Px3.p2.13.m13.1.1.cmml">4</mn><mo id="S2.SS2.SSS0.Px3.p2.13.m13.3.4.2.2" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.4.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px3.p2.13.m13.2.2" mathvariant="normal" xref="S2.SS2.SSS0.Px3.p2.13.m13.2.2.cmml">∞</mi><mo id="S2.SS2.SSS0.Px3.p2.13.m13.3.4.2.3" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.4.1.cmml">,</mo><mn id="S2.SS2.SSS0.Px3.p2.13.m13.3.3" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.3.cmml">7</mn><mo id="S2.SS2.SSS0.Px3.p2.13.m13.3.4.2.4" stretchy="false" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.13.m13.3b"><vector id="S2.SS2.SSS0.Px3.p2.13.m13.3.4.1.cmml" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.4.2"><cn id="S2.SS2.SSS0.Px3.p2.13.m13.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.13.m13.1.1">4</cn><infinity id="S2.SS2.SSS0.Px3.p2.13.m13.2.2.cmml" xref="S2.SS2.SSS0.Px3.p2.13.m13.2.2"></infinity><cn id="S2.SS2.SSS0.Px3.p2.13.m13.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.13.m13.3.3">7</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.13.m13.3c">(4,\infty,7)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.13.m13.3d">( 4 , ∞ , 7 )</annotation></semantics></math> and <math alttext="(\infty,4,\infty)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.14.m14.3"><semantics id="S2.SS2.SSS0.Px3.p2.14.m14.3a"><mrow id="S2.SS2.SSS0.Px3.p2.14.m14.3.4.2" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.4.1.cmml"><mo id="S2.SS2.SSS0.Px3.p2.14.m14.3.4.2.1" stretchy="false" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.4.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px3.p2.14.m14.1.1" mathvariant="normal" xref="S2.SS2.SSS0.Px3.p2.14.m14.1.1.cmml">∞</mi><mo id="S2.SS2.SSS0.Px3.p2.14.m14.3.4.2.2" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.4.1.cmml">,</mo><mn id="S2.SS2.SSS0.Px3.p2.14.m14.2.2" xref="S2.SS2.SSS0.Px3.p2.14.m14.2.2.cmml">4</mn><mo id="S2.SS2.SSS0.Px3.p2.14.m14.3.4.2.3" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.4.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px3.p2.14.m14.3.3" mathvariant="normal" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.3.cmml">∞</mi><mo id="S2.SS2.SSS0.Px3.p2.14.m14.3.4.2.4" stretchy="false" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.14.m14.3b"><vector id="S2.SS2.SSS0.Px3.p2.14.m14.3.4.1.cmml" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.4.2"><infinity id="S2.SS2.SSS0.Px3.p2.14.m14.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.14.m14.1.1"></infinity><cn id="S2.SS2.SSS0.Px3.p2.14.m14.2.2.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.14.m14.2.2">4</cn><infinity id="S2.SS2.SSS0.Px3.p2.14.m14.3.3.cmml" xref="S2.SS2.SSS0.Px3.p2.14.m14.3.3"></infinity></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.14.m14.3c">(\infty,4,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.14.m14.3d">( ∞ , 4 , ∞ )</annotation></semantics></math>, and they both yield a value less than or equal to 5. Notice that <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.15.m15.1"><semantics id="S2.SS2.SSS0.Px3.p2.15.m15.1a"><msub id="S2.SS2.SSS0.Px3.p2.15.m15.1.1" xref="S2.SS2.SSS0.Px3.p2.15.m15.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.15.m15.1.1.2" xref="S2.SS2.SSS0.Px3.p2.15.m15.1.1.2.cmml">σ</mi><mn id="S2.SS2.SSS0.Px3.p2.15.m15.1.1.3" xref="S2.SS2.SSS0.Px3.p2.15.m15.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.15.m15.1b"><apply id="S2.SS2.SSS0.Px3.p2.15.m15.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.15.m15.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.15.m15.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.15.m15.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.15.m15.1.1.2">𝜎</ci><cn id="S2.SS2.SSS0.Px3.p2.15.m15.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.15.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.15.m15.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.15.m15.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> has to loop once on <math alttext="v_{6}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.16.m16.1"><semantics id="S2.SS2.SSS0.Px3.p2.16.m16.1a"><msub id="S2.SS2.SSS0.Px3.p2.16.m16.1.1" xref="S2.SS2.SSS0.Px3.p2.16.m16.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p2.16.m16.1.1.2" xref="S2.SS2.SSS0.Px3.p2.16.m16.1.1.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.16.m16.1.1.3" xref="S2.SS2.SSS0.Px3.p2.16.m16.1.1.3.cmml">6</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.16.m16.1b"><apply id="S2.SS2.SSS0.Px3.p2.16.m16.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.16.m16.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.16.m16.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.16.m16.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.16.m16.1.1.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.16.m16.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.16.m16.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.16.m16.1c">v_{6}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.16.m16.1d">italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT</annotation></semantics></math>, i.e., it is not memoryless<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>One can prove that there exists no memoryless solution.</span></span></span>, otherwise the consistent play <math alttext="v_{0}v_{6}v_{7}v_{8}^{\omega}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.17.m17.1"><semantics id="S2.SS2.SSS0.Px3.p2.17.m17.1a"><mrow id="S2.SS2.SSS0.Px3.p2.17.m17.1.1" 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id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.3" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.3.cmml">7</mn></msub><mo id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.1b" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.1.cmml">⁢</mo><msubsup id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.cmml"><mi id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.2" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.3" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.3.cmml">8</mn><mi id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.3" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.3.cmml">ω</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.17.m17.1b"><apply id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1"><times id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.1"></times><apply id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.2.3">0</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.3.3">6</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.1.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.2.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.4.3">7</cn></apply><apply id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.1.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5">superscript</csymbol><apply id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.1.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.2.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.2.3">8</cn></apply><ci id="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.3.cmml" xref="S2.SS2.SSS0.Px3.p2.17.m17.1.1.5.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.17.m17.1c">v_{0}v_{6}v_{7}v_{8}^{\omega}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.17.m17.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math> has a Pareto-optimal cost of <math alttext="(3,\infty,\infty)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p2.18.m18.3"><semantics id="S2.SS2.SSS0.Px3.p2.18.m18.3a"><mrow id="S2.SS2.SSS0.Px3.p2.18.m18.3.4.2" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.4.1.cmml"><mo id="S2.SS2.SSS0.Px3.p2.18.m18.3.4.2.1" stretchy="false" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.4.1.cmml">(</mo><mn id="S2.SS2.SSS0.Px3.p2.18.m18.1.1" xref="S2.SS2.SSS0.Px3.p2.18.m18.1.1.cmml">3</mn><mo id="S2.SS2.SSS0.Px3.p2.18.m18.3.4.2.2" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.4.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px3.p2.18.m18.2.2" mathvariant="normal" xref="S2.SS2.SSS0.Px3.p2.18.m18.2.2.cmml">∞</mi><mo id="S2.SS2.SSS0.Px3.p2.18.m18.3.4.2.3" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.4.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px3.p2.18.m18.3.3" mathvariant="normal" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.3.cmml">∞</mi><mo id="S2.SS2.SSS0.Px3.p2.18.m18.3.4.2.4" stretchy="false" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p2.18.m18.3b"><vector id="S2.SS2.SSS0.Px3.p2.18.m18.3.4.1.cmml" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.4.2"><cn id="S2.SS2.SSS0.Px3.p2.18.m18.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p2.18.m18.1.1">3</cn><infinity id="S2.SS2.SSS0.Px3.p2.18.m18.2.2.cmml" xref="S2.SS2.SSS0.Px3.p2.18.m18.2.2"></infinity><infinity id="S2.SS2.SSS0.Px3.p2.18.m18.3.3.cmml" xref="S2.SS2.SSS0.Px3.p2.18.m18.3.3"></infinity></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p2.18.m18.3c">(3,\infty,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p2.18.m18.3d">( 3 , ∞ , ∞ )</annotation></semantics></math> and an infinite value.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px3.p3"> <p class="ltx_p" id="S2.SS2.SSS0.Px3.p3.6">Interestingly, the Boolean version of this game does not admit any solution. In this case, given a target, the player’s goal is simply to visit it (and not to minimize the cost to reach it). That is, the Boolean version is equivalent to the quantitative one with all weights and the bound <math alttext="B" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p3.1.m1.1"><semantics id="S2.SS2.SSS0.Px3.p3.1.m1.1a"><mi id="S2.SS2.SSS0.Px3.p3.1.m1.1.1" xref="S2.SS2.SSS0.Px3.p3.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p3.1.m1.1b"><ci id="S2.SS2.SSS0.Px3.p3.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p3.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p3.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p3.1.m1.1d">italic_B</annotation></semantics></math> put to zero. In the example, the play <math alttext="v_{0}v_{1}v_{2}^{\omega}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p3.2.m2.1"><semantics id="S2.SS2.SSS0.Px3.p3.2.m2.1a"><mrow id="S2.SS2.SSS0.Px3.p3.2.m2.1.1" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.2" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.3" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.1" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.1.cmml">⁢</mo><msub id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.2" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.3" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.1a" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.1.cmml">⁢</mo><msubsup id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.cmml"><mi id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.2" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.2.cmml">v</mi><mn id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.3" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.3.cmml">2</mn><mi id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.3" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.3.cmml">ω</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p3.2.m2.1b"><apply id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1"><times id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.1"></times><apply id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.2.3">0</cn></apply><apply id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.3.3">1</cn></apply><apply id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.1.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4">superscript</csymbol><apply id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.1.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.2.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.2">𝑣</ci><cn id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.2.3">2</cn></apply><ci id="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.3.cmml" xref="S2.SS2.SSS0.Px3.p3.2.m2.1.1.4.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p3.2.m2.1c">v_{0}v_{1}v_{2}^{\omega}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p3.2.m2.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math> is Pareto-optimal (with visits to <math alttext="T_{1}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p3.3.m3.1"><semantics id="S2.SS2.SSS0.Px3.p3.3.m3.1a"><msub id="S2.SS2.SSS0.Px3.p3.3.m3.1.1" xref="S2.SS2.SSS0.Px3.p3.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p3.3.m3.1.1.2" xref="S2.SS2.SSS0.Px3.p3.3.m3.1.1.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p3.3.m3.1.1.3" xref="S2.SS2.SSS0.Px3.p3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p3.3.m3.1b"><apply id="S2.SS2.SSS0.Px3.p3.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p3.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p3.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p3.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p3.3.m3.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p3.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p3.3.m3.1c">T_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p3.3.m3.1d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="T_{3}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p3.4.m4.1"><semantics id="S2.SS2.SSS0.Px3.p3.4.m4.1a"><msub id="S2.SS2.SSS0.Px3.p3.4.m4.1.1" xref="S2.SS2.SSS0.Px3.p3.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p3.4.m4.1.1.2" xref="S2.SS2.SSS0.Px3.p3.4.m4.1.1.2.cmml">T</mi><mn id="S2.SS2.SSS0.Px3.p3.4.m4.1.1.3" xref="S2.SS2.SSS0.Px3.p3.4.m4.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p3.4.m4.1b"><apply id="S2.SS2.SSS0.Px3.p3.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p3.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p3.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p3.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p3.4.m4.1.1.2">𝑇</ci><cn id="S2.SS2.SSS0.Px3.p3.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p3.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p3.4.m4.1c">T_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p3.4.m4.1d">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>), whatever the strategy of Player <math alttext="0" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p3.5.m5.1"><semantics id="S2.SS2.SSS0.Px3.p3.5.m5.1a"><mn id="S2.SS2.SSS0.Px3.p3.5.m5.1.1" xref="S2.SS2.SSS0.Px3.p3.5.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p3.5.m5.1b"><cn id="S2.SS2.SSS0.Px3.p3.5.m5.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p3.5.m5.1.1">0</cn></annotation-xml></semantics></math>, and this play does not visit Player <math alttext="0" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p3.6.m6.1"><semantics id="S2.SS2.SSS0.Px3.p3.6.m6.1a"><mn id="S2.SS2.SSS0.Px3.p3.6.m6.1.1" xref="S2.SS2.SSS0.Px3.p3.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p3.6.m6.1b"><cn id="S2.SS2.SSS0.Px3.p3.6.m6.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p3.6.m6.1.1">0</cn></annotation-xml></semantics></math>’s target.</p> </div> </section> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Tools</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">We here present two tools used in the proofs for solving the SPS problem.</p> </div> <section class="ltx_paragraph" id="S2.SS3.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Witnesses.</h4> <div class="ltx_para" id="S2.SS3.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p1.10">An important tool is the concept of witness <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite>. Given a solution <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.1.m1.1"><semantics id="S2.SS3.SSS0.Px1.p1.1.m1.1a"><msub id="S2.SS3.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p1.1.m1.1.1.2" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.1.m1.1.1.3" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.1.m1.1b"><apply id="S2.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, for all <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.2.m2.1"><semantics id="S2.SS3.SSS0.Px1.p1.2.m2.1a"><mrow id="S2.SS3.SSS0.Px1.p1.2.m2.1.1" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.2" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml">c</mi><mo id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.1" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.2" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.2.cmml">C</mi><msub id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.2" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.3" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.2.m2.1b"><apply id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1"><in id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.1"></in><ci id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.2">𝑐</ci><apply id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.2">𝐶</ci><apply id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.2.m2.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.2.m2.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, we can choose arbitrarily a play <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.3.m3.1"><semantics id="S2.SS3.SSS0.Px1.p1.3.m3.1a"><mi id="S2.SS3.SSS0.Px1.p1.3.m3.1.1" xref="S2.SS3.SSS0.Px1.p1.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.3.m3.1b"><ci id="S2.SS3.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.3.m3.1d">italic_ρ</annotation></semantics></math> called <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px1.p1.10.1">witness</em> of the cost <math alttext="c" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.4.m4.1"><semantics id="S2.SS3.SSS0.Px1.p1.4.m4.1a"><mi id="S2.SS3.SSS0.Px1.p1.4.m4.1.1" xref="S2.SS3.SSS0.Px1.p1.4.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.4.m4.1b"><ci id="S2.SS3.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.4.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.4.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.4.m4.1d">italic_c</annotation></semantics></math> such that <math alttext="\textsf{cost}({\rho})=c" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.5.m5.1"><semantics id="S2.SS3.SSS0.Px1.p1.5.m5.1a"><mrow id="S2.SS3.SSS0.Px1.p1.5.m5.1.2" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.cmml"><mrow id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.2" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.2a.cmml">cost</mtext><mo id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.1" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.3.2" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.cmml"><mo id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.cmml">(</mo><mi id="S2.SS3.SSS0.Px1.p1.5.m5.1.1" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.1.cmml">ρ</mi><mo id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.1" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.1.cmml">=</mo><mi id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.3" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.5.m5.1b"><apply id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2"><eq id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.1"></eq><apply id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2"><times id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.1"></times><ci id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.2.2">cost</mtext></ci><ci id="S2.SS3.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.1">𝜌</ci></apply><ci id="S2.SS3.SSS0.Px1.p1.5.m5.1.2.3.cmml" xref="S2.SS3.SSS0.Px1.p1.5.m5.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.5.m5.1c">\textsf{cost}({\rho})=c</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.5.m5.1d">cost ( italic_ρ ) = italic_c</annotation></semantics></math>. The set of all chosen<span class="ltx_note ltx_role_footnote" id="footnote4"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>Note that the witness set is not necessarily unique.</span></span></span> witnesses is denoted by <math alttext="\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.6.m6.1"><semantics id="S2.SS3.SSS0.Px1.p1.6.m6.1a"><msub id="S2.SS3.SSS0.Px1.p1.6.m6.1.1" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.2" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.2" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.3" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.6.m6.1b"><apply id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.6.m6.1c">\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.6.m6.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, whose size is the size of <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.7.m7.1"><semantics id="S2.SS3.SSS0.Px1.p1.7.m7.1a"><msub id="S2.SS3.SSS0.Px1.p1.7.m7.1.1" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.2" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.2.cmml">C</mi><msub id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.2" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.3" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.7.m7.1b"><apply id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.2">𝐶</ci><apply id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.7.m7.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.7.m7.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.7.m7.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.8.m8.1"><semantics id="S2.SS3.SSS0.Px1.p1.8.m8.1a"><msub id="S2.SS3.SSS0.Px1.p1.8.m8.1.1" xref="S2.SS3.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p1.8.m8.1.1.2" xref="S2.SS3.SSS0.Px1.p1.8.m8.1.1.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.8.m8.1.1.3" xref="S2.SS3.SSS0.Px1.p1.8.m8.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.8.m8.1b"><apply id="S2.SS3.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.8.m8.1.1.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.8.m8.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.8.m8.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.8.m8.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution, the value of each witness is below <math alttext="B" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.9.m9.1"><semantics id="S2.SS3.SSS0.Px1.p1.9.m9.1a"><mi id="S2.SS3.SSS0.Px1.p1.9.m9.1.1" xref="S2.SS3.SSS0.Px1.p1.9.m9.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.9.m9.1b"><ci id="S2.SS3.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.9.m9.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.9.m9.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.9.m9.1d">italic_B</annotation></semantics></math>. We define the <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px1.p1.10.2">length</em> of a witness <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.10.m10.1"><semantics id="S2.SS3.SSS0.Px1.p1.10.m10.1a"><mi id="S2.SS3.SSS0.Px1.p1.10.m10.1.1" xref="S2.SS3.SSS0.Px1.p1.10.m10.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.10.m10.1b"><ci id="S2.SS3.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.10.m10.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.10.m10.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.10.m10.1d">italic_ρ</annotation></semantics></math> as the length</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <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="\textsf{length}({\rho})=\min\{|h|\mid h\rho\wedge\textsf{cost}({h})=\textsf{% cost}({\rho})\wedge\textsf{val}(h)=\textsf{val}(\rho)\}." class="ltx_Math" display="block" id="S2.Ex2.m1.8"><semantics id="S2.Ex2.m1.8a"><mrow id="S2.Ex2.m1.8.8.1" xref="S2.Ex2.m1.8.8.1.1.cmml"><mrow id="S2.Ex2.m1.8.8.1.1" xref="S2.Ex2.m1.8.8.1.1.cmml"><mrow id="S2.Ex2.m1.8.8.1.1.3" xref="S2.Ex2.m1.8.8.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex2.m1.8.8.1.1.3.2" xref="S2.Ex2.m1.8.8.1.1.3.2a.cmml">length</mtext><mo id="S2.Ex2.m1.8.8.1.1.3.1" xref="S2.Ex2.m1.8.8.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.8.8.1.1.3.3.2" xref="S2.Ex2.m1.8.8.1.1.3.cmml"><mo id="S2.Ex2.m1.8.8.1.1.3.3.2.1" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.3.cmml">(</mo><mi id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1.cmml">ρ</mi><mo id="S2.Ex2.m1.8.8.1.1.3.3.2.2" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.8.8.1.1.2" xref="S2.Ex2.m1.8.8.1.1.2.cmml">=</mo><mrow 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xref="S2.Ex2.m1.4.4.cmml">ρ</mi><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.2.3.2.2" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.1" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.1.cmml">∧</mo><mrow id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.2" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.2a.cmml">val</mtext><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.1" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.3.2" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.cmml"><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.3.2.1" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.cmml">(</mo><mi id="S2.Ex2.m1.5.5" xref="S2.Ex2.m1.5.5.cmml">h</mi><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.3.2.2" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.5" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.5.cmml">=</mo><mrow id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.2" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.2a.cmml">val</mtext><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.1" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.3.2" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.cmml"><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.3.2.1" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.cmml">(</mo><mi id="S2.Ex2.m1.6.6" xref="S2.Ex2.m1.6.6.cmml">ρ</mi><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.3.2.2" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex2.m1.8.8.1.1.1.1.1.3" stretchy="false" xref="S2.Ex2.m1.8.8.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><mo id="S2.Ex2.m1.8.8.1.2" lspace="0em" xref="S2.Ex2.m1.8.8.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.8b"><apply 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id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.2.2.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.2.2">cost</mtext></ci><ci id="S2.Ex2.m1.4.4.cmml" xref="S2.Ex2.m1.4.4">𝜌</ci></apply><apply id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3"><times id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.1.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.1"></times><ci id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.2a.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.2.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.4.3.2">val</mtext></ci><ci id="S2.Ex2.m1.5.5.cmml" xref="S2.Ex2.m1.5.5">ℎ</ci></apply></apply></apply><apply id="S2.Ex2.m1.8.8.1.1.1.1.1.1c.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1"><eq id="S2.Ex2.m1.8.8.1.1.1.1.1.1.5.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2308.09443v2#S2.Ex2.m1.8.8.1.1.1.1.1.1.4.cmml" id="S2.Ex2.m1.8.8.1.1.1.1.1.1d.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1"></share><apply id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6"><times id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.1.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.1"></times><ci id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.2a.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.2.cmml" xref="S2.Ex2.m1.8.8.1.1.1.1.1.1.6.2">val</mtext></ci><ci id="S2.Ex2.m1.6.6.cmml" xref="S2.Ex2.m1.6.6">𝜌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.8c">\textsf{length}({\rho})=\min\{|h|\mid h\rho\wedge\textsf{cost}({h})=\textsf{% cost}({\rho})\wedge\textsf{val}(h)=\textsf{val}(\rho)\}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.8d">length ( italic_ρ ) = roman_min { | italic_h | ∣ italic_h italic_ρ ∧ cost ( italic_h ) = cost ( italic_ρ ) ∧ val ( italic_h ) = val ( italic_ρ ) } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p1.18">Hence it is the length of the shortest history <math alttext="h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.11.m1.1"><semantics id="S2.SS3.SSS0.Px1.p1.11.m1.1a"><mi id="S2.SS3.SSS0.Px1.p1.11.m1.1.1" xref="S2.SS3.SSS0.Px1.p1.11.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.11.m1.1b"><ci id="S2.SS3.SSS0.Px1.p1.11.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.11.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.11.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.11.m1.1d">italic_h</annotation></semantics></math> that visits the same targets as <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.12.m2.1"><semantics id="S2.SS3.SSS0.Px1.p1.12.m2.1a"><mi id="S2.SS3.SSS0.Px1.p1.12.m2.1.1" xref="S2.SS3.SSS0.Px1.p1.12.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.12.m2.1b"><ci id="S2.SS3.SSS0.Px1.p1.12.m2.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.12.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.12.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.12.m2.1d">italic_ρ</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px1.p1.18.1">length</em> of <math alttext="\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.13.m3.1"><semantics id="S2.SS3.SSS0.Px1.p1.13.m3.1a"><msub id="S2.SS3.SSS0.Px1.p1.13.m3.1.1" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.2" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.2" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.3" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.13.m3.1b"><apply id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.13.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.13.m3.1c">\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.13.m3.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, denoted by <math alttext="\textsf{length}({\textsf{Wit}_{\sigma_{0}}})" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.14.m4.1"><semantics id="S2.SS3.SSS0.Px1.p1.14.m4.1a"><mrow id="S2.SS3.SSS0.Px1.p1.14.m4.1.1" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.3" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.3a.cmml">length</mtext><mo id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.2" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.2" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.3" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.14.m4.1b"><apply id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1"><times id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.2"></times><ci id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.3a.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.3">length</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.14.m4.1.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.14.m4.1c">\textsf{length}({\textsf{Wit}_{\sigma_{0}}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.14.m4.1d">length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>, is equal to <math alttext="\Sigma_{\rho\in\textsf{Wit}_{\sigma_{0}}}\textsf{length}({\rho})" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.15.m5.1"><semantics id="S2.SS3.SSS0.Px1.p1.15.m5.1a"><mrow id="S2.SS3.SSS0.Px1.p1.15.m5.1.2" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.cmml"><msub id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.cmml"><mi id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.2" mathvariant="normal" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.2.cmml">Σ</mi><mrow id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.2" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.2.cmml">ρ</mi><mo id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.1" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.1.cmml">∈</mo><msub id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.2" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.2" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.3" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.3.cmml">0</mn></msub></msub></mrow></msub><mo id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.1" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.1.cmml">⁢</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.3" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.3a.cmml">length</mtext><mo id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.1a" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.4.2" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.cmml"><mo id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.4.2.1" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.cmml">(</mo><mi id="S2.SS3.SSS0.Px1.p1.15.m5.1.1" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.1.cmml">ρ</mi><mo id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.4.2.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.15.m5.1b"><apply id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2"><times id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.1"></times><apply id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.2">Σ</ci><apply id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3"><in id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.1"></in><ci id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.2">𝜌</ci><apply id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.2.cmml" mathsize="70%" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.2.3.3.3.3">0</cn></apply></apply></apply></apply><ci id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.3a.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.15.m5.1.2.3.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.2.3">length</mtext></ci><ci id="S2.SS3.SSS0.Px1.p1.15.m5.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.15.m5.1.1">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.15.m5.1c">\Sigma_{\rho\in\textsf{Wit}_{\sigma_{0}}}\textsf{length}({\rho})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.15.m5.1d">roman_Σ start_POSTSUBSCRIPT italic_ρ ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT length ( italic_ρ )</annotation></semantics></math>. Moreover, given <math alttext="h\in\textsf{Hist}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.16.m6.1"><semantics id="S2.SS3.SSS0.Px1.p1.16.m6.1a"><mrow id="S2.SS3.SSS0.Px1.p1.16.m6.1.1" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.2" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.1" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.1.cmml">∈</mo><msub id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.2" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.2a.cmml">Hist</mtext><msub id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.2" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.3" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.16.m6.1b"><apply id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1"><in id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.1"></in><ci id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.2">ℎ</ci><apply id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.2">Hist</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.16.m6.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.16.m6.1c">h\in\textsf{Hist}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.16.m6.1d">italic_h ∈ Hist start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, we write <math alttext="\textsf{Wit}_{\sigma_{0}}(h)" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.17.m7.1"><semantics id="S2.SS3.SSS0.Px1.p1.17.m7.1a"><mrow id="S2.SS3.SSS0.Px1.p1.17.m7.1.2" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.cmml"><msub id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.2" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.2" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.3" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.1" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.3.2" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.cmml"><mo id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.cmml">(</mo><mi id="S2.SS3.SSS0.Px1.p1.17.m7.1.1" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.1.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.17.m7.1b"><apply id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2"><times id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.1"></times><apply id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.2.2.3.3">0</cn></apply></apply><ci id="S2.SS3.SSS0.Px1.p1.17.m7.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.17.m7.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.17.m7.1c">\textsf{Wit}_{\sigma_{0}}(h)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.17.m7.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h )</annotation></semantics></math> the set of witnesses for which <math alttext="h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.18.m8.1"><semantics id="S2.SS3.SSS0.Px1.p1.18.m8.1a"><mi id="S2.SS3.SSS0.Px1.p1.18.m8.1.1" xref="S2.SS3.SSS0.Px1.p1.18.m8.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.18.m8.1b"><ci id="S2.SS3.SSS0.Px1.p1.18.m8.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.18.m8.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.18.m8.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.18.m8.1d">italic_h</annotation></semantics></math> is a prefix, i.e.,</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <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="\textsf{Wit}_{\sigma_{0}}(h)=\{\rho\in\textsf{Wit}_{\sigma_{0}}\mid h\rho\}." class="ltx_Math" display="block" id="S2.Ex3.m1.2"><semantics id="S2.Ex3.m1.2a"><mrow id="S2.Ex3.m1.2.2.1" xref="S2.Ex3.m1.2.2.1.1.cmml"><mrow id="S2.Ex3.m1.2.2.1.1" xref="S2.Ex3.m1.2.2.1.1.cmml"><mrow id="S2.Ex3.m1.2.2.1.1.4" xref="S2.Ex3.m1.2.2.1.1.4.cmml"><msub id="S2.Ex3.m1.2.2.1.1.4.2" xref="S2.Ex3.m1.2.2.1.1.4.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex3.m1.2.2.1.1.4.2.2" xref="S2.Ex3.m1.2.2.1.1.4.2.2a.cmml">Wit</mtext><msub id="S2.Ex3.m1.2.2.1.1.4.2.3" xref="S2.Ex3.m1.2.2.1.1.4.2.3.cmml"><mi id="S2.Ex3.m1.2.2.1.1.4.2.3.2" xref="S2.Ex3.m1.2.2.1.1.4.2.3.2.cmml">σ</mi><mn id="S2.Ex3.m1.2.2.1.1.4.2.3.3" xref="S2.Ex3.m1.2.2.1.1.4.2.3.3.cmml">0</mn></msub></msub><mo id="S2.Ex3.m1.2.2.1.1.4.1" xref="S2.Ex3.m1.2.2.1.1.4.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.2.1.1.4.3.2" xref="S2.Ex3.m1.2.2.1.1.4.cmml"><mo id="S2.Ex3.m1.2.2.1.1.4.3.2.1" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.4.cmml">(</mo><mi id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">h</mi><mo id="S2.Ex3.m1.2.2.1.1.4.3.2.2" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.4.cmml">)</mo></mrow></mrow><mo id="S2.Ex3.m1.2.2.1.1.3" xref="S2.Ex3.m1.2.2.1.1.3.cmml">=</mo><mrow id="S2.Ex3.m1.2.2.1.1.2.2" xref="S2.Ex3.m1.2.2.1.1.2.3.cmml"><mo id="S2.Ex3.m1.2.2.1.1.2.2.3" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.2.3.1.cmml">{</mo><mrow id="S2.Ex3.m1.2.2.1.1.1.1.1" xref="S2.Ex3.m1.2.2.1.1.1.1.1.cmml"><mi id="S2.Ex3.m1.2.2.1.1.1.1.1.2" xref="S2.Ex3.m1.2.2.1.1.1.1.1.2.cmml">ρ</mi><mo id="S2.Ex3.m1.2.2.1.1.1.1.1.1" xref="S2.Ex3.m1.2.2.1.1.1.1.1.1.cmml">∈</mo><msub id="S2.Ex3.m1.2.2.1.1.1.1.1.3" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex3.m1.2.2.1.1.1.1.1.3.2" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.2a.cmml">Wit</mtext><msub id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.cmml"><mi id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2.cmml">σ</mi><mn id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><mo fence="true" id="S2.Ex3.m1.2.2.1.1.2.2.4" lspace="0em" rspace="0em" xref="S2.Ex3.m1.2.2.1.1.2.3.1.cmml">∣</mo><mrow id="S2.Ex3.m1.2.2.1.1.2.2.2" xref="S2.Ex3.m1.2.2.1.1.2.2.2.cmml"><mi id="S2.Ex3.m1.2.2.1.1.2.2.2.2" xref="S2.Ex3.m1.2.2.1.1.2.2.2.2.cmml">h</mi><mo id="S2.Ex3.m1.2.2.1.1.2.2.2.1" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.cmml">⁢</mo><mi id="S2.Ex3.m1.2.2.1.1.2.2.2.3" xref="S2.Ex3.m1.2.2.1.1.2.2.2.3.cmml">ρ</mi></mrow><mo id="S2.Ex3.m1.2.2.1.1.2.2.5" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S2.Ex3.m1.2.2.1.2" lspace="0em" xref="S2.Ex3.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.2b"><apply id="S2.Ex3.m1.2.2.1.1.cmml" xref="S2.Ex3.m1.2.2.1"><eq id="S2.Ex3.m1.2.2.1.1.3.cmml" xref="S2.Ex3.m1.2.2.1.1.3"></eq><apply id="S2.Ex3.m1.2.2.1.1.4.cmml" xref="S2.Ex3.m1.2.2.1.1.4"><times id="S2.Ex3.m1.2.2.1.1.4.1.cmml" xref="S2.Ex3.m1.2.2.1.1.4.1"></times><apply id="S2.Ex3.m1.2.2.1.1.4.2.cmml" xref="S2.Ex3.m1.2.2.1.1.4.2"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.2.1.1.4.2.1.cmml" xref="S2.Ex3.m1.2.2.1.1.4.2">subscript</csymbol><ci id="S2.Ex3.m1.2.2.1.1.4.2.2a.cmml" xref="S2.Ex3.m1.2.2.1.1.4.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex3.m1.2.2.1.1.4.2.2.cmml" xref="S2.Ex3.m1.2.2.1.1.4.2.2">Wit</mtext></ci><apply id="S2.Ex3.m1.2.2.1.1.4.2.3.cmml" xref="S2.Ex3.m1.2.2.1.1.4.2.3"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.2.1.1.4.2.3.1.cmml" xref="S2.Ex3.m1.2.2.1.1.4.2.3">subscript</csymbol><ci id="S2.Ex3.m1.2.2.1.1.4.2.3.2.cmml" xref="S2.Ex3.m1.2.2.1.1.4.2.3.2">𝜎</ci><cn id="S2.Ex3.m1.2.2.1.1.4.2.3.3.cmml" type="integer" xref="S2.Ex3.m1.2.2.1.1.4.2.3.3">0</cn></apply></apply><ci id="S2.Ex3.m1.1.1.cmml" xref="S2.Ex3.m1.1.1">ℎ</ci></apply><apply id="S2.Ex3.m1.2.2.1.1.2.3.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2"><csymbol cd="latexml" id="S2.Ex3.m1.2.2.1.1.2.3.1.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.3">conditional-set</csymbol><apply id="S2.Ex3.m1.2.2.1.1.1.1.1.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1"><in id="S2.Ex3.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.1"></in><ci id="S2.Ex3.m1.2.2.1.1.1.1.1.2.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.2">𝜌</ci><apply id="S2.Ex3.m1.2.2.1.1.1.1.1.3.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex3.m1.2.2.1.1.1.1.1.3.2a.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex3.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.2">Wit</mtext></ci><apply id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.1.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2">𝜎</ci><cn id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3">0</cn></apply></apply></apply><apply id="S2.Ex3.m1.2.2.1.1.2.2.2.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2"><times id="S2.Ex3.m1.2.2.1.1.2.2.2.1.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1"></times><ci id="S2.Ex3.m1.2.2.1.1.2.2.2.2.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.2">ℎ</ci><ci id="S2.Ex3.m1.2.2.1.1.2.2.2.3.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.3">𝜌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.2c">\textsf{Wit}_{\sigma_{0}}(h)=\{\rho\in\textsf{Wit}_{\sigma_{0}}\mid h\rho\}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.2d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h ) = { italic_ρ ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∣ italic_h italic_ρ } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p1.22">Notice that <math alttext="\textsf{Wit}_{\sigma_{0}}(h)=\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.19.m1.1"><semantics id="S2.SS3.SSS0.Px1.p1.19.m1.1a"><mrow id="S2.SS3.SSS0.Px1.p1.19.m1.1.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.cmml"><mrow id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.cmml"><msub id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.3" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.1" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.3.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.cmml"><mo id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.cmml">(</mo><mi id="S2.SS3.SSS0.Px1.p1.19.m1.1.1" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.1.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.1" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.1.cmml">=</mo><msub id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.2" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.3" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.19.m1.1b"><apply id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2"><eq id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.1"></eq><apply id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2"><times id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.1"></times><apply id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.2.2.3.3">0</cn></apply></apply><ci id="S2.SS3.SSS0.Px1.p1.19.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.1">ℎ</ci></apply><apply id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.19.m1.1.2.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.19.m1.1c">\textsf{Wit}_{\sigma_{0}}(h)=\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.19.m1.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h ) = Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> when <math alttext="h=v_{0}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.20.m2.1"><semantics id="S2.SS3.SSS0.Px1.p1.20.m2.1a"><mrow id="S2.SS3.SSS0.Px1.p1.20.m2.1.1" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.2" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.1" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.1.cmml">=</mo><msub id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.2" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.3" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.20.m2.1b"><apply id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1"><eq id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.1"></eq><ci id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.2">ℎ</ci><apply id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.20.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.20.m2.1c">h=v_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.20.m2.1d">italic_h = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, and that the size of <math alttext="\textsf{Wit}_{\sigma_{0}}(h)" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.21.m3.1"><semantics id="S2.SS3.SSS0.Px1.p1.21.m3.1a"><mrow id="S2.SS3.SSS0.Px1.p1.21.m3.1.2" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.cmml"><msub id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.2" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.cmml"><mi id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.2" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.3" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.1" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.3.2" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.cmml"><mo id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.cmml">(</mo><mi id="S2.SS3.SSS0.Px1.p1.21.m3.1.1" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.1.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.21.m3.1b"><apply id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2"><times id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.1"></times><apply id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.1.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.2a.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.2.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.1.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.2.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.2.2.3.3">0</cn></apply></apply><ci id="S2.SS3.SSS0.Px1.p1.21.m3.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.21.m3.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.21.m3.1c">\textsf{Wit}_{\sigma_{0}}(h)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.21.m3.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h )</annotation></semantics></math> decreases as the size of <math alttext="h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p1.22.m4.1"><semantics id="S2.SS3.SSS0.Px1.p1.22.m4.1a"><mi id="S2.SS3.SSS0.Px1.p1.22.m4.1.1" xref="S2.SS3.SSS0.Px1.p1.22.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.22.m4.1b"><ci id="S2.SS3.SSS0.Px1.p1.22.m4.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.22.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.22.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p1.22.m4.1d">italic_h</annotation></semantics></math> increases, until it contains a single play or becomes empty.</p> </div> <div class="ltx_para" id="S2.SS3.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p2.9">It is useful to see the set <math alttext="\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.1.m1.1"><semantics id="S2.SS3.SSS0.Px1.p2.1.m1.1a"><msub id="S2.SS3.SSS0.Px1.p2.1.m1.1.1" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.2" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.1.m1.1b"><apply id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.2a.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.1.m1.1c">\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.1.m1.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> as a tree composed of <math alttext="|\textsf{Wit}_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.2.m2.1"><semantics id="S2.SS3.SSS0.Px1.p2.2.m2.1a"><mrow id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.2.cmml"><mo id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.2.1.cmml">|</mo><msub id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.2" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.3" stretchy="false" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.2.m2.1b"><apply id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1"><abs id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.2"></abs><apply id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.2a.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p2.2.m2.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.2.m2.1c">|\textsf{Wit}_{\sigma_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.2.m2.1d">| Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math> infinite branches (corresponding to the witnesses). The following notions about this tree will be useful. We say that a history <math alttext="h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.3.m3.1"><semantics id="S2.SS3.SSS0.Px1.p2.3.m3.1a"><mi id="S2.SS3.SSS0.Px1.p2.3.m3.1.1" xref="S2.SS3.SSS0.Px1.p2.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.3.m3.1b"><ci id="S2.SS3.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.3.m3.1d">italic_h</annotation></semantics></math> is a <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px1.p2.9.1">branching point</em> if there are two witnesses whose greatest common prefix is <math alttext="h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.4.m4.1"><semantics id="S2.SS3.SSS0.Px1.p2.4.m4.1a"><mi id="S2.SS3.SSS0.Px1.p2.4.m4.1.1" xref="S2.SS3.SSS0.Px1.p2.4.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.4.m4.1b"><ci id="S2.SS3.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.4.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.4.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.4.m4.1d">italic_h</annotation></semantics></math>, that is, there exists <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.5.m5.1"><semantics id="S2.SS3.SSS0.Px1.p2.5.m5.1a"><mrow id="S2.SS3.SSS0.Px1.p2.5.m5.1.1" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.5.m5.1.1.2" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1.2.cmml">v</mi><mo id="S2.SS3.SSS0.Px1.p2.5.m5.1.1.1" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1.1.cmml">∈</mo><mi id="S2.SS3.SSS0.Px1.p2.5.m5.1.1.3" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.5.m5.1b"><apply id="S2.SS3.SSS0.Px1.p2.5.m5.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1"><in id="S2.SS3.SSS0.Px1.p2.5.m5.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1.1"></in><ci id="S2.SS3.SSS0.Px1.p2.5.m5.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1.2">𝑣</ci><ci id="S2.SS3.SSS0.Px1.p2.5.m5.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.5.m5.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.5.m5.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.5.m5.1d">italic_v ∈ italic_V</annotation></semantics></math> such that <math alttext="0<|\textsf{Wit}_{\sigma_{0}}(hv)|<|\textsf{Wit}_{\sigma_{0}}(h)|" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.6.m6.3"><semantics id="S2.SS3.SSS0.Px1.p2.6.m6.3a"><mrow id="S2.SS3.SSS0.Px1.p2.6.m6.3.3" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.cmml"><mn id="S2.SS3.SSS0.Px1.p2.6.m6.3.3.4" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.4.cmml">0</mn><mo id="S2.SS3.SSS0.Px1.p2.6.m6.3.3.5" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.5.cmml"><</mo><mrow id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.2.cmml"><mo id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.2.1.cmml">|</mo><mrow id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.3.2" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.3.3" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.2" xref="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p2.6.m6.2.2.1.1.1.1.1" 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id="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3.cmml" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3.2.cmml" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p2.6.m6.3.3.2.1.1.2.3.3">0</cn></apply></apply><ci id="S2.SS3.SSS0.Px1.p2.6.m6.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.6.m6.1.1">ℎ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.6.m6.3c">0<|\textsf{Wit}_{\sigma_{0}}(hv)|<|\textsf{Wit}_{\sigma_{0}}(h)|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.6.m6.3d">0 < | Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h italic_v ) | < | Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h ) |</annotation></semantics></math>. Given a witness <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.7.m7.1"><semantics id="S2.SS3.SSS0.Px1.p2.7.m7.1a"><mi id="S2.SS3.SSS0.Px1.p2.7.m7.1.1" xref="S2.SS3.SSS0.Px1.p2.7.m7.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.7.m7.1b"><ci id="S2.SS3.SSS0.Px1.p2.7.m7.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.7.m7.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.7.m7.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.7.m7.1d">italic_ρ</annotation></semantics></math>, we define the following equivalence relations <math alttext="\sim" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.8.m8.1"><semantics id="S2.SS3.SSS0.Px1.p2.8.m8.1a"><mo id="S2.SS3.SSS0.Px1.p2.8.m8.1.1" xref="S2.SS3.SSS0.Px1.p2.8.m8.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.8.m8.1b"><csymbol cd="latexml" id="S2.SS3.SSS0.Px1.p2.8.m8.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.8.m8.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.8.m8.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.8.m8.1d">∼</annotation></semantics></math> on histories that are prefixes of <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.9.m9.1"><semantics id="S2.SS3.SSS0.Px1.p2.9.m9.1a"><mi id="S2.SS3.SSS0.Px1.p2.9.m9.1.1" xref="S2.SS3.SSS0.Px1.p2.9.m9.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.9.m9.1b"><ci id="S2.SS3.SSS0.Px1.p2.9.m9.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.9.m9.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.9.m9.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.9.m9.1d">italic_ρ</annotation></semantics></math>:</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex4"> <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="h\sim h^{\prime}\quad\Leftrightarrow\quad(\textsf{val}(h),\textsf{cost}({h}),% \textsf{Wit}_{\sigma_{0}}(h))=(\textsf{val}(h^{\prime}),\textsf{cost}({h^{% \prime}}),\textsf{Wit}_{\sigma_{0}}(h^{\prime}))." class="ltx_Math" display="block" id="S2.Ex4.m1.5"><semantics id="S2.Ex4.m1.5a"><mrow id="S2.Ex4.m1.5.5.1"><mrow id="S2.Ex4.m1.5.5.1.1.2" xref="S2.Ex4.m1.5.5.1.1.3.cmml"><mrow id="S2.Ex4.m1.5.5.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.1.1.cmml"><mi id="S2.Ex4.m1.5.5.1.1.1.1.3" xref="S2.Ex4.m1.5.5.1.1.1.1.3.cmml">h</mi><mo id="S2.Ex4.m1.5.5.1.1.1.1.2" xref="S2.Ex4.m1.5.5.1.1.1.1.2.cmml">∼</mo><mrow id="S2.Ex4.m1.5.5.1.1.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.1.1.1.2.cmml"><msup id="S2.Ex4.m1.5.5.1.1.1.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m1.5.5.1.1.1.1.1.1.1.2" xref="S2.Ex4.m1.5.5.1.1.1.1.1.1.1.2.cmml">h</mi><mo id="S2.Ex4.m1.5.5.1.1.1.1.1.1.1.3" xref="S2.Ex4.m1.5.5.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mspace id="S2.Ex4.m1.5.5.1.1.1.1.1.1.2" width="1em" xref="S2.Ex4.m1.5.5.1.1.1.1.1.2.cmml"></mspace><mo id="S2.Ex4.m1.4.4" stretchy="false" xref="S2.Ex4.m1.4.4.cmml">⇔</mo></mrow></mrow><mspace id="S2.Ex4.m1.5.5.1.1.2.3" width="1em" xref="S2.Ex4.m1.5.5.1.1.3a.cmml"></mspace><mrow id="S2.Ex4.m1.5.5.1.1.2.2" xref="S2.Ex4.m1.5.5.1.1.2.2.cmml"><mrow id="S2.Ex4.m1.5.5.1.1.2.2.3.3" xref="S2.Ex4.m1.5.5.1.1.2.2.3.4.cmml"><mo id="S2.Ex4.m1.5.5.1.1.2.2.3.3.4" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.3.4.cmml">(</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.1.1.1" xref="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.2" xref="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.2a.cmml">val</mtext><mo id="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.3.2" xref="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.cmml"><mo id="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.cmml">(</mo><mi id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml">h</mi><mo id="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.5.5.1.1.2.2.3.3.5" xref="S2.Ex4.m1.5.5.1.1.2.2.3.4.cmml">,</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.2.2.2" xref="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.2" xref="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.2a.cmml">cost</mtext><mo id="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.1" xref="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.3.2" xref="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.cmml"><mo id="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.3.2.1" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.cmml">(</mo><mi id="S2.Ex4.m1.2.2" xref="S2.Ex4.m1.2.2.cmml">h</mi><mo id="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.3.2.2" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.5.5.1.1.2.2.3.3.6" xref="S2.Ex4.m1.5.5.1.1.2.2.3.4.cmml">,</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.cmml"><msub id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.2" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.2a.cmml">Wit</mtext><msub id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.3" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.3.cmml"><mi id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.3.2" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.3.2.cmml">σ</mi><mn id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.3.3" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.2.3.3.cmml">0</mn></msub></msub><mo id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.1" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.3.2" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.cmml"><mo id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.3.2.1" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.cmml">(</mo><mi id="S2.Ex4.m1.3.3" xref="S2.Ex4.m1.3.3.cmml">h</mi><mo id="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.3.2.2" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.5.5.1.1.2.2.3.3.7" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.3.4.cmml">)</mo></mrow><mo id="S2.Ex4.m1.5.5.1.1.2.2.7" xref="S2.Ex4.m1.5.5.1.1.2.2.7.cmml">=</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.6.3" xref="S2.Ex4.m1.5.5.1.1.2.2.6.4.cmml"><mo id="S2.Ex4.m1.5.5.1.1.2.2.6.3.4" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.6.4.cmml">(</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.3" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.3a.cmml">val</mtext><mo id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.2" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.cmml"><mo id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.cmml">(</mo><msup id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.cmml"><mi id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.2" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.2.cmml">h</mi><mo id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.3" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.2.2.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.5.5.1.1.2.2.6.3.5" xref="S2.Ex4.m1.5.5.1.1.2.2.6.4.cmml">,</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.5.2.2" xref="S2.Ex4.m1.5.5.1.1.2.2.5.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Ex4.m1.5.5.1.1.2.2.5.2.2.3" xref="S2.Ex4.m1.5.5.1.1.2.2.5.2.2.3a.cmml">cost</mtext><mo id="S2.Ex4.m1.5.5.1.1.2.2.5.2.2.2" xref="S2.Ex4.m1.5.5.1.1.2.2.5.2.2.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.5.5.1.1.2.2.5.2.2.1.1" 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xref="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.3.3">subscript</csymbol><ci id="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.3.3.2.cmml" xref="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.3.3.2">𝜎</ci><cn id="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.3.3.3.cmml" type="integer" xref="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.3.3.3">0</cn></apply></apply><apply id="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1.1.cmml" xref="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1"><csymbol cd="ambiguous" id="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1.1.1.cmml" xref="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1">superscript</csymbol><ci id="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1.1.2.cmml" xref="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1.1.2">ℎ</ci><ci id="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1.1.3.cmml" xref="S2.Ex4.m1.5.5.1.1.2.2.6.3.3.1.1.1.3">′</ci></apply></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.5c">h\sim h^{\prime}\quad\Leftrightarrow\quad(\textsf{val}(h),\textsf{cost}({h}),% \textsf{Wit}_{\sigma_{0}}(h))=(\textsf{val}(h^{\prime}),\textsf{cost}({h^{% \prime}}),\textsf{Wit}_{\sigma_{0}}(h^{\prime})).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.5d">italic_h ∼ italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⇔ ( val ( italic_h ) , cost ( italic_h ) , Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h ) ) = ( val ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , cost ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p2.22">Notice that if <math alttext="h\sim h^{\prime}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.10.m1.1"><semantics id="S2.SS3.SSS0.Px1.p2.10.m1.1a"><mrow id="S2.SS3.SSS0.Px1.p2.10.m1.1.1" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.2" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.1" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.1.cmml">∼</mo><msup id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.10.m1.1b"><apply id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1"><csymbol cd="latexml" id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.1">similar-to</csymbol><ci id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.2">ℎ</ci><apply id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3">superscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.2">ℎ</ci><ci id="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px1.p2.10.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.10.m1.1c">h\sim h^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.10.m1.1d">italic_h ∼ italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then either <math alttext="hh^{\prime}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.11.m2.1"><semantics id="S2.SS3.SSS0.Px1.p2.11.m2.1a"><mrow id="S2.SS3.SSS0.Px1.p2.11.m2.1.1" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.2" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.1" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.1.cmml">⁢</mo><msup id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.2" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.3" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.11.m2.1b"><apply id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1"><times id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.1"></times><ci id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.2">ℎ</ci><apply id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3">superscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.2">ℎ</ci><ci id="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px1.p2.11.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.11.m2.1c">hh^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.11.m2.1d">italic_h italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="h^{\prime}h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.12.m3.1"><semantics id="S2.SS3.SSS0.Px1.p2.12.m3.1a"><mrow id="S2.SS3.SSS0.Px1.p2.12.m3.1.1" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.cmml"><msup id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.cmml"><mi id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.2" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.3" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.1" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.3" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.12.m3.1b"><apply id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1"><times id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.1"></times><apply id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2">superscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.2">ℎ</ci><ci id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.3.cmml" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.2.3">′</ci></apply><ci id="S2.SS3.SSS0.Px1.p2.12.m3.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.12.m3.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.12.m3.1c">h^{\prime}h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.12.m3.1d">italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_h</annotation></semantics></math> and no new target is visited and no branching point is crossed from the shortest history to the longest one. We call <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px1.p2.22.1">region</em> of <math alttext="h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.13.m4.1"><semantics id="S2.SS3.SSS0.Px1.p2.13.m4.1a"><mi id="S2.SS3.SSS0.Px1.p2.13.m4.1.1" xref="S2.SS3.SSS0.Px1.p2.13.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.13.m4.1b"><ci id="S2.SS3.SSS0.Px1.p2.13.m4.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.13.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.13.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.13.m4.1d">italic_h</annotation></semantics></math> its equivalence class. This leads to a <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px1.p2.22.2">region decomposition</em> of each witness <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.14.m5.1"><semantics id="S2.SS3.SSS0.Px1.p2.14.m5.1a"><mi id="S2.SS3.SSS0.Px1.p2.14.m5.1.1" xref="S2.SS3.SSS0.Px1.p2.14.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.14.m5.1b"><ci id="S2.SS3.SSS0.Px1.p2.14.m5.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.14.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.14.m5.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.14.m5.1d">italic_ρ</annotation></semantics></math>, such that the first region is the region of the initial vertex <math alttext="v_{0}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.15.m6.1"><semantics id="S2.SS3.SSS0.Px1.p2.15.m6.1a"><msub id="S2.SS3.SSS0.Px1.p2.15.m6.1.1" xref="S2.SS3.SSS0.Px1.p2.15.m6.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.15.m6.1.1.2" xref="S2.SS3.SSS0.Px1.p2.15.m6.1.1.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p2.15.m6.1.1.3" xref="S2.SS3.SSS0.Px1.p2.15.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.15.m6.1b"><apply id="S2.SS3.SSS0.Px1.p2.15.m6.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.15.m6.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.15.m6.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.15.m6.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.15.m6.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.15.m6.1.1.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p2.15.m6.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p2.15.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.15.m6.1c">v_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.15.m6.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and the last region is the region of <math alttext="h\rho" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.16.m7.1"><semantics id="S2.SS3.SSS0.Px1.p2.16.m7.1a"><mrow id="S2.SS3.SSS0.Px1.p2.16.m7.1.1" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.16.m7.1.1.2" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.16.m7.1.1.1" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS0.Px1.p2.16.m7.1.1.3" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.16.m7.1b"><apply id="S2.SS3.SSS0.Px1.p2.16.m7.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1"><times id="S2.SS3.SSS0.Px1.p2.16.m7.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1.1"></times><ci id="S2.SS3.SSS0.Px1.p2.16.m7.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1.2">ℎ</ci><ci id="S2.SS3.SSS0.Px1.p2.16.m7.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.16.m7.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.16.m7.1c">h\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.16.m7.1d">italic_h italic_ρ</annotation></semantics></math> such that <math alttext="|h|=\textsf{length}({\rho})" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.17.m8.2"><semantics id="S2.SS3.SSS0.Px1.p2.17.m8.2a"><mrow id="S2.SS3.SSS0.Px1.p2.17.m8.2.3" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.cmml"><mrow id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.2" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.1.cmml"><mo id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.2.1" stretchy="false" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.1.1.cmml">|</mo><mi id="S2.SS3.SSS0.Px1.p2.17.m8.1.1" xref="S2.SS3.SSS0.Px1.p2.17.m8.1.1.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.2.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.1.1.cmml">|</mo></mrow><mo id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.1" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.1.cmml">=</mo><mrow id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.2" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.2a.cmml">length</mtext><mo id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.1" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.3.2" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.cmml"><mo id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.3.2.1" stretchy="false" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.cmml">(</mo><mi id="S2.SS3.SSS0.Px1.p2.17.m8.2.2" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.2.cmml">ρ</mi><mo id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.3.2.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.17.m8.2b"><apply id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3"><eq id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.1"></eq><apply id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.1.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.2"><abs id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.2.2.1"></abs><ci id="S2.SS3.SSS0.Px1.p2.17.m8.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.1.1">ℎ</ci></apply><apply id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3"><times id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.1"></times><ci id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.3.3.2">length</mtext></ci><ci id="S2.SS3.SSS0.Px1.p2.17.m8.2.2.cmml" xref="S2.SS3.SSS0.Px1.p2.17.m8.2.2">𝜌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.17.m8.2c">|h|=\textsf{length}({\rho})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.17.m8.2d">| italic_h | = length ( italic_ρ )</annotation></semantics></math>. Finally, a <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px1.p2.22.3">deviation</em> is a history <math alttext="hv" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.18.m9.1"><semantics id="S2.SS3.SSS0.Px1.p2.18.m9.1a"><mrow id="S2.SS3.SSS0.Px1.p2.18.m9.1.1" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.18.m9.1.1.2" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.18.m9.1.1.1" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS0.Px1.p2.18.m9.1.1.3" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.18.m9.1b"><apply id="S2.SS3.SSS0.Px1.p2.18.m9.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1"><times id="S2.SS3.SSS0.Px1.p2.18.m9.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1.1"></times><ci id="S2.SS3.SSS0.Px1.p2.18.m9.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1.2">ℎ</ci><ci id="S2.SS3.SSS0.Px1.p2.18.m9.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.18.m9.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.18.m9.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.18.m9.1d">italic_h italic_v</annotation></semantics></math> with <math alttext="h\in\textsf{Hist}^{1}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.19.m10.1"><semantics id="S2.SS3.SSS0.Px1.p2.19.m10.1a"><mrow id="S2.SS3.SSS0.Px1.p2.19.m10.1.1" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.2" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.1" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.1.cmml">∈</mo><msup id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.2" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.2a.cmml">Hist</mtext><mn id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.3" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.3.cmml">1</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.19.m10.1b"><apply id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1"><in id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.1"></in><ci id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.2">ℎ</ci><apply id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3">superscript</csymbol><ci id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.2">Hist</mtext></ci><cn id="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p2.19.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.19.m10.1c">h\in\textsf{Hist}^{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.19.m10.1d">italic_h ∈ Hist start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.20.m11.1"><semantics id="S2.SS3.SSS0.Px1.p2.20.m11.1a"><mrow id="S2.SS3.SSS0.Px1.p2.20.m11.1.1" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.20.m11.1.1.2" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1.2.cmml">v</mi><mo id="S2.SS3.SSS0.Px1.p2.20.m11.1.1.1" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1.1.cmml">∈</mo><mi id="S2.SS3.SSS0.Px1.p2.20.m11.1.1.3" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.20.m11.1b"><apply id="S2.SS3.SSS0.Px1.p2.20.m11.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1"><in id="S2.SS3.SSS0.Px1.p2.20.m11.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1.1"></in><ci id="S2.SS3.SSS0.Px1.p2.20.m11.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1.2">𝑣</ci><ci id="S2.SS3.SSS0.Px1.p2.20.m11.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.20.m11.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.20.m11.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.20.m11.1d">italic_v ∈ italic_V</annotation></semantics></math>, such that <math alttext="h" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.21.m12.1"><semantics id="S2.SS3.SSS0.Px1.p2.21.m12.1a"><mi id="S2.SS3.SSS0.Px1.p2.21.m12.1.1" xref="S2.SS3.SSS0.Px1.p2.21.m12.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.21.m12.1b"><ci id="S2.SS3.SSS0.Px1.p2.21.m12.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.21.m12.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.21.m12.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.21.m12.1d">italic_h</annotation></semantics></math> is prefix of some witness, but <math alttext="hv" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.22.m13.1"><semantics id="S2.SS3.SSS0.Px1.p2.22.m13.1a"><mrow id="S2.SS3.SSS0.Px1.p2.22.m13.1.1" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p2.22.m13.1.1.2" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px1.p2.22.m13.1.1.1" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS0.Px1.p2.22.m13.1.1.3" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.22.m13.1b"><apply id="S2.SS3.SSS0.Px1.p2.22.m13.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1"><times id="S2.SS3.SSS0.Px1.p2.22.m13.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1.1"></times><ci id="S2.SS3.SSS0.Px1.p2.22.m13.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1.2">ℎ</ci><ci id="S2.SS3.SSS0.Px1.p2.22.m13.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p2.22.m13.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.22.m13.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.22.m13.1d">italic_h italic_v</annotation></semantics></math> is prefix of no witness.</p> </div> <div class="ltx_para" id="S2.SS3.SSS0.Px1.p3"> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p3.13">We illustrate these different notions on the previous example and its solution <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.1.m1.1"><semantics id="S2.SS3.SSS0.Px1.p3.1.m1.1a"><msub id="S2.SS3.SSS0.Px1.p3.1.m1.1.1" xref="S2.SS3.SSS0.Px1.p3.1.m1.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p3.1.m1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.1.m1.1.1.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p3.1.m1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p3.1.m1.1b"><apply id="S2.SS3.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.1.m1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.1.m1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.1.m1.1.1.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. A set of witnesses is <math alttext="\textsf{Wit}_{\sigma_{0}}=\{v_{0}v_{1}(v_{3}v_{4})^{\omega},v_{0}v_{6}v_{6}v_{% 7}v_{9}^{\omega}\}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.2.m2.2"><semantics id="S2.SS3.SSS0.Px1.p3.2.m2.2a"><mrow id="S2.SS3.SSS0.Px1.p3.2.m2.2.2" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.cmml"><msub id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.2" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.3" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.3.2" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.3.3" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.4.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.3" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.3.cmml">=</mo><mrow id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2" 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xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.5">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.5.2.cmml" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.5.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.5.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.5.3">7</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.cmml" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.1.cmml" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6">superscript</csymbol><apply id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.2.cmml" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.2.1.cmml" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.2.2.cmml" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.2.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.2.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.2.3">9</cn></apply><ci id="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.3.cmml" xref="S2.SS3.SSS0.Px1.p3.2.m2.2.2.2.2.2.6.3">𝜔</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.2.m2.2c">\textsf{Wit}_{\sigma_{0}}=\{v_{0}v_{1}(v_{3}v_{4})^{\omega},v_{0}v_{6}v_{6}v_{% 7}v_{9}^{\omega}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.2.m2.2d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT }</annotation></semantics></math> depicted on Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.F1" title="Figure 1 ‣ Example. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">1</span></a> (right part). We have that <math alttext="\textsf{length}({v_{0}v_{6}v_{6}v_{7}v_{9}^{\omega}})=|v_{0}v_{6}v_{6}v_{7}v_{% 9}|=4" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.3.m3.2"><semantics id="S2.SS3.SSS0.Px1.p3.3.m3.2a"><mrow id="S2.SS3.SSS0.Px1.p3.3.m3.2.2" xref="S2.SS3.SSS0.Px1.p3.3.m3.2.2.cmml"><mrow id="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1" xref="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.3a.cmml">length</mtext><mo id="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.1.2" 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xref="S2.SS3.SSS0.Px1.p3.3.m3.2.2"></share><cn id="S2.SS3.SSS0.Px1.p3.3.m3.2.2.6.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.3.m3.2.2.6">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.3.m3.2c">\textsf{length}({v_{0}v_{6}v_{6}v_{7}v_{9}^{\omega}})=|v_{0}v_{6}v_{6}v_{7}v_{% 9}|=4</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.3.m3.2d">length ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ) = | italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT | = 4</annotation></semantics></math> and <math alttext="\textsf{length}({\textsf{Wit}_{\sigma_{0}}})=7" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.4.m4.1"><semantics id="S2.SS3.SSS0.Px1.p3.4.m4.1a"><mrow id="S2.SS3.SSS0.Px1.p3.4.m4.1.1" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.cmml"><mrow id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.3a.cmml">length</mtext><mo id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.2" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.2.cmml">=</mo><mn id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.3" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.3.cmml">7</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p3.4.m4.1b"><apply id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1"><eq id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.2"></eq><apply id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1"><times id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.2"></times><ci id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.3a.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.3">length</mtext></ci><apply id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.2a.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><cn id="S2.SS3.SSS0.Px1.p3.4.m4.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.4.m4.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.4.m4.1c">\textsf{length}({\textsf{Wit}_{\sigma_{0}}})=7</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.4.m4.1d">length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = 7</annotation></semantics></math>. Moreover, <math alttext="\textsf{Wit}_{\sigma_{0}}(v_{0})=\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.5.m5.1"><semantics id="S2.SS3.SSS0.Px1.p3.5.m5.1a"><mrow id="S2.SS3.SSS0.Px1.p3.5.m5.1.1" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.cmml"><mrow id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.2" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.3" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.2" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.2.cmml">=</mo><msub id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.2" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.2" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.3" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p3.5.m5.1b"><apply id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1"><eq id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.2"></eq><apply id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1"><times id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.2"></times><apply id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.3.3.3">0</cn></apply></apply><apply id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.5.m5.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.5.m5.1c">\textsf{Wit}_{\sigma_{0}}(v_{0})=\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.5.m5.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\textsf{Wit}_{\sigma_{0}}(v_{0}v_{1})=\{v_{0}v_{1}(v_{3}v_{4})^{\omega}\}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.6.m6.2"><semantics id="S2.SS3.SSS0.Px1.p3.6.m6.2a"><mrow id="S2.SS3.SSS0.Px1.p3.6.m6.2.2" xref="S2.SS3.SSS0.Px1.p3.6.m6.2.2.cmml"><mrow id="S2.SS3.SSS0.Px1.p3.6.m6.1.1.1" xref="S2.SS3.SSS0.Px1.p3.6.m6.1.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p3.6.m6.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.6.m6.1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.6.m6.1.1.1.3.2" 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xref="S2.SS3.SSS0.Px1.p3.6.m6.2.2.2.1.1.1.3">𝜔</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.6.m6.2c">\textsf{Wit}_{\sigma_{0}}(v_{0}v_{1})=\{v_{0}v_{1}(v_{3}v_{4})^{\omega}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.6.m6.2d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = { italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT }</annotation></semantics></math>, and <math alttext="\textsf{Wit}_{\sigma_{0}}(v_{0}v_{1}v_{2})=\emptyset" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.7.m7.1"><semantics id="S2.SS3.SSS0.Px1.p3.7.m7.1a"><mrow id="S2.SS3.SSS0.Px1.p3.7.m7.1.1" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.cmml"><mrow id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.2a.cmml">Wit</mtext><msub id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.3" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.3.cmml">0</mn></msub></msub><mo id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.cmml"><mi id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.3" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.1" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.3" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.1a" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.cmml"><mi id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.3" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.3.cmml">2</mn></msub></mrow><mo id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.2" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.2.cmml">=</mo><mi id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.3" mathvariant="normal" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p3.7.m7.1b"><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1"><eq id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.2"></eq><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1"><times id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.2"></times><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.2a.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.2">Wit</mtext></ci><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.2">𝜎</ci><cn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.3.3.3">0</cn></apply></apply><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1"><times id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.1"></times><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.2.3">0</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.3.3">1</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.1.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.2.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.1.1.1.1.4.3">2</cn></apply></apply></apply><emptyset id="S2.SS3.SSS0.Px1.p3.7.m7.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.7.m7.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.7.m7.1c">\textsf{Wit}_{\sigma_{0}}(v_{0}v_{1}v_{2})=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.7.m7.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ∅</annotation></semantics></math>. The initial vertex <math alttext="v_{0}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.8.m8.1"><semantics id="S2.SS3.SSS0.Px1.p3.8.m8.1a"><msub id="S2.SS3.SSS0.Px1.p3.8.m8.1.1" xref="S2.SS3.SSS0.Px1.p3.8.m8.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p3.8.m8.1.1.2" xref="S2.SS3.SSS0.Px1.p3.8.m8.1.1.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.8.m8.1.1.3" xref="S2.SS3.SSS0.Px1.p3.8.m8.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p3.8.m8.1b"><apply id="S2.SS3.SSS0.Px1.p3.8.m8.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.8.m8.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.8.m8.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.8.m8.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.8.m8.1.1.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.8.m8.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.8.m8.1c">v_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.8.m8.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a branching point, <math alttext="v_{0}v_{1}v_{2}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.9.m9.1"><semantics id="S2.SS3.SSS0.Px1.p3.9.m9.1a"><mrow id="S2.SS3.SSS0.Px1.p3.9.m9.1.1" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.cmml"><mi id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.2" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.3" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.1" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.2" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.3" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.1a" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.cmml"><mi id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.2" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.3" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p3.9.m9.1b"><apply id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1"><times id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.1"></times><apply id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.2.3">0</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.3.3">1</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.1.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.2.cmml" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.9.m9.1.1.4.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.9.m9.1c">v_{0}v_{1}v_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.9.m9.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is a deviation, and the region decomposition of the witness <math alttext="v_{0}v_{6}v_{6}v_{7}v_{9}^{\omega}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p3.10.m10.1"><semantics id="S2.SS3.SSS0.Px1.p3.10.m10.1a"><mrow id="S2.SS3.SSS0.Px1.p3.10.m10.1.1" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.cmml"><msub id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.cmml"><mi id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.2" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.3" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.2" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.3" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.3.cmml">6</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1a" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.cmml"><mi id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.2" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.3" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.3.cmml">6</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1b" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.cmml"><mi id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.2" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.3" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.3.cmml">7</mn></msub><mo id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1c" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1.cmml">⁢</mo><msubsup id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.cmml"><mi id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.2" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.2.cmml">v</mi><mn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.3" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.3.cmml">9</mn><mi id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.3" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.3.cmml">ω</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p3.10.m10.1b"><apply id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1"><times id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.1"></times><apply id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.2.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.2.3">0</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.3.3">6</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.2.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.4.3">6</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.2.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.5.3">7</cn></apply><apply id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6">superscript</csymbol><apply id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.1.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6">subscript</csymbol><ci id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.2.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.2">𝑣</ci><cn id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.3.cmml" type="integer" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.2.3">9</cn></apply><ci id="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.3.cmml" xref="S2.SS3.SSS0.Px1.p3.10.m10.1.1.6.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p3.10.m10.1c">v_{0}v_{6}v_{6}v_{7}v_{9}^{\omega}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.10.m10.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math> is equal to <math 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encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p3.13.m13.2d">{ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∣ italic_k ≥ 1 }</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS3.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Reduction to binary arenas.</h4> <div class="ltx_para" id="S2.SS3.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS3.SSS0.Px2.p1.5">Working with general arenas requires to deal with the parameter <math alttext="W" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.1.m1.1"><semantics id="S2.SS3.SSS0.Px2.p1.1.m1.1a"><mi id="S2.SS3.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.1.m1.1b"><ci id="S2.SS3.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.1.m1.1d">italic_W</annotation></semantics></math> in most of the proofs. To simplify the arguments, we reduce the SPS problem to <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px2.p1.5.1">binary</em> arenas, by replacing each edge with a weight <math alttext="w\geq 2" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.2.m2.1"><semantics id="S2.SS3.SSS0.Px2.p1.2.m2.1a"><mrow id="S2.SS3.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.2" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.2.cmml">w</mi><mo id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.1" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.1.cmml">≥</mo><mn id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.3" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.2.m2.1b"><apply id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1"><geq id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.1"></geq><ci id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.2">𝑤</ci><cn id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.2.m2.1c">w\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.2.m2.1d">italic_w ≥ 2</annotation></semantics></math> by a path of <math alttext="w" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.3.m3.1"><semantics id="S2.SS3.SSS0.Px2.p1.3.m3.1a"><mi id="S2.SS3.SSS0.Px2.p1.3.m3.1.1" xref="S2.SS3.SSS0.Px2.p1.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.3.m3.1b"><ci id="S2.SS3.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.3.m3.1d">italic_w</annotation></semantics></math> edges of weight <math alttext="1" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.4.m4.1"><semantics id="S2.SS3.SSS0.Px2.p1.4.m4.1a"><mn id="S2.SS3.SSS0.Px2.p1.4.m4.1.1" xref="S2.SS3.SSS0.Px2.p1.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.4.m4.1b"><cn id="S2.SS3.SSS0.Px2.p1.4.m4.1.1.cmml" type="integer" xref="S2.SS3.SSS0.Px2.p1.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.4.m4.1d">1</annotation></semantics></math>. This (standard) reduction is exponential, but only in the size of the binary encoding of <math alttext="W" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.5.m5.1"><semantics id="S2.SS3.SSS0.Px2.p1.5.m5.1a"><mi id="S2.SS3.SSS0.Px2.p1.5.m5.1.1" xref="S2.SS3.SSS0.Px2.p1.5.m5.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.5.m5.1b"><ci id="S2.SS3.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.5.m5.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.5.m5.1c">W</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.5.m5.1d">italic_W</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.1.1.1">Lemma 2.2</span></span></h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem2.p1.4.4">Let <math alttext="G=(A,T_{0},\ldots,T_{t})" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.1.m1.4"><semantics id="S2.Thmtheorem2.p1.1.1.m1.4a"><mrow id="S2.Thmtheorem2.p1.1.1.m1.4.4" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.cmml"><mi id="S2.Thmtheorem2.p1.1.1.m1.4.4.4" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.4.cmml">G</mi><mo id="S2.Thmtheorem2.p1.1.1.m1.4.4.3" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.3.cmml">=</mo><mrow id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml"><mo id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.3" stretchy="false" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml">(</mo><mi id="S2.Thmtheorem2.p1.1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml">A</mi><mo id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.4" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml">,</mo><msub id="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1" xref="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2" xref="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.cmml">T</mi><mn id="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3" xref="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.cmml">0</mn></msub><mo id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.5" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml">,</mo><mi id="S2.Thmtheorem2.p1.1.1.m1.2.2" mathvariant="normal" xref="S2.Thmtheorem2.p1.1.1.m1.2.2.cmml">…</mi><mo id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.6" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml">,</mo><msub id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.cmml"><mi id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.cmml">T</mi><mi id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.3" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.3.cmml">t</mi></msub><mo id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.7" stretchy="false" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.1.m1.4b"><apply id="S2.Thmtheorem2.p1.1.1.m1.4.4.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4"><eq id="S2.Thmtheorem2.p1.1.1.m1.4.4.3.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.3"></eq><ci id="S2.Thmtheorem2.p1.1.1.m1.4.4.4.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.4">𝐺</ci><vector id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2"><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1">𝐴</ci><apply id="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2">𝑇</ci><cn id="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3">0</cn></apply><ci id="S2.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.2.2">…</ci><apply id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2">𝑇</ci><ci id="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.3.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.3">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.1.m1.4c">G=(A,T_{0},\ldots,T_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.1.m1.4d">italic_G = ( italic_A , italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> be an SP game and <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.2.m2.1"><semantics id="S2.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem2.p1.2.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">B</mi><mo id="S2.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.2.m2.1b"><apply id="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.1.1"><in id="S2.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.1"></in><ci id="S2.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.2">𝐵</ci><ci id="S2.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>. Then one can construct in exponential time an SP game <math alttext="G^{\prime}=(A^{\prime},T_{0},\ldots,T_{t})" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.3.3.m3.4"><semantics id="S2.Thmtheorem2.p1.3.3.m3.4a"><mrow id="S2.Thmtheorem2.p1.3.3.m3.4.4" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.cmml"><msup id="S2.Thmtheorem2.p1.3.3.m3.4.4.5" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.5.cmml"><mi id="S2.Thmtheorem2.p1.3.3.m3.4.4.5.2" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.5.2.cmml">G</mi><mo id="S2.Thmtheorem2.p1.3.3.m3.4.4.5.3" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.5.3.cmml">′</mo></msup><mo id="S2.Thmtheorem2.p1.3.3.m3.4.4.4" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.4.cmml">=</mo><mrow id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml"><mo id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.4" stretchy="false" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml">(</mo><msup id="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1" xref="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.2" xref="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.3" xref="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.5" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml">,</mo><msub id="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2" xref="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.cmml"><mi id="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.2" xref="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.2.cmml">T</mi><mn id="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.3" xref="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.6" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml">,</mo><mi id="S2.Thmtheorem2.p1.3.3.m3.1.1" mathvariant="normal" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml">…</mi><mo id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.7" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml">,</mo><msub id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.cmml"><mi id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.2" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.2.cmml">T</mi><mi id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.3" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.3.cmml">t</mi></msub><mo id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.8" stretchy="false" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.3.3.m3.4b"><apply id="S2.Thmtheorem2.p1.3.3.m3.4.4.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4"><eq id="S2.Thmtheorem2.p1.3.3.m3.4.4.4.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.4"></eq><apply id="S2.Thmtheorem2.p1.3.3.m3.4.4.5.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.5"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.3.3.m3.4.4.5.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.5">superscript</csymbol><ci id="S2.Thmtheorem2.p1.3.3.m3.4.4.5.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.5.2">𝐺</ci><ci id="S2.Thmtheorem2.p1.3.3.m3.4.4.5.3.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.5.3">′</ci></apply><vector id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3"><apply id="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.2">𝐴</ci><ci id="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.3">′</ci></apply><apply id="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.2">𝑇</ci><cn id="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem2.p1.3.3.m3.3.3.2.2.2.3">0</cn></apply><ci id="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1">…</ci><apply id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3">subscript</csymbol><ci id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.2">𝑇</ci><ci id="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.3.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.4.4.3.3.3.3">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.3.3.m3.4c">G^{\prime}=(A^{\prime},T_{0},\ldots,T_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.3.3.m3.4d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> with a binary arena <math alttext="A^{\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.4.4.m4.1"><semantics id="S2.Thmtheorem2.p1.4.4.m4.1a"><msup id="S2.Thmtheorem2.p1.4.4.m4.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.4.4.m4.1b"><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1">superscript</csymbol><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2">𝐴</ci><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.4.4.m4.1c">A^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.4.4.m4.1d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that</span></p> <ul class="ltx_itemize" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.1">the set of vertices </span><math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><msup id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml"><mi id="S2.I1.i1.p1.1.m1.1.1.2" xref="S2.I1.i1.p1.1.m1.1.1.2.cmml">V</mi><mo id="S2.I1.i1.p1.1.m1.1.1.3" xref="S2.I1.i1.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><apply id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.I1.i1.p1.1.m1.1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1">superscript</csymbol><ci id="S2.I1.i1.p1.1.m1.1.1.2.cmml" xref="S2.I1.i1.p1.1.m1.1.1.2">𝑉</ci><ci id="S2.I1.i1.p1.1.m1.1.1.3.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.2"> of </span><math alttext="A^{\prime}" class="ltx_Math" display="inline" id="S2.I1.i1.p1.2.m2.1"><semantics id="S2.I1.i1.p1.2.m2.1a"><msup id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml"><mi id="S2.I1.i1.p1.2.m2.1.1.2" xref="S2.I1.i1.p1.2.m2.1.1.2.cmml">A</mi><mo id="S2.I1.i1.p1.2.m2.1.1.3" xref="S2.I1.i1.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.1b"><apply id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.I1.i1.p1.2.m2.1.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1">superscript</csymbol><ci id="S2.I1.i1.p1.2.m2.1.1.2.cmml" xref="S2.I1.i1.p1.2.m2.1.1.2">𝐴</ci><ci id="S2.I1.i1.p1.2.m2.1.1.3.cmml" xref="S2.I1.i1.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.1c">A^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.2.m2.1d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.3"> contains </span><math alttext="V" class="ltx_Math" display="inline" id="S2.I1.i1.p1.3.m3.1"><semantics id="S2.I1.i1.p1.3.m3.1a"><mi id="S2.I1.i1.p1.3.m3.1.1" xref="S2.I1.i1.p1.3.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.3.m3.1b"><ci id="S2.I1.i1.p1.3.m3.1.1.cmml" xref="S2.I1.i1.p1.3.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.3.m3.1d">italic_V</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.4"> and has size </span><math alttext="|V^{\prime}|\leq|V|\cdot W" class="ltx_Math" display="inline" id="S2.I1.i1.p1.4.m4.2"><semantics id="S2.I1.i1.p1.4.m4.2a"><mrow id="S2.I1.i1.p1.4.m4.2.2" xref="S2.I1.i1.p1.4.m4.2.2.cmml"><mrow id="S2.I1.i1.p1.4.m4.2.2.1.1" xref="S2.I1.i1.p1.4.m4.2.2.1.2.cmml"><mo id="S2.I1.i1.p1.4.m4.2.2.1.1.2" stretchy="false" xref="S2.I1.i1.p1.4.m4.2.2.1.2.1.cmml">|</mo><msup id="S2.I1.i1.p1.4.m4.2.2.1.1.1" xref="S2.I1.i1.p1.4.m4.2.2.1.1.1.cmml"><mi id="S2.I1.i1.p1.4.m4.2.2.1.1.1.2" xref="S2.I1.i1.p1.4.m4.2.2.1.1.1.2.cmml">V</mi><mo id="S2.I1.i1.p1.4.m4.2.2.1.1.1.3" xref="S2.I1.i1.p1.4.m4.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.I1.i1.p1.4.m4.2.2.1.1.3" stretchy="false" xref="S2.I1.i1.p1.4.m4.2.2.1.2.1.cmml">|</mo></mrow><mo id="S2.I1.i1.p1.4.m4.2.2.2" xref="S2.I1.i1.p1.4.m4.2.2.2.cmml">≤</mo><mrow id="S2.I1.i1.p1.4.m4.2.2.3" xref="S2.I1.i1.p1.4.m4.2.2.3.cmml"><mrow id="S2.I1.i1.p1.4.m4.2.2.3.2.2" xref="S2.I1.i1.p1.4.m4.2.2.3.2.1.cmml"><mo id="S2.I1.i1.p1.4.m4.2.2.3.2.2.1" stretchy="false" xref="S2.I1.i1.p1.4.m4.2.2.3.2.1.1.cmml">|</mo><mi id="S2.I1.i1.p1.4.m4.1.1" xref="S2.I1.i1.p1.4.m4.1.1.cmml">V</mi><mo id="S2.I1.i1.p1.4.m4.2.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.I1.i1.p1.4.m4.2.2.3.2.1.1.cmml">|</mo></mrow><mo id="S2.I1.i1.p1.4.m4.2.2.3.1" rspace="0.222em" xref="S2.I1.i1.p1.4.m4.2.2.3.1.cmml">⋅</mo><mi id="S2.I1.i1.p1.4.m4.2.2.3.3" xref="S2.I1.i1.p1.4.m4.2.2.3.3.cmml">W</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.4.m4.2b"><apply id="S2.I1.i1.p1.4.m4.2.2.cmml" xref="S2.I1.i1.p1.4.m4.2.2"><leq id="S2.I1.i1.p1.4.m4.2.2.2.cmml" xref="S2.I1.i1.p1.4.m4.2.2.2"></leq><apply id="S2.I1.i1.p1.4.m4.2.2.1.2.cmml" xref="S2.I1.i1.p1.4.m4.2.2.1.1"><abs id="S2.I1.i1.p1.4.m4.2.2.1.2.1.cmml" xref="S2.I1.i1.p1.4.m4.2.2.1.1.2"></abs><apply id="S2.I1.i1.p1.4.m4.2.2.1.1.1.cmml" xref="S2.I1.i1.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i1.p1.4.m4.2.2.1.1.1.1.cmml" xref="S2.I1.i1.p1.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S2.I1.i1.p1.4.m4.2.2.1.1.1.2.cmml" xref="S2.I1.i1.p1.4.m4.2.2.1.1.1.2">𝑉</ci><ci id="S2.I1.i1.p1.4.m4.2.2.1.1.1.3.cmml" xref="S2.I1.i1.p1.4.m4.2.2.1.1.1.3">′</ci></apply></apply><apply id="S2.I1.i1.p1.4.m4.2.2.3.cmml" xref="S2.I1.i1.p1.4.m4.2.2.3"><ci id="S2.I1.i1.p1.4.m4.2.2.3.1.cmml" xref="S2.I1.i1.p1.4.m4.2.2.3.1">⋅</ci><apply id="S2.I1.i1.p1.4.m4.2.2.3.2.1.cmml" xref="S2.I1.i1.p1.4.m4.2.2.3.2.2"><abs id="S2.I1.i1.p1.4.m4.2.2.3.2.1.1.cmml" xref="S2.I1.i1.p1.4.m4.2.2.3.2.2.1"></abs><ci id="S2.I1.i1.p1.4.m4.1.1.cmml" xref="S2.I1.i1.p1.4.m4.1.1">𝑉</ci></apply><ci id="S2.I1.i1.p1.4.m4.2.2.3.3.cmml" xref="S2.I1.i1.p1.4.m4.2.2.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.4.m4.2c">|V^{\prime}|\leq|V|\cdot W</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.4.m4.2d">| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≤ | italic_V | ⋅ italic_W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.5">,</span></p> </div> </li> <li class="ltx_item" id="S2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.7"><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.1">there exists a one-to-one correspondance </span><math alttext="g" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.1"><semantics id="S2.I1.i2.p1.1.m1.1a"><mi id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.1b"><ci id="S2.I1.i2.p1.1.m1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.1d">italic_g</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.2"> between the set of strategies on </span><math alttext="A" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.1"><semantics id="S2.I1.i2.p1.2.m2.1a"><mi id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.1b"><ci id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.1d">italic_A</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.3"> and the set of strategies on </span><math alttext="A^{\prime}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.3.m3.1"><semantics id="S2.I1.i2.p1.3.m3.1a"><msup id="S2.I1.i2.p1.3.m3.1.1" xref="S2.I1.i2.p1.3.m3.1.1.cmml"><mi id="S2.I1.i2.p1.3.m3.1.1.2" xref="S2.I1.i2.p1.3.m3.1.1.2.cmml">A</mi><mo id="S2.I1.i2.p1.3.m3.1.1.3" xref="S2.I1.i2.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.3.m3.1b"><apply id="S2.I1.i2.p1.3.m3.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.3.m3.1.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1">superscript</csymbol><ci id="S2.I1.i2.p1.3.m3.1.1.2.cmml" xref="S2.I1.i2.p1.3.m3.1.1.2">𝐴</ci><ci id="S2.I1.i2.p1.3.m3.1.1.3.cmml" xref="S2.I1.i2.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.3.m3.1c">A^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.3.m3.1d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.4">. Moreover </span><math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.4.m4.1"><semantics id="S2.I1.i2.p1.4.m4.1a"><msub id="S2.I1.i2.p1.4.m4.1.1" xref="S2.I1.i2.p1.4.m4.1.1.cmml"><mi id="S2.I1.i2.p1.4.m4.1.1.2" xref="S2.I1.i2.p1.4.m4.1.1.2.cmml">σ</mi><mn id="S2.I1.i2.p1.4.m4.1.1.3" xref="S2.I1.i2.p1.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.4.m4.1b"><apply id="S2.I1.i2.p1.4.m4.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.4.m4.1.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.4.m4.1.1.2.cmml" xref="S2.I1.i2.p1.4.m4.1.1.2">𝜎</ci><cn id="S2.I1.i2.p1.4.m4.1.1.3.cmml" type="integer" xref="S2.I1.i2.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.4.m4.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.5"> is a solution in </span><math alttext="\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S2.I1.i2.p1.5.m5.2"><semantics id="S2.I1.i2.p1.5.m5.2a"><mrow id="S2.I1.i2.p1.5.m5.2.3" xref="S2.I1.i2.p1.5.m5.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S2.I1.i2.p1.5.m5.2.3.2" xref="S2.I1.i2.p1.5.m5.2.3.2a.cmml">SPS</mtext><mo id="S2.I1.i2.p1.5.m5.2.3.1" xref="S2.I1.i2.p1.5.m5.2.3.1.cmml">⁢</mo><mrow id="S2.I1.i2.p1.5.m5.2.3.3.2" xref="S2.I1.i2.p1.5.m5.2.3.3.1.cmml"><mo id="S2.I1.i2.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S2.I1.i2.p1.5.m5.2.3.3.1.cmml">(</mo><mi id="S2.I1.i2.p1.5.m5.1.1" xref="S2.I1.i2.p1.5.m5.1.1.cmml">G</mi><mo id="S2.I1.i2.p1.5.m5.2.3.3.2.2" xref="S2.I1.i2.p1.5.m5.2.3.3.1.cmml">,</mo><mi id="S2.I1.i2.p1.5.m5.2.2" xref="S2.I1.i2.p1.5.m5.2.2.cmml">B</mi><mo id="S2.I1.i2.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S2.I1.i2.p1.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.5.m5.2b"><apply id="S2.I1.i2.p1.5.m5.2.3.cmml" xref="S2.I1.i2.p1.5.m5.2.3"><times id="S2.I1.i2.p1.5.m5.2.3.1.cmml" xref="S2.I1.i2.p1.5.m5.2.3.1"></times><ci id="S2.I1.i2.p1.5.m5.2.3.2a.cmml" xref="S2.I1.i2.p1.5.m5.2.3.2"><mtext class="ltx_mathvariant_italic" id="S2.I1.i2.p1.5.m5.2.3.2.cmml" xref="S2.I1.i2.p1.5.m5.2.3.2">SPS</mtext></ci><interval closure="open" id="S2.I1.i2.p1.5.m5.2.3.3.1.cmml" xref="S2.I1.i2.p1.5.m5.2.3.3.2"><ci id="S2.I1.i2.p1.5.m5.1.1.cmml" xref="S2.I1.i2.p1.5.m5.1.1">𝐺</ci><ci id="S2.I1.i2.p1.5.m5.2.2.cmml" xref="S2.I1.i2.p1.5.m5.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.5.m5.2c">\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.5.m5.2d">SPS ( italic_G , italic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.6"> if and only if </span><math alttext="g(\sigma_{0})" class="ltx_Math" display="inline" id="S2.I1.i2.p1.6.m6.1"><semantics id="S2.I1.i2.p1.6.m6.1a"><mrow id="S2.I1.i2.p1.6.m6.1.1" xref="S2.I1.i2.p1.6.m6.1.1.cmml"><mi id="S2.I1.i2.p1.6.m6.1.1.3" xref="S2.I1.i2.p1.6.m6.1.1.3.cmml">g</mi><mo id="S2.I1.i2.p1.6.m6.1.1.2" xref="S2.I1.i2.p1.6.m6.1.1.2.cmml">⁢</mo><mrow id="S2.I1.i2.p1.6.m6.1.1.1.1" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.cmml"><mo id="S2.I1.i2.p1.6.m6.1.1.1.1.2" stretchy="false" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.cmml">(</mo><msub id="S2.I1.i2.p1.6.m6.1.1.1.1.1" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.cmml"><mi id="S2.I1.i2.p1.6.m6.1.1.1.1.1.2" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.2.cmml">σ</mi><mn id="S2.I1.i2.p1.6.m6.1.1.1.1.1.3" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.I1.i2.p1.6.m6.1.1.1.1.3" stretchy="false" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.6.m6.1b"><apply id="S2.I1.i2.p1.6.m6.1.1.cmml" xref="S2.I1.i2.p1.6.m6.1.1"><times id="S2.I1.i2.p1.6.m6.1.1.2.cmml" xref="S2.I1.i2.p1.6.m6.1.1.2"></times><ci id="S2.I1.i2.p1.6.m6.1.1.3.cmml" xref="S2.I1.i2.p1.6.m6.1.1.3">𝑔</ci><apply id="S2.I1.i2.p1.6.m6.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.6.m6.1.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.6.m6.1.1.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.6.m6.1.1.1.1.1.2.cmml" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.2">𝜎</ci><cn id="S2.I1.i2.p1.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S2.I1.i2.p1.6.m6.1.1.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.6.m6.1c">g(\sigma_{0})</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.6.m6.1d">italic_g ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.7"> is a solution in </span><math alttext="\mbox{SPS}({G^{\prime}},{B})" class="ltx_Math" display="inline" id="S2.I1.i2.p1.7.m7.2"><semantics id="S2.I1.i2.p1.7.m7.2a"><mrow id="S2.I1.i2.p1.7.m7.2.2" xref="S2.I1.i2.p1.7.m7.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.I1.i2.p1.7.m7.2.2.3" xref="S2.I1.i2.p1.7.m7.2.2.3a.cmml">SPS</mtext><mo id="S2.I1.i2.p1.7.m7.2.2.2" xref="S2.I1.i2.p1.7.m7.2.2.2.cmml">⁢</mo><mrow id="S2.I1.i2.p1.7.m7.2.2.1.1" xref="S2.I1.i2.p1.7.m7.2.2.1.2.cmml"><mo id="S2.I1.i2.p1.7.m7.2.2.1.1.2" stretchy="false" xref="S2.I1.i2.p1.7.m7.2.2.1.2.cmml">(</mo><msup id="S2.I1.i2.p1.7.m7.2.2.1.1.1" xref="S2.I1.i2.p1.7.m7.2.2.1.1.1.cmml"><mi id="S2.I1.i2.p1.7.m7.2.2.1.1.1.2" xref="S2.I1.i2.p1.7.m7.2.2.1.1.1.2.cmml">G</mi><mo id="S2.I1.i2.p1.7.m7.2.2.1.1.1.3" xref="S2.I1.i2.p1.7.m7.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.I1.i2.p1.7.m7.2.2.1.1.3" xref="S2.I1.i2.p1.7.m7.2.2.1.2.cmml">,</mo><mi id="S2.I1.i2.p1.7.m7.1.1" xref="S2.I1.i2.p1.7.m7.1.1.cmml">B</mi><mo id="S2.I1.i2.p1.7.m7.2.2.1.1.4" stretchy="false" xref="S2.I1.i2.p1.7.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.7.m7.2b"><apply id="S2.I1.i2.p1.7.m7.2.2.cmml" xref="S2.I1.i2.p1.7.m7.2.2"><times id="S2.I1.i2.p1.7.m7.2.2.2.cmml" xref="S2.I1.i2.p1.7.m7.2.2.2"></times><ci id="S2.I1.i2.p1.7.m7.2.2.3a.cmml" xref="S2.I1.i2.p1.7.m7.2.2.3"><mtext class="ltx_mathvariant_italic" id="S2.I1.i2.p1.7.m7.2.2.3.cmml" xref="S2.I1.i2.p1.7.m7.2.2.3">SPS</mtext></ci><interval closure="open" id="S2.I1.i2.p1.7.m7.2.2.1.2.cmml" xref="S2.I1.i2.p1.7.m7.2.2.1.1"><apply id="S2.I1.i2.p1.7.m7.2.2.1.1.1.cmml" xref="S2.I1.i2.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.7.m7.2.2.1.1.1.1.cmml" xref="S2.I1.i2.p1.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S2.I1.i2.p1.7.m7.2.2.1.1.1.2.cmml" xref="S2.I1.i2.p1.7.m7.2.2.1.1.1.2">𝐺</ci><ci id="S2.I1.i2.p1.7.m7.2.2.1.1.1.3.cmml" xref="S2.I1.i2.p1.7.m7.2.2.1.1.1.3">′</ci></apply><ci id="S2.I1.i2.p1.7.m7.1.1.cmml" xref="S2.I1.i2.p1.7.m7.1.1">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.7.m7.2c">\mbox{SPS}({G^{\prime}},{B})</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.7.m7.2d">SPS ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.7.8">.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S2.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.1.1.1">Proof 2.3</span></span></h6> <div class="ltx_para" id="S2.Thmtheorem3.p1"> <p class="ltx_p" id="S2.Thmtheorem3.p1.22"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p1.22.22">Let <math alttext="X=\{e\in E\mid w(e)\geq 2\}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.1.1.m1.3"><semantics id="S2.Thmtheorem3.p1.1.1.m1.3a"><mrow id="S2.Thmtheorem3.p1.1.1.m1.3.3" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.cmml"><mi id="S2.Thmtheorem3.p1.1.1.m1.3.3.4" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.4.cmml">X</mi><mo id="S2.Thmtheorem3.p1.1.1.m1.3.3.3" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.3.cmml">=</mo><mrow id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.3.cmml"><mo id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.3" stretchy="false" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.3.1.cmml">{</mo><mrow id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.2" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.2.cmml">e</mi><mo id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.1" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.3" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.3.cmml">E</mi></mrow><mo fence="true" id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.4" lspace="0em" rspace="0em" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.3.1.cmml">∣</mo><mrow id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.cmml"><mrow id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.cmml"><mi id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.2" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.2.cmml">w</mi><mo id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.1" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.3.2" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.cmml"><mo id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.cmml">(</mo><mi id="S2.Thmtheorem3.p1.1.1.m1.1.1" xref="S2.Thmtheorem3.p1.1.1.m1.1.1.cmml">e</mi><mo id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.1" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.1.cmml">≥</mo><mn id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.3" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.3.cmml">2</mn></mrow><mo id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.5" stretchy="false" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.1.1.m1.3b"><apply id="S2.Thmtheorem3.p1.1.1.m1.3.3.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3"><eq id="S2.Thmtheorem3.p1.1.1.m1.3.3.3.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.3"></eq><ci id="S2.Thmtheorem3.p1.1.1.m1.3.3.4.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.4">𝑋</ci><apply id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.3.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.3.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.3">conditional-set</csymbol><apply id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1"><in id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.1"></in><ci id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.2">𝑒</ci><ci id="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.2.2.1.1.1.3">𝐸</ci></apply><apply id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2"><geq id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.1"></geq><apply id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2"><times id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.1"></times><ci id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.2.2">𝑤</ci><ci id="S2.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.1">𝑒</ci></apply><cn id="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.1.1.m1.3.3.2.2.2.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.1.1.m1.3c">X=\{e\in E\mid w(e)\geq 2\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.1.1.m1.3d">italic_X = { italic_e ∈ italic_E ∣ italic_w ( italic_e ) ≥ 2 }</annotation></semantics></math> be the set of edges of <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.2.2.m2.1"><semantics id="S2.Thmtheorem3.p1.2.2.m2.1a"><mi id="S2.Thmtheorem3.p1.2.2.m2.1.1" xref="S2.Thmtheorem3.p1.2.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.2.2.m2.1b"><ci id="S2.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem3.p1.2.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.2.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.2.2.m2.1d">italic_A</annotation></semantics></math> with weight at least 2. For each <math alttext="e\in X" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.3.3.m3.1"><semantics id="S2.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem3.p1.3.3.m3.1.1" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">e</mi><mo id="S2.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.3.3.m3.1b"><apply id="S2.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1"><in id="S2.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.1"></in><ci id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2">𝑒</ci><ci id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.3.3.m3.1c">e\in X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.3.3.m3.1d">italic_e ∈ italic_X</annotation></semantics></math>, we replace <math alttext="e" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.4.4.m4.1"><semantics id="S2.Thmtheorem3.p1.4.4.m4.1a"><mi id="S2.Thmtheorem3.p1.4.4.m4.1.1" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.4.4.m4.1b"><ci id="S2.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.4.4.m4.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.4.4.m4.1d">italic_e</annotation></semantics></math> by a succession of <math alttext="w(e)-1" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.5.5.m5.1"><semantics id="S2.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S2.Thmtheorem3.p1.5.5.m5.1.2" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.cmml"><mrow id="S2.Thmtheorem3.p1.5.5.m5.1.2.2" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.cmml"><mi id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.2" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.2.cmml">w</mi><mo id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.1" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.3.2" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.cmml"><mo id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.cmml">(</mo><mi id="S2.Thmtheorem3.p1.5.5.m5.1.1" xref="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml">e</mi><mo id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p1.5.5.m5.1.2.1" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.1.cmml">−</mo><mn id="S2.Thmtheorem3.p1.5.5.m5.1.2.3" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.5.5.m5.1b"><apply id="S2.Thmtheorem3.p1.5.5.m5.1.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.2"><minus id="S2.Thmtheorem3.p1.5.5.m5.1.2.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.1"></minus><apply id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2"><times id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.1"></times><ci id="S2.Thmtheorem3.p1.5.5.m5.1.2.2.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.2.2">𝑤</ci><ci id="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.1">𝑒</ci></apply><cn id="S2.Thmtheorem3.p1.5.5.m5.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.5.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.5.5.m5.1c">w(e)-1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.5.5.m5.1d">italic_w ( italic_e ) - 1</annotation></semantics></math> new vertices linked by <math alttext="w(e)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.6.6.m6.1"><semantics id="S2.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem3.p1.6.6.m6.1.2" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.cmml"><mi id="S2.Thmtheorem3.p1.6.6.m6.1.2.2" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.2.cmml">w</mi><mo id="S2.Thmtheorem3.p1.6.6.m6.1.2.1" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.6.6.m6.1.2.3.2" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.cmml"><mo id="S2.Thmtheorem3.p1.6.6.m6.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.cmml">(</mo><mi id="S2.Thmtheorem3.p1.6.6.m6.1.1" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.cmml">e</mi><mo id="S2.Thmtheorem3.p1.6.6.m6.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.6.6.m6.1b"><apply id="S2.Thmtheorem3.p1.6.6.m6.1.2.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.2"><times id="S2.Thmtheorem3.p1.6.6.m6.1.2.1.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.1"></times><ci id="S2.Thmtheorem3.p1.6.6.m6.1.2.2.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.2.2">𝑤</ci><ci id="S2.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.6.6.m6.1c">w(e)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.6.6.m6.1d">italic_w ( italic_e )</annotation></semantics></math> new edges of weight 1, and for each <math alttext="e\in E\setminus X" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.7.7.m7.1"><semantics id="S2.Thmtheorem3.p1.7.7.m7.1a"><mrow id="S2.Thmtheorem3.p1.7.7.m7.1.1" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.cmml"><mi id="S2.Thmtheorem3.p1.7.7.m7.1.1.2" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.2.cmml">e</mi><mo id="S2.Thmtheorem3.p1.7.7.m7.1.1.1" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.1.cmml">∈</mo><mrow id="S2.Thmtheorem3.p1.7.7.m7.1.1.3" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2.cmml">E</mi><mo id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.1" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.1.cmml">∖</mo><mi id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.3" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.7.7.m7.1b"><apply id="S2.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1"><in id="S2.Thmtheorem3.p1.7.7.m7.1.1.1.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.1"></in><ci id="S2.Thmtheorem3.p1.7.7.m7.1.1.2.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.2">𝑒</ci><apply id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3"><setdiff id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.1"></setdiff><ci id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2">𝐸</ci><ci id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.3.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.7.7.m7.1c">e\in E\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.7.7.m7.1d">italic_e ∈ italic_E ∖ italic_X</annotation></semantics></math>, we keep <math alttext="e" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.8.8.m8.1"><semantics id="S2.Thmtheorem3.p1.8.8.m8.1a"><mi id="S2.Thmtheorem3.p1.8.8.m8.1.1" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.8.8.m8.1b"><ci id="S2.Thmtheorem3.p1.8.8.m8.1.1.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.8.8.m8.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.8.8.m8.1d">italic_e</annotation></semantics></math> unmodified. In this way, we get a directed graph <math alttext="(V^{\prime},E^{\prime})" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.9.9.m9.2"><semantics id="S2.Thmtheorem3.p1.9.9.m9.2a"><mrow id="S2.Thmtheorem3.p1.9.9.m9.2.2.2" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.3.cmml"><mo id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.3" stretchy="false" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.3.cmml">(</mo><msup id="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.2" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.3" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.4" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.3.cmml">,</mo><msup id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.cmml"><mi id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.2" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.2.cmml">E</mi><mo id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.3" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.5" stretchy="false" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.9.9.m9.2b"><interval closure="open" id="S2.Thmtheorem3.p1.9.9.m9.2.2.3.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2"><apply id="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.1.1.3">′</ci></apply><apply id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.2">𝐸</ci><ci id="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.3.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.9.9.m9.2c">(V^{\prime},E^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.9.9.m9.2d">( italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> with a weight function <math alttext="w^{\prime}\colon E^{\prime}\rightarrow\{0,1\}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.10.10.m10.2"><semantics id="S2.Thmtheorem3.p1.10.10.m10.2a"><mrow id="S2.Thmtheorem3.p1.10.10.m10.2.3" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.cmml"><msup id="S2.Thmtheorem3.p1.10.10.m10.2.3.2" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.2.cmml"><mi id="S2.Thmtheorem3.p1.10.10.m10.2.3.2.2" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.2.2.cmml">w</mi><mo id="S2.Thmtheorem3.p1.10.10.m10.2.3.2.3" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.10.10.m10.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.1.cmml">:</mo><mrow id="S2.Thmtheorem3.p1.10.10.m10.2.3.3" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.cmml"><msup id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.cmml"><mi id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.2" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.2.cmml">E</mi><mo id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.3" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.1" stretchy="false" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.1.cmml">→</mo><mrow id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.2" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.1.cmml"><mo id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.1.cmml">{</mo><mn id="S2.Thmtheorem3.p1.10.10.m10.1.1" xref="S2.Thmtheorem3.p1.10.10.m10.1.1.cmml">0</mn><mo id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.2.2" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.1.cmml">,</mo><mn id="S2.Thmtheorem3.p1.10.10.m10.2.2" xref="S2.Thmtheorem3.p1.10.10.m10.2.2.cmml">1</mn><mo id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.2.3" stretchy="false" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.10.10.m10.2b"><apply id="S2.Thmtheorem3.p1.10.10.m10.2.3.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3"><ci id="S2.Thmtheorem3.p1.10.10.m10.2.3.1.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.1">:</ci><apply id="S2.Thmtheorem3.p1.10.10.m10.2.3.2.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.10.10.m10.2.3.2.1.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.10.10.m10.2.3.2.2.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.2.2">𝑤</ci><ci id="S2.Thmtheorem3.p1.10.10.m10.2.3.2.3.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.2.3">′</ci></apply><apply id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3"><ci id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.1.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.1">→</ci><apply id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.1.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.2.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.2">𝐸</ci><ci id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.3.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.2.3">′</ci></apply><set id="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.1.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.2.3.3.3.2"><cn id="S2.Thmtheorem3.p1.10.10.m10.1.1.cmml" type="integer" xref="S2.Thmtheorem3.p1.10.10.m10.1.1">0</cn><cn id="S2.Thmtheorem3.p1.10.10.m10.2.2.cmml" type="integer" xref="S2.Thmtheorem3.p1.10.10.m10.2.2">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.10.10.m10.2c">w^{\prime}\colon E^{\prime}\rightarrow\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.10.10.m10.2d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> and with a size <math alttext="|V^{\prime}|\leq|V|\cdot W" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.11.11.m11.2"><semantics id="S2.Thmtheorem3.p1.11.11.m11.2a"><mrow id="S2.Thmtheorem3.p1.11.11.m11.2.2" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.cmml"><mrow id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.2.cmml"><mo id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.2.1.cmml">|</mo><msup id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.2" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.3" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.2.1.cmml">|</mo></mrow><mo id="S2.Thmtheorem3.p1.11.11.m11.2.2.2" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.2.cmml">≤</mo><mrow id="S2.Thmtheorem3.p1.11.11.m11.2.2.3" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.cmml"><mrow id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.2" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.1.cmml"><mo id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.1.1.cmml">|</mo><mi id="S2.Thmtheorem3.p1.11.11.m11.1.1" xref="S2.Thmtheorem3.p1.11.11.m11.1.1.cmml">V</mi><mo id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.1.1.cmml">|</mo></mrow><mo id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.1" rspace="0.222em" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.1.cmml">⋅</mo><mi id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.3" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.3.cmml">W</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.11.11.m11.2b"><apply id="S2.Thmtheorem3.p1.11.11.m11.2.2.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2"><leq id="S2.Thmtheorem3.p1.11.11.m11.2.2.2.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.2"></leq><apply id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.2.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1"><abs id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.2.1.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.2"></abs><apply id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.1.1.1.3">′</ci></apply></apply><apply id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3"><ci id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.1.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.1">⋅</ci><apply id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.1.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.2"><abs id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.1.1.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.2.2.1"></abs><ci id="S2.Thmtheorem3.p1.11.11.m11.1.1.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.1.1">𝑉</ci></apply><ci id="S2.Thmtheorem3.p1.11.11.m11.2.2.3.3.cmml" xref="S2.Thmtheorem3.p1.11.11.m11.2.2.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.11.11.m11.2c">|V^{\prime}|\leq|V|\cdot W</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.11.11.m11.2d">| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≤ | italic_V | ⋅ italic_W</annotation></semantics></math>. Notice that if <math alttext="X=\emptyset" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.12.12.m12.1"><semantics id="S2.Thmtheorem3.p1.12.12.m12.1a"><mrow id="S2.Thmtheorem3.p1.12.12.m12.1.1" xref="S2.Thmtheorem3.p1.12.12.m12.1.1.cmml"><mi id="S2.Thmtheorem3.p1.12.12.m12.1.1.2" xref="S2.Thmtheorem3.p1.12.12.m12.1.1.2.cmml">X</mi><mo id="S2.Thmtheorem3.p1.12.12.m12.1.1.1" xref="S2.Thmtheorem3.p1.12.12.m12.1.1.1.cmml">=</mo><mi id="S2.Thmtheorem3.p1.12.12.m12.1.1.3" mathvariant="normal" xref="S2.Thmtheorem3.p1.12.12.m12.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.12.12.m12.1b"><apply id="S2.Thmtheorem3.p1.12.12.m12.1.1.cmml" xref="S2.Thmtheorem3.p1.12.12.m12.1.1"><eq id="S2.Thmtheorem3.p1.12.12.m12.1.1.1.cmml" xref="S2.Thmtheorem3.p1.12.12.m12.1.1.1"></eq><ci id="S2.Thmtheorem3.p1.12.12.m12.1.1.2.cmml" xref="S2.Thmtheorem3.p1.12.12.m12.1.1.2">𝑋</ci><emptyset id="S2.Thmtheorem3.p1.12.12.m12.1.1.3.cmml" xref="S2.Thmtheorem3.p1.12.12.m12.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.12.12.m12.1c">X=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.12.12.m12.1d">italic_X = ∅</annotation></semantics></math>, the arena is already binary. 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xref="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.cmml"><mi id="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.2" xref="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.2.cmml">v</mi><mn id="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.3" xref="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.12" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.7.cmml">,</mo><msup id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.cmml"><mi id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.2" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.2.cmml">w</mi><mo id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.3" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.13" stretchy="false" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.7.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.13.13.m13.6b"><apply id="S2.Thmtheorem3.p1.13.13.m13.6.6.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6"><eq id="S2.Thmtheorem3.p1.13.13.m13.6.6.7.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.7"></eq><apply id="S2.Thmtheorem3.p1.13.13.m13.6.6.8.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.8"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.6.6.8.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.8">superscript</csymbol><ci id="S2.Thmtheorem3.p1.13.13.m13.6.6.8.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.8.2">𝐴</ci><ci id="S2.Thmtheorem3.p1.13.13.m13.6.6.8.3.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.8.3">′</ci></apply><vector id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.7.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6"><apply id="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.1.1.1.1.1.3">′</ci></apply><apply id="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2">subscript</csymbol><apply id="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.2.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.2.3.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.2.3">′</ci></apply><cn id="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.13.13.m13.2.2.2.2.2.3">0</cn></apply><apply id="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3">subscript</csymbol><apply id="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.2.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3">superscript</csymbol><ci id="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.2.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.2.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.2.3.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.2.3">′</ci></apply><cn id="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.13.13.m13.3.3.3.3.3.3">1</cn></apply><apply id="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4">superscript</csymbol><ci id="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4.2">𝐸</ci><ci id="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4.3.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.4.4.4.4.4.3">′</ci></apply><apply id="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5">subscript</csymbol><ci id="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.2">𝑣</ci><cn id="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.13.13.m13.5.5.5.5.5.3">0</cn></apply><apply id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.1.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6">superscript</csymbol><ci id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.2.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.2">𝑤</ci><ci id="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.3.cmml" xref="S2.Thmtheorem3.p1.13.13.m13.6.6.6.6.6.3">′</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.13.13.m13.6c">A^{\prime}=(V^{\prime},V^{\prime}_{0},V^{\prime}_{1},E^{\prime},v_{0},w^{% \prime})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.13.13.m13.6d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> has the same initial vertex <math alttext="v_{0}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.14.14.m14.1"><semantics id="S2.Thmtheorem3.p1.14.14.m14.1a"><msub id="S2.Thmtheorem3.p1.14.14.m14.1.1" xref="S2.Thmtheorem3.p1.14.14.m14.1.1.cmml"><mi id="S2.Thmtheorem3.p1.14.14.m14.1.1.2" xref="S2.Thmtheorem3.p1.14.14.m14.1.1.2.cmml">v</mi><mn id="S2.Thmtheorem3.p1.14.14.m14.1.1.3" xref="S2.Thmtheorem3.p1.14.14.m14.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.14.14.m14.1b"><apply id="S2.Thmtheorem3.p1.14.14.m14.1.1.cmml" xref="S2.Thmtheorem3.p1.14.14.m14.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.14.14.m14.1.1.1.cmml" xref="S2.Thmtheorem3.p1.14.14.m14.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.p1.14.14.m14.1.1.2.cmml" xref="S2.Thmtheorem3.p1.14.14.m14.1.1.2">𝑣</ci><cn id="S2.Thmtheorem3.p1.14.14.m14.1.1.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.14.14.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.14.14.m14.1c">v_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.14.14.m14.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> as <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.15.15.m15.1"><semantics id="S2.Thmtheorem3.p1.15.15.m15.1a"><mi id="S2.Thmtheorem3.p1.15.15.m15.1.1" xref="S2.Thmtheorem3.p1.15.15.m15.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.15.15.m15.1b"><ci id="S2.Thmtheorem3.p1.15.15.m15.1.1.cmml" xref="S2.Thmtheorem3.p1.15.15.m15.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.15.15.m15.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.15.15.m15.1d">italic_A</annotation></semantics></math>, and a partition <math alttext="V^{\prime}_{0}\cup V^{\prime}_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.16.16.m16.1"><semantics id="S2.Thmtheorem3.p1.16.16.m16.1a"><mrow id="S2.Thmtheorem3.p1.16.16.m16.1.1" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.cmml"><msubsup id="S2.Thmtheorem3.p1.16.16.m16.1.1.2" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.2" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.2.cmml">V</mi><mn id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.3" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2.3.cmml">0</mn><mo id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.3" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S2.Thmtheorem3.p1.16.16.m16.1.1.1" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.1.cmml">∪</mo><msubsup id="S2.Thmtheorem3.p1.16.16.m16.1.1.3" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.2" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.2.cmml">V</mi><mn id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.3" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3.3.cmml">1</mn><mo id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.3" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.16.16.m16.1b"><apply id="S2.Thmtheorem3.p1.16.16.m16.1.1.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1"><union id="S2.Thmtheorem3.p1.16.16.m16.1.1.1.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.1"></union><apply id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2">subscript</csymbol><apply id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.1.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.2.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.3.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2.2.3">′</ci></apply><cn id="S2.Thmtheorem3.p1.16.16.m16.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.2.3">0</cn></apply><apply id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3">subscript</csymbol><apply id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.1.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.2.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.3.cmml" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3.2.3">′</ci></apply><cn id="S2.Thmtheorem3.p1.16.16.m16.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.16.16.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.16.16.m16.1c">V^{\prime}_{0}\cup V^{\prime}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.16.16.m16.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.17.17.m17.1"><semantics id="S2.Thmtheorem3.p1.17.17.m17.1a"><msup id="S2.Thmtheorem3.p1.17.17.m17.1.1" xref="S2.Thmtheorem3.p1.17.17.m17.1.1.cmml"><mi id="S2.Thmtheorem3.p1.17.17.m17.1.1.2" xref="S2.Thmtheorem3.p1.17.17.m17.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem3.p1.17.17.m17.1.1.3" xref="S2.Thmtheorem3.p1.17.17.m17.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.17.17.m17.1b"><apply id="S2.Thmtheorem3.p1.17.17.m17.1.1.cmml" xref="S2.Thmtheorem3.p1.17.17.m17.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.17.17.m17.1.1.1.cmml" xref="S2.Thmtheorem3.p1.17.17.m17.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p1.17.17.m17.1.1.2.cmml" xref="S2.Thmtheorem3.p1.17.17.m17.1.1.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.17.17.m17.1.1.3.cmml" xref="S2.Thmtheorem3.p1.17.17.m17.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.17.17.m17.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.17.17.m17.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that each new vertex is added<span class="ltx_note ltx_role_footnote" id="footnote5"><sup class="ltx_note_mark">5</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">5</sup><span class="ltx_tag ltx_tag_note"><span class="ltx_text ltx_font_upright" id="footnote5.1.1.1">5</span></span><span class="ltx_text ltx_font_upright" id="footnote5.5">Each new vertex could be added to </span><math alttext="V_{1}" class="ltx_Math" display="inline" id="footnote5.m1.1"><semantics id="footnote5.m1.1b"><msub id="footnote5.m1.1.1" xref="footnote5.m1.1.1.cmml"><mi id="footnote5.m1.1.1.2" xref="footnote5.m1.1.1.2.cmml">V</mi><mn id="footnote5.m1.1.1.3" xref="footnote5.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="footnote5.m1.1c"><apply id="footnote5.m1.1.1.cmml" xref="footnote5.m1.1.1"><csymbol cd="ambiguous" id="footnote5.m1.1.1.1.cmml" xref="footnote5.m1.1.1">subscript</csymbol><ci id="footnote5.m1.1.1.2.cmml" xref="footnote5.m1.1.1.2">𝑉</ci><cn id="footnote5.m1.1.1.3.cmml" type="integer" xref="footnote5.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote5.m1.1d">V_{1}</annotation><annotation encoding="application/x-llamapun" id="footnote5.m1.1e">italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_upright" id="footnote5.6"> as it has a unique successor.</span></span></span></span> to <math alttext="V_{0}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.18.18.m18.1"><semantics id="S2.Thmtheorem3.p1.18.18.m18.1a"><msub id="S2.Thmtheorem3.p1.18.18.m18.1.1" xref="S2.Thmtheorem3.p1.18.18.m18.1.1.cmml"><mi id="S2.Thmtheorem3.p1.18.18.m18.1.1.2" xref="S2.Thmtheorem3.p1.18.18.m18.1.1.2.cmml">V</mi><mn id="S2.Thmtheorem3.p1.18.18.m18.1.1.3" xref="S2.Thmtheorem3.p1.18.18.m18.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.18.18.m18.1b"><apply id="S2.Thmtheorem3.p1.18.18.m18.1.1.cmml" xref="S2.Thmtheorem3.p1.18.18.m18.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.18.18.m18.1.1.1.cmml" xref="S2.Thmtheorem3.p1.18.18.m18.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.p1.18.18.m18.1.1.2.cmml" xref="S2.Thmtheorem3.p1.18.18.m18.1.1.2">𝑉</ci><cn id="S2.Thmtheorem3.p1.18.18.m18.1.1.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.18.18.m18.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.18.18.m18.1c">V_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.18.18.m18.1d">italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, hence <math alttext="V_{0}\subseteq V^{\prime}_{0}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.19.19.m19.1"><semantics id="S2.Thmtheorem3.p1.19.19.m19.1a"><mrow id="S2.Thmtheorem3.p1.19.19.m19.1.1" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.cmml"><msub id="S2.Thmtheorem3.p1.19.19.m19.1.1.2" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.19.19.m19.1.1.2.2" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.2.2.cmml">V</mi><mn id="S2.Thmtheorem3.p1.19.19.m19.1.1.2.3" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p1.19.19.m19.1.1.1" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.1.cmml">⊆</mo><msubsup id="S2.Thmtheorem3.p1.19.19.m19.1.1.3" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.2" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.2.cmml">V</mi><mn id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.3" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3.3.cmml">0</mn><mo id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.3" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.19.19.m19.1b"><apply id="S2.Thmtheorem3.p1.19.19.m19.1.1.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1"><subset id="S2.Thmtheorem3.p1.19.19.m19.1.1.1.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.1"></subset><apply id="S2.Thmtheorem3.p1.19.19.m19.1.1.2.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.19.19.m19.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem3.p1.19.19.m19.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.2.2">𝑉</ci><cn id="S2.Thmtheorem3.p1.19.19.m19.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.2.3">0</cn></apply><apply id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3">subscript</csymbol><apply id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.1.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.2.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.3.cmml" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3.2.3">′</ci></apply><cn id="S2.Thmtheorem3.p1.19.19.m19.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.19.19.m19.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.19.19.m19.1c">V_{0}\subseteq V^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.19.19.m19.1d">italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊆ italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="V_{1}=V^{\prime}_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.20.20.m20.1"><semantics id="S2.Thmtheorem3.p1.20.20.m20.1a"><mrow id="S2.Thmtheorem3.p1.20.20.m20.1.1" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.cmml"><msub id="S2.Thmtheorem3.p1.20.20.m20.1.1.2" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.20.20.m20.1.1.2.2" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.2.2.cmml">V</mi><mn id="S2.Thmtheorem3.p1.20.20.m20.1.1.2.3" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.2.3.cmml">1</mn></msub><mo id="S2.Thmtheorem3.p1.20.20.m20.1.1.1" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.1.cmml">=</mo><msubsup id="S2.Thmtheorem3.p1.20.20.m20.1.1.3" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.2" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.2.cmml">V</mi><mn id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.3" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3.3.cmml">1</mn><mo id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.3" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.20.20.m20.1b"><apply id="S2.Thmtheorem3.p1.20.20.m20.1.1.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1"><eq id="S2.Thmtheorem3.p1.20.20.m20.1.1.1.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.1"></eq><apply id="S2.Thmtheorem3.p1.20.20.m20.1.1.2.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.20.20.m20.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem3.p1.20.20.m20.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.2.2">𝑉</ci><cn id="S2.Thmtheorem3.p1.20.20.m20.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.2.3">1</cn></apply><apply id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3">subscript</csymbol><apply id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.1.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.2.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.3.cmml" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3.2.3">′</ci></apply><cn id="S2.Thmtheorem3.p1.20.20.m20.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.20.20.m20.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.20.20.m20.1c">V_{1}=V^{\prime}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.20.20.m20.1d">italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. The new SP game <math alttext="G^{\prime}=(A^{\prime},T_{0},\ldots,T_{t})" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.21.21.m21.4"><semantics id="S2.Thmtheorem3.p1.21.21.m21.4a"><mrow id="S2.Thmtheorem3.p1.21.21.m21.4.4" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.cmml"><msup id="S2.Thmtheorem3.p1.21.21.m21.4.4.5" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.5.cmml"><mi id="S2.Thmtheorem3.p1.21.21.m21.4.4.5.2" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.5.2.cmml">G</mi><mo id="S2.Thmtheorem3.p1.21.21.m21.4.4.5.3" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.5.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.21.21.m21.4.4.4" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.4.cmml">=</mo><mrow id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.4.cmml"><mo id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.4" stretchy="false" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.4.cmml">(</mo><msup id="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1" xref="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.2" xref="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.3" xref="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.5" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.4.cmml">,</mo><msub id="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2" xref="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.cmml"><mi id="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.2" xref="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.2.cmml">T</mi><mn id="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.3" xref="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.6" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.4.cmml">,</mo><mi id="S2.Thmtheorem3.p1.21.21.m21.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.21.21.m21.1.1.cmml">…</mi><mo id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.7" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.4.cmml">,</mo><msub id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.cmml"><mi id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.2" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.2.cmml">T</mi><mi id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.3" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.3.cmml">t</mi></msub><mo id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.8" stretchy="false" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.21.21.m21.4b"><apply id="S2.Thmtheorem3.p1.21.21.m21.4.4.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4"><eq id="S2.Thmtheorem3.p1.21.21.m21.4.4.4.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.4"></eq><apply id="S2.Thmtheorem3.p1.21.21.m21.4.4.5.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.5"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.21.21.m21.4.4.5.1.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.5">superscript</csymbol><ci id="S2.Thmtheorem3.p1.21.21.m21.4.4.5.2.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.5.2">𝐺</ci><ci id="S2.Thmtheorem3.p1.21.21.m21.4.4.5.3.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.5.3">′</ci></apply><vector id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.4.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3"><apply id="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.2">𝐴</ci><ci id="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.2.2.1.1.1.3">′</ci></apply><apply id="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.2">𝑇</ci><cn id="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.21.21.m21.3.3.2.2.2.3">0</cn></apply><ci id="S2.Thmtheorem3.p1.21.21.m21.1.1.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.1.1">…</ci><apply id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.1.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3">subscript</csymbol><ci id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.2.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.2">𝑇</ci><ci id="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.3.cmml" xref="S2.Thmtheorem3.p1.21.21.m21.4.4.3.3.3.3">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.21.21.m21.4c">G^{\prime}=(A^{\prime},T_{0},\ldots,T_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.21.21.m21.4d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> keeps the same targets as in <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.22.22.m22.1"><semantics id="S2.Thmtheorem3.p1.22.22.m22.1a"><mi id="S2.Thmtheorem3.p1.22.22.m22.1.1" xref="S2.Thmtheorem3.p1.22.22.m22.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.22.22.m22.1b"><ci id="S2.Thmtheorem3.p1.22.22.m22.1.1.cmml" xref="S2.Thmtheorem3.p1.22.22.m22.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.22.22.m22.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.22.22.m22.1d">italic_G</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S2.Thmtheorem3.p2"> <p class="ltx_p" id="S2.Thmtheorem3.p2.17"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p2.17.17">Clearly, there is a trivial bijection <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.1.1.m1.1"><semantics id="S2.Thmtheorem3.p2.1.1.m1.1a"><mi id="S2.Thmtheorem3.p2.1.1.m1.1.1" xref="S2.Thmtheorem3.p2.1.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.1.1.m1.1b"><ci id="S2.Thmtheorem3.p2.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.1.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.1.1.m1.1d">italic_f</annotation></semantics></math> from <math alttext="\textsf{Play}_{A}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.2.2.m2.1"><semantics id="S2.Thmtheorem3.p2.2.2.m2.1a"><msub id="S2.Thmtheorem3.p2.2.2.m2.1.1" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Thmtheorem3.p2.2.2.m2.1.1.2" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.2a.cmml">Play</mtext><mi id="S2.Thmtheorem3.p2.2.2.m2.1.1.3" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.2.2.m2.1b"><apply id="S2.Thmtheorem3.p2.2.2.m2.1.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.p2.2.2.m2.1.1.2a.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Thmtheorem3.p2.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.2">Play</mtext></ci><ci id="S2.Thmtheorem3.p2.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.2.2.m2.1c">\textsf{Play}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.2.2.m2.1d">Play start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\textsf{Play}_{A^{\prime}}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.3.3.m3.1"><semantics id="S2.Thmtheorem3.p2.3.3.m3.1a"><msub id="S2.Thmtheorem3.p2.3.3.m3.1.1" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Thmtheorem3.p2.3.3.m3.1.1.2" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.2a.cmml">Play</mtext><msup id="S2.Thmtheorem3.p2.3.3.m3.1.1.3" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem3.p2.3.3.m3.1.1.3.2" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3.2.cmml">A</mi><mo id="S2.Thmtheorem3.p2.3.3.m3.1.1.3.3" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.3.3.m3.1b"><apply id="S2.Thmtheorem3.p2.3.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.p2.3.3.m3.1.1.2a.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.Thmtheorem3.p2.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.2">Play</mtext></ci><apply id="S2.Thmtheorem3.p2.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.3.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem3.p2.3.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3.2">𝐴</ci><ci id="S2.Thmtheorem3.p2.3.3.m3.1.1.3.3.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.3.3.m3.1c">\textsf{Play}_{A^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.3.3.m3.1d">Play start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Indeed, it suffices to replace all edges <math alttext="e" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.4.4.m4.1"><semantics id="S2.Thmtheorem3.p2.4.4.m4.1a"><mi id="S2.Thmtheorem3.p2.4.4.m4.1.1" xref="S2.Thmtheorem3.p2.4.4.m4.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.4.4.m4.1b"><ci id="S2.Thmtheorem3.p2.4.4.m4.1.1.cmml" xref="S2.Thmtheorem3.p2.4.4.m4.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.4.4.m4.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.4.4.m4.1d">italic_e</annotation></semantics></math> of a play with weight <math alttext="w(e)\geq 2" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.5.5.m5.1"><semantics id="S2.Thmtheorem3.p2.5.5.m5.1a"><mrow id="S2.Thmtheorem3.p2.5.5.m5.1.2" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.cmml"><mrow id="S2.Thmtheorem3.p2.5.5.m5.1.2.2" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.cmml"><mi id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.2" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.2.cmml">w</mi><mo id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.1" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.3.2" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.cmml"><mo id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.cmml">(</mo><mi id="S2.Thmtheorem3.p2.5.5.m5.1.1" xref="S2.Thmtheorem3.p2.5.5.m5.1.1.cmml">e</mi><mo id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p2.5.5.m5.1.2.1" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.1.cmml">≥</mo><mn id="S2.Thmtheorem3.p2.5.5.m5.1.2.3" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.5.5.m5.1b"><apply id="S2.Thmtheorem3.p2.5.5.m5.1.2.cmml" xref="S2.Thmtheorem3.p2.5.5.m5.1.2"><geq id="S2.Thmtheorem3.p2.5.5.m5.1.2.1.cmml" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.1"></geq><apply id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.cmml" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2"><times id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.1.cmml" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.1"></times><ci id="S2.Thmtheorem3.p2.5.5.m5.1.2.2.2.cmml" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.2.2">𝑤</ci><ci id="S2.Thmtheorem3.p2.5.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p2.5.5.m5.1.1">𝑒</ci></apply><cn id="S2.Thmtheorem3.p2.5.5.m5.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.5.5.m5.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.5.5.m5.1c">w(e)\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.5.5.m5.1d">italic_w ( italic_e ) ≥ 2</annotation></semantics></math> by the corresponding new path composed of <math alttext="w(e)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.6.6.m6.1"><semantics id="S2.Thmtheorem3.p2.6.6.m6.1a"><mrow id="S2.Thmtheorem3.p2.6.6.m6.1.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.cmml"><mi id="S2.Thmtheorem3.p2.6.6.m6.1.2.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.cmml">w</mi><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.1" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.cmml"><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.cmml">(</mo><mi id="S2.Thmtheorem3.p2.6.6.m6.1.1" xref="S2.Thmtheorem3.p2.6.6.m6.1.1.cmml">e</mi><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.6.6.m6.1b"><apply id="S2.Thmtheorem3.p2.6.6.m6.1.2.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2"><times id="S2.Thmtheorem3.p2.6.6.m6.1.2.1.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.1"></times><ci id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2">𝑤</ci><ci id="S2.Thmtheorem3.p2.6.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.1">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.6.6.m6.1c">w(e)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.6.6.m6.1d">italic_w ( italic_e )</annotation></semantics></math> edges of weight 1. This bijection preserves the cost and the value of the plays. Moreover, there also exists a bijection <math alttext="g" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.7.7.m7.1"><semantics id="S2.Thmtheorem3.p2.7.7.m7.1a"><mi id="S2.Thmtheorem3.p2.7.7.m7.1.1" xref="S2.Thmtheorem3.p2.7.7.m7.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.7.7.m7.1b"><ci id="S2.Thmtheorem3.p2.7.7.m7.1.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.7.7.m7.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.7.7.m7.1d">italic_g</annotation></semantics></math> from the set of strategies on <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.8.8.m8.1"><semantics id="S2.Thmtheorem3.p2.8.8.m8.1a"><mi id="S2.Thmtheorem3.p2.8.8.m8.1.1" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.8.8.m8.1b"><ci id="S2.Thmtheorem3.p2.8.8.m8.1.1.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.8.8.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.8.8.m8.1d">italic_A</annotation></semantics></math> to the set of strategies on <math alttext="A^{\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.9.9.m9.1"><semantics id="S2.Thmtheorem3.p2.9.9.m9.1a"><msup id="S2.Thmtheorem3.p2.9.9.m9.1.1" xref="S2.Thmtheorem3.p2.9.9.m9.1.1.cmml"><mi id="S2.Thmtheorem3.p2.9.9.m9.1.1.2" xref="S2.Thmtheorem3.p2.9.9.m9.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem3.p2.9.9.m9.1.1.3" xref="S2.Thmtheorem3.p2.9.9.m9.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.9.9.m9.1b"><apply id="S2.Thmtheorem3.p2.9.9.m9.1.1.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.9.9.m9.1.1.1.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p2.9.9.m9.1.1.2.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.1.1.2">𝐴</ci><ci id="S2.Thmtheorem3.p2.9.9.m9.1.1.3.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.9.9.m9.1c">A^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.9.9.m9.1d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> as each new vertex has a unique successor. Notice that <math alttext="g" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.10.10.m10.1"><semantics id="S2.Thmtheorem3.p2.10.10.m10.1a"><mi id="S2.Thmtheorem3.p2.10.10.m10.1.1" xref="S2.Thmtheorem3.p2.10.10.m10.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.10.10.m10.1b"><ci id="S2.Thmtheorem3.p2.10.10.m10.1.1.cmml" xref="S2.Thmtheorem3.p2.10.10.m10.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.10.10.m10.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.10.10.m10.1d">italic_g</annotation></semantics></math> is coherent with <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.11.11.m11.1"><semantics id="S2.Thmtheorem3.p2.11.11.m11.1a"><mi id="S2.Thmtheorem3.p2.11.11.m11.1.1" xref="S2.Thmtheorem3.p2.11.11.m11.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.11.11.m11.1b"><ci id="S2.Thmtheorem3.p2.11.11.m11.1.1.cmml" xref="S2.Thmtheorem3.p2.11.11.m11.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.11.11.m11.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.11.11.m11.1d">italic_f</annotation></semantics></math>, i.e., <math alttext="\textsf{out}(g(\sigma_{0}),g(\sigma_{1}))=f(\textsf{out}(\sigma_{0},\sigma_{1}))" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.12.12.m12.3"><semantics id="S2.Thmtheorem3.p2.12.12.m12.3a"><mrow id="S2.Thmtheorem3.p2.12.12.m12.3.3" xref="S2.Thmtheorem3.p2.12.12.m12.3.3.cmml"><mrow id="S2.Thmtheorem3.p2.12.12.m12.2.2.2" xref="S2.Thmtheorem3.p2.12.12.m12.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S2.Thmtheorem3.p2.12.12.m12.2.2.2.4" xref="S2.Thmtheorem3.p2.12.12.m12.2.2.2.4a.cmml">out</mtext><mo id="S2.Thmtheorem3.p2.12.12.m12.2.2.2.3" xref="S2.Thmtheorem3.p2.12.12.m12.2.2.2.3.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.12.12.m12.2.2.2.2.2" xref="S2.Thmtheorem3.p2.12.12.m12.2.2.2.2.3.cmml"><mo id="S2.Thmtheorem3.p2.12.12.m12.2.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem3.p2.12.12.m12.2.2.2.2.3.cmml">(</mo><mrow id="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1" xref="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.3" xref="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.3.cmml">g</mi><mo id="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.2" xref="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.1.1" xref="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.1.1.1" xref="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p2.12.12.m12.1.1.1.1.1.1.1.1.1.2" 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id="S2.Thmtheorem3.p2.12.12.m12.3c">\textsf{out}(g(\sigma_{0}),g(\sigma_{1}))=f(\textsf{out}(\sigma_{0},\sigma_{1}))</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.12.12.m12.3d">out ( italic_g ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , italic_g ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) = italic_f ( out ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) )</annotation></semantics></math> for all <math alttext="\sigma_{0}\in\Sigma_{0}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.13.13.m13.1"><semantics id="S2.Thmtheorem3.p2.13.13.m13.1a"><mrow id="S2.Thmtheorem3.p2.13.13.m13.1.1" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.cmml"><msub id="S2.Thmtheorem3.p2.13.13.m13.1.1.2" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.2.cmml"><mi id="S2.Thmtheorem3.p2.13.13.m13.1.1.2.2" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.2.2.cmml">σ</mi><mn id="S2.Thmtheorem3.p2.13.13.m13.1.1.2.3" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p2.13.13.m13.1.1.1" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.1.cmml">∈</mo><msub id="S2.Thmtheorem3.p2.13.13.m13.1.1.3" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.3.cmml"><mi id="S2.Thmtheorem3.p2.13.13.m13.1.1.3.2" mathvariant="normal" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.3.2.cmml">Σ</mi><mn id="S2.Thmtheorem3.p2.13.13.m13.1.1.3.3" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.13.13.m13.1b"><apply id="S2.Thmtheorem3.p2.13.13.m13.1.1.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1"><in id="S2.Thmtheorem3.p2.13.13.m13.1.1.1.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.1"></in><apply id="S2.Thmtheorem3.p2.13.13.m13.1.1.2.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.13.13.m13.1.1.2.1.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem3.p2.13.13.m13.1.1.2.2.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.2.2">𝜎</ci><cn id="S2.Thmtheorem3.p2.13.13.m13.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.2.3">0</cn></apply><apply id="S2.Thmtheorem3.p2.13.13.m13.1.1.3.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.13.13.m13.1.1.3.1.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem3.p2.13.13.m13.1.1.3.2.cmml" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.3.2">Σ</ci><cn id="S2.Thmtheorem3.p2.13.13.m13.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.13.13.m13.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.13.13.m13.1c">\sigma_{0}\in\Sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.13.13.m13.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and all <math alttext="\sigma_{1}\in\Sigma_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.14.14.m14.1"><semantics id="S2.Thmtheorem3.p2.14.14.m14.1a"><mrow id="S2.Thmtheorem3.p2.14.14.m14.1.1" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.cmml"><msub id="S2.Thmtheorem3.p2.14.14.m14.1.1.2" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.2.cmml"><mi id="S2.Thmtheorem3.p2.14.14.m14.1.1.2.2" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.2.2.cmml">σ</mi><mn id="S2.Thmtheorem3.p2.14.14.m14.1.1.2.3" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.2.3.cmml">1</mn></msub><mo id="S2.Thmtheorem3.p2.14.14.m14.1.1.1" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.1.cmml">∈</mo><msub id="S2.Thmtheorem3.p2.14.14.m14.1.1.3" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.3.cmml"><mi id="S2.Thmtheorem3.p2.14.14.m14.1.1.3.2" mathvariant="normal" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.3.2.cmml">Σ</mi><mn id="S2.Thmtheorem3.p2.14.14.m14.1.1.3.3" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.14.14.m14.1b"><apply id="S2.Thmtheorem3.p2.14.14.m14.1.1.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1"><in id="S2.Thmtheorem3.p2.14.14.m14.1.1.1.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.1"></in><apply id="S2.Thmtheorem3.p2.14.14.m14.1.1.2.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.14.14.m14.1.1.2.1.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem3.p2.14.14.m14.1.1.2.2.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.2.2">𝜎</ci><cn id="S2.Thmtheorem3.p2.14.14.m14.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.2.3">1</cn></apply><apply id="S2.Thmtheorem3.p2.14.14.m14.1.1.3.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.14.14.m14.1.1.3.1.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem3.p2.14.14.m14.1.1.3.2.cmml" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.3.2">Σ</ci><cn id="S2.Thmtheorem3.p2.14.14.m14.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.14.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.14.14.m14.1c">\sigma_{1}\in\Sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.14.14.m14.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, for all <math alttext="\sigma_{0}\in\Sigma_{0}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.15.15.m15.1"><semantics id="S2.Thmtheorem3.p2.15.15.m15.1a"><mrow id="S2.Thmtheorem3.p2.15.15.m15.1.1" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.cmml"><msub id="S2.Thmtheorem3.p2.15.15.m15.1.1.2" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.2.cmml"><mi id="S2.Thmtheorem3.p2.15.15.m15.1.1.2.2" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.2.2.cmml">σ</mi><mn id="S2.Thmtheorem3.p2.15.15.m15.1.1.2.3" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p2.15.15.m15.1.1.1" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.1.cmml">∈</mo><msub id="S2.Thmtheorem3.p2.15.15.m15.1.1.3" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.3.cmml"><mi id="S2.Thmtheorem3.p2.15.15.m15.1.1.3.2" mathvariant="normal" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.3.2.cmml">Σ</mi><mn id="S2.Thmtheorem3.p2.15.15.m15.1.1.3.3" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.15.15.m15.1b"><apply id="S2.Thmtheorem3.p2.15.15.m15.1.1.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1"><in id="S2.Thmtheorem3.p2.15.15.m15.1.1.1.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.1"></in><apply id="S2.Thmtheorem3.p2.15.15.m15.1.1.2.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.15.15.m15.1.1.2.1.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem3.p2.15.15.m15.1.1.2.2.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.2.2">𝜎</ci><cn id="S2.Thmtheorem3.p2.15.15.m15.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.2.3">0</cn></apply><apply id="S2.Thmtheorem3.p2.15.15.m15.1.1.3.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.15.15.m15.1.1.3.1.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem3.p2.15.15.m15.1.1.3.2.cmml" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.3.2">Σ</ci><cn id="S2.Thmtheorem3.p2.15.15.m15.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.15.15.m15.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.15.15.m15.1c">\sigma_{0}\in\Sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.15.15.m15.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, we get that <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.16.16.m16.2"><semantics id="S2.Thmtheorem3.p2.16.16.m16.2a"><mrow id="S2.Thmtheorem3.p2.16.16.m16.2.3" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.cmml"><msub id="S2.Thmtheorem3.p2.16.16.m16.2.3.2" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.2.cmml"><mi id="S2.Thmtheorem3.p2.16.16.m16.2.3.2.2" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.2.2.cmml">σ</mi><mn id="S2.Thmtheorem3.p2.16.16.m16.2.3.2.3" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p2.16.16.m16.2.3.1" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.1.cmml">∈</mo><mrow id="S2.Thmtheorem3.p2.16.16.m16.2.3.3" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.2" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.2a.cmml">SPS</mtext><mo id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.1" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.2" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.1.cmml"><mo id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem3.p2.16.16.m16.1.1" xref="S2.Thmtheorem3.p2.16.16.m16.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.2.2" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem3.p2.16.16.m16.2.2" xref="S2.Thmtheorem3.p2.16.16.m16.2.2.cmml">B</mi><mo id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.2.3" stretchy="false" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.16.16.m16.2b"><apply id="S2.Thmtheorem3.p2.16.16.m16.2.3.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3"><in id="S2.Thmtheorem3.p2.16.16.m16.2.3.1.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.1"></in><apply id="S2.Thmtheorem3.p2.16.16.m16.2.3.2.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.16.16.m16.2.3.2.1.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.2">subscript</csymbol><ci id="S2.Thmtheorem3.p2.16.16.m16.2.3.2.2.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.2.2">𝜎</ci><cn id="S2.Thmtheorem3.p2.16.16.m16.2.3.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.2.3">0</cn></apply><apply id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3"><times id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.1.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.1"></times><ci id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.2a.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.2.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.1.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.3.3.3.2"><ci id="S2.Thmtheorem3.p2.16.16.m16.1.1.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.1.1">𝐺</ci><ci id="S2.Thmtheorem3.p2.16.16.m16.2.2.cmml" xref="S2.Thmtheorem3.p2.16.16.m16.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.16.16.m16.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.16.16.m16.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> if and only if <math alttext="g(\sigma_{0})\in\mbox{SPS}({G^{\prime}},{B})" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.17.17.m17.3"><semantics id="S2.Thmtheorem3.p2.17.17.m17.3a"><mrow id="S2.Thmtheorem3.p2.17.17.m17.3.3" xref="S2.Thmtheorem3.p2.17.17.m17.3.3.cmml"><mrow id="S2.Thmtheorem3.p2.17.17.m17.2.2.1" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.cmml"><mi id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.3" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.3.cmml">g</mi><mo id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.2" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.cmml">(</mo><msub id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.2" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.2.cmml">σ</mi><mn id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.3" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p2.17.17.m17.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p2.17.17.m17.3.3.3" xref="S2.Thmtheorem3.p2.17.17.m17.3.3.3.cmml">∈</mo><mrow id="S2.Thmtheorem3.p2.17.17.m17.3.3.2" xref="S2.Thmtheorem3.p2.17.17.m17.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p2.17.17.m17.3.3.2.3" xref="S2.Thmtheorem3.p2.17.17.m17.3.3.2.3a.cmml">SPS</mtext><mo id="S2.Thmtheorem3.p2.17.17.m17.3.3.2.2" xref="S2.Thmtheorem3.p2.17.17.m17.3.3.2.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.17.17.m17.3.3.2.1.1" xref="S2.Thmtheorem3.p2.17.17.m17.3.3.2.1.2.cmml"><mo id="S2.Thmtheorem3.p2.17.17.m17.3.3.2.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p2.17.17.m17.3.3.2.1.2.cmml">(</mo><msup 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id="S2.Thmtheorem3.p2.17.17.m17.3c">g(\sigma_{0})\in\mbox{SPS}({G^{\prime}},{B})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.17.17.m17.3d">italic_g ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∈ SPS ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B )</annotation></semantics></math>. </span></p> </div> </div> <div class="ltx_para" id="S2.SS3.SSS0.Px2.p2"> <p class="ltx_p" id="S2.SS3.SSS0.Px2.p2.5">The transformation of the arena <math alttext="A" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p2.1.m1.1"><semantics id="S2.SS3.SSS0.Px2.p2.1.m1.1a"><mi id="S2.SS3.SSS0.Px2.p2.1.m1.1.1" xref="S2.SS3.SSS0.Px2.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.1.m1.1b"><ci id="S2.SS3.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p2.1.m1.1d">italic_A</annotation></semantics></math> into a binary arena <math alttext="A^{\prime}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p2.2.m2.1"><semantics id="S2.SS3.SSS0.Px2.p2.2.m2.1a"><msup id="S2.SS3.SSS0.Px2.p2.2.m2.1.1" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p2.2.m2.1.1.2" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1.2.cmml">A</mi><mo id="S2.SS3.SSS0.Px2.p2.2.m2.1.1.3" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.2.m2.1b"><apply id="S2.SS3.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1">superscript</csymbol><ci id="S2.SS3.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1.2">𝐴</ci><ci id="S2.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.2.m2.1c">A^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p2.2.m2.1d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> has consequences on the size of the SPS problem instance. Since the weights are encoded in binary, the size <math alttext="|V^{\prime}|" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p2.3.m3.1"><semantics id="S2.SS3.SSS0.Px2.p2.3.m3.1a"><mrow id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2.cmml"><mo id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2.1.cmml">|</mo><msup id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.2" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.2.cmml">V</mi><mo id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.3" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.3" stretchy="false" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.3.m3.1b"><apply id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1"><abs id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2.1.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.2"></abs><apply id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1">superscript</csymbol><ci id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.2">𝑉</ci><ci id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.3.m3.1c">|V^{\prime}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p2.3.m3.1d">| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT |</annotation></semantics></math> could be exponential in the size <math alttext="|V|" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p2.4.m4.1"><semantics id="S2.SS3.SSS0.Px2.p2.4.m4.1a"><mrow id="S2.SS3.SSS0.Px2.p2.4.m4.1.2.2" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.2.1.cmml"><mo id="S2.SS3.SSS0.Px2.p2.4.m4.1.2.2.1" stretchy="false" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.2.1.1.cmml">|</mo><mi id="S2.SS3.SSS0.Px2.p2.4.m4.1.1" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.cmml">V</mi><mo id="S2.SS3.SSS0.Px2.p2.4.m4.1.2.2.2" stretchy="false" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.4.m4.1b"><apply id="S2.SS3.SSS0.Px2.p2.4.m4.1.2.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.2.2"><abs id="S2.SS3.SSS0.Px2.p2.4.m4.1.2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.2.2.1"></abs><ci id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.4.m4.1c">|V|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p2.4.m4.1d">| italic_V |</annotation></semantics></math> of the original instance. However, this will have <em class="ltx_emph ltx_font_italic" id="S2.SS3.SSS0.Px2.p2.5.1">no impact</em> on our main result because <math alttext="|V|" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p2.5.m5.1"><semantics id="S2.SS3.SSS0.Px2.p2.5.m5.1a"><mrow id="S2.SS3.SSS0.Px2.p2.5.m5.1.2.2" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.2.1.cmml"><mo id="S2.SS3.SSS0.Px2.p2.5.m5.1.2.2.1" stretchy="false" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.2.1.1.cmml">|</mo><mi id="S2.SS3.SSS0.Px2.p2.5.m5.1.1" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.1.cmml">V</mi><mo id="S2.SS3.SSS0.Px2.p2.5.m5.1.2.2.2" stretchy="false" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.5.m5.1b"><apply id="S2.SS3.SSS0.Px2.p2.5.m5.1.2.1.cmml" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.2.2"><abs id="S2.SS3.SSS0.Px2.p2.5.m5.1.2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.2.2.1"></abs><ci id="S2.SS3.SSS0.Px2.p2.5.m5.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.1">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.5.m5.1c">|V|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p2.5.m5.1d">| italic_V |</annotation></semantics></math> never appears in the exponent in our calculations (this will be detailed in the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem1" title="Theorem 2.1 ‣ Stackelberg-Pareto synthesis problem. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.1</span></a>).</p> </div> <div class="ltx_para" id="S2.SS3.SSS0.Px2.p3"> <p class="ltx_p" id="S2.SS3.SSS0.Px2.p3.4">Note that along any history <math alttext="h=h_{0}\ldots h_{\ell}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p3.1.m1.1"><semantics id="S2.SS3.SSS0.Px2.p3.1.m1.1a"><mrow id="S2.SS3.SSS0.Px2.p3.1.m1.1.1" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.2" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.2.cmml">h</mi><mo id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.1" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.cmml"><msub id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.cmml"><mi id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.2" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.2.cmml">h</mi><mn id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.3" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.1" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.3" mathvariant="normal" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.3.cmml">…</mi><mo id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.1a" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.cmml"><mi id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.2" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.2.cmml">h</mi><mi id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.3" mathvariant="normal" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.3.cmml">ℓ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p3.1.m1.1b"><apply id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1"><eq id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.1"></eq><ci id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.2">ℎ</ci><apply id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3"><times id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.1"></times><apply id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.1.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.2.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.2">ℎ</ci><cn id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.3.cmml" type="integer" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.2.3">0</cn></apply><ci id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.3">…</ci><apply id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.1.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.2.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.2">ℎ</ci><ci id="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.3.cmml" xref="S2.SS3.SSS0.Px2.p3.1.m1.1.1.3.4.3">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p3.1.m1.1c">h=h_{0}\ldots h_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p3.1.m1.1d">italic_h = italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT … italic_h start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> in a binary arena, the total weight increases by a weight of 0 or 1 from <math alttext="w(h_{0})" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p3.2.m2.1"><semantics id="S2.SS3.SSS0.Px2.p3.2.m2.1a"><mrow id="S2.SS3.SSS0.Px2.p3.2.m2.1.1" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.3" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.3.cmml">w</mi><mo id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.2" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.2" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.2.cmml">h</mi><mn id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.3" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p3.2.m2.1b"><apply id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1"><times id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.2"></times><ci id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.3">𝑤</ci><apply id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.2">ℎ</ci><cn id="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS3.SSS0.Px2.p3.2.m2.1.1.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p3.2.m2.1c">w(h_{0})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p3.2.m2.1d">italic_w ( italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math> to <math alttext="w(h)" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p3.3.m3.1"><semantics id="S2.SS3.SSS0.Px2.p3.3.m3.1a"><mrow id="S2.SS3.SSS0.Px2.p3.3.m3.1.2" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.cmml"><mi id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.2" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.2.cmml">w</mi><mo id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.1" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.3.2" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.cmml"><mo id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.cmml">(</mo><mi id="S2.SS3.SSS0.Px2.p3.3.m3.1.1" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.1.cmml">h</mi><mo id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p3.3.m3.1b"><apply id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.cmml" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2"><times id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.1.cmml" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.1"></times><ci id="S2.SS3.SSS0.Px2.p3.3.m3.1.2.2.cmml" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.2.2">𝑤</ci><ci id="S2.SS3.SSS0.Px2.p3.3.m3.1.1.cmml" xref="S2.SS3.SSS0.Px2.p3.3.m3.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p3.3.m3.1c">w(h)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p3.3.m3.1d">italic_w ( italic_h )</annotation></semantics></math>, that is <math alttext="\{w(h_{[0,k]})\mid 0\leq k\leq\ell\}=\{0,1,\ldots,w(h)\}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p3.4.m4.9"><semantics id="S2.SS3.SSS0.Px2.p3.4.m4.9a"><mrow id="S2.SS3.SSS0.Px2.p3.4.m4.9.9" xref="S2.SS3.SSS0.Px2.p3.4.m4.9.9.cmml"><mrow id="S2.SS3.SSS0.Px2.p3.4.m4.8.8.2.2" xref="S2.SS3.SSS0.Px2.p3.4.m4.8.8.2.3.cmml"><mo id="S2.SS3.SSS0.Px2.p3.4.m4.8.8.2.2.3" stretchy="false" xref="S2.SS3.SSS0.Px2.p3.4.m4.8.8.2.3.1.cmml">{</mo><mrow id="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1" xref="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.3" xref="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.3.cmml">w</mi><mo id="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.2" xref="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1" xref="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1.1" xref="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1.1.2" xref="S2.SS3.SSS0.Px2.p3.4.m4.7.7.1.1.1.1.1.1.2.cmml">h</mi><mrow id="S2.SS3.SSS0.Px2.p3.4.m4.2.2.2.4" 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italic_k ≤ roman_ℓ } = { 0 , 1 , … , italic_w ( italic_h ) }</annotation></semantics></math>. This property will be used in some proofs.</p> </div> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Improving a Solution</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">In order to solve efficiently the SPS problem, we would like to work with solutions resulting in small costs. To do so, in this section, we introduce some techniques that allow to modify a solution to the SPS problem into a solution with smaller Pareto-optimal costs.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Order on Strategies and Subgames</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.18">Given two strategies <math alttext="\sigma_{0},\sigma_{0}^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.2"><semantics id="S3.SS1.p1.1.m1.2a"><mrow id="S3.SS1.p1.1.m1.2.2.2" xref="S3.SS1.p1.1.m1.2.2.3.cmml"><msub id="S3.SS1.p1.1.m1.1.1.1.1" xref="S3.SS1.p1.1.m1.1.1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.1.1.2.cmml">σ</mi><mn id="S3.SS1.p1.1.m1.1.1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.1.1.3.cmml">0</mn></msub><mo id="S3.SS1.p1.1.m1.2.2.2.3" xref="S3.SS1.p1.1.m1.2.2.3.cmml">,</mo><msubsup id="S3.SS1.p1.1.m1.2.2.2.2" 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xref="S3.SS1.p1.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.1.m1.2.2.2.2.2.1.cmml" xref="S3.SS1.p1.1.m1.2.2.2.2">subscript</csymbol><ci id="S3.SS1.p1.1.m1.2.2.2.2.2.2.cmml" xref="S3.SS1.p1.1.m1.2.2.2.2.2.2">𝜎</ci><cn id="S3.SS1.p1.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.p1.1.m1.2.2.2.2.2.3">0</cn></apply><ci id="S3.SS1.p1.1.m1.2.2.2.2.3.cmml" xref="S3.SS1.p1.1.m1.2.2.2.2.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.2c">\sigma_{0},\sigma_{0}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for Player 0, we define <math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" 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"><msubsup id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml"><mi id="S3.SS1.p1.2.m2.1.1.2.2.2" xref="S3.SS1.p1.2.m2.1.1.2.2.2.cmml">σ</mi><mn id="S3.SS1.p1.2.m2.1.1.2.2.3" xref="S3.SS1.p1.2.m2.1.1.2.2.3.cmml">0</mn><mo id="S3.SS1.p1.2.m2.1.1.2.3" xref="S3.SS1.p1.2.m2.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.p1.2.m2.1.1.1" xref="S3.SS1.p1.2.m2.1.1.1.cmml">⪯</mo><msub id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml"><mi id="S3.SS1.p1.2.m2.1.1.3.2" xref="S3.SS1.p1.2.m2.1.1.3.2.cmml">σ</mi><mn id="S3.SS1.p1.2.m2.1.1.3.3" xref="S3.SS1.p1.2.m2.1.1.3.3.cmml">0</mn></msub></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"><csymbol cd="latexml" id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1">precedes-or-equals</csymbol><apply id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.2.1.cmml" xref="S3.SS1.p1.2.m2.1.1.2">superscript</csymbol><apply id="S3.SS1.p1.2.m2.1.1.2.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.2.2.1.cmml" xref="S3.SS1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.2.m2.1.1.2.2.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2.2.2">𝜎</ci><cn id="S3.SS1.p1.2.m2.1.1.2.2.3.cmml" type="integer" xref="S3.SS1.p1.2.m2.1.1.2.2.3">0</cn></apply><ci id="S3.SS1.p1.2.m2.1.1.2.3.cmml" xref="S3.SS1.p1.2.m2.1.1.2.3">′</ci></apply><apply id="S3.SS1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.3.1.cmml" xref="S3.SS1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.2.m2.1.1.3.2.cmml" xref="S3.SS1.p1.2.m2.1.1.3.2">𝜎</ci><cn id="S3.SS1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> if for all <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mrow 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">c</mi><mo id="S3.SS1.p1.3.m3.1.1.1" xref="S3.SS1.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml"><mi id="S3.SS1.p1.3.m3.1.1.3.2" xref="S3.SS1.p1.3.m3.1.1.3.2.cmml">C</mi><msub id="S3.SS1.p1.3.m3.1.1.3.3" xref="S3.SS1.p1.3.m3.1.1.3.3.cmml"><mi id="S3.SS1.p1.3.m3.1.1.3.3.2" xref="S3.SS1.p1.3.m3.1.1.3.3.2.cmml">σ</mi><mn id="S3.SS1.p1.3.m3.1.1.3.3.3" xref="S3.SS1.p1.3.m3.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><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"><in id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1.1"></in><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">𝑐</ci><apply id="S3.SS1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.3.m3.1.1.3.1.cmml" xref="S3.SS1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.3.m3.1.1.3.2.cmml" xref="S3.SS1.p1.3.m3.1.1.3.2">𝐶</ci><apply id="S3.SS1.p1.3.m3.1.1.3.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.3.m3.1.1.3.3.1.cmml" xref="S3.SS1.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p1.3.m3.1.1.3.3.2.cmml" xref="S3.SS1.p1.3.m3.1.1.3.3.2">𝜎</ci><cn id="S3.SS1.p1.3.m3.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p1.3.m3.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, there exists <math alttext="c^{\prime}\in C_{\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mrow id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml"><msup id="S3.SS1.p1.4.m4.1.1.2" xref="S3.SS1.p1.4.m4.1.1.2.cmml"><mi id="S3.SS1.p1.4.m4.1.1.2.2" xref="S3.SS1.p1.4.m4.1.1.2.2.cmml">c</mi><mo id="S3.SS1.p1.4.m4.1.1.2.3" xref="S3.SS1.p1.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.p1.4.m4.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S3.SS1.p1.4.m4.1.1.3" xref="S3.SS1.p1.4.m4.1.1.3.cmml"><mi id="S3.SS1.p1.4.m4.1.1.3.2" xref="S3.SS1.p1.4.m4.1.1.3.2.cmml">C</mi><msubsup id="S3.SS1.p1.4.m4.1.1.3.3" xref="S3.SS1.p1.4.m4.1.1.3.3.cmml"><mi id="S3.SS1.p1.4.m4.1.1.3.3.2.2" xref="S3.SS1.p1.4.m4.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.SS1.p1.4.m4.1.1.3.3.2.3" xref="S3.SS1.p1.4.m4.1.1.3.3.2.3.cmml">0</mn><mo id="S3.SS1.p1.4.m4.1.1.3.3.3" xref="S3.SS1.p1.4.m4.1.1.3.3.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><apply id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1"><in id="S3.SS1.p1.4.m4.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"></in><apply id="S3.SS1.p1.4.m4.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.2.1.cmml" xref="S3.SS1.p1.4.m4.1.1.2">superscript</csymbol><ci id="S3.SS1.p1.4.m4.1.1.2.2.cmml" xref="S3.SS1.p1.4.m4.1.1.2.2">𝑐</ci><ci id="S3.SS1.p1.4.m4.1.1.2.3.cmml" xref="S3.SS1.p1.4.m4.1.1.2.3">′</ci></apply><apply id="S3.SS1.p1.4.m4.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.3.1.cmml" xref="S3.SS1.p1.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.4.m4.1.1.3.2.cmml" xref="S3.SS1.p1.4.m4.1.1.3.2">𝐶</ci><apply id="S3.SS1.p1.4.m4.1.1.3.3.cmml" xref="S3.SS1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.3.3.1.cmml" xref="S3.SS1.p1.4.m4.1.1.3.3">superscript</csymbol><apply id="S3.SS1.p1.4.m4.1.1.3.3.2.cmml" xref="S3.SS1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.3.3.2.1.cmml" xref="S3.SS1.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p1.4.m4.1.1.3.3.2.2.cmml" xref="S3.SS1.p1.4.m4.1.1.3.3.2.2">𝜎</ci><cn id="S3.SS1.p1.4.m4.1.1.3.3.2.3.cmml" type="integer" xref="S3.SS1.p1.4.m4.1.1.3.3.2.3">0</cn></apply><ci id="S3.SS1.p1.4.m4.1.1.3.3.3.cmml" xref="S3.SS1.p1.4.m4.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">c^{\prime}\in C_{\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c^{\prime}\leq c" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mrow id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml"><msup id="S3.SS1.p1.5.m5.1.1.2" xref="S3.SS1.p1.5.m5.1.1.2.cmml"><mi id="S3.SS1.p1.5.m5.1.1.2.2" xref="S3.SS1.p1.5.m5.1.1.2.2.cmml">c</mi><mo id="S3.SS1.p1.5.m5.1.1.2.3" xref="S3.SS1.p1.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.p1.5.m5.1.1.1" xref="S3.SS1.p1.5.m5.1.1.1.cmml">≤</mo><mi id="S3.SS1.p1.5.m5.1.1.3" xref="S3.SS1.p1.5.m5.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><apply id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1"><leq id="S3.SS1.p1.5.m5.1.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1.1"></leq><apply id="S3.SS1.p1.5.m5.1.1.2.cmml" xref="S3.SS1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.1.1.2.1.cmml" xref="S3.SS1.p1.5.m5.1.1.2">superscript</csymbol><ci id="S3.SS1.p1.5.m5.1.1.2.2.cmml" xref="S3.SS1.p1.5.m5.1.1.2.2">𝑐</ci><ci id="S3.SS1.p1.5.m5.1.1.2.3.cmml" xref="S3.SS1.p1.5.m5.1.1.2.3">′</ci></apply><ci id="S3.SS1.p1.5.m5.1.1.3.cmml" xref="S3.SS1.p1.5.m5.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">c^{\prime}\leq c</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_c</annotation></semantics></math>, and we say that <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.1"><semantics id="S3.SS1.p1.6.m6.1a"><msubsup id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml"><mi id="S3.SS1.p1.6.m6.1.1.2.2" xref="S3.SS1.p1.6.m6.1.1.2.2.cmml">σ</mi><mn id="S3.SS1.p1.6.m6.1.1.3" xref="S3.SS1.p1.6.m6.1.1.3.cmml">0</mn><mo id="S3.SS1.p1.6.m6.1.1.2.3" xref="S3.SS1.p1.6.m6.1.1.2.3.cmml">′</mo></msubsup><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"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1">subscript</csymbol><apply id="S3.SS1.p1.6.m6.1.1.2.cmml" xref="S3.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.2.1.cmml" xref="S3.SS1.p1.6.m6.1.1">superscript</csymbol><ci id="S3.SS1.p1.6.m6.1.1.2.2.cmml" xref="S3.SS1.p1.6.m6.1.1.2.2">𝜎</ci><ci id="S3.SS1.p1.6.m6.1.1.2.3.cmml" xref="S3.SS1.p1.6.m6.1.1.2.3">′</ci></apply><cn id="S3.SS1.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.SS1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S3.SS1.p1.18.1">better</em> than <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m7.1"><semantics id="S3.SS1.p1.7.m7.1a"><msub id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml"><mi id="S3.SS1.p1.7.m7.1.1.2" xref="S3.SS1.p1.7.m7.1.1.2.cmml">σ</mi><mn id="S3.SS1.p1.7.m7.1.1.3" xref="S3.SS1.p1.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.1b"><apply id="S3.SS1.p1.7.m7.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m7.1.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS1.p1.7.m7.1.1.2.cmml" xref="S3.SS1.p1.7.m7.1.1.2">𝜎</ci><cn id="S3.SS1.p1.7.m7.1.1.3.cmml" type="integer" xref="S3.SS1.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m7.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. This relation <math alttext="\preceq" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m8.1"><semantics id="S3.SS1.p1.8.m8.1a"><mo id="S3.SS1.p1.8.m8.1.1" xref="S3.SS1.p1.8.m8.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m8.1b"><csymbol cd="latexml" id="S3.SS1.p1.8.m8.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m8.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m8.1d">⪯</annotation></semantics></math> on strategies is a preorder (it is reflexive and transitive). Given two strategies <math alttext="\sigma_{0},\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m9.2"><semantics id="S3.SS1.p1.9.m9.2a"><mrow id="S3.SS1.p1.9.m9.2.2.2" xref="S3.SS1.p1.9.m9.2.2.3.cmml"><msub id="S3.SS1.p1.9.m9.1.1.1.1" xref="S3.SS1.p1.9.m9.1.1.1.1.cmml"><mi id="S3.SS1.p1.9.m9.1.1.1.1.2" xref="S3.SS1.p1.9.m9.1.1.1.1.2.cmml">σ</mi><mn id="S3.SS1.p1.9.m9.1.1.1.1.3" xref="S3.SS1.p1.9.m9.1.1.1.1.3.cmml">0</mn></msub><mo id="S3.SS1.p1.9.m9.2.2.2.3" xref="S3.SS1.p1.9.m9.2.2.3.cmml">,</mo><msubsup id="S3.SS1.p1.9.m9.2.2.2.2" xref="S3.SS1.p1.9.m9.2.2.2.2.cmml"><mi id="S3.SS1.p1.9.m9.2.2.2.2.2.2" xref="S3.SS1.p1.9.m9.2.2.2.2.2.2.cmml">σ</mi><mn id="S3.SS1.p1.9.m9.2.2.2.2.3" xref="S3.SS1.p1.9.m9.2.2.2.2.3.cmml">0</mn><mo id="S3.SS1.p1.9.m9.2.2.2.2.2.3" xref="S3.SS1.p1.9.m9.2.2.2.2.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m9.2b"><list id="S3.SS1.p1.9.m9.2.2.3.cmml" xref="S3.SS1.p1.9.m9.2.2.2"><apply id="S3.SS1.p1.9.m9.1.1.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m9.1.1.1.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.9.m9.1.1.1.1.2.cmml" xref="S3.SS1.p1.9.m9.1.1.1.1.2">𝜎</ci><cn id="S3.SS1.p1.9.m9.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.9.m9.1.1.1.1.3">0</cn></apply><apply id="S3.SS1.p1.9.m9.2.2.2.2.cmml" xref="S3.SS1.p1.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m9.2.2.2.2.1.cmml" xref="S3.SS1.p1.9.m9.2.2.2.2">subscript</csymbol><apply id="S3.SS1.p1.9.m9.2.2.2.2.2.cmml" xref="S3.SS1.p1.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m9.2.2.2.2.2.1.cmml" xref="S3.SS1.p1.9.m9.2.2.2.2">superscript</csymbol><ci id="S3.SS1.p1.9.m9.2.2.2.2.2.2.cmml" xref="S3.SS1.p1.9.m9.2.2.2.2.2.2">𝜎</ci><ci id="S3.SS1.p1.9.m9.2.2.2.2.2.3.cmml" xref="S3.SS1.p1.9.m9.2.2.2.2.2.3">′</ci></apply><cn id="S3.SS1.p1.9.m9.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.p1.9.m9.2.2.2.2.3">0</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m9.2c">\sigma_{0},\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.9.m9.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\sigma^{\prime}_{0}\preceq\sigma_{0}" 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"><msubsup id="S3.SS1.p1.10.m10.1.1.2" xref="S3.SS1.p1.10.m10.1.1.2.cmml"><mi id="S3.SS1.p1.10.m10.1.1.2.2.2" xref="S3.SS1.p1.10.m10.1.1.2.2.2.cmml">σ</mi><mn id="S3.SS1.p1.10.m10.1.1.2.3" xref="S3.SS1.p1.10.m10.1.1.2.3.cmml">0</mn><mo id="S3.SS1.p1.10.m10.1.1.2.2.3" xref="S3.SS1.p1.10.m10.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.p1.10.m10.1.1.1" xref="S3.SS1.p1.10.m10.1.1.1.cmml">⪯</mo><msub id="S3.SS1.p1.10.m10.1.1.3" xref="S3.SS1.p1.10.m10.1.1.3.cmml"><mi id="S3.SS1.p1.10.m10.1.1.3.2" xref="S3.SS1.p1.10.m10.1.1.3.2.cmml">σ</mi><mn id="S3.SS1.p1.10.m10.1.1.3.3" xref="S3.SS1.p1.10.m10.1.1.3.3.cmml">0</mn></msub></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"><csymbol cd="latexml" id="S3.SS1.p1.10.m10.1.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1.1">precedes-or-equals</csymbol><apply id="S3.SS1.p1.10.m10.1.1.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.1.1.2.1.cmml" xref="S3.SS1.p1.10.m10.1.1.2">subscript</csymbol><apply id="S3.SS1.p1.10.m10.1.1.2.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.1.1.2.2.1.cmml" xref="S3.SS1.p1.10.m10.1.1.2">superscript</csymbol><ci id="S3.SS1.p1.10.m10.1.1.2.2.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2.2.2">𝜎</ci><ci id="S3.SS1.p1.10.m10.1.1.2.2.3.cmml" xref="S3.SS1.p1.10.m10.1.1.2.2.3">′</ci></apply><cn id="S3.SS1.p1.10.m10.1.1.2.3.cmml" type="integer" xref="S3.SS1.p1.10.m10.1.1.2.3">0</cn></apply><apply id="S3.SS1.p1.10.m10.1.1.3.cmml" xref="S3.SS1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.1.1.3.1.cmml" xref="S3.SS1.p1.10.m10.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.10.m10.1.1.3.2.cmml" xref="S3.SS1.p1.10.m10.1.1.3.2">𝜎</ci><cn id="S3.SS1.p1.10.m10.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.10.m10.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m10.1c">\sigma^{\prime}_{0}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m10.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, we define <math alttext="\sigma^{\prime}_{0}\prec\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.11.m11.1"><semantics id="S3.SS1.p1.11.m11.1a"><mrow id="S3.SS1.p1.11.m11.1.1" xref="S3.SS1.p1.11.m11.1.1.cmml"><msubsup id="S3.SS1.p1.11.m11.1.1.2" xref="S3.SS1.p1.11.m11.1.1.2.cmml"><mi id="S3.SS1.p1.11.m11.1.1.2.2.2" xref="S3.SS1.p1.11.m11.1.1.2.2.2.cmml">σ</mi><mn id="S3.SS1.p1.11.m11.1.1.2.3" xref="S3.SS1.p1.11.m11.1.1.2.3.cmml">0</mn><mo id="S3.SS1.p1.11.m11.1.1.2.2.3" xref="S3.SS1.p1.11.m11.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.p1.11.m11.1.1.1" xref="S3.SS1.p1.11.m11.1.1.1.cmml">≺</mo><msub id="S3.SS1.p1.11.m11.1.1.3" xref="S3.SS1.p1.11.m11.1.1.3.cmml"><mi id="S3.SS1.p1.11.m11.1.1.3.2" xref="S3.SS1.p1.11.m11.1.1.3.2.cmml">σ</mi><mn id="S3.SS1.p1.11.m11.1.1.3.3" xref="S3.SS1.p1.11.m11.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.11.m11.1b"><apply id="S3.SS1.p1.11.m11.1.1.cmml" xref="S3.SS1.p1.11.m11.1.1"><csymbol cd="latexml" id="S3.SS1.p1.11.m11.1.1.1.cmml" xref="S3.SS1.p1.11.m11.1.1.1">precedes</csymbol><apply id="S3.SS1.p1.11.m11.1.1.2.cmml" xref="S3.SS1.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m11.1.1.2.1.cmml" xref="S3.SS1.p1.11.m11.1.1.2">subscript</csymbol><apply id="S3.SS1.p1.11.m11.1.1.2.2.cmml" xref="S3.SS1.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m11.1.1.2.2.1.cmml" xref="S3.SS1.p1.11.m11.1.1.2">superscript</csymbol><ci id="S3.SS1.p1.11.m11.1.1.2.2.2.cmml" xref="S3.SS1.p1.11.m11.1.1.2.2.2">𝜎</ci><ci id="S3.SS1.p1.11.m11.1.1.2.2.3.cmml" xref="S3.SS1.p1.11.m11.1.1.2.2.3">′</ci></apply><cn id="S3.SS1.p1.11.m11.1.1.2.3.cmml" type="integer" xref="S3.SS1.p1.11.m11.1.1.2.3">0</cn></apply><apply id="S3.SS1.p1.11.m11.1.1.3.cmml" xref="S3.SS1.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m11.1.1.3.1.cmml" xref="S3.SS1.p1.11.m11.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.11.m11.1.1.3.2.cmml" xref="S3.SS1.p1.11.m11.1.1.3.2">𝜎</ci><cn id="S3.SS1.p1.11.m11.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.11.m11.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.11.m11.1c">\sigma^{\prime}_{0}\prec\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.11.m11.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> when <math alttext="C_{\sigma^{\prime}_{0}}\neq C_{\sigma_{0}}" 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"><msub id="S3.SS1.p1.12.m12.1.1.2" xref="S3.SS1.p1.12.m12.1.1.2.cmml"><mi id="S3.SS1.p1.12.m12.1.1.2.2" xref="S3.SS1.p1.12.m12.1.1.2.2.cmml">C</mi><msubsup id="S3.SS1.p1.12.m12.1.1.2.3" xref="S3.SS1.p1.12.m12.1.1.2.3.cmml"><mi id="S3.SS1.p1.12.m12.1.1.2.3.2.2" xref="S3.SS1.p1.12.m12.1.1.2.3.2.2.cmml">σ</mi><mn id="S3.SS1.p1.12.m12.1.1.2.3.3" xref="S3.SS1.p1.12.m12.1.1.2.3.3.cmml">0</mn><mo id="S3.SS1.p1.12.m12.1.1.2.3.2.3" xref="S3.SS1.p1.12.m12.1.1.2.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.SS1.p1.12.m12.1.1.1" xref="S3.SS1.p1.12.m12.1.1.1.cmml">≠</mo><msub id="S3.SS1.p1.12.m12.1.1.3" xref="S3.SS1.p1.12.m12.1.1.3.cmml"><mi id="S3.SS1.p1.12.m12.1.1.3.2" xref="S3.SS1.p1.12.m12.1.1.3.2.cmml">C</mi><msub id="S3.SS1.p1.12.m12.1.1.3.3" xref="S3.SS1.p1.12.m12.1.1.3.3.cmml"><mi id="S3.SS1.p1.12.m12.1.1.3.3.2" xref="S3.SS1.p1.12.m12.1.1.3.3.2.cmml">σ</mi><mn id="S3.SS1.p1.12.m12.1.1.3.3.3" xref="S3.SS1.p1.12.m12.1.1.3.3.3.cmml">0</mn></msub></msub></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"><neq id="S3.SS1.p1.12.m12.1.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1.1"></neq><apply id="S3.SS1.p1.12.m12.1.1.2.cmml" xref="S3.SS1.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.12.m12.1.1.2.1.cmml" xref="S3.SS1.p1.12.m12.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.12.m12.1.1.2.2.cmml" xref="S3.SS1.p1.12.m12.1.1.2.2">𝐶</ci><apply id="S3.SS1.p1.12.m12.1.1.2.3.cmml" xref="S3.SS1.p1.12.m12.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.12.m12.1.1.2.3.1.cmml" xref="S3.SS1.p1.12.m12.1.1.2.3">subscript</csymbol><apply id="S3.SS1.p1.12.m12.1.1.2.3.2.cmml" xref="S3.SS1.p1.12.m12.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.12.m12.1.1.2.3.2.1.cmml" xref="S3.SS1.p1.12.m12.1.1.2.3">superscript</csymbol><ci id="S3.SS1.p1.12.m12.1.1.2.3.2.2.cmml" xref="S3.SS1.p1.12.m12.1.1.2.3.2.2">𝜎</ci><ci id="S3.SS1.p1.12.m12.1.1.2.3.2.3.cmml" xref="S3.SS1.p1.12.m12.1.1.2.3.2.3">′</ci></apply><cn id="S3.SS1.p1.12.m12.1.1.2.3.3.cmml" type="integer" xref="S3.SS1.p1.12.m12.1.1.2.3.3">0</cn></apply></apply><apply id="S3.SS1.p1.12.m12.1.1.3.cmml" xref="S3.SS1.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.12.m12.1.1.3.1.cmml" xref="S3.SS1.p1.12.m12.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.12.m12.1.1.3.2.cmml" xref="S3.SS1.p1.12.m12.1.1.3.2">𝐶</ci><apply id="S3.SS1.p1.12.m12.1.1.3.3.cmml" xref="S3.SS1.p1.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.12.m12.1.1.3.3.1.cmml" xref="S3.SS1.p1.12.m12.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p1.12.m12.1.1.3.3.2.cmml" xref="S3.SS1.p1.12.m12.1.1.3.3.2">𝜎</ci><cn id="S3.SS1.p1.12.m12.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p1.12.m12.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.12.m12.1c">C_{\sigma^{\prime}_{0}}\neq C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.12.m12.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\sigma^{\prime}_{0}\simeq\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.13.m13.1"><semantics id="S3.SS1.p1.13.m13.1a"><mrow id="S3.SS1.p1.13.m13.1.1" xref="S3.SS1.p1.13.m13.1.1.cmml"><msubsup id="S3.SS1.p1.13.m13.1.1.2" xref="S3.SS1.p1.13.m13.1.1.2.cmml"><mi id="S3.SS1.p1.13.m13.1.1.2.2.2" xref="S3.SS1.p1.13.m13.1.1.2.2.2.cmml">σ</mi><mn id="S3.SS1.p1.13.m13.1.1.2.3" xref="S3.SS1.p1.13.m13.1.1.2.3.cmml">0</mn><mo id="S3.SS1.p1.13.m13.1.1.2.2.3" xref="S3.SS1.p1.13.m13.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.p1.13.m13.1.1.1" xref="S3.SS1.p1.13.m13.1.1.1.cmml">≃</mo><msub id="S3.SS1.p1.13.m13.1.1.3" xref="S3.SS1.p1.13.m13.1.1.3.cmml"><mi id="S3.SS1.p1.13.m13.1.1.3.2" xref="S3.SS1.p1.13.m13.1.1.3.2.cmml">σ</mi><mn id="S3.SS1.p1.13.m13.1.1.3.3" xref="S3.SS1.p1.13.m13.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.13.m13.1b"><apply id="S3.SS1.p1.13.m13.1.1.cmml" xref="S3.SS1.p1.13.m13.1.1"><csymbol cd="latexml" id="S3.SS1.p1.13.m13.1.1.1.cmml" xref="S3.SS1.p1.13.m13.1.1.1">similar-to-or-equals</csymbol><apply id="S3.SS1.p1.13.m13.1.1.2.cmml" xref="S3.SS1.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m13.1.1.2.1.cmml" xref="S3.SS1.p1.13.m13.1.1.2">subscript</csymbol><apply id="S3.SS1.p1.13.m13.1.1.2.2.cmml" xref="S3.SS1.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m13.1.1.2.2.1.cmml" xref="S3.SS1.p1.13.m13.1.1.2">superscript</csymbol><ci id="S3.SS1.p1.13.m13.1.1.2.2.2.cmml" xref="S3.SS1.p1.13.m13.1.1.2.2.2">𝜎</ci><ci id="S3.SS1.p1.13.m13.1.1.2.2.3.cmml" xref="S3.SS1.p1.13.m13.1.1.2.2.3">′</ci></apply><cn id="S3.SS1.p1.13.m13.1.1.2.3.cmml" type="integer" xref="S3.SS1.p1.13.m13.1.1.2.3">0</cn></apply><apply id="S3.SS1.p1.13.m13.1.1.3.cmml" xref="S3.SS1.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m13.1.1.3.1.cmml" xref="S3.SS1.p1.13.m13.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.13.m13.1.1.3.2.cmml" xref="S3.SS1.p1.13.m13.1.1.3.2">𝜎</ci><cn id="S3.SS1.p1.13.m13.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.13.m13.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.13.m13.1c">\sigma^{\prime}_{0}\simeq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.13.m13.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≃ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> when <math alttext="C_{\sigma^{\prime}_{0}}=C_{\sigma_{0}}" 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"><msub id="S3.SS1.p1.14.m14.1.1.2" xref="S3.SS1.p1.14.m14.1.1.2.cmml"><mi id="S3.SS1.p1.14.m14.1.1.2.2" xref="S3.SS1.p1.14.m14.1.1.2.2.cmml">C</mi><msubsup id="S3.SS1.p1.14.m14.1.1.2.3" xref="S3.SS1.p1.14.m14.1.1.2.3.cmml"><mi id="S3.SS1.p1.14.m14.1.1.2.3.2.2" xref="S3.SS1.p1.14.m14.1.1.2.3.2.2.cmml">σ</mi><mn id="S3.SS1.p1.14.m14.1.1.2.3.3" xref="S3.SS1.p1.14.m14.1.1.2.3.3.cmml">0</mn><mo id="S3.SS1.p1.14.m14.1.1.2.3.2.3" xref="S3.SS1.p1.14.m14.1.1.2.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.SS1.p1.14.m14.1.1.1" xref="S3.SS1.p1.14.m14.1.1.1.cmml">=</mo><msub 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" xref="S3.SS1.p1.14.m14.1.1.3.2.cmml">C</mi><msub id="S3.SS1.p1.14.m14.1.1.3.3" xref="S3.SS1.p1.14.m14.1.1.3.3.cmml"><mi id="S3.SS1.p1.14.m14.1.1.3.3.2" xref="S3.SS1.p1.14.m14.1.1.3.3.2.cmml">σ</mi><mn id="S3.SS1.p1.14.m14.1.1.3.3.3" xref="S3.SS1.p1.14.m14.1.1.3.3.3.cmml">0</mn></msub></msub></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"><eq id="S3.SS1.p1.14.m14.1.1.1.cmml" xref="S3.SS1.p1.14.m14.1.1.1"></eq><apply id="S3.SS1.p1.14.m14.1.1.2.cmml" xref="S3.SS1.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.14.m14.1.1.2.1.cmml" xref="S3.SS1.p1.14.m14.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.14.m14.1.1.2.2.cmml" xref="S3.SS1.p1.14.m14.1.1.2.2">𝐶</ci><apply id="S3.SS1.p1.14.m14.1.1.2.3.cmml" xref="S3.SS1.p1.14.m14.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.14.m14.1.1.2.3.1.cmml" xref="S3.SS1.p1.14.m14.1.1.2.3">subscript</csymbol><apply id="S3.SS1.p1.14.m14.1.1.2.3.2.cmml" xref="S3.SS1.p1.14.m14.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.14.m14.1.1.2.3.2.1.cmml" xref="S3.SS1.p1.14.m14.1.1.2.3">superscript</csymbol><ci id="S3.SS1.p1.14.m14.1.1.2.3.2.2.cmml" xref="S3.SS1.p1.14.m14.1.1.2.3.2.2">𝜎</ci><ci id="S3.SS1.p1.14.m14.1.1.2.3.2.3.cmml" xref="S3.SS1.p1.14.m14.1.1.2.3.2.3">′</ci></apply><cn id="S3.SS1.p1.14.m14.1.1.2.3.3.cmml" type="integer" xref="S3.SS1.p1.14.m14.1.1.2.3.3">0</cn></apply></apply><apply id="S3.SS1.p1.14.m14.1.1.3.cmml" xref="S3.SS1.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.14.m14.1.1.3.1.cmml" xref="S3.SS1.p1.14.m14.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.14.m14.1.1.3.2.cmml" xref="S3.SS1.p1.14.m14.1.1.3.2">𝐶</ci><apply id="S3.SS1.p1.14.m14.1.1.3.3.cmml" xref="S3.SS1.p1.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.14.m14.1.1.3.3.1.cmml" xref="S3.SS1.p1.14.m14.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p1.14.m14.1.1.3.3.2.cmml" xref="S3.SS1.p1.14.m14.1.1.3.3.2">𝜎</ci><cn id="S3.SS1.p1.14.m14.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p1.14.m14.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.14.m14.1c">C_{\sigma^{\prime}_{0}}=C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.14.m14.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. In the sequel, we modify solutions <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.15.m15.1"><semantics id="S3.SS1.p1.15.m15.1a"><msub id="S3.SS1.p1.15.m15.1.1" xref="S3.SS1.p1.15.m15.1.1.cmml"><mi id="S3.SS1.p1.15.m15.1.1.2" xref="S3.SS1.p1.15.m15.1.1.2.cmml">σ</mi><mn id="S3.SS1.p1.15.m15.1.1.3" xref="S3.SS1.p1.15.m15.1.1.3.cmml">0</mn></msub><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"><csymbol cd="ambiguous" id="S3.SS1.p1.15.m15.1.1.1.cmml" xref="S3.SS1.p1.15.m15.1.1">subscript</csymbol><ci id="S3.SS1.p1.15.m15.1.1.2.cmml" xref="S3.SS1.p1.15.m15.1.1.2">𝜎</ci><cn id="S3.SS1.p1.15.m15.1.1.3.cmml" type="integer" xref="S3.SS1.p1.15.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.15.m15.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.15.m15.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> to the SPS problem to get better solutions <math alttext="\sigma^{\prime}_{0}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.16.m16.1"><semantics id="S3.SS1.p1.16.m16.1a"><mrow id="S3.SS1.p1.16.m16.1.1" xref="S3.SS1.p1.16.m16.1.1.cmml"><msubsup id="S3.SS1.p1.16.m16.1.1.2" xref="S3.SS1.p1.16.m16.1.1.2.cmml"><mi id="S3.SS1.p1.16.m16.1.1.2.2.2" xref="S3.SS1.p1.16.m16.1.1.2.2.2.cmml">σ</mi><mn id="S3.SS1.p1.16.m16.1.1.2.3" xref="S3.SS1.p1.16.m16.1.1.2.3.cmml">0</mn><mo id="S3.SS1.p1.16.m16.1.1.2.2.3" xref="S3.SS1.p1.16.m16.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.p1.16.m16.1.1.1" xref="S3.SS1.p1.16.m16.1.1.1.cmml">⪯</mo><msub id="S3.SS1.p1.16.m16.1.1.3" xref="S3.SS1.p1.16.m16.1.1.3.cmml"><mi id="S3.SS1.p1.16.m16.1.1.3.2" xref="S3.SS1.p1.16.m16.1.1.3.2.cmml">σ</mi><mn id="S3.SS1.p1.16.m16.1.1.3.3" xref="S3.SS1.p1.16.m16.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.16.m16.1b"><apply id="S3.SS1.p1.16.m16.1.1.cmml" xref="S3.SS1.p1.16.m16.1.1"><csymbol cd="latexml" id="S3.SS1.p1.16.m16.1.1.1.cmml" xref="S3.SS1.p1.16.m16.1.1.1">precedes-or-equals</csymbol><apply id="S3.SS1.p1.16.m16.1.1.2.cmml" xref="S3.SS1.p1.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.16.m16.1.1.2.1.cmml" xref="S3.SS1.p1.16.m16.1.1.2">subscript</csymbol><apply id="S3.SS1.p1.16.m16.1.1.2.2.cmml" xref="S3.SS1.p1.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.16.m16.1.1.2.2.1.cmml" xref="S3.SS1.p1.16.m16.1.1.2">superscript</csymbol><ci id="S3.SS1.p1.16.m16.1.1.2.2.2.cmml" xref="S3.SS1.p1.16.m16.1.1.2.2.2">𝜎</ci><ci id="S3.SS1.p1.16.m16.1.1.2.2.3.cmml" xref="S3.SS1.p1.16.m16.1.1.2.2.3">′</ci></apply><cn id="S3.SS1.p1.16.m16.1.1.2.3.cmml" type="integer" xref="S3.SS1.p1.16.m16.1.1.2.3">0</cn></apply><apply id="S3.SS1.p1.16.m16.1.1.3.cmml" xref="S3.SS1.p1.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.16.m16.1.1.3.1.cmml" xref="S3.SS1.p1.16.m16.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.16.m16.1.1.3.2.cmml" xref="S3.SS1.p1.16.m16.1.1.3.2">𝜎</ci><cn id="S3.SS1.p1.16.m16.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.16.m16.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.16.m16.1c">\sigma^{\prime}_{0}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.16.m16.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, and we say that <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.17.m17.1"><semantics id="S3.SS1.p1.17.m17.1a"><msubsup id="S3.SS1.p1.17.m17.1.1" xref="S3.SS1.p1.17.m17.1.1.cmml"><mi id="S3.SS1.p1.17.m17.1.1.2.2" xref="S3.SS1.p1.17.m17.1.1.2.2.cmml">σ</mi><mn id="S3.SS1.p1.17.m17.1.1.3" xref="S3.SS1.p1.17.m17.1.1.3.cmml">0</mn><mo id="S3.SS1.p1.17.m17.1.1.2.3" xref="S3.SS1.p1.17.m17.1.1.2.3.cmml">′</mo></msubsup><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"><csymbol cd="ambiguous" id="S3.SS1.p1.17.m17.1.1.1.cmml" xref="S3.SS1.p1.17.m17.1.1">subscript</csymbol><apply id="S3.SS1.p1.17.m17.1.1.2.cmml" xref="S3.SS1.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.17.m17.1.1.2.1.cmml" xref="S3.SS1.p1.17.m17.1.1">superscript</csymbol><ci id="S3.SS1.p1.17.m17.1.1.2.2.cmml" xref="S3.SS1.p1.17.m17.1.1.2.2">𝜎</ci><ci id="S3.SS1.p1.17.m17.1.1.2.3.cmml" xref="S3.SS1.p1.17.m17.1.1.2.3">′</ci></apply><cn id="S3.SS1.p1.17.m17.1.1.3.cmml" type="integer" xref="S3.SS1.p1.17.m17.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.17.m17.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.17.m17.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S3.SS1.p1.18.2">improves</em> the given solution <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.18.m18.1"><semantics id="S3.SS1.p1.18.m18.1a"><msub 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><mn id="S3.SS1.p1.18.m18.1.1.3" xref="S3.SS1.p1.18.m18.1.1.3.cmml">0</mn></msub><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">subscript</csymbol><ci id="S3.SS1.p1.18.m18.1.1.2.cmml" xref="S3.SS1.p1.18.m18.1.1.2">𝜎</ci><cn id="S3.SS1.p1.18.m18.1.1.3.cmml" type="integer" xref="S3.SS1.p1.18.m18.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.18.m18.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.18.m18.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.19">A <em class="ltx_emph ltx_font_italic" id="S3.SS1.p2.19.1">subgame</em> of an SP game <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_G</annotation></semantics></math> is a couple <math alttext="(G,h)" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.2"><semantics id="S3.SS1.p2.2.m2.2a"><mrow id="S3.SS1.p2.2.m2.2.3.2" xref="S3.SS1.p2.2.m2.2.3.1.cmml"><mo id="S3.SS1.p2.2.m2.2.3.2.1" stretchy="false" xref="S3.SS1.p2.2.m2.2.3.1.cmml">(</mo><mi id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml">G</mi><mo id="S3.SS1.p2.2.m2.2.3.2.2" xref="S3.SS1.p2.2.m2.2.3.1.cmml">,</mo><mi id="S3.SS1.p2.2.m2.2.2" xref="S3.SS1.p2.2.m2.2.2.cmml">h</mi><mo id="S3.SS1.p2.2.m2.2.3.2.3" stretchy="false" xref="S3.SS1.p2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.2b"><interval closure="open" id="S3.SS1.p2.2.m2.2.3.1.cmml" xref="S3.SS1.p2.2.m2.2.3.2"><ci id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1">𝐺</ci><ci id="S3.SS1.p2.2.m2.2.2.cmml" xref="S3.SS1.p2.2.m2.2.2">ℎ</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.2c">(G,h)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.2d">( italic_G , italic_h )</annotation></semantics></math>, denoted <math alttext="G_{|h}" class="ltx_math_unparsed" 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"><mi id="S3.SS1.p2.3.m3.1.1.2">G</mi><mrow id="S3.SS1.p2.3.m3.1.1.3"><mo fence="false" id="S3.SS1.p2.3.m3.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p2.3.m3.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="h\in\textsf{Hist}" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.1"><semantics id="S3.SS1.p2.4.m4.1a"><mrow id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml"><mi id="S3.SS1.p2.4.m4.1.1.2" xref="S3.SS1.p2.4.m4.1.1.2.cmml">h</mi><mo id="S3.SS1.p2.4.m4.1.1.1" xref="S3.SS1.p2.4.m4.1.1.1.cmml">∈</mo><mtext class="ltx_mathvariant_sans-serif" id="S3.SS1.p2.4.m4.1.1.3" xref="S3.SS1.p2.4.m4.1.1.3a.cmml">Hist</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.1b"><apply id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1"><in id="S3.SS1.p2.4.m4.1.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1.1"></in><ci id="S3.SS1.p2.4.m4.1.1.2.cmml" xref="S3.SS1.p2.4.m4.1.1.2">ℎ</ci><ci id="S3.SS1.p2.4.m4.1.1.3a.cmml" xref="S3.SS1.p2.4.m4.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.SS1.p2.4.m4.1.1.3.cmml" xref="S3.SS1.p2.4.m4.1.1.3">Hist</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.1c">h\in\textsf{Hist}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">italic_h ∈ Hist</annotation></semantics></math>. In the same way that <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.1"><semantics id="S3.SS1.p2.5.m5.1a"><mi id="S3.SS1.p2.5.m5.1.1" xref="S3.SS1.p2.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m5.1b"><ci id="S3.SS1.p2.5.m5.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m5.1d">italic_G</annotation></semantics></math> can be seen as the set of its plays, <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p2.6.m6.1"><semantics id="S3.SS1.p2.6.m6.1a"><msub id="S3.SS1.p2.6.m6.1.1"><mi id="S3.SS1.p2.6.m6.1.1.2">G</mi><mrow id="S3.SS1.p2.6.m6.1.1.3"><mo fence="false" id="S3.SS1.p2.6.m6.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p2.6.m6.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m6.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m6.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> is seen as the restriction of <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p2.7.m7.1"><semantics id="S3.SS1.p2.7.m7.1a"><mi id="S3.SS1.p2.7.m7.1.1" xref="S3.SS1.p2.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.7.m7.1b"><ci id="S3.SS1.p2.7.m7.1.1.cmml" xref="S3.SS1.p2.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.7.m7.1d">italic_G</annotation></semantics></math> to plays with the prefix <math alttext="h" class="ltx_Math" display="inline" id="S3.SS1.p2.8.m8.1"><semantics id="S3.SS1.p2.8.m8.1a"><mi id="S3.SS1.p2.8.m8.1.1" xref="S3.SS1.p2.8.m8.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.8.m8.1b"><ci id="S3.SS1.p2.8.m8.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.8.m8.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.8.m8.1d">italic_h</annotation></semantics></math> (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.F2" title="Figure 2 ‣ 3.1 Order on Strategies and Subgames ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2</span></a>). In particular, we have <math alttext="G_{|v_{0}}=G" class="ltx_math_unparsed" display="inline" id="S3.SS1.p2.9.m9.1"><semantics id="S3.SS1.p2.9.m9.1a"><mrow id="S3.SS1.p2.9.m9.1.1"><msub id="S3.SS1.p2.9.m9.1.1.2"><mi id="S3.SS1.p2.9.m9.1.1.2.2">G</mi><mrow id="S3.SS1.p2.9.m9.1.1.2.3"><mo fence="false" id="S3.SS1.p2.9.m9.1.1.2.3.1" rspace="0.167em" stretchy="false">|</mo><msub id="S3.SS1.p2.9.m9.1.1.2.3.2"><mi id="S3.SS1.p2.9.m9.1.1.2.3.2.2">v</mi><mn id="S3.SS1.p2.9.m9.1.1.2.3.2.3">0</mn></msub></mrow></msub><mo id="S3.SS1.p2.9.m9.1.1.1">=</mo><mi id="S3.SS1.p2.9.m9.1.1.3">G</mi></mrow><annotation encoding="application/x-tex" id="S3.SS1.p2.9.m9.1b">G_{|v_{0}}=G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.9.m9.1c">italic_G start_POSTSUBSCRIPT | italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_G</annotation></semantics></math> where <math alttext="v_{0}" class="ltx_Math" display="inline" id="S3.SS1.p2.10.m10.1"><semantics id="S3.SS1.p2.10.m10.1a"><msub id="S3.SS1.p2.10.m10.1.1" xref="S3.SS1.p2.10.m10.1.1.cmml"><mi id="S3.SS1.p2.10.m10.1.1.2" xref="S3.SS1.p2.10.m10.1.1.2.cmml">v</mi><mn id="S3.SS1.p2.10.m10.1.1.3" xref="S3.SS1.p2.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.10.m10.1b"><apply id="S3.SS1.p2.10.m10.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.10.m10.1.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1">subscript</csymbol><ci id="S3.SS1.p2.10.m10.1.1.2.cmml" xref="S3.SS1.p2.10.m10.1.1.2">𝑣</ci><cn id="S3.SS1.p2.10.m10.1.1.3.cmml" type="integer" xref="S3.SS1.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.10.m10.1c">v_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.10.m10.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the initial vertex of <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p2.11.m11.1"><semantics id="S3.SS1.p2.11.m11.1a"><mi id="S3.SS1.p2.11.m11.1.1" xref="S3.SS1.p2.11.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.11.m11.1b"><ci id="S3.SS1.p2.11.m11.1.1.cmml" xref="S3.SS1.p2.11.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.11.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.11.m11.1d">italic_G</annotation></semantics></math>. The value and cost of a play <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS1.p2.12.m12.1"><semantics id="S3.SS1.p2.12.m12.1a"><mi id="S3.SS1.p2.12.m12.1.1" xref="S3.SS1.p2.12.m12.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.12.m12.1b"><ci id="S3.SS1.p2.12.m12.1.1.cmml" xref="S3.SS1.p2.12.m12.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.12.m12.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.12.m12.1d">italic_ρ</annotation></semantics></math> in <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p2.13.m13.1"><semantics id="S3.SS1.p2.13.m13.1a"><msub id="S3.SS1.p2.13.m13.1.1"><mi id="S3.SS1.p2.13.m13.1.1.2">G</mi><mrow id="S3.SS1.p2.13.m13.1.1.3"><mo fence="false" id="S3.SS1.p2.13.m13.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p2.13.m13.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p2.13.m13.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.13.m13.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> are the same as those of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS1.p2.14.m14.1"><semantics id="S3.SS1.p2.14.m14.1a"><mi id="S3.SS1.p2.14.m14.1.1" xref="S3.SS1.p2.14.m14.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.14.m14.1b"><ci id="S3.SS1.p2.14.m14.1.1.cmml" xref="S3.SS1.p2.14.m14.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.14.m14.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.14.m14.1d">italic_ρ</annotation></semantics></math> as a play in <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p2.15.m15.1"><semantics id="S3.SS1.p2.15.m15.1a"><mi id="S3.SS1.p2.15.m15.1.1" xref="S3.SS1.p2.15.m15.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.15.m15.1b"><ci id="S3.SS1.p2.15.m15.1.1.cmml" xref="S3.SS1.p2.15.m15.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.15.m15.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.15.m15.1d">italic_G</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S3.SS1.p2.19.2">dimension</em> of <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p2.16.m16.1"><semantics id="S3.SS1.p2.16.m16.1a"><msub id="S3.SS1.p2.16.m16.1.1"><mi id="S3.SS1.p2.16.m16.1.1.2">G</mi><mrow id="S3.SS1.p2.16.m16.1.1.3"><mo fence="false" id="S3.SS1.p2.16.m16.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p2.16.m16.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p2.16.m16.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.16.m16.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> is the dimension of <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p2.17.m17.1"><semantics id="S3.SS1.p2.17.m17.1a"><mi id="S3.SS1.p2.17.m17.1.1" xref="S3.SS1.p2.17.m17.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.17.m17.1b"><ci id="S3.SS1.p2.17.m17.1.1.cmml" xref="S3.SS1.p2.17.m17.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.17.m17.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.17.m17.1d">italic_G</annotation></semantics></math> minus the number of targets visited<span class="ltx_note ltx_role_footnote" id="footnote6"><sup class="ltx_note_mark">6</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">6</sup><span class="ltx_tag ltx_tag_note">6</span>Notice that we do not include <math alttext="\Last{h}" class="ltx_Math" display="inline" id="footnote6.m1.1"><semantics id="footnote6.m1.1b"><mrow id="footnote6.m1.1.1" xref="footnote6.m1.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="footnote6.m1.1.1.2" xref="footnote6.m1.1.1.2b.cmml"><mtext id="footnote6.m1.1.1.2b" xref="footnote6.m1.1.1.2b.cmml">\Last</mtext></merror><mo id="footnote6.m1.1.1.1" xref="footnote6.m1.1.1.1.cmml">⁢</mo><mi id="footnote6.m1.1.1.3" xref="footnote6.m1.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m1.1c"><apply id="footnote6.m1.1.1.cmml" xref="footnote6.m1.1.1"><times id="footnote6.m1.1.1.1.cmml" xref="footnote6.m1.1.1.1"></times><ci id="footnote6.m1.1.1.2b.cmml" xref="footnote6.m1.1.1.2"><merror class="ltx_ERROR undefined undefined" id="footnote6.m1.1.1.2.cmml" xref="footnote6.m1.1.1.2"><mtext id="footnote6.m1.1.1.2a.cmml" xref="footnote6.m1.1.1.2">\Last</mtext></merror></ci><ci id="footnote6.m1.1.1.3.cmml" xref="footnote6.m1.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m1.1d">\Last{h}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m1.1e">italic_h</annotation></semantics></math> in <math alttext="h^{\prime}" class="ltx_Math" display="inline" id="footnote6.m2.1"><semantics id="footnote6.m2.1b"><msup id="footnote6.m2.1.1" xref="footnote6.m2.1.1.cmml"><mi id="footnote6.m2.1.1.2" xref="footnote6.m2.1.1.2.cmml">h</mi><mo id="footnote6.m2.1.1.3" xref="footnote6.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="footnote6.m2.1c"><apply id="footnote6.m2.1.1.cmml" xref="footnote6.m2.1.1"><csymbol cd="ambiguous" id="footnote6.m2.1.1.1.cmml" xref="footnote6.m2.1.1">superscript</csymbol><ci id="footnote6.m2.1.1.2.cmml" xref="footnote6.m2.1.1.2">ℎ</ci><ci id="footnote6.m2.1.1.3.cmml" xref="footnote6.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m2.1d">h^{\prime}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m2.1e">italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, as it can be seen as the initial vertex of <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="footnote6.m3.1"><semantics id="footnote6.m3.1b"><msub id="footnote6.m3.1.1"><mi id="footnote6.m3.1.1.2">G</mi><mrow id="footnote6.m3.1.1.3"><mo fence="false" id="footnote6.m3.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="footnote6.m3.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="footnote6.m3.1c">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m3.1d">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math>.</span></span></span> by <math alttext="h^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.p2.18.m18.1"><semantics id="S3.SS1.p2.18.m18.1a"><msup id="S3.SS1.p2.18.m18.1.1" xref="S3.SS1.p2.18.m18.1.1.cmml"><mi id="S3.SS1.p2.18.m18.1.1.2" xref="S3.SS1.p2.18.m18.1.1.2.cmml">h</mi><mo id="S3.SS1.p2.18.m18.1.1.3" xref="S3.SS1.p2.18.m18.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.18.m18.1b"><apply id="S3.SS1.p2.18.m18.1.1.cmml" xref="S3.SS1.p2.18.m18.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.18.m18.1.1.1.cmml" xref="S3.SS1.p2.18.m18.1.1">superscript</csymbol><ci id="S3.SS1.p2.18.m18.1.1.2.cmml" xref="S3.SS1.p2.18.m18.1.1.2">ℎ</ci><ci id="S3.SS1.p2.18.m18.1.1.3.cmml" xref="S3.SS1.p2.18.m18.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.18.m18.1c">h^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.18.m18.1d">italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="h^{\prime}\Last{h}=h" class="ltx_Math" display="inline" id="S3.SS1.p2.19.m19.1"><semantics id="S3.SS1.p2.19.m19.1a"><mrow id="S3.SS1.p2.19.m19.1.1" xref="S3.SS1.p2.19.m19.1.1.cmml"><mrow id="S3.SS1.p2.19.m19.1.1.2" xref="S3.SS1.p2.19.m19.1.1.2.cmml"><msup id="S3.SS1.p2.19.m19.1.1.2.2" xref="S3.SS1.p2.19.m19.1.1.2.2.cmml"><mi id="S3.SS1.p2.19.m19.1.1.2.2.2" xref="S3.SS1.p2.19.m19.1.1.2.2.2.cmml">h</mi><mo id="S3.SS1.p2.19.m19.1.1.2.2.3" xref="S3.SS1.p2.19.m19.1.1.2.2.3.cmml">′</mo></msup><mo id="S3.SS1.p2.19.m19.1.1.2.1" xref="S3.SS1.p2.19.m19.1.1.2.1.cmml">⁢</mo><merror class="ltx_ERROR undefined undefined" id="S3.SS1.p2.19.m19.1.1.2.3" xref="S3.SS1.p2.19.m19.1.1.2.3b.cmml"><mtext id="S3.SS1.p2.19.m19.1.1.2.3a" xref="S3.SS1.p2.19.m19.1.1.2.3b.cmml">\Last</mtext></merror><mo id="S3.SS1.p2.19.m19.1.1.2.1a" xref="S3.SS1.p2.19.m19.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.p2.19.m19.1.1.2.4" xref="S3.SS1.p2.19.m19.1.1.2.4.cmml">h</mi></mrow><mo id="S3.SS1.p2.19.m19.1.1.1" xref="S3.SS1.p2.19.m19.1.1.1.cmml">=</mo><mi id="S3.SS1.p2.19.m19.1.1.3" xref="S3.SS1.p2.19.m19.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.19.m19.1b"><apply id="S3.SS1.p2.19.m19.1.1.cmml" xref="S3.SS1.p2.19.m19.1.1"><eq id="S3.SS1.p2.19.m19.1.1.1.cmml" xref="S3.SS1.p2.19.m19.1.1.1"></eq><apply id="S3.SS1.p2.19.m19.1.1.2.cmml" xref="S3.SS1.p2.19.m19.1.1.2"><times id="S3.SS1.p2.19.m19.1.1.2.1.cmml" xref="S3.SS1.p2.19.m19.1.1.2.1"></times><apply id="S3.SS1.p2.19.m19.1.1.2.2.cmml" xref="S3.SS1.p2.19.m19.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.19.m19.1.1.2.2.1.cmml" xref="S3.SS1.p2.19.m19.1.1.2.2">superscript</csymbol><ci id="S3.SS1.p2.19.m19.1.1.2.2.2.cmml" xref="S3.SS1.p2.19.m19.1.1.2.2.2">ℎ</ci><ci id="S3.SS1.p2.19.m19.1.1.2.2.3.cmml" xref="S3.SS1.p2.19.m19.1.1.2.2.3">′</ci></apply><ci id="S3.SS1.p2.19.m19.1.1.2.3b.cmml" xref="S3.SS1.p2.19.m19.1.1.2.3"><merror class="ltx_ERROR undefined undefined" id="S3.SS1.p2.19.m19.1.1.2.3.cmml" xref="S3.SS1.p2.19.m19.1.1.2.3"><mtext id="S3.SS1.p2.19.m19.1.1.2.3a.cmml" xref="S3.SS1.p2.19.m19.1.1.2.3">\Last</mtext></merror></ci><ci id="S3.SS1.p2.19.m19.1.1.2.4.cmml" xref="S3.SS1.p2.19.m19.1.1.2.4">ℎ</ci></apply><ci id="S3.SS1.p2.19.m19.1.1.3.cmml" xref="S3.SS1.p2.19.m19.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.19.m19.1c">h^{\prime}\Last{h}=h</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.19.m19.1d">italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_h = italic_h</annotation></semantics></math>.</p> </div> <figure 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stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 474.05 -445.64)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 253.22)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 257.96)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="9.46" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.88"><math alttext="G" class="ltx_Math" display="inline" id="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.F2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_G</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 417.94 -227.64" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 419.26 -546.76)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 310.66)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 312.97)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="14.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="17.5"><math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><mi id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">G</mi><mrow id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3"><mo fence="false" id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g></g></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>A game <math alttext="G" class="ltx_Math" display="inline" id="S3.F2.5.m1.1"><semantics id="S3.F2.5.m1.1b"><mi id="S3.F2.5.m1.1.1" xref="S3.F2.5.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.F2.5.m1.1c"><ci id="S3.F2.5.m1.1.1.cmml" xref="S3.F2.5.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.5.m1.1d">G</annotation><annotation encoding="application/x-llamapun" id="S3.F2.5.m1.1e">italic_G</annotation></semantics></math> seen as the set of its plays (containing <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F2.6.m2.1"><semantics id="S3.F2.6.m2.1b"><mi id="S3.F2.6.m2.1.1" xref="S3.F2.6.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F2.6.m2.1c"><ci id="S3.F2.6.m2.1.1.cmml" xref="S3.F2.6.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.6.m2.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F2.6.m2.1e">italic_ρ</annotation></semantics></math> and <math alttext="\rho^{\prime}" class="ltx_Math" display="inline" id="S3.F2.7.m3.1"><semantics id="S3.F2.7.m3.1b"><msup id="S3.F2.7.m3.1.1" xref="S3.F2.7.m3.1.1.cmml"><mi id="S3.F2.7.m3.1.1.2" xref="S3.F2.7.m3.1.1.2.cmml">ρ</mi><mo id="S3.F2.7.m3.1.1.3" xref="S3.F2.7.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.F2.7.m3.1c"><apply id="S3.F2.7.m3.1.1.cmml" xref="S3.F2.7.m3.1.1"><csymbol cd="ambiguous" id="S3.F2.7.m3.1.1.1.cmml" xref="S3.F2.7.m3.1.1">superscript</csymbol><ci id="S3.F2.7.m3.1.1.2.cmml" xref="S3.F2.7.m3.1.1.2">𝜌</ci><ci id="S3.F2.7.m3.1.1.3.cmml" xref="S3.F2.7.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.7.m3.1d">\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.7.m3.1e">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>), and its subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.F2.8.m4.1"><semantics id="S3.F2.8.m4.1b"><msub id="S3.F2.8.m4.1.1"><mi id="S3.F2.8.m4.1.1.2">G</mi><mrow id="S3.F2.8.m4.1.1.3"><mo fence="false" id="S3.F2.8.m4.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.F2.8.m4.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.F2.8.m4.1c">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.8.m4.1d">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math></figcaption> </figure> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.22">A strategy for Player <math alttext="0" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mn id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><cn id="S3.SS1.p3.1.m1.1.1.cmml" type="integer" xref="S3.SS1.p3.1.m1.1.1">0</cn></annotation-xml></semantics></math> on <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><msub id="S3.SS1.p3.2.m2.1.1"><mi id="S3.SS1.p3.2.m2.1.1.2">G</mi><mrow id="S3.SS1.p3.2.m2.1.1.3"><mo fence="false" id="S3.SS1.p3.2.m2.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p3.2.m2.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> is a strategy <math alttext="\tau_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><msub id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><mi id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml">τ</mi><mn id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2">𝜏</ci><cn id="S3.SS1.p3.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">\tau_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> that is only defined for the histories <math alttext="h^{\prime}\in\textsf{Hist}" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mrow id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml"><msup id="S3.SS1.p3.4.m4.1.1.2" xref="S3.SS1.p3.4.m4.1.1.2.cmml"><mi id="S3.SS1.p3.4.m4.1.1.2.2" xref="S3.SS1.p3.4.m4.1.1.2.2.cmml">h</mi><mo id="S3.SS1.p3.4.m4.1.1.2.3" xref="S3.SS1.p3.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.p3.4.m4.1.1.1" xref="S3.SS1.p3.4.m4.1.1.1.cmml">∈</mo><mtext class="ltx_mathvariant_sans-serif" id="S3.SS1.p3.4.m4.1.1.3" xref="S3.SS1.p3.4.m4.1.1.3a.cmml">Hist</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><apply id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1"><in id="S3.SS1.p3.4.m4.1.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1.1"></in><apply id="S3.SS1.p3.4.m4.1.1.2.cmml" xref="S3.SS1.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p3.4.m4.1.1.2.1.cmml" xref="S3.SS1.p3.4.m4.1.1.2">superscript</csymbol><ci id="S3.SS1.p3.4.m4.1.1.2.2.cmml" xref="S3.SS1.p3.4.m4.1.1.2.2">ℎ</ci><ci id="S3.SS1.p3.4.m4.1.1.2.3.cmml" xref="S3.SS1.p3.4.m4.1.1.2.3">′</ci></apply><ci id="S3.SS1.p3.4.m4.1.1.3a.cmml" xref="S3.SS1.p3.4.m4.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.SS1.p3.4.m4.1.1.3.cmml" xref="S3.SS1.p3.4.m4.1.1.3">Hist</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">h^{\prime}\in\textsf{Hist}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Hist</annotation></semantics></math> such that <math alttext="hh^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.p3.5.m5.1"><semantics id="S3.SS1.p3.5.m5.1a"><mrow id="S3.SS1.p3.5.m5.1.1" xref="S3.SS1.p3.5.m5.1.1.cmml"><mi id="S3.SS1.p3.5.m5.1.1.2" xref="S3.SS1.p3.5.m5.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.5.m5.1.1.1" xref="S3.SS1.p3.5.m5.1.1.1.cmml">⁢</mo><msup id="S3.SS1.p3.5.m5.1.1.3" xref="S3.SS1.p3.5.m5.1.1.3.cmml"><mi id="S3.SS1.p3.5.m5.1.1.3.2" xref="S3.SS1.p3.5.m5.1.1.3.2.cmml">h</mi><mo id="S3.SS1.p3.5.m5.1.1.3.3" xref="S3.SS1.p3.5.m5.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.5.m5.1b"><apply id="S3.SS1.p3.5.m5.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1"><times id="S3.SS1.p3.5.m5.1.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1.1"></times><ci id="S3.SS1.p3.5.m5.1.1.2.cmml" xref="S3.SS1.p3.5.m5.1.1.2">ℎ</ci><apply id="S3.SS1.p3.5.m5.1.1.3.cmml" xref="S3.SS1.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.5.m5.1.1.3.1.cmml" xref="S3.SS1.p3.5.m5.1.1.3">superscript</csymbol><ci id="S3.SS1.p3.5.m5.1.1.3.2.cmml" xref="S3.SS1.p3.5.m5.1.1.3.2">ℎ</ci><ci id="S3.SS1.p3.5.m5.1.1.3.3.cmml" xref="S3.SS1.p3.5.m5.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.5.m5.1c">hh^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.5.m5.1d">italic_h italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We denote <math alttext="\Sigma_{0|h}" class="ltx_Math" display="inline" id="S3.SS1.p3.6.m6.1"><semantics id="S3.SS1.p3.6.m6.1a"><msub id="S3.SS1.p3.6.m6.1.1" xref="S3.SS1.p3.6.m6.1.1.cmml"><mi id="S3.SS1.p3.6.m6.1.1.2" mathvariant="normal" xref="S3.SS1.p3.6.m6.1.1.2.cmml">Σ</mi><mrow id="S3.SS1.p3.6.m6.1.1.3" xref="S3.SS1.p3.6.m6.1.1.3.cmml"><mn id="S3.SS1.p3.6.m6.1.1.3.2" xref="S3.SS1.p3.6.m6.1.1.3.2.cmml">0</mn><mo fence="false" id="S3.SS1.p3.6.m6.1.1.3.1" xref="S3.SS1.p3.6.m6.1.1.3.1.cmml">|</mo><mi id="S3.SS1.p3.6.m6.1.1.3.3" xref="S3.SS1.p3.6.m6.1.1.3.3.cmml">h</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.6.m6.1b"><apply id="S3.SS1.p3.6.m6.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.6.m6.1.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.p3.6.m6.1.1.2.cmml" xref="S3.SS1.p3.6.m6.1.1.2">Σ</ci><apply id="S3.SS1.p3.6.m6.1.1.3.cmml" xref="S3.SS1.p3.6.m6.1.1.3"><csymbol cd="latexml" id="S3.SS1.p3.6.m6.1.1.3.1.cmml" xref="S3.SS1.p3.6.m6.1.1.3.1">conditional</csymbol><cn id="S3.SS1.p3.6.m6.1.1.3.2.cmml" type="integer" xref="S3.SS1.p3.6.m6.1.1.3.2">0</cn><ci id="S3.SS1.p3.6.m6.1.1.3.3.cmml" xref="S3.SS1.p3.6.m6.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.6.m6.1c">\Sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.6.m6.1d">roman_Σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> the set of those strategies. Given a strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.7.m7.1"><semantics id="S3.SS1.p3.7.m7.1a"><msub id="S3.SS1.p3.7.m7.1.1" xref="S3.SS1.p3.7.m7.1.1.cmml"><mi id="S3.SS1.p3.7.m7.1.1.2" xref="S3.SS1.p3.7.m7.1.1.2.cmml">σ</mi><mn id="S3.SS1.p3.7.m7.1.1.3" xref="S3.SS1.p3.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.7.m7.1b"><apply id="S3.SS1.p3.7.m7.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.7.m7.1.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1">subscript</csymbol><ci id="S3.SS1.p3.7.m7.1.1.2.cmml" xref="S3.SS1.p3.7.m7.1.1.2">𝜎</ci><cn id="S3.SS1.p3.7.m7.1.1.3.cmml" type="integer" xref="S3.SS1.p3.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.7.m7.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.7.m7.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> for Player 0 in <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p3.8.m8.1"><semantics id="S3.SS1.p3.8.m8.1a"><mi id="S3.SS1.p3.8.m8.1.1" xref="S3.SS1.p3.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.8.m8.1b"><ci id="S3.SS1.p3.8.m8.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.8.m8.1d">italic_G</annotation></semantics></math> and <math alttext="h\in\textsf{Hist}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.SS1.p3.9.m9.1"><semantics id="S3.SS1.p3.9.m9.1a"><mrow id="S3.SS1.p3.9.m9.1.1" xref="S3.SS1.p3.9.m9.1.1.cmml"><mi id="S3.SS1.p3.9.m9.1.1.2" xref="S3.SS1.p3.9.m9.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.9.m9.1.1.1" xref="S3.SS1.p3.9.m9.1.1.1.cmml">∈</mo><msub id="S3.SS1.p3.9.m9.1.1.3" xref="S3.SS1.p3.9.m9.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.SS1.p3.9.m9.1.1.3.2" xref="S3.SS1.p3.9.m9.1.1.3.2a.cmml">Hist</mtext><msub id="S3.SS1.p3.9.m9.1.1.3.3" xref="S3.SS1.p3.9.m9.1.1.3.3.cmml"><mi id="S3.SS1.p3.9.m9.1.1.3.3.2" xref="S3.SS1.p3.9.m9.1.1.3.3.2.cmml">σ</mi><mn id="S3.SS1.p3.9.m9.1.1.3.3.3" xref="S3.SS1.p3.9.m9.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.9.m9.1b"><apply id="S3.SS1.p3.9.m9.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1"><in id="S3.SS1.p3.9.m9.1.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1.1"></in><ci id="S3.SS1.p3.9.m9.1.1.2.cmml" xref="S3.SS1.p3.9.m9.1.1.2">ℎ</ci><apply id="S3.SS1.p3.9.m9.1.1.3.cmml" xref="S3.SS1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.9.m9.1.1.3.1.cmml" xref="S3.SS1.p3.9.m9.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.9.m9.1.1.3.2a.cmml" xref="S3.SS1.p3.9.m9.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.SS1.p3.9.m9.1.1.3.2.cmml" xref="S3.SS1.p3.9.m9.1.1.3.2">Hist</mtext></ci><apply id="S3.SS1.p3.9.m9.1.1.3.3.cmml" xref="S3.SS1.p3.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p3.9.m9.1.1.3.3.1.cmml" xref="S3.SS1.p3.9.m9.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p3.9.m9.1.1.3.3.2.cmml" xref="S3.SS1.p3.9.m9.1.1.3.3.2">𝜎</ci><cn id="S3.SS1.p3.9.m9.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p3.9.m9.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.9.m9.1c">h\in\textsf{Hist}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.9.m9.1d">italic_h ∈ Hist start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, we denote the restriction of <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.10.m10.1"><semantics id="S3.SS1.p3.10.m10.1a"><msub id="S3.SS1.p3.10.m10.1.1" xref="S3.SS1.p3.10.m10.1.1.cmml"><mi id="S3.SS1.p3.10.m10.1.1.2" xref="S3.SS1.p3.10.m10.1.1.2.cmml">σ</mi><mn id="S3.SS1.p3.10.m10.1.1.3" xref="S3.SS1.p3.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.10.m10.1b"><apply id="S3.SS1.p3.10.m10.1.1.cmml" xref="S3.SS1.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.10.m10.1.1.1.cmml" xref="S3.SS1.p3.10.m10.1.1">subscript</csymbol><ci id="S3.SS1.p3.10.m10.1.1.2.cmml" xref="S3.SS1.p3.10.m10.1.1.2">𝜎</ci><cn id="S3.SS1.p3.10.m10.1.1.3.cmml" type="integer" xref="S3.SS1.p3.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.10.m10.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.10.m10.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p3.11.m11.1"><semantics id="S3.SS1.p3.11.m11.1a"><msub id="S3.SS1.p3.11.m11.1.1"><mi id="S3.SS1.p3.11.m11.1.1.2">G</mi><mrow id="S3.SS1.p3.11.m11.1.1.3"><mo fence="false" id="S3.SS1.p3.11.m11.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p3.11.m11.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p3.11.m11.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.11.m11.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> by the strategy <math alttext="\sigma_{0}{}_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p3.12.m12.1"><semantics id="S3.SS1.p3.12.m12.1a"><mrow id="S3.SS1.p3.12.m12.1b"><msub id="S3.SS1.p3.12.m12.1.1"><mi id="S3.SS1.p3.12.m12.1.1.2">σ</mi><mn id="S3.SS1.p3.12.m12.1.1.3">0</mn></msub><msub id="S3.SS1.p3.12.m12.1.2"><mi id="S3.SS1.p3.12.m12.1.2a"></mi><mrow id="S3.SS1.p3.12.m12.1.2.1"><mo fence="false" id="S3.SS1.p3.12.m12.1.2.1.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p3.12.m12.1.2.1.2">h</mi></mrow></msub></mrow><annotation encoding="application/x-tex" id="S3.SS1.p3.12.m12.1c">\sigma_{0}{}_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.12.m12.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_FLOATSUBSCRIPT | italic_h end_FLOATSUBSCRIPT</annotation></semantics></math>. Moreover, given <math alttext="\tau_{0}\in\Sigma_{0|h}" class="ltx_Math" display="inline" id="S3.SS1.p3.13.m13.1"><semantics id="S3.SS1.p3.13.m13.1a"><mrow id="S3.SS1.p3.13.m13.1.1" xref="S3.SS1.p3.13.m13.1.1.cmml"><msub id="S3.SS1.p3.13.m13.1.1.2" xref="S3.SS1.p3.13.m13.1.1.2.cmml"><mi id="S3.SS1.p3.13.m13.1.1.2.2" xref="S3.SS1.p3.13.m13.1.1.2.2.cmml">τ</mi><mn id="S3.SS1.p3.13.m13.1.1.2.3" xref="S3.SS1.p3.13.m13.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.13.m13.1.1.1" xref="S3.SS1.p3.13.m13.1.1.1.cmml">∈</mo><msub id="S3.SS1.p3.13.m13.1.1.3" xref="S3.SS1.p3.13.m13.1.1.3.cmml"><mi id="S3.SS1.p3.13.m13.1.1.3.2" mathvariant="normal" xref="S3.SS1.p3.13.m13.1.1.3.2.cmml">Σ</mi><mrow id="S3.SS1.p3.13.m13.1.1.3.3" xref="S3.SS1.p3.13.m13.1.1.3.3.cmml"><mn id="S3.SS1.p3.13.m13.1.1.3.3.2" xref="S3.SS1.p3.13.m13.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S3.SS1.p3.13.m13.1.1.3.3.1" xref="S3.SS1.p3.13.m13.1.1.3.3.1.cmml">|</mo><mi id="S3.SS1.p3.13.m13.1.1.3.3.3" xref="S3.SS1.p3.13.m13.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.13.m13.1b"><apply id="S3.SS1.p3.13.m13.1.1.cmml" xref="S3.SS1.p3.13.m13.1.1"><in id="S3.SS1.p3.13.m13.1.1.1.cmml" xref="S3.SS1.p3.13.m13.1.1.1"></in><apply id="S3.SS1.p3.13.m13.1.1.2.cmml" xref="S3.SS1.p3.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p3.13.m13.1.1.2.1.cmml" xref="S3.SS1.p3.13.m13.1.1.2">subscript</csymbol><ci id="S3.SS1.p3.13.m13.1.1.2.2.cmml" xref="S3.SS1.p3.13.m13.1.1.2.2">𝜏</ci><cn id="S3.SS1.p3.13.m13.1.1.2.3.cmml" type="integer" xref="S3.SS1.p3.13.m13.1.1.2.3">0</cn></apply><apply id="S3.SS1.p3.13.m13.1.1.3.cmml" xref="S3.SS1.p3.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.13.m13.1.1.3.1.cmml" xref="S3.SS1.p3.13.m13.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.13.m13.1.1.3.2.cmml" xref="S3.SS1.p3.13.m13.1.1.3.2">Σ</ci><apply id="S3.SS1.p3.13.m13.1.1.3.3.cmml" xref="S3.SS1.p3.13.m13.1.1.3.3"><csymbol cd="latexml" id="S3.SS1.p3.13.m13.1.1.3.3.1.cmml" xref="S3.SS1.p3.13.m13.1.1.3.3.1">conditional</csymbol><cn id="S3.SS1.p3.13.m13.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p3.13.m13.1.1.3.3.2">0</cn><ci id="S3.SS1.p3.13.m13.1.1.3.3.3.cmml" xref="S3.SS1.p3.13.m13.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.13.m13.1c">\tau_{0}\in\Sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.13.m13.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math>, we can define a new strategy <math alttext="\sigma_{0}[h\rightarrow\tau_{0}]" class="ltx_Math" display="inline" id="S3.SS1.p3.14.m14.1"><semantics id="S3.SS1.p3.14.m14.1a"><mrow id="S3.SS1.p3.14.m14.1.1" xref="S3.SS1.p3.14.m14.1.1.cmml"><msub id="S3.SS1.p3.14.m14.1.1.3" xref="S3.SS1.p3.14.m14.1.1.3.cmml"><mi id="S3.SS1.p3.14.m14.1.1.3.2" xref="S3.SS1.p3.14.m14.1.1.3.2.cmml">σ</mi><mn id="S3.SS1.p3.14.m14.1.1.3.3" xref="S3.SS1.p3.14.m14.1.1.3.3.cmml">0</mn></msub><mo id="S3.SS1.p3.14.m14.1.1.2" xref="S3.SS1.p3.14.m14.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.p3.14.m14.1.1.1.1" xref="S3.SS1.p3.14.m14.1.1.1.2.cmml"><mo id="S3.SS1.p3.14.m14.1.1.1.1.2" stretchy="false" xref="S3.SS1.p3.14.m14.1.1.1.2.1.cmml">[</mo><mrow id="S3.SS1.p3.14.m14.1.1.1.1.1" xref="S3.SS1.p3.14.m14.1.1.1.1.1.cmml"><mi id="S3.SS1.p3.14.m14.1.1.1.1.1.2" xref="S3.SS1.p3.14.m14.1.1.1.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.14.m14.1.1.1.1.1.1" stretchy="false" xref="S3.SS1.p3.14.m14.1.1.1.1.1.1.cmml">→</mo><msub id="S3.SS1.p3.14.m14.1.1.1.1.1.3" xref="S3.SS1.p3.14.m14.1.1.1.1.1.3.cmml"><mi id="S3.SS1.p3.14.m14.1.1.1.1.1.3.2" xref="S3.SS1.p3.14.m14.1.1.1.1.1.3.2.cmml">τ</mi><mn id="S3.SS1.p3.14.m14.1.1.1.1.1.3.3" xref="S3.SS1.p3.14.m14.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S3.SS1.p3.14.m14.1.1.1.1.3" stretchy="false" xref="S3.SS1.p3.14.m14.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.14.m14.1b"><apply id="S3.SS1.p3.14.m14.1.1.cmml" xref="S3.SS1.p3.14.m14.1.1"><times id="S3.SS1.p3.14.m14.1.1.2.cmml" xref="S3.SS1.p3.14.m14.1.1.2"></times><apply id="S3.SS1.p3.14.m14.1.1.3.cmml" xref="S3.SS1.p3.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.14.m14.1.1.3.1.cmml" xref="S3.SS1.p3.14.m14.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.14.m14.1.1.3.2.cmml" xref="S3.SS1.p3.14.m14.1.1.3.2">𝜎</ci><cn id="S3.SS1.p3.14.m14.1.1.3.3.cmml" type="integer" xref="S3.SS1.p3.14.m14.1.1.3.3">0</cn></apply><apply id="S3.SS1.p3.14.m14.1.1.1.2.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1"><csymbol cd="latexml" id="S3.SS1.p3.14.m14.1.1.1.2.1.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.SS1.p3.14.m14.1.1.1.1.1.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1.1"><ci id="S3.SS1.p3.14.m14.1.1.1.1.1.1.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1.1.1">→</ci><ci id="S3.SS1.p3.14.m14.1.1.1.1.1.2.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1.1.2">ℎ</ci><apply id="S3.SS1.p3.14.m14.1.1.1.1.1.3.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.14.m14.1.1.1.1.1.3.1.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.14.m14.1.1.1.1.1.3.2.cmml" xref="S3.SS1.p3.14.m14.1.1.1.1.1.3.2">𝜏</ci><cn id="S3.SS1.p3.14.m14.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.p3.14.m14.1.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.14.m14.1c">\sigma_{0}[h\rightarrow\tau_{0}]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.14.m14.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_h → italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math> from <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.15.m15.1"><semantics id="S3.SS1.p3.15.m15.1a"><msub id="S3.SS1.p3.15.m15.1.1" xref="S3.SS1.p3.15.m15.1.1.cmml"><mi id="S3.SS1.p3.15.m15.1.1.2" xref="S3.SS1.p3.15.m15.1.1.2.cmml">σ</mi><mn id="S3.SS1.p3.15.m15.1.1.3" xref="S3.SS1.p3.15.m15.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.15.m15.1b"><apply id="S3.SS1.p3.15.m15.1.1.cmml" xref="S3.SS1.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.15.m15.1.1.1.cmml" xref="S3.SS1.p3.15.m15.1.1">subscript</csymbol><ci id="S3.SS1.p3.15.m15.1.1.2.cmml" xref="S3.SS1.p3.15.m15.1.1.2">𝜎</ci><cn id="S3.SS1.p3.15.m15.1.1.3.cmml" type="integer" xref="S3.SS1.p3.15.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.15.m15.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.15.m15.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> as the strategy on <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.p3.16.m16.1"><semantics id="S3.SS1.p3.16.m16.1a"><mi id="S3.SS1.p3.16.m16.1.1" xref="S3.SS1.p3.16.m16.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.16.m16.1b"><ci id="S3.SS1.p3.16.m16.1.1.cmml" xref="S3.SS1.p3.16.m16.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.16.m16.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.16.m16.1d">italic_G</annotation></semantics></math> which consists in playing the strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.17.m17.1"><semantics id="S3.SS1.p3.17.m17.1a"><msub id="S3.SS1.p3.17.m17.1.1" xref="S3.SS1.p3.17.m17.1.1.cmml"><mi id="S3.SS1.p3.17.m17.1.1.2" xref="S3.SS1.p3.17.m17.1.1.2.cmml">σ</mi><mn id="S3.SS1.p3.17.m17.1.1.3" xref="S3.SS1.p3.17.m17.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.17.m17.1b"><apply id="S3.SS1.p3.17.m17.1.1.cmml" xref="S3.SS1.p3.17.m17.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.17.m17.1.1.1.cmml" xref="S3.SS1.p3.17.m17.1.1">subscript</csymbol><ci id="S3.SS1.p3.17.m17.1.1.2.cmml" xref="S3.SS1.p3.17.m17.1.1.2">𝜎</ci><cn id="S3.SS1.p3.17.m17.1.1.3.cmml" type="integer" xref="S3.SS1.p3.17.m17.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.17.m17.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.17.m17.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> everywhere, except in the subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p3.18.m18.1"><semantics id="S3.SS1.p3.18.m18.1a"><msub id="S3.SS1.p3.18.m18.1.1"><mi id="S3.SS1.p3.18.m18.1.1.2">G</mi><mrow id="S3.SS1.p3.18.m18.1.1.3"><mo fence="false" id="S3.SS1.p3.18.m18.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p3.18.m18.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p3.18.m18.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.18.m18.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="\tau_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.19.m19.1"><semantics id="S3.SS1.p3.19.m19.1a"><msub id="S3.SS1.p3.19.m19.1.1" xref="S3.SS1.p3.19.m19.1.1.cmml"><mi id="S3.SS1.p3.19.m19.1.1.2" xref="S3.SS1.p3.19.m19.1.1.2.cmml">τ</mi><mn id="S3.SS1.p3.19.m19.1.1.3" xref="S3.SS1.p3.19.m19.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.19.m19.1b"><apply id="S3.SS1.p3.19.m19.1.1.cmml" xref="S3.SS1.p3.19.m19.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.19.m19.1.1.1.cmml" xref="S3.SS1.p3.19.m19.1.1">subscript</csymbol><ci id="S3.SS1.p3.19.m19.1.1.2.cmml" xref="S3.SS1.p3.19.m19.1.1.2">𝜏</ci><cn id="S3.SS1.p3.19.m19.1.1.3.cmml" type="integer" xref="S3.SS1.p3.19.m19.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.19.m19.1c">\tau_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.19.m19.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is played. That is, <math alttext="\sigma_{0}[h\rightarrow\tau_{0}](h^{\prime})=\sigma_{0}(h^{\prime})" class="ltx_Math" display="inline" id="S3.SS1.p3.20.m20.3"><semantics id="S3.SS1.p3.20.m20.3a"><mrow id="S3.SS1.p3.20.m20.3.3" xref="S3.SS1.p3.20.m20.3.3.cmml"><mrow id="S3.SS1.p3.20.m20.2.2.2" xref="S3.SS1.p3.20.m20.2.2.2.cmml"><msub id="S3.SS1.p3.20.m20.2.2.2.4" xref="S3.SS1.p3.20.m20.2.2.2.4.cmml"><mi id="S3.SS1.p3.20.m20.2.2.2.4.2" xref="S3.SS1.p3.20.m20.2.2.2.4.2.cmml">σ</mi><mn id="S3.SS1.p3.20.m20.2.2.2.4.3" xref="S3.SS1.p3.20.m20.2.2.2.4.3.cmml">0</mn></msub><mo id="S3.SS1.p3.20.m20.2.2.2.3" xref="S3.SS1.p3.20.m20.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS1.p3.20.m20.1.1.1.1.1" xref="S3.SS1.p3.20.m20.1.1.1.1.2.cmml"><mo id="S3.SS1.p3.20.m20.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p3.20.m20.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.SS1.p3.20.m20.1.1.1.1.1.1" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p3.20.m20.1.1.1.1.1.1.2" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.20.m20.1.1.1.1.1.1.1" stretchy="false" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.1.cmml">→</mo><msub id="S3.SS1.p3.20.m20.1.1.1.1.1.1.3" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.cmml"><mi id="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.2" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.2.cmml">τ</mi><mn id="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.3" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S3.SS1.p3.20.m20.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p3.20.m20.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.SS1.p3.20.m20.2.2.2.3a" xref="S3.SS1.p3.20.m20.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS1.p3.20.m20.2.2.2.2.1" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.cmml"><mo id="S3.SS1.p3.20.m20.2.2.2.2.1.2" stretchy="false" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.cmml">(</mo><msup id="S3.SS1.p3.20.m20.2.2.2.2.1.1" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.cmml"><mi id="S3.SS1.p3.20.m20.2.2.2.2.1.1.2" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.20.m20.2.2.2.2.1.1.3" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.p3.20.m20.2.2.2.2.1.3" stretchy="false" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p3.20.m20.3.3.4" xref="S3.SS1.p3.20.m20.3.3.4.cmml">=</mo><mrow id="S3.SS1.p3.20.m20.3.3.3" xref="S3.SS1.p3.20.m20.3.3.3.cmml"><msub id="S3.SS1.p3.20.m20.3.3.3.3" xref="S3.SS1.p3.20.m20.3.3.3.3.cmml"><mi id="S3.SS1.p3.20.m20.3.3.3.3.2" xref="S3.SS1.p3.20.m20.3.3.3.3.2.cmml">σ</mi><mn id="S3.SS1.p3.20.m20.3.3.3.3.3" xref="S3.SS1.p3.20.m20.3.3.3.3.3.cmml">0</mn></msub><mo id="S3.SS1.p3.20.m20.3.3.3.2" xref="S3.SS1.p3.20.m20.3.3.3.2.cmml">⁢</mo><mrow id="S3.SS1.p3.20.m20.3.3.3.1.1" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.cmml"><mo id="S3.SS1.p3.20.m20.3.3.3.1.1.2" stretchy="false" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.cmml">(</mo><msup id="S3.SS1.p3.20.m20.3.3.3.1.1.1" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.cmml"><mi id="S3.SS1.p3.20.m20.3.3.3.1.1.1.2" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.20.m20.3.3.3.1.1.1.3" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.p3.20.m20.3.3.3.1.1.3" stretchy="false" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.20.m20.3b"><apply id="S3.SS1.p3.20.m20.3.3.cmml" xref="S3.SS1.p3.20.m20.3.3"><eq id="S3.SS1.p3.20.m20.3.3.4.cmml" xref="S3.SS1.p3.20.m20.3.3.4"></eq><apply id="S3.SS1.p3.20.m20.2.2.2.cmml" xref="S3.SS1.p3.20.m20.2.2.2"><times id="S3.SS1.p3.20.m20.2.2.2.3.cmml" xref="S3.SS1.p3.20.m20.2.2.2.3"></times><apply id="S3.SS1.p3.20.m20.2.2.2.4.cmml" xref="S3.SS1.p3.20.m20.2.2.2.4"><csymbol cd="ambiguous" id="S3.SS1.p3.20.m20.2.2.2.4.1.cmml" xref="S3.SS1.p3.20.m20.2.2.2.4">subscript</csymbol><ci id="S3.SS1.p3.20.m20.2.2.2.4.2.cmml" xref="S3.SS1.p3.20.m20.2.2.2.4.2">𝜎</ci><cn id="S3.SS1.p3.20.m20.2.2.2.4.3.cmml" type="integer" xref="S3.SS1.p3.20.m20.2.2.2.4.3">0</cn></apply><apply id="S3.SS1.p3.20.m20.1.1.1.1.2.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1"><csymbol cd="latexml" id="S3.SS1.p3.20.m20.1.1.1.1.2.1.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.SS1.p3.20.m20.1.1.1.1.1.1.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1"><ci id="S3.SS1.p3.20.m20.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.1">→</ci><ci id="S3.SS1.p3.20.m20.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.2">ℎ</ci><apply id="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.1.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.2.cmml" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.2">𝜏</ci><cn id="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.p3.20.m20.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S3.SS1.p3.20.m20.2.2.2.2.1.1.cmml" xref="S3.SS1.p3.20.m20.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.p3.20.m20.2.2.2.2.1.1.1.cmml" xref="S3.SS1.p3.20.m20.2.2.2.2.1">superscript</csymbol><ci id="S3.SS1.p3.20.m20.2.2.2.2.1.1.2.cmml" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.2">ℎ</ci><ci id="S3.SS1.p3.20.m20.2.2.2.2.1.1.3.cmml" xref="S3.SS1.p3.20.m20.2.2.2.2.1.1.3">′</ci></apply></apply><apply id="S3.SS1.p3.20.m20.3.3.3.cmml" xref="S3.SS1.p3.20.m20.3.3.3"><times id="S3.SS1.p3.20.m20.3.3.3.2.cmml" xref="S3.SS1.p3.20.m20.3.3.3.2"></times><apply id="S3.SS1.p3.20.m20.3.3.3.3.cmml" xref="S3.SS1.p3.20.m20.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p3.20.m20.3.3.3.3.1.cmml" xref="S3.SS1.p3.20.m20.3.3.3.3">subscript</csymbol><ci id="S3.SS1.p3.20.m20.3.3.3.3.2.cmml" xref="S3.SS1.p3.20.m20.3.3.3.3.2">𝜎</ci><cn id="S3.SS1.p3.20.m20.3.3.3.3.3.cmml" type="integer" xref="S3.SS1.p3.20.m20.3.3.3.3.3">0</cn></apply><apply id="S3.SS1.p3.20.m20.3.3.3.1.1.1.cmml" xref="S3.SS1.p3.20.m20.3.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.20.m20.3.3.3.1.1.1.1.cmml" xref="S3.SS1.p3.20.m20.3.3.3.1.1">superscript</csymbol><ci id="S3.SS1.p3.20.m20.3.3.3.1.1.1.2.cmml" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.2">ℎ</ci><ci id="S3.SS1.p3.20.m20.3.3.3.1.1.1.3.cmml" xref="S3.SS1.p3.20.m20.3.3.3.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.20.m20.3c">\sigma_{0}[h\rightarrow\tau_{0}](h^{\prime})=\sigma_{0}(h^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.20.m20.3d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_h → italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> if <math alttext="h\not h^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.p3.21.m21.1"><semantics id="S3.SS1.p3.21.m21.1a"><mrow id="S3.SS1.p3.21.m21.1.1" xref="S3.SS1.p3.21.m21.1.1.cmml"><mi id="S3.SS1.p3.21.m21.1.1.2" xref="S3.SS1.p3.21.m21.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.21.m21.1.1.1" xref="S3.SS1.p3.21.m21.1.1.1.cmml">⁢</mo><msup id="S3.SS1.p3.21.m21.1.1.3" xref="S3.SS1.p3.21.m21.1.1.3.cmml"><mi class="ltx_mathvariant_italic" id="S3.SS1.p3.21.m21.1.1.3.2" mathvariant="italic" xref="S3.SS1.p3.21.m21.1.1.3.2.cmml">h̸</mi><mo id="S3.SS1.p3.21.m21.1.1.3.3" xref="S3.SS1.p3.21.m21.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.21.m21.1b"><apply id="S3.SS1.p3.21.m21.1.1.cmml" xref="S3.SS1.p3.21.m21.1.1"><times id="S3.SS1.p3.21.m21.1.1.1.cmml" xref="S3.SS1.p3.21.m21.1.1.1"></times><ci id="S3.SS1.p3.21.m21.1.1.2.cmml" xref="S3.SS1.p3.21.m21.1.1.2">ℎ</ci><apply id="S3.SS1.p3.21.m21.1.1.3.cmml" xref="S3.SS1.p3.21.m21.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.21.m21.1.1.3.1.cmml" xref="S3.SS1.p3.21.m21.1.1.3">superscript</csymbol><ci id="S3.SS1.p3.21.m21.1.1.3.2.cmml" xref="S3.SS1.p3.21.m21.1.1.3.2">italic-h̸</ci><ci id="S3.SS1.p3.21.m21.1.1.3.3.cmml" xref="S3.SS1.p3.21.m21.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.21.m21.1c">h\not h^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.21.m21.1d">italic_h italic_h̸ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\sigma_{0}[h\rightarrow\tau_{0}](h^{\prime})=\tau_{0}(h^{\prime})" class="ltx_Math" display="inline" id="S3.SS1.p3.22.m22.3"><semantics id="S3.SS1.p3.22.m22.3a"><mrow id="S3.SS1.p3.22.m22.3.3" xref="S3.SS1.p3.22.m22.3.3.cmml"><mrow id="S3.SS1.p3.22.m22.2.2.2" xref="S3.SS1.p3.22.m22.2.2.2.cmml"><msub id="S3.SS1.p3.22.m22.2.2.2.4" xref="S3.SS1.p3.22.m22.2.2.2.4.cmml"><mi id="S3.SS1.p3.22.m22.2.2.2.4.2" xref="S3.SS1.p3.22.m22.2.2.2.4.2.cmml">σ</mi><mn id="S3.SS1.p3.22.m22.2.2.2.4.3" xref="S3.SS1.p3.22.m22.2.2.2.4.3.cmml">0</mn></msub><mo id="S3.SS1.p3.22.m22.2.2.2.3" xref="S3.SS1.p3.22.m22.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS1.p3.22.m22.1.1.1.1.1" xref="S3.SS1.p3.22.m22.1.1.1.1.2.cmml"><mo id="S3.SS1.p3.22.m22.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p3.22.m22.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.SS1.p3.22.m22.1.1.1.1.1.1" xref="S3.SS1.p3.22.m22.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p3.22.m22.1.1.1.1.1.1.2" xref="S3.SS1.p3.22.m22.1.1.1.1.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.22.m22.1.1.1.1.1.1.1" stretchy="false" xref="S3.SS1.p3.22.m22.1.1.1.1.1.1.1.cmml">→</mo><msub id="S3.SS1.p3.22.m22.1.1.1.1.1.1.3" xref="S3.SS1.p3.22.m22.1.1.1.1.1.1.3.cmml"><mi id="S3.SS1.p3.22.m22.1.1.1.1.1.1.3.2" xref="S3.SS1.p3.22.m22.1.1.1.1.1.1.3.2.cmml">τ</mi><mn id="S3.SS1.p3.22.m22.1.1.1.1.1.1.3.3" xref="S3.SS1.p3.22.m22.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S3.SS1.p3.22.m22.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p3.22.m22.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.SS1.p3.22.m22.2.2.2.3a" xref="S3.SS1.p3.22.m22.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS1.p3.22.m22.2.2.2.2.1" xref="S3.SS1.p3.22.m22.2.2.2.2.1.1.cmml"><mo id="S3.SS1.p3.22.m22.2.2.2.2.1.2" stretchy="false" xref="S3.SS1.p3.22.m22.2.2.2.2.1.1.cmml">(</mo><msup id="S3.SS1.p3.22.m22.2.2.2.2.1.1" xref="S3.SS1.p3.22.m22.2.2.2.2.1.1.cmml"><mi id="S3.SS1.p3.22.m22.2.2.2.2.1.1.2" xref="S3.SS1.p3.22.m22.2.2.2.2.1.1.2.cmml">h</mi><mo id="S3.SS1.p3.22.m22.2.2.2.2.1.1.3" xref="S3.SS1.p3.22.m22.2.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.p3.22.m22.2.2.2.2.1.3" stretchy="false" xref="S3.SS1.p3.22.m22.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p3.22.m22.3.3.4" xref="S3.SS1.p3.22.m22.3.3.4.cmml">=</mo><mrow id="S3.SS1.p3.22.m22.3.3.3" xref="S3.SS1.p3.22.m22.3.3.3.cmml"><msub id="S3.SS1.p3.22.m22.3.3.3.3" xref="S3.SS1.p3.22.m22.3.3.3.3.cmml"><mi id="S3.SS1.p3.22.m22.3.3.3.3.2" xref="S3.SS1.p3.22.m22.3.3.3.3.2.cmml">τ</mi><mn id="S3.SS1.p3.22.m22.3.3.3.3.3" xref="S3.SS1.p3.22.m22.3.3.3.3.3.cmml">0</mn></msub><mo id="S3.SS1.p3.22.m22.3.3.3.2" xref="S3.SS1.p3.22.m22.3.3.3.2.cmml">⁢</mo><mrow 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xref="S3.SS1.p3.22.m22.3.3.3.3.2">𝜏</ci><cn id="S3.SS1.p3.22.m22.3.3.3.3.3.cmml" type="integer" xref="S3.SS1.p3.22.m22.3.3.3.3.3">0</cn></apply><apply id="S3.SS1.p3.22.m22.3.3.3.1.1.1.cmml" xref="S3.SS1.p3.22.m22.3.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.22.m22.3.3.3.1.1.1.1.cmml" xref="S3.SS1.p3.22.m22.3.3.3.1.1">superscript</csymbol><ci id="S3.SS1.p3.22.m22.3.3.3.1.1.1.2.cmml" xref="S3.SS1.p3.22.m22.3.3.3.1.1.1.2">ℎ</ci><ci id="S3.SS1.p3.22.m22.3.3.3.1.1.1.3.cmml" xref="S3.SS1.p3.22.m22.3.3.3.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.22.m22.3c">\sigma_{0}[h\rightarrow\tau_{0}](h^{\prime})=\tau_{0}(h^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.22.m22.3d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_h → italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> otherwise.</p> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.5">As done with <math alttext="\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.2"><semantics id="S3.SS1.p4.1.m1.2a"><mrow id="S3.SS1.p4.1.m1.2.3" xref="S3.SS1.p4.1.m1.2.3.cmml"><mtext id="S3.SS1.p4.1.m1.2.3.2" xref="S3.SS1.p4.1.m1.2.3.2a.cmml">SPS</mtext><mo id="S3.SS1.p4.1.m1.2.3.1" xref="S3.SS1.p4.1.m1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p4.1.m1.2.3.3.2" xref="S3.SS1.p4.1.m1.2.3.3.1.cmml"><mo id="S3.SS1.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S3.SS1.p4.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml">G</mi><mo id="S3.SS1.p4.1.m1.2.3.3.2.2" xref="S3.SS1.p4.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.SS1.p4.1.m1.2.2" xref="S3.SS1.p4.1.m1.2.2.cmml">B</mi><mo id="S3.SS1.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S3.SS1.p4.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.2b"><apply id="S3.SS1.p4.1.m1.2.3.cmml" xref="S3.SS1.p4.1.m1.2.3"><times id="S3.SS1.p4.1.m1.2.3.1.cmml" xref="S3.SS1.p4.1.m1.2.3.1"></times><ci id="S3.SS1.p4.1.m1.2.3.2a.cmml" xref="S3.SS1.p4.1.m1.2.3.2"><mtext id="S3.SS1.p4.1.m1.2.3.2.cmml" xref="S3.SS1.p4.1.m1.2.3.2">SPS</mtext></ci><interval closure="open" id="S3.SS1.p4.1.m1.2.3.3.1.cmml" xref="S3.SS1.p4.1.m1.2.3.3.2"><ci id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1">𝐺</ci><ci id="S3.SS1.p4.1.m1.2.2.cmml" xref="S3.SS1.p4.1.m1.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.2c">\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.2d">SPS ( italic_G , italic_B )</annotation></semantics></math>, we denote by <math alttext="\mbox{SPS}({G_{|h}},{B})" class="ltx_math_unparsed" display="inline" id="S3.SS1.p4.2.m2.2"><semantics id="S3.SS1.p4.2.m2.2a"><mrow id="S3.SS1.p4.2.m2.2.2"><mtext id="S3.SS1.p4.2.m2.2.2.3">SPS</mtext><mo id="S3.SS1.p4.2.m2.2.2.2">⁢</mo><mrow id="S3.SS1.p4.2.m2.2.2.1.1"><mo id="S3.SS1.p4.2.m2.2.2.1.1.2" stretchy="false">(</mo><msub id="S3.SS1.p4.2.m2.2.2.1.1.1"><mi id="S3.SS1.p4.2.m2.2.2.1.1.1.2">G</mi><mrow id="S3.SS1.p4.2.m2.2.2.1.1.1.3"><mo fence="false" id="S3.SS1.p4.2.m2.2.2.1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p4.2.m2.2.2.1.1.1.3.2">h</mi></mrow></msub><mo id="S3.SS1.p4.2.m2.2.2.1.1.3">,</mo><mi id="S3.SS1.p4.2.m2.1.1">B</mi><mo id="S3.SS1.p4.2.m2.2.2.1.1.4" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.2b">\mbox{SPS}({G_{|h}},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.2c">SPS ( italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT , italic_B )</annotation></semantics></math> the set of all solutions <math alttext="\tau_{0}\in\Sigma_{0|h}" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.1"><semantics id="S3.SS1.p4.3.m3.1a"><mrow id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml"><msub id="S3.SS1.p4.3.m3.1.1.2" xref="S3.SS1.p4.3.m3.1.1.2.cmml"><mi id="S3.SS1.p4.3.m3.1.1.2.2" xref="S3.SS1.p4.3.m3.1.1.2.2.cmml">τ</mi><mn id="S3.SS1.p4.3.m3.1.1.2.3" xref="S3.SS1.p4.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS1.p4.3.m3.1.1.1" xref="S3.SS1.p4.3.m3.1.1.1.cmml">∈</mo><msub id="S3.SS1.p4.3.m3.1.1.3" xref="S3.SS1.p4.3.m3.1.1.3.cmml"><mi id="S3.SS1.p4.3.m3.1.1.3.2" mathvariant="normal" xref="S3.SS1.p4.3.m3.1.1.3.2.cmml">Σ</mi><mrow id="S3.SS1.p4.3.m3.1.1.3.3" xref="S3.SS1.p4.3.m3.1.1.3.3.cmml"><mn id="S3.SS1.p4.3.m3.1.1.3.3.2" xref="S3.SS1.p4.3.m3.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S3.SS1.p4.3.m3.1.1.3.3.1" xref="S3.SS1.p4.3.m3.1.1.3.3.1.cmml">|</mo><mi id="S3.SS1.p4.3.m3.1.1.3.3.3" xref="S3.SS1.p4.3.m3.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.1b"><apply id="S3.SS1.p4.3.m3.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1"><in id="S3.SS1.p4.3.m3.1.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1.1"></in><apply id="S3.SS1.p4.3.m3.1.1.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p4.3.m3.1.1.2.1.cmml" xref="S3.SS1.p4.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS1.p4.3.m3.1.1.2.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2.2">𝜏</ci><cn id="S3.SS1.p4.3.m3.1.1.2.3.cmml" type="integer" xref="S3.SS1.p4.3.m3.1.1.2.3">0</cn></apply><apply id="S3.SS1.p4.3.m3.1.1.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p4.3.m3.1.1.3.1.cmml" xref="S3.SS1.p4.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS1.p4.3.m3.1.1.3.2.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2">Σ</ci><apply id="S3.SS1.p4.3.m3.1.1.3.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3.3"><csymbol cd="latexml" id="S3.SS1.p4.3.m3.1.1.3.3.1.cmml" xref="S3.SS1.p4.3.m3.1.1.3.3.1">conditional</csymbol><cn id="S3.SS1.p4.3.m3.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p4.3.m3.1.1.3.3.2">0</cn><ci id="S3.SS1.p4.3.m3.1.1.3.3.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.1c">\tau_{0}\in\Sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> to the SPS problem for the subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p4.4.m4.1"><semantics id="S3.SS1.p4.4.m4.1a"><msub id="S3.SS1.p4.4.m4.1.1"><mi id="S3.SS1.p4.4.m4.1.1.2">G</mi><mrow id="S3.SS1.p4.4.m4.1.1.3"><mo fence="false" id="S3.SS1.p4.4.m4.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.SS1.p4.4.m4.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.SS1.p4.4.m4.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.4.m4.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> and the bound <math alttext="B" class="ltx_Math" display="inline" id="S3.SS1.p4.5.m5.1"><semantics id="S3.SS1.p4.5.m5.1a"><mi id="S3.SS1.p4.5.m5.1.1" xref="S3.SS1.p4.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.5.m5.1b"><ci id="S3.SS1.p4.5.m5.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.5.m5.1d">italic_B</annotation></semantics></math>.</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>Improving a Solution in a Subgame</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">A natural way to improve a strategy is to improve it on a subgame. Moreover, if it is a solution to the SPS problem, it is also the case for the improved strategy.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Lemma 3.1</span></span></h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.11"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.11.11">Let <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.1.m1.1a"><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.1b"><ci id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be an SP game, <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.2.m2.1"><semantics id="S3.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem1.p1.2.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">B</mi><mo id="S3.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.2.2.m2.1b"><apply id="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1"><in id="S3.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2">𝐵</ci><ci id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.3.m3.2"><semantics id="S3.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem1.p1.3.3.m3.2.3" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.cmml"><msub id="S3.Thmtheorem1.p1.3.3.m3.2.3.2" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.2.cmml"><mi id="S3.Thmtheorem1.p1.3.3.m3.2.3.2.2" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem1.p1.3.3.m3.2.3.2.3" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem1.p1.3.3.m3.2.3.1" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem1.p1.3.3.m3.2.3.3" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.2" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.2a.cmml">SPS</mtext><mo id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.1" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.2" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml"><mo id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml">G</mi><mo id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.2.2" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.3.3.m3.2.2" xref="S3.Thmtheorem1.p1.3.3.m3.2.2.cmml">B</mi><mo id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.3.m3.2b"><apply id="S3.Thmtheorem1.p1.3.3.m3.2.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3"><in id="S3.Thmtheorem1.p1.3.3.m3.2.3.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.1"></in><apply id="S3.Thmtheorem1.p1.3.3.m3.2.3.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.3.3.m3.2.3.2.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.3.3.m3.2.3.2.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.2.2">𝜎</ci><cn id="S3.Thmtheorem1.p1.3.3.m3.2.3.2.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.2.3">0</cn></apply><apply id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3"><times id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.1"></times><ci id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.2a.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.3.3.3.2"><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1">𝐺</ci><ci id="S3.Thmtheorem1.p1.3.3.m3.2.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.3.m3.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.3.m3.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. Consider a history <math alttext="h\in\textsf{Hist}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.4.4.m4.1"><semantics id="S3.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem1.p1.4.4.m4.1.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.2.cmml">h</mi><mo id="S3.Thmtheorem1.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem1.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.2a.cmml">Hist</mtext><msub id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.3" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.4.4.m4.1b"><apply id="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1"><in id="S3.Thmtheorem1.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.2">ℎ</ci><apply id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.2a.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.2">Hist</mtext></ci><apply id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.4.m4.1c">h\in\textsf{Hist}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.4.m4.1d">italic_h ∈ Hist start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and a strategy <math alttext="\tau_{0}\in\Sigma_{0|h}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.5.5.m5.1"><semantics id="S3.Thmtheorem1.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem1.p1.5.5.m5.1.1" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.cmml"><msub id="S3.Thmtheorem1.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2.cmml"><mi id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.2" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2.2.cmml">τ</mi><mn id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.3" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem1.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem1.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.2" mathvariant="normal" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.2.cmml">Σ</mi><mrow id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.cmml"><mn id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.1" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.1.cmml">|</mo><mi id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.5.5.m5.1b"><apply id="S3.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1"><in id="S3.Thmtheorem1.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.1"></in><apply id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2.2">𝜏</ci><cn id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2.3">0</cn></apply><apply id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.2">Σ</ci><apply id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.1">conditional</csymbol><cn id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.2">0</cn><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.5.5.m5.1c">\tau_{0}\in\Sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.5.5.m5.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> in the subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem1.p1.6.6.m6.1"><semantics id="S3.Thmtheorem1.p1.6.6.m6.1a"><msub id="S3.Thmtheorem1.p1.6.6.m6.1.1"><mi id="S3.Thmtheorem1.p1.6.6.m6.1.1.2">G</mi><mrow id="S3.Thmtheorem1.p1.6.6.m6.1.1.3"><mo fence="false" id="S3.Thmtheorem1.p1.6.6.m6.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem1.p1.6.6.m6.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.6.6.m6.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.6.6.m6.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\tau_{0}\prec\sigma_{0|h}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.7.7.m7.1"><semantics id="S3.Thmtheorem1.p1.7.7.m7.1a"><mrow id="S3.Thmtheorem1.p1.7.7.m7.1.1" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.cmml"><msub id="S3.Thmtheorem1.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2.cmml"><mi id="S3.Thmtheorem1.p1.7.7.m7.1.1.2.2" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2.2.cmml">τ</mi><mn id="S3.Thmtheorem1.p1.7.7.m7.1.1.2.3" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem1.p1.7.7.m7.1.1.1" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.1.cmml">≺</mo><msub id="S3.Thmtheorem1.p1.7.7.m7.1.1.3" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.2" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.2.cmml">σ</mi><mrow id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.cmml"><mn id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.2" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.1" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.1.cmml">|</mo><mi id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.3" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.7.7.m7.1b"><apply id="S3.Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.1">precedes</csymbol><apply id="S3.Thmtheorem1.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.7.7.m7.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.7.7.m7.1.1.2.2.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2.2">𝜏</ci><cn id="S3.Thmtheorem1.p1.7.7.m7.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2.3">0</cn></apply><apply id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.2">𝜎</ci><apply id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.1.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.1">conditional</csymbol><cn id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.2">0</cn><ci id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.3.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.7.7.m7.1c">\tau_{0}\prec\sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.7.7.m7.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tau_{0}\in\mbox{SPS}({G_{|h}},{B})" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem1.p1.8.8.m8.2"><semantics id="S3.Thmtheorem1.p1.8.8.m8.2a"><mrow id="S3.Thmtheorem1.p1.8.8.m8.2.2"><msub id="S3.Thmtheorem1.p1.8.8.m8.2.2.3"><mi id="S3.Thmtheorem1.p1.8.8.m8.2.2.3.2">τ</mi><mn id="S3.Thmtheorem1.p1.8.8.m8.2.2.3.3">0</mn></msub><mo id="S3.Thmtheorem1.p1.8.8.m8.2.2.2">∈</mo><mrow id="S3.Thmtheorem1.p1.8.8.m8.2.2.1"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.3">SPS</mtext><mo id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.2">⁢</mo><mrow id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1"><mo id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.2" stretchy="false">(</mo><msub id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.1"><mi id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.1.2">G</mi><mrow id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.1.3"><mo fence="false" id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.1.3.2">h</mi></mrow></msub><mo id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.3">,</mo><mi id="S3.Thmtheorem1.p1.8.8.m8.1.1">B</mi><mo id="S3.Thmtheorem1.p1.8.8.m8.2.2.1.1.1.4" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.8.8.m8.2b">\tau_{0}\in\mbox{SPS}({G_{|h}},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.8.8.m8.2c">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT , italic_B )</annotation></semantics></math>. Then the strategy <math alttext="\sigma_{0}^{\prime}=\sigma_{0}[h\rightarrow\tau_{0}]" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.9.9.m9.1"><semantics id="S3.Thmtheorem1.p1.9.9.m9.1a"><mrow id="S3.Thmtheorem1.p1.9.9.m9.1.1" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.cmml"><msubsup id="S3.Thmtheorem1.p1.9.9.m9.1.1.3" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.2" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.3" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.3.cmml">0</mn><mo id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.3" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem1.p1.9.9.m9.1.1.2" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.2.cmml">=</mo><mrow id="S3.Thmtheorem1.p1.9.9.m9.1.1.1" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.cmml"><msub id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.2" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.3" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.3.cmml">0</mn></msub><mo id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.2" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.2.cmml"><mo id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.2" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.2.cmml">h</mi><mo id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.1.cmml">→</mo><msub id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.2" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.2.cmml">τ</mi><mn id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.3" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.9.9.m9.1b"><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1"><eq id="S3.Thmtheorem1.p1.9.9.m9.1.1.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.2"></eq><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3">superscript</csymbol><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.2">𝜎</ci><cn id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3.2.3">0</cn></apply><ci id="S3.Thmtheorem1.p1.9.9.m9.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.3.3">′</ci></apply><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1"><times id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.2"></times><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.3.3">0</cn></apply><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1"><ci id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.1">→</ci><ci id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.2">ℎ</ci><apply id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.2">𝜏</ci><cn id="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.9.9.m9.1c">\sigma_{0}^{\prime}=\sigma_{0}[h\rightarrow\tau_{0}]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.9.9.m9.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_h → italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math> is a solution in <math alttext="\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.10.10.m10.2"><semantics id="S3.Thmtheorem1.p1.10.10.m10.2a"><mrow id="S3.Thmtheorem1.p1.10.10.m10.2.3" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem1.p1.10.10.m10.2.3.2" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.2a.cmml">SPS</mtext><mo id="S3.Thmtheorem1.p1.10.10.m10.2.3.1" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.10.10.m10.2.3.3.2" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.3.1.cmml"><mo id="S3.Thmtheorem1.p1.10.10.m10.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem1.p1.10.10.m10.1.1" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.cmml">G</mi><mo id="S3.Thmtheorem1.p1.10.10.m10.2.3.3.2.2" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.10.10.m10.2.2" xref="S3.Thmtheorem1.p1.10.10.m10.2.2.cmml">B</mi><mo id="S3.Thmtheorem1.p1.10.10.m10.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.10.10.m10.2b"><apply id="S3.Thmtheorem1.p1.10.10.m10.2.3.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.2.3"><times id="S3.Thmtheorem1.p1.10.10.m10.2.3.1.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.1"></times><ci id="S3.Thmtheorem1.p1.10.10.m10.2.3.2a.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.2"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem1.p1.10.10.m10.2.3.2.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.2">SPS</mtext></ci><interval closure="open" id="S3.Thmtheorem1.p1.10.10.m10.2.3.3.1.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.2.3.3.2"><ci id="S3.Thmtheorem1.p1.10.10.m10.1.1.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.1.1">𝐺</ci><ci id="S3.Thmtheorem1.p1.10.10.m10.2.2.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.10.10.m10.2c">\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.10.10.m10.2d">SPS ( italic_G , italic_B )</annotation></semantics></math> and <math alttext="\sigma_{0}^{\prime}\prec\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.11.11.m11.1"><semantics id="S3.Thmtheorem1.p1.11.11.m11.1a"><mrow id="S3.Thmtheorem1.p1.11.11.m11.1.1" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.cmml"><msubsup id="S3.Thmtheorem1.p1.11.11.m11.1.1.2" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2.cmml"><mi id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.2" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.3" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.3.cmml">0</mn><mo id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.3" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem1.p1.11.11.m11.1.1.1" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.1.cmml">≺</mo><msub id="S3.Thmtheorem1.p1.11.11.m11.1.1.3" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.11.11.m11.1.1.3.2" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem1.p1.11.11.m11.1.1.3.3" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.11.11.m11.1b"><apply id="S3.Thmtheorem1.p1.11.11.m11.1.1.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.11.11.m11.1.1.1.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.1">precedes</csymbol><apply id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2">superscript</csymbol><apply id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.1.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.2.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.2">𝜎</ci><cn id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2.2.3">0</cn></apply><ci id="S3.Thmtheorem1.p1.11.11.m11.1.1.2.3.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem1.p1.11.11.m11.1.1.3.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.11.11.m11.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.11.11.m11.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem1.p1.11.11.m11.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.11.11.m11.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.11.11.m11.1c">\sigma_{0}^{\prime}\prec\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.11.11.m11.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Proof 3.2</span></span></h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.7"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.7.7">Let us first prove that <math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem2.p1.1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml"><msubsup id="S3.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2.cmml"><mi id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.2" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.3" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.3" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem2.p1.1.1.m1.1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">⪯</mo><msub id="S3.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.1.m1.1b"><apply id="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1"><csymbol cd="latexml" id="S3.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.1">precedes-or-equals</csymbol><apply id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2">superscript</csymbol><apply id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p1.1.1.m1.1.1.2.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.1.m1.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, that is, for all <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem2.p1.2.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.2" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.3" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.2.m2.1b"><apply id="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1"><in id="S3.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.2">𝑐</ci><apply id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.2.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.2.m2.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.2.m2.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, there exists <math alttext="c^{\prime}\in C_{\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem2.p1.3.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml"><msup id="S3.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2.cmml"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1.2.2" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem2.p1.3.3.m3.1.1.2.3" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml">C</mi><msubsup id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.2" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.3" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.3" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.3.m3.1b"><apply id="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1"><in id="S3.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.1"></in><apply id="S3.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.3.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3">superscript</csymbol><apply id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.3.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.3.m3.1c">c^{\prime}\in C_{\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.3.m3.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c^{\prime}\leq c" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.4.m4.1"><semantics id="S3.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem2.p1.4.4.m4.1.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml"><msup id="S3.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2.cmml"><mi id="S3.Thmtheorem2.p1.4.4.m4.1.1.2.2" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem2.p1.4.4.m4.1.1.2.3" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.1.cmml">≤</mo><mi id="S3.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.4.4.m4.1b"><apply id="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1"><leq id="S3.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.1"></leq><apply id="S3.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.4.4.m4.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.4.4.m4.1.1.2.2.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem2.p1.4.4.m4.1.1.2.3.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2.3">′</ci></apply><ci id="S3.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.4.m4.1c">c^{\prime}\leq c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.4.m4.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_c</annotation></semantics></math>. Let <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.5.m5.1"><semantics id="S3.Thmtheorem2.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem2.p1.5.5.m5.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem2.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.cmml"><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.5.5.m5.1b"><apply id="S3.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1"><in id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1"></in><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.2">𝑐</ci><apply id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.5.m5.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.5.m5.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\rho\in\textsf{Play}_{G,\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.6.6.m6.2"><semantics id="S3.Thmtheorem2.p1.6.6.m6.2a"><mrow id="S3.Thmtheorem2.p1.6.6.m6.2.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.cmml"><mi id="S3.Thmtheorem2.p1.6.6.m6.2.3.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p1.6.6.m6.2.3.1" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p1.6.6.m6.2.3.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2a.cmml">Play</mtext><mrow id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml"><mi id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml">G</mi><mo id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml">,</mo><msub id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.cmml"><mi id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.3.cmml">0</mn></msub></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.6.6.m6.2b"><apply id="S3.Thmtheorem2.p1.6.6.m6.2.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3"><in id="S3.Thmtheorem2.p1.6.6.m6.2.3.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.1"></in><ci id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2">𝜌</ci><apply id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2a.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2">Play</mtext></ci><list id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2"><ci id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1">𝐺</ci><apply id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1">subscript</csymbol><ci id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.3">0</cn></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.6.6.m6.2c">\rho\in\textsf{Play}_{G,\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.6.6.m6.2d">italic_ρ ∈ Play start_POSTSUBSCRIPT italic_G , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be such that <math alttext="\textsf{cost}({\rho})=c" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.7.7.m7.1"><semantics id="S3.Thmtheorem2.p1.7.7.m7.1a"><mrow id="S3.Thmtheorem2.p1.7.7.m7.1.2" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.cmml"><mrow id="S3.Thmtheorem2.p1.7.7.m7.1.2.2" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.2" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.2a.cmml">cost</mtext><mo id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.1" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.3.2" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.cmml"><mo id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem2.p1.7.7.m7.1.1" xref="S3.Thmtheorem2.p1.7.7.m7.1.1.cmml">ρ</mi><mo id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p1.7.7.m7.1.2.1" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.1.cmml">=</mo><mi id="S3.Thmtheorem2.p1.7.7.m7.1.2.3" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.7.7.m7.1b"><apply id="S3.Thmtheorem2.p1.7.7.m7.1.2.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.2"><eq id="S3.Thmtheorem2.p1.7.7.m7.1.2.1.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.1"></eq><apply id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2"><times id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.1.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.1"></times><ci id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.2a.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p1.7.7.m7.1.2.2.2.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.2.2">cost</mtext></ci><ci id="S3.Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.1">𝜌</ci></apply><ci id="S3.Thmtheorem2.p1.7.7.m7.1.2.3.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.7.7.m7.1c">\textsf{cost}({\rho})=c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.7.7.m7.1d">cost ( italic_ρ ) = italic_c</annotation></semantics></math>.</span></p> <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.6"><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.1">If </span><math alttext="h\not\rho" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mrow id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml">h</mi><mo id="S3.I1.i1.p1.1.m1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_mathvariant_italic" id="S3.I1.i1.p1.1.m1.1.1.3" mathvariant="italic" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml">ρ̸</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><times id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1"></times><ci id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">ℎ</ci><ci id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3">italic-ρ̸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">h\not\rho</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_h italic_ρ̸</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.2">, then </span><math alttext="\rho" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><mi id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><ci id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.3"> is also consistent with </span><math alttext="\sigma_{0}^{\prime}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.3.m3.1"><semantics id="S3.I1.i1.p1.3.m3.1a"><msubsup id="S3.I1.i1.p1.3.m3.1.1" xref="S3.I1.i1.p1.3.m3.1.1.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.2.2" xref="S3.I1.i1.p1.3.m3.1.1.2.2.cmml">σ</mi><mn id="S3.I1.i1.p1.3.m3.1.1.2.3" xref="S3.I1.i1.p1.3.m3.1.1.2.3.cmml">0</mn><mo id="S3.I1.i1.p1.3.m3.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.3.m3.1b"><apply id="S3.I1.i1.p1.3.m3.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1">superscript</csymbol><apply id="S3.I1.i1.p1.3.m3.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.2.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.3.m3.1.1.2.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2.2">𝜎</ci><cn id="S3.I1.i1.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S3.I1.i1.p1.3.m3.1.1.2.3">0</cn></apply><ci id="S3.I1.i1.p1.3.m3.1.1.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">\sigma_{0}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.4">, thus there exists </span><math alttext="c^{\prime}\in C_{\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.4.m4.1"><semantics id="S3.I1.i1.p1.4.m4.1a"><mrow id="S3.I1.i1.p1.4.m4.1.1" xref="S3.I1.i1.p1.4.m4.1.1.cmml"><msup id="S3.I1.i1.p1.4.m4.1.1.2" xref="S3.I1.i1.p1.4.m4.1.1.2.cmml"><mi id="S3.I1.i1.p1.4.m4.1.1.2.2" xref="S3.I1.i1.p1.4.m4.1.1.2.2.cmml">c</mi><mo id="S3.I1.i1.p1.4.m4.1.1.2.3" xref="S3.I1.i1.p1.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S3.I1.i1.p1.4.m4.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S3.I1.i1.p1.4.m4.1.1.3" xref="S3.I1.i1.p1.4.m4.1.1.3.cmml"><mi id="S3.I1.i1.p1.4.m4.1.1.3.2" xref="S3.I1.i1.p1.4.m4.1.1.3.2.cmml">C</mi><msubsup id="S3.I1.i1.p1.4.m4.1.1.3.3" xref="S3.I1.i1.p1.4.m4.1.1.3.3.cmml"><mi id="S3.I1.i1.p1.4.m4.1.1.3.3.2.2" xref="S3.I1.i1.p1.4.m4.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.I1.i1.p1.4.m4.1.1.3.3.2.3" xref="S3.I1.i1.p1.4.m4.1.1.3.3.2.3.cmml">0</mn><mo id="S3.I1.i1.p1.4.m4.1.1.3.3.3" xref="S3.I1.i1.p1.4.m4.1.1.3.3.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.4.m4.1b"><apply id="S3.I1.i1.p1.4.m4.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1"><in id="S3.I1.i1.p1.4.m4.1.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1"></in><apply id="S3.I1.i1.p1.4.m4.1.1.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.4.m4.1.1.2.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.2">superscript</csymbol><ci id="S3.I1.i1.p1.4.m4.1.1.2.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.2.2">𝑐</ci><ci id="S3.I1.i1.p1.4.m4.1.1.2.3.cmml" xref="S3.I1.i1.p1.4.m4.1.1.2.3">′</ci></apply><apply id="S3.I1.i1.p1.4.m4.1.1.3.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.4.m4.1.1.3.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3">subscript</csymbol><ci id="S3.I1.i1.p1.4.m4.1.1.3.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3.2">𝐶</ci><apply id="S3.I1.i1.p1.4.m4.1.1.3.3.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.4.m4.1.1.3.3.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3.3">superscript</csymbol><apply id="S3.I1.i1.p1.4.m4.1.1.3.3.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.4.m4.1.1.3.3.2.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S3.I1.i1.p1.4.m4.1.1.3.3.2.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3.3.2.2">𝜎</ci><cn id="S3.I1.i1.p1.4.m4.1.1.3.3.2.3.cmml" type="integer" xref="S3.I1.i1.p1.4.m4.1.1.3.3.2.3">0</cn></apply><ci id="S3.I1.i1.p1.4.m4.1.1.3.3.3.cmml" xref="S3.I1.i1.p1.4.m4.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.4.m4.1c">c^{\prime}\in C_{\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.4.m4.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.5"> such that </span><math alttext="c^{\prime}\leq c=\textsf{cost}({\rho})" class="ltx_Math" display="inline" id="S3.I1.i1.p1.5.m5.1"><semantics id="S3.I1.i1.p1.5.m5.1a"><mrow id="S3.I1.i1.p1.5.m5.1.2" xref="S3.I1.i1.p1.5.m5.1.2.cmml"><msup id="S3.I1.i1.p1.5.m5.1.2.2" xref="S3.I1.i1.p1.5.m5.1.2.2.cmml"><mi id="S3.I1.i1.p1.5.m5.1.2.2.2" xref="S3.I1.i1.p1.5.m5.1.2.2.2.cmml">c</mi><mo id="S3.I1.i1.p1.5.m5.1.2.2.3" xref="S3.I1.i1.p1.5.m5.1.2.2.3.cmml">′</mo></msup><mo id="S3.I1.i1.p1.5.m5.1.2.3" xref="S3.I1.i1.p1.5.m5.1.2.3.cmml">≤</mo><mi id="S3.I1.i1.p1.5.m5.1.2.4" xref="S3.I1.i1.p1.5.m5.1.2.4.cmml">c</mi><mo id="S3.I1.i1.p1.5.m5.1.2.5" xref="S3.I1.i1.p1.5.m5.1.2.5.cmml">=</mo><mrow id="S3.I1.i1.p1.5.m5.1.2.6" xref="S3.I1.i1.p1.5.m5.1.2.6.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I1.i1.p1.5.m5.1.2.6.2" xref="S3.I1.i1.p1.5.m5.1.2.6.2a.cmml">cost</mtext><mo id="S3.I1.i1.p1.5.m5.1.2.6.1" xref="S3.I1.i1.p1.5.m5.1.2.6.1.cmml">⁢</mo><mrow id="S3.I1.i1.p1.5.m5.1.2.6.3.2" xref="S3.I1.i1.p1.5.m5.1.2.6.cmml"><mo id="S3.I1.i1.p1.5.m5.1.2.6.3.2.1" stretchy="false" xref="S3.I1.i1.p1.5.m5.1.2.6.cmml">(</mo><mi id="S3.I1.i1.p1.5.m5.1.1" xref="S3.I1.i1.p1.5.m5.1.1.cmml">ρ</mi><mo id="S3.I1.i1.p1.5.m5.1.2.6.3.2.2" stretchy="false" xref="S3.I1.i1.p1.5.m5.1.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.5.m5.1b"><apply id="S3.I1.i1.p1.5.m5.1.2.cmml" xref="S3.I1.i1.p1.5.m5.1.2"><and id="S3.I1.i1.p1.5.m5.1.2a.cmml" xref="S3.I1.i1.p1.5.m5.1.2"></and><apply id="S3.I1.i1.p1.5.m5.1.2b.cmml" xref="S3.I1.i1.p1.5.m5.1.2"><leq id="S3.I1.i1.p1.5.m5.1.2.3.cmml" xref="S3.I1.i1.p1.5.m5.1.2.3"></leq><apply id="S3.I1.i1.p1.5.m5.1.2.2.cmml" xref="S3.I1.i1.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.5.m5.1.2.2.1.cmml" xref="S3.I1.i1.p1.5.m5.1.2.2">superscript</csymbol><ci id="S3.I1.i1.p1.5.m5.1.2.2.2.cmml" xref="S3.I1.i1.p1.5.m5.1.2.2.2">𝑐</ci><ci id="S3.I1.i1.p1.5.m5.1.2.2.3.cmml" xref="S3.I1.i1.p1.5.m5.1.2.2.3">′</ci></apply><ci id="S3.I1.i1.p1.5.m5.1.2.4.cmml" xref="S3.I1.i1.p1.5.m5.1.2.4">𝑐</ci></apply><apply id="S3.I1.i1.p1.5.m5.1.2c.cmml" xref="S3.I1.i1.p1.5.m5.1.2"><eq id="S3.I1.i1.p1.5.m5.1.2.5.cmml" xref="S3.I1.i1.p1.5.m5.1.2.5"></eq><share href="https://arxiv.org/html/2308.09443v2#S3.I1.i1.p1.5.m5.1.2.4.cmml" id="S3.I1.i1.p1.5.m5.1.2d.cmml" xref="S3.I1.i1.p1.5.m5.1.2"></share><apply id="S3.I1.i1.p1.5.m5.1.2.6.cmml" xref="S3.I1.i1.p1.5.m5.1.2.6"><times id="S3.I1.i1.p1.5.m5.1.2.6.1.cmml" xref="S3.I1.i1.p1.5.m5.1.2.6.1"></times><ci id="S3.I1.i1.p1.5.m5.1.2.6.2a.cmml" xref="S3.I1.i1.p1.5.m5.1.2.6.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I1.i1.p1.5.m5.1.2.6.2.cmml" xref="S3.I1.i1.p1.5.m5.1.2.6.2">cost</mtext></ci><ci id="S3.I1.i1.p1.5.m5.1.1.cmml" xref="S3.I1.i1.p1.5.m5.1.1">𝜌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.5.m5.1c">c^{\prime}\leq c=\textsf{cost}({\rho})</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.5.m5.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_c = cost ( italic_ρ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.6"> by definition of </span><math alttext="C_{\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.6.m6.1"><semantics id="S3.I1.i1.p1.6.m6.1a"><msub id="S3.I1.i1.p1.6.m6.1.1" xref="S3.I1.i1.p1.6.m6.1.1.cmml"><mi id="S3.I1.i1.p1.6.m6.1.1.2" xref="S3.I1.i1.p1.6.m6.1.1.2.cmml">C</mi><msubsup id="S3.I1.i1.p1.6.m6.1.1.3" xref="S3.I1.i1.p1.6.m6.1.1.3.cmml"><mi id="S3.I1.i1.p1.6.m6.1.1.3.2.2" xref="S3.I1.i1.p1.6.m6.1.1.3.2.2.cmml">σ</mi><mn id="S3.I1.i1.p1.6.m6.1.1.3.2.3" xref="S3.I1.i1.p1.6.m6.1.1.3.2.3.cmml">0</mn><mo id="S3.I1.i1.p1.6.m6.1.1.3.3" xref="S3.I1.i1.p1.6.m6.1.1.3.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.6.m6.1b"><apply id="S3.I1.i1.p1.6.m6.1.1.cmml" xref="S3.I1.i1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.1.1.1.cmml" xref="S3.I1.i1.p1.6.m6.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.6.m6.1.1.2.cmml" xref="S3.I1.i1.p1.6.m6.1.1.2">𝐶</ci><apply id="S3.I1.i1.p1.6.m6.1.1.3.cmml" xref="S3.I1.i1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.1.1.3.1.cmml" xref="S3.I1.i1.p1.6.m6.1.1.3">superscript</csymbol><apply id="S3.I1.i1.p1.6.m6.1.1.3.2.cmml" xref="S3.I1.i1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.1.1.3.2.1.cmml" xref="S3.I1.i1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S3.I1.i1.p1.6.m6.1.1.3.2.2.cmml" xref="S3.I1.i1.p1.6.m6.1.1.3.2.2">𝜎</ci><cn id="S3.I1.i1.p1.6.m6.1.1.3.2.3.cmml" type="integer" xref="S3.I1.i1.p1.6.m6.1.1.3.2.3">0</cn></apply><ci id="S3.I1.i1.p1.6.m6.1.1.3.3.cmml" xref="S3.I1.i1.p1.6.m6.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.6.m6.1c">C_{\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.6.m6.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.7">.</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.20"><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.1">Otherwise, </span><math alttext="h\rho" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><mrow id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml">h</mi><mo id="S3.I1.i2.p1.1.m1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><times id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1"></times><ci id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">ℎ</ci><ci id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">h\rho</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_h italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.2">. Hence </span><math alttext="\rho" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><mi id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><ci id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.3"> is a play in the subgame </span><math alttext="{G_{|h}}" class="ltx_math_unparsed" display="inline" id="S3.I1.i2.p1.3.m3.1"><semantics id="S3.I1.i2.p1.3.m3.1a"><msub id="S3.I1.i2.p1.3.m3.1.1"><mi id="S3.I1.i2.p1.3.m3.1.1.2">G</mi><mrow id="S3.I1.i2.p1.3.m3.1.1.3"><mo fence="false" id="S3.I1.i2.p1.3.m3.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.I1.i2.p1.3.m3.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.I1.i2.p1.3.m3.1b">{G_{|h}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.3.m3.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.4"> with a Pareto-optimal cost </span><math alttext="c\in C_{\sigma_{0|h}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.4.m4.1"><semantics id="S3.I1.i2.p1.4.m4.1a"><mrow id="S3.I1.i2.p1.4.m4.1.1" xref="S3.I1.i2.p1.4.m4.1.1.cmml"><mi id="S3.I1.i2.p1.4.m4.1.1.2" xref="S3.I1.i2.p1.4.m4.1.1.2.cmml">c</mi><mo id="S3.I1.i2.p1.4.m4.1.1.1" xref="S3.I1.i2.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S3.I1.i2.p1.4.m4.1.1.3" xref="S3.I1.i2.p1.4.m4.1.1.3.cmml"><mi id="S3.I1.i2.p1.4.m4.1.1.3.2" xref="S3.I1.i2.p1.4.m4.1.1.3.2.cmml">C</mi><msub id="S3.I1.i2.p1.4.m4.1.1.3.3" xref="S3.I1.i2.p1.4.m4.1.1.3.3.cmml"><mi id="S3.I1.i2.p1.4.m4.1.1.3.3.2" xref="S3.I1.i2.p1.4.m4.1.1.3.3.2.cmml">σ</mi><mrow id="S3.I1.i2.p1.4.m4.1.1.3.3.3" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3.cmml"><mn id="S3.I1.i2.p1.4.m4.1.1.3.3.3.2" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3.2.cmml">0</mn><mo fence="false" id="S3.I1.i2.p1.4.m4.1.1.3.3.3.1" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3.1.cmml">|</mo><mi id="S3.I1.i2.p1.4.m4.1.1.3.3.3.3" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3.3.cmml">h</mi></mrow></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.4.m4.1b"><apply id="S3.I1.i2.p1.4.m4.1.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1"><in id="S3.I1.i2.p1.4.m4.1.1.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1.1"></in><ci id="S3.I1.i2.p1.4.m4.1.1.2.cmml" xref="S3.I1.i2.p1.4.m4.1.1.2">𝑐</ci><apply id="S3.I1.i2.p1.4.m4.1.1.3.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.4.m4.1.1.3.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.4.m4.1.1.3.2.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3.2">𝐶</ci><apply id="S3.I1.i2.p1.4.m4.1.1.3.3.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.4.m4.1.1.3.3.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S3.I1.i2.p1.4.m4.1.1.3.3.2.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3.3.2">𝜎</ci><apply id="S3.I1.i2.p1.4.m4.1.1.3.3.3.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3"><csymbol cd="latexml" id="S3.I1.i2.p1.4.m4.1.1.3.3.3.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3.1">conditional</csymbol><cn id="S3.I1.i2.p1.4.m4.1.1.3.3.3.2.cmml" type="integer" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3.2">0</cn><ci id="S3.I1.i2.p1.4.m4.1.1.3.3.3.3.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3.3.3.3">ℎ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.4.m4.1c">c\in C_{\sigma_{0|h}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.4.m4.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.5">. By hypothesis, we have </span><math alttext="\tau_{0}\prec\sigma_{0|h}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.5.m5.1"><semantics id="S3.I1.i2.p1.5.m5.1a"><mrow id="S3.I1.i2.p1.5.m5.1.1" xref="S3.I1.i2.p1.5.m5.1.1.cmml"><msub id="S3.I1.i2.p1.5.m5.1.1.2" xref="S3.I1.i2.p1.5.m5.1.1.2.cmml"><mi id="S3.I1.i2.p1.5.m5.1.1.2.2" xref="S3.I1.i2.p1.5.m5.1.1.2.2.cmml">τ</mi><mn id="S3.I1.i2.p1.5.m5.1.1.2.3" xref="S3.I1.i2.p1.5.m5.1.1.2.3.cmml">0</mn></msub><mo id="S3.I1.i2.p1.5.m5.1.1.1" xref="S3.I1.i2.p1.5.m5.1.1.1.cmml">≺</mo><msub id="S3.I1.i2.p1.5.m5.1.1.3" xref="S3.I1.i2.p1.5.m5.1.1.3.cmml"><mi id="S3.I1.i2.p1.5.m5.1.1.3.2" xref="S3.I1.i2.p1.5.m5.1.1.3.2.cmml">σ</mi><mrow id="S3.I1.i2.p1.5.m5.1.1.3.3" xref="S3.I1.i2.p1.5.m5.1.1.3.3.cmml"><mn id="S3.I1.i2.p1.5.m5.1.1.3.3.2" xref="S3.I1.i2.p1.5.m5.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S3.I1.i2.p1.5.m5.1.1.3.3.1" xref="S3.I1.i2.p1.5.m5.1.1.3.3.1.cmml">|</mo><mi id="S3.I1.i2.p1.5.m5.1.1.3.3.3" xref="S3.I1.i2.p1.5.m5.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.5.m5.1b"><apply id="S3.I1.i2.p1.5.m5.1.1.cmml" xref="S3.I1.i2.p1.5.m5.1.1"><csymbol cd="latexml" id="S3.I1.i2.p1.5.m5.1.1.1.cmml" xref="S3.I1.i2.p1.5.m5.1.1.1">precedes</csymbol><apply id="S3.I1.i2.p1.5.m5.1.1.2.cmml" xref="S3.I1.i2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.5.m5.1.1.2.1.cmml" xref="S3.I1.i2.p1.5.m5.1.1.2">subscript</csymbol><ci id="S3.I1.i2.p1.5.m5.1.1.2.2.cmml" xref="S3.I1.i2.p1.5.m5.1.1.2.2">𝜏</ci><cn id="S3.I1.i2.p1.5.m5.1.1.2.3.cmml" type="integer" xref="S3.I1.i2.p1.5.m5.1.1.2.3">0</cn></apply><apply id="S3.I1.i2.p1.5.m5.1.1.3.cmml" xref="S3.I1.i2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.5.m5.1.1.3.1.cmml" xref="S3.I1.i2.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.5.m5.1.1.3.2.cmml" xref="S3.I1.i2.p1.5.m5.1.1.3.2">𝜎</ci><apply id="S3.I1.i2.p1.5.m5.1.1.3.3.cmml" xref="S3.I1.i2.p1.5.m5.1.1.3.3"><csymbol cd="latexml" id="S3.I1.i2.p1.5.m5.1.1.3.3.1.cmml" xref="S3.I1.i2.p1.5.m5.1.1.3.3.1">conditional</csymbol><cn id="S3.I1.i2.p1.5.m5.1.1.3.3.2.cmml" type="integer" xref="S3.I1.i2.p1.5.m5.1.1.3.3.2">0</cn><ci id="S3.I1.i2.p1.5.m5.1.1.3.3.3.cmml" xref="S3.I1.i2.p1.5.m5.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.5.m5.1c">\tau_{0}\prec\sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.5.m5.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.6">, therefore there exists </span><math alttext="c^{\prime}\in C_{\tau_{0}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.6.m6.1"><semantics id="S3.I1.i2.p1.6.m6.1a"><mrow id="S3.I1.i2.p1.6.m6.1.1" xref="S3.I1.i2.p1.6.m6.1.1.cmml"><msup id="S3.I1.i2.p1.6.m6.1.1.2" xref="S3.I1.i2.p1.6.m6.1.1.2.cmml"><mi id="S3.I1.i2.p1.6.m6.1.1.2.2" xref="S3.I1.i2.p1.6.m6.1.1.2.2.cmml">c</mi><mo id="S3.I1.i2.p1.6.m6.1.1.2.3" xref="S3.I1.i2.p1.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S3.I1.i2.p1.6.m6.1.1.1" xref="S3.I1.i2.p1.6.m6.1.1.1.cmml">∈</mo><msub id="S3.I1.i2.p1.6.m6.1.1.3" xref="S3.I1.i2.p1.6.m6.1.1.3.cmml"><mi id="S3.I1.i2.p1.6.m6.1.1.3.2" xref="S3.I1.i2.p1.6.m6.1.1.3.2.cmml">C</mi><msub id="S3.I1.i2.p1.6.m6.1.1.3.3" xref="S3.I1.i2.p1.6.m6.1.1.3.3.cmml"><mi id="S3.I1.i2.p1.6.m6.1.1.3.3.2" xref="S3.I1.i2.p1.6.m6.1.1.3.3.2.cmml">τ</mi><mn id="S3.I1.i2.p1.6.m6.1.1.3.3.3" xref="S3.I1.i2.p1.6.m6.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.6.m6.1b"><apply id="S3.I1.i2.p1.6.m6.1.1.cmml" xref="S3.I1.i2.p1.6.m6.1.1"><in id="S3.I1.i2.p1.6.m6.1.1.1.cmml" xref="S3.I1.i2.p1.6.m6.1.1.1"></in><apply id="S3.I1.i2.p1.6.m6.1.1.2.cmml" xref="S3.I1.i2.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.6.m6.1.1.2.1.cmml" xref="S3.I1.i2.p1.6.m6.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.6.m6.1.1.2.2.cmml" xref="S3.I1.i2.p1.6.m6.1.1.2.2">𝑐</ci><ci id="S3.I1.i2.p1.6.m6.1.1.2.3.cmml" xref="S3.I1.i2.p1.6.m6.1.1.2.3">′</ci></apply><apply id="S3.I1.i2.p1.6.m6.1.1.3.cmml" xref="S3.I1.i2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.6.m6.1.1.3.1.cmml" xref="S3.I1.i2.p1.6.m6.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.6.m6.1.1.3.2.cmml" xref="S3.I1.i2.p1.6.m6.1.1.3.2">𝐶</ci><apply id="S3.I1.i2.p1.6.m6.1.1.3.3.cmml" xref="S3.I1.i2.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.6.m6.1.1.3.3.1.cmml" xref="S3.I1.i2.p1.6.m6.1.1.3.3">subscript</csymbol><ci id="S3.I1.i2.p1.6.m6.1.1.3.3.2.cmml" xref="S3.I1.i2.p1.6.m6.1.1.3.3.2">𝜏</ci><cn id="S3.I1.i2.p1.6.m6.1.1.3.3.3.cmml" type="integer" xref="S3.I1.i2.p1.6.m6.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.6.m6.1c">c^{\prime}\in C_{\tau_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.6.m6.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.7"> such that </span><math alttext="c^{\prime}\leq c" class="ltx_Math" display="inline" id="S3.I1.i2.p1.7.m7.1"><semantics id="S3.I1.i2.p1.7.m7.1a"><mrow id="S3.I1.i2.p1.7.m7.1.1" xref="S3.I1.i2.p1.7.m7.1.1.cmml"><msup id="S3.I1.i2.p1.7.m7.1.1.2" xref="S3.I1.i2.p1.7.m7.1.1.2.cmml"><mi id="S3.I1.i2.p1.7.m7.1.1.2.2" xref="S3.I1.i2.p1.7.m7.1.1.2.2.cmml">c</mi><mo id="S3.I1.i2.p1.7.m7.1.1.2.3" xref="S3.I1.i2.p1.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S3.I1.i2.p1.7.m7.1.1.1" xref="S3.I1.i2.p1.7.m7.1.1.1.cmml">≤</mo><mi id="S3.I1.i2.p1.7.m7.1.1.3" xref="S3.I1.i2.p1.7.m7.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.7.m7.1b"><apply id="S3.I1.i2.p1.7.m7.1.1.cmml" xref="S3.I1.i2.p1.7.m7.1.1"><leq id="S3.I1.i2.p1.7.m7.1.1.1.cmml" xref="S3.I1.i2.p1.7.m7.1.1.1"></leq><apply id="S3.I1.i2.p1.7.m7.1.1.2.cmml" xref="S3.I1.i2.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.7.m7.1.1.2.1.cmml" xref="S3.I1.i2.p1.7.m7.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.7.m7.1.1.2.2.cmml" xref="S3.I1.i2.p1.7.m7.1.1.2.2">𝑐</ci><ci id="S3.I1.i2.p1.7.m7.1.1.2.3.cmml" xref="S3.I1.i2.p1.7.m7.1.1.2.3">′</ci></apply><ci id="S3.I1.i2.p1.7.m7.1.1.3.cmml" xref="S3.I1.i2.p1.7.m7.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.7.m7.1c">c^{\prime}\leq c</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.7.m7.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.8"> (for some </span><math alttext="c" class="ltx_Math" display="inline" id="S3.I1.i2.p1.8.m8.1"><semantics id="S3.I1.i2.p1.8.m8.1a"><mi id="S3.I1.i2.p1.8.m8.1.1" xref="S3.I1.i2.p1.8.m8.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.8.m8.1b"><ci id="S3.I1.i2.p1.8.m8.1.1.cmml" xref="S3.I1.i2.p1.8.m8.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.8.m8.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.8.m8.1d">italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.9">, the inequality is strict: </span><math alttext="c^{\prime}<c" class="ltx_Math" display="inline" id="S3.I1.i2.p1.9.m9.1"><semantics id="S3.I1.i2.p1.9.m9.1a"><mrow id="S3.I1.i2.p1.9.m9.1.1" xref="S3.I1.i2.p1.9.m9.1.1.cmml"><msup id="S3.I1.i2.p1.9.m9.1.1.2" xref="S3.I1.i2.p1.9.m9.1.1.2.cmml"><mi id="S3.I1.i2.p1.9.m9.1.1.2.2" xref="S3.I1.i2.p1.9.m9.1.1.2.2.cmml">c</mi><mo id="S3.I1.i2.p1.9.m9.1.1.2.3" xref="S3.I1.i2.p1.9.m9.1.1.2.3.cmml">′</mo></msup><mo id="S3.I1.i2.p1.9.m9.1.1.1" xref="S3.I1.i2.p1.9.m9.1.1.1.cmml"><</mo><mi id="S3.I1.i2.p1.9.m9.1.1.3" xref="S3.I1.i2.p1.9.m9.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.9.m9.1b"><apply id="S3.I1.i2.p1.9.m9.1.1.cmml" xref="S3.I1.i2.p1.9.m9.1.1"><lt id="S3.I1.i2.p1.9.m9.1.1.1.cmml" xref="S3.I1.i2.p1.9.m9.1.1.1"></lt><apply id="S3.I1.i2.p1.9.m9.1.1.2.cmml" xref="S3.I1.i2.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.9.m9.1.1.2.1.cmml" xref="S3.I1.i2.p1.9.m9.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.9.m9.1.1.2.2.cmml" xref="S3.I1.i2.p1.9.m9.1.1.2.2">𝑐</ci><ci id="S3.I1.i2.p1.9.m9.1.1.2.3.cmml" xref="S3.I1.i2.p1.9.m9.1.1.2.3">′</ci></apply><ci id="S3.I1.i2.p1.9.m9.1.1.3.cmml" xref="S3.I1.i2.p1.9.m9.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.9.m9.1c">c^{\prime}<c</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.9.m9.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.10">). As </span><math alttext="c^{\prime}\in C_{\tau_{0}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.10.m10.1"><semantics id="S3.I1.i2.p1.10.m10.1a"><mrow id="S3.I1.i2.p1.10.m10.1.1" xref="S3.I1.i2.p1.10.m10.1.1.cmml"><msup id="S3.I1.i2.p1.10.m10.1.1.2" xref="S3.I1.i2.p1.10.m10.1.1.2.cmml"><mi id="S3.I1.i2.p1.10.m10.1.1.2.2" xref="S3.I1.i2.p1.10.m10.1.1.2.2.cmml">c</mi><mo id="S3.I1.i2.p1.10.m10.1.1.2.3" xref="S3.I1.i2.p1.10.m10.1.1.2.3.cmml">′</mo></msup><mo id="S3.I1.i2.p1.10.m10.1.1.1" xref="S3.I1.i2.p1.10.m10.1.1.1.cmml">∈</mo><msub id="S3.I1.i2.p1.10.m10.1.1.3" xref="S3.I1.i2.p1.10.m10.1.1.3.cmml"><mi id="S3.I1.i2.p1.10.m10.1.1.3.2" xref="S3.I1.i2.p1.10.m10.1.1.3.2.cmml">C</mi><msub id="S3.I1.i2.p1.10.m10.1.1.3.3" xref="S3.I1.i2.p1.10.m10.1.1.3.3.cmml"><mi id="S3.I1.i2.p1.10.m10.1.1.3.3.2" xref="S3.I1.i2.p1.10.m10.1.1.3.3.2.cmml">τ</mi><mn id="S3.I1.i2.p1.10.m10.1.1.3.3.3" xref="S3.I1.i2.p1.10.m10.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.10.m10.1b"><apply id="S3.I1.i2.p1.10.m10.1.1.cmml" xref="S3.I1.i2.p1.10.m10.1.1"><in id="S3.I1.i2.p1.10.m10.1.1.1.cmml" xref="S3.I1.i2.p1.10.m10.1.1.1"></in><apply id="S3.I1.i2.p1.10.m10.1.1.2.cmml" xref="S3.I1.i2.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.10.m10.1.1.2.1.cmml" xref="S3.I1.i2.p1.10.m10.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.10.m10.1.1.2.2.cmml" xref="S3.I1.i2.p1.10.m10.1.1.2.2">𝑐</ci><ci id="S3.I1.i2.p1.10.m10.1.1.2.3.cmml" xref="S3.I1.i2.p1.10.m10.1.1.2.3">′</ci></apply><apply id="S3.I1.i2.p1.10.m10.1.1.3.cmml" xref="S3.I1.i2.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.10.m10.1.1.3.1.cmml" xref="S3.I1.i2.p1.10.m10.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.10.m10.1.1.3.2.cmml" xref="S3.I1.i2.p1.10.m10.1.1.3.2">𝐶</ci><apply id="S3.I1.i2.p1.10.m10.1.1.3.3.cmml" xref="S3.I1.i2.p1.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.10.m10.1.1.3.3.1.cmml" xref="S3.I1.i2.p1.10.m10.1.1.3.3">subscript</csymbol><ci id="S3.I1.i2.p1.10.m10.1.1.3.3.2.cmml" xref="S3.I1.i2.p1.10.m10.1.1.3.3.2">𝜏</ci><cn id="S3.I1.i2.p1.10.m10.1.1.3.3.3.cmml" type="integer" xref="S3.I1.i2.p1.10.m10.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.10.m10.1c">c^{\prime}\in C_{\tau_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.10.m10.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.11">, there exists </span><math alttext="\rho^{\prime}\in\textsf{Play}_{G_{|h},\tau_{0}}" class="ltx_math_unparsed" display="inline" id="S3.I1.i2.p1.11.m11.2"><semantics id="S3.I1.i2.p1.11.m11.2a"><mrow id="S3.I1.i2.p1.11.m11.2.3"><msup id="S3.I1.i2.p1.11.m11.2.3.2"><mi id="S3.I1.i2.p1.11.m11.2.3.2.2">ρ</mi><mo id="S3.I1.i2.p1.11.m11.2.3.2.3">′</mo></msup><mo id="S3.I1.i2.p1.11.m11.2.3.1">∈</mo><msub id="S3.I1.i2.p1.11.m11.2.3.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I1.i2.p1.11.m11.2.3.3.2">Play</mtext><mrow id="S3.I1.i2.p1.11.m11.2.2.2.2"><msub id="S3.I1.i2.p1.11.m11.1.1.1.1.1"><mi id="S3.I1.i2.p1.11.m11.1.1.1.1.1.2">G</mi><mrow id="S3.I1.i2.p1.11.m11.1.1.1.1.1.3"><mo fence="false" id="S3.I1.i2.p1.11.m11.1.1.1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.I1.i2.p1.11.m11.1.1.1.1.1.3.2">h</mi></mrow></msub><mo id="S3.I1.i2.p1.11.m11.2.2.2.2.3">,</mo><msub id="S3.I1.i2.p1.11.m11.2.2.2.2.2"><mi id="S3.I1.i2.p1.11.m11.2.2.2.2.2.2">τ</mi><mn id="S3.I1.i2.p1.11.m11.2.2.2.2.2.3">0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex" id="S3.I1.i2.p1.11.m11.2b">\rho^{\prime}\in\textsf{Play}_{G_{|h},\tau_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.11.m11.2c">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.12"> such that </span><math alttext="h\rho^{\prime}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.12.m12.1"><semantics id="S3.I1.i2.p1.12.m12.1a"><mrow id="S3.I1.i2.p1.12.m12.1.1" xref="S3.I1.i2.p1.12.m12.1.1.cmml"><mi id="S3.I1.i2.p1.12.m12.1.1.2" xref="S3.I1.i2.p1.12.m12.1.1.2.cmml">h</mi><mo id="S3.I1.i2.p1.12.m12.1.1.1" xref="S3.I1.i2.p1.12.m12.1.1.1.cmml">⁢</mo><msup id="S3.I1.i2.p1.12.m12.1.1.3" xref="S3.I1.i2.p1.12.m12.1.1.3.cmml"><mi id="S3.I1.i2.p1.12.m12.1.1.3.2" xref="S3.I1.i2.p1.12.m12.1.1.3.2.cmml">ρ</mi><mo id="S3.I1.i2.p1.12.m12.1.1.3.3" xref="S3.I1.i2.p1.12.m12.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.12.m12.1b"><apply id="S3.I1.i2.p1.12.m12.1.1.cmml" xref="S3.I1.i2.p1.12.m12.1.1"><times id="S3.I1.i2.p1.12.m12.1.1.1.cmml" xref="S3.I1.i2.p1.12.m12.1.1.1"></times><ci id="S3.I1.i2.p1.12.m12.1.1.2.cmml" xref="S3.I1.i2.p1.12.m12.1.1.2">ℎ</ci><apply id="S3.I1.i2.p1.12.m12.1.1.3.cmml" xref="S3.I1.i2.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.12.m12.1.1.3.1.cmml" xref="S3.I1.i2.p1.12.m12.1.1.3">superscript</csymbol><ci id="S3.I1.i2.p1.12.m12.1.1.3.2.cmml" xref="S3.I1.i2.p1.12.m12.1.1.3.2">𝜌</ci><ci id="S3.I1.i2.p1.12.m12.1.1.3.3.cmml" xref="S3.I1.i2.p1.12.m12.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.12.m12.1c">h\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.12.m12.1d">italic_h italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.13"> and </span><math alttext="\textsf{cost}({\rho^{\prime}})=c^{\prime}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.13.m13.1"><semantics id="S3.I1.i2.p1.13.m13.1a"><mrow id="S3.I1.i2.p1.13.m13.1.1" xref="S3.I1.i2.p1.13.m13.1.1.cmml"><mrow id="S3.I1.i2.p1.13.m13.1.1.1" xref="S3.I1.i2.p1.13.m13.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I1.i2.p1.13.m13.1.1.1.3" xref="S3.I1.i2.p1.13.m13.1.1.1.3a.cmml">cost</mtext><mo id="S3.I1.i2.p1.13.m13.1.1.1.2" xref="S3.I1.i2.p1.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S3.I1.i2.p1.13.m13.1.1.1.1.1" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.cmml"><mo id="S3.I1.i2.p1.13.m13.1.1.1.1.1.2" stretchy="false" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I1.i2.p1.13.m13.1.1.1.1.1.1" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.cmml"><mi id="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.2" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.3" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I1.i2.p1.13.m13.1.1.1.1.1.3" stretchy="false" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I1.i2.p1.13.m13.1.1.2" xref="S3.I1.i2.p1.13.m13.1.1.2.cmml">=</mo><msup id="S3.I1.i2.p1.13.m13.1.1.3" xref="S3.I1.i2.p1.13.m13.1.1.3.cmml"><mi id="S3.I1.i2.p1.13.m13.1.1.3.2" xref="S3.I1.i2.p1.13.m13.1.1.3.2.cmml">c</mi><mo id="S3.I1.i2.p1.13.m13.1.1.3.3" xref="S3.I1.i2.p1.13.m13.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.13.m13.1b"><apply id="S3.I1.i2.p1.13.m13.1.1.cmml" xref="S3.I1.i2.p1.13.m13.1.1"><eq id="S3.I1.i2.p1.13.m13.1.1.2.cmml" xref="S3.I1.i2.p1.13.m13.1.1.2"></eq><apply id="S3.I1.i2.p1.13.m13.1.1.1.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1"><times id="S3.I1.i2.p1.13.m13.1.1.1.2.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1.2"></times><ci id="S3.I1.i2.p1.13.m13.1.1.1.3a.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I1.i2.p1.13.m13.1.1.1.3.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1.3">cost</mtext></ci><apply id="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1">superscript</csymbol><ci id="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.2.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.2">𝜌</ci><ci id="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.3.cmml" xref="S3.I1.i2.p1.13.m13.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S3.I1.i2.p1.13.m13.1.1.3.cmml" xref="S3.I1.i2.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.13.m13.1.1.3.1.cmml" xref="S3.I1.i2.p1.13.m13.1.1.3">superscript</csymbol><ci id="S3.I1.i2.p1.13.m13.1.1.3.2.cmml" xref="S3.I1.i2.p1.13.m13.1.1.3.2">𝑐</ci><ci id="S3.I1.i2.p1.13.m13.1.1.3.3.cmml" xref="S3.I1.i2.p1.13.m13.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.13.m13.1c">\textsf{cost}({\rho^{\prime}})=c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.13.m13.1d">cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.14">. By definition of </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.14.m14.1"><semantics id="S3.I1.i2.p1.14.m14.1a"><msubsup id="S3.I1.i2.p1.14.m14.1.1" xref="S3.I1.i2.p1.14.m14.1.1.cmml"><mi id="S3.I1.i2.p1.14.m14.1.1.2.2" xref="S3.I1.i2.p1.14.m14.1.1.2.2.cmml">σ</mi><mn id="S3.I1.i2.p1.14.m14.1.1.3" xref="S3.I1.i2.p1.14.m14.1.1.3.cmml">0</mn><mo id="S3.I1.i2.p1.14.m14.1.1.2.3" xref="S3.I1.i2.p1.14.m14.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.14.m14.1b"><apply id="S3.I1.i2.p1.14.m14.1.1.cmml" xref="S3.I1.i2.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.14.m14.1.1.1.cmml" xref="S3.I1.i2.p1.14.m14.1.1">subscript</csymbol><apply id="S3.I1.i2.p1.14.m14.1.1.2.cmml" xref="S3.I1.i2.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.14.m14.1.1.2.1.cmml" xref="S3.I1.i2.p1.14.m14.1.1">superscript</csymbol><ci id="S3.I1.i2.p1.14.m14.1.1.2.2.cmml" xref="S3.I1.i2.p1.14.m14.1.1.2.2">𝜎</ci><ci id="S3.I1.i2.p1.14.m14.1.1.2.3.cmml" xref="S3.I1.i2.p1.14.m14.1.1.2.3">′</ci></apply><cn id="S3.I1.i2.p1.14.m14.1.1.3.cmml" type="integer" xref="S3.I1.i2.p1.14.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.14.m14.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.14.m14.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.15">, we also have that </span><math alttext="\rho^{\prime}\in\textsf{Play}_{G,\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.15.m15.2"><semantics id="S3.I1.i2.p1.15.m15.2a"><mrow id="S3.I1.i2.p1.15.m15.2.3" xref="S3.I1.i2.p1.15.m15.2.3.cmml"><msup id="S3.I1.i2.p1.15.m15.2.3.2" xref="S3.I1.i2.p1.15.m15.2.3.2.cmml"><mi id="S3.I1.i2.p1.15.m15.2.3.2.2" xref="S3.I1.i2.p1.15.m15.2.3.2.2.cmml">ρ</mi><mo id="S3.I1.i2.p1.15.m15.2.3.2.3" xref="S3.I1.i2.p1.15.m15.2.3.2.3.cmml">′</mo></msup><mo id="S3.I1.i2.p1.15.m15.2.3.1" xref="S3.I1.i2.p1.15.m15.2.3.1.cmml">∈</mo><msub id="S3.I1.i2.p1.15.m15.2.3.3" xref="S3.I1.i2.p1.15.m15.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I1.i2.p1.15.m15.2.3.3.2" xref="S3.I1.i2.p1.15.m15.2.3.3.2a.cmml">Play</mtext><mrow id="S3.I1.i2.p1.15.m15.2.2.2.2" xref="S3.I1.i2.p1.15.m15.2.2.2.3.cmml"><mi id="S3.I1.i2.p1.15.m15.1.1.1.1" xref="S3.I1.i2.p1.15.m15.1.1.1.1.cmml">G</mi><mo id="S3.I1.i2.p1.15.m15.2.2.2.2.2" xref="S3.I1.i2.p1.15.m15.2.2.2.3.cmml">,</mo><msubsup id="S3.I1.i2.p1.15.m15.2.2.2.2.1" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1.cmml"><mi id="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.2" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.2.cmml">σ</mi><mn id="S3.I1.i2.p1.15.m15.2.2.2.2.1.3" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1.3.cmml">0</mn><mo id="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.3" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.3.cmml">′</mo></msubsup></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.15.m15.2b"><apply id="S3.I1.i2.p1.15.m15.2.3.cmml" xref="S3.I1.i2.p1.15.m15.2.3"><in id="S3.I1.i2.p1.15.m15.2.3.1.cmml" xref="S3.I1.i2.p1.15.m15.2.3.1"></in><apply id="S3.I1.i2.p1.15.m15.2.3.2.cmml" xref="S3.I1.i2.p1.15.m15.2.3.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.15.m15.2.3.2.1.cmml" xref="S3.I1.i2.p1.15.m15.2.3.2">superscript</csymbol><ci id="S3.I1.i2.p1.15.m15.2.3.2.2.cmml" xref="S3.I1.i2.p1.15.m15.2.3.2.2">𝜌</ci><ci id="S3.I1.i2.p1.15.m15.2.3.2.3.cmml" xref="S3.I1.i2.p1.15.m15.2.3.2.3">′</ci></apply><apply id="S3.I1.i2.p1.15.m15.2.3.3.cmml" xref="S3.I1.i2.p1.15.m15.2.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.15.m15.2.3.3.1.cmml" xref="S3.I1.i2.p1.15.m15.2.3.3">subscript</csymbol><ci id="S3.I1.i2.p1.15.m15.2.3.3.2a.cmml" xref="S3.I1.i2.p1.15.m15.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I1.i2.p1.15.m15.2.3.3.2.cmml" xref="S3.I1.i2.p1.15.m15.2.3.3.2">Play</mtext></ci><list id="S3.I1.i2.p1.15.m15.2.2.2.3.cmml" xref="S3.I1.i2.p1.15.m15.2.2.2.2"><ci id="S3.I1.i2.p1.15.m15.1.1.1.1.cmml" xref="S3.I1.i2.p1.15.m15.1.1.1.1">𝐺</ci><apply id="S3.I1.i2.p1.15.m15.2.2.2.2.1.cmml" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.15.m15.2.2.2.2.1.1.cmml" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1">subscript</csymbol><apply id="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.cmml" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.1.cmml" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1">superscript</csymbol><ci id="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.2.cmml" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.2">𝜎</ci><ci id="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.3.cmml" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1.2.3">′</ci></apply><cn id="S3.I1.i2.p1.15.m15.2.2.2.2.1.3.cmml" type="integer" xref="S3.I1.i2.p1.15.m15.2.2.2.2.1.3">0</cn></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.15.m15.2c">\rho^{\prime}\in\textsf{Play}_{G,\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.15.m15.2d">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_G , italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.16">. Hence, by definition of </span><math alttext="C_{\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.16.m16.1"><semantics id="S3.I1.i2.p1.16.m16.1a"><msub id="S3.I1.i2.p1.16.m16.1.1" xref="S3.I1.i2.p1.16.m16.1.1.cmml"><mi id="S3.I1.i2.p1.16.m16.1.1.2" xref="S3.I1.i2.p1.16.m16.1.1.2.cmml">C</mi><msubsup id="S3.I1.i2.p1.16.m16.1.1.3" xref="S3.I1.i2.p1.16.m16.1.1.3.cmml"><mi id="S3.I1.i2.p1.16.m16.1.1.3.2.2" xref="S3.I1.i2.p1.16.m16.1.1.3.2.2.cmml">σ</mi><mn id="S3.I1.i2.p1.16.m16.1.1.3.2.3" xref="S3.I1.i2.p1.16.m16.1.1.3.2.3.cmml">0</mn><mo id="S3.I1.i2.p1.16.m16.1.1.3.3" xref="S3.I1.i2.p1.16.m16.1.1.3.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.16.m16.1b"><apply id="S3.I1.i2.p1.16.m16.1.1.cmml" xref="S3.I1.i2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.16.m16.1.1.1.cmml" xref="S3.I1.i2.p1.16.m16.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.16.m16.1.1.2.cmml" xref="S3.I1.i2.p1.16.m16.1.1.2">𝐶</ci><apply id="S3.I1.i2.p1.16.m16.1.1.3.cmml" xref="S3.I1.i2.p1.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.16.m16.1.1.3.1.cmml" xref="S3.I1.i2.p1.16.m16.1.1.3">superscript</csymbol><apply id="S3.I1.i2.p1.16.m16.1.1.3.2.cmml" xref="S3.I1.i2.p1.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.16.m16.1.1.3.2.1.cmml" xref="S3.I1.i2.p1.16.m16.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.16.m16.1.1.3.2.2.cmml" xref="S3.I1.i2.p1.16.m16.1.1.3.2.2">𝜎</ci><cn id="S3.I1.i2.p1.16.m16.1.1.3.2.3.cmml" type="integer" xref="S3.I1.i2.p1.16.m16.1.1.3.2.3">0</cn></apply><ci id="S3.I1.i2.p1.16.m16.1.1.3.3.cmml" xref="S3.I1.i2.p1.16.m16.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.16.m16.1c">C_{\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.16.m16.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.17">, there exists </span><math alttext="c^{\prime\prime}\in C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.17.m17.1"><semantics id="S3.I1.i2.p1.17.m17.1a"><mrow id="S3.I1.i2.p1.17.m17.1.1" xref="S3.I1.i2.p1.17.m17.1.1.cmml"><msup id="S3.I1.i2.p1.17.m17.1.1.2" xref="S3.I1.i2.p1.17.m17.1.1.2.cmml"><mi id="S3.I1.i2.p1.17.m17.1.1.2.2" xref="S3.I1.i2.p1.17.m17.1.1.2.2.cmml">c</mi><mo id="S3.I1.i2.p1.17.m17.1.1.2.3" xref="S3.I1.i2.p1.17.m17.1.1.2.3.cmml">′′</mo></msup><mo id="S3.I1.i2.p1.17.m17.1.1.1" xref="S3.I1.i2.p1.17.m17.1.1.1.cmml">∈</mo><msub id="S3.I1.i2.p1.17.m17.1.1.3" xref="S3.I1.i2.p1.17.m17.1.1.3.cmml"><mi id="S3.I1.i2.p1.17.m17.1.1.3.2" xref="S3.I1.i2.p1.17.m17.1.1.3.2.cmml">C</mi><msubsup id="S3.I1.i2.p1.17.m17.1.1.3.3" xref="S3.I1.i2.p1.17.m17.1.1.3.3.cmml"><mi id="S3.I1.i2.p1.17.m17.1.1.3.3.2.2" xref="S3.I1.i2.p1.17.m17.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.I1.i2.p1.17.m17.1.1.3.3.3" xref="S3.I1.i2.p1.17.m17.1.1.3.3.3.cmml">0</mn><mo id="S3.I1.i2.p1.17.m17.1.1.3.3.2.3" xref="S3.I1.i2.p1.17.m17.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.17.m17.1b"><apply id="S3.I1.i2.p1.17.m17.1.1.cmml" xref="S3.I1.i2.p1.17.m17.1.1"><in id="S3.I1.i2.p1.17.m17.1.1.1.cmml" xref="S3.I1.i2.p1.17.m17.1.1.1"></in><apply id="S3.I1.i2.p1.17.m17.1.1.2.cmml" xref="S3.I1.i2.p1.17.m17.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.17.m17.1.1.2.1.cmml" xref="S3.I1.i2.p1.17.m17.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.17.m17.1.1.2.2.cmml" xref="S3.I1.i2.p1.17.m17.1.1.2.2">𝑐</ci><ci id="S3.I1.i2.p1.17.m17.1.1.2.3.cmml" xref="S3.I1.i2.p1.17.m17.1.1.2.3">′′</ci></apply><apply id="S3.I1.i2.p1.17.m17.1.1.3.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.17.m17.1.1.3.1.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.17.m17.1.1.3.2.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3.2">𝐶</ci><apply id="S3.I1.i2.p1.17.m17.1.1.3.3.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.17.m17.1.1.3.3.1.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3.3">subscript</csymbol><apply id="S3.I1.i2.p1.17.m17.1.1.3.3.2.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.17.m17.1.1.3.3.2.1.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3.3">superscript</csymbol><ci id="S3.I1.i2.p1.17.m17.1.1.3.3.2.2.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3.3.2.2">𝜎</ci><ci id="S3.I1.i2.p1.17.m17.1.1.3.3.2.3.cmml" xref="S3.I1.i2.p1.17.m17.1.1.3.3.2.3">′</ci></apply><cn id="S3.I1.i2.p1.17.m17.1.1.3.3.3.cmml" type="integer" xref="S3.I1.i2.p1.17.m17.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.17.m17.1c">c^{\prime\prime}\in C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.17.m17.1d">italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.18"> such that </span><math alttext="c^{\prime\prime}\leq c^{\prime}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.18.m18.1"><semantics id="S3.I1.i2.p1.18.m18.1a"><mrow id="S3.I1.i2.p1.18.m18.1.1" xref="S3.I1.i2.p1.18.m18.1.1.cmml"><msup id="S3.I1.i2.p1.18.m18.1.1.2" xref="S3.I1.i2.p1.18.m18.1.1.2.cmml"><mi id="S3.I1.i2.p1.18.m18.1.1.2.2" xref="S3.I1.i2.p1.18.m18.1.1.2.2.cmml">c</mi><mo id="S3.I1.i2.p1.18.m18.1.1.2.3" xref="S3.I1.i2.p1.18.m18.1.1.2.3.cmml">′′</mo></msup><mo id="S3.I1.i2.p1.18.m18.1.1.1" xref="S3.I1.i2.p1.18.m18.1.1.1.cmml">≤</mo><msup id="S3.I1.i2.p1.18.m18.1.1.3" xref="S3.I1.i2.p1.18.m18.1.1.3.cmml"><mi id="S3.I1.i2.p1.18.m18.1.1.3.2" xref="S3.I1.i2.p1.18.m18.1.1.3.2.cmml">c</mi><mo id="S3.I1.i2.p1.18.m18.1.1.3.3" xref="S3.I1.i2.p1.18.m18.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.18.m18.1b"><apply id="S3.I1.i2.p1.18.m18.1.1.cmml" xref="S3.I1.i2.p1.18.m18.1.1"><leq id="S3.I1.i2.p1.18.m18.1.1.1.cmml" xref="S3.I1.i2.p1.18.m18.1.1.1"></leq><apply id="S3.I1.i2.p1.18.m18.1.1.2.cmml" xref="S3.I1.i2.p1.18.m18.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.18.m18.1.1.2.1.cmml" xref="S3.I1.i2.p1.18.m18.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.18.m18.1.1.2.2.cmml" xref="S3.I1.i2.p1.18.m18.1.1.2.2">𝑐</ci><ci id="S3.I1.i2.p1.18.m18.1.1.2.3.cmml" xref="S3.I1.i2.p1.18.m18.1.1.2.3">′′</ci></apply><apply id="S3.I1.i2.p1.18.m18.1.1.3.cmml" xref="S3.I1.i2.p1.18.m18.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.18.m18.1.1.3.1.cmml" xref="S3.I1.i2.p1.18.m18.1.1.3">superscript</csymbol><ci id="S3.I1.i2.p1.18.m18.1.1.3.2.cmml" xref="S3.I1.i2.p1.18.m18.1.1.3.2">𝑐</ci><ci id="S3.I1.i2.p1.18.m18.1.1.3.3.cmml" xref="S3.I1.i2.p1.18.m18.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.18.m18.1c">c^{\prime\prime}\leq c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.18.m18.1d">italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≤ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.19">, and thus </span><math alttext="c^{\prime\prime}\leq c" class="ltx_Math" display="inline" id="S3.I1.i2.p1.19.m19.1"><semantics id="S3.I1.i2.p1.19.m19.1a"><mrow id="S3.I1.i2.p1.19.m19.1.1" xref="S3.I1.i2.p1.19.m19.1.1.cmml"><msup id="S3.I1.i2.p1.19.m19.1.1.2" xref="S3.I1.i2.p1.19.m19.1.1.2.cmml"><mi id="S3.I1.i2.p1.19.m19.1.1.2.2" xref="S3.I1.i2.p1.19.m19.1.1.2.2.cmml">c</mi><mo id="S3.I1.i2.p1.19.m19.1.1.2.3" xref="S3.I1.i2.p1.19.m19.1.1.2.3.cmml">′′</mo></msup><mo id="S3.I1.i2.p1.19.m19.1.1.1" xref="S3.I1.i2.p1.19.m19.1.1.1.cmml">≤</mo><mi id="S3.I1.i2.p1.19.m19.1.1.3" xref="S3.I1.i2.p1.19.m19.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.19.m19.1b"><apply id="S3.I1.i2.p1.19.m19.1.1.cmml" xref="S3.I1.i2.p1.19.m19.1.1"><leq id="S3.I1.i2.p1.19.m19.1.1.1.cmml" xref="S3.I1.i2.p1.19.m19.1.1.1"></leq><apply id="S3.I1.i2.p1.19.m19.1.1.2.cmml" xref="S3.I1.i2.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.19.m19.1.1.2.1.cmml" xref="S3.I1.i2.p1.19.m19.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.19.m19.1.1.2.2.cmml" xref="S3.I1.i2.p1.19.m19.1.1.2.2">𝑐</ci><ci id="S3.I1.i2.p1.19.m19.1.1.2.3.cmml" xref="S3.I1.i2.p1.19.m19.1.1.2.3">′′</ci></apply><ci id="S3.I1.i2.p1.19.m19.1.1.3.cmml" xref="S3.I1.i2.p1.19.m19.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.19.m19.1c">c^{\prime\prime}\leq c</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.19.m19.1d">italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≤ italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.20">. We have thus proved that </span><math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.20.m20.1"><semantics id="S3.I1.i2.p1.20.m20.1a"><mrow id="S3.I1.i2.p1.20.m20.1.1" xref="S3.I1.i2.p1.20.m20.1.1.cmml"><msubsup id="S3.I1.i2.p1.20.m20.1.1.2" xref="S3.I1.i2.p1.20.m20.1.1.2.cmml"><mi id="S3.I1.i2.p1.20.m20.1.1.2.2.2" xref="S3.I1.i2.p1.20.m20.1.1.2.2.2.cmml">σ</mi><mn id="S3.I1.i2.p1.20.m20.1.1.2.2.3" xref="S3.I1.i2.p1.20.m20.1.1.2.2.3.cmml">0</mn><mo id="S3.I1.i2.p1.20.m20.1.1.2.3" xref="S3.I1.i2.p1.20.m20.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.I1.i2.p1.20.m20.1.1.1" xref="S3.I1.i2.p1.20.m20.1.1.1.cmml">⪯</mo><msub id="S3.I1.i2.p1.20.m20.1.1.3" xref="S3.I1.i2.p1.20.m20.1.1.3.cmml"><mi id="S3.I1.i2.p1.20.m20.1.1.3.2" xref="S3.I1.i2.p1.20.m20.1.1.3.2.cmml">σ</mi><mn id="S3.I1.i2.p1.20.m20.1.1.3.3" xref="S3.I1.i2.p1.20.m20.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.20.m20.1b"><apply id="S3.I1.i2.p1.20.m20.1.1.cmml" xref="S3.I1.i2.p1.20.m20.1.1"><csymbol cd="latexml" id="S3.I1.i2.p1.20.m20.1.1.1.cmml" xref="S3.I1.i2.p1.20.m20.1.1.1">precedes-or-equals</csymbol><apply id="S3.I1.i2.p1.20.m20.1.1.2.cmml" xref="S3.I1.i2.p1.20.m20.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.20.m20.1.1.2.1.cmml" xref="S3.I1.i2.p1.20.m20.1.1.2">superscript</csymbol><apply id="S3.I1.i2.p1.20.m20.1.1.2.2.cmml" xref="S3.I1.i2.p1.20.m20.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.20.m20.1.1.2.2.1.cmml" xref="S3.I1.i2.p1.20.m20.1.1.2">subscript</csymbol><ci id="S3.I1.i2.p1.20.m20.1.1.2.2.2.cmml" xref="S3.I1.i2.p1.20.m20.1.1.2.2.2">𝜎</ci><cn id="S3.I1.i2.p1.20.m20.1.1.2.2.3.cmml" type="integer" xref="S3.I1.i2.p1.20.m20.1.1.2.2.3">0</cn></apply><ci id="S3.I1.i2.p1.20.m20.1.1.2.3.cmml" xref="S3.I1.i2.p1.20.m20.1.1.2.3">′</ci></apply><apply id="S3.I1.i2.p1.20.m20.1.1.3.cmml" xref="S3.I1.i2.p1.20.m20.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.20.m20.1.1.3.1.cmml" xref="S3.I1.i2.p1.20.m20.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.20.m20.1.1.3.2.cmml" xref="S3.I1.i2.p1.20.m20.1.1.3.2">𝜎</ci><cn id="S3.I1.i2.p1.20.m20.1.1.3.3.cmml" type="integer" xref="S3.I1.i2.p1.20.m20.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.20.m20.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.20.m20.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.20.21">.</span></p> </div> </li> </ul> <p class="ltx_p" id="S3.Thmtheorem2.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.9.2">Notice that we have <math alttext="\sigma_{0}^{\prime}\prec\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.8.1.m1.1"><semantics id="S3.Thmtheorem2.p1.8.1.m1.1a"><mrow id="S3.Thmtheorem2.p1.8.1.m1.1.1" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.cmml"><msubsup id="S3.Thmtheorem2.p1.8.1.m1.1.1.2" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2.cmml"><mi id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.2" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.3" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.3" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem2.p1.8.1.m1.1.1.1" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.1.cmml">≺</mo><msub id="S3.Thmtheorem2.p1.8.1.m1.1.1.3" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.8.1.m1.1.1.3.2" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p1.8.1.m1.1.1.3.3" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.8.1.m1.1b"><apply id="S3.Thmtheorem2.p1.8.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1"><csymbol cd="latexml" id="S3.Thmtheorem2.p1.8.1.m1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.1">precedes</csymbol><apply id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2">superscript</csymbol><apply id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.2.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p1.8.1.m1.1.1.2.3.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p1.8.1.m1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.8.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.8.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p1.8.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.8.1.m1.1c">\sigma_{0}^{\prime}\prec\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.8.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, by the arguments of the second item, as <math alttext="\tau_{0}\prec\sigma_{0|h}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.9.2.m2.1"><semantics id="S3.Thmtheorem2.p1.9.2.m2.1a"><mrow id="S3.Thmtheorem2.p1.9.2.m2.1.1" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.cmml"><msub id="S3.Thmtheorem2.p1.9.2.m2.1.1.2" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.2.cmml"><mi id="S3.Thmtheorem2.p1.9.2.m2.1.1.2.2" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.2.2.cmml">τ</mi><mn id="S3.Thmtheorem2.p1.9.2.m2.1.1.2.3" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem2.p1.9.2.m2.1.1.1" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.1.cmml">≺</mo><msub id="S3.Thmtheorem2.p1.9.2.m2.1.1.3" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.2" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.2.cmml">σ</mi><mrow id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.cmml"><mn id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.2" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.1" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.1.cmml">|</mo><mi id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.3" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.9.2.m2.1b"><apply id="S3.Thmtheorem2.p1.9.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1"><csymbol cd="latexml" id="S3.Thmtheorem2.p1.9.2.m2.1.1.1.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.1">precedes</csymbol><apply id="S3.Thmtheorem2.p1.9.2.m2.1.1.2.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.9.2.m2.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem2.p1.9.2.m2.1.1.2.2.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.2.2">𝜏</ci><cn id="S3.Thmtheorem2.p1.9.2.m2.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.2.3">0</cn></apply><apply id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.2">𝜎</ci><apply id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3"><csymbol cd="latexml" id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.1">conditional</csymbol><cn id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.2">0</cn><ci id="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.3.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.9.2.m2.1c">\tau_{0}\prec\sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.9.2.m2.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p2"> <p class="ltx_p" id="S3.Thmtheorem2.p2.28"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p2.28.28">Let us now show that <math alttext="\sigma^{\prime}_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.1.1.m1.2"><semantics id="S3.Thmtheorem2.p2.1.1.m1.2a"><mrow id="S3.Thmtheorem2.p2.1.1.m1.2.3" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.cmml"><msubsup id="S3.Thmtheorem2.p2.1.1.m1.2.3.2" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2.cmml"><mi id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.2" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.3" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.3" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem2.p2.1.1.m1.2.3.1" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem2.p2.1.1.m1.2.3.3" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.2" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.2a.cmml">SPS</mtext><mo id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.1" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.2" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.1.cmml"><mo id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem2.p2.1.1.m1.1.1" xref="S3.Thmtheorem2.p2.1.1.m1.1.1.cmml">G</mi><mo id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.2.2" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem2.p2.1.1.m1.2.2" xref="S3.Thmtheorem2.p2.1.1.m1.2.2.cmml">B</mi><mo id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.1.1.m1.2b"><apply id="S3.Thmtheorem2.p2.1.1.m1.2.3.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3"><in id="S3.Thmtheorem2.p2.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.1"></in><apply id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.1.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2">subscript</csymbol><apply id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.1.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.2.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.2">𝜎</ci><ci id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.3.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2.2.3">′</ci></apply><cn id="S3.Thmtheorem2.p2.1.1.m1.2.3.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.2.3">0</cn></apply><apply id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3"><times id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.1"></times><ci id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.2a.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.2.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.1.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.3.3.3.2"><ci id="S3.Thmtheorem2.p2.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.1.1">𝐺</ci><ci id="S3.Thmtheorem2.p2.1.1.m1.2.2.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.1.1.m1.2c">\sigma^{\prime}_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.1.1.m1.2d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math>. Let <math alttext="\rho^{\prime}\in\textsf{Play}_{G,\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.2.2.m2.2"><semantics id="S3.Thmtheorem2.p2.2.2.m2.2a"><mrow id="S3.Thmtheorem2.p2.2.2.m2.2.3" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.cmml"><msup id="S3.Thmtheorem2.p2.2.2.m2.2.3.2" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.2.cmml"><mi id="S3.Thmtheorem2.p2.2.2.m2.2.3.2.2" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.2.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.2.2.m2.2.3.2.3" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.2.2.m2.2.3.1" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.2.2.m2.2.3.3" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.2.2.m2.2.3.3.2" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.3.2a.cmml">Play</mtext><mrow id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.3.cmml"><mi id="S3.Thmtheorem2.p2.2.2.m2.1.1.1.1" xref="S3.Thmtheorem2.p2.2.2.m2.1.1.1.1.cmml">G</mi><mo id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.2" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.3.cmml">,</mo><msubsup id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.cmml"><mi id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.2" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.3" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.3" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.3.cmml">′</mo></msubsup></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.2.2.m2.2b"><apply id="S3.Thmtheorem2.p2.2.2.m2.2.3.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3"><in id="S3.Thmtheorem2.p2.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.1"></in><apply id="S3.Thmtheorem2.p2.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.2.2.m2.2.3.2.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p2.2.2.m2.2.3.2.2.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.2.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.2.2.m2.2.3.2.3.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p2.2.2.m2.2.3.3.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.2.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.2.2.m2.2.3.3.2a.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.2.2.m2.2.3.3.2.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.3.3.2">Play</mtext></ci><list id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.3.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2"><ci id="S3.Thmtheorem2.p2.2.2.m2.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.1.1.1.1">𝐺</ci><apply id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1">superscript</csymbol><apply id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1">subscript</csymbol><ci id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.2.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.3.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.2.2.2.2.1.3">′</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.2.2.m2.2c">\rho^{\prime}\in\textsf{Play}_{G,\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.2.2.m2.2d">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_G , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be such that <math alttext="c^{\prime}=\textsf{cost}({\rho^{\prime}})\in C_{\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.3.3.m3.1"><semantics id="S3.Thmtheorem2.p2.3.3.m3.1a"><mrow id="S3.Thmtheorem2.p2.3.3.m3.1.1" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.cmml"><msup id="S3.Thmtheorem2.p2.3.3.m3.1.1.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.3.3.m3.1.1.3.2" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.3.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.4" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.4.cmml">=</mo><mrow id="S3.Thmtheorem2.p2.3.3.m3.1.1.1" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.3a.cmml">cost</mtext><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.2" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.2" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.5" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.5.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.3.3.m3.1.1.6" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.cmml"><mi id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.2" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.2.cmml">C</mi><msubsup id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.cmml"><mi id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.2" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.3.3.m3.1b"><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1"><and id="S3.Thmtheorem2.p2.3.3.m3.1.1a.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1"></and><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1b.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1"><eq id="S3.Thmtheorem2.p2.3.3.m3.1.1.4.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.4"></eq><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3.3">′</ci></apply><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1"><times id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.2"></times><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.3a.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.3">cost</mtext></ci><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1c.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1"><in id="S3.Thmtheorem2.p2.3.3.m3.1.1.5.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.5"></in><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem2.p2.3.3.m3.1.1.1.cmml" id="S3.Thmtheorem2.p2.3.3.m3.1.1d.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1"></share><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6">subscript</csymbol><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.2.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.2">𝐶</ci><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3">superscript</csymbol><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.2.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.3.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.6.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.3.3.m3.1c">c^{\prime}=\textsf{cost}({\rho^{\prime}})\in C_{\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.3.3.m3.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. We have to prove that <math alttext="\textsf{val}(\rho^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.4.4.m4.1"><semantics id="S3.Thmtheorem2.p2.4.4.m4.1a"><mrow id="S3.Thmtheorem2.p2.4.4.m4.1.1" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.cmml"><mrow id="S3.Thmtheorem2.p2.4.4.m4.1.1.1" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.3" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.3a.cmml">val</mtext><mo id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.2" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.2" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.3" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p2.4.4.m4.1.1.2" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.2.cmml">≤</mo><mi id="S3.Thmtheorem2.p2.4.4.m4.1.1.3" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.4.4.m4.1b"><apply id="S3.Thmtheorem2.p2.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1"><leq id="S3.Thmtheorem2.p2.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.2"></leq><apply id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1"><times id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.2"></times><ci id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.3a.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.3">val</mtext></ci><apply id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.Thmtheorem2.p2.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.4.4.m4.1c">\textsf{val}(\rho^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.4.4.m4.1d">val ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math>. If <math alttext="h\not\rho^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.5.5.m5.1"><semantics id="S3.Thmtheorem2.p2.5.5.m5.1a"><mrow id="S3.Thmtheorem2.p2.5.5.m5.1.1" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem2.p2.5.5.m5.1.1.2" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.2.cmml">h</mi><mo id="S3.Thmtheorem2.p2.5.5.m5.1.1.1" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.1.cmml">⁢</mo><msup id="S3.Thmtheorem2.p2.5.5.m5.1.1.3" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3.cmml"><mi class="ltx_mathvariant_italic" id="S3.Thmtheorem2.p2.5.5.m5.1.1.3.2" mathvariant="italic" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3.2.cmml">ρ̸</mi><mo id="S3.Thmtheorem2.p2.5.5.m5.1.1.3.3" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.5.5.m5.1b"><apply id="S3.Thmtheorem2.p2.5.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1"><times id="S3.Thmtheorem2.p2.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.1"></times><ci id="S3.Thmtheorem2.p2.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.2">ℎ</ci><apply id="S3.Thmtheorem2.p2.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p2.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3.2">italic-ρ̸</ci><ci id="S3.Thmtheorem2.p2.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.5.5.m5.1c">h\not\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.5.5.m5.1d">italic_h italic_ρ̸ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="\rho^{\prime}\in\textsf{Play}_{G,\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.6.6.m6.2"><semantics id="S3.Thmtheorem2.p2.6.6.m6.2a"><mrow id="S3.Thmtheorem2.p2.6.6.m6.2.3" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.cmml"><msup id="S3.Thmtheorem2.p2.6.6.m6.2.3.2" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.2.cmml"><mi id="S3.Thmtheorem2.p2.6.6.m6.2.3.2.2" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.2.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.6.6.m6.2.3.2.3" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.6.6.m6.2.3.1" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.6.6.m6.2.3.3" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.6.6.m6.2.3.3.2" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.3.2a.cmml">Play</mtext><mrow id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.3.cmml"><mi id="S3.Thmtheorem2.p2.6.6.m6.1.1.1.1" xref="S3.Thmtheorem2.p2.6.6.m6.1.1.1.1.cmml">G</mi><mo id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.2" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.3.cmml">,</mo><msub id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.cmml"><mi id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.2" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.3" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.3.cmml">0</mn></msub></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.6.6.m6.2b"><apply id="S3.Thmtheorem2.p2.6.6.m6.2.3.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3"><in id="S3.Thmtheorem2.p2.6.6.m6.2.3.1.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.1"></in><apply id="S3.Thmtheorem2.p2.6.6.m6.2.3.2.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.6.6.m6.2.3.2.1.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p2.6.6.m6.2.3.2.2.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.2.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.6.6.m6.2.3.2.3.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p2.6.6.m6.2.3.3.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.6.6.m6.2.3.3.1.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.6.6.m6.2.3.3.2a.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.6.6.m6.2.3.3.2.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.3.3.2">Play</mtext></ci><list id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.3.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2"><ci id="S3.Thmtheorem2.p2.6.6.m6.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.1.1.1.1">𝐺</ci><apply id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1">subscript</csymbol><ci id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.2.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.6.6.m6.2.2.2.2.1.3">0</cn></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.6.6.m6.2c">\rho^{\prime}\in\textsf{Play}_{G,\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.6.6.m6.2d">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_G , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, thus there exists <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.7.7.m7.1"><semantics id="S3.Thmtheorem2.p2.7.7.m7.1a"><mrow id="S3.Thmtheorem2.p2.7.7.m7.1.1" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem2.p2.7.7.m7.1.1.2" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.7.7.m7.1.1.1" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.7.7.m7.1.1.3" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.2" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.cmml"><mi id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.2" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.3" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.7.7.m7.1b"><apply id="S3.Thmtheorem2.p2.7.7.m7.1.1.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1"><in id="S3.Thmtheorem2.p2.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.1"></in><ci id="S3.Thmtheorem2.p2.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.2">𝑐</ci><apply id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.2.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.7.7.m7.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.7.7.m7.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c\leq c^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.8.8.m8.1"><semantics id="S3.Thmtheorem2.p2.8.8.m8.1a"><mrow id="S3.Thmtheorem2.p2.8.8.m8.1.1" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem2.p2.8.8.m8.1.1.2" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.8.8.m8.1.1.1" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.1.cmml">≤</mo><msup id="S3.Thmtheorem2.p2.8.8.m8.1.1.3" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.8.8.m8.1.1.3.2" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.3.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.8.8.m8.1.1.3.3" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.8.8.m8.1b"><apply id="S3.Thmtheorem2.p2.8.8.m8.1.1.cmml" xref="S3.Thmtheorem2.p2.8.8.m8.1.1"><leq id="S3.Thmtheorem2.p2.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.1"></leq><ci id="S3.Thmtheorem2.p2.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.2">𝑐</ci><apply id="S3.Thmtheorem2.p2.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.8.8.m8.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p2.8.8.m8.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.3.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.8.8.m8.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.8.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.8.8.m8.1c">c\leq c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.8.8.m8.1d">italic_c ≤ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> by definition of <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.9.9.m9.1"><semantics id="S3.Thmtheorem2.p2.9.9.m9.1a"><msub id="S3.Thmtheorem2.p2.9.9.m9.1.1" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.cmml"><mi id="S3.Thmtheorem2.p2.9.9.m9.1.1.2" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.2.cmml">C</mi><msub id="S3.Thmtheorem2.p2.9.9.m9.1.1.3" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.9.9.m9.1.1.3.2" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.9.9.m9.1.1.3.3" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.9.9.m9.1b"><apply id="S3.Thmtheorem2.p2.9.9.m9.1.1.cmml" xref="S3.Thmtheorem2.p2.9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.9.9.m9.1.1.1.cmml" xref="S3.Thmtheorem2.p2.9.9.m9.1.1">subscript</csymbol><ci id="S3.Thmtheorem2.p2.9.9.m9.1.1.2.cmml" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.2">𝐶</ci><apply id="S3.Thmtheorem2.p2.9.9.m9.1.1.3.cmml" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.9.9.m9.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.9.9.m9.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.9.9.m9.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.9.9.m9.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.9.9.m9.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.9.9.m9.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.10.10.m10.1"><semantics id="S3.Thmtheorem2.p2.10.10.m10.1a"><mrow id="S3.Thmtheorem2.p2.10.10.m10.1.1" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.cmml"><msubsup id="S3.Thmtheorem2.p2.10.10.m10.1.1.2" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2.cmml"><mi id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.2" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.3" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.3" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem2.p2.10.10.m10.1.1.1" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.1.cmml">⪯</mo><msub id="S3.Thmtheorem2.p2.10.10.m10.1.1.3" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.10.10.m10.1.1.3.2" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.10.10.m10.1.1.3.3" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.10.10.m10.1b"><apply id="S3.Thmtheorem2.p2.10.10.m10.1.1.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1"><csymbol cd="latexml" id="S3.Thmtheorem2.p2.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.1">precedes-or-equals</csymbol><apply id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.1.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2">superscript</csymbol><apply id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.1.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.2.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p2.10.10.m10.1.1.2.3.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p2.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.10.10.m10.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.10.10.m10.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.10.10.m10.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.10.10.m10.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.10.10.m10.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.10.10.m10.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> by the first part of the proof, it follows that <math alttext="c=c^{\prime}=\textsf{cost}({\rho^{\prime}})\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.11.11.m11.1"><semantics id="S3.Thmtheorem2.p2.11.11.m11.1a"><mrow id="S3.Thmtheorem2.p2.11.11.m11.1.1" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.cmml"><mi id="S3.Thmtheorem2.p2.11.11.m11.1.1.3" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.3.cmml">c</mi><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.4" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.4.cmml">=</mo><msup id="S3.Thmtheorem2.p2.11.11.m11.1.1.5" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.5.cmml"><mi id="S3.Thmtheorem2.p2.11.11.m11.1.1.5.2" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.5.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.5.3" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.5.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.6" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.6.cmml">=</mo><mrow id="S3.Thmtheorem2.p2.11.11.m11.1.1.1" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.3" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.3a.cmml">cost</mtext><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.2" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.2" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.3" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p2.11.11.m11.1.1.7" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.7.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.11.11.m11.1.1.8" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.cmml"><mi id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.2" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.2.cmml">C</mi><msub id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.cmml"><mi id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.2" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.3" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.11.11.m11.1b"><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1"><and id="S3.Thmtheorem2.p2.11.11.m11.1.1a.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1"></and><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1b.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1"><eq id="S3.Thmtheorem2.p2.11.11.m11.1.1.4.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.4"></eq><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.3.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.3">𝑐</ci><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1.5.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.5"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.11.11.m11.1.1.5.1.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.5">superscript</csymbol><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.5.2.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.5.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.5.3.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.5.3">′</ci></apply></apply><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1c.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1"><eq id="S3.Thmtheorem2.p2.11.11.m11.1.1.6.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.6"></eq><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem2.p2.11.11.m11.1.1.5.cmml" id="S3.Thmtheorem2.p2.11.11.m11.1.1d.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1"></share><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1"><times id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.2"></times><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.3a.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.3">cost</mtext></ci><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1e.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1"><in id="S3.Thmtheorem2.p2.11.11.m11.1.1.7.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.7"></in><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem2.p2.11.11.m11.1.1.1.cmml" id="S3.Thmtheorem2.p2.11.11.m11.1.1f.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1"></share><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.1.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8">subscript</csymbol><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.2.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.2">𝐶</ci><apply id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.1.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.2.cmml" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.11.11.m11.1.1.8.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.11.11.m11.1c">c=c^{\prime}=\textsf{cost}({\rho^{\prime}})\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.11.11.m11.1d">italic_c = italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Now, recall that <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.12.12.m12.1"><semantics id="S3.Thmtheorem2.p2.12.12.m12.1a"><msub id="S3.Thmtheorem2.p2.12.12.m12.1.1" xref="S3.Thmtheorem2.p2.12.12.m12.1.1.cmml"><mi id="S3.Thmtheorem2.p2.12.12.m12.1.1.2" xref="S3.Thmtheorem2.p2.12.12.m12.1.1.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.12.12.m12.1.1.3" xref="S3.Thmtheorem2.p2.12.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.12.12.m12.1b"><apply id="S3.Thmtheorem2.p2.12.12.m12.1.1.cmml" xref="S3.Thmtheorem2.p2.12.12.m12.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.12.12.m12.1.1.1.cmml" xref="S3.Thmtheorem2.p2.12.12.m12.1.1">subscript</csymbol><ci id="S3.Thmtheorem2.p2.12.12.m12.1.1.2.cmml" xref="S3.Thmtheorem2.p2.12.12.m12.1.1.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.12.12.m12.1.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.12.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.12.12.m12.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.12.12.m12.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution in SPS(<math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.13.13.m13.1"><semantics id="S3.Thmtheorem2.p2.13.13.m13.1a"><mi id="S3.Thmtheorem2.p2.13.13.m13.1.1" xref="S3.Thmtheorem2.p2.13.13.m13.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.13.13.m13.1b"><ci id="S3.Thmtheorem2.p2.13.13.m13.1.1.cmml" xref="S3.Thmtheorem2.p2.13.13.m13.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.13.13.m13.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.13.13.m13.1d">italic_G</annotation></semantics></math>,<math alttext="B" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.14.14.m14.1"><semantics id="S3.Thmtheorem2.p2.14.14.m14.1a"><mi id="S3.Thmtheorem2.p2.14.14.m14.1.1" xref="S3.Thmtheorem2.p2.14.14.m14.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.14.14.m14.1b"><ci id="S3.Thmtheorem2.p2.14.14.m14.1.1.cmml" xref="S3.Thmtheorem2.p2.14.14.m14.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.14.14.m14.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.14.14.m14.1d">italic_B</annotation></semantics></math>), implying that <math alttext="\textsf{val}(\rho^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.15.15.m15.1"><semantics id="S3.Thmtheorem2.p2.15.15.m15.1a"><mrow id="S3.Thmtheorem2.p2.15.15.m15.1.1" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.cmml"><mrow id="S3.Thmtheorem2.p2.15.15.m15.1.1.1" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.3" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.3a.cmml">val</mtext><mo id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.2" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.2" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.3" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p2.15.15.m15.1.1.2" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.2.cmml">≤</mo><mi id="S3.Thmtheorem2.p2.15.15.m15.1.1.3" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.15.15.m15.1b"><apply id="S3.Thmtheorem2.p2.15.15.m15.1.1.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1"><leq id="S3.Thmtheorem2.p2.15.15.m15.1.1.2.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.2"></leq><apply id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1"><times id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.2"></times><ci id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.3a.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.3">val</mtext></ci><apply id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.Thmtheorem2.p2.15.15.m15.1.1.3.cmml" xref="S3.Thmtheorem2.p2.15.15.m15.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.15.15.m15.1c">\textsf{val}(\rho^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.15.15.m15.1d">val ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math>. If <math alttext="h\rho^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.16.16.m16.1"><semantics id="S3.Thmtheorem2.p2.16.16.m16.1a"><mrow id="S3.Thmtheorem2.p2.16.16.m16.1.1" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.cmml"><mi id="S3.Thmtheorem2.p2.16.16.m16.1.1.2" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.2.cmml">h</mi><mo id="S3.Thmtheorem2.p2.16.16.m16.1.1.1" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.1.cmml">⁢</mo><msup id="S3.Thmtheorem2.p2.16.16.m16.1.1.3" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.16.16.m16.1.1.3.2" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.3.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.16.16.m16.1.1.3.3" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.16.16.m16.1b"><apply id="S3.Thmtheorem2.p2.16.16.m16.1.1.cmml" xref="S3.Thmtheorem2.p2.16.16.m16.1.1"><times id="S3.Thmtheorem2.p2.16.16.m16.1.1.1.cmml" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.1"></times><ci id="S3.Thmtheorem2.p2.16.16.m16.1.1.2.cmml" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.2">ℎ</ci><apply id="S3.Thmtheorem2.p2.16.16.m16.1.1.3.cmml" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.16.16.m16.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p2.16.16.m16.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.3.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.16.16.m16.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.16.16.m16.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.16.16.m16.1c">h\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.16.16.m16.1d">italic_h italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="\rho^{\prime}\in\textsf{Play}_{G_{|h},\tau_{0}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p2.17.17.m17.2"><semantics id="S3.Thmtheorem2.p2.17.17.m17.2a"><mrow id="S3.Thmtheorem2.p2.17.17.m17.2.3"><msup id="S3.Thmtheorem2.p2.17.17.m17.2.3.2"><mi id="S3.Thmtheorem2.p2.17.17.m17.2.3.2.2">ρ</mi><mo id="S3.Thmtheorem2.p2.17.17.m17.2.3.2.3">′</mo></msup><mo id="S3.Thmtheorem2.p2.17.17.m17.2.3.1">∈</mo><msub id="S3.Thmtheorem2.p2.17.17.m17.2.3.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.17.17.m17.2.3.3.2">Play</mtext><mrow id="S3.Thmtheorem2.p2.17.17.m17.2.2.2.2"><msub id="S3.Thmtheorem2.p2.17.17.m17.1.1.1.1.1"><mi id="S3.Thmtheorem2.p2.17.17.m17.1.1.1.1.1.2">G</mi><mrow id="S3.Thmtheorem2.p2.17.17.m17.1.1.1.1.1.3"><mo fence="false" id="S3.Thmtheorem2.p2.17.17.m17.1.1.1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p2.17.17.m17.1.1.1.1.1.3.2">h</mi></mrow></msub><mo id="S3.Thmtheorem2.p2.17.17.m17.2.2.2.2.3">,</mo><msub id="S3.Thmtheorem2.p2.17.17.m17.2.2.2.2.2"><mi id="S3.Thmtheorem2.p2.17.17.m17.2.2.2.2.2.2">τ</mi><mn id="S3.Thmtheorem2.p2.17.17.m17.2.2.2.2.2.3">0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.17.17.m17.2b">\rho^{\prime}\in\textsf{Play}_{G_{|h},\tau_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.17.17.m17.2c">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. As <math alttext="c^{\prime}\in C_{\sigma_{0}^{\prime}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.18.18.m18.1"><semantics id="S3.Thmtheorem2.p2.18.18.m18.1a"><mrow id="S3.Thmtheorem2.p2.18.18.m18.1.1" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.cmml"><msup id="S3.Thmtheorem2.p2.18.18.m18.1.1.2" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.2.cmml"><mi id="S3.Thmtheorem2.p2.18.18.m18.1.1.2.2" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.18.18.m18.1.1.2.3" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.18.18.m18.1.1.1" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.18.18.m18.1.1.3" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.2" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.2.cmml">C</mi><msubsup id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.cmml"><mi id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.2" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.3" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.3" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.18.18.m18.1b"><apply id="S3.Thmtheorem2.p2.18.18.m18.1.1.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1"><in id="S3.Thmtheorem2.p2.18.18.m18.1.1.1.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.1"></in><apply id="S3.Thmtheorem2.p2.18.18.m18.1.1.2.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.18.18.m18.1.1.2.1.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem2.p2.18.18.m18.1.1.2.2.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.18.18.m18.1.1.2.3.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3">superscript</csymbol><apply id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.3.cmml" xref="S3.Thmtheorem2.p2.18.18.m18.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.18.18.m18.1c">c^{\prime}\in C_{\sigma_{0}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.18.18.m18.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, we have <math alttext="c^{\prime}\in C_{\tau_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.19.19.m19.1"><semantics id="S3.Thmtheorem2.p2.19.19.m19.1a"><mrow id="S3.Thmtheorem2.p2.19.19.m19.1.1" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.cmml"><msup id="S3.Thmtheorem2.p2.19.19.m19.1.1.2" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.2.cmml"><mi id="S3.Thmtheorem2.p2.19.19.m19.1.1.2.2" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.19.19.m19.1.1.2.3" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.19.19.m19.1.1.1" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.19.19.m19.1.1.3" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.2" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.cmml"><mi id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.2" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.2.cmml">τ</mi><mn id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.3" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.19.19.m19.1b"><apply id="S3.Thmtheorem2.p2.19.19.m19.1.1.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1"><in id="S3.Thmtheorem2.p2.19.19.m19.1.1.1.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.1"></in><apply id="S3.Thmtheorem2.p2.19.19.m19.1.1.2.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.19.19.m19.1.1.2.1.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem2.p2.19.19.m19.1.1.2.2.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.19.19.m19.1.1.2.3.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.2.cmml" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.2">𝜏</ci><cn id="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.19.19.m19.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.19.19.m19.1c">c^{\prime}\in C_{\tau_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.19.19.m19.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> (it is not possible to have <math alttext="c^{\prime\prime}\in C_{\tau_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.20.20.m20.1"><semantics id="S3.Thmtheorem2.p2.20.20.m20.1a"><mrow id="S3.Thmtheorem2.p2.20.20.m20.1.1" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.cmml"><msup id="S3.Thmtheorem2.p2.20.20.m20.1.1.2" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.2.cmml"><mi id="S3.Thmtheorem2.p2.20.20.m20.1.1.2.2" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.20.20.m20.1.1.2.3" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.2.3.cmml">′′</mo></msup><mo id="S3.Thmtheorem2.p2.20.20.m20.1.1.1" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem2.p2.20.20.m20.1.1.3" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.2" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.cmml"><mi id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.2" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.2.cmml">τ</mi><mn id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.3" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.20.20.m20.1b"><apply id="S3.Thmtheorem2.p2.20.20.m20.1.1.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1"><in id="S3.Thmtheorem2.p2.20.20.m20.1.1.1.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.1"></in><apply id="S3.Thmtheorem2.p2.20.20.m20.1.1.2.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.20.20.m20.1.1.2.1.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem2.p2.20.20.m20.1.1.2.2.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.20.20.m20.1.1.2.3.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.2.3">′′</ci></apply><apply id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.1.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.2.cmml" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.2">𝜏</ci><cn id="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.20.20.m20.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.20.20.m20.1c">c^{\prime\prime}\in C_{\tau_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.20.20.m20.1d">italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c^{\prime\prime}<c^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.21.21.m21.1"><semantics id="S3.Thmtheorem2.p2.21.21.m21.1a"><mrow id="S3.Thmtheorem2.p2.21.21.m21.1.1" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.cmml"><msup id="S3.Thmtheorem2.p2.21.21.m21.1.1.2" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.2.cmml"><mi id="S3.Thmtheorem2.p2.21.21.m21.1.1.2.2" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.21.21.m21.1.1.2.3" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.2.3.cmml">′′</mo></msup><mo id="S3.Thmtheorem2.p2.21.21.m21.1.1.1" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.1.cmml"><</mo><msup id="S3.Thmtheorem2.p2.21.21.m21.1.1.3" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.3.cmml"><mi id="S3.Thmtheorem2.p2.21.21.m21.1.1.3.2" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.3.2.cmml">c</mi><mo id="S3.Thmtheorem2.p2.21.21.m21.1.1.3.3" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.21.21.m21.1b"><apply id="S3.Thmtheorem2.p2.21.21.m21.1.1.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1"><lt id="S3.Thmtheorem2.p2.21.21.m21.1.1.1.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.1"></lt><apply id="S3.Thmtheorem2.p2.21.21.m21.1.1.2.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.21.21.m21.1.1.2.1.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem2.p2.21.21.m21.1.1.2.2.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.21.21.m21.1.1.2.3.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.2.3">′′</ci></apply><apply id="S3.Thmtheorem2.p2.21.21.m21.1.1.3.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.21.21.m21.1.1.3.1.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p2.21.21.m21.1.1.3.2.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.3.2">𝑐</ci><ci id="S3.Thmtheorem2.p2.21.21.m21.1.1.3.3.cmml" xref="S3.Thmtheorem2.p2.21.21.m21.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.21.21.m21.1c">c^{\prime\prime}<c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.21.21.m21.1d">italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> by definition of <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.22.22.m22.1"><semantics id="S3.Thmtheorem2.p2.22.22.m22.1a"><msubsup id="S3.Thmtheorem2.p2.22.22.m22.1.1" xref="S3.Thmtheorem2.p2.22.22.m22.1.1.cmml"><mi id="S3.Thmtheorem2.p2.22.22.m22.1.1.2.2" xref="S3.Thmtheorem2.p2.22.22.m22.1.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.22.22.m22.1.1.3" xref="S3.Thmtheorem2.p2.22.22.m22.1.1.3.cmml">0</mn><mo id="S3.Thmtheorem2.p2.22.22.m22.1.1.2.3" xref="S3.Thmtheorem2.p2.22.22.m22.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.22.22.m22.1b"><apply id="S3.Thmtheorem2.p2.22.22.m22.1.1.cmml" xref="S3.Thmtheorem2.p2.22.22.m22.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.22.22.m22.1.1.1.cmml" xref="S3.Thmtheorem2.p2.22.22.m22.1.1">subscript</csymbol><apply id="S3.Thmtheorem2.p2.22.22.m22.1.1.2.cmml" xref="S3.Thmtheorem2.p2.22.22.m22.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.22.22.m22.1.1.2.1.cmml" xref="S3.Thmtheorem2.p2.22.22.m22.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.22.22.m22.1.1.2.2.cmml" xref="S3.Thmtheorem2.p2.22.22.m22.1.1.2.2">𝜎</ci><ci id="S3.Thmtheorem2.p2.22.22.m22.1.1.2.3.cmml" xref="S3.Thmtheorem2.p2.22.22.m22.1.1.2.3">′</ci></apply><cn id="S3.Thmtheorem2.p2.22.22.m22.1.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.22.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.22.22.m22.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.22.22.m22.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>). As <math alttext="\tau_{0}\in\mbox{SPS}({G_{|h}},{B})" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p2.23.23.m23.2"><semantics id="S3.Thmtheorem2.p2.23.23.m23.2a"><mrow id="S3.Thmtheorem2.p2.23.23.m23.2.2"><msub id="S3.Thmtheorem2.p2.23.23.m23.2.2.3"><mi id="S3.Thmtheorem2.p2.23.23.m23.2.2.3.2">τ</mi><mn id="S3.Thmtheorem2.p2.23.23.m23.2.2.3.3">0</mn></msub><mo id="S3.Thmtheorem2.p2.23.23.m23.2.2.2">∈</mo><mrow id="S3.Thmtheorem2.p2.23.23.m23.2.2.1"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.3">SPS</mtext><mo id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.2">⁢</mo><mrow id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1"><mo id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.2" stretchy="false">(</mo><msub id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.1"><mi id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.1.2">G</mi><mrow id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.1.3"><mo fence="false" id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.1.3.2">h</mi></mrow></msub><mo id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.3">,</mo><mi id="S3.Thmtheorem2.p2.23.23.m23.1.1">B</mi><mo id="S3.Thmtheorem2.p2.23.23.m23.2.2.1.1.1.4" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.23.23.m23.2b">\tau_{0}\in\mbox{SPS}({G_{|h}},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.23.23.m23.2c">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT , italic_B )</annotation></semantics></math>, it follows that <math alttext="\textsf{val}(\rho^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.24.24.m24.1"><semantics id="S3.Thmtheorem2.p2.24.24.m24.1a"><mrow id="S3.Thmtheorem2.p2.24.24.m24.1.1" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.cmml"><mrow id="S3.Thmtheorem2.p2.24.24.m24.1.1.1" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.3" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.3a.cmml">val</mtext><mo id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.2" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.2" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.3" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p2.24.24.m24.1.1.2" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.2.cmml">≤</mo><mi id="S3.Thmtheorem2.p2.24.24.m24.1.1.3" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.24.24.m24.1b"><apply id="S3.Thmtheorem2.p2.24.24.m24.1.1.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1"><leq id="S3.Thmtheorem2.p2.24.24.m24.1.1.2.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.2"></leq><apply id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1"><times id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.2"></times><ci id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.3a.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.3">val</mtext></ci><apply id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.Thmtheorem2.p2.24.24.m24.1.1.3.cmml" xref="S3.Thmtheorem2.p2.24.24.m24.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.24.24.m24.1c">\textsf{val}(\rho^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.24.24.m24.1d">val ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math>. In any case, <math alttext="\textsf{val}(\rho^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.25.25.m25.1"><semantics id="S3.Thmtheorem2.p2.25.25.m25.1a"><mrow id="S3.Thmtheorem2.p2.25.25.m25.1.1" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.cmml"><mrow id="S3.Thmtheorem2.p2.25.25.m25.1.1.1" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.3" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.3a.cmml">val</mtext><mo id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.2" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.2" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.3" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p2.25.25.m25.1.1.2" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.2.cmml">≤</mo><mi id="S3.Thmtheorem2.p2.25.25.m25.1.1.3" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.25.25.m25.1b"><apply id="S3.Thmtheorem2.p2.25.25.m25.1.1.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1"><leq id="S3.Thmtheorem2.p2.25.25.m25.1.1.2.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.2"></leq><apply id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1"><times id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.2"></times><ci id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.3a.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.3">val</mtext></ci><apply id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.2">𝜌</ci><ci id="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.Thmtheorem2.p2.25.25.m25.1.1.3.cmml" xref="S3.Thmtheorem2.p2.25.25.m25.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.25.25.m25.1c">\textsf{val}(\rho^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.25.25.m25.1d">val ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math> showing that <math alttext="\sigma_{0}^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.26.26.m26.1"><semantics id="S3.Thmtheorem2.p2.26.26.m26.1a"><msubsup id="S3.Thmtheorem2.p2.26.26.m26.1.1" xref="S3.Thmtheorem2.p2.26.26.m26.1.1.cmml"><mi id="S3.Thmtheorem2.p2.26.26.m26.1.1.2.2" xref="S3.Thmtheorem2.p2.26.26.m26.1.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem2.p2.26.26.m26.1.1.2.3" xref="S3.Thmtheorem2.p2.26.26.m26.1.1.2.3.cmml">0</mn><mo id="S3.Thmtheorem2.p2.26.26.m26.1.1.3" xref="S3.Thmtheorem2.p2.26.26.m26.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.26.26.m26.1b"><apply id="S3.Thmtheorem2.p2.26.26.m26.1.1.cmml" xref="S3.Thmtheorem2.p2.26.26.m26.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.26.26.m26.1.1.1.cmml" xref="S3.Thmtheorem2.p2.26.26.m26.1.1">superscript</csymbol><apply id="S3.Thmtheorem2.p2.26.26.m26.1.1.2.cmml" xref="S3.Thmtheorem2.p2.26.26.m26.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.26.26.m26.1.1.2.1.cmml" xref="S3.Thmtheorem2.p2.26.26.m26.1.1">subscript</csymbol><ci id="S3.Thmtheorem2.p2.26.26.m26.1.1.2.2.cmml" xref="S3.Thmtheorem2.p2.26.26.m26.1.1.2.2">𝜎</ci><cn id="S3.Thmtheorem2.p2.26.26.m26.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem2.p2.26.26.m26.1.1.2.3">0</cn></apply><ci id="S3.Thmtheorem2.p2.26.26.m26.1.1.3.cmml" xref="S3.Thmtheorem2.p2.26.26.m26.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.26.26.m26.1c">\sigma_{0}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.26.26.m26.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a solution to the SPS problem in SPS(<math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.27.27.m27.1"><semantics id="S3.Thmtheorem2.p2.27.27.m27.1a"><mi id="S3.Thmtheorem2.p2.27.27.m27.1.1" xref="S3.Thmtheorem2.p2.27.27.m27.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.27.27.m27.1b"><ci id="S3.Thmtheorem2.p2.27.27.m27.1.1.cmml" xref="S3.Thmtheorem2.p2.27.27.m27.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.27.27.m27.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.27.27.m27.1d">italic_G</annotation></semantics></math>,<math alttext="B" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.28.28.m28.1"><semantics id="S3.Thmtheorem2.p2.28.28.m28.1a"><mi id="S3.Thmtheorem2.p2.28.28.m28.1.1" xref="S3.Thmtheorem2.p2.28.28.m28.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.28.28.m28.1b"><ci id="S3.Thmtheorem2.p2.28.28.m28.1.1.cmml" xref="S3.Thmtheorem2.p2.28.28.m28.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.28.28.m28.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.28.28.m28.1d">italic_B</annotation></semantics></math>).</span></p> </div> </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>Eliminating a Cycle in a Witness</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">Another way to improve solutions to the SPS problem is to delete some particular cycles occurring in witnesses as explained in the next lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.1.1.1">Lemma 3.3</span></span></h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.5.5">Let <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.1.m1.1a"><mi id="S3.Thmtheorem3.p1.1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.1.1.m1.1b"><ci id="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be an SP game, <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.2.m2.1"><semantics id="S3.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem3.p1.2.2.m2.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">B</mi><mo id="S3.Thmtheorem3.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.2.2.m2.1b"><apply id="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1"><in id="S3.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2">𝐵</ci><ci id="S3.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.2.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.3.3.m3.2"><semantics id="S3.Thmtheorem3.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem3.p1.3.3.m3.2.3" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.cmml"><msub id="S3.Thmtheorem3.p1.3.3.m3.2.3.2" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.2.cmml"><mi id="S3.Thmtheorem3.p1.3.3.m3.2.3.2.2" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.3.3.m3.2.3.2.3" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem3.p1.3.3.m3.2.3.1" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem3.p1.3.3.m3.2.3.3" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.2" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.2a.cmml">SPS</mtext><mo id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.1" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.2" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.1.cmml"><mo id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem3.p1.3.3.m3.1.1" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml">G</mi><mo id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.2.2" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem3.p1.3.3.m3.2.2" xref="S3.Thmtheorem3.p1.3.3.m3.2.2.cmml">B</mi><mo id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.2.3" stretchy="false" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.3.3.m3.2b"><apply id="S3.Thmtheorem3.p1.3.3.m3.2.3.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3"><in id="S3.Thmtheorem3.p1.3.3.m3.2.3.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.1"></in><apply id="S3.Thmtheorem3.p1.3.3.m3.2.3.2.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.3.3.m3.2.3.2.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem3.p1.3.3.m3.2.3.2.2.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.2.2">𝜎</ci><cn id="S3.Thmtheorem3.p1.3.3.m3.2.3.2.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.2.3">0</cn></apply><apply id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3"><times id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.1"></times><ci id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.2a.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.2.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.3.3.3.2"><ci id="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1">𝐺</ci><ci id="S3.Thmtheorem3.p1.3.3.m3.2.2.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.3.3.m3.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.3.3.m3.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. Suppose that in a witness <math alttext="\rho=\rho_{0}\rho_{1}\ldots\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.4.4.m4.1"><semantics id="S3.Thmtheorem3.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem3.p1.4.4.m4.1.1" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.2.cmml">ρ</mi><mo id="S3.Thmtheorem3.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.3.cmml">=</mo><mrow id="S3.Thmtheorem3.p1.4.4.m4.1.1.4" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.cmml"><msub id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.cmml"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.2" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.2.cmml">ρ</mi><mn id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.3" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.1" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.1.cmml">⁢</mo><msub id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.cmml"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.2" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.2.cmml">ρ</mi><mn id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.3" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.3.cmml">1</mn></msub><mo id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.1a" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.1.cmml">⁢</mo><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.4" mathvariant="normal" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.4.cmml">…</mi></mrow><mo id="S3.Thmtheorem3.p1.4.4.m4.1.1.5" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.5.cmml">∈</mo><msub id="S3.Thmtheorem3.p1.4.4.m4.1.1.6" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.2" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.2a.cmml">Wit</mtext><msub id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.cmml"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.2" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.3" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.4.4.m4.1b"><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1"><and id="S3.Thmtheorem3.p1.4.4.m4.1.1a.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1"></and><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1b.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1"><eq id="S3.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.3"></eq><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.2">𝜌</ci><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4"><times id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.1"></times><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2">subscript</csymbol><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.2.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.2">𝜌</ci><cn id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.2.3">0</cn></apply><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3">subscript</csymbol><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.2.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.2">𝜌</ci><cn id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.3.3">1</cn></apply><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.4.4.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.4.4">…</ci></apply></apply><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1c.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1"><in id="S3.Thmtheorem3.p1.4.4.m4.1.1.5.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.5"></in><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3.p1.4.4.m4.1.1.4.cmml" id="S3.Thmtheorem3.p1.4.4.m4.1.1d.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1"></share><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6">subscript</csymbol><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.2a.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.2.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.2">Wit</mtext></ci><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3">subscript</csymbol><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.2.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.2">𝜎</ci><cn id="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.6.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.4.4.m4.1c">\rho=\rho_{0}\rho_{1}\ldots\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.4.4.m4.1d">italic_ρ = italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, there exist <math alttext="m,n\in{\rm Nature}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.5.5.m5.2"><semantics id="S3.Thmtheorem3.p1.5.5.m5.2a"><mrow id="S3.Thmtheorem3.p1.5.5.m5.2.3" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.cmml"><mrow id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.2" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.1.cmml"><mi id="S3.Thmtheorem3.p1.5.5.m5.1.1" xref="S3.Thmtheorem3.p1.5.5.m5.1.1.cmml">m</mi><mo id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.2.1" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem3.p1.5.5.m5.2.2" xref="S3.Thmtheorem3.p1.5.5.m5.2.2.cmml">n</mi></mrow><mo id="S3.Thmtheorem3.p1.5.5.m5.2.3.1" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.1.cmml">∈</mo><mi id="S3.Thmtheorem3.p1.5.5.m5.2.3.3" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.5.5.m5.2b"><apply id="S3.Thmtheorem3.p1.5.5.m5.2.3.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3"><in id="S3.Thmtheorem3.p1.5.5.m5.2.3.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.1"></in><list id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.2"><ci id="S3.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.1.1">𝑚</ci><ci id="S3.Thmtheorem3.p1.5.5.m5.2.2.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.2">𝑛</ci></list><ci id="S3.Thmtheorem3.p1.5.5.m5.2.3.3.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.5.5.m5.2c">m,n\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.5.5.m5.2d">italic_m , italic_n ∈ roman_Nature</annotation></semantics></math> such that</span></p> <ul class="ltx_itemize" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.2"><math alttext="m<n<\textsf{length}({\rho})" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.1"><semantics id="S3.I2.i1.p1.1.m1.1a"><mrow id="S3.I2.i1.p1.1.m1.1.2" xref="S3.I2.i1.p1.1.m1.1.2.cmml"><mi id="S3.I2.i1.p1.1.m1.1.2.2" xref="S3.I2.i1.p1.1.m1.1.2.2.cmml">m</mi><mo id="S3.I2.i1.p1.1.m1.1.2.3" xref="S3.I2.i1.p1.1.m1.1.2.3.cmml"><</mo><mi id="S3.I2.i1.p1.1.m1.1.2.4" xref="S3.I2.i1.p1.1.m1.1.2.4.cmml">n</mi><mo id="S3.I2.i1.p1.1.m1.1.2.5" xref="S3.I2.i1.p1.1.m1.1.2.5.cmml"><</mo><mrow id="S3.I2.i1.p1.1.m1.1.2.6" xref="S3.I2.i1.p1.1.m1.1.2.6.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I2.i1.p1.1.m1.1.2.6.2" xref="S3.I2.i1.p1.1.m1.1.2.6.2a.cmml">length</mtext><mo id="S3.I2.i1.p1.1.m1.1.2.6.1" xref="S3.I2.i1.p1.1.m1.1.2.6.1.cmml">⁢</mo><mrow id="S3.I2.i1.p1.1.m1.1.2.6.3.2" xref="S3.I2.i1.p1.1.m1.1.2.6.cmml"><mo id="S3.I2.i1.p1.1.m1.1.2.6.3.2.1" stretchy="false" xref="S3.I2.i1.p1.1.m1.1.2.6.cmml">(</mo><mi id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml">ρ</mi><mo id="S3.I2.i1.p1.1.m1.1.2.6.3.2.2" stretchy="false" xref="S3.I2.i1.p1.1.m1.1.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.1b"><apply id="S3.I2.i1.p1.1.m1.1.2.cmml" xref="S3.I2.i1.p1.1.m1.1.2"><and id="S3.I2.i1.p1.1.m1.1.2a.cmml" xref="S3.I2.i1.p1.1.m1.1.2"></and><apply id="S3.I2.i1.p1.1.m1.1.2b.cmml" xref="S3.I2.i1.p1.1.m1.1.2"><lt id="S3.I2.i1.p1.1.m1.1.2.3.cmml" xref="S3.I2.i1.p1.1.m1.1.2.3"></lt><ci id="S3.I2.i1.p1.1.m1.1.2.2.cmml" xref="S3.I2.i1.p1.1.m1.1.2.2">𝑚</ci><ci id="S3.I2.i1.p1.1.m1.1.2.4.cmml" xref="S3.I2.i1.p1.1.m1.1.2.4">𝑛</ci></apply><apply id="S3.I2.i1.p1.1.m1.1.2c.cmml" xref="S3.I2.i1.p1.1.m1.1.2"><lt id="S3.I2.i1.p1.1.m1.1.2.5.cmml" xref="S3.I2.i1.p1.1.m1.1.2.5"></lt><share href="https://arxiv.org/html/2308.09443v2#S3.I2.i1.p1.1.m1.1.2.4.cmml" id="S3.I2.i1.p1.1.m1.1.2d.cmml" xref="S3.I2.i1.p1.1.m1.1.2"></share><apply id="S3.I2.i1.p1.1.m1.1.2.6.cmml" xref="S3.I2.i1.p1.1.m1.1.2.6"><times id="S3.I2.i1.p1.1.m1.1.2.6.1.cmml" xref="S3.I2.i1.p1.1.m1.1.2.6.1"></times><ci id="S3.I2.i1.p1.1.m1.1.2.6.2a.cmml" xref="S3.I2.i1.p1.1.m1.1.2.6.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I2.i1.p1.1.m1.1.2.6.2.cmml" xref="S3.I2.i1.p1.1.m1.1.2.6.2">length</mtext></ci><ci id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1">𝜌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.1c">m<n<\textsf{length}({\rho})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.1d">italic_m < italic_n < length ( italic_ρ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i1.p1.2.1"> and </span><math alttext="\rho_{m}=\rho_{n}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><mrow id="S3.I2.i1.p1.2.m2.1.1" xref="S3.I2.i1.p1.2.m2.1.1.cmml"><msub id="S3.I2.i1.p1.2.m2.1.1.2" xref="S3.I2.i1.p1.2.m2.1.1.2.cmml"><mi id="S3.I2.i1.p1.2.m2.1.1.2.2" xref="S3.I2.i1.p1.2.m2.1.1.2.2.cmml">ρ</mi><mi id="S3.I2.i1.p1.2.m2.1.1.2.3" xref="S3.I2.i1.p1.2.m2.1.1.2.3.cmml">m</mi></msub><mo id="S3.I2.i1.p1.2.m2.1.1.1" xref="S3.I2.i1.p1.2.m2.1.1.1.cmml">=</mo><msub id="S3.I2.i1.p1.2.m2.1.1.3" xref="S3.I2.i1.p1.2.m2.1.1.3.cmml"><mi id="S3.I2.i1.p1.2.m2.1.1.3.2" xref="S3.I2.i1.p1.2.m2.1.1.3.2.cmml">ρ</mi><mi id="S3.I2.i1.p1.2.m2.1.1.3.3" xref="S3.I2.i1.p1.2.m2.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.1b"><apply id="S3.I2.i1.p1.2.m2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1"><eq id="S3.I2.i1.p1.2.m2.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1"></eq><apply id="S3.I2.i1.p1.2.m2.1.1.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.1.1.2.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S3.I2.i1.p1.2.m2.1.1.2.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2.2">𝜌</ci><ci id="S3.I2.i1.p1.2.m2.1.1.2.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2.3">𝑚</ci></apply><apply id="S3.I2.i1.p1.2.m2.1.1.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.1.1.3.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.I2.i1.p1.2.m2.1.1.3.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3.2">𝜌</ci><ci id="S3.I2.i1.p1.2.m2.1.1.3.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1c">\rho_{m}=\rho_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1d">italic_ρ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i1.p1.2.2">,</span></p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.2"><math alttext="\rho_{\leq m}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><msub id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.2" xref="S3.I2.i2.p1.1.m1.1.1.2.cmml">ρ</mi><mrow id="S3.I2.i2.p1.1.m1.1.1.3" xref="S3.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.3.2" xref="S3.I2.i2.p1.1.m1.1.1.3.2.cmml"></mi><mo id="S3.I2.i2.p1.1.m1.1.1.3.1" xref="S3.I2.i2.p1.1.m1.1.1.3.1.cmml">≤</mo><mi id="S3.I2.i2.p1.1.m1.1.1.3.3" xref="S3.I2.i2.p1.1.m1.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><apply id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2">𝜌</ci><apply id="S3.I2.i2.p1.1.m1.1.1.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3"><leq id="S3.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.1"></leq><csymbol cd="latexml" id="S3.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.2">absent</csymbol><ci id="S3.I2.i2.p1.1.m1.1.1.3.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">\rho_{\leq m}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i2.p1.2.1"> and </span><math alttext="\rho_{\leq n}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><msub id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.2.cmml">ρ</mi><mrow id="S3.I2.i2.p1.2.m2.1.1.3" xref="S3.I2.i2.p1.2.m2.1.1.3.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.3.2" xref="S3.I2.i2.p1.2.m2.1.1.3.2.cmml"></mi><mo id="S3.I2.i2.p1.2.m2.1.1.3.1" xref="S3.I2.i2.p1.2.m2.1.1.3.1.cmml">≤</mo><mi id="S3.I2.i2.p1.2.m2.1.1.3.3" xref="S3.I2.i2.p1.2.m2.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.2">𝜌</ci><apply id="S3.I2.i2.p1.2.m2.1.1.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3"><leq id="S3.I2.i2.p1.2.m2.1.1.3.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3.1"></leq><csymbol cd="latexml" id="S3.I2.i2.p1.2.m2.1.1.3.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3.2">absent</csymbol><ci id="S3.I2.i2.p1.2.m2.1.1.3.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">\rho_{\leq n}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_n end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i2.p1.2.2"> belong to the same region, and</span></p> </div> </li> <li class="ltx_item" id="S3.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i3.p1"> <p class="ltx_p" id="S3.I2.i3.p1.2"><span class="ltx_text ltx_font_italic" id="S3.I2.i3.p1.2.1">if </span><math alttext="\textsf{val}(\rho_{\leq m})=\infty" class="ltx_Math" display="inline" id="S3.I2.i3.p1.1.m1.1"><semantics id="S3.I2.i3.p1.1.m1.1a"><mrow id="S3.I2.i3.p1.1.m1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.cmml"><mrow id="S3.I2.i3.p1.1.m1.1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I2.i3.p1.1.m1.1.1.1.3" xref="S3.I2.i3.p1.1.m1.1.1.1.3a.cmml">val</mtext><mo id="S3.I2.i3.p1.1.m1.1.1.1.2" xref="S3.I2.i3.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.I2.i3.p1.1.m1.1.1.1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.I2.i3.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.2" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.2.cmml">ρ</mi><mrow id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.2" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.2.cmml"></mi><mo id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.1" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.1.cmml">≤</mo><mi id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.3" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.3.cmml">m</mi></mrow></msub><mo id="S3.I2.i3.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I2.i3.p1.1.m1.1.1.2" xref="S3.I2.i3.p1.1.m1.1.1.2.cmml">=</mo><mi id="S3.I2.i3.p1.1.m1.1.1.3" mathvariant="normal" xref="S3.I2.i3.p1.1.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.1.m1.1b"><apply id="S3.I2.i3.p1.1.m1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1"><eq id="S3.I2.i3.p1.1.m1.1.1.2.cmml" xref="S3.I2.i3.p1.1.m1.1.1.2"></eq><apply id="S3.I2.i3.p1.1.m1.1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1"><times id="S3.I2.i3.p1.1.m1.1.1.1.2.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.2"></times><ci id="S3.I2.i3.p1.1.m1.1.1.1.3a.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I2.i3.p1.1.m1.1.1.1.3.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.3">val</mtext></ci><apply id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.2">𝜌</ci><apply id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3"><leq id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.1"></leq><csymbol cd="latexml" id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.2">absent</csymbol><ci id="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><infinity id="S3.I2.i3.p1.1.m1.1.1.3.cmml" xref="S3.I2.i3.p1.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.1.m1.1c">\textsf{val}(\rho_{\leq m})=\infty</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.1.m1.1d">val ( italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT ) = ∞</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i3.p1.2.2">, then the weight </span><math alttext="w(\rho_{[m,n]})" class="ltx_Math" display="inline" id="S3.I2.i3.p1.2.m2.3"><semantics id="S3.I2.i3.p1.2.m2.3a"><mrow id="S3.I2.i3.p1.2.m2.3.3" xref="S3.I2.i3.p1.2.m2.3.3.cmml"><mi id="S3.I2.i3.p1.2.m2.3.3.3" xref="S3.I2.i3.p1.2.m2.3.3.3.cmml">w</mi><mo id="S3.I2.i3.p1.2.m2.3.3.2" xref="S3.I2.i3.p1.2.m2.3.3.2.cmml">⁢</mo><mrow id="S3.I2.i3.p1.2.m2.3.3.1.1" xref="S3.I2.i3.p1.2.m2.3.3.1.1.1.cmml"><mo id="S3.I2.i3.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S3.I2.i3.p1.2.m2.3.3.1.1.1.cmml">(</mo><msub id="S3.I2.i3.p1.2.m2.3.3.1.1.1" xref="S3.I2.i3.p1.2.m2.3.3.1.1.1.cmml"><mi id="S3.I2.i3.p1.2.m2.3.3.1.1.1.2" xref="S3.I2.i3.p1.2.m2.3.3.1.1.1.2.cmml">ρ</mi><mrow id="S3.I2.i3.p1.2.m2.2.2.2.4" xref="S3.I2.i3.p1.2.m2.2.2.2.3.cmml"><mo id="S3.I2.i3.p1.2.m2.2.2.2.4.1" stretchy="false" xref="S3.I2.i3.p1.2.m2.2.2.2.3.cmml">[</mo><mi id="S3.I2.i3.p1.2.m2.1.1.1.1" xref="S3.I2.i3.p1.2.m2.1.1.1.1.cmml">m</mi><mo id="S3.I2.i3.p1.2.m2.2.2.2.4.2" xref="S3.I2.i3.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.I2.i3.p1.2.m2.2.2.2.2" xref="S3.I2.i3.p1.2.m2.2.2.2.2.cmml">n</mi><mo id="S3.I2.i3.p1.2.m2.2.2.2.4.3" stretchy="false" xref="S3.I2.i3.p1.2.m2.2.2.2.3.cmml">]</mo></mrow></msub><mo id="S3.I2.i3.p1.2.m2.3.3.1.1.3" stretchy="false" xref="S3.I2.i3.p1.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.2.m2.3b"><apply id="S3.I2.i3.p1.2.m2.3.3.cmml" xref="S3.I2.i3.p1.2.m2.3.3"><times id="S3.I2.i3.p1.2.m2.3.3.2.cmml" xref="S3.I2.i3.p1.2.m2.3.3.2"></times><ci id="S3.I2.i3.p1.2.m2.3.3.3.cmml" xref="S3.I2.i3.p1.2.m2.3.3.3">𝑤</ci><apply id="S3.I2.i3.p1.2.m2.3.3.1.1.1.cmml" xref="S3.I2.i3.p1.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S3.I2.i3.p1.2.m2.3.3.1.1.1.1.cmml" xref="S3.I2.i3.p1.2.m2.3.3.1.1">subscript</csymbol><ci id="S3.I2.i3.p1.2.m2.3.3.1.1.1.2.cmml" xref="S3.I2.i3.p1.2.m2.3.3.1.1.1.2">𝜌</ci><interval closure="closed" id="S3.I2.i3.p1.2.m2.2.2.2.3.cmml" xref="S3.I2.i3.p1.2.m2.2.2.2.4"><ci id="S3.I2.i3.p1.2.m2.1.1.1.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1.1.1">𝑚</ci><ci id="S3.I2.i3.p1.2.m2.2.2.2.2.cmml" xref="S3.I2.i3.p1.2.m2.2.2.2.2">𝑛</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.2.m2.3c">w(\rho_{[m,n]})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.2.m2.3d">italic_w ( italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i3.p1.2.3"> is null.</span></p> </div> </li> </ul> <p class="ltx_p" id="S3.Thmtheorem3.p1.12"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.12.7">Then the strategy <math alttext="\sigma_{0}^{\prime}=\sigma_{0}[\rho_{\leq m}\rightarrow\sigma_{0|\rho_{\leq n}}]" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.6.1.m1.1"><semantics id="S3.Thmtheorem3.p1.6.1.m1.1a"><mrow id="S3.Thmtheorem3.p1.6.1.m1.1.1" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.cmml"><msubsup id="S3.Thmtheorem3.p1.6.1.m1.1.1.3" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.6.1.m1.1.1.3.2.2" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.6.1.m1.1.1.3.2.3" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.3.2.3.cmml">0</mn><mo id="S3.Thmtheorem3.p1.6.1.m1.1.1.3.3" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.3.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem3.p1.6.1.m1.1.1.2" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.2.cmml">=</mo><mrow id="S3.Thmtheorem3.p1.6.1.m1.1.1.1" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.cmml"><msub id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.3" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.3.2" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.3.3" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.3.3.cmml">0</mn></msub><mo id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.2" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.2.cmml"><mo id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.cmml"><msub id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.cmml"><mi id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.2" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.2.cmml">ρ</mi><mrow id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3" 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xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1">→</ci><apply id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.2">𝜌</ci><apply id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3"><leq id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3.1"></leq><csymbol cd="latexml" id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3.2.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3.2">absent</csymbol><ci id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3.3.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2.3.3">𝑚</ci></apply></apply><apply id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.2">𝜎</ci><apply id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3"><csymbol cd="latexml" id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.1.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.1">conditional</csymbol><cn id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.2">0</cn><apply id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.1.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3">subscript</csymbol><ci id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.2.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.2">𝜌</ci><apply id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3"><leq id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3.1.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3.1"></leq><csymbol cd="latexml" id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3.2.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3.2">absent</csymbol><ci id="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3.3.cmml" xref="S3.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3.3.3.3.3">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.6.1.m1.1c">\sigma_{0}^{\prime}=\sigma_{0}[\rho_{\leq m}\rightarrow\sigma_{0|\rho_{\leq n}}]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.6.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT → italic_σ start_POSTSUBSCRIPT 0 | italic_ρ start_POSTSUBSCRIPT ≤ italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ]</annotation></semantics></math> is a solution in <math alttext="\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.7.2.m2.2"><semantics id="S3.Thmtheorem3.p1.7.2.m2.2a"><mrow id="S3.Thmtheorem3.p1.7.2.m2.2.3" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem3.p1.7.2.m2.2.3.2" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.2a.cmml">SPS</mtext><mo id="S3.Thmtheorem3.p1.7.2.m2.2.3.1" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.7.2.m2.2.3.3.2" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.3.1.cmml"><mo id="S3.Thmtheorem3.p1.7.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem3.p1.7.2.m2.1.1" xref="S3.Thmtheorem3.p1.7.2.m2.1.1.cmml">G</mi><mo id="S3.Thmtheorem3.p1.7.2.m2.2.3.3.2.2" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem3.p1.7.2.m2.2.2" xref="S3.Thmtheorem3.p1.7.2.m2.2.2.cmml">B</mi><mo id="S3.Thmtheorem3.p1.7.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.7.2.m2.2b"><apply id="S3.Thmtheorem3.p1.7.2.m2.2.3.cmml" xref="S3.Thmtheorem3.p1.7.2.m2.2.3"><times id="S3.Thmtheorem3.p1.7.2.m2.2.3.1.cmml" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.1"></times><ci id="S3.Thmtheorem3.p1.7.2.m2.2.3.2a.cmml" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem3.p1.7.2.m2.2.3.2.cmml" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.2">SPS</mtext></ci><interval closure="open" id="S3.Thmtheorem3.p1.7.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem3.p1.7.2.m2.2.3.3.2"><ci id="S3.Thmtheorem3.p1.7.2.m2.1.1.cmml" xref="S3.Thmtheorem3.p1.7.2.m2.1.1">𝐺</ci><ci id="S3.Thmtheorem3.p1.7.2.m2.2.2.cmml" xref="S3.Thmtheorem3.p1.7.2.m2.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.7.2.m2.2c">\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.7.2.m2.2d">SPS ( italic_G , italic_B )</annotation></semantics></math> such that <math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.8.3.m3.1"><semantics id="S3.Thmtheorem3.p1.8.3.m3.1a"><mrow id="S3.Thmtheorem3.p1.8.3.m3.1.1" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.cmml"><msubsup id="S3.Thmtheorem3.p1.8.3.m3.1.1.2" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2.cmml"><mi id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.2" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.3" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.3.cmml">0</mn><mo id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.3" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem3.p1.8.3.m3.1.1.1" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.1.cmml">⪯</mo><msub id="S3.Thmtheorem3.p1.8.3.m3.1.1.3" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.8.3.m3.1.1.3.2" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.8.3.m3.1.1.3.3" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.8.3.m3.1b"><apply id="S3.Thmtheorem3.p1.8.3.m3.1.1.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1"><csymbol cd="latexml" id="S3.Thmtheorem3.p1.8.3.m3.1.1.1.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.1">precedes-or-equals</csymbol><apply id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2">superscript</csymbol><apply id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.1.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.2.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.2">𝜎</ci><cn id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2.2.3">0</cn></apply><ci id="S3.Thmtheorem3.p1.8.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem3.p1.8.3.m3.1.1.3.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.8.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem3.p1.8.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem3.p1.8.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.8.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.8.3.m3.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.8.3.m3.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, if <math alttext="\sigma_{0}^{\prime}\simeq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.9.4.m4.1"><semantics id="S3.Thmtheorem3.p1.9.4.m4.1a"><mrow id="S3.Thmtheorem3.p1.9.4.m4.1.1" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.cmml"><msubsup id="S3.Thmtheorem3.p1.9.4.m4.1.1.2" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2.cmml"><mi id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.2" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.3" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.3.cmml">0</mn><mo id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.3" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem3.p1.9.4.m4.1.1.1" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.1.cmml">≃</mo><msub id="S3.Thmtheorem3.p1.9.4.m4.1.1.3" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.9.4.m4.1.1.3.2" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.9.4.m4.1.1.3.3" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.9.4.m4.1b"><apply id="S3.Thmtheorem3.p1.9.4.m4.1.1.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1"><csymbol cd="latexml" id="S3.Thmtheorem3.p1.9.4.m4.1.1.1.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.1">similar-to-or-equals</csymbol><apply id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.1.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2">superscript</csymbol><apply id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.1.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.2.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.2">𝜎</ci><cn id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2.2.3">0</cn></apply><ci id="S3.Thmtheorem3.p1.9.4.m4.1.1.2.3.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem3.p1.9.4.m4.1.1.3.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.9.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem3.p1.9.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem3.p1.9.4.m4.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.9.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.9.4.m4.1c">\sigma_{0}^{\prime}\simeq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.9.4.m4.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≃ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.10.5.m5.1"><semantics id="S3.Thmtheorem3.p1.10.5.m5.1a"><msubsup id="S3.Thmtheorem3.p1.10.5.m5.1.1" xref="S3.Thmtheorem3.p1.10.5.m5.1.1.cmml"><mi id="S3.Thmtheorem3.p1.10.5.m5.1.1.2.2" xref="S3.Thmtheorem3.p1.10.5.m5.1.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.10.5.m5.1.1.3" xref="S3.Thmtheorem3.p1.10.5.m5.1.1.3.cmml">0</mn><mo id="S3.Thmtheorem3.p1.10.5.m5.1.1.2.3" xref="S3.Thmtheorem3.p1.10.5.m5.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.10.5.m5.1b"><apply id="S3.Thmtheorem3.p1.10.5.m5.1.1.cmml" xref="S3.Thmtheorem3.p1.10.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.10.5.m5.1.1.1.cmml" xref="S3.Thmtheorem3.p1.10.5.m5.1.1">subscript</csymbol><apply id="S3.Thmtheorem3.p1.10.5.m5.1.1.2.cmml" xref="S3.Thmtheorem3.p1.10.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.10.5.m5.1.1.2.1.cmml" xref="S3.Thmtheorem3.p1.10.5.m5.1.1">superscript</csymbol><ci id="S3.Thmtheorem3.p1.10.5.m5.1.1.2.2.cmml" xref="S3.Thmtheorem3.p1.10.5.m5.1.1.2.2">𝜎</ci><ci id="S3.Thmtheorem3.p1.10.5.m5.1.1.2.3.cmml" xref="S3.Thmtheorem3.p1.10.5.m5.1.1.2.3">′</ci></apply><cn id="S3.Thmtheorem3.p1.10.5.m5.1.1.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.10.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.10.5.m5.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.10.5.m5.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> has a witness set <math alttext="\textsf{Wit}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.11.6.m6.1"><semantics id="S3.Thmtheorem3.p1.11.6.m6.1a"><msub id="S3.Thmtheorem3.p1.11.6.m6.1.1" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.11.6.m6.1.1.2" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.2a.cmml">Wit</mtext><msubsup id="S3.Thmtheorem3.p1.11.6.m6.1.1.3" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.2" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.3" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3.3.cmml">0</mn><mo id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.3" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.11.6.m6.1b"><apply id="S3.Thmtheorem3.p1.11.6.m6.1.1.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.11.6.m6.1.1.1.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.11.6.m6.1.1.2a.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.11.6.m6.1.1.2.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.2">Wit</mtext></ci><apply id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3">subscript</csymbol><apply id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.1.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.2.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.2">𝜎</ci><ci id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.3.cmml" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3.2.3">′</ci></apply><cn id="S3.Thmtheorem3.p1.11.6.m6.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.11.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.11.6.m6.1c">\textsf{Wit}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.11.6.m6.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\textsf{length}({\textsf{Wit}_{\sigma^{\prime}_{0}}})<\textsf{length}({\textsf% {Wit}_{\sigma_{0}}})" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.12.7.m7.2"><semantics id="S3.Thmtheorem3.p1.12.7.m7.2a"><mrow id="S3.Thmtheorem3.p1.12.7.m7.2.2" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.cmml"><mrow id="S3.Thmtheorem3.p1.12.7.m7.1.1.1" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.3" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.3a.cmml">length</mtext><mo id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.2" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.2" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.2a.cmml">Wit</mtext><msubsup id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.2" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.3" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.3.cmml">0</mn><mo id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.3" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem3.p1.12.7.m7.2.2.3" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.3.cmml"><</mo><mrow id="S3.Thmtheorem3.p1.12.7.m7.2.2.2" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.3" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.3a.cmml">length</mtext><mo id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.2" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.2" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.2a.cmml">Wit</mtext><msub id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.2" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.3" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.3" stretchy="false" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.12.7.m7.2b"><apply id="S3.Thmtheorem3.p1.12.7.m7.2.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2"><lt id="S3.Thmtheorem3.p1.12.7.m7.2.2.3.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.3"></lt><apply id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1"><times id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.2"></times><ci id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.3a.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.3">length</mtext></ci><apply id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.2a.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.2">Wit</mtext></ci><apply id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3">subscript</csymbol><apply id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.2">𝜎</ci><ci id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.3.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.2.3">′</ci></apply><cn id="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.12.7.m7.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2"><times id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.2"></times><ci id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.3a.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.3.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.3">length</mtext></ci><apply id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.2a.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.2">Wit</mtext></ci><apply id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.12.7.m7.2.2.2.1.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.12.7.m7.2c">\textsf{length}({\textsf{Wit}_{\sigma^{\prime}_{0}}})<\textsf{length}({\textsf% {Wit}_{\sigma_{0}}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.12.7.m7.2d">length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) < length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.14">Given a witness <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><mi id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><ci id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">italic_ρ</annotation></semantics></math>, the first condition of the lemma means that <math alttext="\rho_{[m,n]}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.2"><semantics id="S3.SS3.p2.2.m2.2a"><msub id="S3.SS3.p2.2.m2.2.3" xref="S3.SS3.p2.2.m2.2.3.cmml"><mi id="S3.SS3.p2.2.m2.2.3.2" xref="S3.SS3.p2.2.m2.2.3.2.cmml">ρ</mi><mrow id="S3.SS3.p2.2.m2.2.2.2.4" xref="S3.SS3.p2.2.m2.2.2.2.3.cmml"><mo id="S3.SS3.p2.2.m2.2.2.2.4.1" stretchy="false" xref="S3.SS3.p2.2.m2.2.2.2.3.cmml">[</mo><mi id="S3.SS3.p2.2.m2.1.1.1.1" xref="S3.SS3.p2.2.m2.1.1.1.1.cmml">m</mi><mo id="S3.SS3.p2.2.m2.2.2.2.4.2" xref="S3.SS3.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p2.2.m2.2.2.2.2" xref="S3.SS3.p2.2.m2.2.2.2.2.cmml">n</mi><mo id="S3.SS3.p2.2.m2.2.2.2.4.3" stretchy="false" xref="S3.SS3.p2.2.m2.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.2b"><apply id="S3.SS3.p2.2.m2.2.3.cmml" xref="S3.SS3.p2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.2.m2.2.3.1.cmml" xref="S3.SS3.p2.2.m2.2.3">subscript</csymbol><ci id="S3.SS3.p2.2.m2.2.3.2.cmml" xref="S3.SS3.p2.2.m2.2.3.2">𝜌</ci><interval closure="closed" id="S3.SS3.p2.2.m2.2.2.2.3.cmml" xref="S3.SS3.p2.2.m2.2.2.2.4"><ci id="S3.SS3.p2.2.m2.1.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1.1.1">𝑚</ci><ci id="S3.SS3.p2.2.m2.2.2.2.2.cmml" xref="S3.SS3.p2.2.m2.2.2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.2c">\rho_{[m,n]}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.2d">italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT</annotation></semantics></math> is a cycle and that it appears before the last visit of a target by <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m3.1"><semantics id="S3.SS3.p2.3.m3.1a"><mi id="S3.SS3.p2.3.m3.1.1" xref="S3.SS3.p2.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.3.m3.1b"><ci id="S3.SS3.p2.3.m3.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.3.m3.1d">italic_ρ</annotation></semantics></math>. The second one says that <math alttext="\rho_{\leq m}\sim\rho_{\leq n}" class="ltx_Math" display="inline" id="S3.SS3.p2.4.m4.1"><semantics id="S3.SS3.p2.4.m4.1a"><mrow id="S3.SS3.p2.4.m4.1.1" xref="S3.SS3.p2.4.m4.1.1.cmml"><msub id="S3.SS3.p2.4.m4.1.1.2" xref="S3.SS3.p2.4.m4.1.1.2.cmml"><mi id="S3.SS3.p2.4.m4.1.1.2.2" xref="S3.SS3.p2.4.m4.1.1.2.2.cmml">ρ</mi><mrow id="S3.SS3.p2.4.m4.1.1.2.3" xref="S3.SS3.p2.4.m4.1.1.2.3.cmml"><mi id="S3.SS3.p2.4.m4.1.1.2.3.2" xref="S3.SS3.p2.4.m4.1.1.2.3.2.cmml"></mi><mo id="S3.SS3.p2.4.m4.1.1.2.3.1" xref="S3.SS3.p2.4.m4.1.1.2.3.1.cmml">≤</mo><mi id="S3.SS3.p2.4.m4.1.1.2.3.3" xref="S3.SS3.p2.4.m4.1.1.2.3.3.cmml">m</mi></mrow></msub><mo id="S3.SS3.p2.4.m4.1.1.1" xref="S3.SS3.p2.4.m4.1.1.1.cmml">∼</mo><msub id="S3.SS3.p2.4.m4.1.1.3" xref="S3.SS3.p2.4.m4.1.1.3.cmml"><mi id="S3.SS3.p2.4.m4.1.1.3.2" xref="S3.SS3.p2.4.m4.1.1.3.2.cmml">ρ</mi><mrow id="S3.SS3.p2.4.m4.1.1.3.3" xref="S3.SS3.p2.4.m4.1.1.3.3.cmml"><mi id="S3.SS3.p2.4.m4.1.1.3.3.2" xref="S3.SS3.p2.4.m4.1.1.3.3.2.cmml"></mi><mo id="S3.SS3.p2.4.m4.1.1.3.3.1" xref="S3.SS3.p2.4.m4.1.1.3.3.1.cmml">≤</mo><mi id="S3.SS3.p2.4.m4.1.1.3.3.3" xref="S3.SS3.p2.4.m4.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.4.m4.1b"><apply id="S3.SS3.p2.4.m4.1.1.cmml" xref="S3.SS3.p2.4.m4.1.1"><csymbol cd="latexml" id="S3.SS3.p2.4.m4.1.1.1.cmml" xref="S3.SS3.p2.4.m4.1.1.1">similar-to</csymbol><apply id="S3.SS3.p2.4.m4.1.1.2.cmml" xref="S3.SS3.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p2.4.m4.1.1.2.1.cmml" xref="S3.SS3.p2.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS3.p2.4.m4.1.1.2.2.cmml" xref="S3.SS3.p2.4.m4.1.1.2.2">𝜌</ci><apply id="S3.SS3.p2.4.m4.1.1.2.3.cmml" xref="S3.SS3.p2.4.m4.1.1.2.3"><leq id="S3.SS3.p2.4.m4.1.1.2.3.1.cmml" xref="S3.SS3.p2.4.m4.1.1.2.3.1"></leq><csymbol cd="latexml" id="S3.SS3.p2.4.m4.1.1.2.3.2.cmml" xref="S3.SS3.p2.4.m4.1.1.2.3.2">absent</csymbol><ci id="S3.SS3.p2.4.m4.1.1.2.3.3.cmml" xref="S3.SS3.p2.4.m4.1.1.2.3.3">𝑚</ci></apply></apply><apply id="S3.SS3.p2.4.m4.1.1.3.cmml" xref="S3.SS3.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p2.4.m4.1.1.3.1.cmml" xref="S3.SS3.p2.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS3.p2.4.m4.1.1.3.2.cmml" xref="S3.SS3.p2.4.m4.1.1.3.2">𝜌</ci><apply id="S3.SS3.p2.4.m4.1.1.3.3.cmml" xref="S3.SS3.p2.4.m4.1.1.3.3"><leq id="S3.SS3.p2.4.m4.1.1.3.3.1.cmml" xref="S3.SS3.p2.4.m4.1.1.3.3.1"></leq><csymbol cd="latexml" id="S3.SS3.p2.4.m4.1.1.3.3.2.cmml" xref="S3.SS3.p2.4.m4.1.1.3.3.2">absent</csymbol><ci id="S3.SS3.p2.4.m4.1.1.3.3.3.cmml" xref="S3.SS3.p2.4.m4.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.4.m4.1c">\rho_{\leq m}\sim\rho_{\leq n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.4.m4.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT ∼ italic_ρ start_POSTSUBSCRIPT ≤ italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, i.e., no new target is visited and no branching point is crossed from history <math alttext="\rho_{\leq m}" class="ltx_Math" display="inline" id="S3.SS3.p2.5.m5.1"><semantics id="S3.SS3.p2.5.m5.1a"><msub id="S3.SS3.p2.5.m5.1.1" xref="S3.SS3.p2.5.m5.1.1.cmml"><mi id="S3.SS3.p2.5.m5.1.1.2" xref="S3.SS3.p2.5.m5.1.1.2.cmml">ρ</mi><mrow id="S3.SS3.p2.5.m5.1.1.3" xref="S3.SS3.p2.5.m5.1.1.3.cmml"><mi id="S3.SS3.p2.5.m5.1.1.3.2" xref="S3.SS3.p2.5.m5.1.1.3.2.cmml"></mi><mo id="S3.SS3.p2.5.m5.1.1.3.1" xref="S3.SS3.p2.5.m5.1.1.3.1.cmml">≤</mo><mi id="S3.SS3.p2.5.m5.1.1.3.3" xref="S3.SS3.p2.5.m5.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.5.m5.1b"><apply id="S3.SS3.p2.5.m5.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.1.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.p2.5.m5.1.1.2.cmml" xref="S3.SS3.p2.5.m5.1.1.2">𝜌</ci><apply id="S3.SS3.p2.5.m5.1.1.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3"><leq id="S3.SS3.p2.5.m5.1.1.3.1.cmml" xref="S3.SS3.p2.5.m5.1.1.3.1"></leq><csymbol cd="latexml" id="S3.SS3.p2.5.m5.1.1.3.2.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2">absent</csymbol><ci id="S3.SS3.p2.5.m5.1.1.3.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.5.m5.1c">\rho_{\leq m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.5.m5.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT</annotation></semantics></math> to history <math alttext="\rho_{\leq n}" class="ltx_Math" display="inline" id="S3.SS3.p2.6.m6.1"><semantics id="S3.SS3.p2.6.m6.1a"><msub id="S3.SS3.p2.6.m6.1.1" xref="S3.SS3.p2.6.m6.1.1.cmml"><mi id="S3.SS3.p2.6.m6.1.1.2" xref="S3.SS3.p2.6.m6.1.1.2.cmml">ρ</mi><mrow id="S3.SS3.p2.6.m6.1.1.3" xref="S3.SS3.p2.6.m6.1.1.3.cmml"><mi id="S3.SS3.p2.6.m6.1.1.3.2" xref="S3.SS3.p2.6.m6.1.1.3.2.cmml"></mi><mo id="S3.SS3.p2.6.m6.1.1.3.1" xref="S3.SS3.p2.6.m6.1.1.3.1.cmml">≤</mo><mi id="S3.SS3.p2.6.m6.1.1.3.3" xref="S3.SS3.p2.6.m6.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.6.m6.1b"><apply id="S3.SS3.p2.6.m6.1.1.cmml" xref="S3.SS3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.6.m6.1.1.1.cmml" xref="S3.SS3.p2.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.p2.6.m6.1.1.2.cmml" xref="S3.SS3.p2.6.m6.1.1.2">𝜌</ci><apply id="S3.SS3.p2.6.m6.1.1.3.cmml" xref="S3.SS3.p2.6.m6.1.1.3"><leq id="S3.SS3.p2.6.m6.1.1.3.1.cmml" xref="S3.SS3.p2.6.m6.1.1.3.1"></leq><csymbol cd="latexml" id="S3.SS3.p2.6.m6.1.1.3.2.cmml" xref="S3.SS3.p2.6.m6.1.1.3.2">absent</csymbol><ci id="S3.SS3.p2.6.m6.1.1.3.3.cmml" xref="S3.SS3.p2.6.m6.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.6.m6.1c">\rho_{\leq n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.6.m6.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. The third one says that if <math alttext="\rho_{\leq m}" class="ltx_Math" display="inline" id="S3.SS3.p2.7.m7.1"><semantics id="S3.SS3.p2.7.m7.1a"><msub id="S3.SS3.p2.7.m7.1.1" xref="S3.SS3.p2.7.m7.1.1.cmml"><mi id="S3.SS3.p2.7.m7.1.1.2" xref="S3.SS3.p2.7.m7.1.1.2.cmml">ρ</mi><mrow id="S3.SS3.p2.7.m7.1.1.3" xref="S3.SS3.p2.7.m7.1.1.3.cmml"><mi id="S3.SS3.p2.7.m7.1.1.3.2" xref="S3.SS3.p2.7.m7.1.1.3.2.cmml"></mi><mo id="S3.SS3.p2.7.m7.1.1.3.1" xref="S3.SS3.p2.7.m7.1.1.3.1.cmml">≤</mo><mi id="S3.SS3.p2.7.m7.1.1.3.3" xref="S3.SS3.p2.7.m7.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.7.m7.1b"><apply id="S3.SS3.p2.7.m7.1.1.cmml" xref="S3.SS3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.7.m7.1.1.1.cmml" xref="S3.SS3.p2.7.m7.1.1">subscript</csymbol><ci id="S3.SS3.p2.7.m7.1.1.2.cmml" xref="S3.SS3.p2.7.m7.1.1.2">𝜌</ci><apply id="S3.SS3.p2.7.m7.1.1.3.cmml" xref="S3.SS3.p2.7.m7.1.1.3"><leq id="S3.SS3.p2.7.m7.1.1.3.1.cmml" xref="S3.SS3.p2.7.m7.1.1.3.1"></leq><csymbol cd="latexml" id="S3.SS3.p2.7.m7.1.1.3.2.cmml" xref="S3.SS3.p2.7.m7.1.1.3.2">absent</csymbol><ci id="S3.SS3.p2.7.m7.1.1.3.3.cmml" xref="S3.SS3.p2.7.m7.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.7.m7.1c">\rho_{\leq m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.7.m7.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT</annotation></semantics></math> does not visit Player <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.p2.8.m8.1"><semantics id="S3.SS3.p2.8.m8.1a"><mn id="S3.SS3.p2.8.m8.1.1" xref="S3.SS3.p2.8.m8.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.8.m8.1b"><cn id="S3.SS3.p2.8.m8.1.1.cmml" type="integer" xref="S3.SS3.p2.8.m8.1.1">0</cn></annotation-xml></semantics></math>’s target, then the cycle <math alttext="\rho_{[m,n]}" class="ltx_Math" display="inline" id="S3.SS3.p2.9.m9.2"><semantics id="S3.SS3.p2.9.m9.2a"><msub id="S3.SS3.p2.9.m9.2.3" xref="S3.SS3.p2.9.m9.2.3.cmml"><mi id="S3.SS3.p2.9.m9.2.3.2" xref="S3.SS3.p2.9.m9.2.3.2.cmml">ρ</mi><mrow id="S3.SS3.p2.9.m9.2.2.2.4" xref="S3.SS3.p2.9.m9.2.2.2.3.cmml"><mo id="S3.SS3.p2.9.m9.2.2.2.4.1" stretchy="false" xref="S3.SS3.p2.9.m9.2.2.2.3.cmml">[</mo><mi id="S3.SS3.p2.9.m9.1.1.1.1" xref="S3.SS3.p2.9.m9.1.1.1.1.cmml">m</mi><mo id="S3.SS3.p2.9.m9.2.2.2.4.2" xref="S3.SS3.p2.9.m9.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p2.9.m9.2.2.2.2" xref="S3.SS3.p2.9.m9.2.2.2.2.cmml">n</mi><mo id="S3.SS3.p2.9.m9.2.2.2.4.3" stretchy="false" xref="S3.SS3.p2.9.m9.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.9.m9.2b"><apply id="S3.SS3.p2.9.m9.2.3.cmml" xref="S3.SS3.p2.9.m9.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.9.m9.2.3.1.cmml" xref="S3.SS3.p2.9.m9.2.3">subscript</csymbol><ci id="S3.SS3.p2.9.m9.2.3.2.cmml" xref="S3.SS3.p2.9.m9.2.3.2">𝜌</ci><interval closure="closed" id="S3.SS3.p2.9.m9.2.2.2.3.cmml" xref="S3.SS3.p2.9.m9.2.2.2.4"><ci id="S3.SS3.p2.9.m9.1.1.1.1.cmml" xref="S3.SS3.p2.9.m9.1.1.1.1">𝑚</ci><ci id="S3.SS3.p2.9.m9.2.2.2.2.cmml" xref="S3.SS3.p2.9.m9.2.2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.9.m9.2c">\rho_{[m,n]}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.9.m9.2d">italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT</annotation></semantics></math> must have a null weight. The new strategy <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.SS3.p2.10.m10.1"><semantics id="S3.SS3.p2.10.m10.1a"><msubsup id="S3.SS3.p2.10.m10.1.1" xref="S3.SS3.p2.10.m10.1.1.cmml"><mi id="S3.SS3.p2.10.m10.1.1.2.2" xref="S3.SS3.p2.10.m10.1.1.2.2.cmml">σ</mi><mn id="S3.SS3.p2.10.m10.1.1.3" xref="S3.SS3.p2.10.m10.1.1.3.cmml">0</mn><mo id="S3.SS3.p2.10.m10.1.1.2.3" xref="S3.SS3.p2.10.m10.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.10.m10.1b"><apply id="S3.SS3.p2.10.m10.1.1.cmml" xref="S3.SS3.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.10.m10.1.1.1.cmml" xref="S3.SS3.p2.10.m10.1.1">subscript</csymbol><apply id="S3.SS3.p2.10.m10.1.1.2.cmml" xref="S3.SS3.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.10.m10.1.1.2.1.cmml" xref="S3.SS3.p2.10.m10.1.1">superscript</csymbol><ci id="S3.SS3.p2.10.m10.1.1.2.2.cmml" xref="S3.SS3.p2.10.m10.1.1.2.2">𝜎</ci><ci id="S3.SS3.p2.10.m10.1.1.2.3.cmml" xref="S3.SS3.p2.10.m10.1.1.2.3">′</ci></apply><cn id="S3.SS3.p2.10.m10.1.1.3.cmml" type="integer" xref="S3.SS3.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.10.m10.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.10.m10.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is obtained from <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS3.p2.11.m11.1"><semantics id="S3.SS3.p2.11.m11.1a"><msub id="S3.SS3.p2.11.m11.1.1" xref="S3.SS3.p2.11.m11.1.1.cmml"><mi id="S3.SS3.p2.11.m11.1.1.2" xref="S3.SS3.p2.11.m11.1.1.2.cmml">σ</mi><mn id="S3.SS3.p2.11.m11.1.1.3" xref="S3.SS3.p2.11.m11.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.11.m11.1b"><apply id="S3.SS3.p2.11.m11.1.1.cmml" xref="S3.SS3.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.11.m11.1.1.1.cmml" xref="S3.SS3.p2.11.m11.1.1">subscript</csymbol><ci id="S3.SS3.p2.11.m11.1.1.2.cmml" xref="S3.SS3.p2.11.m11.1.1.2">𝜎</ci><cn id="S3.SS3.p2.11.m11.1.1.3.cmml" type="integer" xref="S3.SS3.p2.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.11.m11.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.11.m11.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> by playing after <math alttext="\rho_{\leq m}" class="ltx_Math" display="inline" id="S3.SS3.p2.12.m12.1"><semantics id="S3.SS3.p2.12.m12.1a"><msub id="S3.SS3.p2.12.m12.1.1" xref="S3.SS3.p2.12.m12.1.1.cmml"><mi id="S3.SS3.p2.12.m12.1.1.2" xref="S3.SS3.p2.12.m12.1.1.2.cmml">ρ</mi><mrow id="S3.SS3.p2.12.m12.1.1.3" xref="S3.SS3.p2.12.m12.1.1.3.cmml"><mi id="S3.SS3.p2.12.m12.1.1.3.2" xref="S3.SS3.p2.12.m12.1.1.3.2.cmml"></mi><mo id="S3.SS3.p2.12.m12.1.1.3.1" xref="S3.SS3.p2.12.m12.1.1.3.1.cmml">≤</mo><mi id="S3.SS3.p2.12.m12.1.1.3.3" xref="S3.SS3.p2.12.m12.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.12.m12.1b"><apply id="S3.SS3.p2.12.m12.1.1.cmml" xref="S3.SS3.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.12.m12.1.1.1.cmml" xref="S3.SS3.p2.12.m12.1.1">subscript</csymbol><ci id="S3.SS3.p2.12.m12.1.1.2.cmml" xref="S3.SS3.p2.12.m12.1.1.2">𝜌</ci><apply id="S3.SS3.p2.12.m12.1.1.3.cmml" xref="S3.SS3.p2.12.m12.1.1.3"><leq id="S3.SS3.p2.12.m12.1.1.3.1.cmml" xref="S3.SS3.p2.12.m12.1.1.3.1"></leq><csymbol cd="latexml" id="S3.SS3.p2.12.m12.1.1.3.2.cmml" xref="S3.SS3.p2.12.m12.1.1.3.2">absent</csymbol><ci id="S3.SS3.p2.12.m12.1.1.3.3.cmml" xref="S3.SS3.p2.12.m12.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.12.m12.1c">\rho_{\leq m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.12.m12.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT</annotation></semantics></math> as playing after <math alttext="\rho_{\leq n}" class="ltx_Math" display="inline" id="S3.SS3.p2.13.m13.1"><semantics id="S3.SS3.p2.13.m13.1a"><msub id="S3.SS3.p2.13.m13.1.1" xref="S3.SS3.p2.13.m13.1.1.cmml"><mi id="S3.SS3.p2.13.m13.1.1.2" xref="S3.SS3.p2.13.m13.1.1.2.cmml">ρ</mi><mrow id="S3.SS3.p2.13.m13.1.1.3" xref="S3.SS3.p2.13.m13.1.1.3.cmml"><mi id="S3.SS3.p2.13.m13.1.1.3.2" xref="S3.SS3.p2.13.m13.1.1.3.2.cmml"></mi><mo id="S3.SS3.p2.13.m13.1.1.3.1" xref="S3.SS3.p2.13.m13.1.1.3.1.cmml">≤</mo><mi id="S3.SS3.p2.13.m13.1.1.3.3" xref="S3.SS3.p2.13.m13.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.13.m13.1b"><apply id="S3.SS3.p2.13.m13.1.1.cmml" xref="S3.SS3.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.13.m13.1.1.1.cmml" xref="S3.SS3.p2.13.m13.1.1">subscript</csymbol><ci id="S3.SS3.p2.13.m13.1.1.2.cmml" xref="S3.SS3.p2.13.m13.1.1.2">𝜌</ci><apply id="S3.SS3.p2.13.m13.1.1.3.cmml" xref="S3.SS3.p2.13.m13.1.1.3"><leq id="S3.SS3.p2.13.m13.1.1.3.1.cmml" xref="S3.SS3.p2.13.m13.1.1.3.1"></leq><csymbol cd="latexml" id="S3.SS3.p2.13.m13.1.1.3.2.cmml" xref="S3.SS3.p2.13.m13.1.1.3.2">absent</csymbol><ci id="S3.SS3.p2.13.m13.1.1.3.3.cmml" xref="S3.SS3.p2.13.m13.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.13.m13.1c">\rho_{\leq n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.13.m13.1d">italic_ρ start_POSTSUBSCRIPT ≤ italic_n end_POSTSUBSCRIPT</annotation></semantics></math> (thus deleting the cycle <math alttext="\rho_{[m,n]}" class="ltx_Math" display="inline" id="S3.SS3.p2.14.m14.2"><semantics id="S3.SS3.p2.14.m14.2a"><msub id="S3.SS3.p2.14.m14.2.3" xref="S3.SS3.p2.14.m14.2.3.cmml"><mi id="S3.SS3.p2.14.m14.2.3.2" xref="S3.SS3.p2.14.m14.2.3.2.cmml">ρ</mi><mrow id="S3.SS3.p2.14.m14.2.2.2.4" xref="S3.SS3.p2.14.m14.2.2.2.3.cmml"><mo id="S3.SS3.p2.14.m14.2.2.2.4.1" stretchy="false" xref="S3.SS3.p2.14.m14.2.2.2.3.cmml">[</mo><mi id="S3.SS3.p2.14.m14.1.1.1.1" xref="S3.SS3.p2.14.m14.1.1.1.1.cmml">m</mi><mo id="S3.SS3.p2.14.m14.2.2.2.4.2" xref="S3.SS3.p2.14.m14.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p2.14.m14.2.2.2.2" xref="S3.SS3.p2.14.m14.2.2.2.2.cmml">n</mi><mo id="S3.SS3.p2.14.m14.2.2.2.4.3" stretchy="false" xref="S3.SS3.p2.14.m14.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.14.m14.2b"><apply id="S3.SS3.p2.14.m14.2.3.cmml" xref="S3.SS3.p2.14.m14.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.14.m14.2.3.1.cmml" xref="S3.SS3.p2.14.m14.2.3">subscript</csymbol><ci id="S3.SS3.p2.14.m14.2.3.2.cmml" xref="S3.SS3.p2.14.m14.2.3.2">𝜌</ci><interval closure="closed" id="S3.SS3.p2.14.m14.2.2.2.3.cmml" xref="S3.SS3.p2.14.m14.2.2.2.4"><ci id="S3.SS3.p2.14.m14.1.1.1.1.cmml" xref="S3.SS3.p2.14.m14.1.1.1.1">𝑚</ci><ci id="S3.SS3.p2.14.m14.2.2.2.2.cmml" xref="S3.SS3.p2.14.m14.2.2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.14.m14.2c">\rho_{[m,n]}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.14.m14.2d">italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT</annotation></semantics></math>). See Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.F3" title="Figure 3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F3"><svg class="ltx_picture ltx_centering" height="529.78" id="S3.F3.pic1" overflow="visible" version="1.1" width="257.17"><g transform="translate(0,529.78) matrix(1 0 0 -1 0 0) translate(-224.65,0) translate(0,544.52)"><g fill="#000000" stroke="#000000" stroke-opacity="1" stroke-width="1.5pt"><path d="M 359.47 -28.27 L 469.46 -221.33" style="fill:none"></path></g><g fill="#000000" stroke="#000000"><g stroke-width="1.5pt"><path d="M 334.44 -28.13 L 225.68 -221.08" style="fill:none"></path></g><g stroke-width="1.5pt"><path d="M 359.47 -28.27 C 359.47 -21.38 353.88 -15.78 346.99 -15.78 L 346.93 -15.78 C 340.04 -15.78 334.44 -21.38 334.44 -28.27 L 334.44 -28.27 C 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id="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜌</ci><ci id="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\rho_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g></g></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Illustration of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a></figcaption> </figure> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.1">From now on, we say that we can <em class="ltx_emph ltx_font_italic" id="S3.SS3.p3.1.1">eliminate cycles</em> according to this lemma<span class="ltx_note ltx_role_footnote" id="footnote7"><sup class="ltx_note_mark">7</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">7</sup><span class="ltx_tag ltx_tag_note">7</span>These are the cycles satisfying the lemma, and not just any cycle.</span></span></span> without explicitly building the new strategy. We also say that a solution <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><msub id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml">σ</mi><mn id="S3.SS3.p3.1.m1.1.1.3" xref="S3.SS3.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2">𝜎</ci><cn id="S3.SS3.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S3.SS3.p3.1.2">without cycles</em> if it does not satisfy the hypotheses of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>, i.e., if it is impossible to eliminate cycles to get a better solution.</p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Proof 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2"> </span>(Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>)</h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.9.9">Let <math alttext="g=\rho_{\leq m}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">g</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">=</mo><msub id="S3.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml">ρ</mi><mrow id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.2" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.2.cmml"></mi><mo id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.1.cmml">≤</mo><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.3" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.3.cmml">m</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.1b"><apply id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1"><eq id="S3.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.1"></eq><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.2">𝑔</ci><apply id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2">𝜌</ci><apply id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3"><leq id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.1"></leq><csymbol cd="latexml" id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.2">absent</csymbol><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.1c">g=\rho_{\leq m}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.1d">italic_g = italic_ρ start_POSTSUBSCRIPT ≤ italic_m end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="h=\rho_{\leq n}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">h</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">=</mo><msub id="S3.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2.cmml">ρ</mi><mrow id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.2.cmml"></mi><mo id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.1.cmml">≤</mo><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1"><eq id="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1"></eq><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2">ℎ</ci><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2">𝜌</ci><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3"><leq id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.1"></leq><csymbol cd="latexml" id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.2">absent</csymbol><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.1c">h=\rho_{\leq n}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.1d">italic_h = italic_ρ start_POSTSUBSCRIPT ≤ italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. We introduce the following notation: for each play <math alttext="\pi=\pi_{0}\pi_{1}\ldots\in\textsf{Play}_{G}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.3.m3.1"><semantics id="S3.Thmtheorem4.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem4.p1.3.3.m3.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.cmml">=</mo><mrow id="S3.Thmtheorem4.p1.3.3.m3.1.1.4" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.cmml"><msub id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.2.cmml">π</mi><mn id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.1.cmml">⁢</mo><msub id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.2.cmml">π</mi><mn id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.3.cmml">1</mn></msub><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.1a" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.4" mathvariant="normal" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.4.cmml">…</mi></mrow><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.5" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.5.cmml">∈</mo><msub id="S3.Thmtheorem4.p1.3.3.m3.1.1.6" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p1.3.3.m3.1.1.6.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6.2a.cmml">Play</mtext><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.6.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6.3.cmml">G</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.3.3.m3.1b"><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"><and id="S3.Thmtheorem4.p1.3.3.m3.1.1a.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"></and><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1b.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"><eq id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3"></eq><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.2">𝜋</ci><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4"><times id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.1"></times><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2">subscript</csymbol><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.2">𝜋</ci><cn id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.2.3">0</cn></apply><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3">subscript</csymbol><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.2">𝜋</ci><cn id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.3.3">1</cn></apply><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.4.4.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.4.4">…</ci></apply></apply><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1c.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"><in id="S3.Thmtheorem4.p1.3.3.m3.1.1.5.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.5"></in><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem4.p1.3.3.m3.1.1.4.cmml" id="S3.Thmtheorem4.p1.3.3.m3.1.1d.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"></share><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.6.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.3.3.m3.1.1.6.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6">subscript</csymbol><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.6.2a.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p1.3.3.m3.1.1.6.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6.2">Play</mtext></ci><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.6.3.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.6.3">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.3.m3.1c">\pi=\pi_{0}\pi_{1}\ldots\in\textsf{Play}_{G}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.3.m3.1d">italic_π = italic_π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … ∈ Play start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="h\pi" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.4.m4.1"><semantics id="S3.Thmtheorem4.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem4.p1.4.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">h</mi><mo id="S3.Thmtheorem4.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.4.m4.1b"><apply id="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1"><times id="S3.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.1"></times><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2">ℎ</ci><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.4.m4.1c">h\pi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.4.m4.1d">italic_h italic_π</annotation></semantics></math>, we denote by <math alttext="\bar{\pi}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.5.5.m5.1"><semantics id="S3.Thmtheorem4.p1.5.5.m5.1a"><mover accent="true" id="S3.Thmtheorem4.p1.5.5.m5.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem4.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.5.5.m5.1b"><apply id="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1"><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.1">¯</ci><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.2">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.5.5.m5.1c">\bar{\pi}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.5.5.m5.1d">over¯ start_ARG italic_π end_ARG</annotation></semantics></math> the play <math alttext="g\pi_{\geq n+1}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.6.6.m6.1"><semantics id="S3.Thmtheorem4.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem4.p1.6.6.m6.1.1" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1.2" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.2.cmml">g</mi><mo id="S3.Thmtheorem4.p1.6.6.m6.1.1.1" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.1.cmml">⁢</mo><msub id="S3.Thmtheorem4.p1.6.6.m6.1.1.3" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.cmml"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.2" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.2.cmml">π</mi><mrow id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.2" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.2.cmml"></mi><mo id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.1" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.1.cmml">≥</mo><mrow id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.cmml"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.2" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.2.cmml">n</mi><mo id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.1" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.1.cmml">+</mo><mn id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.3" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.3.cmml">1</mn></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.6.6.m6.1b"><apply id="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1"><times id="S3.Thmtheorem4.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.1"></times><ci id="S3.Thmtheorem4.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.2">𝑔</ci><apply id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.2">𝜋</ci><apply id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3"><geq id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.1"></geq><csymbol cd="latexml" id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.2">absent</csymbol><apply id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3"><plus id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.1"></plus><ci id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.2.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.2">𝑛</ci><cn id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.6.6.m6.1c">g\pi_{\geq n+1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.6.6.m6.1d">italic_g italic_π start_POSTSUBSCRIPT ≥ italic_n + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, that is, we delete the cycle <math alttext="\rho_{[m,n]}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.7.7.m7.2"><semantics id="S3.Thmtheorem4.p1.7.7.m7.2a"><msub id="S3.Thmtheorem4.p1.7.7.m7.2.3" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.cmml"><mi id="S3.Thmtheorem4.p1.7.7.m7.2.3.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.2.cmml">ρ</mi><mrow id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.4" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml"><mo id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.4.1" stretchy="false" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml">[</mo><mi id="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1" xref="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1.cmml">m</mi><mo id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.4.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.cmml">n</mi><mo id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.4.3" stretchy="false" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.7.7.m7.2b"><apply id="S3.Thmtheorem4.p1.7.7.m7.2.3.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.7.7.m7.2.3.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3">subscript</csymbol><ci id="S3.Thmtheorem4.p1.7.7.m7.2.3.2.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.2">𝜌</ci><interval closure="closed" id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.4"><ci id="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1">𝑚</ci><ci id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.7.7.m7.2c">\rho_{[m,n]}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.7.7.m7.2d">italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\pi" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.8.8.m8.1"><semantics id="S3.Thmtheorem4.p1.8.8.m8.1a"><mi id="S3.Thmtheorem4.p1.8.8.m8.1.1" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.8.8.m8.1b"><ci id="S3.Thmtheorem4.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.8.8.m8.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.8.8.m8.1d">italic_π</annotation></semantics></math> (we only keep the vertex <math alttext="\rho_{m}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.9.9.m9.1"><semantics id="S3.Thmtheorem4.p1.9.9.m9.1a"><msub id="S3.Thmtheorem4.p1.9.9.m9.1.1" xref="S3.Thmtheorem4.p1.9.9.m9.1.1.cmml"><mi id="S3.Thmtheorem4.p1.9.9.m9.1.1.2" xref="S3.Thmtheorem4.p1.9.9.m9.1.1.2.cmml">ρ</mi><mi id="S3.Thmtheorem4.p1.9.9.m9.1.1.3" xref="S3.Thmtheorem4.p1.9.9.m9.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.9.9.m9.1b"><apply id="S3.Thmtheorem4.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.9.9.m9.1.1.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p1.9.9.m9.1.1.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.1.1.2">𝜌</ci><ci id="S3.Thmtheorem4.p1.9.9.m9.1.1.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.9.9.m9.1c">\rho_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.9.9.m9.1d">italic_ρ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>).</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem4.p2"> <p class="ltx_p" id="S3.Thmtheorem4.p2.25"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p2.25.25">We first prove that <math alttext="\sigma^{\prime}_{0}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.1.1.m1.1"><semantics id="S3.Thmtheorem4.p2.1.1.m1.1a"><mrow id="S3.Thmtheorem4.p2.1.1.m1.1.1" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.cmml"><msubsup id="S3.Thmtheorem4.p2.1.1.m1.1.1.2" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.2" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.3" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.3" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem4.p2.1.1.m1.1.1.1" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.1.cmml">⪯</mo><msub id="S3.Thmtheorem4.p2.1.1.m1.1.1.3" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.1.1.m1.1.1.3.2" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.1.1.m1.1.1.3.3" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.1.1.m1.1b"><apply id="S3.Thmtheorem4.p2.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1"><csymbol cd="latexml" id="S3.Thmtheorem4.p2.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.1">precedes-or-equals</csymbol><apply id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2">subscript</csymbol><apply id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.1.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.2.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.3.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.1.1.m1.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.2.3">0</cn></apply><apply id="S3.Thmtheorem4.p2.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p2.1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.1.1.m1.1c">\sigma^{\prime}_{0}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.2.2.m2.1"><semantics id="S3.Thmtheorem4.p2.2.2.m2.1a"><mrow id="S3.Thmtheorem4.p2.2.2.m2.1.1" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem4.p2.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.2.2.m2.1.1.1" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p2.2.2.m2.1.1.3" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.2" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.2" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.3" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.2.2.m2.1b"><apply id="S3.Thmtheorem4.p2.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1"><in id="S3.Thmtheorem4.p2.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem4.p2.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.2">𝑐</ci><apply id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.2.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.2.2.m2.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.2.2.m2.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\pi\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.3.3.m3.1"><semantics id="S3.Thmtheorem4.p2.3.3.m3.1a"><mrow id="S3.Thmtheorem4.p2.3.3.m3.1.1" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem4.p2.3.3.m3.1.1.2" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p2.3.3.m3.1.1.1" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p2.3.3.m3.1.1.3" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.2" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.2a.cmml">Wit</mtext><msub id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.2" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.3" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.3.3.m3.1b"><apply id="S3.Thmtheorem4.p2.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1"><in id="S3.Thmtheorem4.p2.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem4.p2.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.2">𝜋</ci><apply id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.2a.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.2">Wit</mtext></ci><apply id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.3.3.m3.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.3.3.m3.1c">\pi\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.3.3.m3.1d">italic_π ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be a witness with <math alttext="\textsf{cost}({\pi})=c" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.4.4.m4.1"><semantics id="S3.Thmtheorem4.p2.4.4.m4.1a"><mrow id="S3.Thmtheorem4.p2.4.4.m4.1.2" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.cmml"><mrow id="S3.Thmtheorem4.p2.4.4.m4.1.2.2" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.2" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.2a.cmml">cost</mtext><mo id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.1" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.3.2" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.cmml"><mo id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem4.p2.4.4.m4.1.1" xref="S3.Thmtheorem4.p2.4.4.m4.1.1.cmml">π</mi><mo id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p2.4.4.m4.1.2.1" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.1.cmml">=</mo><mi id="S3.Thmtheorem4.p2.4.4.m4.1.2.3" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.4.4.m4.1b"><apply id="S3.Thmtheorem4.p2.4.4.m4.1.2.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.2"><eq id="S3.Thmtheorem4.p2.4.4.m4.1.2.1.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.1"></eq><apply id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2"><times id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.1.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.1"></times><ci id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.2a.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.4.4.m4.1.2.2.2.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.2.2">cost</mtext></ci><ci id="S3.Thmtheorem4.p2.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.1">𝜋</ci></apply><ci id="S3.Thmtheorem4.p2.4.4.m4.1.2.3.cmml" xref="S3.Thmtheorem4.p2.4.4.m4.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.4.4.m4.1c">\textsf{cost}({\pi})=c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.4.4.m4.1d">cost ( italic_π ) = italic_c</annotation></semantics></math>. Let us prove that there exists <math alttext="c^{\prime}\in C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.5.5.m5.1"><semantics id="S3.Thmtheorem4.p2.5.5.m5.1a"><mrow id="S3.Thmtheorem4.p2.5.5.m5.1.1" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.cmml"><msup id="S3.Thmtheorem4.p2.5.5.m5.1.1.2" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.5.5.m5.1.1.2.2" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.5.5.m5.1.1.2.3" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p2.5.5.m5.1.1.1" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p2.5.5.m5.1.1.3" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.2" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.2" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.3" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.5.5.m5.1b"><apply id="S3.Thmtheorem4.p2.5.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1"><in id="S3.Thmtheorem4.p2.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.1"></in><apply id="S3.Thmtheorem4.p2.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.5.5.m5.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.5.5.m5.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p2.5.5.m5.1.1.2.3.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3">subscript</csymbol><apply id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.3.cmml" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.5.5.m5.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.5.5.m5.1c">c^{\prime}\in C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.5.5.m5.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c^{\prime}\leq c" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.6.6.m6.1"><semantics id="S3.Thmtheorem4.p2.6.6.m6.1a"><mrow id="S3.Thmtheorem4.p2.6.6.m6.1.1" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.cmml"><msup id="S3.Thmtheorem4.p2.6.6.m6.1.1.2" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.6.6.m6.1.1.2.2" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.6.6.m6.1.1.2.3" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p2.6.6.m6.1.1.1" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.1.cmml">≤</mo><mi id="S3.Thmtheorem4.p2.6.6.m6.1.1.3" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.6.6.m6.1b"><apply id="S3.Thmtheorem4.p2.6.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p2.6.6.m6.1.1"><leq id="S3.Thmtheorem4.p2.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.1"></leq><apply id="S3.Thmtheorem4.p2.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.6.6.m6.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.6.6.m6.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p2.6.6.m6.1.1.2.3.cmml" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.2.3">′</ci></apply><ci id="S3.Thmtheorem4.p2.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem4.p2.6.6.m6.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.6.6.m6.1c">c^{\prime}\leq c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.6.6.m6.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_c</annotation></semantics></math>. If <math alttext="g\not\pi" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.7.7.m7.1"><semantics id="S3.Thmtheorem4.p2.7.7.m7.1a"><mrow id="S3.Thmtheorem4.p2.7.7.m7.1.1" xref="S3.Thmtheorem4.p2.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem4.p2.7.7.m7.1.1.2" xref="S3.Thmtheorem4.p2.7.7.m7.1.1.2.cmml">g</mi><mo id="S3.Thmtheorem4.p2.7.7.m7.1.1.1" xref="S3.Thmtheorem4.p2.7.7.m7.1.1.1.cmml">⁢</mo><mi class="ltx_mathvariant_italic" id="S3.Thmtheorem4.p2.7.7.m7.1.1.3" mathvariant="italic" xref="S3.Thmtheorem4.p2.7.7.m7.1.1.3.cmml">π̸</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.7.7.m7.1b"><apply id="S3.Thmtheorem4.p2.7.7.m7.1.1.cmml" xref="S3.Thmtheorem4.p2.7.7.m7.1.1"><times id="S3.Thmtheorem4.p2.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem4.p2.7.7.m7.1.1.1"></times><ci id="S3.Thmtheorem4.p2.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem4.p2.7.7.m7.1.1.2">𝑔</ci><ci id="S3.Thmtheorem4.p2.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem4.p2.7.7.m7.1.1.3">italic-π̸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.7.7.m7.1c">g\not\pi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.7.7.m7.1d">italic_g italic_π̸</annotation></semantics></math>, then <math alttext="\pi\in\textsf{Play}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.8.8.m8.1"><semantics id="S3.Thmtheorem4.p2.8.8.m8.1a"><mrow id="S3.Thmtheorem4.p2.8.8.m8.1.1" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem4.p2.8.8.m8.1.1.2" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p2.8.8.m8.1.1.1" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p2.8.8.m8.1.1.3" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.2" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.2a.cmml">Play</mtext><msubsup id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.2" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.3" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.3" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.8.8.m8.1b"><apply id="S3.Thmtheorem4.p2.8.8.m8.1.1.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1"><in id="S3.Thmtheorem4.p2.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.1"></in><ci id="S3.Thmtheorem4.p2.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.2">𝜋</ci><apply id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.2a.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.2">Play</mtext></ci><apply id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3">subscript</csymbol><apply id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.3.cmml" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.8.8.m8.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.8.8.m8.1c">\pi\in\textsf{Play}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.8.8.m8.1d">italic_π ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> by definition of <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.9.9.m9.1"><semantics id="S3.Thmtheorem4.p2.9.9.m9.1a"><msubsup id="S3.Thmtheorem4.p2.9.9.m9.1.1" xref="S3.Thmtheorem4.p2.9.9.m9.1.1.cmml"><mi id="S3.Thmtheorem4.p2.9.9.m9.1.1.2.2" xref="S3.Thmtheorem4.p2.9.9.m9.1.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.9.9.m9.1.1.3" xref="S3.Thmtheorem4.p2.9.9.m9.1.1.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.9.9.m9.1.1.2.3" xref="S3.Thmtheorem4.p2.9.9.m9.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.9.9.m9.1b"><apply id="S3.Thmtheorem4.p2.9.9.m9.1.1.cmml" xref="S3.Thmtheorem4.p2.9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.9.9.m9.1.1.1.cmml" xref="S3.Thmtheorem4.p2.9.9.m9.1.1">subscript</csymbol><apply id="S3.Thmtheorem4.p2.9.9.m9.1.1.2.cmml" xref="S3.Thmtheorem4.p2.9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.9.9.m9.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.9.9.m9.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p2.9.9.m9.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.9.9.m9.1.1.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.9.9.m9.1.1.2.3.cmml" xref="S3.Thmtheorem4.p2.9.9.m9.1.1.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.9.9.m9.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.9.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.9.9.m9.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.9.9.m9.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, and thus there exists <math alttext="c^{\prime}\in C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.10.10.m10.1"><semantics id="S3.Thmtheorem4.p2.10.10.m10.1a"><mrow id="S3.Thmtheorem4.p2.10.10.m10.1.1" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.cmml"><msup id="S3.Thmtheorem4.p2.10.10.m10.1.1.2" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.10.10.m10.1.1.2.2" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.10.10.m10.1.1.2.3" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p2.10.10.m10.1.1.1" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p2.10.10.m10.1.1.3" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.2" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.2" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.3" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.3" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.10.10.m10.1b"><apply id="S3.Thmtheorem4.p2.10.10.m10.1.1.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1"><in id="S3.Thmtheorem4.p2.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.1"></in><apply id="S3.Thmtheorem4.p2.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.10.10.m10.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.10.10.m10.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p2.10.10.m10.1.1.2.3.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3">subscript</csymbol><apply id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.3.cmml" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.10.10.m10.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.10.10.m10.1c">c^{\prime}\in C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.10.10.m10.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c^{\prime}\leq c" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.11.11.m11.1"><semantics id="S3.Thmtheorem4.p2.11.11.m11.1a"><mrow id="S3.Thmtheorem4.p2.11.11.m11.1.1" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.cmml"><msup id="S3.Thmtheorem4.p2.11.11.m11.1.1.2" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.11.11.m11.1.1.2.2" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.11.11.m11.1.1.2.3" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p2.11.11.m11.1.1.1" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.1.cmml">≤</mo><mi id="S3.Thmtheorem4.p2.11.11.m11.1.1.3" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.11.11.m11.1b"><apply id="S3.Thmtheorem4.p2.11.11.m11.1.1.cmml" xref="S3.Thmtheorem4.p2.11.11.m11.1.1"><leq id="S3.Thmtheorem4.p2.11.11.m11.1.1.1.cmml" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.1"></leq><apply id="S3.Thmtheorem4.p2.11.11.m11.1.1.2.cmml" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.11.11.m11.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.11.11.m11.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p2.11.11.m11.1.1.2.3.cmml" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.2.3">′</ci></apply><ci id="S3.Thmtheorem4.p2.11.11.m11.1.1.3.cmml" xref="S3.Thmtheorem4.p2.11.11.m11.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.11.11.m11.1c">c^{\prime}\leq c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.11.11.m11.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_c</annotation></semantics></math> by definition of <math alttext="C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.12.12.m12.1"><semantics id="S3.Thmtheorem4.p2.12.12.m12.1a"><msub id="S3.Thmtheorem4.p2.12.12.m12.1.1" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.cmml"><mi id="S3.Thmtheorem4.p2.12.12.m12.1.1.2" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p2.12.12.m12.1.1.3" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.2" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.3" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.3" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.12.12.m12.1b"><apply id="S3.Thmtheorem4.p2.12.12.m12.1.1.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.12.12.m12.1.1.1.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p2.12.12.m12.1.1.2.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.2">𝐶</ci><apply id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3">subscript</csymbol><apply id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.1.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.2.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.3.cmml" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.12.12.m12.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.12.12.m12.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.12.12.m12.1c">C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.12.12.m12.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. If <math alttext="g\pi" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.13.13.m13.1"><semantics id="S3.Thmtheorem4.p2.13.13.m13.1a"><mrow id="S3.Thmtheorem4.p2.13.13.m13.1.1" xref="S3.Thmtheorem4.p2.13.13.m13.1.1.cmml"><mi id="S3.Thmtheorem4.p2.13.13.m13.1.1.2" xref="S3.Thmtheorem4.p2.13.13.m13.1.1.2.cmml">g</mi><mo id="S3.Thmtheorem4.p2.13.13.m13.1.1.1" xref="S3.Thmtheorem4.p2.13.13.m13.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p2.13.13.m13.1.1.3" xref="S3.Thmtheorem4.p2.13.13.m13.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.13.13.m13.1b"><apply id="S3.Thmtheorem4.p2.13.13.m13.1.1.cmml" xref="S3.Thmtheorem4.p2.13.13.m13.1.1"><times id="S3.Thmtheorem4.p2.13.13.m13.1.1.1.cmml" xref="S3.Thmtheorem4.p2.13.13.m13.1.1.1"></times><ci id="S3.Thmtheorem4.p2.13.13.m13.1.1.2.cmml" xref="S3.Thmtheorem4.p2.13.13.m13.1.1.2">𝑔</ci><ci id="S3.Thmtheorem4.p2.13.13.m13.1.1.3.cmml" xref="S3.Thmtheorem4.p2.13.13.m13.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.13.13.m13.1c">g\pi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.13.13.m13.1d">italic_g italic_π</annotation></semantics></math>, as <math alttext="g\sim h" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.14.14.m14.1"><semantics id="S3.Thmtheorem4.p2.14.14.m14.1a"><mrow id="S3.Thmtheorem4.p2.14.14.m14.1.1" xref="S3.Thmtheorem4.p2.14.14.m14.1.1.cmml"><mi id="S3.Thmtheorem4.p2.14.14.m14.1.1.2" xref="S3.Thmtheorem4.p2.14.14.m14.1.1.2.cmml">g</mi><mo id="S3.Thmtheorem4.p2.14.14.m14.1.1.1" xref="S3.Thmtheorem4.p2.14.14.m14.1.1.1.cmml">∼</mo><mi id="S3.Thmtheorem4.p2.14.14.m14.1.1.3" xref="S3.Thmtheorem4.p2.14.14.m14.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.14.14.m14.1b"><apply id="S3.Thmtheorem4.p2.14.14.m14.1.1.cmml" xref="S3.Thmtheorem4.p2.14.14.m14.1.1"><csymbol cd="latexml" id="S3.Thmtheorem4.p2.14.14.m14.1.1.1.cmml" xref="S3.Thmtheorem4.p2.14.14.m14.1.1.1">similar-to</csymbol><ci id="S3.Thmtheorem4.p2.14.14.m14.1.1.2.cmml" xref="S3.Thmtheorem4.p2.14.14.m14.1.1.2">𝑔</ci><ci id="S3.Thmtheorem4.p2.14.14.m14.1.1.3.cmml" xref="S3.Thmtheorem4.p2.14.14.m14.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.14.14.m14.1c">g\sim h</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.14.14.m14.1d">italic_g ∼ italic_h</annotation></semantics></math> by hypothesis, we have that <math alttext="\textsf{Wit}_{\sigma_{0}}(g)=\textsf{Wit}_{\sigma_{0}}(h)" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.15.15.m15.2"><semantics id="S3.Thmtheorem4.p2.15.15.m15.2a"><mrow id="S3.Thmtheorem4.p2.15.15.m15.2.3" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.cmml"><mrow id="S3.Thmtheorem4.p2.15.15.m15.2.3.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.cmml"><msub id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.2a.cmml">Wit</mtext><msub id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.cmml"><mi id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.3" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.3.cmml">0</mn></msub></msub><mo id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.1" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.3.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.cmml"><mo id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.cmml">(</mo><mi id="S3.Thmtheorem4.p2.15.15.m15.1.1" xref="S3.Thmtheorem4.p2.15.15.m15.1.1.cmml">g</mi><mo id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p2.15.15.m15.2.3.1" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.1.cmml">=</mo><mrow id="S3.Thmtheorem4.p2.15.15.m15.2.3.3" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.cmml"><msub id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.2a.cmml">Wit</mtext><msub id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.cmml"><mi id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.3" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.3.cmml">0</mn></msub></msub><mo id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.1" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.3.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.cmml"><mo id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.cmml">(</mo><mi id="S3.Thmtheorem4.p2.15.15.m15.2.2" xref="S3.Thmtheorem4.p2.15.15.m15.2.2.cmml">h</mi><mo id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.15.15.m15.2b"><apply id="S3.Thmtheorem4.p2.15.15.m15.2.3.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3"><eq id="S3.Thmtheorem4.p2.15.15.m15.2.3.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.1"></eq><apply id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2"><times id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.1"></times><apply id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2">subscript</csymbol><ci id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.2a.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.2">Wit</mtext></ci><apply id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.2.2.3.3">0</cn></apply></apply><ci id="S3.Thmtheorem4.p2.15.15.m15.1.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.1.1">𝑔</ci></apply><apply id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3"><times id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.1"></times><apply id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2">subscript</csymbol><ci id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.2a.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.2">Wit</mtext></ci><apply id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.1.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.15.15.m15.2.3.3.2.3.3">0</cn></apply></apply><ci id="S3.Thmtheorem4.p2.15.15.m15.2.2.cmml" xref="S3.Thmtheorem4.p2.15.15.m15.2.2">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.15.15.m15.2c">\textsf{Wit}_{\sigma_{0}}(g)=\textsf{Wit}_{\sigma_{0}}(h)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.15.15.m15.2d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_g ) = Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h )</annotation></semantics></math> and no new target is visited from <math alttext="g" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.16.16.m16.1"><semantics id="S3.Thmtheorem4.p2.16.16.m16.1a"><mi id="S3.Thmtheorem4.p2.16.16.m16.1.1" xref="S3.Thmtheorem4.p2.16.16.m16.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.16.16.m16.1b"><ci id="S3.Thmtheorem4.p2.16.16.m16.1.1.cmml" xref="S3.Thmtheorem4.p2.16.16.m16.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.16.16.m16.1c">g</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.16.16.m16.1d">italic_g</annotation></semantics></math> to <math alttext="h" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.17.17.m17.1"><semantics id="S3.Thmtheorem4.p2.17.17.m17.1a"><mi id="S3.Thmtheorem4.p2.17.17.m17.1.1" xref="S3.Thmtheorem4.p2.17.17.m17.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.17.17.m17.1b"><ci id="S3.Thmtheorem4.p2.17.17.m17.1.1.cmml" xref="S3.Thmtheorem4.p2.17.17.m17.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.17.17.m17.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.17.17.m17.1d">italic_h</annotation></semantics></math>. It follows that <math alttext="h\pi" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.18.18.m18.1"><semantics id="S3.Thmtheorem4.p2.18.18.m18.1a"><mrow id="S3.Thmtheorem4.p2.18.18.m18.1.1" xref="S3.Thmtheorem4.p2.18.18.m18.1.1.cmml"><mi id="S3.Thmtheorem4.p2.18.18.m18.1.1.2" xref="S3.Thmtheorem4.p2.18.18.m18.1.1.2.cmml">h</mi><mo id="S3.Thmtheorem4.p2.18.18.m18.1.1.1" xref="S3.Thmtheorem4.p2.18.18.m18.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p2.18.18.m18.1.1.3" xref="S3.Thmtheorem4.p2.18.18.m18.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.18.18.m18.1b"><apply id="S3.Thmtheorem4.p2.18.18.m18.1.1.cmml" xref="S3.Thmtheorem4.p2.18.18.m18.1.1"><times id="S3.Thmtheorem4.p2.18.18.m18.1.1.1.cmml" xref="S3.Thmtheorem4.p2.18.18.m18.1.1.1"></times><ci id="S3.Thmtheorem4.p2.18.18.m18.1.1.2.cmml" xref="S3.Thmtheorem4.p2.18.18.m18.1.1.2">ℎ</ci><ci id="S3.Thmtheorem4.p2.18.18.m18.1.1.3.cmml" xref="S3.Thmtheorem4.p2.18.18.m18.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.18.18.m18.1c">h\pi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.18.18.m18.1d">italic_h italic_π</annotation></semantics></math> and <math alttext="\bar{\pi}\in\textsf{Play}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.19.19.m19.1"><semantics id="S3.Thmtheorem4.p2.19.19.m19.1a"><mrow id="S3.Thmtheorem4.p2.19.19.m19.1.1" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.cmml"><mover accent="true" id="S3.Thmtheorem4.p2.19.19.m19.1.1.2" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.19.19.m19.1.1.2.2" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.2.2.cmml">π</mi><mo id="S3.Thmtheorem4.p2.19.19.m19.1.1.2.1" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.2.1.cmml">¯</mo></mover><mo id="S3.Thmtheorem4.p2.19.19.m19.1.1.1" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p2.19.19.m19.1.1.3" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.2" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.2a.cmml">Play</mtext><msubsup id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.2" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.3" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.3" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.19.19.m19.1b"><apply id="S3.Thmtheorem4.p2.19.19.m19.1.1.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1"><in id="S3.Thmtheorem4.p2.19.19.m19.1.1.1.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.1"></in><apply id="S3.Thmtheorem4.p2.19.19.m19.1.1.2.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.2"><ci id="S3.Thmtheorem4.p2.19.19.m19.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.2.1">¯</ci><ci id="S3.Thmtheorem4.p2.19.19.m19.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.2.2">𝜋</ci></apply><apply id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.2a.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.2">Play</mtext></ci><apply id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3">subscript</csymbol><apply id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.3.cmml" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.19.19.m19.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.19.19.m19.1c">\bar{\pi}\in\textsf{Play}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.19.19.m19.1d">over¯ start_ARG italic_π end_ARG ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\textsf{cost}({\bar{\pi}})=\textsf{cost}({\pi})-w(\rho_{[m,n]})\leq\textsf{% cost}({\pi})" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.20.20.m20.6"><semantics id="S3.Thmtheorem4.p2.20.20.m20.6a"><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.cmml"><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.3" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3.2a.cmml">cost</mtext><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.1" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.3.2" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.cmml"><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.cmml">(</mo><mover accent="true" id="S3.Thmtheorem4.p2.20.20.m20.3.3" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.cmml"><mi id="S3.Thmtheorem4.p2.20.20.m20.3.3.2" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.2.cmml">π</mi><mo id="S3.Thmtheorem4.p2.20.20.m20.3.3.1" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.1.cmml">¯</mo></mover><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.4" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.4.cmml">=</mo><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.1" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.cmml"><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.2a.cmml">cost</mtext><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.1" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.3.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.cmml"><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.cmml">(</mo><mi id="S3.Thmtheorem4.p2.20.20.m20.4.4" xref="S3.Thmtheorem4.p2.20.20.m20.4.4.cmml">π</mi><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.2.cmml">−</mo><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.cmml"><mi id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.3" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.3.cmml">w</mi><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.2.cmml">ρ</mi><mrow id="S3.Thmtheorem4.p2.20.20.m20.2.2.2.4" xref="S3.Thmtheorem4.p2.20.20.m20.2.2.2.3.cmml"><mo id="S3.Thmtheorem4.p2.20.20.m20.2.2.2.4.1" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.2.2.2.3.cmml">[</mo><mi id="S3.Thmtheorem4.p2.20.20.m20.1.1.1.1" xref="S3.Thmtheorem4.p2.20.20.m20.1.1.1.1.cmml">m</mi><mo id="S3.Thmtheorem4.p2.20.20.m20.2.2.2.4.2" xref="S3.Thmtheorem4.p2.20.20.m20.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem4.p2.20.20.m20.2.2.2.2" xref="S3.Thmtheorem4.p2.20.20.m20.2.2.2.2.cmml">n</mi><mo id="S3.Thmtheorem4.p2.20.20.m20.2.2.2.4.3" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.2.2.2.3.cmml">]</mo></mrow></msub><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.5" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.5.cmml">≤</mo><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.6" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.2a.cmml">cost</mtext><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.1" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.3.2" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.cmml"><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.cmml">(</mo><mi id="S3.Thmtheorem4.p2.20.20.m20.5.5" xref="S3.Thmtheorem4.p2.20.20.m20.5.5.cmml">π</mi><mo id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.20.20.m20.6b"><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6"><and id="S3.Thmtheorem4.p2.20.20.m20.6.6a.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6"></and><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6b.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6"><eq id="S3.Thmtheorem4.p2.20.20.m20.6.6.4.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.4"></eq><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3"><times id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3.1"></times><ci id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.2a.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.20.20.m20.6.6.3.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3.2">cost</mtext></ci><apply id="S3.Thmtheorem4.p2.20.20.m20.3.3.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.3.3.2"><ci id="S3.Thmtheorem4.p2.20.20.m20.3.3.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.1">¯</ci><ci id="S3.Thmtheorem4.p2.20.20.m20.3.3.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.3.3.2">𝜋</ci></apply></apply><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1"><minus id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.2"></minus><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3"><times id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.1"></times><ci id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.2a.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.3.2">cost</mtext></ci><ci id="S3.Thmtheorem4.p2.20.20.m20.4.4.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.4.4">𝜋</ci></apply><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1"><times id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.2"></times><ci id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.3.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.3">𝑤</ci><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.1.1.1.1.1.2">𝜌</ci><interval closure="closed" id="S3.Thmtheorem4.p2.20.20.m20.2.2.2.3.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.2.2.2.4"><ci id="S3.Thmtheorem4.p2.20.20.m20.1.1.1.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.1.1.1.1">𝑚</ci><ci id="S3.Thmtheorem4.p2.20.20.m20.2.2.2.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.2.2.2.2">𝑛</ci></interval></apply></apply></apply></apply><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6c.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6"><leq id="S3.Thmtheorem4.p2.20.20.m20.6.6.5.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.5"></leq><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem4.p2.20.20.m20.6.6.1.cmml" id="S3.Thmtheorem4.p2.20.20.m20.6.6d.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6"></share><apply id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6"><times id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.1.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.1"></times><ci id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.2a.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.20.20.m20.6.6.6.2.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.6.6.6.2">cost</mtext></ci><ci id="S3.Thmtheorem4.p2.20.20.m20.5.5.cmml" xref="S3.Thmtheorem4.p2.20.20.m20.5.5">𝜋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.20.20.m20.6c">\textsf{cost}({\bar{\pi}})=\textsf{cost}({\pi})-w(\rho_{[m,n]})\leq\textsf{% cost}({\pi})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.20.20.m20.6d">cost ( over¯ start_ARG italic_π end_ARG ) = cost ( italic_π ) - italic_w ( italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT ) ≤ cost ( italic_π )</annotation></semantics></math>. By definition of <math alttext="C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.21.21.m21.1"><semantics id="S3.Thmtheorem4.p2.21.21.m21.1a"><msub id="S3.Thmtheorem4.p2.21.21.m21.1.1" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.cmml"><mi id="S3.Thmtheorem4.p2.21.21.m21.1.1.2" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p2.21.21.m21.1.1.3" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.2" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.3" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.3" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.21.21.m21.1b"><apply id="S3.Thmtheorem4.p2.21.21.m21.1.1.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.21.21.m21.1.1.1.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p2.21.21.m21.1.1.2.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.2">𝐶</ci><apply id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3">subscript</csymbol><apply id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.1.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.2.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.3.cmml" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.21.21.m21.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.21.21.m21.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.21.21.m21.1c">C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.21.21.m21.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, there exists <math alttext="c^{\prime}\in C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.22.22.m22.1"><semantics id="S3.Thmtheorem4.p2.22.22.m22.1a"><mrow id="S3.Thmtheorem4.p2.22.22.m22.1.1" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.cmml"><msup id="S3.Thmtheorem4.p2.22.22.m22.1.1.2" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.22.22.m22.1.1.2.2" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.22.22.m22.1.1.2.3" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p2.22.22.m22.1.1.1" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p2.22.22.m22.1.1.3" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.2" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.2" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.3" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.3" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.22.22.m22.1b"><apply id="S3.Thmtheorem4.p2.22.22.m22.1.1.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1"><in id="S3.Thmtheorem4.p2.22.22.m22.1.1.1.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.1"></in><apply id="S3.Thmtheorem4.p2.22.22.m22.1.1.2.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.22.22.m22.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.22.22.m22.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p2.22.22.m22.1.1.2.3.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3">subscript</csymbol><apply id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.3.cmml" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.22.22.m22.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.22.22.m22.1c">c^{\prime}\in C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.22.22.m22.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c^{\prime}\leq\textsf{cost}({\bar{\pi}})" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.23.23.m23.1"><semantics id="S3.Thmtheorem4.p2.23.23.m23.1a"><mrow id="S3.Thmtheorem4.p2.23.23.m23.1.2" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.cmml"><msup id="S3.Thmtheorem4.p2.23.23.m23.1.2.2" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.2.cmml"><mi id="S3.Thmtheorem4.p2.23.23.m23.1.2.2.2" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.23.23.m23.1.2.2.3" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p2.23.23.m23.1.2.1" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.1.cmml">≤</mo><mrow id="S3.Thmtheorem4.p2.23.23.m23.1.2.3" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.2" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3.2a.cmml">cost</mtext><mo id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.1" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.3.2" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.cmml"><mo id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem4.p2.23.23.m23.1.1" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.cmml"><mi id="S3.Thmtheorem4.p2.23.23.m23.1.1.2" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p2.23.23.m23.1.1.1" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.1.cmml">¯</mo></mover><mo id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.23.23.m23.1b"><apply id="S3.Thmtheorem4.p2.23.23.m23.1.2.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2"><leq id="S3.Thmtheorem4.p2.23.23.m23.1.2.1.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.1"></leq><apply id="S3.Thmtheorem4.p2.23.23.m23.1.2.2.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.23.23.m23.1.2.2.1.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.23.23.m23.1.2.2.2.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p2.23.23.m23.1.2.2.3.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3"><times id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.1.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3.1"></times><ci id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.2a.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.23.23.m23.1.2.3.2.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3.2">cost</mtext></ci><apply id="S3.Thmtheorem4.p2.23.23.m23.1.1.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.2.3.3.2"><ci id="S3.Thmtheorem4.p2.23.23.m23.1.1.1.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.1">¯</ci><ci id="S3.Thmtheorem4.p2.23.23.m23.1.1.2.cmml" xref="S3.Thmtheorem4.p2.23.23.m23.1.1.2">𝜋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.23.23.m23.1c">c^{\prime}\leq\textsf{cost}({\bar{\pi}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.23.23.m23.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ cost ( over¯ start_ARG italic_π end_ARG )</annotation></semantics></math>, and therefore <math alttext="c^{\prime}\leq\textsf{cost}({\pi})=c" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.24.24.m24.1"><semantics id="S3.Thmtheorem4.p2.24.24.m24.1a"><mrow id="S3.Thmtheorem4.p2.24.24.m24.1.2" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.cmml"><msup id="S3.Thmtheorem4.p2.24.24.m24.1.2.2" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.2.cmml"><mi id="S3.Thmtheorem4.p2.24.24.m24.1.2.2.2" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p2.24.24.m24.1.2.2.3" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p2.24.24.m24.1.2.3" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.3.cmml">≤</mo><mrow id="S3.Thmtheorem4.p2.24.24.m24.1.2.4" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.2" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.2a.cmml">cost</mtext><mo id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.1" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.3.2" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.cmml"><mo id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.cmml">(</mo><mi id="S3.Thmtheorem4.p2.24.24.m24.1.1" xref="S3.Thmtheorem4.p2.24.24.m24.1.1.cmml">π</mi><mo id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p2.24.24.m24.1.2.5" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.5.cmml">=</mo><mi id="S3.Thmtheorem4.p2.24.24.m24.1.2.6" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.6.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.24.24.m24.1b"><apply id="S3.Thmtheorem4.p2.24.24.m24.1.2.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2"><and id="S3.Thmtheorem4.p2.24.24.m24.1.2a.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2"></and><apply id="S3.Thmtheorem4.p2.24.24.m24.1.2b.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2"><leq id="S3.Thmtheorem4.p2.24.24.m24.1.2.3.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.3"></leq><apply id="S3.Thmtheorem4.p2.24.24.m24.1.2.2.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.24.24.m24.1.2.2.1.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.24.24.m24.1.2.2.2.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p2.24.24.m24.1.2.2.3.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4"><times id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.1.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.1"></times><ci id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.2a.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p2.24.24.m24.1.2.4.2.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.4.2">cost</mtext></ci><ci id="S3.Thmtheorem4.p2.24.24.m24.1.1.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.1">𝜋</ci></apply></apply><apply id="S3.Thmtheorem4.p2.24.24.m24.1.2c.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2"><eq id="S3.Thmtheorem4.p2.24.24.m24.1.2.5.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.5"></eq><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem4.p2.24.24.m24.1.2.4.cmml" id="S3.Thmtheorem4.p2.24.24.m24.1.2d.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2"></share><ci id="S3.Thmtheorem4.p2.24.24.m24.1.2.6.cmml" xref="S3.Thmtheorem4.p2.24.24.m24.1.2.6">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.24.24.m24.1c">c^{\prime}\leq\textsf{cost}({\pi})=c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.24.24.m24.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ cost ( italic_π ) = italic_c</annotation></semantics></math>. Hence <math alttext="\sigma^{\prime}_{0}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p2.25.25.m25.1"><semantics id="S3.Thmtheorem4.p2.25.25.m25.1a"><mrow id="S3.Thmtheorem4.p2.25.25.m25.1.1" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.cmml"><msubsup id="S3.Thmtheorem4.p2.25.25.m25.1.1.2" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2.cmml"><mi id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.2" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.3" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2.3.cmml">0</mn><mo id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.3" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem4.p2.25.25.m25.1.1.1" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.1.cmml">⪯</mo><msub id="S3.Thmtheorem4.p2.25.25.m25.1.1.3" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.3.cmml"><mi id="S3.Thmtheorem4.p2.25.25.m25.1.1.3.2" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p2.25.25.m25.1.1.3.3" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p2.25.25.m25.1b"><apply id="S3.Thmtheorem4.p2.25.25.m25.1.1.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1"><csymbol cd="latexml" id="S3.Thmtheorem4.p2.25.25.m25.1.1.1.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.1">precedes-or-equals</csymbol><apply id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.1.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2">subscript</csymbol><apply id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.1.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.2.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.3.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p2.25.25.m25.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.2.3">0</cn></apply><apply id="S3.Thmtheorem4.p2.25.25.m25.1.1.3.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p2.25.25.m25.1.1.3.1.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p2.25.25.m25.1.1.3.2.cmml" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p2.25.25.m25.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p2.25.25.m25.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p2.25.25.m25.1c">\sigma^{\prime}_{0}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p2.25.25.m25.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem4.p3"> <p class="ltx_p" id="S3.Thmtheorem4.p3.7"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p3.7.7">We then prove that, if <math alttext="\sigma^{\prime}_{0}\simeq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.1.1.m1.1"><semantics id="S3.Thmtheorem4.p3.1.1.m1.1a"><mrow id="S3.Thmtheorem4.p3.1.1.m1.1.1" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.cmml"><msubsup id="S3.Thmtheorem4.p3.1.1.m1.1.1.2" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2.cmml"><mi id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.2" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.3" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2.3.cmml">0</mn><mo id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.3" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem4.p3.1.1.m1.1.1.1" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.1.cmml">≃</mo><msub id="S3.Thmtheorem4.p3.1.1.m1.1.1.3" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem4.p3.1.1.m1.1.1.3.2" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.1.1.m1.1.1.3.3" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.1.1.m1.1b"><apply id="S3.Thmtheorem4.p3.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1"><csymbol cd="latexml" id="S3.Thmtheorem4.p3.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.1">similar-to-or-equals</csymbol><apply id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.1.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2">subscript</csymbol><apply id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.1.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.2.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.3.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p3.1.1.m1.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.2.3">0</cn></apply><apply id="S3.Thmtheorem4.p3.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p3.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p3.1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.1.1.m1.1c">\sigma^{\prime}_{0}\simeq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≃ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> (that is <math alttext="C_{\sigma^{\prime}_{0}}=C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.2.2.m2.1"><semantics id="S3.Thmtheorem4.p3.2.2.m2.1a"><mrow id="S3.Thmtheorem4.p3.2.2.m2.1.1" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.cmml"><msub id="S3.Thmtheorem4.p3.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.cmml"><mi id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.2" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.cmml"><mi id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.2" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.3" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.3" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.Thmtheorem4.p3.2.2.m2.1.1.1" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.1.cmml">=</mo><msub id="S3.Thmtheorem4.p3.2.2.m2.1.1.3" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.2" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.2" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.3" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.2.2.m2.1b"><apply id="S3.Thmtheorem4.p3.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1"><eq id="S3.Thmtheorem4.p3.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.1"></eq><apply id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.1.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.2.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.2">𝐶</ci><apply id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.1.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3">subscript</csymbol><apply id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.1.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3">superscript</csymbol><ci id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.2.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.3.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.2.3.3">0</cn></apply></apply><apply id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.2.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.2.2.m2.1c">C_{\sigma^{\prime}_{0}}=C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.2.2.m2.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>), then <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.3.3.m3.1"><semantics id="S3.Thmtheorem4.p3.3.3.m3.1a"><msubsup id="S3.Thmtheorem4.p3.3.3.m3.1.1" xref="S3.Thmtheorem4.p3.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem4.p3.3.3.m3.1.1.2.2" xref="S3.Thmtheorem4.p3.3.3.m3.1.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.3.3.m3.1.1.3" xref="S3.Thmtheorem4.p3.3.3.m3.1.1.3.cmml">0</mn><mo id="S3.Thmtheorem4.p3.3.3.m3.1.1.2.3" xref="S3.Thmtheorem4.p3.3.3.m3.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.3.3.m3.1b"><apply id="S3.Thmtheorem4.p3.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p3.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem4.p3.3.3.m3.1.1">subscript</csymbol><apply id="S3.Thmtheorem4.p3.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem4.p3.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.3.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem4.p3.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p3.3.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem4.p3.3.3.m3.1.1.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p3.3.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem4.p3.3.3.m3.1.1.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p3.3.3.m3.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.3.3.m3.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.3.3.m3.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> has a witness set <math alttext="\textsf{Wit}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.4.4.m4.1"><semantics id="S3.Thmtheorem4.p3.4.4.m4.1a"><msub id="S3.Thmtheorem4.p3.4.4.m4.1.1" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.4.4.m4.1.1.2" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.2a.cmml">Wit</mtext><msubsup id="S3.Thmtheorem4.p3.4.4.m4.1.1.3" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3.cmml"><mi id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.2" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.3" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.3" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.4.4.m4.1b"><apply id="S3.Thmtheorem4.p3.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p3.4.4.m4.1.1.2a.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.2">Wit</mtext></ci><apply id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3">subscript</csymbol><apply id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.1.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.2.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.3.cmml" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p3.4.4.m4.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.4.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.4.4.m4.1c">\textsf{Wit}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.4.4.m4.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\textsf{length}({\textsf{Wit}_{\sigma^{\prime}_{0}}})<\textsf{length}({\textsf% {Wit}_{\sigma_{0}}})" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.5.5.m5.2"><semantics id="S3.Thmtheorem4.p3.5.5.m5.2a"><mrow id="S3.Thmtheorem4.p3.5.5.m5.2.2" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.cmml"><mrow id="S3.Thmtheorem4.p3.5.5.m5.1.1.1" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.3" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.3a.cmml">length</mtext><mo id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.2" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.2" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.2a.cmml">Wit</mtext><msubsup id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.cmml"><mi id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.2" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.3" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.3" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p3.5.5.m5.2.2.3" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.3.cmml"><</mo><mrow id="S3.Thmtheorem4.p3.5.5.m5.2.2.2" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.3" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.3a.cmml">length</mtext><mo id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.2" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.2" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.2a.cmml">Wit</mtext><msub id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.cmml"><mi id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.2" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.3" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.3" stretchy="false" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.5.5.m5.2b"><apply id="S3.Thmtheorem4.p3.5.5.m5.2.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2"><lt id="S3.Thmtheorem4.p3.5.5.m5.2.2.3.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.3"></lt><apply id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1"><times id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.2"></times><ci id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.3a.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.3.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.3">length</mtext></ci><apply id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.2a.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.2">Wit</mtext></ci><apply id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3">subscript</csymbol><apply id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.3.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.5.5.m5.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2"><times id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.2"></times><ci id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.3a.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.3.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.3">length</mtext></ci><apply id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.2a.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.2">Wit</mtext></ci><apply id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.1.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.2.cmml" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.5.5.m5.2.2.2.1.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.5.5.m5.2c">\textsf{length}({\textsf{Wit}_{\sigma^{\prime}_{0}}})<\textsf{length}({\textsf% {Wit}_{\sigma_{0}}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.5.5.m5.2d">length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) < length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>. Note that we necessarily have <math alttext="w(\rho_{[m,n]}=0)" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.6.6.m6.3"><semantics id="S3.Thmtheorem4.p3.6.6.m6.3a"><mrow id="S3.Thmtheorem4.p3.6.6.m6.3.3" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.cmml"><mi id="S3.Thmtheorem4.p3.6.6.m6.3.3.3" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.3.cmml">w</mi><mo id="S3.Thmtheorem4.p3.6.6.m6.3.3.2" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.2.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.cmml"><mo id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.2" stretchy="false" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.cmml"><msub id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2.cmml"><mi id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2.2" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2.2.cmml">ρ</mi><mrow id="S3.Thmtheorem4.p3.6.6.m6.2.2.2.4" xref="S3.Thmtheorem4.p3.6.6.m6.2.2.2.3.cmml"><mo id="S3.Thmtheorem4.p3.6.6.m6.2.2.2.4.1" stretchy="false" xref="S3.Thmtheorem4.p3.6.6.m6.2.2.2.3.cmml">[</mo><mi id="S3.Thmtheorem4.p3.6.6.m6.1.1.1.1" xref="S3.Thmtheorem4.p3.6.6.m6.1.1.1.1.cmml">m</mi><mo id="S3.Thmtheorem4.p3.6.6.m6.2.2.2.4.2" xref="S3.Thmtheorem4.p3.6.6.m6.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem4.p3.6.6.m6.2.2.2.2" xref="S3.Thmtheorem4.p3.6.6.m6.2.2.2.2.cmml">n</mi><mo id="S3.Thmtheorem4.p3.6.6.m6.2.2.2.4.3" stretchy="false" xref="S3.Thmtheorem4.p3.6.6.m6.2.2.2.3.cmml">]</mo></mrow></msub><mo id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.1" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.1.cmml">=</mo><mn id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.3" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.3.cmml">0</mn></mrow><mo id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.3" stretchy="false" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.6.6.m6.3b"><apply id="S3.Thmtheorem4.p3.6.6.m6.3.3.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3"><times id="S3.Thmtheorem4.p3.6.6.m6.3.3.2.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.2"></times><ci id="S3.Thmtheorem4.p3.6.6.m6.3.3.3.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.3">𝑤</ci><apply id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1"><eq id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.1"></eq><apply id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2.1.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2.2.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.2.2">𝜌</ci><interval closure="closed" id="S3.Thmtheorem4.p3.6.6.m6.2.2.2.3.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.2.2.2.4"><ci id="S3.Thmtheorem4.p3.6.6.m6.1.1.1.1.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.1.1.1.1">𝑚</ci><ci id="S3.Thmtheorem4.p3.6.6.m6.2.2.2.2.cmml" xref="S3.Thmtheorem4.p3.6.6.m6.2.2.2.2">𝑛</ci></interval></apply><cn id="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.6.6.m6.3.3.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.6.6.m6.3c">w(\rho_{[m,n]}=0)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.6.6.m6.3d">italic_w ( italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT = 0 )</annotation></semantics></math>, as <math alttext="C_{\sigma^{\prime}_{0}}=C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.7.7.m7.1"><semantics id="S3.Thmtheorem4.p3.7.7.m7.1a"><mrow id="S3.Thmtheorem4.p3.7.7.m7.1.1" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.cmml"><msub id="S3.Thmtheorem4.p3.7.7.m7.1.1.2" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.cmml"><mi id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.2" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.cmml"><mi id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.2" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.3" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.3" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.Thmtheorem4.p3.7.7.m7.1.1.1" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.1.cmml">=</mo><msub id="S3.Thmtheorem4.p3.7.7.m7.1.1.3" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.cmml"><mi id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.2" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.2.cmml">C</mi><msub id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.2" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.3" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.7.7.m7.1b"><apply id="S3.Thmtheorem4.p3.7.7.m7.1.1.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1"><eq id="S3.Thmtheorem4.p3.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.1"></eq><apply id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.1.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.2.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.2">𝐶</ci><apply id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.1.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3">subscript</csymbol><apply id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.1.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3">superscript</csymbol><ci id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.2.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.3.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.2.3.3">0</cn></apply></apply><apply id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.1.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.2.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.7.7.m7.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.7.7.m7.1c">C_{\sigma^{\prime}_{0}}=C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.7.7.m7.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Let us show that the set</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex5"> <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="\textsf{W}=\{\pi\mid\pi\in\textsf{Wit}_{\sigma_{0}}\mbox{ and }g\not\pi\}\cup% \{\bar{\pi}\mid\pi\in\textsf{Wit}_{\sigma_{0}}\mbox{ and }g\pi\}" class="ltx_Math" display="block" id="S3.Ex5.m1.4"><semantics id="S3.Ex5.m1.4a"><mrow id="S3.Ex5.m1.4.4" xref="S3.Ex5.m1.4.4.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.Ex5.m1.4.4.4" xref="S3.Ex5.m1.4.4.4a.cmml">W</mtext><mo id="S3.Ex5.m1.4.4.3" xref="S3.Ex5.m1.4.4.3.cmml">=</mo><mrow id="S3.Ex5.m1.4.4.2" xref="S3.Ex5.m1.4.4.2.cmml"><mrow id="S3.Ex5.m1.3.3.1.1.1" xref="S3.Ex5.m1.3.3.1.1.2.cmml"><mo id="S3.Ex5.m1.3.3.1.1.1.2" stretchy="false" xref="S3.Ex5.m1.3.3.1.1.2.1.cmml">{</mo><mi id="S3.Ex5.m1.1.1" xref="S3.Ex5.m1.1.1.cmml">π</mi><mo fence="true" id="S3.Ex5.m1.3.3.1.1.1.3" lspace="0em" rspace="0em" xref="S3.Ex5.m1.3.3.1.1.2.1.cmml">∣</mo><mrow id="S3.Ex5.m1.3.3.1.1.1.1" xref="S3.Ex5.m1.3.3.1.1.1.1.cmml"><mi id="S3.Ex5.m1.3.3.1.1.1.1.2" xref="S3.Ex5.m1.3.3.1.1.1.1.2.cmml">π</mi><mo id="S3.Ex5.m1.3.3.1.1.1.1.1" xref="S3.Ex5.m1.3.3.1.1.1.1.1.cmml">∈</mo><mrow id="S3.Ex5.m1.3.3.1.1.1.1.3" xref="S3.Ex5.m1.3.3.1.1.1.1.3.cmml"><msub id="S3.Ex5.m1.3.3.1.1.1.1.3.2" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.Ex5.m1.3.3.1.1.1.1.3.2.2" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2.2a.cmml">Wit</mtext><msub id="S3.Ex5.m1.3.3.1.1.1.1.3.2.3" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2.3.cmml"><mi id="S3.Ex5.m1.3.3.1.1.1.1.3.2.3.2" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2.3.2.cmml">σ</mi><mn id="S3.Ex5.m1.3.3.1.1.1.1.3.2.3.3" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2.3.3.cmml">0</mn></msub></msub><mo id="S3.Ex5.m1.3.3.1.1.1.1.3.1" xref="S3.Ex5.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S3.Ex5.m1.3.3.1.1.1.1.3.3" xref="S3.Ex5.m1.3.3.1.1.1.1.3.3a.cmml"> and </mtext><mo id="S3.Ex5.m1.3.3.1.1.1.1.3.1a" xref="S3.Ex5.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex5.m1.3.3.1.1.1.1.3.4" xref="S3.Ex5.m1.3.3.1.1.1.1.3.4.cmml">g</mi><mo id="S3.Ex5.m1.3.3.1.1.1.1.3.1b" xref="S3.Ex5.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi class="ltx_mathvariant_italic" id="S3.Ex5.m1.3.3.1.1.1.1.3.5" mathvariant="italic" xref="S3.Ex5.m1.3.3.1.1.1.1.3.5.cmml">π̸</mi></mrow></mrow><mo id="S3.Ex5.m1.3.3.1.1.1.4" stretchy="false" xref="S3.Ex5.m1.3.3.1.1.2.1.cmml">}</mo></mrow><mo id="S3.Ex5.m1.4.4.2.3" xref="S3.Ex5.m1.4.4.2.3.cmml">∪</mo><mrow id="S3.Ex5.m1.4.4.2.2.1" xref="S3.Ex5.m1.4.4.2.2.2.cmml"><mo id="S3.Ex5.m1.4.4.2.2.1.2" stretchy="false" xref="S3.Ex5.m1.4.4.2.2.2.1.cmml">{</mo><mover accent="true" id="S3.Ex5.m1.2.2" xref="S3.Ex5.m1.2.2.cmml"><mi id="S3.Ex5.m1.2.2.2" xref="S3.Ex5.m1.2.2.2.cmml">π</mi><mo id="S3.Ex5.m1.2.2.1" xref="S3.Ex5.m1.2.2.1.cmml">¯</mo></mover><mo fence="true" id="S3.Ex5.m1.4.4.2.2.1.3" lspace="0em" rspace="0em" xref="S3.Ex5.m1.4.4.2.2.2.1.cmml">∣</mo><mrow id="S3.Ex5.m1.4.4.2.2.1.1" xref="S3.Ex5.m1.4.4.2.2.1.1.cmml"><mi id="S3.Ex5.m1.4.4.2.2.1.1.2" xref="S3.Ex5.m1.4.4.2.2.1.1.2.cmml">π</mi><mo id="S3.Ex5.m1.4.4.2.2.1.1.1" xref="S3.Ex5.m1.4.4.2.2.1.1.1.cmml">∈</mo><mrow id="S3.Ex5.m1.4.4.2.2.1.1.3" xref="S3.Ex5.m1.4.4.2.2.1.1.3.cmml"><msub id="S3.Ex5.m1.4.4.2.2.1.1.3.2" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.Ex5.m1.4.4.2.2.1.1.3.2.2" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.2a.cmml">Wit</mtext><msub id="S3.Ex5.m1.4.4.2.2.1.1.3.2.3" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.cmml"><mi id="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.2" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.2.cmml">σ</mi><mn id="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.3" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.3.cmml">0</mn></msub></msub><mo id="S3.Ex5.m1.4.4.2.2.1.1.3.1" xref="S3.Ex5.m1.4.4.2.2.1.1.3.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S3.Ex5.m1.4.4.2.2.1.1.3.3" xref="S3.Ex5.m1.4.4.2.2.1.1.3.3a.cmml"> and </mtext><mo id="S3.Ex5.m1.4.4.2.2.1.1.3.1a" xref="S3.Ex5.m1.4.4.2.2.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex5.m1.4.4.2.2.1.1.3.4" xref="S3.Ex5.m1.4.4.2.2.1.1.3.4.cmml">g</mi><mo id="S3.Ex5.m1.4.4.2.2.1.1.3.1b" xref="S3.Ex5.m1.4.4.2.2.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex5.m1.4.4.2.2.1.1.3.5" xref="S3.Ex5.m1.4.4.2.2.1.1.3.5.cmml">π</mi></mrow></mrow><mo id="S3.Ex5.m1.4.4.2.2.1.4" stretchy="false" xref="S3.Ex5.m1.4.4.2.2.2.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex5.m1.4b"><apply id="S3.Ex5.m1.4.4.cmml" xref="S3.Ex5.m1.4.4"><eq id="S3.Ex5.m1.4.4.3.cmml" xref="S3.Ex5.m1.4.4.3"></eq><ci id="S3.Ex5.m1.4.4.4a.cmml" xref="S3.Ex5.m1.4.4.4"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.Ex5.m1.4.4.4.cmml" xref="S3.Ex5.m1.4.4.4">W</mtext></ci><apply id="S3.Ex5.m1.4.4.2.cmml" xref="S3.Ex5.m1.4.4.2"><union id="S3.Ex5.m1.4.4.2.3.cmml" xref="S3.Ex5.m1.4.4.2.3"></union><apply id="S3.Ex5.m1.3.3.1.1.2.cmml" xref="S3.Ex5.m1.3.3.1.1.1"><csymbol cd="latexml" id="S3.Ex5.m1.3.3.1.1.2.1.cmml" xref="S3.Ex5.m1.3.3.1.1.1.2">conditional-set</csymbol><ci id="S3.Ex5.m1.1.1.cmml" xref="S3.Ex5.m1.1.1">𝜋</ci><apply id="S3.Ex5.m1.3.3.1.1.1.1.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1"><in id="S3.Ex5.m1.3.3.1.1.1.1.1.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1.1"></in><ci id="S3.Ex5.m1.3.3.1.1.1.1.2.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1.2">𝜋</ci><apply id="S3.Ex5.m1.3.3.1.1.1.1.3.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1.3"><times id="S3.Ex5.m1.3.3.1.1.1.1.3.1.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1.3.1"></times><apply id="S3.Ex5.m1.3.3.1.1.1.1.3.2.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.1.1.1.1.3.2.1.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2">subscript</csymbol><ci id="S3.Ex5.m1.3.3.1.1.1.1.3.2.2a.cmml" xref="S3.Ex5.m1.3.3.1.1.1.1.3.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.Ex5.m1.3.3.1.1.1.1.3.2.2.cmml" 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xref="S3.Ex5.m1.4.4.2.2.1.2">conditional-set</csymbol><apply id="S3.Ex5.m1.2.2.cmml" xref="S3.Ex5.m1.2.2"><ci id="S3.Ex5.m1.2.2.1.cmml" xref="S3.Ex5.m1.2.2.1">¯</ci><ci id="S3.Ex5.m1.2.2.2.cmml" xref="S3.Ex5.m1.2.2.2">𝜋</ci></apply><apply id="S3.Ex5.m1.4.4.2.2.1.1.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1"><in id="S3.Ex5.m1.4.4.2.2.1.1.1.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.1"></in><ci id="S3.Ex5.m1.4.4.2.2.1.1.2.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.2">𝜋</ci><apply id="S3.Ex5.m1.4.4.2.2.1.1.3.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3"><times id="S3.Ex5.m1.4.4.2.2.1.1.3.1.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.1"></times><apply id="S3.Ex5.m1.4.4.2.2.1.1.3.2.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.4.4.2.2.1.1.3.2.1.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2">subscript</csymbol><ci id="S3.Ex5.m1.4.4.2.2.1.1.3.2.2a.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.Ex5.m1.4.4.2.2.1.1.3.2.2.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.2">Wit</mtext></ci><apply id="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.1.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.3">subscript</csymbol><ci id="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.2.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.2">𝜎</ci><cn id="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.3.cmml" type="integer" xref="S3.Ex5.m1.4.4.2.2.1.1.3.2.3.3">0</cn></apply></apply><ci id="S3.Ex5.m1.4.4.2.2.1.1.3.3a.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.3"><mtext class="ltx_mathvariant_italic" id="S3.Ex5.m1.4.4.2.2.1.1.3.3.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.3"> and </mtext></ci><ci id="S3.Ex5.m1.4.4.2.2.1.1.3.4.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.4">𝑔</ci><ci id="S3.Ex5.m1.4.4.2.2.1.1.3.5.cmml" xref="S3.Ex5.m1.4.4.2.2.1.1.3.5">𝜋</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex5.m1.4c">\textsf{W}=\{\pi\mid\pi\in\textsf{Wit}_{\sigma_{0}}\mbox{ and }g\not\pi\}\cup% \{\bar{\pi}\mid\pi\in\textsf{Wit}_{\sigma_{0}}\mbox{ and }g\pi\}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5.m1.4d">W = { italic_π ∣ italic_π ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and italic_g italic_π̸ } ∪ { over¯ start_ARG italic_π end_ARG ∣ italic_π ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and italic_g italic_π }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem4.p3.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p3.8.1">is the required witness set <math alttext="\textsf{Wit}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p3.8.1.m1.1"><semantics id="S3.Thmtheorem4.p3.8.1.m1.1a"><msub id="S3.Thmtheorem4.p3.8.1.m1.1.1" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.8.1.m1.1.1.2" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.2a.cmml">Wit</mtext><msubsup id="S3.Thmtheorem4.p3.8.1.m1.1.1.3" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.2" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.3" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.3" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p3.8.1.m1.1b"><apply id="S3.Thmtheorem4.p3.8.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.8.1.m1.1.1.1.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p3.8.1.m1.1.1.2a.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p3.8.1.m1.1.1.2.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.2">Wit</mtext></ci><apply id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3">subscript</csymbol><apply id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.1.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.2.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.3.cmml" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p3.8.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p3.8.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p3.8.1.m1.1c">\textsf{Wit}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p3.8.1.m1.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> <ul class="ltx_itemize" id="S3.I3"> <li class="ltx_item" id="S3.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I3.i1.p1"> <p class="ltx_p" id="S3.I3.i1.p1.9"><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.1">First note that in this set </span><span class="ltx_text ltx_markedasmath ltx_font_sansserif ltx_font_italic" id="S3.I3.i1.p1.9.2">W</span><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.3">, </span><math alttext="g\pi" class="ltx_Math" display="inline" id="S3.I3.i1.p1.2.m2.1"><semantics id="S3.I3.i1.p1.2.m2.1a"><mrow id="S3.I3.i1.p1.2.m2.1.1" xref="S3.I3.i1.p1.2.m2.1.1.cmml"><mi id="S3.I3.i1.p1.2.m2.1.1.2" xref="S3.I3.i1.p1.2.m2.1.1.2.cmml">g</mi><mo id="S3.I3.i1.p1.2.m2.1.1.1" xref="S3.I3.i1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.I3.i1.p1.2.m2.1.1.3" xref="S3.I3.i1.p1.2.m2.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.2.m2.1b"><apply id="S3.I3.i1.p1.2.m2.1.1.cmml" xref="S3.I3.i1.p1.2.m2.1.1"><times id="S3.I3.i1.p1.2.m2.1.1.1.cmml" xref="S3.I3.i1.p1.2.m2.1.1.1"></times><ci id="S3.I3.i1.p1.2.m2.1.1.2.cmml" xref="S3.I3.i1.p1.2.m2.1.1.2">𝑔</ci><ci id="S3.I3.i1.p1.2.m2.1.1.3.cmml" xref="S3.I3.i1.p1.2.m2.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.2.m2.1c">g\pi</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.2.m2.1d">italic_g italic_π</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.4"> implies </span><math alttext="h\pi" class="ltx_Math" display="inline" id="S3.I3.i1.p1.3.m3.1"><semantics id="S3.I3.i1.p1.3.m3.1a"><mrow id="S3.I3.i1.p1.3.m3.1.1" xref="S3.I3.i1.p1.3.m3.1.1.cmml"><mi id="S3.I3.i1.p1.3.m3.1.1.2" xref="S3.I3.i1.p1.3.m3.1.1.2.cmml">h</mi><mo id="S3.I3.i1.p1.3.m3.1.1.1" xref="S3.I3.i1.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.I3.i1.p1.3.m3.1.1.3" xref="S3.I3.i1.p1.3.m3.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.3.m3.1b"><apply id="S3.I3.i1.p1.3.m3.1.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1"><times id="S3.I3.i1.p1.3.m3.1.1.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1.1"></times><ci id="S3.I3.i1.p1.3.m3.1.1.2.cmml" xref="S3.I3.i1.p1.3.m3.1.1.2">ℎ</ci><ci id="S3.I3.i1.p1.3.m3.1.1.3.cmml" xref="S3.I3.i1.p1.3.m3.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.3.m3.1c">h\pi</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.3.m3.1d">italic_h italic_π</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.5"> (as already explained in the first part of the proof). Hence, the number of plays in </span><span class="ltx_text ltx_markedasmath ltx_font_sansserif ltx_font_italic" id="S3.I3.i1.p1.9.6">W</span><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.7"> is equal to </span><math alttext="|\textsf{Wit}_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S3.I3.i1.p1.5.m5.1"><semantics id="S3.I3.i1.p1.5.m5.1a"><mrow id="S3.I3.i1.p1.5.m5.1.1.1" xref="S3.I3.i1.p1.5.m5.1.1.2.cmml"><mo id="S3.I3.i1.p1.5.m5.1.1.1.2" stretchy="false" xref="S3.I3.i1.p1.5.m5.1.1.2.1.cmml">|</mo><msub id="S3.I3.i1.p1.5.m5.1.1.1.1" xref="S3.I3.i1.p1.5.m5.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i1.p1.5.m5.1.1.1.1.2" xref="S3.I3.i1.p1.5.m5.1.1.1.1.2a.cmml">Wit</mtext><msub id="S3.I3.i1.p1.5.m5.1.1.1.1.3" xref="S3.I3.i1.p1.5.m5.1.1.1.1.3.cmml"><mi id="S3.I3.i1.p1.5.m5.1.1.1.1.3.2" xref="S3.I3.i1.p1.5.m5.1.1.1.1.3.2.cmml">σ</mi><mn id="S3.I3.i1.p1.5.m5.1.1.1.1.3.3" xref="S3.I3.i1.p1.5.m5.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S3.I3.i1.p1.5.m5.1.1.1.3" stretchy="false" xref="S3.I3.i1.p1.5.m5.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.5.m5.1b"><apply id="S3.I3.i1.p1.5.m5.1.1.2.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1"><abs id="S3.I3.i1.p1.5.m5.1.1.2.1.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.2"></abs><apply id="S3.I3.i1.p1.5.m5.1.1.1.1.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3.I3.i1.p1.5.m5.1.1.1.1.1.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S3.I3.i1.p1.5.m5.1.1.1.1.2a.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i1.p1.5.m5.1.1.1.1.2.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.1.2">Wit</mtext></ci><apply id="S3.I3.i1.p1.5.m5.1.1.1.1.3.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.5.m5.1.1.1.1.3.1.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.1.3">subscript</csymbol><ci id="S3.I3.i1.p1.5.m5.1.1.1.1.3.2.cmml" xref="S3.I3.i1.p1.5.m5.1.1.1.1.3.2">𝜎</ci><cn id="S3.I3.i1.p1.5.m5.1.1.1.1.3.3.cmml" type="integer" xref="S3.I3.i1.p1.5.m5.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.5.m5.1c">|\textsf{Wit}_{\sigma_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.5.m5.1d">| Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.8"> and their costs constitute the </span><math alttext="C_{\sigma_{0}}=C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.6.m6.1"><semantics id="S3.I3.i1.p1.6.m6.1a"><mrow id="S3.I3.i1.p1.6.m6.1.1" xref="S3.I3.i1.p1.6.m6.1.1.cmml"><msub id="S3.I3.i1.p1.6.m6.1.1.2" xref="S3.I3.i1.p1.6.m6.1.1.2.cmml"><mi id="S3.I3.i1.p1.6.m6.1.1.2.2" xref="S3.I3.i1.p1.6.m6.1.1.2.2.cmml">C</mi><msub id="S3.I3.i1.p1.6.m6.1.1.2.3" xref="S3.I3.i1.p1.6.m6.1.1.2.3.cmml"><mi id="S3.I3.i1.p1.6.m6.1.1.2.3.2" xref="S3.I3.i1.p1.6.m6.1.1.2.3.2.cmml">σ</mi><mn id="S3.I3.i1.p1.6.m6.1.1.2.3.3" xref="S3.I3.i1.p1.6.m6.1.1.2.3.3.cmml">0</mn></msub></msub><mo id="S3.I3.i1.p1.6.m6.1.1.1" xref="S3.I3.i1.p1.6.m6.1.1.1.cmml">=</mo><msub id="S3.I3.i1.p1.6.m6.1.1.3" xref="S3.I3.i1.p1.6.m6.1.1.3.cmml"><mi id="S3.I3.i1.p1.6.m6.1.1.3.2" xref="S3.I3.i1.p1.6.m6.1.1.3.2.cmml">C</mi><msubsup id="S3.I3.i1.p1.6.m6.1.1.3.3" xref="S3.I3.i1.p1.6.m6.1.1.3.3.cmml"><mi id="S3.I3.i1.p1.6.m6.1.1.3.3.2.2" xref="S3.I3.i1.p1.6.m6.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.I3.i1.p1.6.m6.1.1.3.3.3" xref="S3.I3.i1.p1.6.m6.1.1.3.3.3.cmml">0</mn><mo id="S3.I3.i1.p1.6.m6.1.1.3.3.2.3" xref="S3.I3.i1.p1.6.m6.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.6.m6.1b"><apply id="S3.I3.i1.p1.6.m6.1.1.cmml" xref="S3.I3.i1.p1.6.m6.1.1"><eq id="S3.I3.i1.p1.6.m6.1.1.1.cmml" xref="S3.I3.i1.p1.6.m6.1.1.1"></eq><apply id="S3.I3.i1.p1.6.m6.1.1.2.cmml" xref="S3.I3.i1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i1.p1.6.m6.1.1.2.1.cmml" xref="S3.I3.i1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S3.I3.i1.p1.6.m6.1.1.2.2.cmml" xref="S3.I3.i1.p1.6.m6.1.1.2.2">𝐶</ci><apply id="S3.I3.i1.p1.6.m6.1.1.2.3.cmml" xref="S3.I3.i1.p1.6.m6.1.1.2.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.6.m6.1.1.2.3.1.cmml" xref="S3.I3.i1.p1.6.m6.1.1.2.3">subscript</csymbol><ci id="S3.I3.i1.p1.6.m6.1.1.2.3.2.cmml" xref="S3.I3.i1.p1.6.m6.1.1.2.3.2">𝜎</ci><cn id="S3.I3.i1.p1.6.m6.1.1.2.3.3.cmml" type="integer" xref="S3.I3.i1.p1.6.m6.1.1.2.3.3">0</cn></apply></apply><apply id="S3.I3.i1.p1.6.m6.1.1.3.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.6.m6.1.1.3.1.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S3.I3.i1.p1.6.m6.1.1.3.2.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3.2">𝐶</ci><apply id="S3.I3.i1.p1.6.m6.1.1.3.3.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.6.m6.1.1.3.3.1.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3.3">subscript</csymbol><apply id="S3.I3.i1.p1.6.m6.1.1.3.3.2.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.6.m6.1.1.3.3.2.1.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3.3">superscript</csymbol><ci id="S3.I3.i1.p1.6.m6.1.1.3.3.2.2.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3.3.2.2">𝜎</ci><ci id="S3.I3.i1.p1.6.m6.1.1.3.3.2.3.cmml" xref="S3.I3.i1.p1.6.m6.1.1.3.3.2.3">′</ci></apply><cn id="S3.I3.i1.p1.6.m6.1.1.3.3.3.cmml" type="integer" xref="S3.I3.i1.p1.6.m6.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.6.m6.1c">C_{\sigma_{0}}=C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.6.m6.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.9">. Moreover, by definition of </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.7.m7.1"><semantics id="S3.I3.i1.p1.7.m7.1a"><msubsup id="S3.I3.i1.p1.7.m7.1.1" xref="S3.I3.i1.p1.7.m7.1.1.cmml"><mi id="S3.I3.i1.p1.7.m7.1.1.2.2" xref="S3.I3.i1.p1.7.m7.1.1.2.2.cmml">σ</mi><mn id="S3.I3.i1.p1.7.m7.1.1.3" xref="S3.I3.i1.p1.7.m7.1.1.3.cmml">0</mn><mo id="S3.I3.i1.p1.7.m7.1.1.2.3" xref="S3.I3.i1.p1.7.m7.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.7.m7.1b"><apply id="S3.I3.i1.p1.7.m7.1.1.cmml" xref="S3.I3.i1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.I3.i1.p1.7.m7.1.1.1.cmml" xref="S3.I3.i1.p1.7.m7.1.1">subscript</csymbol><apply id="S3.I3.i1.p1.7.m7.1.1.2.cmml" xref="S3.I3.i1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.I3.i1.p1.7.m7.1.1.2.1.cmml" xref="S3.I3.i1.p1.7.m7.1.1">superscript</csymbol><ci id="S3.I3.i1.p1.7.m7.1.1.2.2.cmml" xref="S3.I3.i1.p1.7.m7.1.1.2.2">𝜎</ci><ci id="S3.I3.i1.p1.7.m7.1.1.2.3.cmml" xref="S3.I3.i1.p1.7.m7.1.1.2.3">′</ci></apply><cn id="S3.I3.i1.p1.7.m7.1.1.3.cmml" type="integer" xref="S3.I3.i1.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.7.m7.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.7.m7.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.10">, </span><span class="ltx_text ltx_markedasmath ltx_font_sansserif ltx_font_italic" id="S3.I3.i1.p1.9.11">W</span><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.12"> is composed of plays consistent with </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.9.m9.1"><semantics id="S3.I3.i1.p1.9.m9.1a"><msubsup id="S3.I3.i1.p1.9.m9.1.1" xref="S3.I3.i1.p1.9.m9.1.1.cmml"><mi id="S3.I3.i1.p1.9.m9.1.1.2.2" xref="S3.I3.i1.p1.9.m9.1.1.2.2.cmml">σ</mi><mn id="S3.I3.i1.p1.9.m9.1.1.3" xref="S3.I3.i1.p1.9.m9.1.1.3.cmml">0</mn><mo id="S3.I3.i1.p1.9.m9.1.1.2.3" xref="S3.I3.i1.p1.9.m9.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.9.m9.1b"><apply id="S3.I3.i1.p1.9.m9.1.1.cmml" xref="S3.I3.i1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.I3.i1.p1.9.m9.1.1.1.cmml" xref="S3.I3.i1.p1.9.m9.1.1">subscript</csymbol><apply id="S3.I3.i1.p1.9.m9.1.1.2.cmml" xref="S3.I3.i1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.I3.i1.p1.9.m9.1.1.2.1.cmml" xref="S3.I3.i1.p1.9.m9.1.1">superscript</csymbol><ci id="S3.I3.i1.p1.9.m9.1.1.2.2.cmml" xref="S3.I3.i1.p1.9.m9.1.1.2.2">𝜎</ci><ci id="S3.I3.i1.p1.9.m9.1.1.2.3.cmml" xref="S3.I3.i1.p1.9.m9.1.1.2.3">′</ci></apply><cn id="S3.I3.i1.p1.9.m9.1.1.3.cmml" type="integer" xref="S3.I3.i1.p1.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.9.m9.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.9.m9.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.9.13">.</span></p> </div> </li> <li class="ltx_item" id="S3.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I3.i2.p1"> <p class="ltx_p" id="S3.I3.i2.p1.20"><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.1">Second, let us prove that </span><span class="ltx_text ltx_markedasmath ltx_font_sansserif ltx_font_italic" id="S3.I3.i2.p1.20.2">W</span><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.3"> is a witness set for </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.2.m2.1"><semantics id="S3.I3.i2.p1.2.m2.1a"><msubsup id="S3.I3.i2.p1.2.m2.1.1" xref="S3.I3.i2.p1.2.m2.1.1.cmml"><mi id="S3.I3.i2.p1.2.m2.1.1.2.2" xref="S3.I3.i2.p1.2.m2.1.1.2.2.cmml">σ</mi><mn id="S3.I3.i2.p1.2.m2.1.1.3" xref="S3.I3.i2.p1.2.m2.1.1.3.cmml">0</mn><mo id="S3.I3.i2.p1.2.m2.1.1.2.3" xref="S3.I3.i2.p1.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.2.m2.1b"><apply id="S3.I3.i2.p1.2.m2.1.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.2.m2.1.1.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1">subscript</csymbol><apply id="S3.I3.i2.p1.2.m2.1.1.2.cmml" xref="S3.I3.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.2.m2.1.1.2.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.2.m2.1.1.2.2.cmml" xref="S3.I3.i2.p1.2.m2.1.1.2.2">𝜎</ci><ci id="S3.I3.i2.p1.2.m2.1.1.2.3.cmml" xref="S3.I3.i2.p1.2.m2.1.1.2.3">′</ci></apply><cn id="S3.I3.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.I3.i2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.2.m2.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.4">. We have to prove that for any play </span><math alttext="\rho^{\prime}\in\textsf{Play}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.3.m3.1"><semantics id="S3.I3.i2.p1.3.m3.1a"><mrow id="S3.I3.i2.p1.3.m3.1.1" xref="S3.I3.i2.p1.3.m3.1.1.cmml"><msup id="S3.I3.i2.p1.3.m3.1.1.2" xref="S3.I3.i2.p1.3.m3.1.1.2.cmml"><mi id="S3.I3.i2.p1.3.m3.1.1.2.2" xref="S3.I3.i2.p1.3.m3.1.1.2.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.3.m3.1.1.2.3" xref="S3.I3.i2.p1.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.3.m3.1.1.1" xref="S3.I3.i2.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S3.I3.i2.p1.3.m3.1.1.3" xref="S3.I3.i2.p1.3.m3.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.3.m3.1.1.3.2" xref="S3.I3.i2.p1.3.m3.1.1.3.2a.cmml">Play</mtext><msubsup id="S3.I3.i2.p1.3.m3.1.1.3.3" xref="S3.I3.i2.p1.3.m3.1.1.3.3.cmml"><mi id="S3.I3.i2.p1.3.m3.1.1.3.3.2.2" xref="S3.I3.i2.p1.3.m3.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.I3.i2.p1.3.m3.1.1.3.3.3" xref="S3.I3.i2.p1.3.m3.1.1.3.3.3.cmml">0</mn><mo id="S3.I3.i2.p1.3.m3.1.1.3.3.2.3" xref="S3.I3.i2.p1.3.m3.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.3.m3.1b"><apply id="S3.I3.i2.p1.3.m3.1.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1"><in id="S3.I3.i2.p1.3.m3.1.1.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.1"></in><apply id="S3.I3.i2.p1.3.m3.1.1.2.cmml" xref="S3.I3.i2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.3.m3.1.1.2.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.3.m3.1.1.2.2.cmml" xref="S3.I3.i2.p1.3.m3.1.1.2.2">𝜌</ci><ci id="S3.I3.i2.p1.3.m3.1.1.2.3.cmml" xref="S3.I3.i2.p1.3.m3.1.1.2.3">′</ci></apply><apply id="S3.I3.i2.p1.3.m3.1.1.3.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.3.m3.1.1.3.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.I3.i2.p1.3.m3.1.1.3.2a.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.3.m3.1.1.3.2.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.2">Play</mtext></ci><apply id="S3.I3.i2.p1.3.m3.1.1.3.3.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.3.m3.1.1.3.3.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.3">subscript</csymbol><apply id="S3.I3.i2.p1.3.m3.1.1.3.3.2.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.3.m3.1.1.3.3.2.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.3">superscript</csymbol><ci id="S3.I3.i2.p1.3.m3.1.1.3.3.2.2.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.3.2.2">𝜎</ci><ci id="S3.I3.i2.p1.3.m3.1.1.3.3.2.3.cmml" xref="S3.I3.i2.p1.3.m3.1.1.3.3.2.3">′</ci></apply><cn id="S3.I3.i2.p1.3.m3.1.1.3.3.3.cmml" type="integer" xref="S3.I3.i2.p1.3.m3.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.3.m3.1c">\rho^{\prime}\in\textsf{Play}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.3.m3.1d">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.5">, there exists </span><math alttext="\pi^{\prime}\in\textsf{W}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.4.m4.1"><semantics id="S3.I3.i2.p1.4.m4.1a"><mrow id="S3.I3.i2.p1.4.m4.1.1" xref="S3.I3.i2.p1.4.m4.1.1.cmml"><msup id="S3.I3.i2.p1.4.m4.1.1.2" xref="S3.I3.i2.p1.4.m4.1.1.2.cmml"><mi id="S3.I3.i2.p1.4.m4.1.1.2.2" xref="S3.I3.i2.p1.4.m4.1.1.2.2.cmml">π</mi><mo id="S3.I3.i2.p1.4.m4.1.1.2.3" xref="S3.I3.i2.p1.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.4.m4.1.1.1" xref="S3.I3.i2.p1.4.m4.1.1.1.cmml">∈</mo><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.4.m4.1.1.3" xref="S3.I3.i2.p1.4.m4.1.1.3a.cmml">W</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.4.m4.1b"><apply id="S3.I3.i2.p1.4.m4.1.1.cmml" xref="S3.I3.i2.p1.4.m4.1.1"><in id="S3.I3.i2.p1.4.m4.1.1.1.cmml" xref="S3.I3.i2.p1.4.m4.1.1.1"></in><apply id="S3.I3.i2.p1.4.m4.1.1.2.cmml" xref="S3.I3.i2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.4.m4.1.1.2.1.cmml" xref="S3.I3.i2.p1.4.m4.1.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.4.m4.1.1.2.2.cmml" xref="S3.I3.i2.p1.4.m4.1.1.2.2">𝜋</ci><ci id="S3.I3.i2.p1.4.m4.1.1.2.3.cmml" xref="S3.I3.i2.p1.4.m4.1.1.2.3">′</ci></apply><ci id="S3.I3.i2.p1.4.m4.1.1.3a.cmml" xref="S3.I3.i2.p1.4.m4.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.4.m4.1.1.3.cmml" xref="S3.I3.i2.p1.4.m4.1.1.3">W</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.4.m4.1c">\pi^{\prime}\in\textsf{W}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.4.m4.1d">italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.6"> such that </span><math alttext="\textsf{cost}({\pi^{\prime}})\leq\textsf{cost}({\rho^{\prime}})" class="ltx_Math" display="inline" id="S3.I3.i2.p1.5.m5.2"><semantics id="S3.I3.i2.p1.5.m5.2a"><mrow id="S3.I3.i2.p1.5.m5.2.2" xref="S3.I3.i2.p1.5.m5.2.2.cmml"><mrow id="S3.I3.i2.p1.5.m5.1.1.1" xref="S3.I3.i2.p1.5.m5.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.5.m5.1.1.1.3" xref="S3.I3.i2.p1.5.m5.1.1.1.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.5.m5.1.1.1.2" xref="S3.I3.i2.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.5.m5.1.1.1.1.1" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.5.m5.1.1.1.1.1.1" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.2" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.2.cmml">π</mi><mo id="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.3" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.5.m5.2.2.3" xref="S3.I3.i2.p1.5.m5.2.2.3.cmml">≤</mo><mrow id="S3.I3.i2.p1.5.m5.2.2.2" xref="S3.I3.i2.p1.5.m5.2.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.5.m5.2.2.2.3" xref="S3.I3.i2.p1.5.m5.2.2.2.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.5.m5.2.2.2.2" xref="S3.I3.i2.p1.5.m5.2.2.2.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.5.m5.2.2.2.1.1" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.cmml"><mo id="S3.I3.i2.p1.5.m5.2.2.2.1.1.2" stretchy="false" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.5.m5.2.2.2.1.1.1" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.cmml"><mi id="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.2" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.3" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.5.m5.2.2.2.1.1.3" stretchy="false" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.5.m5.2b"><apply id="S3.I3.i2.p1.5.m5.2.2.cmml" xref="S3.I3.i2.p1.5.m5.2.2"><leq id="S3.I3.i2.p1.5.m5.2.2.3.cmml" xref="S3.I3.i2.p1.5.m5.2.2.3"></leq><apply id="S3.I3.i2.p1.5.m5.1.1.1.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1"><times id="S3.I3.i2.p1.5.m5.1.1.1.2.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1.2"></times><ci id="S3.I3.i2.p1.5.m5.1.1.1.3a.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.5.m5.1.1.1.3.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1.3">cost</mtext></ci><apply id="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.2">𝜋</ci><ci id="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S3.I3.i2.p1.5.m5.2.2.2.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2"><times id="S3.I3.i2.p1.5.m5.2.2.2.2.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2.2"></times><ci id="S3.I3.i2.p1.5.m5.2.2.2.3a.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.5.m5.2.2.2.3.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2.3">cost</mtext></ci><apply id="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.2.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.2">𝜌</ci><ci id="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.3.cmml" xref="S3.I3.i2.p1.5.m5.2.2.2.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.5.m5.2c">\textsf{cost}({\pi^{\prime}})\leq\textsf{cost}({\rho^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.5.m5.2d">cost ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.7">. If </span><math alttext="g\not\rho^{\prime}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.6.m6.1"><semantics id="S3.I3.i2.p1.6.m6.1a"><mrow id="S3.I3.i2.p1.6.m6.1.1" xref="S3.I3.i2.p1.6.m6.1.1.cmml"><mi id="S3.I3.i2.p1.6.m6.1.1.2" xref="S3.I3.i2.p1.6.m6.1.1.2.cmml">g</mi><mo id="S3.I3.i2.p1.6.m6.1.1.1" xref="S3.I3.i2.p1.6.m6.1.1.1.cmml">⁢</mo><msup id="S3.I3.i2.p1.6.m6.1.1.3" xref="S3.I3.i2.p1.6.m6.1.1.3.cmml"><mi class="ltx_mathvariant_italic" id="S3.I3.i2.p1.6.m6.1.1.3.2" mathvariant="italic" xref="S3.I3.i2.p1.6.m6.1.1.3.2.cmml">ρ̸</mi><mo id="S3.I3.i2.p1.6.m6.1.1.3.3" xref="S3.I3.i2.p1.6.m6.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.6.m6.1b"><apply id="S3.I3.i2.p1.6.m6.1.1.cmml" xref="S3.I3.i2.p1.6.m6.1.1"><times id="S3.I3.i2.p1.6.m6.1.1.1.cmml" xref="S3.I3.i2.p1.6.m6.1.1.1"></times><ci id="S3.I3.i2.p1.6.m6.1.1.2.cmml" xref="S3.I3.i2.p1.6.m6.1.1.2">𝑔</ci><apply id="S3.I3.i2.p1.6.m6.1.1.3.cmml" xref="S3.I3.i2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.6.m6.1.1.3.1.cmml" xref="S3.I3.i2.p1.6.m6.1.1.3">superscript</csymbol><ci id="S3.I3.i2.p1.6.m6.1.1.3.2.cmml" xref="S3.I3.i2.p1.6.m6.1.1.3.2">italic-ρ̸</ci><ci id="S3.I3.i2.p1.6.m6.1.1.3.3.cmml" xref="S3.I3.i2.p1.6.m6.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.6.m6.1c">g\not\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.6.m6.1d">italic_g italic_ρ̸ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.8">, then </span><math alttext="\rho^{\prime}\in\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.7.m7.1"><semantics id="S3.I3.i2.p1.7.m7.1a"><mrow id="S3.I3.i2.p1.7.m7.1.1" xref="S3.I3.i2.p1.7.m7.1.1.cmml"><msup id="S3.I3.i2.p1.7.m7.1.1.2" xref="S3.I3.i2.p1.7.m7.1.1.2.cmml"><mi id="S3.I3.i2.p1.7.m7.1.1.2.2" xref="S3.I3.i2.p1.7.m7.1.1.2.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.7.m7.1.1.2.3" xref="S3.I3.i2.p1.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.7.m7.1.1.1" xref="S3.I3.i2.p1.7.m7.1.1.1.cmml">∈</mo><msub id="S3.I3.i2.p1.7.m7.1.1.3" xref="S3.I3.i2.p1.7.m7.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.7.m7.1.1.3.2" xref="S3.I3.i2.p1.7.m7.1.1.3.2a.cmml">Play</mtext><msub id="S3.I3.i2.p1.7.m7.1.1.3.3" xref="S3.I3.i2.p1.7.m7.1.1.3.3.cmml"><mi id="S3.I3.i2.p1.7.m7.1.1.3.3.2" xref="S3.I3.i2.p1.7.m7.1.1.3.3.2.cmml">σ</mi><mn id="S3.I3.i2.p1.7.m7.1.1.3.3.3" xref="S3.I3.i2.p1.7.m7.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.7.m7.1b"><apply id="S3.I3.i2.p1.7.m7.1.1.cmml" xref="S3.I3.i2.p1.7.m7.1.1"><in id="S3.I3.i2.p1.7.m7.1.1.1.cmml" xref="S3.I3.i2.p1.7.m7.1.1.1"></in><apply id="S3.I3.i2.p1.7.m7.1.1.2.cmml" xref="S3.I3.i2.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.7.m7.1.1.2.1.cmml" xref="S3.I3.i2.p1.7.m7.1.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.7.m7.1.1.2.2.cmml" xref="S3.I3.i2.p1.7.m7.1.1.2.2">𝜌</ci><ci id="S3.I3.i2.p1.7.m7.1.1.2.3.cmml" xref="S3.I3.i2.p1.7.m7.1.1.2.3">′</ci></apply><apply id="S3.I3.i2.p1.7.m7.1.1.3.cmml" xref="S3.I3.i2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.7.m7.1.1.3.1.cmml" xref="S3.I3.i2.p1.7.m7.1.1.3">subscript</csymbol><ci id="S3.I3.i2.p1.7.m7.1.1.3.2a.cmml" xref="S3.I3.i2.p1.7.m7.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.7.m7.1.1.3.2.cmml" xref="S3.I3.i2.p1.7.m7.1.1.3.2">Play</mtext></ci><apply id="S3.I3.i2.p1.7.m7.1.1.3.3.cmml" xref="S3.I3.i2.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.7.m7.1.1.3.3.1.cmml" xref="S3.I3.i2.p1.7.m7.1.1.3.3">subscript</csymbol><ci id="S3.I3.i2.p1.7.m7.1.1.3.3.2.cmml" xref="S3.I3.i2.p1.7.m7.1.1.3.3.2">𝜎</ci><cn id="S3.I3.i2.p1.7.m7.1.1.3.3.3.cmml" type="integer" xref="S3.I3.i2.p1.7.m7.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.7.m7.1c">\rho^{\prime}\in\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.7.m7.1d">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.9"> and there exists </span><math alttext="\pi\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.8.m8.1"><semantics id="S3.I3.i2.p1.8.m8.1a"><mrow id="S3.I3.i2.p1.8.m8.1.1" xref="S3.I3.i2.p1.8.m8.1.1.cmml"><mi id="S3.I3.i2.p1.8.m8.1.1.2" xref="S3.I3.i2.p1.8.m8.1.1.2.cmml">π</mi><mo id="S3.I3.i2.p1.8.m8.1.1.1" xref="S3.I3.i2.p1.8.m8.1.1.1.cmml">∈</mo><msub id="S3.I3.i2.p1.8.m8.1.1.3" xref="S3.I3.i2.p1.8.m8.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.8.m8.1.1.3.2" xref="S3.I3.i2.p1.8.m8.1.1.3.2a.cmml">Wit</mtext><msub id="S3.I3.i2.p1.8.m8.1.1.3.3" xref="S3.I3.i2.p1.8.m8.1.1.3.3.cmml"><mi id="S3.I3.i2.p1.8.m8.1.1.3.3.2" xref="S3.I3.i2.p1.8.m8.1.1.3.3.2.cmml">σ</mi><mn id="S3.I3.i2.p1.8.m8.1.1.3.3.3" xref="S3.I3.i2.p1.8.m8.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.8.m8.1b"><apply id="S3.I3.i2.p1.8.m8.1.1.cmml" xref="S3.I3.i2.p1.8.m8.1.1"><in id="S3.I3.i2.p1.8.m8.1.1.1.cmml" xref="S3.I3.i2.p1.8.m8.1.1.1"></in><ci id="S3.I3.i2.p1.8.m8.1.1.2.cmml" xref="S3.I3.i2.p1.8.m8.1.1.2">𝜋</ci><apply id="S3.I3.i2.p1.8.m8.1.1.3.cmml" xref="S3.I3.i2.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.8.m8.1.1.3.1.cmml" xref="S3.I3.i2.p1.8.m8.1.1.3">subscript</csymbol><ci id="S3.I3.i2.p1.8.m8.1.1.3.2a.cmml" xref="S3.I3.i2.p1.8.m8.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.8.m8.1.1.3.2.cmml" xref="S3.I3.i2.p1.8.m8.1.1.3.2">Wit</mtext></ci><apply id="S3.I3.i2.p1.8.m8.1.1.3.3.cmml" xref="S3.I3.i2.p1.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.8.m8.1.1.3.3.1.cmml" xref="S3.I3.i2.p1.8.m8.1.1.3.3">subscript</csymbol><ci id="S3.I3.i2.p1.8.m8.1.1.3.3.2.cmml" xref="S3.I3.i2.p1.8.m8.1.1.3.3.2">𝜎</ci><cn id="S3.I3.i2.p1.8.m8.1.1.3.3.3.cmml" type="integer" xref="S3.I3.i2.p1.8.m8.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.8.m8.1c">\pi\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.8.m8.1d">italic_π ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.10"> such that </span><math alttext="\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{\prime}})" class="ltx_Math" display="inline" id="S3.I3.i2.p1.9.m9.2"><semantics id="S3.I3.i2.p1.9.m9.2a"><mrow id="S3.I3.i2.p1.9.m9.2.2" xref="S3.I3.i2.p1.9.m9.2.2.cmml"><mrow id="S3.I3.i2.p1.9.m9.2.2.3" xref="S3.I3.i2.p1.9.m9.2.2.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.9.m9.2.2.3.2" xref="S3.I3.i2.p1.9.m9.2.2.3.2a.cmml">cost</mtext><mo id="S3.I3.i2.p1.9.m9.2.2.3.1" xref="S3.I3.i2.p1.9.m9.2.2.3.1.cmml">⁢</mo><mrow id="S3.I3.i2.p1.9.m9.2.2.3.3.2" xref="S3.I3.i2.p1.9.m9.2.2.3.cmml"><mo id="S3.I3.i2.p1.9.m9.2.2.3.3.2.1" stretchy="false" xref="S3.I3.i2.p1.9.m9.2.2.3.cmml">(</mo><mi id="S3.I3.i2.p1.9.m9.1.1" xref="S3.I3.i2.p1.9.m9.1.1.cmml">π</mi><mo id="S3.I3.i2.p1.9.m9.2.2.3.3.2.2" stretchy="false" xref="S3.I3.i2.p1.9.m9.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.9.m9.2.2.2" xref="S3.I3.i2.p1.9.m9.2.2.2.cmml">≤</mo><mrow id="S3.I3.i2.p1.9.m9.2.2.1" xref="S3.I3.i2.p1.9.m9.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.9.m9.2.2.1.3" xref="S3.I3.i2.p1.9.m9.2.2.1.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.9.m9.2.2.1.2" xref="S3.I3.i2.p1.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.9.m9.2.2.1.1.1" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.9.m9.2.2.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.9.m9.2.2.1.1.1.1" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.2" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.3" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.9.m9.2.2.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.9.m9.2b"><apply id="S3.I3.i2.p1.9.m9.2.2.cmml" xref="S3.I3.i2.p1.9.m9.2.2"><leq id="S3.I3.i2.p1.9.m9.2.2.2.cmml" xref="S3.I3.i2.p1.9.m9.2.2.2"></leq><apply id="S3.I3.i2.p1.9.m9.2.2.3.cmml" xref="S3.I3.i2.p1.9.m9.2.2.3"><times id="S3.I3.i2.p1.9.m9.2.2.3.1.cmml" xref="S3.I3.i2.p1.9.m9.2.2.3.1"></times><ci id="S3.I3.i2.p1.9.m9.2.2.3.2a.cmml" xref="S3.I3.i2.p1.9.m9.2.2.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.9.m9.2.2.3.2.cmml" xref="S3.I3.i2.p1.9.m9.2.2.3.2">cost</mtext></ci><ci id="S3.I3.i2.p1.9.m9.1.1.cmml" xref="S3.I3.i2.p1.9.m9.1.1">𝜋</ci></apply><apply id="S3.I3.i2.p1.9.m9.2.2.1.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1"><times id="S3.I3.i2.p1.9.m9.2.2.1.2.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1.2"></times><ci id="S3.I3.i2.p1.9.m9.2.2.1.3a.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.9.m9.2.2.1.3.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1.3">cost</mtext></ci><apply id="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.2">𝜌</ci><ci id="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.3.cmml" xref="S3.I3.i2.p1.9.m9.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.9.m9.2c">\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.9.m9.2d">cost ( italic_π ) ≤ cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.11">. By definition of </span><span class="ltx_text ltx_markedasmath ltx_font_sansserif ltx_font_italic" id="S3.I3.i2.p1.20.12">W</span><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.13"> and as </span><math alttext="w(\rho_{[m,n]}=0)" class="ltx_Math" display="inline" id="S3.I3.i2.p1.11.m11.3"><semantics id="S3.I3.i2.p1.11.m11.3a"><mrow id="S3.I3.i2.p1.11.m11.3.3" xref="S3.I3.i2.p1.11.m11.3.3.cmml"><mi id="S3.I3.i2.p1.11.m11.3.3.3" xref="S3.I3.i2.p1.11.m11.3.3.3.cmml">w</mi><mo id="S3.I3.i2.p1.11.m11.3.3.2" xref="S3.I3.i2.p1.11.m11.3.3.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.11.m11.3.3.1.1" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.cmml"><mo id="S3.I3.i2.p1.11.m11.3.3.1.1.2" stretchy="false" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.cmml">(</mo><mrow id="S3.I3.i2.p1.11.m11.3.3.1.1.1" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.cmml"><msub id="S3.I3.i2.p1.11.m11.3.3.1.1.1.2" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.2.cmml"><mi id="S3.I3.i2.p1.11.m11.3.3.1.1.1.2.2" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.2.2.cmml">ρ</mi><mrow id="S3.I3.i2.p1.11.m11.2.2.2.4" xref="S3.I3.i2.p1.11.m11.2.2.2.3.cmml"><mo id="S3.I3.i2.p1.11.m11.2.2.2.4.1" stretchy="false" xref="S3.I3.i2.p1.11.m11.2.2.2.3.cmml">[</mo><mi id="S3.I3.i2.p1.11.m11.1.1.1.1" xref="S3.I3.i2.p1.11.m11.1.1.1.1.cmml">m</mi><mo id="S3.I3.i2.p1.11.m11.2.2.2.4.2" xref="S3.I3.i2.p1.11.m11.2.2.2.3.cmml">,</mo><mi id="S3.I3.i2.p1.11.m11.2.2.2.2" xref="S3.I3.i2.p1.11.m11.2.2.2.2.cmml">n</mi><mo id="S3.I3.i2.p1.11.m11.2.2.2.4.3" stretchy="false" xref="S3.I3.i2.p1.11.m11.2.2.2.3.cmml">]</mo></mrow></msub><mo id="S3.I3.i2.p1.11.m11.3.3.1.1.1.1" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.1.cmml">=</mo><mn id="S3.I3.i2.p1.11.m11.3.3.1.1.1.3" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.3.cmml">0</mn></mrow><mo id="S3.I3.i2.p1.11.m11.3.3.1.1.3" stretchy="false" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.11.m11.3b"><apply id="S3.I3.i2.p1.11.m11.3.3.cmml" xref="S3.I3.i2.p1.11.m11.3.3"><times id="S3.I3.i2.p1.11.m11.3.3.2.cmml" xref="S3.I3.i2.p1.11.m11.3.3.2"></times><ci id="S3.I3.i2.p1.11.m11.3.3.3.cmml" xref="S3.I3.i2.p1.11.m11.3.3.3">𝑤</ci><apply id="S3.I3.i2.p1.11.m11.3.3.1.1.1.cmml" xref="S3.I3.i2.p1.11.m11.3.3.1.1"><eq id="S3.I3.i2.p1.11.m11.3.3.1.1.1.1.cmml" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.1"></eq><apply id="S3.I3.i2.p1.11.m11.3.3.1.1.1.2.cmml" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.11.m11.3.3.1.1.1.2.1.cmml" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.2">subscript</csymbol><ci id="S3.I3.i2.p1.11.m11.3.3.1.1.1.2.2.cmml" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.2.2">𝜌</ci><interval closure="closed" id="S3.I3.i2.p1.11.m11.2.2.2.3.cmml" xref="S3.I3.i2.p1.11.m11.2.2.2.4"><ci id="S3.I3.i2.p1.11.m11.1.1.1.1.cmml" xref="S3.I3.i2.p1.11.m11.1.1.1.1">𝑚</ci><ci id="S3.I3.i2.p1.11.m11.2.2.2.2.cmml" xref="S3.I3.i2.p1.11.m11.2.2.2.2">𝑛</ci></interval></apply><cn id="S3.I3.i2.p1.11.m11.3.3.1.1.1.3.cmml" type="integer" xref="S3.I3.i2.p1.11.m11.3.3.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.11.m11.3c">w(\rho_{[m,n]}=0)</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.11.m11.3d">italic_w ( italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT = 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.14">, there exists </span><math alttext="\pi^{\prime}\in\textsf{W}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.12.m12.1"><semantics id="S3.I3.i2.p1.12.m12.1a"><mrow id="S3.I3.i2.p1.12.m12.1.1" xref="S3.I3.i2.p1.12.m12.1.1.cmml"><msup id="S3.I3.i2.p1.12.m12.1.1.2" xref="S3.I3.i2.p1.12.m12.1.1.2.cmml"><mi id="S3.I3.i2.p1.12.m12.1.1.2.2" xref="S3.I3.i2.p1.12.m12.1.1.2.2.cmml">π</mi><mo id="S3.I3.i2.p1.12.m12.1.1.2.3" xref="S3.I3.i2.p1.12.m12.1.1.2.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.12.m12.1.1.1" xref="S3.I3.i2.p1.12.m12.1.1.1.cmml">∈</mo><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.12.m12.1.1.3" xref="S3.I3.i2.p1.12.m12.1.1.3a.cmml">W</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.12.m12.1b"><apply id="S3.I3.i2.p1.12.m12.1.1.cmml" xref="S3.I3.i2.p1.12.m12.1.1"><in id="S3.I3.i2.p1.12.m12.1.1.1.cmml" xref="S3.I3.i2.p1.12.m12.1.1.1"></in><apply id="S3.I3.i2.p1.12.m12.1.1.2.cmml" xref="S3.I3.i2.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.12.m12.1.1.2.1.cmml" xref="S3.I3.i2.p1.12.m12.1.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.12.m12.1.1.2.2.cmml" xref="S3.I3.i2.p1.12.m12.1.1.2.2">𝜋</ci><ci id="S3.I3.i2.p1.12.m12.1.1.2.3.cmml" xref="S3.I3.i2.p1.12.m12.1.1.2.3">′</ci></apply><ci id="S3.I3.i2.p1.12.m12.1.1.3a.cmml" xref="S3.I3.i2.p1.12.m12.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.12.m12.1.1.3.cmml" xref="S3.I3.i2.p1.12.m12.1.1.3">W</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.12.m12.1c">\pi^{\prime}\in\textsf{W}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.12.m12.1d">italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.15"> such that </span><math alttext="\textsf{cost}({\pi^{\prime}})=\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{% \prime}})" class="ltx_Math" display="inline" id="S3.I3.i2.p1.13.m13.3"><semantics id="S3.I3.i2.p1.13.m13.3a"><mrow id="S3.I3.i2.p1.13.m13.3.3" xref="S3.I3.i2.p1.13.m13.3.3.cmml"><mrow id="S3.I3.i2.p1.13.m13.2.2.1" xref="S3.I3.i2.p1.13.m13.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.13.m13.2.2.1.3" xref="S3.I3.i2.p1.13.m13.2.2.1.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.13.m13.2.2.1.2" xref="S3.I3.i2.p1.13.m13.2.2.1.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.13.m13.2.2.1.1.1" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.13.m13.2.2.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.13.m13.2.2.1.1.1.1" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.2" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.2.cmml">π</mi><mo id="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.3" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.13.m13.2.2.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.13.m13.3.3.4" xref="S3.I3.i2.p1.13.m13.3.3.4.cmml">=</mo><mrow id="S3.I3.i2.p1.13.m13.3.3.5" xref="S3.I3.i2.p1.13.m13.3.3.5.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.13.m13.3.3.5.2" xref="S3.I3.i2.p1.13.m13.3.3.5.2a.cmml">cost</mtext><mo id="S3.I3.i2.p1.13.m13.3.3.5.1" xref="S3.I3.i2.p1.13.m13.3.3.5.1.cmml">⁢</mo><mrow id="S3.I3.i2.p1.13.m13.3.3.5.3.2" xref="S3.I3.i2.p1.13.m13.3.3.5.cmml"><mo id="S3.I3.i2.p1.13.m13.3.3.5.3.2.1" stretchy="false" xref="S3.I3.i2.p1.13.m13.3.3.5.cmml">(</mo><mi id="S3.I3.i2.p1.13.m13.1.1" xref="S3.I3.i2.p1.13.m13.1.1.cmml">π</mi><mo id="S3.I3.i2.p1.13.m13.3.3.5.3.2.2" stretchy="false" xref="S3.I3.i2.p1.13.m13.3.3.5.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.13.m13.3.3.6" xref="S3.I3.i2.p1.13.m13.3.3.6.cmml">≤</mo><mrow id="S3.I3.i2.p1.13.m13.3.3.2" xref="S3.I3.i2.p1.13.m13.3.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.13.m13.3.3.2.3" xref="S3.I3.i2.p1.13.m13.3.3.2.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.13.m13.3.3.2.2" xref="S3.I3.i2.p1.13.m13.3.3.2.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.13.m13.3.3.2.1.1" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.cmml"><mo id="S3.I3.i2.p1.13.m13.3.3.2.1.1.2" stretchy="false" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.13.m13.3.3.2.1.1.1" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.cmml"><mi id="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.2" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.3" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.13.m13.3.3.2.1.1.3" stretchy="false" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.13.m13.3b"><apply id="S3.I3.i2.p1.13.m13.3.3.cmml" xref="S3.I3.i2.p1.13.m13.3.3"><and id="S3.I3.i2.p1.13.m13.3.3a.cmml" xref="S3.I3.i2.p1.13.m13.3.3"></and><apply id="S3.I3.i2.p1.13.m13.3.3b.cmml" xref="S3.I3.i2.p1.13.m13.3.3"><eq id="S3.I3.i2.p1.13.m13.3.3.4.cmml" xref="S3.I3.i2.p1.13.m13.3.3.4"></eq><apply id="S3.I3.i2.p1.13.m13.2.2.1.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1"><times id="S3.I3.i2.p1.13.m13.2.2.1.2.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1.2"></times><ci id="S3.I3.i2.p1.13.m13.2.2.1.3a.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.13.m13.2.2.1.3.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1.3">cost</mtext></ci><apply id="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.2">𝜋</ci><ci id="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.3.cmml" xref="S3.I3.i2.p1.13.m13.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.I3.i2.p1.13.m13.3.3.5.cmml" xref="S3.I3.i2.p1.13.m13.3.3.5"><times id="S3.I3.i2.p1.13.m13.3.3.5.1.cmml" xref="S3.I3.i2.p1.13.m13.3.3.5.1"></times><ci id="S3.I3.i2.p1.13.m13.3.3.5.2a.cmml" xref="S3.I3.i2.p1.13.m13.3.3.5.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.13.m13.3.3.5.2.cmml" xref="S3.I3.i2.p1.13.m13.3.3.5.2">cost</mtext></ci><ci id="S3.I3.i2.p1.13.m13.1.1.cmml" xref="S3.I3.i2.p1.13.m13.1.1">𝜋</ci></apply></apply><apply id="S3.I3.i2.p1.13.m13.3.3c.cmml" xref="S3.I3.i2.p1.13.m13.3.3"><leq id="S3.I3.i2.p1.13.m13.3.3.6.cmml" xref="S3.I3.i2.p1.13.m13.3.3.6"></leq><share href="https://arxiv.org/html/2308.09443v2#S3.I3.i2.p1.13.m13.3.3.5.cmml" id="S3.I3.i2.p1.13.m13.3.3d.cmml" xref="S3.I3.i2.p1.13.m13.3.3"></share><apply id="S3.I3.i2.p1.13.m13.3.3.2.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2"><times id="S3.I3.i2.p1.13.m13.3.3.2.2.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2.2"></times><ci id="S3.I3.i2.p1.13.m13.3.3.2.3a.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.13.m13.3.3.2.3.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2.3">cost</mtext></ci><apply id="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.2.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.2">𝜌</ci><ci id="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.3.cmml" xref="S3.I3.i2.p1.13.m13.3.3.2.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.13.m13.3c">\textsf{cost}({\pi^{\prime}})=\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{% \prime}})</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.13.m13.3d">cost ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = cost ( italic_π ) ≤ cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.16">. If </span><math alttext="g\rho^{\prime}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.14.m14.1"><semantics id="S3.I3.i2.p1.14.m14.1a"><mrow id="S3.I3.i2.p1.14.m14.1.1" xref="S3.I3.i2.p1.14.m14.1.1.cmml"><mi id="S3.I3.i2.p1.14.m14.1.1.2" xref="S3.I3.i2.p1.14.m14.1.1.2.cmml">g</mi><mo id="S3.I3.i2.p1.14.m14.1.1.1" xref="S3.I3.i2.p1.14.m14.1.1.1.cmml">⁢</mo><msup id="S3.I3.i2.p1.14.m14.1.1.3" xref="S3.I3.i2.p1.14.m14.1.1.3.cmml"><mi id="S3.I3.i2.p1.14.m14.1.1.3.2" xref="S3.I3.i2.p1.14.m14.1.1.3.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.14.m14.1.1.3.3" xref="S3.I3.i2.p1.14.m14.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.14.m14.1b"><apply id="S3.I3.i2.p1.14.m14.1.1.cmml" xref="S3.I3.i2.p1.14.m14.1.1"><times id="S3.I3.i2.p1.14.m14.1.1.1.cmml" xref="S3.I3.i2.p1.14.m14.1.1.1"></times><ci id="S3.I3.i2.p1.14.m14.1.1.2.cmml" xref="S3.I3.i2.p1.14.m14.1.1.2">𝑔</ci><apply id="S3.I3.i2.p1.14.m14.1.1.3.cmml" xref="S3.I3.i2.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.14.m14.1.1.3.1.cmml" xref="S3.I3.i2.p1.14.m14.1.1.3">superscript</csymbol><ci id="S3.I3.i2.p1.14.m14.1.1.3.2.cmml" xref="S3.I3.i2.p1.14.m14.1.1.3.2">𝜌</ci><ci id="S3.I3.i2.p1.14.m14.1.1.3.3.cmml" xref="S3.I3.i2.p1.14.m14.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.14.m14.1c">g\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.14.m14.1d">italic_g italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.17">, we consider </span><math alttext="\rho^{\prime\prime}\in\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.15.m15.1"><semantics id="S3.I3.i2.p1.15.m15.1a"><mrow id="S3.I3.i2.p1.15.m15.1.1" xref="S3.I3.i2.p1.15.m15.1.1.cmml"><msup id="S3.I3.i2.p1.15.m15.1.1.2" xref="S3.I3.i2.p1.15.m15.1.1.2.cmml"><mi id="S3.I3.i2.p1.15.m15.1.1.2.2" xref="S3.I3.i2.p1.15.m15.1.1.2.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.15.m15.1.1.2.3" xref="S3.I3.i2.p1.15.m15.1.1.2.3.cmml">′′</mo></msup><mo id="S3.I3.i2.p1.15.m15.1.1.1" xref="S3.I3.i2.p1.15.m15.1.1.1.cmml">∈</mo><msub id="S3.I3.i2.p1.15.m15.1.1.3" xref="S3.I3.i2.p1.15.m15.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.15.m15.1.1.3.2" xref="S3.I3.i2.p1.15.m15.1.1.3.2a.cmml">Play</mtext><msub id="S3.I3.i2.p1.15.m15.1.1.3.3" xref="S3.I3.i2.p1.15.m15.1.1.3.3.cmml"><mi id="S3.I3.i2.p1.15.m15.1.1.3.3.2" xref="S3.I3.i2.p1.15.m15.1.1.3.3.2.cmml">σ</mi><mn id="S3.I3.i2.p1.15.m15.1.1.3.3.3" xref="S3.I3.i2.p1.15.m15.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.15.m15.1b"><apply id="S3.I3.i2.p1.15.m15.1.1.cmml" xref="S3.I3.i2.p1.15.m15.1.1"><in id="S3.I3.i2.p1.15.m15.1.1.1.cmml" xref="S3.I3.i2.p1.15.m15.1.1.1"></in><apply id="S3.I3.i2.p1.15.m15.1.1.2.cmml" xref="S3.I3.i2.p1.15.m15.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.15.m15.1.1.2.1.cmml" xref="S3.I3.i2.p1.15.m15.1.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.15.m15.1.1.2.2.cmml" xref="S3.I3.i2.p1.15.m15.1.1.2.2">𝜌</ci><ci id="S3.I3.i2.p1.15.m15.1.1.2.3.cmml" xref="S3.I3.i2.p1.15.m15.1.1.2.3">′′</ci></apply><apply id="S3.I3.i2.p1.15.m15.1.1.3.cmml" xref="S3.I3.i2.p1.15.m15.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.15.m15.1.1.3.1.cmml" xref="S3.I3.i2.p1.15.m15.1.1.3">subscript</csymbol><ci id="S3.I3.i2.p1.15.m15.1.1.3.2a.cmml" xref="S3.I3.i2.p1.15.m15.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.15.m15.1.1.3.2.cmml" xref="S3.I3.i2.p1.15.m15.1.1.3.2">Play</mtext></ci><apply id="S3.I3.i2.p1.15.m15.1.1.3.3.cmml" xref="S3.I3.i2.p1.15.m15.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.15.m15.1.1.3.3.1.cmml" xref="S3.I3.i2.p1.15.m15.1.1.3.3">subscript</csymbol><ci id="S3.I3.i2.p1.15.m15.1.1.3.3.2.cmml" xref="S3.I3.i2.p1.15.m15.1.1.3.3.2">𝜎</ci><cn id="S3.I3.i2.p1.15.m15.1.1.3.3.3.cmml" type="integer" xref="S3.I3.i2.p1.15.m15.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.15.m15.1c">\rho^{\prime\prime}\in\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.15.m15.1d">italic_ρ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.18"> such that </span><math alttext="\bar{\rho}^{\prime\prime}=\rho^{\prime}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.16.m16.1"><semantics id="S3.I3.i2.p1.16.m16.1a"><mrow id="S3.I3.i2.p1.16.m16.1.1" xref="S3.I3.i2.p1.16.m16.1.1.cmml"><msup id="S3.I3.i2.p1.16.m16.1.1.2" xref="S3.I3.i2.p1.16.m16.1.1.2.cmml"><mover accent="true" id="S3.I3.i2.p1.16.m16.1.1.2.2" xref="S3.I3.i2.p1.16.m16.1.1.2.2.cmml"><mi id="S3.I3.i2.p1.16.m16.1.1.2.2.2" xref="S3.I3.i2.p1.16.m16.1.1.2.2.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.16.m16.1.1.2.2.1" xref="S3.I3.i2.p1.16.m16.1.1.2.2.1.cmml">¯</mo></mover><mo id="S3.I3.i2.p1.16.m16.1.1.2.3" xref="S3.I3.i2.p1.16.m16.1.1.2.3.cmml">′′</mo></msup><mo id="S3.I3.i2.p1.16.m16.1.1.1" xref="S3.I3.i2.p1.16.m16.1.1.1.cmml">=</mo><msup id="S3.I3.i2.p1.16.m16.1.1.3" xref="S3.I3.i2.p1.16.m16.1.1.3.cmml"><mi id="S3.I3.i2.p1.16.m16.1.1.3.2" xref="S3.I3.i2.p1.16.m16.1.1.3.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.16.m16.1.1.3.3" xref="S3.I3.i2.p1.16.m16.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.16.m16.1b"><apply id="S3.I3.i2.p1.16.m16.1.1.cmml" xref="S3.I3.i2.p1.16.m16.1.1"><eq id="S3.I3.i2.p1.16.m16.1.1.1.cmml" xref="S3.I3.i2.p1.16.m16.1.1.1"></eq><apply id="S3.I3.i2.p1.16.m16.1.1.2.cmml" xref="S3.I3.i2.p1.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.16.m16.1.1.2.1.cmml" xref="S3.I3.i2.p1.16.m16.1.1.2">superscript</csymbol><apply id="S3.I3.i2.p1.16.m16.1.1.2.2.cmml" xref="S3.I3.i2.p1.16.m16.1.1.2.2"><ci id="S3.I3.i2.p1.16.m16.1.1.2.2.1.cmml" xref="S3.I3.i2.p1.16.m16.1.1.2.2.1">¯</ci><ci id="S3.I3.i2.p1.16.m16.1.1.2.2.2.cmml" xref="S3.I3.i2.p1.16.m16.1.1.2.2.2">𝜌</ci></apply><ci id="S3.I3.i2.p1.16.m16.1.1.2.3.cmml" xref="S3.I3.i2.p1.16.m16.1.1.2.3">′′</ci></apply><apply id="S3.I3.i2.p1.16.m16.1.1.3.cmml" xref="S3.I3.i2.p1.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.16.m16.1.1.3.1.cmml" xref="S3.I3.i2.p1.16.m16.1.1.3">superscript</csymbol><ci id="S3.I3.i2.p1.16.m16.1.1.3.2.cmml" xref="S3.I3.i2.p1.16.m16.1.1.3.2">𝜌</ci><ci id="S3.I3.i2.p1.16.m16.1.1.3.3.cmml" xref="S3.I3.i2.p1.16.m16.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.16.m16.1c">\bar{\rho}^{\prime\prime}=\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.16.m16.1d">over¯ start_ARG italic_ρ end_ARG start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.19">. Again there exists </span><math alttext="\pi\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.17.m17.1"><semantics id="S3.I3.i2.p1.17.m17.1a"><mrow id="S3.I3.i2.p1.17.m17.1.1" xref="S3.I3.i2.p1.17.m17.1.1.cmml"><mi id="S3.I3.i2.p1.17.m17.1.1.2" xref="S3.I3.i2.p1.17.m17.1.1.2.cmml">π</mi><mo id="S3.I3.i2.p1.17.m17.1.1.1" xref="S3.I3.i2.p1.17.m17.1.1.1.cmml">∈</mo><msub id="S3.I3.i2.p1.17.m17.1.1.3" xref="S3.I3.i2.p1.17.m17.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.17.m17.1.1.3.2" xref="S3.I3.i2.p1.17.m17.1.1.3.2a.cmml">Wit</mtext><msub id="S3.I3.i2.p1.17.m17.1.1.3.3" xref="S3.I3.i2.p1.17.m17.1.1.3.3.cmml"><mi id="S3.I3.i2.p1.17.m17.1.1.3.3.2" xref="S3.I3.i2.p1.17.m17.1.1.3.3.2.cmml">σ</mi><mn id="S3.I3.i2.p1.17.m17.1.1.3.3.3" xref="S3.I3.i2.p1.17.m17.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.17.m17.1b"><apply id="S3.I3.i2.p1.17.m17.1.1.cmml" xref="S3.I3.i2.p1.17.m17.1.1"><in id="S3.I3.i2.p1.17.m17.1.1.1.cmml" xref="S3.I3.i2.p1.17.m17.1.1.1"></in><ci id="S3.I3.i2.p1.17.m17.1.1.2.cmml" xref="S3.I3.i2.p1.17.m17.1.1.2">𝜋</ci><apply id="S3.I3.i2.p1.17.m17.1.1.3.cmml" xref="S3.I3.i2.p1.17.m17.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.17.m17.1.1.3.1.cmml" xref="S3.I3.i2.p1.17.m17.1.1.3">subscript</csymbol><ci id="S3.I3.i2.p1.17.m17.1.1.3.2a.cmml" xref="S3.I3.i2.p1.17.m17.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.17.m17.1.1.3.2.cmml" xref="S3.I3.i2.p1.17.m17.1.1.3.2">Wit</mtext></ci><apply id="S3.I3.i2.p1.17.m17.1.1.3.3.cmml" xref="S3.I3.i2.p1.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.17.m17.1.1.3.3.1.cmml" xref="S3.I3.i2.p1.17.m17.1.1.3.3">subscript</csymbol><ci id="S3.I3.i2.p1.17.m17.1.1.3.3.2.cmml" xref="S3.I3.i2.p1.17.m17.1.1.3.3.2">𝜎</ci><cn id="S3.I3.i2.p1.17.m17.1.1.3.3.3.cmml" type="integer" xref="S3.I3.i2.p1.17.m17.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.17.m17.1c">\pi\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.17.m17.1d">italic_π ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.20"> such that </span><math alttext="\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{\prime\prime}})=\textsf{cost}({% \rho^{\prime}})" class="ltx_Math" display="inline" id="S3.I3.i2.p1.18.m18.3"><semantics id="S3.I3.i2.p1.18.m18.3a"><mrow id="S3.I3.i2.p1.18.m18.3.3" xref="S3.I3.i2.p1.18.m18.3.3.cmml"><mrow id="S3.I3.i2.p1.18.m18.3.3.4" xref="S3.I3.i2.p1.18.m18.3.3.4.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.18.m18.3.3.4.2" xref="S3.I3.i2.p1.18.m18.3.3.4.2a.cmml">cost</mtext><mo id="S3.I3.i2.p1.18.m18.3.3.4.1" xref="S3.I3.i2.p1.18.m18.3.3.4.1.cmml">⁢</mo><mrow id="S3.I3.i2.p1.18.m18.3.3.4.3.2" xref="S3.I3.i2.p1.18.m18.3.3.4.cmml"><mo id="S3.I3.i2.p1.18.m18.3.3.4.3.2.1" stretchy="false" xref="S3.I3.i2.p1.18.m18.3.3.4.cmml">(</mo><mi id="S3.I3.i2.p1.18.m18.1.1" xref="S3.I3.i2.p1.18.m18.1.1.cmml">π</mi><mo id="S3.I3.i2.p1.18.m18.3.3.4.3.2.2" stretchy="false" xref="S3.I3.i2.p1.18.m18.3.3.4.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.18.m18.3.3.5" xref="S3.I3.i2.p1.18.m18.3.3.5.cmml">≤</mo><mrow id="S3.I3.i2.p1.18.m18.2.2.1" xref="S3.I3.i2.p1.18.m18.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.18.m18.2.2.1.3" xref="S3.I3.i2.p1.18.m18.2.2.1.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.18.m18.2.2.1.2" xref="S3.I3.i2.p1.18.m18.2.2.1.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.18.m18.2.2.1.1.1" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.18.m18.2.2.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.18.m18.2.2.1.1.1.1" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.2" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.3" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.3.cmml">′′</mo></msup><mo id="S3.I3.i2.p1.18.m18.2.2.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.18.m18.3.3.6" xref="S3.I3.i2.p1.18.m18.3.3.6.cmml">=</mo><mrow id="S3.I3.i2.p1.18.m18.3.3.2" xref="S3.I3.i2.p1.18.m18.3.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.18.m18.3.3.2.3" xref="S3.I3.i2.p1.18.m18.3.3.2.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.18.m18.3.3.2.2" xref="S3.I3.i2.p1.18.m18.3.3.2.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.18.m18.3.3.2.1.1" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.cmml"><mo id="S3.I3.i2.p1.18.m18.3.3.2.1.1.2" stretchy="false" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.18.m18.3.3.2.1.1.1" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.cmml"><mi id="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.2" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.3" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.18.m18.3.3.2.1.1.3" stretchy="false" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.18.m18.3b"><apply id="S3.I3.i2.p1.18.m18.3.3.cmml" xref="S3.I3.i2.p1.18.m18.3.3"><and id="S3.I3.i2.p1.18.m18.3.3a.cmml" xref="S3.I3.i2.p1.18.m18.3.3"></and><apply id="S3.I3.i2.p1.18.m18.3.3b.cmml" xref="S3.I3.i2.p1.18.m18.3.3"><leq id="S3.I3.i2.p1.18.m18.3.3.5.cmml" xref="S3.I3.i2.p1.18.m18.3.3.5"></leq><apply id="S3.I3.i2.p1.18.m18.3.3.4.cmml" xref="S3.I3.i2.p1.18.m18.3.3.4"><times id="S3.I3.i2.p1.18.m18.3.3.4.1.cmml" xref="S3.I3.i2.p1.18.m18.3.3.4.1"></times><ci id="S3.I3.i2.p1.18.m18.3.3.4.2a.cmml" xref="S3.I3.i2.p1.18.m18.3.3.4.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.18.m18.3.3.4.2.cmml" xref="S3.I3.i2.p1.18.m18.3.3.4.2">cost</mtext></ci><ci id="S3.I3.i2.p1.18.m18.1.1.cmml" xref="S3.I3.i2.p1.18.m18.1.1">𝜋</ci></apply><apply id="S3.I3.i2.p1.18.m18.2.2.1.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1"><times id="S3.I3.i2.p1.18.m18.2.2.1.2.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1.2"></times><ci id="S3.I3.i2.p1.18.m18.2.2.1.3a.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.18.m18.2.2.1.3.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1.3">cost</mtext></ci><apply id="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.2">𝜌</ci><ci id="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.3.cmml" xref="S3.I3.i2.p1.18.m18.2.2.1.1.1.1.3">′′</ci></apply></apply></apply><apply id="S3.I3.i2.p1.18.m18.3.3c.cmml" xref="S3.I3.i2.p1.18.m18.3.3"><eq id="S3.I3.i2.p1.18.m18.3.3.6.cmml" xref="S3.I3.i2.p1.18.m18.3.3.6"></eq><share href="https://arxiv.org/html/2308.09443v2#S3.I3.i2.p1.18.m18.2.2.1.cmml" id="S3.I3.i2.p1.18.m18.3.3d.cmml" xref="S3.I3.i2.p1.18.m18.3.3"></share><apply id="S3.I3.i2.p1.18.m18.3.3.2.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2"><times id="S3.I3.i2.p1.18.m18.3.3.2.2.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2.2"></times><ci id="S3.I3.i2.p1.18.m18.3.3.2.3a.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.18.m18.3.3.2.3.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2.3">cost</mtext></ci><apply id="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.2.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.2">𝜌</ci><ci id="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.3.cmml" xref="S3.I3.i2.p1.18.m18.3.3.2.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.18.m18.3c">\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{\prime\prime}})=\textsf{cost}({% \rho^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.18.m18.3d">cost ( italic_π ) ≤ cost ( italic_ρ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.21">, and there exists </span><math alttext="\pi^{\prime}\in\textsf{W}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.19.m19.1"><semantics id="S3.I3.i2.p1.19.m19.1a"><mrow id="S3.I3.i2.p1.19.m19.1.1" xref="S3.I3.i2.p1.19.m19.1.1.cmml"><msup id="S3.I3.i2.p1.19.m19.1.1.2" xref="S3.I3.i2.p1.19.m19.1.1.2.cmml"><mi id="S3.I3.i2.p1.19.m19.1.1.2.2" xref="S3.I3.i2.p1.19.m19.1.1.2.2.cmml">π</mi><mo id="S3.I3.i2.p1.19.m19.1.1.2.3" xref="S3.I3.i2.p1.19.m19.1.1.2.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.19.m19.1.1.1" xref="S3.I3.i2.p1.19.m19.1.1.1.cmml">∈</mo><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.19.m19.1.1.3" xref="S3.I3.i2.p1.19.m19.1.1.3a.cmml">W</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.19.m19.1b"><apply id="S3.I3.i2.p1.19.m19.1.1.cmml" xref="S3.I3.i2.p1.19.m19.1.1"><in id="S3.I3.i2.p1.19.m19.1.1.1.cmml" xref="S3.I3.i2.p1.19.m19.1.1.1"></in><apply id="S3.I3.i2.p1.19.m19.1.1.2.cmml" xref="S3.I3.i2.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.19.m19.1.1.2.1.cmml" xref="S3.I3.i2.p1.19.m19.1.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.19.m19.1.1.2.2.cmml" xref="S3.I3.i2.p1.19.m19.1.1.2.2">𝜋</ci><ci id="S3.I3.i2.p1.19.m19.1.1.2.3.cmml" xref="S3.I3.i2.p1.19.m19.1.1.2.3">′</ci></apply><ci id="S3.I3.i2.p1.19.m19.1.1.3a.cmml" xref="S3.I3.i2.p1.19.m19.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.19.m19.1.1.3.cmml" xref="S3.I3.i2.p1.19.m19.1.1.3">W</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.19.m19.1c">\pi^{\prime}\in\textsf{W}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.19.m19.1d">italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.22"> such that </span><math alttext="\textsf{cost}({\pi^{\prime}})=\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{% \prime}})" class="ltx_Math" display="inline" id="S3.I3.i2.p1.20.m20.3"><semantics id="S3.I3.i2.p1.20.m20.3a"><mrow id="S3.I3.i2.p1.20.m20.3.3" xref="S3.I3.i2.p1.20.m20.3.3.cmml"><mrow id="S3.I3.i2.p1.20.m20.2.2.1" xref="S3.I3.i2.p1.20.m20.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.20.m20.2.2.1.3" xref="S3.I3.i2.p1.20.m20.2.2.1.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.20.m20.2.2.1.2" xref="S3.I3.i2.p1.20.m20.2.2.1.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.20.m20.2.2.1.1.1" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.20.m20.2.2.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.20.m20.2.2.1.1.1.1" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.2" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.2.cmml">π</mi><mo id="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.3" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.20.m20.2.2.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.20.m20.3.3.4" xref="S3.I3.i2.p1.20.m20.3.3.4.cmml">=</mo><mrow id="S3.I3.i2.p1.20.m20.3.3.5" xref="S3.I3.i2.p1.20.m20.3.3.5.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.20.m20.3.3.5.2" xref="S3.I3.i2.p1.20.m20.3.3.5.2a.cmml">cost</mtext><mo id="S3.I3.i2.p1.20.m20.3.3.5.1" xref="S3.I3.i2.p1.20.m20.3.3.5.1.cmml">⁢</mo><mrow id="S3.I3.i2.p1.20.m20.3.3.5.3.2" xref="S3.I3.i2.p1.20.m20.3.3.5.cmml"><mo id="S3.I3.i2.p1.20.m20.3.3.5.3.2.1" stretchy="false" xref="S3.I3.i2.p1.20.m20.3.3.5.cmml">(</mo><mi id="S3.I3.i2.p1.20.m20.1.1" xref="S3.I3.i2.p1.20.m20.1.1.cmml">π</mi><mo id="S3.I3.i2.p1.20.m20.3.3.5.3.2.2" stretchy="false" xref="S3.I3.i2.p1.20.m20.3.3.5.cmml">)</mo></mrow></mrow><mo id="S3.I3.i2.p1.20.m20.3.3.6" xref="S3.I3.i2.p1.20.m20.3.3.6.cmml">≤</mo><mrow id="S3.I3.i2.p1.20.m20.3.3.2" xref="S3.I3.i2.p1.20.m20.3.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.20.m20.3.3.2.3" xref="S3.I3.i2.p1.20.m20.3.3.2.3a.cmml">cost</mtext><mo id="S3.I3.i2.p1.20.m20.3.3.2.2" xref="S3.I3.i2.p1.20.m20.3.3.2.2.cmml">⁢</mo><mrow id="S3.I3.i2.p1.20.m20.3.3.2.1.1" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.cmml"><mo id="S3.I3.i2.p1.20.m20.3.3.2.1.1.2" stretchy="false" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.cmml">(</mo><msup id="S3.I3.i2.p1.20.m20.3.3.2.1.1.1" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.cmml"><mi id="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.2" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.2.cmml">ρ</mi><mo id="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.3" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.3.cmml">′</mo></msup><mo id="S3.I3.i2.p1.20.m20.3.3.2.1.1.3" stretchy="false" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.20.m20.3b"><apply id="S3.I3.i2.p1.20.m20.3.3.cmml" xref="S3.I3.i2.p1.20.m20.3.3"><and id="S3.I3.i2.p1.20.m20.3.3a.cmml" xref="S3.I3.i2.p1.20.m20.3.3"></and><apply id="S3.I3.i2.p1.20.m20.3.3b.cmml" xref="S3.I3.i2.p1.20.m20.3.3"><eq id="S3.I3.i2.p1.20.m20.3.3.4.cmml" xref="S3.I3.i2.p1.20.m20.3.3.4"></eq><apply id="S3.I3.i2.p1.20.m20.2.2.1.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1"><times id="S3.I3.i2.p1.20.m20.2.2.1.2.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1.2"></times><ci id="S3.I3.i2.p1.20.m20.2.2.1.3a.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.20.m20.2.2.1.3.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1.3">cost</mtext></ci><apply id="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.2">𝜋</ci><ci id="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.3.cmml" xref="S3.I3.i2.p1.20.m20.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.I3.i2.p1.20.m20.3.3.5.cmml" xref="S3.I3.i2.p1.20.m20.3.3.5"><times id="S3.I3.i2.p1.20.m20.3.3.5.1.cmml" xref="S3.I3.i2.p1.20.m20.3.3.5.1"></times><ci id="S3.I3.i2.p1.20.m20.3.3.5.2a.cmml" xref="S3.I3.i2.p1.20.m20.3.3.5.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.20.m20.3.3.5.2.cmml" xref="S3.I3.i2.p1.20.m20.3.3.5.2">cost</mtext></ci><ci id="S3.I3.i2.p1.20.m20.1.1.cmml" xref="S3.I3.i2.p1.20.m20.1.1">𝜋</ci></apply></apply><apply id="S3.I3.i2.p1.20.m20.3.3c.cmml" xref="S3.I3.i2.p1.20.m20.3.3"><leq id="S3.I3.i2.p1.20.m20.3.3.6.cmml" xref="S3.I3.i2.p1.20.m20.3.3.6"></leq><share href="https://arxiv.org/html/2308.09443v2#S3.I3.i2.p1.20.m20.3.3.5.cmml" id="S3.I3.i2.p1.20.m20.3.3d.cmml" xref="S3.I3.i2.p1.20.m20.3.3"></share><apply id="S3.I3.i2.p1.20.m20.3.3.2.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2"><times id="S3.I3.i2.p1.20.m20.3.3.2.2.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2.2"></times><ci id="S3.I3.i2.p1.20.m20.3.3.2.3a.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i2.p1.20.m20.3.3.2.3.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2.3">cost</mtext></ci><apply id="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.1.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1">superscript</csymbol><ci id="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.2.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.2">𝜌</ci><ci id="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.3.cmml" xref="S3.I3.i2.p1.20.m20.3.3.2.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.20.m20.3c">\textsf{cost}({\pi^{\prime}})=\textsf{cost}({\pi})\leq\textsf{cost}({\rho^{% \prime}})</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.20.m20.3d">cost ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = cost ( italic_π ) ≤ cost ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.20.23">.</span></p> </div> </li> <li class="ltx_item" id="S3.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I3.i3.p1"> <p class="ltx_p" id="S3.I3.i3.p1.2"><span class="ltx_text ltx_font_italic" id="S3.I3.i3.p1.2.1">Third, by definition of </span><span class="ltx_text ltx_markedasmath ltx_font_sansserif ltx_font_italic" id="S3.I3.i3.p1.2.2">W</span><span class="ltx_text ltx_font_italic" id="S3.I3.i3.p1.2.3">, we clearly have </span><math alttext="\textsf{length}({\textsf{W}})<\textsf{length}({\textsf{Wit}_{\sigma_{0}}})" class="ltx_Math" display="inline" id="S3.I3.i3.p1.2.m2.2"><semantics id="S3.I3.i3.p1.2.m2.2a"><mrow id="S3.I3.i3.p1.2.m2.2.2" xref="S3.I3.i3.p1.2.m2.2.2.cmml"><mrow id="S3.I3.i3.p1.2.m2.2.2.3" xref="S3.I3.i3.p1.2.m2.2.2.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.2.2.3.2" xref="S3.I3.i3.p1.2.m2.2.2.3.2a.cmml">length</mtext><mo id="S3.I3.i3.p1.2.m2.2.2.3.1" xref="S3.I3.i3.p1.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="S3.I3.i3.p1.2.m2.2.2.3.3.2" xref="S3.I3.i3.p1.2.m2.1.1a.cmml"><mo id="S3.I3.i3.p1.2.m2.2.2.3.3.2.1" stretchy="false" xref="S3.I3.i3.p1.2.m2.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.1.1" xref="S3.I3.i3.p1.2.m2.1.1.cmml">W</mtext><mo id="S3.I3.i3.p1.2.m2.2.2.3.3.2.2" stretchy="false" xref="S3.I3.i3.p1.2.m2.1.1a.cmml">)</mo></mrow></mrow><mo id="S3.I3.i3.p1.2.m2.2.2.2" xref="S3.I3.i3.p1.2.m2.2.2.2.cmml"><</mo><mrow id="S3.I3.i3.p1.2.m2.2.2.1" xref="S3.I3.i3.p1.2.m2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.2.2.1.3" xref="S3.I3.i3.p1.2.m2.2.2.1.3a.cmml">length</mtext><mo id="S3.I3.i3.p1.2.m2.2.2.1.2" xref="S3.I3.i3.p1.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S3.I3.i3.p1.2.m2.2.2.1.1.1" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.cmml"><mo id="S3.I3.i3.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.cmml">(</mo><msub id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.2" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.2a.cmml">Wit</mtext><msub id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.cmml"><mi id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.2" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.2.cmml">σ</mi><mn id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.3" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S3.I3.i3.p1.2.m2.2.2.1.1.1.3" stretchy="false" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i3.p1.2.m2.2b"><apply id="S3.I3.i3.p1.2.m2.2.2.cmml" xref="S3.I3.i3.p1.2.m2.2.2"><lt id="S3.I3.i3.p1.2.m2.2.2.2.cmml" xref="S3.I3.i3.p1.2.m2.2.2.2"></lt><apply id="S3.I3.i3.p1.2.m2.2.2.3.cmml" xref="S3.I3.i3.p1.2.m2.2.2.3"><times id="S3.I3.i3.p1.2.m2.2.2.3.1.cmml" xref="S3.I3.i3.p1.2.m2.2.2.3.1"></times><ci id="S3.I3.i3.p1.2.m2.2.2.3.2a.cmml" xref="S3.I3.i3.p1.2.m2.2.2.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.2.2.3.2.cmml" xref="S3.I3.i3.p1.2.m2.2.2.3.2">length</mtext></ci><ci id="S3.I3.i3.p1.2.m2.1.1a.cmml" xref="S3.I3.i3.p1.2.m2.2.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.1.1.cmml" xref="S3.I3.i3.p1.2.m2.1.1">W</mtext></ci></apply><apply id="S3.I3.i3.p1.2.m2.2.2.1.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1"><times id="S3.I3.i3.p1.2.m2.2.2.1.2.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.2"></times><ci id="S3.I3.i3.p1.2.m2.2.2.1.3a.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.2.2.1.3.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.3">length</mtext></ci><apply id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.2a.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.2">Wit</mtext></ci><apply id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.1.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3">subscript</csymbol><ci id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.2.cmml" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.2">𝜎</ci><cn id="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S3.I3.i3.p1.2.m2.2.2.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i3.p1.2.m2.2c">\textsf{length}({\textsf{W}})<\textsf{length}({\textsf{Wit}_{\sigma_{0}}})</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i3.p1.2.m2.2d">length ( W ) < length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i3.p1.2.4">.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S3.Thmtheorem4.p4"> <p class="ltx_p" id="S3.Thmtheorem4.p4.14"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p4.14.14">Finally, we prove that <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.1.1.m1.1"><semantics id="S3.Thmtheorem4.p4.1.1.m1.1a"><msubsup id="S3.Thmtheorem4.p4.1.1.m1.1.1" xref="S3.Thmtheorem4.p4.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem4.p4.1.1.m1.1.1.2.2" xref="S3.Thmtheorem4.p4.1.1.m1.1.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.1.1.m1.1.1.3" xref="S3.Thmtheorem4.p4.1.1.m1.1.1.3.cmml">0</mn><mo id="S3.Thmtheorem4.p4.1.1.m1.1.1.2.3" xref="S3.Thmtheorem4.p4.1.1.m1.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.1.1.m1.1b"><apply id="S3.Thmtheorem4.p4.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p4.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.1.1.m1.1.1">subscript</csymbol><apply id="S3.Thmtheorem4.p4.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.1.1.m1.1.1.2.1.cmml" xref="S3.Thmtheorem4.p4.1.1.m1.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p4.1.1.m1.1.1.2.2.cmml" xref="S3.Thmtheorem4.p4.1.1.m1.1.1.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p4.1.1.m1.1.1.2.3.cmml" xref="S3.Thmtheorem4.p4.1.1.m1.1.1.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p4.1.1.m1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.1.1.m1.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution to the SPS problem. We have to show that each play <math alttext="\pi^{\prime}\in\textsf{Play}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.2.2.m2.1"><semantics id="S3.Thmtheorem4.p4.2.2.m2.1a"><mrow id="S3.Thmtheorem4.p4.2.2.m2.1.1" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.cmml"><msup id="S3.Thmtheorem4.p4.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.2.cmml"><mi id="S3.Thmtheorem4.p4.2.2.m2.1.1.2.2" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.2.2.cmml">π</mi><mo id="S3.Thmtheorem4.p4.2.2.m2.1.1.2.3" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p4.2.2.m2.1.1.1" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p4.2.2.m2.1.1.3" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.2" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.2a.cmml">Play</mtext><msubsup id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.2" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.3" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.3" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.2.2.m2.1b"><apply id="S3.Thmtheorem4.p4.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1"><in id="S3.Thmtheorem4.p4.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.1"></in><apply id="S3.Thmtheorem4.p4.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.2.2.m2.1.1.2.1.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p4.2.2.m2.1.1.2.2.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.2.2">𝜋</ci><ci id="S3.Thmtheorem4.p4.2.2.m2.1.1.2.3.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.2a.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.2">Play</mtext></ci><apply id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3">subscript</csymbol><apply id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.3.cmml" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.2.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.2.2.m2.1c">\pi^{\prime}\in\textsf{Play}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.2.2.m2.1d">italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with a cost <math alttext="c^{\prime}\in C_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.3.3.m3.1"><semantics id="S3.Thmtheorem4.p4.3.3.m3.1a"><mrow id="S3.Thmtheorem4.p4.3.3.m3.1.1" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.cmml"><msup id="S3.Thmtheorem4.p4.3.3.m3.1.1.2" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.2.cmml"><mi id="S3.Thmtheorem4.p4.3.3.m3.1.1.2.2" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.2.2.cmml">c</mi><mo id="S3.Thmtheorem4.p4.3.3.m3.1.1.2.3" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p4.3.3.m3.1.1.1" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p4.3.3.m3.1.1.3" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.2" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.2.cmml">C</mi><msubsup id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.2" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.3" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.3.cmml">0</mn><mo id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.3" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.3.3.m3.1b"><apply id="S3.Thmtheorem4.p4.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1"><in id="S3.Thmtheorem4.p4.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.1"></in><apply id="S3.Thmtheorem4.p4.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.3.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p4.3.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.2.2">𝑐</ci><ci id="S3.Thmtheorem4.p4.3.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.2">𝐶</ci><apply id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3">subscript</csymbol><apply id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.3.cmml" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.3.3.m3.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.3.3.m3.1c">c^{\prime}\in C_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.3.3.m3.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> has a value <math alttext="\textsf{val}(\pi^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.4.4.m4.1"><semantics id="S3.Thmtheorem4.p4.4.4.m4.1a"><mrow id="S3.Thmtheorem4.p4.4.4.m4.1.1" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.cmml"><mrow id="S3.Thmtheorem4.p4.4.4.m4.1.1.1" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.3" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.3a.cmml">val</mtext><mo id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.2" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.2" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.3" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p4.4.4.m4.1.1.2" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.2.cmml">≤</mo><mi id="S3.Thmtheorem4.p4.4.4.m4.1.1.3" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.4.4.m4.1b"><apply id="S3.Thmtheorem4.p4.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1"><leq id="S3.Thmtheorem4.p4.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.2"></leq><apply id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1"><times id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.2"></times><ci id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.3a.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.3.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.3">val</mtext></ci><apply id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.2">𝜋</ci><ci id="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.Thmtheorem4.p4.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem4.p4.4.4.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.4.4.m4.1c">\textsf{val}(\pi^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.4.4.m4.1d">val ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math>. If <math alttext="g\not\pi^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.5.5.m5.1"><semantics id="S3.Thmtheorem4.p4.5.5.m5.1a"><mrow id="S3.Thmtheorem4.p4.5.5.m5.1.1" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem4.p4.5.5.m5.1.1.2" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.2.cmml">g</mi><mo id="S3.Thmtheorem4.p4.5.5.m5.1.1.1" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.1.cmml">⁢</mo><msup id="S3.Thmtheorem4.p4.5.5.m5.1.1.3" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.3.cmml"><mi class="ltx_mathvariant_italic" id="S3.Thmtheorem4.p4.5.5.m5.1.1.3.2" mathvariant="italic" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.3.2.cmml">π̸</mi><mo id="S3.Thmtheorem4.p4.5.5.m5.1.1.3.3" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.5.5.m5.1b"><apply id="S3.Thmtheorem4.p4.5.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p4.5.5.m5.1.1"><times id="S3.Thmtheorem4.p4.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.1"></times><ci id="S3.Thmtheorem4.p4.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.2">𝑔</ci><apply id="S3.Thmtheorem4.p4.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p4.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.3.2">italic-π̸</ci><ci id="S3.Thmtheorem4.p4.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem4.p4.5.5.m5.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.5.5.m5.1c">g\not\pi^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.5.5.m5.1d">italic_g italic_π̸ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="\pi^{\prime}\in\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.6.6.m6.1"><semantics id="S3.Thmtheorem4.p4.6.6.m6.1a"><mrow id="S3.Thmtheorem4.p4.6.6.m6.1.1" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.cmml"><msup id="S3.Thmtheorem4.p4.6.6.m6.1.1.2" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.2.cmml"><mi id="S3.Thmtheorem4.p4.6.6.m6.1.1.2.2" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.2.2.cmml">π</mi><mo id="S3.Thmtheorem4.p4.6.6.m6.1.1.2.3" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p4.6.6.m6.1.1.1" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem4.p4.6.6.m6.1.1.3" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.2" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.2a.cmml">Play</mtext><msub id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.2" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.3" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.6.6.m6.1b"><apply id="S3.Thmtheorem4.p4.6.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1"><in id="S3.Thmtheorem4.p4.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.1"></in><apply id="S3.Thmtheorem4.p4.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.6.6.m6.1.1.2.1.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p4.6.6.m6.1.1.2.2.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.2.2">𝜋</ci><ci id="S3.Thmtheorem4.p4.6.6.m6.1.1.2.3.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.1.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.2a.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.2.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.2">Play</mtext></ci><apply id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.6.6.m6.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.6.6.m6.1c">\pi^{\prime}\in\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.6.6.m6.1d">italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Notice that <math alttext="c^{\prime}=\textsf{cost}({\pi^{\prime}})\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.7.7.m7.1"><semantics id="S3.Thmtheorem4.p4.7.7.m7.1a"><mrow id="S3.Thmtheorem4.p4.7.7.m7.1.1" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.cmml"><msup id="S3.Thmtheorem4.p4.7.7.m7.1.1.3" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.3.cmml"><mi id="S3.Thmtheorem4.p4.7.7.m7.1.1.3.2" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.3.2.cmml">c</mi><mo id="S3.Thmtheorem4.p4.7.7.m7.1.1.3.3" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.3.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p4.7.7.m7.1.1.4" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.4.cmml">=</mo><mrow id="S3.Thmtheorem4.p4.7.7.m7.1.1.1" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.3" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.3a.cmml">cost</mtext><mo id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.2" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.2" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.3" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p4.7.7.m7.1.1.5" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.5.cmml">∈</mo><msub id="S3.Thmtheorem4.p4.7.7.m7.1.1.6" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.cmml"><mi id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.2" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.2.cmml">C</mi><msub id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.cmml"><mi id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.2" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.3" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.7.7.m7.1b"><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1"><and id="S3.Thmtheorem4.p4.7.7.m7.1.1a.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1"></and><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1b.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1"><eq id="S3.Thmtheorem4.p4.7.7.m7.1.1.4.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.4"></eq><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.7.7.m7.1.1.3.1.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p4.7.7.m7.1.1.3.2.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.3.2">𝑐</ci><ci id="S3.Thmtheorem4.p4.7.7.m7.1.1.3.3.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.3.3">′</ci></apply><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1"><times id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.2"></times><ci id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.3a.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.3.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.3">cost</mtext></ci><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.2">𝜋</ci><ci id="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1c.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1"><in id="S3.Thmtheorem4.p4.7.7.m7.1.1.5.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.5"></in><share href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem4.p4.7.7.m7.1.1.1.cmml" id="S3.Thmtheorem4.p4.7.7.m7.1.1d.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1"></share><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.1.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6">subscript</csymbol><ci id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.2.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.2">𝐶</ci><apply id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.1.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3">subscript</csymbol><ci id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.2.cmml" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.7.7.m7.1.1.6.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.7.7.m7.1c">c^{\prime}=\textsf{cost}({\pi^{\prime}})\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.7.7.m7.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = cost ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> because <math alttext="\sigma^{\prime}_{0}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.8.8.m8.1"><semantics id="S3.Thmtheorem4.p4.8.8.m8.1a"><mrow id="S3.Thmtheorem4.p4.8.8.m8.1.1" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.cmml"><msubsup id="S3.Thmtheorem4.p4.8.8.m8.1.1.2" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2.cmml"><mi id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.2" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.3" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2.3.cmml">0</mn><mo id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.3" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.Thmtheorem4.p4.8.8.m8.1.1.1" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.1.cmml">⪯</mo><msub id="S3.Thmtheorem4.p4.8.8.m8.1.1.3" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.3.cmml"><mi id="S3.Thmtheorem4.p4.8.8.m8.1.1.3.2" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.3.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.8.8.m8.1.1.3.3" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.8.8.m8.1b"><apply id="S3.Thmtheorem4.p4.8.8.m8.1.1.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1"><csymbol cd="latexml" id="S3.Thmtheorem4.p4.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.1">precedes-or-equals</csymbol><apply id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.1.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2">subscript</csymbol><apply id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.1.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.2.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.3.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p4.8.8.m8.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.2.3">0</cn></apply><apply id="S3.Thmtheorem4.p4.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.8.8.m8.1.1.3.1.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem4.p4.8.8.m8.1.1.3.2.cmml" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.3.2">𝜎</ci><cn id="S3.Thmtheorem4.p4.8.8.m8.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.8.8.m8.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.8.8.m8.1c">\sigma^{\prime}_{0}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.8.8.m8.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, as <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.9.9.m9.2"><semantics id="S3.Thmtheorem4.p4.9.9.m9.2a"><mrow id="S3.Thmtheorem4.p4.9.9.m9.2.3" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.cmml"><msub id="S3.Thmtheorem4.p4.9.9.m9.2.3.2" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.2.cmml"><mi id="S3.Thmtheorem4.p4.9.9.m9.2.3.2.2" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.9.9.m9.2.3.2.3" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.2.3.cmml">0</mn></msub><mo id="S3.Thmtheorem4.p4.9.9.m9.2.3.1" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem4.p4.9.9.m9.2.3.3" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.2" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.2a.cmml">SPS</mtext><mo id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.1" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.2" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.1.cmml"><mo id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem4.p4.9.9.m9.1.1" xref="S3.Thmtheorem4.p4.9.9.m9.1.1.cmml">G</mi><mo id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.2.2" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem4.p4.9.9.m9.2.2" xref="S3.Thmtheorem4.p4.9.9.m9.2.2.cmml">B</mi><mo id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.2.3" stretchy="false" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.9.9.m9.2b"><apply id="S3.Thmtheorem4.p4.9.9.m9.2.3.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3"><in id="S3.Thmtheorem4.p4.9.9.m9.2.3.1.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.1"></in><apply id="S3.Thmtheorem4.p4.9.9.m9.2.3.2.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.9.9.m9.2.3.2.1.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem4.p4.9.9.m9.2.3.2.2.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.2.2">𝜎</ci><cn id="S3.Thmtheorem4.p4.9.9.m9.2.3.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.2.3">0</cn></apply><apply id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3"><times id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.1.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.1"></times><ci id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.2a.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.2.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.1.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.3.3.3.2"><ci id="S3.Thmtheorem4.p4.9.9.m9.1.1.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.1.1">𝐺</ci><ci id="S3.Thmtheorem4.p4.9.9.m9.2.2.cmml" xref="S3.Thmtheorem4.p4.9.9.m9.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.9.9.m9.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.9.9.m9.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math>, we get that <math alttext="\textsf{val}(\pi^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.10.10.m10.1"><semantics id="S3.Thmtheorem4.p4.10.10.m10.1a"><mrow id="S3.Thmtheorem4.p4.10.10.m10.1.1" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.cmml"><mrow id="S3.Thmtheorem4.p4.10.10.m10.1.1.1" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.3" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.3a.cmml">val</mtext><mo id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.2" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.2" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.2.cmml">π</mi><mo id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.3" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p4.10.10.m10.1.1.2" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.2.cmml">≤</mo><mi id="S3.Thmtheorem4.p4.10.10.m10.1.1.3" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.10.10.m10.1b"><apply id="S3.Thmtheorem4.p4.10.10.m10.1.1.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1"><leq id="S3.Thmtheorem4.p4.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.2"></leq><apply id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1"><times id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.2"></times><ci id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.3a.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.3.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.3">val</mtext></ci><apply id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.2">𝜋</ci><ci id="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.Thmtheorem4.p4.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem4.p4.10.10.m10.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.10.10.m10.1c">\textsf{val}(\pi^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.10.10.m10.1d">val ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math>. If <math alttext="g\pi^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.11.11.m11.1"><semantics id="S3.Thmtheorem4.p4.11.11.m11.1a"><mrow id="S3.Thmtheorem4.p4.11.11.m11.1.1" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.cmml"><mi id="S3.Thmtheorem4.p4.11.11.m11.1.1.2" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.2.cmml">g</mi><mo id="S3.Thmtheorem4.p4.11.11.m11.1.1.1" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.1.cmml">⁢</mo><msup id="S3.Thmtheorem4.p4.11.11.m11.1.1.3" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.3.cmml"><mi id="S3.Thmtheorem4.p4.11.11.m11.1.1.3.2" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.3.2.cmml">π</mi><mo id="S3.Thmtheorem4.p4.11.11.m11.1.1.3.3" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.11.11.m11.1b"><apply id="S3.Thmtheorem4.p4.11.11.m11.1.1.cmml" xref="S3.Thmtheorem4.p4.11.11.m11.1.1"><times id="S3.Thmtheorem4.p4.11.11.m11.1.1.1.cmml" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.1"></times><ci id="S3.Thmtheorem4.p4.11.11.m11.1.1.2.cmml" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.2">𝑔</ci><apply id="S3.Thmtheorem4.p4.11.11.m11.1.1.3.cmml" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.11.11.m11.1.1.3.1.cmml" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p4.11.11.m11.1.1.3.2.cmml" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.3.2">𝜋</ci><ci id="S3.Thmtheorem4.p4.11.11.m11.1.1.3.3.cmml" xref="S3.Thmtheorem4.p4.11.11.m11.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.11.11.m11.1c">g\pi^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.11.11.m11.1d">italic_g italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then either <math alttext="\textsf{val}(g)<\infty" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.12.12.m12.1"><semantics id="S3.Thmtheorem4.p4.12.12.m12.1a"><mrow id="S3.Thmtheorem4.p4.12.12.m12.1.2" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.cmml"><mrow id="S3.Thmtheorem4.p4.12.12.m12.1.2.2" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.2" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.2a.cmml">val</mtext><mo id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.1" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.3.2" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.cmml"><mo id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem4.p4.12.12.m12.1.1" xref="S3.Thmtheorem4.p4.12.12.m12.1.1.cmml">g</mi><mo id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p4.12.12.m12.1.2.1" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.1.cmml"><</mo><mi id="S3.Thmtheorem4.p4.12.12.m12.1.2.3" mathvariant="normal" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.12.12.m12.1b"><apply id="S3.Thmtheorem4.p4.12.12.m12.1.2.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.2"><lt id="S3.Thmtheorem4.p4.12.12.m12.1.2.1.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.1"></lt><apply id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2"><times id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.1.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.1"></times><ci id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.2a.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.12.12.m12.1.2.2.2.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.2.2">val</mtext></ci><ci id="S3.Thmtheorem4.p4.12.12.m12.1.1.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.1">𝑔</ci></apply><infinity id="S3.Thmtheorem4.p4.12.12.m12.1.2.3.cmml" xref="S3.Thmtheorem4.p4.12.12.m12.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.12.12.m12.1c">\textsf{val}(g)<\infty</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.12.12.m12.1d">val ( italic_g ) < ∞</annotation></semantics></math>, or <math alttext="\textsf{val}(g)=\infty" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.13.13.m13.1"><semantics id="S3.Thmtheorem4.p4.13.13.m13.1a"><mrow id="S3.Thmtheorem4.p4.13.13.m13.1.2" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.cmml"><mrow id="S3.Thmtheorem4.p4.13.13.m13.1.2.2" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.2" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.2a.cmml">val</mtext><mo id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.1" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.3.2" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.cmml"><mo id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem4.p4.13.13.m13.1.1" xref="S3.Thmtheorem4.p4.13.13.m13.1.1.cmml">g</mi><mo id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p4.13.13.m13.1.2.1" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.1.cmml">=</mo><mi id="S3.Thmtheorem4.p4.13.13.m13.1.2.3" mathvariant="normal" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.13.13.m13.1b"><apply id="S3.Thmtheorem4.p4.13.13.m13.1.2.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.2"><eq id="S3.Thmtheorem4.p4.13.13.m13.1.2.1.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.1"></eq><apply id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2"><times id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.1.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.1"></times><ci id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.2a.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S3.Thmtheorem4.p4.13.13.m13.1.2.2.2.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.2.2">val</mtext></ci><ci id="S3.Thmtheorem4.p4.13.13.m13.1.1.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.1">𝑔</ci></apply><infinity id="S3.Thmtheorem4.p4.13.13.m13.1.2.3.cmml" xref="S3.Thmtheorem4.p4.13.13.m13.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.13.13.m13.1c">\textsf{val}(g)=\infty</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.13.13.m13.1d">val ( italic_g ) = ∞</annotation></semantics></math> in which case the weight of <math alttext="\rho_{[m,n]}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.14.14.m14.2"><semantics id="S3.Thmtheorem4.p4.14.14.m14.2a"><msub id="S3.Thmtheorem4.p4.14.14.m14.2.3" xref="S3.Thmtheorem4.p4.14.14.m14.2.3.cmml"><mi id="S3.Thmtheorem4.p4.14.14.m14.2.3.2" xref="S3.Thmtheorem4.p4.14.14.m14.2.3.2.cmml">ρ</mi><mrow id="S3.Thmtheorem4.p4.14.14.m14.2.2.2.4" xref="S3.Thmtheorem4.p4.14.14.m14.2.2.2.3.cmml"><mo id="S3.Thmtheorem4.p4.14.14.m14.2.2.2.4.1" stretchy="false" xref="S3.Thmtheorem4.p4.14.14.m14.2.2.2.3.cmml">[</mo><mi id="S3.Thmtheorem4.p4.14.14.m14.1.1.1.1" xref="S3.Thmtheorem4.p4.14.14.m14.1.1.1.1.cmml">m</mi><mo id="S3.Thmtheorem4.p4.14.14.m14.2.2.2.4.2" xref="S3.Thmtheorem4.p4.14.14.m14.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem4.p4.14.14.m14.2.2.2.2" xref="S3.Thmtheorem4.p4.14.14.m14.2.2.2.2.cmml">n</mi><mo id="S3.Thmtheorem4.p4.14.14.m14.2.2.2.4.3" stretchy="false" xref="S3.Thmtheorem4.p4.14.14.m14.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.14.14.m14.2b"><apply id="S3.Thmtheorem4.p4.14.14.m14.2.3.cmml" xref="S3.Thmtheorem4.p4.14.14.m14.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.14.14.m14.2.3.1.cmml" xref="S3.Thmtheorem4.p4.14.14.m14.2.3">subscript</csymbol><ci id="S3.Thmtheorem4.p4.14.14.m14.2.3.2.cmml" xref="S3.Thmtheorem4.p4.14.14.m14.2.3.2">𝜌</ci><interval closure="closed" id="S3.Thmtheorem4.p4.14.14.m14.2.2.2.3.cmml" xref="S3.Thmtheorem4.p4.14.14.m14.2.2.2.4"><ci id="S3.Thmtheorem4.p4.14.14.m14.1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.14.14.m14.1.1.1.1">𝑚</ci><ci id="S3.Thmtheorem4.p4.14.14.m14.2.2.2.2.cmml" xref="S3.Thmtheorem4.p4.14.14.m14.2.2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.14.14.m14.2c">\rho_{[m,n]}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.14.14.m14.2d">italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT</annotation></semantics></math> is null by hypothesis.</span></p> <ul class="ltx_itemize" id="S3.I4"> <li class="ltx_item" id="S3.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I4.i1.p1"> <p class="ltx_p" id="S3.I4.i1.p1.5"><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.5.1">In the first case, </span><math alttext="\textsf{val}(g)\leq B" class="ltx_Math" display="inline" id="S3.I4.i1.p1.1.m1.1"><semantics id="S3.I4.i1.p1.1.m1.1a"><mrow id="S3.I4.i1.p1.1.m1.1.2" xref="S3.I4.i1.p1.1.m1.1.2.cmml"><mrow id="S3.I4.i1.p1.1.m1.1.2.2" xref="S3.I4.i1.p1.1.m1.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i1.p1.1.m1.1.2.2.2" xref="S3.I4.i1.p1.1.m1.1.2.2.2a.cmml">val</mtext><mo id="S3.I4.i1.p1.1.m1.1.2.2.1" xref="S3.I4.i1.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.I4.i1.p1.1.m1.1.2.2.3.2" xref="S3.I4.i1.p1.1.m1.1.2.2.cmml"><mo id="S3.I4.i1.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S3.I4.i1.p1.1.m1.1.2.2.cmml">(</mo><mi id="S3.I4.i1.p1.1.m1.1.1" xref="S3.I4.i1.p1.1.m1.1.1.cmml">g</mi><mo id="S3.I4.i1.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S3.I4.i1.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I4.i1.p1.1.m1.1.2.1" xref="S3.I4.i1.p1.1.m1.1.2.1.cmml">≤</mo><mi id="S3.I4.i1.p1.1.m1.1.2.3" xref="S3.I4.i1.p1.1.m1.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.1.m1.1b"><apply id="S3.I4.i1.p1.1.m1.1.2.cmml" xref="S3.I4.i1.p1.1.m1.1.2"><leq id="S3.I4.i1.p1.1.m1.1.2.1.cmml" xref="S3.I4.i1.p1.1.m1.1.2.1"></leq><apply id="S3.I4.i1.p1.1.m1.1.2.2.cmml" xref="S3.I4.i1.p1.1.m1.1.2.2"><times id="S3.I4.i1.p1.1.m1.1.2.2.1.cmml" xref="S3.I4.i1.p1.1.m1.1.2.2.1"></times><ci id="S3.I4.i1.p1.1.m1.1.2.2.2a.cmml" xref="S3.I4.i1.p1.1.m1.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i1.p1.1.m1.1.2.2.2.cmml" xref="S3.I4.i1.p1.1.m1.1.2.2.2">val</mtext></ci><ci id="S3.I4.i1.p1.1.m1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1">𝑔</ci></apply><ci id="S3.I4.i1.p1.1.m1.1.2.3.cmml" xref="S3.I4.i1.p1.1.m1.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.1.m1.1c">\textsf{val}(g)\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.1.m1.1d">val ( italic_g ) ≤ italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.5.2"> because </span><math alttext="g" class="ltx_Math" display="inline" id="S3.I4.i1.p1.2.m2.1"><semantics id="S3.I4.i1.p1.2.m2.1a"><mi id="S3.I4.i1.p1.2.m2.1.1" xref="S3.I4.i1.p1.2.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.2.m2.1b"><ci id="S3.I4.i1.p1.2.m2.1.1.cmml" xref="S3.I4.i1.p1.2.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.2.m2.1c">g</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.2.m2.1d">italic_g</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.5.3"> is prefix of the witness </span><math alttext="\rho\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.I4.i1.p1.3.m3.1"><semantics id="S3.I4.i1.p1.3.m3.1a"><mrow id="S3.I4.i1.p1.3.m3.1.1" xref="S3.I4.i1.p1.3.m3.1.1.cmml"><mi id="S3.I4.i1.p1.3.m3.1.1.2" xref="S3.I4.i1.p1.3.m3.1.1.2.cmml">ρ</mi><mo id="S3.I4.i1.p1.3.m3.1.1.1" xref="S3.I4.i1.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S3.I4.i1.p1.3.m3.1.1.3" xref="S3.I4.i1.p1.3.m3.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i1.p1.3.m3.1.1.3.2" xref="S3.I4.i1.p1.3.m3.1.1.3.2a.cmml">Wit</mtext><msub id="S3.I4.i1.p1.3.m3.1.1.3.3" xref="S3.I4.i1.p1.3.m3.1.1.3.3.cmml"><mi id="S3.I4.i1.p1.3.m3.1.1.3.3.2" xref="S3.I4.i1.p1.3.m3.1.1.3.3.2.cmml">σ</mi><mn id="S3.I4.i1.p1.3.m3.1.1.3.3.3" xref="S3.I4.i1.p1.3.m3.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.3.m3.1b"><apply id="S3.I4.i1.p1.3.m3.1.1.cmml" xref="S3.I4.i1.p1.3.m3.1.1"><in id="S3.I4.i1.p1.3.m3.1.1.1.cmml" xref="S3.I4.i1.p1.3.m3.1.1.1"></in><ci id="S3.I4.i1.p1.3.m3.1.1.2.cmml" xref="S3.I4.i1.p1.3.m3.1.1.2">𝜌</ci><apply id="S3.I4.i1.p1.3.m3.1.1.3.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I4.i1.p1.3.m3.1.1.3.1.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.I4.i1.p1.3.m3.1.1.3.2a.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i1.p1.3.m3.1.1.3.2.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3.2">Wit</mtext></ci><apply id="S3.I4.i1.p1.3.m3.1.1.3.3.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.I4.i1.p1.3.m3.1.1.3.3.1.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.I4.i1.p1.3.m3.1.1.3.3.2.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3.3.2">𝜎</ci><cn id="S3.I4.i1.p1.3.m3.1.1.3.3.3.cmml" type="integer" xref="S3.I4.i1.p1.3.m3.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.3.m3.1c">\rho\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.3.m3.1d">italic_ρ ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.5.4"> and </span><math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.I4.i1.p1.4.m4.2"><semantics id="S3.I4.i1.p1.4.m4.2a"><mrow id="S3.I4.i1.p1.4.m4.2.3" xref="S3.I4.i1.p1.4.m4.2.3.cmml"><msub id="S3.I4.i1.p1.4.m4.2.3.2" xref="S3.I4.i1.p1.4.m4.2.3.2.cmml"><mi id="S3.I4.i1.p1.4.m4.2.3.2.2" xref="S3.I4.i1.p1.4.m4.2.3.2.2.cmml">σ</mi><mn id="S3.I4.i1.p1.4.m4.2.3.2.3" xref="S3.I4.i1.p1.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S3.I4.i1.p1.4.m4.2.3.1" xref="S3.I4.i1.p1.4.m4.2.3.1.cmml">∈</mo><mrow id="S3.I4.i1.p1.4.m4.2.3.3" xref="S3.I4.i1.p1.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.I4.i1.p1.4.m4.2.3.3.2" xref="S3.I4.i1.p1.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S3.I4.i1.p1.4.m4.2.3.3.1" xref="S3.I4.i1.p1.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S3.I4.i1.p1.4.m4.2.3.3.3.2" xref="S3.I4.i1.p1.4.m4.2.3.3.3.1.cmml"><mo id="S3.I4.i1.p1.4.m4.2.3.3.3.2.1" stretchy="false" xref="S3.I4.i1.p1.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S3.I4.i1.p1.4.m4.1.1" xref="S3.I4.i1.p1.4.m4.1.1.cmml">G</mi><mo id="S3.I4.i1.p1.4.m4.2.3.3.3.2.2" xref="S3.I4.i1.p1.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S3.I4.i1.p1.4.m4.2.2" xref="S3.I4.i1.p1.4.m4.2.2.cmml">B</mi><mo id="S3.I4.i1.p1.4.m4.2.3.3.3.2.3" stretchy="false" xref="S3.I4.i1.p1.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.4.m4.2b"><apply id="S3.I4.i1.p1.4.m4.2.3.cmml" xref="S3.I4.i1.p1.4.m4.2.3"><in id="S3.I4.i1.p1.4.m4.2.3.1.cmml" xref="S3.I4.i1.p1.4.m4.2.3.1"></in><apply id="S3.I4.i1.p1.4.m4.2.3.2.cmml" xref="S3.I4.i1.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S3.I4.i1.p1.4.m4.2.3.2.1.cmml" xref="S3.I4.i1.p1.4.m4.2.3.2">subscript</csymbol><ci id="S3.I4.i1.p1.4.m4.2.3.2.2.cmml" xref="S3.I4.i1.p1.4.m4.2.3.2.2">𝜎</ci><cn id="S3.I4.i1.p1.4.m4.2.3.2.3.cmml" type="integer" xref="S3.I4.i1.p1.4.m4.2.3.2.3">0</cn></apply><apply id="S3.I4.i1.p1.4.m4.2.3.3.cmml" xref="S3.I4.i1.p1.4.m4.2.3.3"><times id="S3.I4.i1.p1.4.m4.2.3.3.1.cmml" xref="S3.I4.i1.p1.4.m4.2.3.3.1"></times><ci id="S3.I4.i1.p1.4.m4.2.3.3.2a.cmml" xref="S3.I4.i1.p1.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S3.I4.i1.p1.4.m4.2.3.3.2.cmml" xref="S3.I4.i1.p1.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S3.I4.i1.p1.4.m4.2.3.3.3.1.cmml" xref="S3.I4.i1.p1.4.m4.2.3.3.3.2"><ci id="S3.I4.i1.p1.4.m4.1.1.cmml" xref="S3.I4.i1.p1.4.m4.1.1">𝐺</ci><ci id="S3.I4.i1.p1.4.m4.2.2.cmml" xref="S3.I4.i1.p1.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.4.m4.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.5.5">. This shows that </span><math alttext="\textsf{val}(\pi^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.I4.i1.p1.5.m5.1"><semantics id="S3.I4.i1.p1.5.m5.1a"><mrow id="S3.I4.i1.p1.5.m5.1.1" xref="S3.I4.i1.p1.5.m5.1.1.cmml"><mrow id="S3.I4.i1.p1.5.m5.1.1.1" xref="S3.I4.i1.p1.5.m5.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i1.p1.5.m5.1.1.1.3" xref="S3.I4.i1.p1.5.m5.1.1.1.3a.cmml">val</mtext><mo id="S3.I4.i1.p1.5.m5.1.1.1.2" xref="S3.I4.i1.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.I4.i1.p1.5.m5.1.1.1.1.1" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.I4.i1.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I4.i1.p1.5.m5.1.1.1.1.1.1" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.2" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.2.cmml">π</mi><mo id="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.3" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I4.i1.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I4.i1.p1.5.m5.1.1.2" xref="S3.I4.i1.p1.5.m5.1.1.2.cmml">≤</mo><mi id="S3.I4.i1.p1.5.m5.1.1.3" xref="S3.I4.i1.p1.5.m5.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.5.m5.1b"><apply id="S3.I4.i1.p1.5.m5.1.1.cmml" xref="S3.I4.i1.p1.5.m5.1.1"><leq id="S3.I4.i1.p1.5.m5.1.1.2.cmml" xref="S3.I4.i1.p1.5.m5.1.1.2"></leq><apply id="S3.I4.i1.p1.5.m5.1.1.1.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1"><times id="S3.I4.i1.p1.5.m5.1.1.1.2.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1.2"></times><ci id="S3.I4.i1.p1.5.m5.1.1.1.3a.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i1.p1.5.m5.1.1.1.3.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1.3">val</mtext></ci><apply id="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.2">𝜋</ci><ci id="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.I4.i1.p1.5.m5.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.I4.i1.p1.5.m5.1.1.3.cmml" xref="S3.I4.i1.p1.5.m5.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.5.m5.1c">\textsf{val}(\pi^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.5.m5.1d">val ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.5.6">.</span></p> </div> </li> <li class="ltx_item" id="S3.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I4.i2.p1"> <p class="ltx_p" id="S3.I4.i2.p1.21"><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.1">In the second case, we consider </span><math alttext="\pi\in\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.1.m1.1"><semantics id="S3.I4.i2.p1.1.m1.1a"><mrow id="S3.I4.i2.p1.1.m1.1.1" xref="S3.I4.i2.p1.1.m1.1.1.cmml"><mi id="S3.I4.i2.p1.1.m1.1.1.2" xref="S3.I4.i2.p1.1.m1.1.1.2.cmml">π</mi><mo id="S3.I4.i2.p1.1.m1.1.1.1" xref="S3.I4.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S3.I4.i2.p1.1.m1.1.1.3" xref="S3.I4.i2.p1.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.1.m1.1.1.3.2" xref="S3.I4.i2.p1.1.m1.1.1.3.2a.cmml">Play</mtext><msub id="S3.I4.i2.p1.1.m1.1.1.3.3" xref="S3.I4.i2.p1.1.m1.1.1.3.3.cmml"><mi id="S3.I4.i2.p1.1.m1.1.1.3.3.2" xref="S3.I4.i2.p1.1.m1.1.1.3.3.2.cmml">σ</mi><mn id="S3.I4.i2.p1.1.m1.1.1.3.3.3" xref="S3.I4.i2.p1.1.m1.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.1.m1.1b"><apply id="S3.I4.i2.p1.1.m1.1.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1"><in id="S3.I4.i2.p1.1.m1.1.1.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.1"></in><ci id="S3.I4.i2.p1.1.m1.1.1.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2">𝜋</ci><apply id="S3.I4.i2.p1.1.m1.1.1.3.cmml" xref="S3.I4.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.1.m1.1.1.3.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.I4.i2.p1.1.m1.1.1.3.2a.cmml" xref="S3.I4.i2.p1.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.1.m1.1.1.3.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.3.2">Play</mtext></ci><apply id="S3.I4.i2.p1.1.m1.1.1.3.3.cmml" xref="S3.I4.i2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.1.m1.1.1.3.3.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.I4.i2.p1.1.m1.1.1.3.3.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.3.3.2">𝜎</ci><cn id="S3.I4.i2.p1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S3.I4.i2.p1.1.m1.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.1.m1.1c">\pi\in\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.1.m1.1d">italic_π ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.2"> such that </span><math alttext="\bar{\pi}=\pi^{\prime}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.2.m2.1"><semantics id="S3.I4.i2.p1.2.m2.1a"><mrow id="S3.I4.i2.p1.2.m2.1.1" xref="S3.I4.i2.p1.2.m2.1.1.cmml"><mover accent="true" id="S3.I4.i2.p1.2.m2.1.1.2" xref="S3.I4.i2.p1.2.m2.1.1.2.cmml"><mi id="S3.I4.i2.p1.2.m2.1.1.2.2" xref="S3.I4.i2.p1.2.m2.1.1.2.2.cmml">π</mi><mo id="S3.I4.i2.p1.2.m2.1.1.2.1" xref="S3.I4.i2.p1.2.m2.1.1.2.1.cmml">¯</mo></mover><mo id="S3.I4.i2.p1.2.m2.1.1.1" xref="S3.I4.i2.p1.2.m2.1.1.1.cmml">=</mo><msup id="S3.I4.i2.p1.2.m2.1.1.3" xref="S3.I4.i2.p1.2.m2.1.1.3.cmml"><mi id="S3.I4.i2.p1.2.m2.1.1.3.2" xref="S3.I4.i2.p1.2.m2.1.1.3.2.cmml">π</mi><mo id="S3.I4.i2.p1.2.m2.1.1.3.3" xref="S3.I4.i2.p1.2.m2.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.2.m2.1b"><apply id="S3.I4.i2.p1.2.m2.1.1.cmml" xref="S3.I4.i2.p1.2.m2.1.1"><eq id="S3.I4.i2.p1.2.m2.1.1.1.cmml" xref="S3.I4.i2.p1.2.m2.1.1.1"></eq><apply id="S3.I4.i2.p1.2.m2.1.1.2.cmml" xref="S3.I4.i2.p1.2.m2.1.1.2"><ci id="S3.I4.i2.p1.2.m2.1.1.2.1.cmml" xref="S3.I4.i2.p1.2.m2.1.1.2.1">¯</ci><ci id="S3.I4.i2.p1.2.m2.1.1.2.2.cmml" xref="S3.I4.i2.p1.2.m2.1.1.2.2">𝜋</ci></apply><apply id="S3.I4.i2.p1.2.m2.1.1.3.cmml" xref="S3.I4.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.2.m2.1.1.3.1.cmml" xref="S3.I4.i2.p1.2.m2.1.1.3">superscript</csymbol><ci id="S3.I4.i2.p1.2.m2.1.1.3.2.cmml" xref="S3.I4.i2.p1.2.m2.1.1.3.2">𝜋</ci><ci id="S3.I4.i2.p1.2.m2.1.1.3.3.cmml" xref="S3.I4.i2.p1.2.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.2.m2.1c">\bar{\pi}=\pi^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.2.m2.1d">over¯ start_ARG italic_π end_ARG = italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.3">. Notice that </span><math alttext="\textsf{cost}({\pi})=c^{\prime}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.3.m3.1"><semantics id="S3.I4.i2.p1.3.m3.1a"><mrow id="S3.I4.i2.p1.3.m3.1.2" xref="S3.I4.i2.p1.3.m3.1.2.cmml"><mrow id="S3.I4.i2.p1.3.m3.1.2.2" xref="S3.I4.i2.p1.3.m3.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.3.m3.1.2.2.2" xref="S3.I4.i2.p1.3.m3.1.2.2.2a.cmml">cost</mtext><mo id="S3.I4.i2.p1.3.m3.1.2.2.1" xref="S3.I4.i2.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.3.m3.1.2.2.3.2" xref="S3.I4.i2.p1.3.m3.1.2.2.cmml"><mo id="S3.I4.i2.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.I4.i2.p1.3.m3.1.2.2.cmml">(</mo><mi id="S3.I4.i2.p1.3.m3.1.1" xref="S3.I4.i2.p1.3.m3.1.1.cmml">π</mi><mo id="S3.I4.i2.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.I4.i2.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I4.i2.p1.3.m3.1.2.1" xref="S3.I4.i2.p1.3.m3.1.2.1.cmml">=</mo><msup id="S3.I4.i2.p1.3.m3.1.2.3" xref="S3.I4.i2.p1.3.m3.1.2.3.cmml"><mi id="S3.I4.i2.p1.3.m3.1.2.3.2" xref="S3.I4.i2.p1.3.m3.1.2.3.2.cmml">c</mi><mo id="S3.I4.i2.p1.3.m3.1.2.3.3" xref="S3.I4.i2.p1.3.m3.1.2.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.3.m3.1b"><apply id="S3.I4.i2.p1.3.m3.1.2.cmml" xref="S3.I4.i2.p1.3.m3.1.2"><eq id="S3.I4.i2.p1.3.m3.1.2.1.cmml" xref="S3.I4.i2.p1.3.m3.1.2.1"></eq><apply id="S3.I4.i2.p1.3.m3.1.2.2.cmml" xref="S3.I4.i2.p1.3.m3.1.2.2"><times id="S3.I4.i2.p1.3.m3.1.2.2.1.cmml" xref="S3.I4.i2.p1.3.m3.1.2.2.1"></times><ci id="S3.I4.i2.p1.3.m3.1.2.2.2a.cmml" xref="S3.I4.i2.p1.3.m3.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.3.m3.1.2.2.2.cmml" xref="S3.I4.i2.p1.3.m3.1.2.2.2">cost</mtext></ci><ci id="S3.I4.i2.p1.3.m3.1.1.cmml" xref="S3.I4.i2.p1.3.m3.1.1">𝜋</ci></apply><apply id="S3.I4.i2.p1.3.m3.1.2.3.cmml" xref="S3.I4.i2.p1.3.m3.1.2.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.3.m3.1.2.3.1.cmml" xref="S3.I4.i2.p1.3.m3.1.2.3">superscript</csymbol><ci id="S3.I4.i2.p1.3.m3.1.2.3.2.cmml" xref="S3.I4.i2.p1.3.m3.1.2.3.2">𝑐</ci><ci id="S3.I4.i2.p1.3.m3.1.2.3.3.cmml" xref="S3.I4.i2.p1.3.m3.1.2.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.3.m3.1c">\textsf{cost}({\pi})=c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.3.m3.1d">cost ( italic_π ) = italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.4"> because </span><math alttext="\rho_{[m,n]}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.4.m4.2"><semantics id="S3.I4.i2.p1.4.m4.2a"><msub id="S3.I4.i2.p1.4.m4.2.3" xref="S3.I4.i2.p1.4.m4.2.3.cmml"><mi id="S3.I4.i2.p1.4.m4.2.3.2" xref="S3.I4.i2.p1.4.m4.2.3.2.cmml">ρ</mi><mrow id="S3.I4.i2.p1.4.m4.2.2.2.4" xref="S3.I4.i2.p1.4.m4.2.2.2.3.cmml"><mo id="S3.I4.i2.p1.4.m4.2.2.2.4.1" stretchy="false" xref="S3.I4.i2.p1.4.m4.2.2.2.3.cmml">[</mo><mi id="S3.I4.i2.p1.4.m4.1.1.1.1" xref="S3.I4.i2.p1.4.m4.1.1.1.1.cmml">m</mi><mo id="S3.I4.i2.p1.4.m4.2.2.2.4.2" xref="S3.I4.i2.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.I4.i2.p1.4.m4.2.2.2.2" xref="S3.I4.i2.p1.4.m4.2.2.2.2.cmml">n</mi><mo id="S3.I4.i2.p1.4.m4.2.2.2.4.3" stretchy="false" xref="S3.I4.i2.p1.4.m4.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.4.m4.2b"><apply id="S3.I4.i2.p1.4.m4.2.3.cmml" xref="S3.I4.i2.p1.4.m4.2.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.4.m4.2.3.1.cmml" xref="S3.I4.i2.p1.4.m4.2.3">subscript</csymbol><ci id="S3.I4.i2.p1.4.m4.2.3.2.cmml" xref="S3.I4.i2.p1.4.m4.2.3.2">𝜌</ci><interval closure="closed" id="S3.I4.i2.p1.4.m4.2.2.2.3.cmml" xref="S3.I4.i2.p1.4.m4.2.2.2.4"><ci id="S3.I4.i2.p1.4.m4.1.1.1.1.cmml" xref="S3.I4.i2.p1.4.m4.1.1.1.1">𝑚</ci><ci id="S3.I4.i2.p1.4.m4.2.2.2.2.cmml" xref="S3.I4.i2.p1.4.m4.2.2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.4.m4.2c">\rho_{[m,n]}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.4.m4.2d">italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.5"> has a null weight and </span><math alttext="g\sim h" class="ltx_Math" display="inline" id="S3.I4.i2.p1.5.m5.1"><semantics id="S3.I4.i2.p1.5.m5.1a"><mrow id="S3.I4.i2.p1.5.m5.1.1" xref="S3.I4.i2.p1.5.m5.1.1.cmml"><mi id="S3.I4.i2.p1.5.m5.1.1.2" xref="S3.I4.i2.p1.5.m5.1.1.2.cmml">g</mi><mo id="S3.I4.i2.p1.5.m5.1.1.1" xref="S3.I4.i2.p1.5.m5.1.1.1.cmml">∼</mo><mi id="S3.I4.i2.p1.5.m5.1.1.3" xref="S3.I4.i2.p1.5.m5.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.5.m5.1b"><apply id="S3.I4.i2.p1.5.m5.1.1.cmml" xref="S3.I4.i2.p1.5.m5.1.1"><csymbol cd="latexml" id="S3.I4.i2.p1.5.m5.1.1.1.cmml" xref="S3.I4.i2.p1.5.m5.1.1.1">similar-to</csymbol><ci id="S3.I4.i2.p1.5.m5.1.1.2.cmml" xref="S3.I4.i2.p1.5.m5.1.1.2">𝑔</ci><ci id="S3.I4.i2.p1.5.m5.1.1.3.cmml" xref="S3.I4.i2.p1.5.m5.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.5.m5.1c">g\sim h</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.5.m5.1d">italic_g ∼ italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.6"> (in particular, </span><math alttext="\textsf{cost}({g})=\textsf{cost}({h})" class="ltx_Math" display="inline" id="S3.I4.i2.p1.6.m6.2"><semantics id="S3.I4.i2.p1.6.m6.2a"><mrow id="S3.I4.i2.p1.6.m6.2.3" xref="S3.I4.i2.p1.6.m6.2.3.cmml"><mrow id="S3.I4.i2.p1.6.m6.2.3.2" xref="S3.I4.i2.p1.6.m6.2.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.6.m6.2.3.2.2" xref="S3.I4.i2.p1.6.m6.2.3.2.2a.cmml">cost</mtext><mo id="S3.I4.i2.p1.6.m6.2.3.2.1" xref="S3.I4.i2.p1.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.6.m6.2.3.2.3.2" xref="S3.I4.i2.p1.6.m6.2.3.2.cmml"><mo id="S3.I4.i2.p1.6.m6.2.3.2.3.2.1" stretchy="false" xref="S3.I4.i2.p1.6.m6.2.3.2.cmml">(</mo><mi id="S3.I4.i2.p1.6.m6.1.1" xref="S3.I4.i2.p1.6.m6.1.1.cmml">g</mi><mo id="S3.I4.i2.p1.6.m6.2.3.2.3.2.2" stretchy="false" xref="S3.I4.i2.p1.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.I4.i2.p1.6.m6.2.3.1" xref="S3.I4.i2.p1.6.m6.2.3.1.cmml">=</mo><mrow id="S3.I4.i2.p1.6.m6.2.3.3" xref="S3.I4.i2.p1.6.m6.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.6.m6.2.3.3.2" xref="S3.I4.i2.p1.6.m6.2.3.3.2a.cmml">cost</mtext><mo id="S3.I4.i2.p1.6.m6.2.3.3.1" xref="S3.I4.i2.p1.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.6.m6.2.3.3.3.2" xref="S3.I4.i2.p1.6.m6.2.3.3.cmml"><mo id="S3.I4.i2.p1.6.m6.2.3.3.3.2.1" stretchy="false" xref="S3.I4.i2.p1.6.m6.2.3.3.cmml">(</mo><mi id="S3.I4.i2.p1.6.m6.2.2" xref="S3.I4.i2.p1.6.m6.2.2.cmml">h</mi><mo id="S3.I4.i2.p1.6.m6.2.3.3.3.2.2" stretchy="false" xref="S3.I4.i2.p1.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.6.m6.2b"><apply id="S3.I4.i2.p1.6.m6.2.3.cmml" xref="S3.I4.i2.p1.6.m6.2.3"><eq id="S3.I4.i2.p1.6.m6.2.3.1.cmml" xref="S3.I4.i2.p1.6.m6.2.3.1"></eq><apply id="S3.I4.i2.p1.6.m6.2.3.2.cmml" xref="S3.I4.i2.p1.6.m6.2.3.2"><times id="S3.I4.i2.p1.6.m6.2.3.2.1.cmml" xref="S3.I4.i2.p1.6.m6.2.3.2.1"></times><ci id="S3.I4.i2.p1.6.m6.2.3.2.2a.cmml" xref="S3.I4.i2.p1.6.m6.2.3.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.6.m6.2.3.2.2.cmml" xref="S3.I4.i2.p1.6.m6.2.3.2.2">cost</mtext></ci><ci id="S3.I4.i2.p1.6.m6.1.1.cmml" xref="S3.I4.i2.p1.6.m6.1.1">𝑔</ci></apply><apply id="S3.I4.i2.p1.6.m6.2.3.3.cmml" xref="S3.I4.i2.p1.6.m6.2.3.3"><times id="S3.I4.i2.p1.6.m6.2.3.3.1.cmml" xref="S3.I4.i2.p1.6.m6.2.3.3.1"></times><ci id="S3.I4.i2.p1.6.m6.2.3.3.2a.cmml" xref="S3.I4.i2.p1.6.m6.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.6.m6.2.3.3.2.cmml" xref="S3.I4.i2.p1.6.m6.2.3.3.2">cost</mtext></ci><ci id="S3.I4.i2.p1.6.m6.2.2.cmml" xref="S3.I4.i2.p1.6.m6.2.2">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.6.m6.2c">\textsf{cost}({g})=\textsf{cost}({h})</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.6.m6.2d">cost ( italic_g ) = cost ( italic_h )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.7">, that is, no new Player 1’s target is visited from </span><math alttext="g" class="ltx_Math" display="inline" id="S3.I4.i2.p1.7.m7.1"><semantics id="S3.I4.i2.p1.7.m7.1a"><mi id="S3.I4.i2.p1.7.m7.1.1" xref="S3.I4.i2.p1.7.m7.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.7.m7.1b"><ci id="S3.I4.i2.p1.7.m7.1.1.cmml" xref="S3.I4.i2.p1.7.m7.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.7.m7.1c">g</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.7.m7.1d">italic_g</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.8"> to </span><math alttext="h" class="ltx_Math" display="inline" id="S3.I4.i2.p1.8.m8.1"><semantics id="S3.I4.i2.p1.8.m8.1a"><mi id="S3.I4.i2.p1.8.m8.1.1" xref="S3.I4.i2.p1.8.m8.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.8.m8.1b"><ci id="S3.I4.i2.p1.8.m8.1.1.cmml" xref="S3.I4.i2.p1.8.m8.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.8.m8.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.8.m8.1d">italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.9">). We get that </span><math alttext="c^{\prime}=\textsf{cost}({\pi})\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.9.m9.1"><semantics id="S3.I4.i2.p1.9.m9.1a"><mrow id="S3.I4.i2.p1.9.m9.1.2" xref="S3.I4.i2.p1.9.m9.1.2.cmml"><msup id="S3.I4.i2.p1.9.m9.1.2.2" xref="S3.I4.i2.p1.9.m9.1.2.2.cmml"><mi id="S3.I4.i2.p1.9.m9.1.2.2.2" xref="S3.I4.i2.p1.9.m9.1.2.2.2.cmml">c</mi><mo id="S3.I4.i2.p1.9.m9.1.2.2.3" xref="S3.I4.i2.p1.9.m9.1.2.2.3.cmml">′</mo></msup><mo id="S3.I4.i2.p1.9.m9.1.2.3" xref="S3.I4.i2.p1.9.m9.1.2.3.cmml">=</mo><mrow id="S3.I4.i2.p1.9.m9.1.2.4" xref="S3.I4.i2.p1.9.m9.1.2.4.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.9.m9.1.2.4.2" xref="S3.I4.i2.p1.9.m9.1.2.4.2a.cmml">cost</mtext><mo id="S3.I4.i2.p1.9.m9.1.2.4.1" xref="S3.I4.i2.p1.9.m9.1.2.4.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.9.m9.1.2.4.3.2" xref="S3.I4.i2.p1.9.m9.1.2.4.cmml"><mo id="S3.I4.i2.p1.9.m9.1.2.4.3.2.1" stretchy="false" xref="S3.I4.i2.p1.9.m9.1.2.4.cmml">(</mo><mi id="S3.I4.i2.p1.9.m9.1.1" xref="S3.I4.i2.p1.9.m9.1.1.cmml">π</mi><mo id="S3.I4.i2.p1.9.m9.1.2.4.3.2.2" stretchy="false" xref="S3.I4.i2.p1.9.m9.1.2.4.cmml">)</mo></mrow></mrow><mo id="S3.I4.i2.p1.9.m9.1.2.5" xref="S3.I4.i2.p1.9.m9.1.2.5.cmml">∈</mo><msub id="S3.I4.i2.p1.9.m9.1.2.6" xref="S3.I4.i2.p1.9.m9.1.2.6.cmml"><mi id="S3.I4.i2.p1.9.m9.1.2.6.2" xref="S3.I4.i2.p1.9.m9.1.2.6.2.cmml">C</mi><msub id="S3.I4.i2.p1.9.m9.1.2.6.3" xref="S3.I4.i2.p1.9.m9.1.2.6.3.cmml"><mi id="S3.I4.i2.p1.9.m9.1.2.6.3.2" xref="S3.I4.i2.p1.9.m9.1.2.6.3.2.cmml">σ</mi><mn id="S3.I4.i2.p1.9.m9.1.2.6.3.3" xref="S3.I4.i2.p1.9.m9.1.2.6.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.9.m9.1b"><apply id="S3.I4.i2.p1.9.m9.1.2.cmml" xref="S3.I4.i2.p1.9.m9.1.2"><and id="S3.I4.i2.p1.9.m9.1.2a.cmml" xref="S3.I4.i2.p1.9.m9.1.2"></and><apply id="S3.I4.i2.p1.9.m9.1.2b.cmml" xref="S3.I4.i2.p1.9.m9.1.2"><eq id="S3.I4.i2.p1.9.m9.1.2.3.cmml" xref="S3.I4.i2.p1.9.m9.1.2.3"></eq><apply id="S3.I4.i2.p1.9.m9.1.2.2.cmml" xref="S3.I4.i2.p1.9.m9.1.2.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.9.m9.1.2.2.1.cmml" xref="S3.I4.i2.p1.9.m9.1.2.2">superscript</csymbol><ci id="S3.I4.i2.p1.9.m9.1.2.2.2.cmml" xref="S3.I4.i2.p1.9.m9.1.2.2.2">𝑐</ci><ci id="S3.I4.i2.p1.9.m9.1.2.2.3.cmml" xref="S3.I4.i2.p1.9.m9.1.2.2.3">′</ci></apply><apply id="S3.I4.i2.p1.9.m9.1.2.4.cmml" xref="S3.I4.i2.p1.9.m9.1.2.4"><times id="S3.I4.i2.p1.9.m9.1.2.4.1.cmml" xref="S3.I4.i2.p1.9.m9.1.2.4.1"></times><ci id="S3.I4.i2.p1.9.m9.1.2.4.2a.cmml" xref="S3.I4.i2.p1.9.m9.1.2.4.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.9.m9.1.2.4.2.cmml" xref="S3.I4.i2.p1.9.m9.1.2.4.2">cost</mtext></ci><ci id="S3.I4.i2.p1.9.m9.1.1.cmml" xref="S3.I4.i2.p1.9.m9.1.1">𝜋</ci></apply></apply><apply id="S3.I4.i2.p1.9.m9.1.2c.cmml" xref="S3.I4.i2.p1.9.m9.1.2"><in id="S3.I4.i2.p1.9.m9.1.2.5.cmml" xref="S3.I4.i2.p1.9.m9.1.2.5"></in><share href="https://arxiv.org/html/2308.09443v2#S3.I4.i2.p1.9.m9.1.2.4.cmml" id="S3.I4.i2.p1.9.m9.1.2d.cmml" xref="S3.I4.i2.p1.9.m9.1.2"></share><apply id="S3.I4.i2.p1.9.m9.1.2.6.cmml" xref="S3.I4.i2.p1.9.m9.1.2.6"><csymbol cd="ambiguous" id="S3.I4.i2.p1.9.m9.1.2.6.1.cmml" xref="S3.I4.i2.p1.9.m9.1.2.6">subscript</csymbol><ci id="S3.I4.i2.p1.9.m9.1.2.6.2.cmml" xref="S3.I4.i2.p1.9.m9.1.2.6.2">𝐶</ci><apply id="S3.I4.i2.p1.9.m9.1.2.6.3.cmml" xref="S3.I4.i2.p1.9.m9.1.2.6.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.9.m9.1.2.6.3.1.cmml" xref="S3.I4.i2.p1.9.m9.1.2.6.3">subscript</csymbol><ci id="S3.I4.i2.p1.9.m9.1.2.6.3.2.cmml" xref="S3.I4.i2.p1.9.m9.1.2.6.3.2">𝜎</ci><cn id="S3.I4.i2.p1.9.m9.1.2.6.3.3.cmml" type="integer" xref="S3.I4.i2.p1.9.m9.1.2.6.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.9.m9.1c">c^{\prime}=\textsf{cost}({\pi})\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.9.m9.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = cost ( italic_π ) ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.10"> since </span><math alttext="\sigma^{\prime}_{0}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.10.m10.1"><semantics id="S3.I4.i2.p1.10.m10.1a"><mrow id="S3.I4.i2.p1.10.m10.1.1" xref="S3.I4.i2.p1.10.m10.1.1.cmml"><msubsup id="S3.I4.i2.p1.10.m10.1.1.2" xref="S3.I4.i2.p1.10.m10.1.1.2.cmml"><mi id="S3.I4.i2.p1.10.m10.1.1.2.2.2" xref="S3.I4.i2.p1.10.m10.1.1.2.2.2.cmml">σ</mi><mn id="S3.I4.i2.p1.10.m10.1.1.2.3" xref="S3.I4.i2.p1.10.m10.1.1.2.3.cmml">0</mn><mo id="S3.I4.i2.p1.10.m10.1.1.2.2.3" xref="S3.I4.i2.p1.10.m10.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.I4.i2.p1.10.m10.1.1.1" xref="S3.I4.i2.p1.10.m10.1.1.1.cmml">⪯</mo><msub id="S3.I4.i2.p1.10.m10.1.1.3" xref="S3.I4.i2.p1.10.m10.1.1.3.cmml"><mi id="S3.I4.i2.p1.10.m10.1.1.3.2" xref="S3.I4.i2.p1.10.m10.1.1.3.2.cmml">σ</mi><mn id="S3.I4.i2.p1.10.m10.1.1.3.3" xref="S3.I4.i2.p1.10.m10.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.10.m10.1b"><apply id="S3.I4.i2.p1.10.m10.1.1.cmml" xref="S3.I4.i2.p1.10.m10.1.1"><csymbol cd="latexml" id="S3.I4.i2.p1.10.m10.1.1.1.cmml" xref="S3.I4.i2.p1.10.m10.1.1.1">precedes-or-equals</csymbol><apply id="S3.I4.i2.p1.10.m10.1.1.2.cmml" xref="S3.I4.i2.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.10.m10.1.1.2.1.cmml" xref="S3.I4.i2.p1.10.m10.1.1.2">subscript</csymbol><apply id="S3.I4.i2.p1.10.m10.1.1.2.2.cmml" xref="S3.I4.i2.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.10.m10.1.1.2.2.1.cmml" xref="S3.I4.i2.p1.10.m10.1.1.2">superscript</csymbol><ci id="S3.I4.i2.p1.10.m10.1.1.2.2.2.cmml" xref="S3.I4.i2.p1.10.m10.1.1.2.2.2">𝜎</ci><ci id="S3.I4.i2.p1.10.m10.1.1.2.2.3.cmml" xref="S3.I4.i2.p1.10.m10.1.1.2.2.3">′</ci></apply><cn id="S3.I4.i2.p1.10.m10.1.1.2.3.cmml" type="integer" xref="S3.I4.i2.p1.10.m10.1.1.2.3">0</cn></apply><apply id="S3.I4.i2.p1.10.m10.1.1.3.cmml" xref="S3.I4.i2.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.10.m10.1.1.3.1.cmml" xref="S3.I4.i2.p1.10.m10.1.1.3">subscript</csymbol><ci id="S3.I4.i2.p1.10.m10.1.1.3.2.cmml" xref="S3.I4.i2.p1.10.m10.1.1.3.2">𝜎</ci><cn id="S3.I4.i2.p1.10.m10.1.1.3.3.cmml" type="integer" xref="S3.I4.i2.p1.10.m10.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.10.m10.1c">\sigma^{\prime}_{0}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.10.m10.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.11">. It follows that </span><math alttext="\textsf{val}(\pi)\leq B" class="ltx_Math" display="inline" id="S3.I4.i2.p1.11.m11.1"><semantics id="S3.I4.i2.p1.11.m11.1a"><mrow id="S3.I4.i2.p1.11.m11.1.2" xref="S3.I4.i2.p1.11.m11.1.2.cmml"><mrow id="S3.I4.i2.p1.11.m11.1.2.2" xref="S3.I4.i2.p1.11.m11.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.11.m11.1.2.2.2" xref="S3.I4.i2.p1.11.m11.1.2.2.2a.cmml">val</mtext><mo id="S3.I4.i2.p1.11.m11.1.2.2.1" xref="S3.I4.i2.p1.11.m11.1.2.2.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.11.m11.1.2.2.3.2" xref="S3.I4.i2.p1.11.m11.1.2.2.cmml"><mo id="S3.I4.i2.p1.11.m11.1.2.2.3.2.1" stretchy="false" xref="S3.I4.i2.p1.11.m11.1.2.2.cmml">(</mo><mi id="S3.I4.i2.p1.11.m11.1.1" xref="S3.I4.i2.p1.11.m11.1.1.cmml">π</mi><mo id="S3.I4.i2.p1.11.m11.1.2.2.3.2.2" stretchy="false" xref="S3.I4.i2.p1.11.m11.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I4.i2.p1.11.m11.1.2.1" xref="S3.I4.i2.p1.11.m11.1.2.1.cmml">≤</mo><mi id="S3.I4.i2.p1.11.m11.1.2.3" xref="S3.I4.i2.p1.11.m11.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.11.m11.1b"><apply id="S3.I4.i2.p1.11.m11.1.2.cmml" xref="S3.I4.i2.p1.11.m11.1.2"><leq id="S3.I4.i2.p1.11.m11.1.2.1.cmml" xref="S3.I4.i2.p1.11.m11.1.2.1"></leq><apply id="S3.I4.i2.p1.11.m11.1.2.2.cmml" xref="S3.I4.i2.p1.11.m11.1.2.2"><times id="S3.I4.i2.p1.11.m11.1.2.2.1.cmml" xref="S3.I4.i2.p1.11.m11.1.2.2.1"></times><ci id="S3.I4.i2.p1.11.m11.1.2.2.2a.cmml" xref="S3.I4.i2.p1.11.m11.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.11.m11.1.2.2.2.cmml" xref="S3.I4.i2.p1.11.m11.1.2.2.2">val</mtext></ci><ci id="S3.I4.i2.p1.11.m11.1.1.cmml" xref="S3.I4.i2.p1.11.m11.1.1">𝜋</ci></apply><ci id="S3.I4.i2.p1.11.m11.1.2.3.cmml" xref="S3.I4.i2.p1.11.m11.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.11.m11.1c">\textsf{val}(\pi)\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.11.m11.1d">val ( italic_π ) ≤ italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.12"> as </span><math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.I4.i2.p1.12.m12.2"><semantics id="S3.I4.i2.p1.12.m12.2a"><mrow id="S3.I4.i2.p1.12.m12.2.3" xref="S3.I4.i2.p1.12.m12.2.3.cmml"><msub id="S3.I4.i2.p1.12.m12.2.3.2" xref="S3.I4.i2.p1.12.m12.2.3.2.cmml"><mi id="S3.I4.i2.p1.12.m12.2.3.2.2" xref="S3.I4.i2.p1.12.m12.2.3.2.2.cmml">σ</mi><mn id="S3.I4.i2.p1.12.m12.2.3.2.3" xref="S3.I4.i2.p1.12.m12.2.3.2.3.cmml">0</mn></msub><mo id="S3.I4.i2.p1.12.m12.2.3.1" xref="S3.I4.i2.p1.12.m12.2.3.1.cmml">∈</mo><mrow id="S3.I4.i2.p1.12.m12.2.3.3" xref="S3.I4.i2.p1.12.m12.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.I4.i2.p1.12.m12.2.3.3.2" xref="S3.I4.i2.p1.12.m12.2.3.3.2a.cmml">SPS</mtext><mo id="S3.I4.i2.p1.12.m12.2.3.3.1" xref="S3.I4.i2.p1.12.m12.2.3.3.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.12.m12.2.3.3.3.2" xref="S3.I4.i2.p1.12.m12.2.3.3.3.1.cmml"><mo id="S3.I4.i2.p1.12.m12.2.3.3.3.2.1" stretchy="false" xref="S3.I4.i2.p1.12.m12.2.3.3.3.1.cmml">(</mo><mi id="S3.I4.i2.p1.12.m12.1.1" xref="S3.I4.i2.p1.12.m12.1.1.cmml">G</mi><mo id="S3.I4.i2.p1.12.m12.2.3.3.3.2.2" xref="S3.I4.i2.p1.12.m12.2.3.3.3.1.cmml">,</mo><mi id="S3.I4.i2.p1.12.m12.2.2" xref="S3.I4.i2.p1.12.m12.2.2.cmml">B</mi><mo id="S3.I4.i2.p1.12.m12.2.3.3.3.2.3" stretchy="false" xref="S3.I4.i2.p1.12.m12.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.12.m12.2b"><apply id="S3.I4.i2.p1.12.m12.2.3.cmml" xref="S3.I4.i2.p1.12.m12.2.3"><in id="S3.I4.i2.p1.12.m12.2.3.1.cmml" xref="S3.I4.i2.p1.12.m12.2.3.1"></in><apply id="S3.I4.i2.p1.12.m12.2.3.2.cmml" xref="S3.I4.i2.p1.12.m12.2.3.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.12.m12.2.3.2.1.cmml" xref="S3.I4.i2.p1.12.m12.2.3.2">subscript</csymbol><ci id="S3.I4.i2.p1.12.m12.2.3.2.2.cmml" xref="S3.I4.i2.p1.12.m12.2.3.2.2">𝜎</ci><cn id="S3.I4.i2.p1.12.m12.2.3.2.3.cmml" type="integer" xref="S3.I4.i2.p1.12.m12.2.3.2.3">0</cn></apply><apply id="S3.I4.i2.p1.12.m12.2.3.3.cmml" xref="S3.I4.i2.p1.12.m12.2.3.3"><times id="S3.I4.i2.p1.12.m12.2.3.3.1.cmml" xref="S3.I4.i2.p1.12.m12.2.3.3.1"></times><ci id="S3.I4.i2.p1.12.m12.2.3.3.2a.cmml" xref="S3.I4.i2.p1.12.m12.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S3.I4.i2.p1.12.m12.2.3.3.2.cmml" xref="S3.I4.i2.p1.12.m12.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S3.I4.i2.p1.12.m12.2.3.3.3.1.cmml" xref="S3.I4.i2.p1.12.m12.2.3.3.3.2"><ci id="S3.I4.i2.p1.12.m12.1.1.cmml" xref="S3.I4.i2.p1.12.m12.1.1">𝐺</ci><ci id="S3.I4.i2.p1.12.m12.2.2.cmml" xref="S3.I4.i2.p1.12.m12.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.12.m12.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.12.m12.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.13">. As </span><math alttext="g\sim h" class="ltx_Math" display="inline" id="S3.I4.i2.p1.13.m13.1"><semantics id="S3.I4.i2.p1.13.m13.1a"><mrow id="S3.I4.i2.p1.13.m13.1.1" xref="S3.I4.i2.p1.13.m13.1.1.cmml"><mi id="S3.I4.i2.p1.13.m13.1.1.2" xref="S3.I4.i2.p1.13.m13.1.1.2.cmml">g</mi><mo id="S3.I4.i2.p1.13.m13.1.1.1" xref="S3.I4.i2.p1.13.m13.1.1.1.cmml">∼</mo><mi id="S3.I4.i2.p1.13.m13.1.1.3" xref="S3.I4.i2.p1.13.m13.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.13.m13.1b"><apply id="S3.I4.i2.p1.13.m13.1.1.cmml" xref="S3.I4.i2.p1.13.m13.1.1"><csymbol cd="latexml" id="S3.I4.i2.p1.13.m13.1.1.1.cmml" xref="S3.I4.i2.p1.13.m13.1.1.1">similar-to</csymbol><ci id="S3.I4.i2.p1.13.m13.1.1.2.cmml" xref="S3.I4.i2.p1.13.m13.1.1.2">𝑔</ci><ci id="S3.I4.i2.p1.13.m13.1.1.3.cmml" xref="S3.I4.i2.p1.13.m13.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.13.m13.1c">g\sim h</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.13.m13.1d">italic_g ∼ italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.14"> (in particular, </span><math alttext="\textsf{val}(g)=\textsf{val}(h)" class="ltx_Math" display="inline" id="S3.I4.i2.p1.14.m14.2"><semantics id="S3.I4.i2.p1.14.m14.2a"><mrow id="S3.I4.i2.p1.14.m14.2.3" xref="S3.I4.i2.p1.14.m14.2.3.cmml"><mrow id="S3.I4.i2.p1.14.m14.2.3.2" xref="S3.I4.i2.p1.14.m14.2.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.14.m14.2.3.2.2" xref="S3.I4.i2.p1.14.m14.2.3.2.2a.cmml">val</mtext><mo id="S3.I4.i2.p1.14.m14.2.3.2.1" xref="S3.I4.i2.p1.14.m14.2.3.2.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.14.m14.2.3.2.3.2" xref="S3.I4.i2.p1.14.m14.2.3.2.cmml"><mo id="S3.I4.i2.p1.14.m14.2.3.2.3.2.1" stretchy="false" xref="S3.I4.i2.p1.14.m14.2.3.2.cmml">(</mo><mi id="S3.I4.i2.p1.14.m14.1.1" xref="S3.I4.i2.p1.14.m14.1.1.cmml">g</mi><mo id="S3.I4.i2.p1.14.m14.2.3.2.3.2.2" stretchy="false" xref="S3.I4.i2.p1.14.m14.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.I4.i2.p1.14.m14.2.3.1" xref="S3.I4.i2.p1.14.m14.2.3.1.cmml">=</mo><mrow id="S3.I4.i2.p1.14.m14.2.3.3" xref="S3.I4.i2.p1.14.m14.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.14.m14.2.3.3.2" xref="S3.I4.i2.p1.14.m14.2.3.3.2a.cmml">val</mtext><mo id="S3.I4.i2.p1.14.m14.2.3.3.1" xref="S3.I4.i2.p1.14.m14.2.3.3.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.14.m14.2.3.3.3.2" xref="S3.I4.i2.p1.14.m14.2.3.3.cmml"><mo id="S3.I4.i2.p1.14.m14.2.3.3.3.2.1" stretchy="false" xref="S3.I4.i2.p1.14.m14.2.3.3.cmml">(</mo><mi id="S3.I4.i2.p1.14.m14.2.2" xref="S3.I4.i2.p1.14.m14.2.2.cmml">h</mi><mo id="S3.I4.i2.p1.14.m14.2.3.3.3.2.2" stretchy="false" xref="S3.I4.i2.p1.14.m14.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.14.m14.2b"><apply id="S3.I4.i2.p1.14.m14.2.3.cmml" xref="S3.I4.i2.p1.14.m14.2.3"><eq id="S3.I4.i2.p1.14.m14.2.3.1.cmml" xref="S3.I4.i2.p1.14.m14.2.3.1"></eq><apply id="S3.I4.i2.p1.14.m14.2.3.2.cmml" xref="S3.I4.i2.p1.14.m14.2.3.2"><times id="S3.I4.i2.p1.14.m14.2.3.2.1.cmml" xref="S3.I4.i2.p1.14.m14.2.3.2.1"></times><ci id="S3.I4.i2.p1.14.m14.2.3.2.2a.cmml" xref="S3.I4.i2.p1.14.m14.2.3.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.14.m14.2.3.2.2.cmml" xref="S3.I4.i2.p1.14.m14.2.3.2.2">val</mtext></ci><ci id="S3.I4.i2.p1.14.m14.1.1.cmml" xref="S3.I4.i2.p1.14.m14.1.1">𝑔</ci></apply><apply id="S3.I4.i2.p1.14.m14.2.3.3.cmml" xref="S3.I4.i2.p1.14.m14.2.3.3"><times id="S3.I4.i2.p1.14.m14.2.3.3.1.cmml" xref="S3.I4.i2.p1.14.m14.2.3.3.1"></times><ci id="S3.I4.i2.p1.14.m14.2.3.3.2a.cmml" xref="S3.I4.i2.p1.14.m14.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.14.m14.2.3.3.2.cmml" xref="S3.I4.i2.p1.14.m14.2.3.3.2">val</mtext></ci><ci id="S3.I4.i2.p1.14.m14.2.2.cmml" xref="S3.I4.i2.p1.14.m14.2.2">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.14.m14.2c">\textsf{val}(g)=\textsf{val}(h)</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.14.m14.2d">val ( italic_g ) = val ( italic_h )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.15">), as </span><math alttext="\textsf{val}(g)=\infty" class="ltx_Math" display="inline" id="S3.I4.i2.p1.15.m15.1"><semantics id="S3.I4.i2.p1.15.m15.1a"><mrow id="S3.I4.i2.p1.15.m15.1.2" xref="S3.I4.i2.p1.15.m15.1.2.cmml"><mrow id="S3.I4.i2.p1.15.m15.1.2.2" xref="S3.I4.i2.p1.15.m15.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.15.m15.1.2.2.2" xref="S3.I4.i2.p1.15.m15.1.2.2.2a.cmml">val</mtext><mo id="S3.I4.i2.p1.15.m15.1.2.2.1" xref="S3.I4.i2.p1.15.m15.1.2.2.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.15.m15.1.2.2.3.2" xref="S3.I4.i2.p1.15.m15.1.2.2.cmml"><mo id="S3.I4.i2.p1.15.m15.1.2.2.3.2.1" stretchy="false" xref="S3.I4.i2.p1.15.m15.1.2.2.cmml">(</mo><mi id="S3.I4.i2.p1.15.m15.1.1" xref="S3.I4.i2.p1.15.m15.1.1.cmml">g</mi><mo id="S3.I4.i2.p1.15.m15.1.2.2.3.2.2" stretchy="false" xref="S3.I4.i2.p1.15.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I4.i2.p1.15.m15.1.2.1" xref="S3.I4.i2.p1.15.m15.1.2.1.cmml">=</mo><mi id="S3.I4.i2.p1.15.m15.1.2.3" mathvariant="normal" xref="S3.I4.i2.p1.15.m15.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.15.m15.1b"><apply id="S3.I4.i2.p1.15.m15.1.2.cmml" xref="S3.I4.i2.p1.15.m15.1.2"><eq id="S3.I4.i2.p1.15.m15.1.2.1.cmml" xref="S3.I4.i2.p1.15.m15.1.2.1"></eq><apply id="S3.I4.i2.p1.15.m15.1.2.2.cmml" xref="S3.I4.i2.p1.15.m15.1.2.2"><times id="S3.I4.i2.p1.15.m15.1.2.2.1.cmml" xref="S3.I4.i2.p1.15.m15.1.2.2.1"></times><ci id="S3.I4.i2.p1.15.m15.1.2.2.2a.cmml" xref="S3.I4.i2.p1.15.m15.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.15.m15.1.2.2.2.cmml" xref="S3.I4.i2.p1.15.m15.1.2.2.2">val</mtext></ci><ci id="S3.I4.i2.p1.15.m15.1.1.cmml" xref="S3.I4.i2.p1.15.m15.1.1">𝑔</ci></apply><infinity id="S3.I4.i2.p1.15.m15.1.2.3.cmml" xref="S3.I4.i2.p1.15.m15.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.15.m15.1c">\textsf{val}(g)=\infty</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.15.m15.1d">val ( italic_g ) = ∞</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.16">, </span><math alttext="\pi" class="ltx_Math" display="inline" id="S3.I4.i2.p1.16.m16.1"><semantics id="S3.I4.i2.p1.16.m16.1a"><mi id="S3.I4.i2.p1.16.m16.1.1" xref="S3.I4.i2.p1.16.m16.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.16.m16.1b"><ci id="S3.I4.i2.p1.16.m16.1.1.cmml" xref="S3.I4.i2.p1.16.m16.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.16.m16.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.16.m16.1d">italic_π</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.17"> visits Player </span><math alttext="0" class="ltx_Math" display="inline" id="S3.I4.i2.p1.17.m17.1"><semantics id="S3.I4.i2.p1.17.m17.1a"><mn id="S3.I4.i2.p1.17.m17.1.1" xref="S3.I4.i2.p1.17.m17.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.17.m17.1b"><cn id="S3.I4.i2.p1.17.m17.1.1.cmml" type="integer" xref="S3.I4.i2.p1.17.m17.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.18">’s target after </span><math alttext="h" class="ltx_Math" display="inline" id="S3.I4.i2.p1.18.m18.1"><semantics id="S3.I4.i2.p1.18.m18.1a"><mi id="S3.I4.i2.p1.18.m18.1.1" xref="S3.I4.i2.p1.18.m18.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.18.m18.1b"><ci id="S3.I4.i2.p1.18.m18.1.1.cmml" xref="S3.I4.i2.p1.18.m18.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.18.m18.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.18.m18.1d">italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.19">, thus not along </span><math alttext="\rho_{[m,n]}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.19.m19.2"><semantics id="S3.I4.i2.p1.19.m19.2a"><msub id="S3.I4.i2.p1.19.m19.2.3" xref="S3.I4.i2.p1.19.m19.2.3.cmml"><mi id="S3.I4.i2.p1.19.m19.2.3.2" xref="S3.I4.i2.p1.19.m19.2.3.2.cmml">ρ</mi><mrow id="S3.I4.i2.p1.19.m19.2.2.2.4" xref="S3.I4.i2.p1.19.m19.2.2.2.3.cmml"><mo id="S3.I4.i2.p1.19.m19.2.2.2.4.1" stretchy="false" xref="S3.I4.i2.p1.19.m19.2.2.2.3.cmml">[</mo><mi id="S3.I4.i2.p1.19.m19.1.1.1.1" xref="S3.I4.i2.p1.19.m19.1.1.1.1.cmml">m</mi><mo id="S3.I4.i2.p1.19.m19.2.2.2.4.2" xref="S3.I4.i2.p1.19.m19.2.2.2.3.cmml">,</mo><mi id="S3.I4.i2.p1.19.m19.2.2.2.2" xref="S3.I4.i2.p1.19.m19.2.2.2.2.cmml">n</mi><mo id="S3.I4.i2.p1.19.m19.2.2.2.4.3" stretchy="false" xref="S3.I4.i2.p1.19.m19.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.19.m19.2b"><apply id="S3.I4.i2.p1.19.m19.2.3.cmml" xref="S3.I4.i2.p1.19.m19.2.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.19.m19.2.3.1.cmml" xref="S3.I4.i2.p1.19.m19.2.3">subscript</csymbol><ci id="S3.I4.i2.p1.19.m19.2.3.2.cmml" xref="S3.I4.i2.p1.19.m19.2.3.2">𝜌</ci><interval closure="closed" id="S3.I4.i2.p1.19.m19.2.2.2.3.cmml" xref="S3.I4.i2.p1.19.m19.2.2.2.4"><ci id="S3.I4.i2.p1.19.m19.1.1.1.1.cmml" xref="S3.I4.i2.p1.19.m19.1.1.1.1">𝑚</ci><ci id="S3.I4.i2.p1.19.m19.2.2.2.2.cmml" xref="S3.I4.i2.p1.19.m19.2.2.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.19.m19.2c">\rho_{[m,n]}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.19.m19.2d">italic_ρ start_POSTSUBSCRIPT [ italic_m , italic_n ] end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.20">. It follows that </span><math alttext="\textsf{val}(\pi^{\prime})\leq B" class="ltx_Math" display="inline" id="S3.I4.i2.p1.20.m20.1"><semantics id="S3.I4.i2.p1.20.m20.1a"><mrow id="S3.I4.i2.p1.20.m20.1.1" xref="S3.I4.i2.p1.20.m20.1.1.cmml"><mrow id="S3.I4.i2.p1.20.m20.1.1.1" xref="S3.I4.i2.p1.20.m20.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.20.m20.1.1.1.3" xref="S3.I4.i2.p1.20.m20.1.1.1.3a.cmml">val</mtext><mo id="S3.I4.i2.p1.20.m20.1.1.1.2" xref="S3.I4.i2.p1.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S3.I4.i2.p1.20.m20.1.1.1.1.1" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.cmml"><mo id="S3.I4.i2.p1.20.m20.1.1.1.1.1.2" stretchy="false" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I4.i2.p1.20.m20.1.1.1.1.1.1" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.cmml"><mi id="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.2" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.2.cmml">π</mi><mo id="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.3" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I4.i2.p1.20.m20.1.1.1.1.1.3" stretchy="false" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I4.i2.p1.20.m20.1.1.2" xref="S3.I4.i2.p1.20.m20.1.1.2.cmml">≤</mo><mi id="S3.I4.i2.p1.20.m20.1.1.3" xref="S3.I4.i2.p1.20.m20.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.20.m20.1b"><apply id="S3.I4.i2.p1.20.m20.1.1.cmml" xref="S3.I4.i2.p1.20.m20.1.1"><leq id="S3.I4.i2.p1.20.m20.1.1.2.cmml" xref="S3.I4.i2.p1.20.m20.1.1.2"></leq><apply id="S3.I4.i2.p1.20.m20.1.1.1.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1"><times id="S3.I4.i2.p1.20.m20.1.1.1.2.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1.2"></times><ci id="S3.I4.i2.p1.20.m20.1.1.1.3a.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S3.I4.i2.p1.20.m20.1.1.1.3.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1.3">val</mtext></ci><apply id="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.1.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1">superscript</csymbol><ci id="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.2.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.2">𝜋</ci><ci id="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.3.cmml" xref="S3.I4.i2.p1.20.m20.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.I4.i2.p1.20.m20.1.1.3.cmml" xref="S3.I4.i2.p1.20.m20.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.20.m20.1c">\textsf{val}(\pi^{\prime})\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.20.m20.1d">val ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.21">. Therefore </span><math alttext="\sigma^{\prime}_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S3.I4.i2.p1.21.m21.2"><semantics id="S3.I4.i2.p1.21.m21.2a"><mrow id="S3.I4.i2.p1.21.m21.2.3" xref="S3.I4.i2.p1.21.m21.2.3.cmml"><msubsup id="S3.I4.i2.p1.21.m21.2.3.2" xref="S3.I4.i2.p1.21.m21.2.3.2.cmml"><mi id="S3.I4.i2.p1.21.m21.2.3.2.2.2" xref="S3.I4.i2.p1.21.m21.2.3.2.2.2.cmml">σ</mi><mn id="S3.I4.i2.p1.21.m21.2.3.2.3" xref="S3.I4.i2.p1.21.m21.2.3.2.3.cmml">0</mn><mo id="S3.I4.i2.p1.21.m21.2.3.2.2.3" xref="S3.I4.i2.p1.21.m21.2.3.2.2.3.cmml">′</mo></msubsup><mo id="S3.I4.i2.p1.21.m21.2.3.1" xref="S3.I4.i2.p1.21.m21.2.3.1.cmml">∈</mo><mrow id="S3.I4.i2.p1.21.m21.2.3.3" xref="S3.I4.i2.p1.21.m21.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S3.I4.i2.p1.21.m21.2.3.3.2" xref="S3.I4.i2.p1.21.m21.2.3.3.2a.cmml">SPS</mtext><mo id="S3.I4.i2.p1.21.m21.2.3.3.1" xref="S3.I4.i2.p1.21.m21.2.3.3.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.21.m21.2.3.3.3.2" xref="S3.I4.i2.p1.21.m21.2.3.3.3.1.cmml"><mo id="S3.I4.i2.p1.21.m21.2.3.3.3.2.1" stretchy="false" xref="S3.I4.i2.p1.21.m21.2.3.3.3.1.cmml">(</mo><mi id="S3.I4.i2.p1.21.m21.1.1" xref="S3.I4.i2.p1.21.m21.1.1.cmml">G</mi><mo id="S3.I4.i2.p1.21.m21.2.3.3.3.2.2" xref="S3.I4.i2.p1.21.m21.2.3.3.3.1.cmml">,</mo><mi id="S3.I4.i2.p1.21.m21.2.2" xref="S3.I4.i2.p1.21.m21.2.2.cmml">B</mi><mo id="S3.I4.i2.p1.21.m21.2.3.3.3.2.3" stretchy="false" xref="S3.I4.i2.p1.21.m21.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.21.m21.2b"><apply id="S3.I4.i2.p1.21.m21.2.3.cmml" xref="S3.I4.i2.p1.21.m21.2.3"><in id="S3.I4.i2.p1.21.m21.2.3.1.cmml" xref="S3.I4.i2.p1.21.m21.2.3.1"></in><apply id="S3.I4.i2.p1.21.m21.2.3.2.cmml" xref="S3.I4.i2.p1.21.m21.2.3.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.21.m21.2.3.2.1.cmml" xref="S3.I4.i2.p1.21.m21.2.3.2">subscript</csymbol><apply id="S3.I4.i2.p1.21.m21.2.3.2.2.cmml" xref="S3.I4.i2.p1.21.m21.2.3.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.21.m21.2.3.2.2.1.cmml" xref="S3.I4.i2.p1.21.m21.2.3.2">superscript</csymbol><ci id="S3.I4.i2.p1.21.m21.2.3.2.2.2.cmml" xref="S3.I4.i2.p1.21.m21.2.3.2.2.2">𝜎</ci><ci id="S3.I4.i2.p1.21.m21.2.3.2.2.3.cmml" xref="S3.I4.i2.p1.21.m21.2.3.2.2.3">′</ci></apply><cn id="S3.I4.i2.p1.21.m21.2.3.2.3.cmml" type="integer" xref="S3.I4.i2.p1.21.m21.2.3.2.3">0</cn></apply><apply id="S3.I4.i2.p1.21.m21.2.3.3.cmml" xref="S3.I4.i2.p1.21.m21.2.3.3"><times id="S3.I4.i2.p1.21.m21.2.3.3.1.cmml" xref="S3.I4.i2.p1.21.m21.2.3.3.1"></times><ci id="S3.I4.i2.p1.21.m21.2.3.3.2a.cmml" xref="S3.I4.i2.p1.21.m21.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S3.I4.i2.p1.21.m21.2.3.3.2.cmml" xref="S3.I4.i2.p1.21.m21.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S3.I4.i2.p1.21.m21.2.3.3.3.1.cmml" xref="S3.I4.i2.p1.21.m21.2.3.3.3.2"><ci id="S3.I4.i2.p1.21.m21.1.1.cmml" xref="S3.I4.i2.p1.21.m21.1.1">𝐺</ci><ci id="S3.I4.i2.p1.21.m21.2.2.cmml" xref="S3.I4.i2.p1.21.m21.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.21.m21.2c">\sigma^{\prime}_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.21.m21.2d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.21.22">.</span></p> </div> </li> </ul> <p class="ltx_p" id="S3.Thmtheorem4.p4.15"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p4.15.1">This shows that <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p4.15.1.m1.1"><semantics id="S3.Thmtheorem4.p4.15.1.m1.1a"><msubsup id="S3.Thmtheorem4.p4.15.1.m1.1.1" xref="S3.Thmtheorem4.p4.15.1.m1.1.1.cmml"><mi id="S3.Thmtheorem4.p4.15.1.m1.1.1.2.2" xref="S3.Thmtheorem4.p4.15.1.m1.1.1.2.2.cmml">σ</mi><mn id="S3.Thmtheorem4.p4.15.1.m1.1.1.3" xref="S3.Thmtheorem4.p4.15.1.m1.1.1.3.cmml">0</mn><mo id="S3.Thmtheorem4.p4.15.1.m1.1.1.2.3" xref="S3.Thmtheorem4.p4.15.1.m1.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p4.15.1.m1.1b"><apply id="S3.Thmtheorem4.p4.15.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p4.15.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.15.1.m1.1.1.1.cmml" xref="S3.Thmtheorem4.p4.15.1.m1.1.1">subscript</csymbol><apply id="S3.Thmtheorem4.p4.15.1.m1.1.1.2.cmml" xref="S3.Thmtheorem4.p4.15.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p4.15.1.m1.1.1.2.1.cmml" xref="S3.Thmtheorem4.p4.15.1.m1.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p4.15.1.m1.1.1.2.2.cmml" xref="S3.Thmtheorem4.p4.15.1.m1.1.1.2.2">𝜎</ci><ci id="S3.Thmtheorem4.p4.15.1.m1.1.1.2.3.cmml" xref="S3.Thmtheorem4.p4.15.1.m1.1.1.2.3">′</ci></apply><cn id="S3.Thmtheorem4.p4.15.1.m1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p4.15.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p4.15.1.m1.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p4.15.1.m1.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution to the SPS problem.</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>Bounding Pareto-Optimal Payoffs</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section, we show that if there exists a solution to the SPS problem, then there exists one whose Pareto-optimal costs are exponential in the size of the instance (see Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> below). It is a <em class="ltx_emph ltx_font_italic" id="S4.p1.1.1">crucial step</em> to prove that the SPS problem is in <span class="ltx_text ltx_font_sansserif" id="S4.p1.1.2">NEXPTIME</span>. This is the main contribution of this paper. Its proof is partly based on the lemmas of the previous section.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Theorem 4.1</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.6.6">Let <math alttext="G\in\textsf{BinGames}_{t}" 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">G</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2a.cmml">BinGames</mtext><mi id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml">t</mi></msub></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"><in id="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2">𝐺</ci><apply id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2a.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2">BinGames</mtext></ci><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.1.m1.1c">G\in\textsf{BinGames}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.1.m1.1d">italic_G ∈ BinGames start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> be a binary SP game with dimension <math alttext="t" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.2.m2.1"><semantics id="S4.Thmtheorem1.p1.2.2.m2.1a"><mi id="S4.Thmtheorem1.p1.2.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.2.m2.1b"><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">italic_t</annotation></semantics></math>, <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.3.m3.1"><semantics id="S4.Thmtheorem1.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem1.p1.3.3.m3.1.1" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem1.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.2.cmml">B</mi><mo id="S4.Thmtheorem1.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem1.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.3.m3.1b"><apply id="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1"><in id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1"></in><ci id="S4.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.2">𝐵</ci><ci id="S4.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.3.m3.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.3.m3.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.4.m4.2"><semantics id="S4.Thmtheorem1.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem1.p1.4.4.m4.2.3" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.cmml"><msub id="S4.Thmtheorem1.p1.4.4.m4.2.3.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.2.cmml"><mi id="S4.Thmtheorem1.p1.4.4.m4.2.3.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem1.p1.4.4.m4.2.3.2.3" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem1.p1.4.4.m4.2.3.1" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem1.p1.4.4.m4.2.3.3" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.1" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem1.p1.4.4.m4.1.1" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml">G</mi><mo id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem1.p1.4.4.m4.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.cmml">B</mi><mo id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.4.m4.2b"><apply id="S4.Thmtheorem1.p1.4.4.m4.2.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3"><in id="S4.Thmtheorem1.p1.4.4.m4.2.3.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.1"></in><apply id="S4.Thmtheorem1.p1.4.4.m4.2.3.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.4.m4.2.3.2.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem1.p1.4.4.m4.2.3.2.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem1.p1.4.4.m4.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3"><times id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.1"></times><ci id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.2a.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.3.3.3.2"><ci id="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1">𝐺</ci><ci id="S4.Thmtheorem1.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.4.m4.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. Then there exists a solution <math alttext="\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.5.m5.2"><semantics id="S4.Thmtheorem1.p1.5.5.m5.2a"><mrow id="S4.Thmtheorem1.p1.5.5.m5.2.3" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.cmml"><msubsup id="S4.Thmtheorem1.p1.5.5.m5.2.3.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.cmml"><mi id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.3" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.3.cmml">0</mn><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.3" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.1" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem1.p1.5.5.m5.2.3.3" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.1" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem1.p1.5.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml">G</mi><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem1.p1.5.5.m5.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.2.cmml">B</mi><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.5.m5.2b"><apply id="S4.Thmtheorem1.p1.5.5.m5.2.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3"><in id="S4.Thmtheorem1.p1.5.5.m5.2.3.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.1"></in><apply id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2">superscript</csymbol><apply id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.2">𝜎</ci><cn id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.2.3">0</cn></apply><ci id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.3">′</ci></apply><apply id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3"><times id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.1"></times><ci id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2a.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.2"><ci id="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.1">𝐺</ci><ci id="S4.Thmtheorem1.p1.5.5.m5.2.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.5.m5.2c">\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.5.m5.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> without cycles such that <math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.6.6.m6.1"><semantics id="S4.Thmtheorem1.p1.6.6.m6.1a"><mrow id="S4.Thmtheorem1.p1.6.6.m6.1.1" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml"><msubsup id="S4.Thmtheorem1.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2.cmml"><mi id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.2" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.3" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.3.cmml">0</mn><mo id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.3" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem1.p1.6.6.m6.1.1.1" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.1.cmml">⪯</mo><msub id="S4.Thmtheorem1.p1.6.6.m6.1.1.3" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.3.cmml"><mi id="S4.Thmtheorem1.p1.6.6.m6.1.1.3.2" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem1.p1.6.6.m6.1.1.3.3" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.6.6.m6.1b"><apply id="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1"><csymbol cd="latexml" id="S4.Thmtheorem1.p1.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.1">precedes-or-equals</csymbol><apply id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2">superscript</csymbol><apply id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.2">𝜎</ci><cn id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2.2.3">0</cn></apply><ci id="S4.Thmtheorem1.p1.6.6.m6.1.1.2.3.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.2.3">′</ci></apply><apply id="S4.Thmtheorem1.p1.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.6.6.m6.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem1.p1.6.6.m6.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem1.p1.6.6.m6.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.6.6.m6.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.6.6.m6.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, and</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx2"> <tbody id="S4.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_left ltx_eqn_cell"><math alttext="\displaystyle\forall c^{\prime}\in C_{\sigma_{0}^{\prime}},\forall i\in\{1,% \ldots,t\}:~{}~{}c^{\prime}_{i}\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot% (B+3)~{}~{}\vee~{}~{}c^{\prime}_{i}=\infty." class="ltx_Math" display="inline" id="S4.E2.m1.7"><semantics id="S4.E2.m1.7a"><mrow id="S4.E2.m1.7.7.1" xref="S4.E2.m1.7.7.1.1.cmml"><mrow id="S4.E2.m1.7.7.1.1" xref="S4.E2.m1.7.7.1.1.cmml"><mrow id="S4.E2.m1.7.7.1.1.2.2" xref="S4.E2.m1.7.7.1.1.2.3.cmml"><mrow id="S4.E2.m1.7.7.1.1.1.1.1" xref="S4.E2.m1.7.7.1.1.1.1.1.cmml"><mrow id="S4.E2.m1.7.7.1.1.1.1.1.2" xref="S4.E2.m1.7.7.1.1.1.1.1.2.cmml"><mo id="S4.E2.m1.7.7.1.1.1.1.1.2.1" rspace="0.167em" xref="S4.E2.m1.7.7.1.1.1.1.1.2.1.cmml">∀</mo><msup id="S4.E2.m1.7.7.1.1.1.1.1.2.2" xref="S4.E2.m1.7.7.1.1.1.1.1.2.2.cmml"><mi id="S4.E2.m1.7.7.1.1.1.1.1.2.2.2" xref="S4.E2.m1.7.7.1.1.1.1.1.2.2.2.cmml">c</mi><mo id="S4.E2.m1.7.7.1.1.1.1.1.2.2.3" 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id="S4.E2.m1.7.7.1.1.3.1.3.2.1.cmml" xref="S4.E2.m1.7.7.1.1.3.1.3">superscript</csymbol><ci id="S4.E2.m1.7.7.1.1.3.1.3.2.2.cmml" xref="S4.E2.m1.7.7.1.1.3.1.3.2.2">𝑐</ci><ci id="S4.E2.m1.7.7.1.1.3.1.3.2.3.cmml" xref="S4.E2.m1.7.7.1.1.3.1.3.2.3">′</ci></apply><ci id="S4.E2.m1.7.7.1.1.3.1.3.3.cmml" xref="S4.E2.m1.7.7.1.1.3.1.3.3">𝑖</ci></apply></apply></apply><apply id="S4.E2.m1.7.7.1.1.3c.cmml" xref="S4.E2.m1.7.7.1.1.3"><eq id="S4.E2.m1.7.7.1.1.3.5.cmml" xref="S4.E2.m1.7.7.1.1.3.5"></eq><share href="https://arxiv.org/html/2308.09443v2#S4.E2.m1.7.7.1.1.3.1.cmml" id="S4.E2.m1.7.7.1.1.3d.cmml" xref="S4.E2.m1.7.7.1.1.3"></share><infinity id="S4.E2.m1.7.7.1.1.3.6.cmml" xref="S4.E2.m1.7.7.1.1.3.6"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2.m1.7c">\displaystyle\forall c^{\prime}\in C_{\sigma_{0}^{\prime}},\forall i\in\{1,% \ldots,t\}:~{}~{}c^{\prime}_{i}\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot% (B+3)~{}~{}\vee~{}~{}c^{\prime}_{i}=\infty.</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m1.7d">∀ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , ∀ italic_i ∈ { 1 , … , italic_t } : italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 ) ∨ italic_c start_POSTSUPERSCRIPT ′ 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">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem1.p1.9"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.9.3">In case of any general SP game <math alttext="G\in\textsf{Games}_{t}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.7.1.m1.1"><semantics id="S4.Thmtheorem1.p1.7.1.m1.1a"><mrow id="S4.Thmtheorem1.p1.7.1.m1.1.1" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.7.1.m1.1.1.2" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem1.p1.7.1.m1.1.1.1" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem1.p1.7.1.m1.1.1.3" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem1.p1.7.1.m1.1.1.3.2" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3.2a.cmml">Games</mtext><mi id="S4.Thmtheorem1.p1.7.1.m1.1.1.3.3" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.7.1.m1.1b"><apply id="S4.Thmtheorem1.p1.7.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1"><in id="S4.Thmtheorem1.p1.7.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.1"></in><ci id="S4.Thmtheorem1.p1.7.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.2">𝐺</ci><apply id="S4.Thmtheorem1.p1.7.1.m1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.7.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem1.p1.7.1.m1.1.1.3.2a.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem1.p1.7.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3.2">Games</mtext></ci><ci id="S4.Thmtheorem1.p1.7.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem1.p1.7.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.7.1.m1.1c">G\in\textsf{Games}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.7.1.m1.1d">italic_G ∈ Games start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, the same result holds with <math alttext="|V|" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.8.2.m2.1"><semantics id="S4.Thmtheorem1.p1.8.2.m2.1a"><mrow id="S4.Thmtheorem1.p1.8.2.m2.1.2.2" xref="S4.Thmtheorem1.p1.8.2.m2.1.2.1.cmml"><mo id="S4.Thmtheorem1.p1.8.2.m2.1.2.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.8.2.m2.1.2.1.1.cmml">|</mo><mi id="S4.Thmtheorem1.p1.8.2.m2.1.1" xref="S4.Thmtheorem1.p1.8.2.m2.1.1.cmml">V</mi><mo id="S4.Thmtheorem1.p1.8.2.m2.1.2.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.8.2.m2.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.8.2.m2.1b"><apply id="S4.Thmtheorem1.p1.8.2.m2.1.2.1.cmml" xref="S4.Thmtheorem1.p1.8.2.m2.1.2.2"><abs id="S4.Thmtheorem1.p1.8.2.m2.1.2.1.1.cmml" xref="S4.Thmtheorem1.p1.8.2.m2.1.2.2.1"></abs><ci id="S4.Thmtheorem1.p1.8.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.8.2.m2.1.1">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.8.2.m2.1c">|V|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.8.2.m2.1d">| italic_V |</annotation></semantics></math> replaced by <math alttext="|V|\cdot W" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.9.3.m3.1"><semantics id="S4.Thmtheorem1.p1.9.3.m3.1a"><mrow id="S4.Thmtheorem1.p1.9.3.m3.1.2" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.cmml"><mrow id="S4.Thmtheorem1.p1.9.3.m3.1.2.2.2" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.2.1.cmml"><mo id="S4.Thmtheorem1.p1.9.3.m3.1.2.2.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.2.1.1.cmml">|</mo><mi id="S4.Thmtheorem1.p1.9.3.m3.1.1" xref="S4.Thmtheorem1.p1.9.3.m3.1.1.cmml">V</mi><mo id="S4.Thmtheorem1.p1.9.3.m3.1.2.2.2.2" rspace="0.055em" stretchy="false" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem1.p1.9.3.m3.1.2.1" rspace="0.222em" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.1.cmml">⋅</mo><mi id="S4.Thmtheorem1.p1.9.3.m3.1.2.3" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.9.3.m3.1b"><apply id="S4.Thmtheorem1.p1.9.3.m3.1.2.cmml" xref="S4.Thmtheorem1.p1.9.3.m3.1.2"><ci id="S4.Thmtheorem1.p1.9.3.m3.1.2.1.cmml" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.1">⋅</ci><apply id="S4.Thmtheorem1.p1.9.3.m3.1.2.2.1.cmml" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.2.2"><abs id="S4.Thmtheorem1.p1.9.3.m3.1.2.2.1.1.cmml" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.2.2.1"></abs><ci id="S4.Thmtheorem1.p1.9.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.9.3.m3.1.1">𝑉</ci></apply><ci id="S4.Thmtheorem1.p1.9.3.m3.1.2.3.cmml" xref="S4.Thmtheorem1.p1.9.3.m3.1.2.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.9.3.m3.1c">|V|\cdot W</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.9.3.m3.1d">| italic_V | ⋅ italic_W</annotation></semantics></math> in the inequality.</span></p> </div> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">In view of this result, a solution to the SPS problem is said to be <em class="ltx_emph ltx_font_italic" id="S4.p2.1.1">Pareto-bounded</em> when its Pareto-optimal costs are bounded as stated in the theorem.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.5">The theorem is proved by induction on the dimension <math alttext="t" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mi id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><ci id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_t</annotation></semantics></math>, with the calculation of a <em class="ltx_emph ltx_font_italic" id="S4.p3.4.3">function <math alttext="f({B},{t})" class="ltx_Math" display="inline" id="S4.p3.2.1.m1.2"><semantics id="S4.p3.2.1.m1.2a"><mrow id="S4.p3.2.1.m1.2.3" xref="S4.p3.2.1.m1.2.3.cmml"><mi id="S4.p3.2.1.m1.2.3.2" xref="S4.p3.2.1.m1.2.3.2.cmml">f</mi><mo id="S4.p3.2.1.m1.2.3.1" xref="S4.p3.2.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.p3.2.1.m1.2.3.3.2" xref="S4.p3.2.1.m1.2.3.3.1.cmml"><mo id="S4.p3.2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.p3.2.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.p3.2.1.m1.1.1" xref="S4.p3.2.1.m1.1.1.cmml">B</mi><mo id="S4.p3.2.1.m1.2.3.3.2.2" xref="S4.p3.2.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.p3.2.1.m1.2.2" xref="S4.p3.2.1.m1.2.2.cmml">t</mi><mo id="S4.p3.2.1.m1.2.3.3.2.3" stretchy="false" xref="S4.p3.2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.2.1.m1.2b"><apply id="S4.p3.2.1.m1.2.3.cmml" xref="S4.p3.2.1.m1.2.3"><times id="S4.p3.2.1.m1.2.3.1.cmml" xref="S4.p3.2.1.m1.2.3.1"></times><ci id="S4.p3.2.1.m1.2.3.2.cmml" xref="S4.p3.2.1.m1.2.3.2">𝑓</ci><interval closure="open" id="S4.p3.2.1.m1.2.3.3.1.cmml" xref="S4.p3.2.1.m1.2.3.3.2"><ci id="S4.p3.2.1.m1.1.1.cmml" xref="S4.p3.2.1.m1.1.1">𝐵</ci><ci id="S4.p3.2.1.m1.2.2.cmml" xref="S4.p3.2.1.m1.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.1.m1.2c">f({B},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.1.m1.2d">italic_f ( italic_B , italic_t )</annotation></semantics></math> depending on both <math alttext="B" class="ltx_Math" display="inline" id="S4.p3.3.2.m2.1"><semantics id="S4.p3.3.2.m2.1a"><mi id="S4.p3.3.2.m2.1.1" xref="S4.p3.3.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.p3.3.2.m2.1b"><ci id="S4.p3.3.2.m2.1.1.cmml" xref="S4.p3.3.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.2.m2.1d">italic_B</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="S4.p3.4.3.m3.1"><semantics id="S4.p3.4.3.m3.1a"><mi id="S4.p3.4.3.m3.1.1" xref="S4.p3.4.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.p3.4.3.m3.1b"><ci id="S4.p3.4.3.m3.1.1.cmml" xref="S4.p3.4.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.3.m3.1d">italic_t</annotation></semantics></math></em>, that bounds the components <math alttext="c^{\prime}_{i}\neq\infty" class="ltx_Math" display="inline" id="S4.p3.5.m2.1"><semantics id="S4.p3.5.m2.1a"><mrow id="S4.p3.5.m2.1.1" xref="S4.p3.5.m2.1.1.cmml"><msubsup id="S4.p3.5.m2.1.1.2" xref="S4.p3.5.m2.1.1.2.cmml"><mi id="S4.p3.5.m2.1.1.2.2.2" xref="S4.p3.5.m2.1.1.2.2.2.cmml">c</mi><mi id="S4.p3.5.m2.1.1.2.3" xref="S4.p3.5.m2.1.1.2.3.cmml">i</mi><mo id="S4.p3.5.m2.1.1.2.2.3" xref="S4.p3.5.m2.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S4.p3.5.m2.1.1.1" xref="S4.p3.5.m2.1.1.1.cmml">≠</mo><mi id="S4.p3.5.m2.1.1.3" mathvariant="normal" xref="S4.p3.5.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.5.m2.1b"><apply id="S4.p3.5.m2.1.1.cmml" xref="S4.p3.5.m2.1.1"><neq id="S4.p3.5.m2.1.1.1.cmml" xref="S4.p3.5.m2.1.1.1"></neq><apply id="S4.p3.5.m2.1.1.2.cmml" xref="S4.p3.5.m2.1.1.2"><csymbol cd="ambiguous" id="S4.p3.5.m2.1.1.2.1.cmml" xref="S4.p3.5.m2.1.1.2">subscript</csymbol><apply id="S4.p3.5.m2.1.1.2.2.cmml" xref="S4.p3.5.m2.1.1.2"><csymbol cd="ambiguous" id="S4.p3.5.m2.1.1.2.2.1.cmml" xref="S4.p3.5.m2.1.1.2">superscript</csymbol><ci id="S4.p3.5.m2.1.1.2.2.2.cmml" xref="S4.p3.5.m2.1.1.2.2.2">𝑐</ci><ci id="S4.p3.5.m2.1.1.2.2.3.cmml" xref="S4.p3.5.m2.1.1.2.2.3">′</ci></apply><ci id="S4.p3.5.m2.1.1.2.3.cmml" xref="S4.p3.5.m2.1.1.2.3">𝑖</ci></apply><infinity id="S4.p3.5.m2.1.1.3.cmml" xref="S4.p3.5.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.5.m2.1c">c^{\prime}_{i}\neq\infty</annotation><annotation encoding="application/x-llamapun" id="S4.p3.5.m2.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∞</annotation></semantics></math>. That is, in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>, Equation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E2" title="In Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2</span></a>) is replaced by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx3"> <tbody id="S4.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_left ltx_eqn_cell"><math alttext="\displaystyle\forall c^{\prime}\in C_{\sigma_{0}^{\prime}},\forall i\in\{1,% \ldots,t\}:~{}~{}c^{\prime}_{i}\leq f({B},{t})~{}~{}\vee~{}~{}c^{\prime}_{i}=\infty." class="ltx_Math" display="inline" id="S4.E3.m1.6"><semantics id="S4.E3.m1.6a"><mrow id="S4.E3.m1.6.6.1" xref="S4.E3.m1.6.6.1.1.cmml"><mrow id="S4.E3.m1.6.6.1.1" xref="S4.E3.m1.6.6.1.1.cmml"><mrow id="S4.E3.m1.6.6.1.1.2.2" xref="S4.E3.m1.6.6.1.1.2.3.cmml"><mrow id="S4.E3.m1.6.6.1.1.1.1.1" xref="S4.E3.m1.6.6.1.1.1.1.1.cmml"><mrow id="S4.E3.m1.6.6.1.1.1.1.1.2" xref="S4.E3.m1.6.6.1.1.1.1.1.2.cmml"><mo id="S4.E3.m1.6.6.1.1.1.1.1.2.1" rspace="0.167em" xref="S4.E3.m1.6.6.1.1.1.1.1.2.1.cmml">∀</mo><msup id="S4.E3.m1.6.6.1.1.1.1.1.2.2" xref="S4.E3.m1.6.6.1.1.1.1.1.2.2.cmml"><mi id="S4.E3.m1.6.6.1.1.1.1.1.2.2.2" xref="S4.E3.m1.6.6.1.1.1.1.1.2.2.2.cmml">c</mi><mo id="S4.E3.m1.6.6.1.1.1.1.1.2.2.3" xref="S4.E3.m1.6.6.1.1.1.1.1.2.2.3.cmml">′</mo></msup></mrow><mo id="S4.E3.m1.6.6.1.1.1.1.1.1" xref="S4.E3.m1.6.6.1.1.1.1.1.1.cmml">∈</mo><msub id="S4.E3.m1.6.6.1.1.1.1.1.3" xref="S4.E3.m1.6.6.1.1.1.1.1.3.cmml"><mi id="S4.E3.m1.6.6.1.1.1.1.1.3.2" xref="S4.E3.m1.6.6.1.1.1.1.1.3.2.cmml">C</mi><msubsup id="S4.E3.m1.6.6.1.1.1.1.1.3.3" xref="S4.E3.m1.6.6.1.1.1.1.1.3.3.cmml"><mi id="S4.E3.m1.6.6.1.1.1.1.1.3.3.2.2" xref="S4.E3.m1.6.6.1.1.1.1.1.3.3.2.2.cmml">σ</mi><mn id="S4.E3.m1.6.6.1.1.1.1.1.3.3.2.3" xref="S4.E3.m1.6.6.1.1.1.1.1.3.3.2.3.cmml">0</mn><mo id="S4.E3.m1.6.6.1.1.1.1.1.3.3.3" xref="S4.E3.m1.6.6.1.1.1.1.1.3.3.3.cmml">′</mo></msubsup></msub></mrow><mo id="S4.E3.m1.6.6.1.1.2.2.3" xref="S4.E3.m1.6.6.1.1.2.3a.cmml">,</mo><mrow id="S4.E3.m1.6.6.1.1.2.2.2" xref="S4.E3.m1.6.6.1.1.2.2.2.cmml"><mrow id="S4.E3.m1.6.6.1.1.2.2.2.2" xref="S4.E3.m1.6.6.1.1.2.2.2.2.cmml"><mo id="S4.E3.m1.6.6.1.1.2.2.2.2.1" rspace="0.167em" xref="S4.E3.m1.6.6.1.1.2.2.2.2.1.cmml">∀</mo><mi id="S4.E3.m1.6.6.1.1.2.2.2.2.2" xref="S4.E3.m1.6.6.1.1.2.2.2.2.2.cmml">i</mi></mrow><mo id="S4.E3.m1.6.6.1.1.2.2.2.1" xref="S4.E3.m1.6.6.1.1.2.2.2.1.cmml">∈</mo><mrow id="S4.E3.m1.6.6.1.1.2.2.2.3.2" xref="S4.E3.m1.6.6.1.1.2.2.2.3.1.cmml"><mo id="S4.E3.m1.6.6.1.1.2.2.2.3.2.1" stretchy="false" 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\ldots,t\}:~{}~{}c^{\prime}_{i}\leq f({B},{t})~{}~{}\vee~{}~{}c^{\prime}_{i}=\infty.</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.6d">∀ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , ∀ italic_i ∈ { 1 , … , italic_t } : italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_f ( italic_B , italic_t ) ∨ italic_c start_POSTSUPERSCRIPT ′ 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p3.6">This function is defined by induction on <math alttext="t" class="ltx_Math" display="inline" id="S4.p3.6.m1.1"><semantics id="S4.p3.6.m1.1a"><mi id="S4.p3.6.m1.1.1" xref="S4.p3.6.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.p3.6.m1.1b"><ci id="S4.p3.6.m1.1.1.cmml" xref="S4.p3.6.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.6.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.p3.6.m1.1d">italic_t</annotation></semantics></math> through the proofs, and afterwards made explicit and upper bounded by the bound given in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E2" title="In Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2</span></a>):</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx4"> <tbody id="S4.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 f({B},{t})\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot(B+3)." class="ltx_Math" display="inline" id="S4.E4.m1.6"><semantics id="S4.E4.m1.6a"><mrow id="S4.E4.m1.6.6.1" xref="S4.E4.m1.6.6.1.1.cmml"><mrow id="S4.E4.m1.6.6.1.1" xref="S4.E4.m1.6.6.1.1.cmml"><mrow id="S4.E4.m1.6.6.1.1.3" xref="S4.E4.m1.6.6.1.1.3.cmml"><mi id="S4.E4.m1.6.6.1.1.3.2" xref="S4.E4.m1.6.6.1.1.3.2.cmml">f</mi><mo id="S4.E4.m1.6.6.1.1.3.1" xref="S4.E4.m1.6.6.1.1.3.1.cmml">⁢</mo><mrow id="S4.E4.m1.6.6.1.1.3.3.2" xref="S4.E4.m1.6.6.1.1.3.3.1.cmml"><mo id="S4.E4.m1.6.6.1.1.3.3.2.1" stretchy="false" xref="S4.E4.m1.6.6.1.1.3.3.1.cmml">(</mo><mi id="S4.E4.m1.3.3" xref="S4.E4.m1.3.3.cmml">B</mi><mo id="S4.E4.m1.6.6.1.1.3.3.2.2" xref="S4.E4.m1.6.6.1.1.3.3.1.cmml">,</mo><mi id="S4.E4.m1.4.4" xref="S4.E4.m1.4.4.cmml">t</mi><mo id="S4.E4.m1.6.6.1.1.3.3.2.3" stretchy="false" 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id="S4.E4.m1.6.6.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S4.E4.m1.6.6.1.1.1.2.cmml">⋅</mo><msup id="S4.E4.m1.6.6.1.1.1.4" xref="S4.E4.m1.6.6.1.1.1.4.cmml"><mrow id="S4.E4.m1.6.6.1.1.1.4.2.2" xref="S4.E4.m1.6.6.1.1.1.4.2.1.cmml"><mo id="S4.E4.m1.6.6.1.1.1.4.2.2.1" stretchy="false" xref="S4.E4.m1.6.6.1.1.1.4.2.1.1.cmml">|</mo><mi id="S4.E4.m1.5.5" xref="S4.E4.m1.5.5.cmml">V</mi><mo id="S4.E4.m1.6.6.1.1.1.4.2.2.2" rspace="0.055em" stretchy="false" xref="S4.E4.m1.6.6.1.1.1.4.2.1.1.cmml">|</mo></mrow><mrow id="S4.E4.m1.2.2.1" xref="S4.E4.m1.2.2.1.cmml"><mi id="S4.E4.m1.2.2.1.3" mathvariant="normal" xref="S4.E4.m1.2.2.1.3.cmml">Θ</mi><mo id="S4.E4.m1.2.2.1.2" xref="S4.E4.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.E4.m1.2.2.1.4.2" xref="S4.E4.m1.2.2.1.cmml"><mo id="S4.E4.m1.2.2.1.4.2.1" stretchy="false" xref="S4.E4.m1.2.2.1.cmml">(</mo><mi id="S4.E4.m1.2.2.1.1" xref="S4.E4.m1.2.2.1.1.cmml">t</mi><mo id="S4.E4.m1.2.2.1.4.2.2" stretchy="false" xref="S4.E4.m1.2.2.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.E4.m1.6.6.1.1.1.2a" rspace="0.222em" xref="S4.E4.m1.6.6.1.1.1.2.cmml">⋅</mo><mrow id="S4.E4.m1.6.6.1.1.1.1.1" xref="S4.E4.m1.6.6.1.1.1.1.1.1.cmml"><mo id="S4.E4.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S4.E4.m1.6.6.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E4.m1.6.6.1.1.1.1.1.1" xref="S4.E4.m1.6.6.1.1.1.1.1.1.cmml"><mi id="S4.E4.m1.6.6.1.1.1.1.1.1.2" xref="S4.E4.m1.6.6.1.1.1.1.1.1.2.cmml">B</mi><mo id="S4.E4.m1.6.6.1.1.1.1.1.1.1" xref="S4.E4.m1.6.6.1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.E4.m1.6.6.1.1.1.1.1.1.3" xref="S4.E4.m1.6.6.1.1.1.1.1.1.3.cmml">3</mn></mrow><mo id="S4.E4.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="S4.E4.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E4.m1.6.6.1.2" lspace="0em" xref="S4.E4.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E4.m1.6b"><apply id="S4.E4.m1.6.6.1.1.cmml" xref="S4.E4.m1.6.6.1"><leq id="S4.E4.m1.6.6.1.1.2.cmml" xref="S4.E4.m1.6.6.1.1.2"></leq><apply id="S4.E4.m1.6.6.1.1.3.cmml" xref="S4.E4.m1.6.6.1.1.3"><times id="S4.E4.m1.6.6.1.1.3.1.cmml" xref="S4.E4.m1.6.6.1.1.3.1"></times><ci id="S4.E4.m1.6.6.1.1.3.2.cmml" xref="S4.E4.m1.6.6.1.1.3.2">𝑓</ci><interval closure="open" id="S4.E4.m1.6.6.1.1.3.3.1.cmml" xref="S4.E4.m1.6.6.1.1.3.3.2"><ci id="S4.E4.m1.3.3.cmml" xref="S4.E4.m1.3.3">𝐵</ci><ci id="S4.E4.m1.4.4.cmml" xref="S4.E4.m1.4.4">𝑡</ci></interval></apply><apply id="S4.E4.m1.6.6.1.1.1.cmml" xref="S4.E4.m1.6.6.1.1.1"><ci id="S4.E4.m1.6.6.1.1.1.2.cmml" xref="S4.E4.m1.6.6.1.1.1.2">⋅</ci><apply id="S4.E4.m1.6.6.1.1.1.3.cmml" xref="S4.E4.m1.6.6.1.1.1.3"><csymbol cd="ambiguous" id="S4.E4.m1.6.6.1.1.1.3.1.cmml" xref="S4.E4.m1.6.6.1.1.1.3">superscript</csymbol><cn id="S4.E4.m1.6.6.1.1.1.3.2.cmml" type="integer" xref="S4.E4.m1.6.6.1.1.1.3.2">2</cn><apply id="S4.E4.m1.1.1.1.cmml" xref="S4.E4.m1.1.1.1"><times id="S4.E4.m1.1.1.1.2.cmml" xref="S4.E4.m1.1.1.1.2"></times><ci id="S4.E4.m1.1.1.1.3.cmml" xref="S4.E4.m1.1.1.1.3">Θ</ci><apply id="S4.E4.m1.1.1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.E4.m1.1.1.1.1.1.1.2.cmml" xref="S4.E4.m1.1.1.1.1.1.1.2">𝑡</ci><cn id="S4.E4.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E4.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply><apply id="S4.E4.m1.6.6.1.1.1.4.cmml" xref="S4.E4.m1.6.6.1.1.1.4"><csymbol cd="ambiguous" id="S4.E4.m1.6.6.1.1.1.4.1.cmml" xref="S4.E4.m1.6.6.1.1.1.4">superscript</csymbol><apply id="S4.E4.m1.6.6.1.1.1.4.2.1.cmml" xref="S4.E4.m1.6.6.1.1.1.4.2.2"><abs id="S4.E4.m1.6.6.1.1.1.4.2.1.1.cmml" xref="S4.E4.m1.6.6.1.1.1.4.2.2.1"></abs><ci id="S4.E4.m1.5.5.cmml" xref="S4.E4.m1.5.5">𝑉</ci></apply><apply id="S4.E4.m1.2.2.1.cmml" xref="S4.E4.m1.2.2.1"><times id="S4.E4.m1.2.2.1.2.cmml" xref="S4.E4.m1.2.2.1.2"></times><ci id="S4.E4.m1.2.2.1.3.cmml" xref="S4.E4.m1.2.2.1.3">Θ</ci><ci id="S4.E4.m1.2.2.1.1.cmml" xref="S4.E4.m1.2.2.1.1">𝑡</ci></apply></apply><apply id="S4.E4.m1.6.6.1.1.1.1.1.1.cmml" xref="S4.E4.m1.6.6.1.1.1.1.1"><plus id="S4.E4.m1.6.6.1.1.1.1.1.1.1.cmml" xref="S4.E4.m1.6.6.1.1.1.1.1.1.1"></plus><ci id="S4.E4.m1.6.6.1.1.1.1.1.1.2.cmml" xref="S4.E4.m1.6.6.1.1.1.1.1.1.2">𝐵</ci><cn id="S4.E4.m1.6.6.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E4.m1.6.6.1.1.1.1.1.1.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.6c">\displaystyle f({B},{t})\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot(B+3).</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.6d">italic_f ( italic_B , italic_t ) ≤ 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 ) .</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 class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">The proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> is detailed in the next four subsections for binary SP games; it is then easily adapted to any SP games by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem2" title="Lemma 2.2 ‣ Reduction to binary arenas. ‣ 2.3 Tools ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.2</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>Dimension One</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">We begin the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> with the case <math alttext="t=1" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mrow id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml">t</mi><mo id="S4.SS1.p1.1.m1.1.1.1" xref="S4.SS1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS1.p1.1.m1.1.1.3" xref="S4.SS1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1"><eq id="S4.SS1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1.1"></eq><ci id="S4.SS1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.p1.1.m1.1.1.2">𝑡</ci><cn id="S4.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_t = 1</annotation></semantics></math>. In this case, the order on costs is total.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Lemma 4.2</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.6.6">Let <math alttext="G\in\textsf{BinGames}_{1}" 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">G</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2a.cmml">BinGames</mtext><mn id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">1</mn></msub></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"><in id="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.2">𝐺</ci><apply id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2a.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2">BinGames</mtext></ci><cn id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.1c">G\in\textsf{BinGames}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.1d">italic_G ∈ BinGames start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a binary SP game with dimension <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.2.m2.1a"><mn id="S4.Thmtheorem2.p1.2.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.2.m2.1b"><cn id="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml" type="integer" xref="S4.Thmtheorem2.p1.2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.1d">1</annotation></semantics></math>, <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.3.m3.1"><semantics id="S4.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem2.p1.3.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">B</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.3.m3.1b"><apply id="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1"><in id="S4.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.1"></in><ci id="S4.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.2">𝐵</ci><ci id="S4.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.3.m3.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.3.m3.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.4.m4.2"><semantics id="S4.Thmtheorem2.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem2.p1.4.4.m4.2.3" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.cmml"><msub id="S4.Thmtheorem2.p1.4.4.m4.2.3.2" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.2.cmml"><mi id="S4.Thmtheorem2.p1.4.4.m4.2.3.2.2" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem2.p1.4.4.m4.2.3.2.3" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem2.p1.4.4.m4.2.3.1" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem2.p1.4.4.m4.2.3.3" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.2" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.1" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.2" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem2.p1.4.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml">G</mi><mo id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.2.2" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.4.4.m4.2.2" xref="S4.Thmtheorem2.p1.4.4.m4.2.2.cmml">B</mi><mo id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.4.m4.2b"><apply id="S4.Thmtheorem2.p1.4.4.m4.2.3.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3"><in id="S4.Thmtheorem2.p1.4.4.m4.2.3.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.1"></in><apply id="S4.Thmtheorem2.p1.4.4.m4.2.3.2.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.4.4.m4.2.3.2.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.4.4.m4.2.3.2.2.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem2.p1.4.4.m4.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3"><times id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.1"></times><ci id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.2a.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.2.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.3.3.3.2"><ci id="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1">𝐺</ci><ci id="S4.Thmtheorem2.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.4.m4.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. Then there exists a solution <math alttext="\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.5.m5.2"><semantics id="S4.Thmtheorem2.p1.5.5.m5.2a"><mrow id="S4.Thmtheorem2.p1.5.5.m5.2.3" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.cmml"><msubsup id="S4.Thmtheorem2.p1.5.5.m5.2.3.2" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2.cmml"><mi id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.3" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.3.cmml">0</mn><mo id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.3" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem2.p1.5.5.m5.2.3.1" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem2.p1.5.5.m5.2.3.3" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.2" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.1" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.2" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem2.p1.5.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml">G</mi><mo id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.5.5.m5.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.2.2.cmml">B</mi><mo id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.5.m5.2b"><apply id="S4.Thmtheorem2.p1.5.5.m5.2.3.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3"><in id="S4.Thmtheorem2.p1.5.5.m5.2.3.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.1"></in><apply id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2">superscript</csymbol><apply id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.2.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.2">𝜎</ci><cn id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2.2.3">0</cn></apply><ci id="S4.Thmtheorem2.p1.5.5.m5.2.3.2.3.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.2.3">′</ci></apply><apply id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3"><times id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.1"></times><ci id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.2a.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.2.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.3.3.3.2"><ci id="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.1.1">𝐺</ci><ci id="S4.Thmtheorem2.p1.5.5.m5.2.2.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.5.m5.2c">\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.5.m5.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> without cycles such that <math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.6.6.m6.1"><semantics id="S4.Thmtheorem2.p1.6.6.m6.1a"><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.cmml"><msubsup id="S4.Thmtheorem2.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.cmml"><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.3.cmml">0</mn><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.cmml">⪯</mo><msub id="S4.Thmtheorem2.p1.6.6.m6.1.1.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3.cmml"><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.3.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem2.p1.6.6.m6.1.1.3.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.6.6.m6.1b"><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1"><csymbol cd="latexml" id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1">precedes-or-equals</csymbol><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2">superscript</csymbol><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.2">𝜎</ci><cn id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.3">0</cn></apply><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3">′</ci></apply><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.6.m6.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem2.p1.6.6.m6.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.6.6.m6.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.6.6.m6.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx5"> <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\forall c^{\prime}\in C_{\sigma_{0}^{\prime}}:~{}~{}c^{\prime}% \leq f({B},{1})=B+|V|~{}~{}\vee~{}~{}c^{\prime}=\infty." class="ltx_Math" display="inline" id="S4.E5.m1.4"><semantics id="S4.E5.m1.4a"><mrow id="S4.E5.m1.4.4.1" xref="S4.E5.m1.4.4.1.1.cmml"><mrow id="S4.E5.m1.4.4.1.1" xref="S4.E5.m1.4.4.1.1.cmml"><mrow id="S4.E5.m1.4.4.1.1.2" xref="S4.E5.m1.4.4.1.1.2.cmml"><mrow id="S4.E5.m1.4.4.1.1.2.2" xref="S4.E5.m1.4.4.1.1.2.2.cmml"><mo id="S4.E5.m1.4.4.1.1.2.2.1" rspace="0.167em" xref="S4.E5.m1.4.4.1.1.2.2.1.cmml">∀</mo><msup id="S4.E5.m1.4.4.1.1.2.2.2" xref="S4.E5.m1.4.4.1.1.2.2.2.cmml"><mi id="S4.E5.m1.4.4.1.1.2.2.2.2" xref="S4.E5.m1.4.4.1.1.2.2.2.2.cmml">c</mi><mo id="S4.E5.m1.4.4.1.1.2.2.2.3" xref="S4.E5.m1.4.4.1.1.2.2.2.3.cmml">′</mo></msup></mrow><mo id="S4.E5.m1.4.4.1.1.2.1" xref="S4.E5.m1.4.4.1.1.2.1.cmml">∈</mo><msub id="S4.E5.m1.4.4.1.1.2.3" xref="S4.E5.m1.4.4.1.1.2.3.cmml"><mi 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xref="S4.E5.m1.4.4.1.1.3.7"></eq><share href="https://arxiv.org/html/2308.09443v2#S4.E5.m1.4.4.1.1.3.6.cmml" id="S4.E5.m1.4.4.1.1.3f.cmml" xref="S4.E5.m1.4.4.1.1.3"></share><infinity id="S4.E5.m1.4.4.1.1.3.8.cmml" xref="S4.E5.m1.4.4.1.1.3.8"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5.m1.4c">\displaystyle\forall c^{\prime}\in C_{\sigma_{0}^{\prime}}:~{}~{}c^{\prime}% \leq f({B},{1})=B+|V|~{}~{}\vee~{}~{}c^{\prime}=\infty.</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.4d">∀ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT : italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_f ( italic_B , 1 ) = italic_B + | italic_V | ∨ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∞ .</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> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.2">Notice that <math alttext="f({B},{1})" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.2"><semantics id="S4.SS1.p2.1.m1.2a"><mrow id="S4.SS1.p2.1.m1.2.3" xref="S4.SS1.p2.1.m1.2.3.cmml"><mi id="S4.SS1.p2.1.m1.2.3.2" xref="S4.SS1.p2.1.m1.2.3.2.cmml">f</mi><mo id="S4.SS1.p2.1.m1.2.3.1" xref="S4.SS1.p2.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.1.m1.2.3.3.2" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml"><mo id="S4.SS1.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">B</mi><mo id="S4.SS1.p2.1.m1.2.3.3.2.2" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">,</mo><mn id="S4.SS1.p2.1.m1.2.2" xref="S4.SS1.p2.1.m1.2.2.cmml">1</mn><mo id="S4.SS1.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.2b"><apply id="S4.SS1.p2.1.m1.2.3.cmml" xref="S4.SS1.p2.1.m1.2.3"><times id="S4.SS1.p2.1.m1.2.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.1"></times><ci id="S4.SS1.p2.1.m1.2.3.2.cmml" xref="S4.SS1.p2.1.m1.2.3.2">𝑓</ci><interval closure="open" id="S4.SS1.p2.1.m1.2.3.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.3.2"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝐵</ci><cn id="S4.SS1.p2.1.m1.2.2.cmml" type="integer" xref="S4.SS1.p2.1.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.2c">f({B},{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.2d">italic_f ( italic_B , 1 )</annotation></semantics></math> satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E4" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>) when <math alttext="t=1" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mrow id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml">t</mi><mo id="S4.SS1.p2.2.m2.1.1.1" xref="S4.SS1.p2.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><eq id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1.1"></eq><ci id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2">𝑡</ci><cn id="S4.SS1.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS1.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_t = 1</annotation></semantics></math>. Indeed, we have</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx6"> <tbody id="S4.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({B},{1})\leq|V|\cdot(B+3)\leq 2^{\Theta(t^{2})}\cdot|V|^{% \Theta(t)}\cdot(B+3)." class="ltx_Math" display="inline" id="S4.E6.m1.7"><semantics id="S4.E6.m1.7a"><mrow id="S4.E6.m1.7.7.1" xref="S4.E6.m1.7.7.1.1.cmml"><mrow id="S4.E6.m1.7.7.1.1" xref="S4.E6.m1.7.7.1.1.cmml"><mrow id="S4.E6.m1.7.7.1.1.4" xref="S4.E6.m1.7.7.1.1.4.cmml"><mi id="S4.E6.m1.7.7.1.1.4.2" xref="S4.E6.m1.7.7.1.1.4.2.cmml">f</mi><mo id="S4.E6.m1.7.7.1.1.4.1" xref="S4.E6.m1.7.7.1.1.4.1.cmml">⁢</mo><mrow id="S4.E6.m1.7.7.1.1.4.3.2" xref="S4.E6.m1.7.7.1.1.4.3.1.cmml"><mo id="S4.E6.m1.7.7.1.1.4.3.2.1" stretchy="false" xref="S4.E6.m1.7.7.1.1.4.3.1.cmml">(</mo><mi id="S4.E6.m1.3.3" xref="S4.E6.m1.3.3.cmml">B</mi><mo id="S4.E6.m1.7.7.1.1.4.3.2.2" xref="S4.E6.m1.7.7.1.1.4.3.1.cmml">,</mo><mn id="S4.E6.m1.4.4" xref="S4.E6.m1.4.4.cmml">1</mn><mo id="S4.E6.m1.7.7.1.1.4.3.2.3" stretchy="false" xref="S4.E6.m1.7.7.1.1.4.3.1.cmml">)</mo></mrow></mrow><mo id="S4.E6.m1.7.7.1.1.5" xref="S4.E6.m1.7.7.1.1.5.cmml">≤</mo><mrow id="S4.E6.m1.7.7.1.1.1" xref="S4.E6.m1.7.7.1.1.1.cmml"><mrow id="S4.E6.m1.7.7.1.1.1.3.2" xref="S4.E6.m1.7.7.1.1.1.3.1.cmml"><mo id="S4.E6.m1.7.7.1.1.1.3.2.1" stretchy="false" xref="S4.E6.m1.7.7.1.1.1.3.1.1.cmml">|</mo><mi id="S4.E6.m1.5.5" xref="S4.E6.m1.5.5.cmml">V</mi><mo id="S4.E6.m1.7.7.1.1.1.3.2.2" rspace="0.055em" stretchy="false" xref="S4.E6.m1.7.7.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S4.E6.m1.7.7.1.1.1.2" rspace="0.222em" xref="S4.E6.m1.7.7.1.1.1.2.cmml">⋅</mo><mrow id="S4.E6.m1.7.7.1.1.1.1.1" xref="S4.E6.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S4.E6.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="S4.E6.m1.7.7.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E6.m1.7.7.1.1.1.1.1.1" xref="S4.E6.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S4.E6.m1.7.7.1.1.1.1.1.1.2" xref="S4.E6.m1.7.7.1.1.1.1.1.1.2.cmml">B</mi><mo id="S4.E6.m1.7.7.1.1.1.1.1.1.1" xref="S4.E6.m1.7.7.1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.E6.m1.7.7.1.1.1.1.1.1.3" xref="S4.E6.m1.7.7.1.1.1.1.1.1.3.cmml">3</mn></mrow><mo id="S4.E6.m1.7.7.1.1.1.1.1.3" stretchy="false" xref="S4.E6.m1.7.7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E6.m1.7.7.1.1.6" xref="S4.E6.m1.7.7.1.1.6.cmml">≤</mo><mrow id="S4.E6.m1.7.7.1.1.2" xref="S4.E6.m1.7.7.1.1.2.cmml"><msup id="S4.E6.m1.7.7.1.1.2.3" xref="S4.E6.m1.7.7.1.1.2.3.cmml"><mn id="S4.E6.m1.7.7.1.1.2.3.2" xref="S4.E6.m1.7.7.1.1.2.3.2.cmml">2</mn><mrow id="S4.E6.m1.1.1.1" xref="S4.E6.m1.1.1.1.cmml"><mi id="S4.E6.m1.1.1.1.3" mathvariant="normal" xref="S4.E6.m1.1.1.1.3.cmml">Θ</mi><mo id="S4.E6.m1.1.1.1.2" xref="S4.E6.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E6.m1.1.1.1.1.1" xref="S4.E6.m1.1.1.1.1.1.1.cmml"><mo id="S4.E6.m1.1.1.1.1.1.2" stretchy="false" 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id="S4.E6.m1.7.7.1.1.2.1.1.3" stretchy="false" xref="S4.E6.m1.7.7.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E6.m1.7.7.1.2" lspace="0em" xref="S4.E6.m1.7.7.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.7b"><apply id="S4.E6.m1.7.7.1.1.cmml" xref="S4.E6.m1.7.7.1"><and id="S4.E6.m1.7.7.1.1a.cmml" xref="S4.E6.m1.7.7.1"></and><apply id="S4.E6.m1.7.7.1.1b.cmml" xref="S4.E6.m1.7.7.1"><leq id="S4.E6.m1.7.7.1.1.5.cmml" xref="S4.E6.m1.7.7.1.1.5"></leq><apply id="S4.E6.m1.7.7.1.1.4.cmml" xref="S4.E6.m1.7.7.1.1.4"><times id="S4.E6.m1.7.7.1.1.4.1.cmml" xref="S4.E6.m1.7.7.1.1.4.1"></times><ci id="S4.E6.m1.7.7.1.1.4.2.cmml" xref="S4.E6.m1.7.7.1.1.4.2">𝑓</ci><interval closure="open" id="S4.E6.m1.7.7.1.1.4.3.1.cmml" xref="S4.E6.m1.7.7.1.1.4.3.2"><ci id="S4.E6.m1.3.3.cmml" xref="S4.E6.m1.3.3">𝐵</ci><cn id="S4.E6.m1.4.4.cmml" type="integer" xref="S4.E6.m1.4.4">1</cn></interval></apply><apply id="S4.E6.m1.7.7.1.1.1.cmml" xref="S4.E6.m1.7.7.1.1.1"><ci id="S4.E6.m1.7.7.1.1.1.2.cmml" xref="S4.E6.m1.7.7.1.1.1.2">⋅</ci><apply id="S4.E6.m1.7.7.1.1.1.3.1.cmml" xref="S4.E6.m1.7.7.1.1.1.3.2"><abs id="S4.E6.m1.7.7.1.1.1.3.1.1.cmml" xref="S4.E6.m1.7.7.1.1.1.3.2.1"></abs><ci id="S4.E6.m1.5.5.cmml" xref="S4.E6.m1.5.5">𝑉</ci></apply><apply id="S4.E6.m1.7.7.1.1.1.1.1.1.cmml" xref="S4.E6.m1.7.7.1.1.1.1.1"><plus id="S4.E6.m1.7.7.1.1.1.1.1.1.1.cmml" xref="S4.E6.m1.7.7.1.1.1.1.1.1.1"></plus><ci id="S4.E6.m1.7.7.1.1.1.1.1.1.2.cmml" xref="S4.E6.m1.7.7.1.1.1.1.1.1.2">𝐵</ci><cn id="S4.E6.m1.7.7.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E6.m1.7.7.1.1.1.1.1.1.3">3</cn></apply></apply></apply><apply id="S4.E6.m1.7.7.1.1c.cmml" xref="S4.E6.m1.7.7.1"><leq id="S4.E6.m1.7.7.1.1.6.cmml" xref="S4.E6.m1.7.7.1.1.6"></leq><share href="https://arxiv.org/html/2308.09443v2#S4.E6.m1.7.7.1.1.1.cmml" id="S4.E6.m1.7.7.1.1d.cmml" xref="S4.E6.m1.7.7.1"></share><apply id="S4.E6.m1.7.7.1.1.2.cmml" xref="S4.E6.m1.7.7.1.1.2"><ci id="S4.E6.m1.7.7.1.1.2.2.cmml" xref="S4.E6.m1.7.7.1.1.2.2">⋅</ci><apply id="S4.E6.m1.7.7.1.1.2.3.cmml" xref="S4.E6.m1.7.7.1.1.2.3"><csymbol cd="ambiguous" id="S4.E6.m1.7.7.1.1.2.3.1.cmml" xref="S4.E6.m1.7.7.1.1.2.3">superscript</csymbol><cn id="S4.E6.m1.7.7.1.1.2.3.2.cmml" type="integer" xref="S4.E6.m1.7.7.1.1.2.3.2">2</cn><apply id="S4.E6.m1.1.1.1.cmml" xref="S4.E6.m1.1.1.1"><times id="S4.E6.m1.1.1.1.2.cmml" xref="S4.E6.m1.1.1.1.2"></times><ci id="S4.E6.m1.1.1.1.3.cmml" xref="S4.E6.m1.1.1.1.3">Θ</ci><apply id="S4.E6.m1.1.1.1.1.1.1.cmml" xref="S4.E6.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.1.1.1.1.1.cmml" xref="S4.E6.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.E6.m1.1.1.1.1.1.1.2.cmml" xref="S4.E6.m1.1.1.1.1.1.1.2">𝑡</ci><cn id="S4.E6.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E6.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply><apply id="S4.E6.m1.7.7.1.1.2.4.cmml" xref="S4.E6.m1.7.7.1.1.2.4"><csymbol cd="ambiguous" id="S4.E6.m1.7.7.1.1.2.4.1.cmml" xref="S4.E6.m1.7.7.1.1.2.4">superscript</csymbol><apply id="S4.E6.m1.7.7.1.1.2.4.2.1.cmml" xref="S4.E6.m1.7.7.1.1.2.4.2.2"><abs id="S4.E6.m1.7.7.1.1.2.4.2.1.1.cmml" xref="S4.E6.m1.7.7.1.1.2.4.2.2.1"></abs><ci id="S4.E6.m1.6.6.cmml" xref="S4.E6.m1.6.6">𝑉</ci></apply><apply id="S4.E6.m1.2.2.1.cmml" xref="S4.E6.m1.2.2.1"><times id="S4.E6.m1.2.2.1.2.cmml" xref="S4.E6.m1.2.2.1.2"></times><ci id="S4.E6.m1.2.2.1.3.cmml" xref="S4.E6.m1.2.2.1.3">Θ</ci><ci id="S4.E6.m1.2.2.1.1.cmml" xref="S4.E6.m1.2.2.1.1">𝑡</ci></apply></apply><apply id="S4.E6.m1.7.7.1.1.2.1.1.1.cmml" xref="S4.E6.m1.7.7.1.1.2.1.1"><plus id="S4.E6.m1.7.7.1.1.2.1.1.1.1.cmml" xref="S4.E6.m1.7.7.1.1.2.1.1.1.1"></plus><ci id="S4.E6.m1.7.7.1.1.2.1.1.1.2.cmml" xref="S4.E6.m1.7.7.1.1.2.1.1.1.2">𝐵</ci><cn id="S4.E6.m1.7.7.1.1.2.1.1.1.3.cmml" type="integer" xref="S4.E6.m1.7.7.1.1.2.1.1.1.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.7c">\displaystyle f({B},{1})\leq|V|\cdot(B+3)\leq 2^{\Theta(t^{2})}\cdot|V|^{% \Theta(t)}\cdot(B+3).</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.7d">italic_f ( italic_B , 1 ) ≤ | italic_V | ⋅ ( italic_B + 3 ) ≤ 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 ) .</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> </table> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.2">Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem2" title="Lemma 4.2 ‣ 4.1 Dimension One ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.2</span></a> is proved by showing that if the unique Pareto-optimal cost of <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><msub id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml"><mi id="S4.SS1.p3.1.m1.1.1.2" xref="S4.SS1.p3.1.m1.1.1.2.cmml">σ</mi><mn id="S4.SS1.p3.1.m1.1.1.3" xref="S4.SS1.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><apply id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p3.1.m1.1.1.2.cmml" xref="S4.SS1.p3.1.m1.1.1.2">𝜎</ci><cn id="S4.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is finite but greater than <math alttext="B+|V|" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mrow id="S4.SS1.p3.2.m2.1.2" xref="S4.SS1.p3.2.m2.1.2.cmml"><mi id="S4.SS1.p3.2.m2.1.2.2" xref="S4.SS1.p3.2.m2.1.2.2.cmml">B</mi><mo id="S4.SS1.p3.2.m2.1.2.1" xref="S4.SS1.p3.2.m2.1.2.1.cmml">+</mo><mrow id="S4.SS1.p3.2.m2.1.2.3.2" xref="S4.SS1.p3.2.m2.1.2.3.1.cmml"><mo id="S4.SS1.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS1.p3.2.m2.1.2.3.1.1.cmml">|</mo><mi id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">V</mi><mo id="S4.SS1.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS1.p3.2.m2.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.2.cmml" xref="S4.SS1.p3.2.m2.1.2"><plus id="S4.SS1.p3.2.m2.1.2.1.cmml" xref="S4.SS1.p3.2.m2.1.2.1"></plus><ci id="S4.SS1.p3.2.m2.1.2.2.cmml" xref="S4.SS1.p3.2.m2.1.2.2">𝐵</ci><apply id="S4.SS1.p3.2.m2.1.2.3.1.cmml" xref="S4.SS1.p3.2.m2.1.2.3.2"><abs id="S4.SS1.p3.2.m2.1.2.3.1.1.cmml" xref="S4.SS1.p3.2.m2.1.2.3.2.1"></abs><ci id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">B+|V|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">italic_B + | italic_V |</annotation></semantics></math>, then we can eliminate a cycle according to Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a> and get a better solution.</p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.1.1.1">Proof 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2"> </span>(Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem2" title="Lemma 4.2 ‣ 4.1 Dimension One ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.2</span></a>)</h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.24"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.24.24">Let <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.2"><semantics id="S4.Thmtheorem3.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem3.p1.1.1.m1.2.3" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.cmml"><msub id="S4.Thmtheorem3.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.3" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem3.p1.1.1.m1.2.3.3" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.1" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml">G</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem3.p1.1.1.m1.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.cmml">B</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.2b"><apply id="S4.Thmtheorem3.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3"><in id="S4.Thmtheorem3.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.1"></in><apply id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3"><times id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.1"></times><ci id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.2a.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.3.3.2"><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1">𝐺</ci><ci id="S4.Thmtheorem3.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. Player <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mn id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><cn id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" type="integer" xref="S4.Thmtheorem3.p1.2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">1</annotation></semantics></math> has only one target, thus <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.3.3.m3.1"><semantics id="S4.Thmtheorem3.p1.3.3.m3.1a"><msub id="S4.Thmtheorem3.p1.3.3.m3.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.2" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.3" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.3.3.m3.1b"><apply id="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.2">𝐶</ci><apply id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.3.3.m3.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.3.3.m3.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a singleton, say <math alttext="C_{\sigma_{0}}=\{c\}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.4.4.m4.1"><semantics id="S4.Thmtheorem3.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem3.p1.4.4.m4.1.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.cmml"><msub id="S4.Thmtheorem3.p1.4.4.m4.1.2.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.cmml"><mi id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.2.cmml">C</mi><msub id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.cmml"><mi id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.2.cmml">σ</mi><mn id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.3" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem3.p1.4.4.m4.1.2.1" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.1.cmml">=</mo><mrow id="S4.Thmtheorem3.p1.4.4.m4.1.2.3.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.3.1.cmml"><mo id="S4.Thmtheorem3.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.3.1.cmml">{</mo><mi id="S4.Thmtheorem3.p1.4.4.m4.1.1" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.cmml">c</mi><mo id="S4.Thmtheorem3.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.4.4.m4.1b"><apply id="S4.Thmtheorem3.p1.4.4.m4.1.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2"><eq id="S4.Thmtheorem3.p1.4.4.m4.1.2.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.1"></eq><apply id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.2">𝐶</ci><apply id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.2">𝜎</ci><cn id="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.2.3.3">0</cn></apply></apply><set id="S4.Thmtheorem3.p1.4.4.m4.1.2.3.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.2.3.2"><ci id="S4.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1">𝑐</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.4.4.m4.1c">C_{\sigma_{0}}=\{c\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.4.4.m4.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { italic_c }</annotation></semantics></math>. If <math alttext="c\leq B+|V|" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.5.5.m5.1"><semantics id="S4.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem3.p1.5.5.m5.1.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.1.2.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.2.cmml">c</mi><mo id="S4.Thmtheorem3.p1.5.5.m5.1.2.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.1.cmml">≤</mo><mrow id="S4.Thmtheorem3.p1.5.5.m5.1.2.3" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.2.cmml">B</mi><mo id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.1.cmml">+</mo><mrow id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.1.cmml"><mo id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem3.p1.5.5.m5.1.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.cmml">V</mi><mo id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.1.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.5.5.m5.1b"><apply id="S4.Thmtheorem3.p1.5.5.m5.1.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2"><leq id="S4.Thmtheorem3.p1.5.5.m5.1.2.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.1"></leq><ci id="S4.Thmtheorem3.p1.5.5.m5.1.2.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.2">𝑐</ci><apply id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3"><plus id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.1"></plus><ci id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.2">𝐵</ci><apply id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.2"><abs id="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.1.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.2.3.3.2.1"></abs><ci id="S4.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.5.5.m5.1c">c\leq B+|V|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.5.5.m5.1d">italic_c ≤ italic_B + | italic_V |</annotation></semantics></math> or <math alttext="c=\infty" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.6.6.m6.1"><semantics id="S4.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.cmml"><mi id="S4.Thmtheorem3.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.cmml">=</mo><mi id="S4.Thmtheorem3.p1.6.6.m6.1.1.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.6.6.m6.1b"><apply id="S4.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1"><eq id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1"></eq><ci id="S4.Thmtheorem3.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.2">𝑐</ci><infinity id="S4.Thmtheorem3.p1.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.6.6.m6.1c">c=\infty</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.6.6.m6.1d">italic_c = ∞</annotation></semantics></math>, Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem2" title="Lemma 4.2 ‣ 4.1 Dimension One ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.2</span></a> trivially holds (we eliminate cycles if necessary). Therefore, let us suppose that <math alttext="B+|V|<c<\infty" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.7.7.m7.1"><semantics id="S4.Thmtheorem3.p1.7.7.m7.1a"><mrow id="S4.Thmtheorem3.p1.7.7.m7.1.2" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.cmml"><mrow id="S4.Thmtheorem3.p1.7.7.m7.1.2.2" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.cmml"><mi id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.2" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.2.cmml">B</mi><mo id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.1" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.1.cmml">+</mo><mrow id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.2" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.1.cmml"><mo id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem3.p1.7.7.m7.1.1" xref="S4.Thmtheorem3.p1.7.7.m7.1.1.cmml">V</mi><mo id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.1.1.cmml">|</mo></mrow></mrow><mo id="S4.Thmtheorem3.p1.7.7.m7.1.2.3" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.3.cmml"><</mo><mi id="S4.Thmtheorem3.p1.7.7.m7.1.2.4" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.4.cmml">c</mi><mo id="S4.Thmtheorem3.p1.7.7.m7.1.2.5" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.5.cmml"><</mo><mi id="S4.Thmtheorem3.p1.7.7.m7.1.2.6" mathvariant="normal" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.6.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.7.7.m7.1b"><apply id="S4.Thmtheorem3.p1.7.7.m7.1.2.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2"><and id="S4.Thmtheorem3.p1.7.7.m7.1.2a.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2"></and><apply id="S4.Thmtheorem3.p1.7.7.m7.1.2b.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2"><lt id="S4.Thmtheorem3.p1.7.7.m7.1.2.3.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.3"></lt><apply id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2"><plus id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.1"></plus><ci id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.2.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.2">𝐵</ci><apply id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.1.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.2"><abs id="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.1.1.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.2.3.2.1"></abs><ci id="S4.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.1">𝑉</ci></apply></apply><ci id="S4.Thmtheorem3.p1.7.7.m7.1.2.4.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.4">𝑐</ci></apply><apply id="S4.Thmtheorem3.p1.7.7.m7.1.2c.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2"><lt id="S4.Thmtheorem3.p1.7.7.m7.1.2.5.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.5"></lt><share href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem3.p1.7.7.m7.1.2.4.cmml" id="S4.Thmtheorem3.p1.7.7.m7.1.2d.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2"></share><infinity id="S4.Thmtheorem3.p1.7.7.m7.1.2.6.cmml" xref="S4.Thmtheorem3.p1.7.7.m7.1.2.6"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.7.7.m7.1c">B+|V|<c<\infty</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.7.7.m7.1d">italic_B + | italic_V | < italic_c < ∞</annotation></semantics></math>. Let <math alttext="\rho\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.8.8.m8.1"><semantics id="S4.Thmtheorem3.p1.8.8.m8.1a"><mrow id="S4.Thmtheorem3.p1.8.8.m8.1.1" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.cmml"><mi id="S4.Thmtheorem3.p1.8.8.m8.1.1.2" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.2.cmml">ρ</mi><mo id="S4.Thmtheorem3.p1.8.8.m8.1.1.1" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem3.p1.8.8.m8.1.1.3" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.2" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.cmml"><mi id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.2" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.3" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.8.8.m8.1b"><apply id="S4.Thmtheorem3.p1.8.8.m8.1.1.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1"><in id="S4.Thmtheorem3.p1.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.1"></in><ci id="S4.Thmtheorem3.p1.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.2">𝜌</ci><apply id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.2a.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.2">Wit</mtext></ci><apply id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.1.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.2.cmml" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.8.8.m8.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.8.8.m8.1c">\rho\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.8.8.m8.1d">italic_ρ ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be such that <math alttext="\textsf{cost}({\rho})=c" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.9.9.m9.1"><semantics id="S4.Thmtheorem3.p1.9.9.m9.1a"><mrow id="S4.Thmtheorem3.p1.9.9.m9.1.2" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.cmml"><mrow id="S4.Thmtheorem3.p1.9.9.m9.1.2.2" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.2" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.2a.cmml">cost</mtext><mo id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.1" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.3.2" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.cmml"><mo id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem3.p1.9.9.m9.1.1" xref="S4.Thmtheorem3.p1.9.9.m9.1.1.cmml">ρ</mi><mo id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p1.9.9.m9.1.2.1" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.1.cmml">=</mo><mi id="S4.Thmtheorem3.p1.9.9.m9.1.2.3" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.9.9.m9.1b"><apply id="S4.Thmtheorem3.p1.9.9.m9.1.2.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.2"><eq id="S4.Thmtheorem3.p1.9.9.m9.1.2.1.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.1"></eq><apply id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2"><times id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.1"></times><ci id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.2a.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.9.9.m9.1.2.2.2.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.2.2">cost</mtext></ci><ci id="S4.Thmtheorem3.p1.9.9.m9.1.1.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.1">𝜌</ci></apply><ci id="S4.Thmtheorem3.p1.9.9.m9.1.2.3.cmml" xref="S4.Thmtheorem3.p1.9.9.m9.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.9.9.m9.1c">\textsf{cost}({\rho})=c</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.9.9.m9.1d">cost ( italic_ρ ) = italic_c</annotation></semantics></math>. Let <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.10.10.m10.1"><semantics id="S4.Thmtheorem3.p1.10.10.m10.1a"><mi id="S4.Thmtheorem3.p1.10.10.m10.1.1" xref="S4.Thmtheorem3.p1.10.10.m10.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.10.10.m10.1b"><ci id="S4.Thmtheorem3.p1.10.10.m10.1.1.cmml" xref="S4.Thmtheorem3.p1.10.10.m10.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.10.10.m10.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.10.10.m10.1d">italic_h</annotation></semantics></math> be the history of maximal length such that <math alttext="h\rho" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.11.11.m11.1"><semantics id="S4.Thmtheorem3.p1.11.11.m11.1a"><mrow id="S4.Thmtheorem3.p1.11.11.m11.1.1" xref="S4.Thmtheorem3.p1.11.11.m11.1.1.cmml"><mi id="S4.Thmtheorem3.p1.11.11.m11.1.1.2" xref="S4.Thmtheorem3.p1.11.11.m11.1.1.2.cmml">h</mi><mo id="S4.Thmtheorem3.p1.11.11.m11.1.1.1" xref="S4.Thmtheorem3.p1.11.11.m11.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem3.p1.11.11.m11.1.1.3" xref="S4.Thmtheorem3.p1.11.11.m11.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.11.11.m11.1b"><apply id="S4.Thmtheorem3.p1.11.11.m11.1.1.cmml" xref="S4.Thmtheorem3.p1.11.11.m11.1.1"><times id="S4.Thmtheorem3.p1.11.11.m11.1.1.1.cmml" xref="S4.Thmtheorem3.p1.11.11.m11.1.1.1"></times><ci id="S4.Thmtheorem3.p1.11.11.m11.1.1.2.cmml" xref="S4.Thmtheorem3.p1.11.11.m11.1.1.2">ℎ</ci><ci id="S4.Thmtheorem3.p1.11.11.m11.1.1.3.cmml" xref="S4.Thmtheorem3.p1.11.11.m11.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.11.11.m11.1c">h\rho</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.11.11.m11.1d">italic_h italic_ρ</annotation></semantics></math> and <math alttext="w(h)=B" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.12.12.m12.1"><semantics id="S4.Thmtheorem3.p1.12.12.m12.1a"><mrow id="S4.Thmtheorem3.p1.12.12.m12.1.2" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.cmml"><mrow id="S4.Thmtheorem3.p1.12.12.m12.1.2.2" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.cmml"><mi id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.2" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.2.cmml">w</mi><mo id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.1" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.3.2" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.cmml"><mo id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem3.p1.12.12.m12.1.1" xref="S4.Thmtheorem3.p1.12.12.m12.1.1.cmml">h</mi><mo id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p1.12.12.m12.1.2.1" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.1.cmml">=</mo><mi id="S4.Thmtheorem3.p1.12.12.m12.1.2.3" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.12.12.m12.1b"><apply id="S4.Thmtheorem3.p1.12.12.m12.1.2.cmml" xref="S4.Thmtheorem3.p1.12.12.m12.1.2"><eq id="S4.Thmtheorem3.p1.12.12.m12.1.2.1.cmml" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.1"></eq><apply id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.cmml" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2"><times id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.1"></times><ci id="S4.Thmtheorem3.p1.12.12.m12.1.2.2.2.cmml" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.2.2">𝑤</ci><ci id="S4.Thmtheorem3.p1.12.12.m12.1.1.cmml" xref="S4.Thmtheorem3.p1.12.12.m12.1.1">ℎ</ci></apply><ci id="S4.Thmtheorem3.p1.12.12.m12.1.2.3.cmml" xref="S4.Thmtheorem3.p1.12.12.m12.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.12.12.m12.1c">w(h)=B</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.12.12.m12.1d">italic_w ( italic_h ) = italic_B</annotation></semantics></math>. Notice that <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.13.13.m13.1"><semantics id="S4.Thmtheorem3.p1.13.13.m13.1a"><mi id="S4.Thmtheorem3.p1.13.13.m13.1.1" xref="S4.Thmtheorem3.p1.13.13.m13.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.13.13.m13.1b"><ci id="S4.Thmtheorem3.p1.13.13.m13.1.1.cmml" xref="S4.Thmtheorem3.p1.13.13.m13.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.13.13.m13.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.13.13.m13.1d">italic_h</annotation></semantics></math> exists as the arena is binary and <math alttext="B+|V|<c<\infty" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.14.14.m14.1"><semantics id="S4.Thmtheorem3.p1.14.14.m14.1a"><mrow id="S4.Thmtheorem3.p1.14.14.m14.1.2" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.cmml"><mrow id="S4.Thmtheorem3.p1.14.14.m14.1.2.2" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.cmml"><mi id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.2" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.2.cmml">B</mi><mo id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.1" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.1.cmml">+</mo><mrow id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.2" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.1.cmml"><mo id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem3.p1.14.14.m14.1.1" xref="S4.Thmtheorem3.p1.14.14.m14.1.1.cmml">V</mi><mo id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.1.1.cmml">|</mo></mrow></mrow><mo id="S4.Thmtheorem3.p1.14.14.m14.1.2.3" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.3.cmml"><</mo><mi id="S4.Thmtheorem3.p1.14.14.m14.1.2.4" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.4.cmml">c</mi><mo id="S4.Thmtheorem3.p1.14.14.m14.1.2.5" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.5.cmml"><</mo><mi id="S4.Thmtheorem3.p1.14.14.m14.1.2.6" mathvariant="normal" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.6.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.14.14.m14.1b"><apply id="S4.Thmtheorem3.p1.14.14.m14.1.2.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2"><and id="S4.Thmtheorem3.p1.14.14.m14.1.2a.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2"></and><apply id="S4.Thmtheorem3.p1.14.14.m14.1.2b.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2"><lt id="S4.Thmtheorem3.p1.14.14.m14.1.2.3.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.3"></lt><apply id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2"><plus id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.1"></plus><ci id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.2.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.2">𝐵</ci><apply id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.1.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.2"><abs id="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.1.1.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.2.3.2.1"></abs><ci id="S4.Thmtheorem3.p1.14.14.m14.1.1.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.1">𝑉</ci></apply></apply><ci id="S4.Thmtheorem3.p1.14.14.m14.1.2.4.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.4">𝑐</ci></apply><apply id="S4.Thmtheorem3.p1.14.14.m14.1.2c.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2"><lt id="S4.Thmtheorem3.p1.14.14.m14.1.2.5.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.5"></lt><share href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem3.p1.14.14.m14.1.2.4.cmml" id="S4.Thmtheorem3.p1.14.14.m14.1.2d.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2"></share><infinity id="S4.Thmtheorem3.p1.14.14.m14.1.2.6.cmml" xref="S4.Thmtheorem3.p1.14.14.m14.1.2.6"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.14.14.m14.1c">B+|V|<c<\infty</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.14.14.m14.1d">italic_B + | italic_V | < italic_c < ∞</annotation></semantics></math>. Since <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.15.15.m15.2"><semantics id="S4.Thmtheorem3.p1.15.15.m15.2a"><mrow id="S4.Thmtheorem3.p1.15.15.m15.2.3" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.cmml"><msub id="S4.Thmtheorem3.p1.15.15.m15.2.3.2" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.2.cmml"><mi id="S4.Thmtheorem3.p1.15.15.m15.2.3.2.2" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem3.p1.15.15.m15.2.3.2.3" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem3.p1.15.15.m15.2.3.1" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem3.p1.15.15.m15.2.3.3" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.2" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.1" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.2" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem3.p1.15.15.m15.1.1" xref="S4.Thmtheorem3.p1.15.15.m15.1.1.cmml">G</mi><mo id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.2.2" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem3.p1.15.15.m15.2.2" xref="S4.Thmtheorem3.p1.15.15.m15.2.2.cmml">B</mi><mo id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.15.15.m15.2b"><apply id="S4.Thmtheorem3.p1.15.15.m15.2.3.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3"><in id="S4.Thmtheorem3.p1.15.15.m15.2.3.1.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.1"></in><apply id="S4.Thmtheorem3.p1.15.15.m15.2.3.2.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.15.15.m15.2.3.2.1.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem3.p1.15.15.m15.2.3.2.2.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem3.p1.15.15.m15.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3"><times id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.1"></times><ci id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.2a.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.2.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.1.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.3.3.3.2"><ci id="S4.Thmtheorem3.p1.15.15.m15.1.1.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.1.1">𝐺</ci><ci id="S4.Thmtheorem3.p1.15.15.m15.2.2.cmml" xref="S4.Thmtheorem3.p1.15.15.m15.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.15.15.m15.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.15.15.m15.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> and <math alttext="\rho\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.16.16.m16.1"><semantics id="S4.Thmtheorem3.p1.16.16.m16.1a"><mrow id="S4.Thmtheorem3.p1.16.16.m16.1.1" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.cmml"><mi id="S4.Thmtheorem3.p1.16.16.m16.1.1.2" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.2.cmml">ρ</mi><mo id="S4.Thmtheorem3.p1.16.16.m16.1.1.1" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem3.p1.16.16.m16.1.1.3" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.2" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.cmml"><mi id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.2" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.3" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.16.16.m16.1b"><apply id="S4.Thmtheorem3.p1.16.16.m16.1.1.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1"><in id="S4.Thmtheorem3.p1.16.16.m16.1.1.1.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.1"></in><ci id="S4.Thmtheorem3.p1.16.16.m16.1.1.2.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.2">𝜌</ci><apply id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.2a.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.2">Wit</mtext></ci><apply id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.1.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.2.cmml" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.16.16.m16.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.16.16.m16.1c">\rho\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.16.16.m16.1d">italic_ρ ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, Player 0’s target is visited by <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.17.17.m17.1"><semantics id="S4.Thmtheorem3.p1.17.17.m17.1a"><mi id="S4.Thmtheorem3.p1.17.17.m17.1.1" xref="S4.Thmtheorem3.p1.17.17.m17.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.17.17.m17.1b"><ci id="S4.Thmtheorem3.p1.17.17.m17.1.1.cmml" xref="S4.Thmtheorem3.p1.17.17.m17.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.17.17.m17.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.17.17.m17.1d">italic_h</annotation></semantics></math> and Player <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.18.18.m18.1"><semantics id="S4.Thmtheorem3.p1.18.18.m18.1a"><mn id="S4.Thmtheorem3.p1.18.18.m18.1.1" xref="S4.Thmtheorem3.p1.18.18.m18.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.18.18.m18.1b"><cn id="S4.Thmtheorem3.p1.18.18.m18.1.1.cmml" type="integer" xref="S4.Thmtheorem3.p1.18.18.m18.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.18.18.m18.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.18.18.m18.1d">1</annotation></semantics></math>’s target is visited by <math alttext="\rho" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.19.19.m19.1"><semantics id="S4.Thmtheorem3.p1.19.19.m19.1a"><mi id="S4.Thmtheorem3.p1.19.19.m19.1.1" xref="S4.Thmtheorem3.p1.19.19.m19.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.19.19.m19.1b"><ci id="S4.Thmtheorem3.p1.19.19.m19.1.1.cmml" xref="S4.Thmtheorem3.p1.19.19.m19.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.19.19.m19.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.19.19.m19.1d">italic_ρ</annotation></semantics></math> at least <math alttext="|V|+1" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.20.20.m20.1"><semantics id="S4.Thmtheorem3.p1.20.20.m20.1a"><mrow id="S4.Thmtheorem3.p1.20.20.m20.1.2" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.cmml"><mrow id="S4.Thmtheorem3.p1.20.20.m20.1.2.2.2" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.2.1.cmml"><mo id="S4.Thmtheorem3.p1.20.20.m20.1.2.2.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.2.1.1.cmml">|</mo><mi id="S4.Thmtheorem3.p1.20.20.m20.1.1" xref="S4.Thmtheorem3.p1.20.20.m20.1.1.cmml">V</mi><mo id="S4.Thmtheorem3.p1.20.20.m20.1.2.2.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem3.p1.20.20.m20.1.2.1" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.1.cmml">+</mo><mn id="S4.Thmtheorem3.p1.20.20.m20.1.2.3" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.20.20.m20.1b"><apply id="S4.Thmtheorem3.p1.20.20.m20.1.2.cmml" xref="S4.Thmtheorem3.p1.20.20.m20.1.2"><plus id="S4.Thmtheorem3.p1.20.20.m20.1.2.1.cmml" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.1"></plus><apply id="S4.Thmtheorem3.p1.20.20.m20.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.2.2"><abs id="S4.Thmtheorem3.p1.20.20.m20.1.2.2.1.1.cmml" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.2.2.1"></abs><ci id="S4.Thmtheorem3.p1.20.20.m20.1.1.cmml" xref="S4.Thmtheorem3.p1.20.20.m20.1.1">𝑉</ci></apply><cn id="S4.Thmtheorem3.p1.20.20.m20.1.2.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.20.20.m20.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.20.20.m20.1c">|V|+1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.20.20.m20.1d">| italic_V | + 1</annotation></semantics></math> vertices after <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.21.21.m21.1"><semantics id="S4.Thmtheorem3.p1.21.21.m21.1a"><mi id="S4.Thmtheorem3.p1.21.21.m21.1.1" xref="S4.Thmtheorem3.p1.21.21.m21.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.21.21.m21.1b"><ci id="S4.Thmtheorem3.p1.21.21.m21.1.1.cmml" xref="S4.Thmtheorem3.p1.21.21.m21.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.21.21.m21.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.21.21.m21.1d">italic_h</annotation></semantics></math>. Hence, <math alttext="\rho" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.22.22.m22.1"><semantics id="S4.Thmtheorem3.p1.22.22.m22.1a"><mi id="S4.Thmtheorem3.p1.22.22.m22.1.1" xref="S4.Thmtheorem3.p1.22.22.m22.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.22.22.m22.1b"><ci id="S4.Thmtheorem3.p1.22.22.m22.1.1.cmml" xref="S4.Thmtheorem3.p1.22.22.m22.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.22.22.m22.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.22.22.m22.1d">italic_ρ</annotation></semantics></math> performs a cycle between the two visits, that satisfies the hypotheses of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>. We can thus eliminate this cycle and create a better solution. We repeat this process until <math alttext="c\leq B+|V|" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.23.23.m23.1"><semantics id="S4.Thmtheorem3.p1.23.23.m23.1a"><mrow id="S4.Thmtheorem3.p1.23.23.m23.1.2" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.cmml"><mi id="S4.Thmtheorem3.p1.23.23.m23.1.2.2" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.2.cmml">c</mi><mo id="S4.Thmtheorem3.p1.23.23.m23.1.2.1" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.1.cmml">≤</mo><mrow id="S4.Thmtheorem3.p1.23.23.m23.1.2.3" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.cmml"><mi id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.2" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.2.cmml">B</mi><mo id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.1" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.1.cmml">+</mo><mrow id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.2" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.1.cmml"><mo id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem3.p1.23.23.m23.1.1" xref="S4.Thmtheorem3.p1.23.23.m23.1.1.cmml">V</mi><mo id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.1.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.23.23.m23.1b"><apply id="S4.Thmtheorem3.p1.23.23.m23.1.2.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2"><leq id="S4.Thmtheorem3.p1.23.23.m23.1.2.1.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.1"></leq><ci id="S4.Thmtheorem3.p1.23.23.m23.1.2.2.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.2">𝑐</ci><apply id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3"><plus id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.1.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.1"></plus><ci id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.2.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.2">𝐵</ci><apply id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.2"><abs id="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.1.1.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.2.3.3.2.1"></abs><ci id="S4.Thmtheorem3.p1.23.23.m23.1.1.cmml" xref="S4.Thmtheorem3.p1.23.23.m23.1.1">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.23.23.m23.1c">c\leq B+|V|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.23.23.m23.1d">italic_c ≤ italic_B + | italic_V |</annotation></semantics></math> and the termination is guaranteed by the strict reduction of <math alttext="\textsf{length}({\textsf{Wit}_{\sigma_{0}}})" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.24.24.m24.1"><semantics id="S4.Thmtheorem3.p1.24.24.m24.1a"><mrow id="S4.Thmtheorem3.p1.24.24.m24.1.1" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.24.24.m24.1.1.3" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.3a.cmml">length</mtext><mo id="S4.Thmtheorem3.p1.24.24.m24.1.1.2" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.2" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.2" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.3" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.24.24.m24.1b"><apply id="S4.Thmtheorem3.p1.24.24.m24.1.1.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1"><times id="S4.Thmtheorem3.p1.24.24.m24.1.1.2.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.2"></times><ci id="S4.Thmtheorem3.p1.24.24.m24.1.1.3a.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.24.24.m24.1.1.3.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.3">length</mtext></ci><apply id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.2a.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.2">Wit</mtext></ci><apply id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.24.24.m24.1.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.24.24.m24.1c">\textsf{length}({\textsf{Wit}_{\sigma_{0}}})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.24.24.m24.1d">length ( Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math> (by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>). In this way, Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem2" title="Lemma 4.2 ‣ 4.1 Dimension One ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.2</span></a> is established.</span></p> </div> </div> </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>Bounding the Pareto-Optimal Costs by Induction</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.5">We now proceed to the case of dimension <math alttext="t+1" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mrow id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">t</mi><mo id="S4.SS2.p1.1.m1.1.1.1" xref="S4.SS2.p1.1.m1.1.1.1.cmml">+</mo><mn id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">1</mn></mrow><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"><plus id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1.1"></plus><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝑡</ci><cn id="S4.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_t + 1</annotation></semantics></math>, with <math alttext="t\geq 1" 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">t</mi><mo id="S4.SS2.p1.2.m2.1.1.1" xref="S4.SS2.p1.2.m2.1.1.1.cmml">≥</mo><mn id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml">1</mn></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"><geq id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1.1"></geq><ci id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2">𝑡</ci><cn id="S4.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_t ≥ 1</annotation></semantics></math>. The next lemma is proved by using the <em class="ltx_emph ltx_font_italic" id="S4.SS2.p1.5.1">induction hypothesis</em> (that is, we suppose that Equation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E3" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3</span></a>) holds for dimension <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><mi id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><ci id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_t</annotation></semantics></math>). Recall that <math alttext="c_{min}" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><msub id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.2" xref="S4.SS2.p1.4.m4.1.1.2.cmml">c</mi><mrow id="S4.SS2.p1.4.m4.1.1.3" xref="S4.SS2.p1.4.m4.1.1.3.cmml"><mi id="S4.SS2.p1.4.m4.1.1.3.2" xref="S4.SS2.p1.4.m4.1.1.3.2.cmml">m</mi><mo id="S4.SS2.p1.4.m4.1.1.3.1" xref="S4.SS2.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.4.m4.1.1.3.3" xref="S4.SS2.p1.4.m4.1.1.3.3.cmml">i</mi><mo id="S4.SS2.p1.4.m4.1.1.3.1a" xref="S4.SS2.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.4.m4.1.1.3.4" xref="S4.SS2.p1.4.m4.1.1.3.4.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><apply id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.p1.4.m4.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.2">𝑐</ci><apply id="S4.SS2.p1.4.m4.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3"><times id="S4.SS2.p1.4.m4.1.1.3.1.cmml" xref="S4.SS2.p1.4.m4.1.1.3.1"></times><ci id="S4.SS2.p1.4.m4.1.1.3.2.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2">𝑚</ci><ci id="S4.SS2.p1.4.m4.1.1.3.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3.3">𝑖</ci><ci id="S4.SS2.p1.4.m4.1.1.3.4.cmml" xref="S4.SS2.p1.4.m4.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">c_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.1d">italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is the minimum component of the cost <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.1"><semantics id="S4.SS2.p1.5.m5.1a"><mi id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.1b"><ci id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">italic_c</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Lemma 4.4</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.6.6">Let <math alttext="G\in\textsf{BinGames}_{t+1}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.1.m1.1"><semantics id="S4.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem4.p1.1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2a.cmml">BinGames</mtext><mrow id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml"><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.2" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.2.cmml">t</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.1.cmml">+</mo><mn id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.1.m1.1b"><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1"><in id="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.2">𝐺</ci><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2a.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2">BinGames</mtext></ci><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3"><plus id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.1"></plus><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.2">𝑡</ci><cn id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.1.m1.1c">G\in\textsf{BinGames}_{t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.1.m1.1d">italic_G ∈ BinGames start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a binary SP game with dimension <math alttext="t+1" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem4.p1.2.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.2.m2.1b"><apply id="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1"><plus id="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1"></plus><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2">𝑡</ci><cn id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.2.m2.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.2.m2.1d">italic_t + 1</annotation></semantics></math>, <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.3.3.m3.1"><semantics id="S4.Thmtheorem4.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem4.p1.3.3.m3.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem4.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.2.cmml">B</mi><mo id="S4.Thmtheorem4.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem4.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.3.3.m3.1b"><apply id="S4.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.1.1"><in id="S4.Thmtheorem4.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.1"></in><ci id="S4.Thmtheorem4.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.2">𝐵</ci><ci id="S4.Thmtheorem4.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.3.m3.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.3.m3.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.4.m4.2"><semantics id="S4.Thmtheorem4.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem4.p1.4.4.m4.2.3" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.cmml"><msub id="S4.Thmtheorem4.p1.4.4.m4.2.3.2" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.2.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.2.3.2.2" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem4.p1.4.4.m4.2.3.2.3" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem4.p1.4.4.m4.2.3.1" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem4.p1.4.4.m4.2.3.3" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.2" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.1" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.2" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml">G</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.2.2" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.4.4.m4.2.2" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.cmml">B</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.4.m4.2b"><apply id="S4.Thmtheorem4.p1.4.4.m4.2.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3"><in id="S4.Thmtheorem4.p1.4.4.m4.2.3.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.1"></in><apply id="S4.Thmtheorem4.p1.4.4.m4.2.3.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.4.4.m4.2.3.2.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.4.4.m4.2.3.2.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem4.p1.4.4.m4.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3"><times id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.1"></times><ci id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.2a.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.3.3.3.2"><ci id="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1">𝐺</ci><ci id="S4.Thmtheorem4.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.4.m4.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. Then there exists a solution <math alttext="\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.5.5.m5.2"><semantics id="S4.Thmtheorem4.p1.5.5.m5.2a"><mrow id="S4.Thmtheorem4.p1.5.5.m5.2.3" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.cmml"><msubsup id="S4.Thmtheorem4.p1.5.5.m5.2.3.2" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2.cmml"><mi id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.2" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.3" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.3.cmml">0</mn><mo id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.3" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem4.p1.5.5.m5.2.3.1" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem4.p1.5.5.m5.2.3.3" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.2" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.1" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.2" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.5.5.m5.1.1" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.cmml">G</mi><mo id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.2.2" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.5.5.m5.2.2" xref="S4.Thmtheorem4.p1.5.5.m5.2.2.cmml">B</mi><mo id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.5.5.m5.2b"><apply id="S4.Thmtheorem4.p1.5.5.m5.2.3.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3"><in id="S4.Thmtheorem4.p1.5.5.m5.2.3.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.1"></in><apply id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2">superscript</csymbol><apply id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.2">𝜎</ci><cn id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2.2.3">0</cn></apply><ci id="S4.Thmtheorem4.p1.5.5.m5.2.3.2.3.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.2.3">′</ci></apply><apply id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3"><times id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.1"></times><ci id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.2a.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.3.3.3.2"><ci id="S4.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1">𝐺</ci><ci id="S4.Thmtheorem4.p1.5.5.m5.2.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.5.5.m5.2c">\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.5.5.m5.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> without cycles such that <math alttext="\sigma_{0}^{\prime}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.6.6.m6.1"><semantics id="S4.Thmtheorem4.p1.6.6.m6.1a"><mrow id="S4.Thmtheorem4.p1.6.6.m6.1.1" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.cmml"><msubsup id="S4.Thmtheorem4.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2.cmml"><mi id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.2" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.3" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.3.cmml">0</mn><mo id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.3" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem4.p1.6.6.m6.1.1.1" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.1.cmml">⪯</mo><msub id="S4.Thmtheorem4.p1.6.6.m6.1.1.3" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.3.cmml"><mi id="S4.Thmtheorem4.p1.6.6.m6.1.1.3.2" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem4.p1.6.6.m6.1.1.3.3" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.6.6.m6.1b"><apply id="S4.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1"><csymbol cd="latexml" id="S4.Thmtheorem4.p1.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.1">precedes-or-equals</csymbol><apply id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2">superscript</csymbol><apply id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.1.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.2.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.2">𝜎</ci><cn id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2.2.3">0</cn></apply><ci id="S4.Thmtheorem4.p1.6.6.m6.1.1.2.3.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.2.3">′</ci></apply><apply id="S4.Thmtheorem4.p1.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.6.6.m6.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem4.p1.6.6.m6.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem4.p1.6.6.m6.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.6.6.m6.1c">\sigma_{0}^{\prime}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.6.6.m6.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, and</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx7"> <tbody id="S4.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\forall c^{\prime}\in C_{\sigma_{0}^{\prime}},\forall i\in\{1,% \ldots,t+1\}:~{}~{}c^{\prime}_{i}\leq\max\{c^{\prime}_{min},B\}+1+f({0},{t})~{% }~{}\vee~{}~{}c^{\prime}_{i}=\infty." class="ltx_Math" display="inline" id="S4.E7.m1.7"><semantics id="S4.E7.m1.7a"><mrow id="S4.E7.m1.7.7.1" xref="S4.E7.m1.7.7.1.1.cmml"><mrow id="S4.E7.m1.7.7.1.1" xref="S4.E7.m1.7.7.1.1.cmml"><mrow id="S4.E7.m1.7.7.1.1.2.2" xref="S4.E7.m1.7.7.1.1.2.3.cmml"><mrow id="S4.E7.m1.7.7.1.1.1.1.1" xref="S4.E7.m1.7.7.1.1.1.1.1.cmml"><mrow id="S4.E7.m1.7.7.1.1.1.1.1.2" xref="S4.E7.m1.7.7.1.1.1.1.1.2.cmml"><mo id="S4.E7.m1.7.7.1.1.1.1.1.2.1" rspace="0.167em" xref="S4.E7.m1.7.7.1.1.1.1.1.2.1.cmml">∀</mo><msup id="S4.E7.m1.7.7.1.1.1.1.1.2.2" xref="S4.E7.m1.7.7.1.1.1.1.1.2.2.cmml"><mi id="S4.E7.m1.7.7.1.1.1.1.1.2.2.2" xref="S4.E7.m1.7.7.1.1.1.1.1.2.2.2.cmml">c</mi><mo id="S4.E7.m1.7.7.1.1.1.1.1.2.2.3" xref="S4.E7.m1.7.7.1.1.1.1.1.2.2.3.cmml">′</mo></msup></mrow><mo id="S4.E7.m1.7.7.1.1.1.1.1.1" xref="S4.E7.m1.7.7.1.1.1.1.1.1.cmml">∈</mo><msub id="S4.E7.m1.7.7.1.1.1.1.1.3" xref="S4.E7.m1.7.7.1.1.1.1.1.3.cmml"><mi id="S4.E7.m1.7.7.1.1.1.1.1.3.2" xref="S4.E7.m1.7.7.1.1.1.1.1.3.2.cmml">C</mi><msubsup id="S4.E7.m1.7.7.1.1.1.1.1.3.3" xref="S4.E7.m1.7.7.1.1.1.1.1.3.3.cmml"><mi id="S4.E7.m1.7.7.1.1.1.1.1.3.3.2.2" xref="S4.E7.m1.7.7.1.1.1.1.1.3.3.2.2.cmml">σ</mi><mn id="S4.E7.m1.7.7.1.1.1.1.1.3.3.2.3" xref="S4.E7.m1.7.7.1.1.1.1.1.3.3.2.3.cmml">0</mn><mo id="S4.E7.m1.7.7.1.1.1.1.1.3.3.3" xref="S4.E7.m1.7.7.1.1.1.1.1.3.3.3.cmml">′</mo></msubsup></msub></mrow><mo id="S4.E7.m1.7.7.1.1.2.2.3" xref="S4.E7.m1.7.7.1.1.2.3a.cmml">,</mo><mrow id="S4.E7.m1.7.7.1.1.2.2.2" xref="S4.E7.m1.7.7.1.1.2.2.2.cmml"><mrow id="S4.E7.m1.7.7.1.1.2.2.2.3" xref="S4.E7.m1.7.7.1.1.2.2.2.3.cmml"><mo id="S4.E7.m1.7.7.1.1.2.2.2.3.1" rspace="0.167em" xref="S4.E7.m1.7.7.1.1.2.2.2.3.1.cmml">∀</mo><mi id="S4.E7.m1.7.7.1.1.2.2.2.3.2" xref="S4.E7.m1.7.7.1.1.2.2.2.3.2.cmml">i</mi></mrow><mo id="S4.E7.m1.7.7.1.1.2.2.2.2" xref="S4.E7.m1.7.7.1.1.2.2.2.2.cmml">∈</mo><mrow id="S4.E7.m1.7.7.1.1.2.2.2.1.1" xref="S4.E7.m1.7.7.1.1.2.2.2.1.2.cmml"><mo id="S4.E7.m1.7.7.1.1.2.2.2.1.1.2" stretchy="false" xref="S4.E7.m1.7.7.1.1.2.2.2.1.2.cmml">{</mo><mn id="S4.E7.m1.1.1" xref="S4.E7.m1.1.1.cmml">1</mn><mo id="S4.E7.m1.7.7.1.1.2.2.2.1.1.3" xref="S4.E7.m1.7.7.1.1.2.2.2.1.2.cmml">,</mo><mi id="S4.E7.m1.2.2" mathvariant="normal" xref="S4.E7.m1.2.2.cmml">…</mi><mo id="S4.E7.m1.7.7.1.1.2.2.2.1.1.4" xref="S4.E7.m1.7.7.1.1.2.2.2.1.2.cmml">,</mo><mrow id="S4.E7.m1.7.7.1.1.2.2.2.1.1.1" xref="S4.E7.m1.7.7.1.1.2.2.2.1.1.1.cmml"><mi id="S4.E7.m1.7.7.1.1.2.2.2.1.1.1.2" xref="S4.E7.m1.7.7.1.1.2.2.2.1.1.1.2.cmml">t</mi><mo id="S4.E7.m1.7.7.1.1.2.2.2.1.1.1.1" xref="S4.E7.m1.7.7.1.1.2.2.2.1.1.1.1.cmml">+</mo><mn id="S4.E7.m1.7.7.1.1.2.2.2.1.1.1.3" xref="S4.E7.m1.7.7.1.1.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo 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id="S4.E7.m1.7.7.1.1.3.5.cmml" xref="S4.E7.m1.7.7.1.1.3.5"></eq><share href="https://arxiv.org/html/2308.09443v2#S4.E7.m1.7.7.1.1.3.1.cmml" id="S4.E7.m1.7.7.1.1.3d.cmml" xref="S4.E7.m1.7.7.1.1.3"></share><infinity id="S4.E7.m1.7.7.1.1.3.6.cmml" xref="S4.E7.m1.7.7.1.1.3.6"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.7c">\displaystyle\forall c^{\prime}\in C_{\sigma_{0}^{\prime}},\forall i\in\{1,% \ldots,t+1\}:~{}~{}c^{\prime}_{i}\leq\max\{c^{\prime}_{min},B\}+1+f({0},{t})~{% }~{}\vee~{}~{}c^{\prime}_{i}=\infty.</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.7d">∀ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , ∀ italic_i ∈ { 1 , … , italic_t + 1 } : italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ roman_max { italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_B } + 1 + italic_f ( 0 , italic_t ) ∨ italic_c start_POSTSUPERSCRIPT ′ 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">(7)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.15">The idea of the proof is as follows. If there exists <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mrow id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">c</mi><mo id="S4.SS2.p2.1.m1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.cmml">∈</mo><msub id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml"><mi id="S4.SS2.p2.1.m1.1.1.3.2" xref="S4.SS2.p2.1.m1.1.1.3.2.cmml">C</mi><msub id="S4.SS2.p2.1.m1.1.1.3.3" xref="S4.SS2.p2.1.m1.1.1.3.3.cmml"><mi id="S4.SS2.p2.1.m1.1.1.3.3.2" xref="S4.SS2.p2.1.m1.1.1.3.3.2.cmml">σ</mi><mn id="S4.SS2.p2.1.m1.1.1.3.3.3" xref="S4.SS2.p2.1.m1.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><in id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1"></in><ci id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2">𝑐</ci><apply id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.3.1.cmml" xref="S4.SS2.p2.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.3.2.cmml" xref="S4.SS2.p2.1.m1.1.1.3.2">𝐶</ci><apply id="S4.SS2.p2.1.m1.1.1.3.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.3.3.1.cmml" xref="S4.SS2.p2.1.m1.1.1.3.3">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.3.3.2.cmml" xref="S4.SS2.p2.1.m1.1.1.3.3.2">𝜎</ci><cn id="S4.SS2.p2.1.m1.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that for some <math alttext="i\in\{1,\ldots,t+1\}" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.3"><semantics id="S4.SS2.p2.2.m2.3a"><mrow id="S4.SS2.p2.2.m2.3.3" xref="S4.SS2.p2.2.m2.3.3.cmml"><mi id="S4.SS2.p2.2.m2.3.3.3" xref="S4.SS2.p2.2.m2.3.3.3.cmml">i</mi><mo id="S4.SS2.p2.2.m2.3.3.2" xref="S4.SS2.p2.2.m2.3.3.2.cmml">∈</mo><mrow id="S4.SS2.p2.2.m2.3.3.1.1" xref="S4.SS2.p2.2.m2.3.3.1.2.cmml"><mo id="S4.SS2.p2.2.m2.3.3.1.1.2" stretchy="false" xref="S4.SS2.p2.2.m2.3.3.1.2.cmml">{</mo><mn id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">1</mn><mo id="S4.SS2.p2.2.m2.3.3.1.1.3" xref="S4.SS2.p2.2.m2.3.3.1.2.cmml">,</mo><mi id="S4.SS2.p2.2.m2.2.2" mathvariant="normal" xref="S4.SS2.p2.2.m2.2.2.cmml">…</mi><mo id="S4.SS2.p2.2.m2.3.3.1.1.4" xref="S4.SS2.p2.2.m2.3.3.1.2.cmml">,</mo><mrow id="S4.SS2.p2.2.m2.3.3.1.1.1" xref="S4.SS2.p2.2.m2.3.3.1.1.1.cmml"><mi id="S4.SS2.p2.2.m2.3.3.1.1.1.2" xref="S4.SS2.p2.2.m2.3.3.1.1.1.2.cmml">t</mi><mo id="S4.SS2.p2.2.m2.3.3.1.1.1.1" xref="S4.SS2.p2.2.m2.3.3.1.1.1.1.cmml">+</mo><mn 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xref="S4.SS2.p2.2.m2.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.3c">i\in\{1,\ldots,t+1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.3d">italic_i ∈ { 1 , … , italic_t + 1 }</annotation></semantics></math>, <math alttext="c_{i}" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><msub id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml"><mi id="S4.SS2.p2.3.m3.1.1.2" xref="S4.SS2.p2.3.m3.1.1.2.cmml">c</mi><mi id="S4.SS2.p2.3.m3.1.1.3" xref="S4.SS2.p2.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.1.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2">𝑐</ci><ci id="S4.SS2.p2.3.m3.1.1.3.cmml" xref="S4.SS2.p2.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">c_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> does not satisfy (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E7" title="In Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">7</span></a>), then we consider a witness <math alttext="\rho" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><mi id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><ci id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">italic_ρ</annotation></semantics></math> with <math alttext="\textsf{cost}({\rho})=c" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.1"><semantics id="S4.SS2.p2.5.m5.1a"><mrow id="S4.SS2.p2.5.m5.1.2" xref="S4.SS2.p2.5.m5.1.2.cmml"><mrow id="S4.SS2.p2.5.m5.1.2.2" xref="S4.SS2.p2.5.m5.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.SS2.p2.5.m5.1.2.2.2" xref="S4.SS2.p2.5.m5.1.2.2.2a.cmml">cost</mtext><mo id="S4.SS2.p2.5.m5.1.2.2.1" xref="S4.SS2.p2.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p2.5.m5.1.2.2.3.2" xref="S4.SS2.p2.5.m5.1.2.2.cmml"><mo id="S4.SS2.p2.5.m5.1.2.2.3.2.1" stretchy="false" xref="S4.SS2.p2.5.m5.1.2.2.cmml">(</mo><mi id="S4.SS2.p2.5.m5.1.1" xref="S4.SS2.p2.5.m5.1.1.cmml">ρ</mi><mo id="S4.SS2.p2.5.m5.1.2.2.3.2.2" stretchy="false" xref="S4.SS2.p2.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p2.5.m5.1.2.1" xref="S4.SS2.p2.5.m5.1.2.1.cmml">=</mo><mi id="S4.SS2.p2.5.m5.1.2.3" xref="S4.SS2.p2.5.m5.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.1b"><apply id="S4.SS2.p2.5.m5.1.2.cmml" xref="S4.SS2.p2.5.m5.1.2"><eq id="S4.SS2.p2.5.m5.1.2.1.cmml" xref="S4.SS2.p2.5.m5.1.2.1"></eq><apply id="S4.SS2.p2.5.m5.1.2.2.cmml" xref="S4.SS2.p2.5.m5.1.2.2"><times id="S4.SS2.p2.5.m5.1.2.2.1.cmml" xref="S4.SS2.p2.5.m5.1.2.2.1"></times><ci id="S4.SS2.p2.5.m5.1.2.2.2a.cmml" xref="S4.SS2.p2.5.m5.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.SS2.p2.5.m5.1.2.2.2.cmml" xref="S4.SS2.p2.5.m5.1.2.2.2">cost</mtext></ci><ci id="S4.SS2.p2.5.m5.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1">𝜌</ci></apply><ci id="S4.SS2.p2.5.m5.1.2.3.cmml" xref="S4.SS2.p2.5.m5.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.1c">\textsf{cost}({\rho})=c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.1d">cost ( italic_ρ ) = italic_c</annotation></semantics></math> and the history <math alttext="h" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.1"><semantics id="S4.SS2.p2.6.m6.1a"><mi id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.1b"><ci id="S4.SS2.p2.6.m6.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.1d">italic_h</annotation></semantics></math> of minimal length such that <math alttext="h\rho" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.1"><semantics id="S4.SS2.p2.7.m7.1a"><mrow id="S4.SS2.p2.7.m7.1.1" xref="S4.SS2.p2.7.m7.1.1.cmml"><mi id="S4.SS2.p2.7.m7.1.1.2" xref="S4.SS2.p2.7.m7.1.1.2.cmml">h</mi><mo id="S4.SS2.p2.7.m7.1.1.1" xref="S4.SS2.p2.7.m7.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.7.m7.1.1.3" xref="S4.SS2.p2.7.m7.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.1b"><apply id="S4.SS2.p2.7.m7.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1"><times id="S4.SS2.p2.7.m7.1.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1.1"></times><ci id="S4.SS2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2">ℎ</ci><ci id="S4.SS2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.p2.7.m7.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.1c">h\rho</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.1d">italic_h italic_ρ</annotation></semantics></math> and <math alttext="w(h)=\max\{c_{min},B\}+1" class="ltx_Math" display="inline" id="S4.SS2.p2.8.m8.4"><semantics id="S4.SS2.p2.8.m8.4a"><mrow id="S4.SS2.p2.8.m8.4.4" xref="S4.SS2.p2.8.m8.4.4.cmml"><mrow id="S4.SS2.p2.8.m8.4.4.3" xref="S4.SS2.p2.8.m8.4.4.3.cmml"><mi id="S4.SS2.p2.8.m8.4.4.3.2" xref="S4.SS2.p2.8.m8.4.4.3.2.cmml">w</mi><mo id="S4.SS2.p2.8.m8.4.4.3.1" xref="S4.SS2.p2.8.m8.4.4.3.1.cmml">⁢</mo><mrow id="S4.SS2.p2.8.m8.4.4.3.3.2" xref="S4.SS2.p2.8.m8.4.4.3.cmml"><mo id="S4.SS2.p2.8.m8.4.4.3.3.2.1" stretchy="false" xref="S4.SS2.p2.8.m8.4.4.3.cmml">(</mo><mi id="S4.SS2.p2.8.m8.1.1" xref="S4.SS2.p2.8.m8.1.1.cmml">h</mi><mo id="S4.SS2.p2.8.m8.4.4.3.3.2.2" stretchy="false" xref="S4.SS2.p2.8.m8.4.4.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p2.8.m8.4.4.2" xref="S4.SS2.p2.8.m8.4.4.2.cmml">=</mo><mrow id="S4.SS2.p2.8.m8.4.4.1" xref="S4.SS2.p2.8.m8.4.4.1.cmml"><mrow id="S4.SS2.p2.8.m8.4.4.1.1.1" xref="S4.SS2.p2.8.m8.4.4.1.1.2.cmml"><mi id="S4.SS2.p2.8.m8.2.2" xref="S4.SS2.p2.8.m8.2.2.cmml">max</mi><mo id="S4.SS2.p2.8.m8.4.4.1.1.1a" xref="S4.SS2.p2.8.m8.4.4.1.1.2.cmml">⁡</mo><mrow id="S4.SS2.p2.8.m8.4.4.1.1.1.1" xref="S4.SS2.p2.8.m8.4.4.1.1.2.cmml"><mo id="S4.SS2.p2.8.m8.4.4.1.1.1.1.2" stretchy="false" xref="S4.SS2.p2.8.m8.4.4.1.1.2.cmml">{</mo><msub id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.cmml"><mi id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.2" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.2.cmml">c</mi><mrow id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.cmml"><mi id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.2" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.2.cmml">m</mi><mo id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.1" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.3" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.3.cmml">i</mi><mo id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.1a" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.4" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S4.SS2.p2.8.m8.4.4.1.1.1.1.3" xref="S4.SS2.p2.8.m8.4.4.1.1.2.cmml">,</mo><mi id="S4.SS2.p2.8.m8.3.3" xref="S4.SS2.p2.8.m8.3.3.cmml">B</mi><mo id="S4.SS2.p2.8.m8.4.4.1.1.1.1.4" stretchy="false" xref="S4.SS2.p2.8.m8.4.4.1.1.2.cmml">}</mo></mrow></mrow><mo id="S4.SS2.p2.8.m8.4.4.1.2" xref="S4.SS2.p2.8.m8.4.4.1.2.cmml">+</mo><mn id="S4.SS2.p2.8.m8.4.4.1.3" xref="S4.SS2.p2.8.m8.4.4.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.8.m8.4b"><apply id="S4.SS2.p2.8.m8.4.4.cmml" xref="S4.SS2.p2.8.m8.4.4"><eq id="S4.SS2.p2.8.m8.4.4.2.cmml" xref="S4.SS2.p2.8.m8.4.4.2"></eq><apply id="S4.SS2.p2.8.m8.4.4.3.cmml" xref="S4.SS2.p2.8.m8.4.4.3"><times id="S4.SS2.p2.8.m8.4.4.3.1.cmml" xref="S4.SS2.p2.8.m8.4.4.3.1"></times><ci id="S4.SS2.p2.8.m8.4.4.3.2.cmml" xref="S4.SS2.p2.8.m8.4.4.3.2">𝑤</ci><ci id="S4.SS2.p2.8.m8.1.1.cmml" xref="S4.SS2.p2.8.m8.1.1">ℎ</ci></apply><apply id="S4.SS2.p2.8.m8.4.4.1.cmml" xref="S4.SS2.p2.8.m8.4.4.1"><plus id="S4.SS2.p2.8.m8.4.4.1.2.cmml" xref="S4.SS2.p2.8.m8.4.4.1.2"></plus><apply id="S4.SS2.p2.8.m8.4.4.1.1.2.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1"><max id="S4.SS2.p2.8.m8.2.2.cmml" xref="S4.SS2.p2.8.m8.2.2"></max><apply id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.1.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.2.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.2">𝑐</ci><apply id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3"><times id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.1.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.1"></times><ci id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.2.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.2">𝑚</ci><ci id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.3.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.3">𝑖</ci><ci id="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.4.cmml" xref="S4.SS2.p2.8.m8.4.4.1.1.1.1.1.3.4">𝑛</ci></apply></apply><ci id="S4.SS2.p2.8.m8.3.3.cmml" xref="S4.SS2.p2.8.m8.3.3">𝐵</ci></apply><cn id="S4.SS2.p2.8.m8.4.4.1.3.cmml" type="integer" xref="S4.SS2.p2.8.m8.4.4.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.8.m8.4c">w(h)=\max\{c_{min},B\}+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.8.m8.4d">italic_w ( italic_h ) = roman_max { italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_B } + 1</annotation></semantics></math>. It follows that the subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S4.SS2.p2.9.m9.1"><semantics id="S4.SS2.p2.9.m9.1a"><msub id="S4.SS2.p2.9.m9.1.1"><mi id="S4.SS2.p2.9.m9.1.1.2">G</mi><mrow id="S4.SS2.p2.9.m9.1.1.3"><mo fence="false" id="S4.SS2.p2.9.m9.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.SS2.p2.9.m9.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S4.SS2.p2.9.m9.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.9.m9.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> has a smaller dimension and we can thus apply the induction hypothesis in <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S4.SS2.p2.10.m10.1"><semantics id="S4.SS2.p2.10.m10.1a"><msub id="S4.SS2.p2.10.m10.1.1"><mi id="S4.SS2.p2.10.m10.1.1.2">G</mi><mrow id="S4.SS2.p2.10.m10.1.1.3"><mo fence="false" id="S4.SS2.p2.10.m10.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.SS2.p2.10.m10.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S4.SS2.p2.10.m10.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.10.m10.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="B=0" class="ltx_Math" display="inline" id="S4.SS2.p2.11.m11.1"><semantics id="S4.SS2.p2.11.m11.1a"><mrow id="S4.SS2.p2.11.m11.1.1" xref="S4.SS2.p2.11.m11.1.1.cmml"><mi id="S4.SS2.p2.11.m11.1.1.2" xref="S4.SS2.p2.11.m11.1.1.2.cmml">B</mi><mo id="S4.SS2.p2.11.m11.1.1.1" xref="S4.SS2.p2.11.m11.1.1.1.cmml">=</mo><mn id="S4.SS2.p2.11.m11.1.1.3" xref="S4.SS2.p2.11.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.11.m11.1b"><apply id="S4.SS2.p2.11.m11.1.1.cmml" xref="S4.SS2.p2.11.m11.1.1"><eq id="S4.SS2.p2.11.m11.1.1.1.cmml" xref="S4.SS2.p2.11.m11.1.1.1"></eq><ci id="S4.SS2.p2.11.m11.1.1.2.cmml" xref="S4.SS2.p2.11.m11.1.1.2">𝐵</ci><cn id="S4.SS2.p2.11.m11.1.1.3.cmml" type="integer" xref="S4.SS2.p2.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.11.m11.1c">B=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.11.m11.1d">italic_B = 0</annotation></semantics></math> as <math alttext="h" class="ltx_Math" display="inline" id="S4.SS2.p2.12.m12.1"><semantics id="S4.SS2.p2.12.m12.1a"><mi id="S4.SS2.p2.12.m12.1.1" xref="S4.SS2.p2.12.m12.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.12.m12.1b"><ci id="S4.SS2.p2.12.m12.1.1.cmml" xref="S4.SS2.p2.12.m12.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.12.m12.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.12.m12.1d">italic_h</annotation></semantics></math> has already visited Player <math alttext="0" class="ltx_Math" display="inline" id="S4.SS2.p2.13.m13.1"><semantics id="S4.SS2.p2.13.m13.1a"><mn id="S4.SS2.p2.13.m13.1.1" xref="S4.SS2.p2.13.m13.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.13.m13.1b"><cn id="S4.SS2.p2.13.m13.1.1.cmml" type="integer" xref="S4.SS2.p2.13.m13.1.1">0</cn></annotation-xml></semantics></math>’s target. Hence, by (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E3" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3</span></a>), we get a better solution in the subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S4.SS2.p2.14.m14.1"><semantics id="S4.SS2.p2.14.m14.1a"><msub id="S4.SS2.p2.14.m14.1.1"><mi id="S4.SS2.p2.14.m14.1.1.2">G</mi><mrow id="S4.SS2.p2.14.m14.1.1.3"><mo fence="false" id="S4.SS2.p2.14.m14.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.SS2.p2.14.m14.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S4.SS2.p2.14.m14.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.14.m14.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math>, and then a better solution in the whole game <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.p2.15.m15.1"><semantics id="S4.SS2.p2.15.m15.1a"><mi id="S4.SS2.p2.15.m15.1.1" xref="S4.SS2.p2.15.m15.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.15.m15.1b"><ci id="S4.SS2.p2.15.m15.1.1.cmml" xref="S4.SS2.p2.15.m15.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.15.m15.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.15.m15.1d">italic_G</annotation></semantics></math> by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem1" title="Lemma 3.1 ‣ 3.2 Improving a Solution in a Subgame ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.1</span></a>.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.1">Let us proceed to the formal proof.</p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.1.1.1">Proof 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.2.2"> </span>(Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem4" title="Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.4</span></a>)</h6> <div class="ltx_para" id="S4.Thmtheorem5.p1"> <p class="ltx_p" id="S4.Thmtheorem5.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.4.4">In this proof, we assume that Equation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E3" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3</span></a>) holds for dimension <math alttext="t" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.1.m1.1"><semantics id="S4.Thmtheorem5.p1.1.1.m1.1a"><mi id="S4.Thmtheorem5.p1.1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.1.m1.1b"><ci id="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.1.m1.1d">italic_t</annotation></semantics></math>, by induction hypothesis. Let <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.2.2.m2.2"><semantics id="S4.Thmtheorem5.p1.2.2.m2.2a"><mrow id="S4.Thmtheorem5.p1.2.2.m2.2.3" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.cmml"><msub id="S4.Thmtheorem5.p1.2.2.m2.2.3.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.1" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem5.p1.2.2.m2.2.3.3" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.1" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml">G</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.2.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.2.2.m2.2.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.2.cmml">B</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.2.2.m2.2b"><apply id="S4.Thmtheorem5.p1.2.2.m2.2.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3"><in id="S4.Thmtheorem5.p1.2.2.m2.2.3.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.1"></in><apply id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3"><times id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.1"></times><ci id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.2a.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.3.2"><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1">𝐺</ci><ci id="S4.Thmtheorem5.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.2.2.m2.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.2.2.m2.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. If Equation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E7" title="In Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">7</span></a>) is satisfied, then Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem4" title="Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.4</span></a> holds (we eliminate cycles if necessary). Therefore, let us suppose that there exists <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.3.3.m3.1"><semantics id="S4.Thmtheorem5.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem5.p1.3.3.m3.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem5.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.cmml"><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.2" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml">C</mi><msub id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.cmml"><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.2" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.3" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.3.3.m3.1b"><apply id="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1"><in id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.1"></in><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.2">𝑐</ci><apply id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.2">𝐶</ci><apply id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.3.3.m3.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.3.3.m3.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that for some <math alttext="i\in\{1,\ldots,t+1\}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.4.4.m4.3"><semantics id="S4.Thmtheorem5.p1.4.4.m4.3a"><mrow id="S4.Thmtheorem5.p1.4.4.m4.3.3" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.cmml"><mi id="S4.Thmtheorem5.p1.4.4.m4.3.3.3" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.3.cmml">i</mi><mo id="S4.Thmtheorem5.p1.4.4.m4.3.3.2" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.2.cmml">∈</mo><mrow id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.2.cmml"><mo id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.2.cmml">{</mo><mn id="S4.Thmtheorem5.p1.4.4.m4.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.cmml">1</mn><mo id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.3" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.2.cmml">,</mo><mi id="S4.Thmtheorem5.p1.4.4.m4.2.2" mathvariant="normal" xref="S4.Thmtheorem5.p1.4.4.m4.2.2.cmml">…</mi><mo id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.4" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.2.cmml">,</mo><mrow id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.cmml"><mi id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.2" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.3" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.5" stretchy="false" xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml 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xref="S4.Thmtheorem5.p1.4.4.m4.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.4.4.m4.3c">i\in\{1,\ldots,t+1\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.4.4.m4.3d">italic_i ∈ { 1 , … , italic_t + 1 }</annotation></semantics></math>:</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx8"> <tbody id="S4.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\max\{c_{min},B\}+1+f({0},{t})<c_{i}<\infty." class="ltx_Math" display="inline" id="S4.E8.m1.5"><semantics id="S4.E8.m1.5a"><mrow id="S4.E8.m1.5.5.1" xref="S4.E8.m1.5.5.1.1.cmml"><mrow id="S4.E8.m1.5.5.1.1" xref="S4.E8.m1.5.5.1.1.cmml"><mrow id="S4.E8.m1.5.5.1.1.1" xref="S4.E8.m1.5.5.1.1.1.cmml"><mrow id="S4.E8.m1.5.5.1.1.1.1.1" xref="S4.E8.m1.5.5.1.1.1.1.2.cmml"><mi id="S4.E8.m1.1.1" xref="S4.E8.m1.1.1.cmml">max</mi><mo id="S4.E8.m1.5.5.1.1.1.1.1a" xref="S4.E8.m1.5.5.1.1.1.1.2.cmml">⁡</mo><mrow id="S4.E8.m1.5.5.1.1.1.1.1.1" xref="S4.E8.m1.5.5.1.1.1.1.2.cmml"><mo id="S4.E8.m1.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S4.E8.m1.5.5.1.1.1.1.2.cmml">{</mo><msub id="S4.E8.m1.5.5.1.1.1.1.1.1.1" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.cmml"><mi id="S4.E8.m1.5.5.1.1.1.1.1.1.1.2" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.2.cmml">c</mi><mrow id="S4.E8.m1.5.5.1.1.1.1.1.1.1.3" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.2" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.1" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.3" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.3.cmml">i</mi><mo id="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.1a" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.4" xref="S4.E8.m1.5.5.1.1.1.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S4.E8.m1.5.5.1.1.1.1.1.1.3" xref="S4.E8.m1.5.5.1.1.1.1.2.cmml">,</mo><mi id="S4.E8.m1.2.2" xref="S4.E8.m1.2.2.cmml">B</mi><mo id="S4.E8.m1.5.5.1.1.1.1.1.1.4" stretchy="false" xref="S4.E8.m1.5.5.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S4.E8.m1.5.5.1.1.1.2" xref="S4.E8.m1.5.5.1.1.1.2.cmml">+</mo><mn id="S4.E8.m1.5.5.1.1.1.3" xref="S4.E8.m1.5.5.1.1.1.3.cmml">1</mn><mo id="S4.E8.m1.5.5.1.1.1.2a" xref="S4.E8.m1.5.5.1.1.1.2.cmml">+</mo><mrow id="S4.E8.m1.5.5.1.1.1.4" xref="S4.E8.m1.5.5.1.1.1.4.cmml"><mi id="S4.E8.m1.5.5.1.1.1.4.2" xref="S4.E8.m1.5.5.1.1.1.4.2.cmml">f</mi><mo id="S4.E8.m1.5.5.1.1.1.4.1" xref="S4.E8.m1.5.5.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.E8.m1.5.5.1.1.1.4.3.2" xref="S4.E8.m1.5.5.1.1.1.4.3.1.cmml"><mo id="S4.E8.m1.5.5.1.1.1.4.3.2.1" stretchy="false" xref="S4.E8.m1.5.5.1.1.1.4.3.1.cmml">(</mo><mn id="S4.E8.m1.3.3" xref="S4.E8.m1.3.3.cmml">0</mn><mo id="S4.E8.m1.5.5.1.1.1.4.3.2.2" xref="S4.E8.m1.5.5.1.1.1.4.3.1.cmml">,</mo><mi id="S4.E8.m1.4.4" xref="S4.E8.m1.4.4.cmml">t</mi><mo id="S4.E8.m1.5.5.1.1.1.4.3.2.3" stretchy="false" xref="S4.E8.m1.5.5.1.1.1.4.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E8.m1.5.5.1.1.3" xref="S4.E8.m1.5.5.1.1.3.cmml"><</mo><msub id="S4.E8.m1.5.5.1.1.4" xref="S4.E8.m1.5.5.1.1.4.cmml"><mi id="S4.E8.m1.5.5.1.1.4.2" xref="S4.E8.m1.5.5.1.1.4.2.cmml">c</mi><mi id="S4.E8.m1.5.5.1.1.4.3" xref="S4.E8.m1.5.5.1.1.4.3.cmml">i</mi></msub><mo id="S4.E8.m1.5.5.1.1.5" xref="S4.E8.m1.5.5.1.1.5.cmml"><</mo><mi id="S4.E8.m1.5.5.1.1.6" mathvariant="normal" xref="S4.E8.m1.5.5.1.1.6.cmml">∞</mi></mrow><mo id="S4.E8.m1.5.5.1.2" lspace="0em" xref="S4.E8.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E8.m1.5b"><apply id="S4.E8.m1.5.5.1.1.cmml" xref="S4.E8.m1.5.5.1"><and id="S4.E8.m1.5.5.1.1a.cmml" xref="S4.E8.m1.5.5.1"></and><apply id="S4.E8.m1.5.5.1.1b.cmml" xref="S4.E8.m1.5.5.1"><lt id="S4.E8.m1.5.5.1.1.3.cmml" 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id="S4.E8.m1.5.5.1.1c.cmml" xref="S4.E8.m1.5.5.1"><lt id="S4.E8.m1.5.5.1.1.5.cmml" xref="S4.E8.m1.5.5.1.1.5"></lt><share href="https://arxiv.org/html/2308.09443v2#S4.E8.m1.5.5.1.1.4.cmml" id="S4.E8.m1.5.5.1.1d.cmml" xref="S4.E8.m1.5.5.1"></share><infinity id="S4.E8.m1.5.5.1.1.6.cmml" xref="S4.E8.m1.5.5.1.1.6"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.5c">\displaystyle\max\{c_{min},B\}+1+f({0},{t})<c_{i}<\infty.</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.5d">roman_max { italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_B } + 1 + italic_f ( 0 , italic_t ) < italic_c 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">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem5.p1.7"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.7.3">Let <math alttext="\rho\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.5.1.m1.1"><semantics id="S4.Thmtheorem5.p1.5.1.m1.1a"><mrow id="S4.Thmtheorem5.p1.5.1.m1.1.1" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.cmml"><mi id="S4.Thmtheorem5.p1.5.1.m1.1.1.2" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.2.cmml">ρ</mi><mo id="S4.Thmtheorem5.p1.5.1.m1.1.1.1" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p1.5.1.m1.1.1.3" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.2" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.cmml"><mi id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.2" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.3" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.5.1.m1.1b"><apply id="S4.Thmtheorem5.p1.5.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1"><in id="S4.Thmtheorem5.p1.5.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.1"></in><ci id="S4.Thmtheorem5.p1.5.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.2">𝜌</ci><apply id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.2a.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.2">Wit</mtext></ci><apply id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.5.1.m1.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.5.1.m1.1c">\rho\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.5.1.m1.1d">italic_ρ ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be a witness with <math alttext="\textsf{cost}({\rho})=c" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.6.2.m2.1"><semantics id="S4.Thmtheorem5.p1.6.2.m2.1a"><mrow id="S4.Thmtheorem5.p1.6.2.m2.1.2" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.cmml"><mrow id="S4.Thmtheorem5.p1.6.2.m2.1.2.2" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.2" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.2a.cmml">cost</mtext><mo id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.1" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.3.2" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.cmml"><mo id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem5.p1.6.2.m2.1.1" xref="S4.Thmtheorem5.p1.6.2.m2.1.1.cmml">ρ</mi><mo id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem5.p1.6.2.m2.1.2.1" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.1.cmml">=</mo><mi id="S4.Thmtheorem5.p1.6.2.m2.1.2.3" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.6.2.m2.1b"><apply id="S4.Thmtheorem5.p1.6.2.m2.1.2.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.2"><eq id="S4.Thmtheorem5.p1.6.2.m2.1.2.1.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.1"></eq><apply id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2"><times id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.1.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.1"></times><ci id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.2a.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p1.6.2.m2.1.2.2.2.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.2.2">cost</mtext></ci><ci id="S4.Thmtheorem5.p1.6.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.1">𝜌</ci></apply><ci id="S4.Thmtheorem5.p1.6.2.m2.1.2.3.cmml" xref="S4.Thmtheorem5.p1.6.2.m2.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.6.2.m2.1c">\textsf{cost}({\rho})=c</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.6.2.m2.1d">cost ( italic_ρ ) = italic_c</annotation></semantics></math>. We define the history <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.7.3.m3.1"><semantics id="S4.Thmtheorem5.p1.7.3.m3.1a"><mi id="S4.Thmtheorem5.p1.7.3.m3.1.1" xref="S4.Thmtheorem5.p1.7.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.7.3.m3.1b"><ci id="S4.Thmtheorem5.p1.7.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.7.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.7.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.7.3.m3.1d">italic_h</annotation></semantics></math> of minimal length such that</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx9"> <tbody id="S4.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_align_right ltx_eqn_cell"><math alttext="\displaystyle h\rho\mbox{ and }w(h)=\max\{c_{min},B\}+1." class="ltx_Math" display="inline" id="S4.E9.m1.4"><semantics id="S4.E9.m1.4a"><mrow id="S4.E9.m1.4.4.1" xref="S4.E9.m1.4.4.1.1.cmml"><mrow id="S4.E9.m1.4.4.1.1" xref="S4.E9.m1.4.4.1.1.cmml"><mrow id="S4.E9.m1.4.4.1.1.3" xref="S4.E9.m1.4.4.1.1.3.cmml"><mi id="S4.E9.m1.4.4.1.1.3.2" xref="S4.E9.m1.4.4.1.1.3.2.cmml">h</mi><mo id="S4.E9.m1.4.4.1.1.3.1" xref="S4.E9.m1.4.4.1.1.3.1.cmml">⁢</mo><mi id="S4.E9.m1.4.4.1.1.3.3" xref="S4.E9.m1.4.4.1.1.3.3.cmml">ρ</mi><mo id="S4.E9.m1.4.4.1.1.3.1a" xref="S4.E9.m1.4.4.1.1.3.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S4.E9.m1.4.4.1.1.3.4" xref="S4.E9.m1.4.4.1.1.3.4a.cmml"> and </mtext><mo id="S4.E9.m1.4.4.1.1.3.1b" xref="S4.E9.m1.4.4.1.1.3.1.cmml">⁢</mo><mi id="S4.E9.m1.4.4.1.1.3.5" xref="S4.E9.m1.4.4.1.1.3.5.cmml">w</mi><mo id="S4.E9.m1.4.4.1.1.3.1c" xref="S4.E9.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S4.E9.m1.4.4.1.1.3.6.2" xref="S4.E9.m1.4.4.1.1.3.cmml"><mo id="S4.E9.m1.4.4.1.1.3.6.2.1" stretchy="false" xref="S4.E9.m1.4.4.1.1.3.cmml">(</mo><mi id="S4.E9.m1.1.1" xref="S4.E9.m1.1.1.cmml">h</mi><mo id="S4.E9.m1.4.4.1.1.3.6.2.2" stretchy="false" xref="S4.E9.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow><mo id="S4.E9.m1.4.4.1.1.2" xref="S4.E9.m1.4.4.1.1.2.cmml">=</mo><mrow id="S4.E9.m1.4.4.1.1.1" xref="S4.E9.m1.4.4.1.1.1.cmml"><mrow id="S4.E9.m1.4.4.1.1.1.1.1" xref="S4.E9.m1.4.4.1.1.1.1.2.cmml"><mi id="S4.E9.m1.2.2" xref="S4.E9.m1.2.2.cmml">max</mi><mo id="S4.E9.m1.4.4.1.1.1.1.1a" xref="S4.E9.m1.4.4.1.1.1.1.2.cmml">⁡</mo><mrow id="S4.E9.m1.4.4.1.1.1.1.1.1" xref="S4.E9.m1.4.4.1.1.1.1.2.cmml"><mo id="S4.E9.m1.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S4.E9.m1.4.4.1.1.1.1.2.cmml">{</mo><msub id="S4.E9.m1.4.4.1.1.1.1.1.1.1" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.cmml"><mi id="S4.E9.m1.4.4.1.1.1.1.1.1.1.2" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.2.cmml">c</mi><mrow id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.2" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.2.cmml">m</mi><mo id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.1" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.3" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.3.cmml">i</mi><mo id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.1a" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.4" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S4.E9.m1.4.4.1.1.1.1.1.1.3" xref="S4.E9.m1.4.4.1.1.1.1.2.cmml">,</mo><mi id="S4.E9.m1.3.3" xref="S4.E9.m1.3.3.cmml">B</mi><mo id="S4.E9.m1.4.4.1.1.1.1.1.1.4" stretchy="false" xref="S4.E9.m1.4.4.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S4.E9.m1.4.4.1.1.1.2" xref="S4.E9.m1.4.4.1.1.1.2.cmml">+</mo><mn id="S4.E9.m1.4.4.1.1.1.3" xref="S4.E9.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow><mo id="S4.E9.m1.4.4.1.2" lspace="0em" xref="S4.E9.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E9.m1.4b"><apply id="S4.E9.m1.4.4.1.1.cmml" xref="S4.E9.m1.4.4.1"><eq id="S4.E9.m1.4.4.1.1.2.cmml" xref="S4.E9.m1.4.4.1.1.2"></eq><apply id="S4.E9.m1.4.4.1.1.3.cmml" xref="S4.E9.m1.4.4.1.1.3"><times id="S4.E9.m1.4.4.1.1.3.1.cmml" xref="S4.E9.m1.4.4.1.1.3.1"></times><ci id="S4.E9.m1.4.4.1.1.3.2.cmml" xref="S4.E9.m1.4.4.1.1.3.2">ℎ</ci><ci id="S4.E9.m1.4.4.1.1.3.3.cmml" xref="S4.E9.m1.4.4.1.1.3.3">𝜌</ci><ci id="S4.E9.m1.4.4.1.1.3.4a.cmml" xref="S4.E9.m1.4.4.1.1.3.4"><mtext class="ltx_mathvariant_italic" id="S4.E9.m1.4.4.1.1.3.4.cmml" xref="S4.E9.m1.4.4.1.1.3.4"> and </mtext></ci><ci id="S4.E9.m1.4.4.1.1.3.5.cmml" xref="S4.E9.m1.4.4.1.1.3.5">𝑤</ci><ci id="S4.E9.m1.1.1.cmml" xref="S4.E9.m1.1.1">ℎ</ci></apply><apply id="S4.E9.m1.4.4.1.1.1.cmml" xref="S4.E9.m1.4.4.1.1.1"><plus id="S4.E9.m1.4.4.1.1.1.2.cmml" xref="S4.E9.m1.4.4.1.1.1.2"></plus><apply id="S4.E9.m1.4.4.1.1.1.1.2.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1"><max id="S4.E9.m1.2.2.cmml" xref="S4.E9.m1.2.2"></max><apply id="S4.E9.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E9.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E9.m1.4.4.1.1.1.1.1.1.1.2.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.2">𝑐</ci><apply id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3"><times id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.1"></times><ci id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.2">𝑚</ci><ci id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.3.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.3">𝑖</ci><ci id="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.4.cmml" xref="S4.E9.m1.4.4.1.1.1.1.1.1.1.3.4">𝑛</ci></apply></apply><ci id="S4.E9.m1.3.3.cmml" xref="S4.E9.m1.3.3">𝐵</ci></apply><cn id="S4.E9.m1.4.4.1.1.1.3.cmml" type="integer" xref="S4.E9.m1.4.4.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9.m1.4c">\displaystyle h\rho\mbox{ and }w(h)=\max\{c_{min},B\}+1.</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m1.4d">italic_h italic_ρ and italic_w ( italic_h ) = roman_max { italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_B } + 1 .</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="S4.Thmtheorem5.p1.18"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.18.11">Notice that <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.8.1.m1.1"><semantics id="S4.Thmtheorem5.p1.8.1.m1.1a"><mi id="S4.Thmtheorem5.p1.8.1.m1.1.1" xref="S4.Thmtheorem5.p1.8.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.8.1.m1.1b"><ci id="S4.Thmtheorem5.p1.8.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.8.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.8.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.8.1.m1.1d">italic_h</annotation></semantics></math> exists by definition of <math alttext="c_{i}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.9.2.m2.1"><semantics id="S4.Thmtheorem5.p1.9.2.m2.1a"><msub id="S4.Thmtheorem5.p1.9.2.m2.1.1" xref="S4.Thmtheorem5.p1.9.2.m2.1.1.cmml"><mi id="S4.Thmtheorem5.p1.9.2.m2.1.1.2" xref="S4.Thmtheorem5.p1.9.2.m2.1.1.2.cmml">c</mi><mi id="S4.Thmtheorem5.p1.9.2.m2.1.1.3" xref="S4.Thmtheorem5.p1.9.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.9.2.m2.1b"><apply id="S4.Thmtheorem5.p1.9.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.9.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.9.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.9.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p1.9.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.9.2.m2.1.1.2">𝑐</ci><ci id="S4.Thmtheorem5.p1.9.2.m2.1.1.3.cmml" xref="S4.Thmtheorem5.p1.9.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.9.2.m2.1c">c_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.9.2.m2.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and as the arena is binary. As <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.10.3.m3.2"><semantics id="S4.Thmtheorem5.p1.10.3.m3.2a"><mrow id="S4.Thmtheorem5.p1.10.3.m3.2.3" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.cmml"><msub id="S4.Thmtheorem5.p1.10.3.m3.2.3.2" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.2.cmml"><mi id="S4.Thmtheorem5.p1.10.3.m3.2.3.2.2" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p1.10.3.m3.2.3.2.3" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p1.10.3.m3.2.3.1" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem5.p1.10.3.m3.2.3.3" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.2" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.1" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.2" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem5.p1.10.3.m3.1.1" xref="S4.Thmtheorem5.p1.10.3.m3.1.1.cmml">G</mi><mo id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.2.2" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.10.3.m3.2.2" xref="S4.Thmtheorem5.p1.10.3.m3.2.2.cmml">B</mi><mo id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.10.3.m3.2b"><apply id="S4.Thmtheorem5.p1.10.3.m3.2.3.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3"><in id="S4.Thmtheorem5.p1.10.3.m3.2.3.1.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.1"></in><apply id="S4.Thmtheorem5.p1.10.3.m3.2.3.2.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.10.3.m3.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem5.p1.10.3.m3.2.3.2.2.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem5.p1.10.3.m3.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3"><times id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.1.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.1"></times><ci id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.2a.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.2.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.1.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.3.3.3.2"><ci id="S4.Thmtheorem5.p1.10.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.1.1">𝐺</ci><ci id="S4.Thmtheorem5.p1.10.3.m3.2.2.cmml" xref="S4.Thmtheorem5.p1.10.3.m3.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.10.3.m3.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.10.3.m3.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math>, by definition of <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.11.4.m4.1"><semantics id="S4.Thmtheorem5.p1.11.4.m4.1a"><mi id="S4.Thmtheorem5.p1.11.4.m4.1.1" xref="S4.Thmtheorem5.p1.11.4.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.11.4.m4.1b"><ci id="S4.Thmtheorem5.p1.11.4.m4.1.1.cmml" xref="S4.Thmtheorem5.p1.11.4.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.11.4.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.11.4.m4.1d">italic_h</annotation></semantics></math>, Player 0’s target and at least one target of Player <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.12.5.m5.1"><semantics id="S4.Thmtheorem5.p1.12.5.m5.1a"><mn id="S4.Thmtheorem5.p1.12.5.m5.1.1" xref="S4.Thmtheorem5.p1.12.5.m5.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.12.5.m5.1b"><cn id="S4.Thmtheorem5.p1.12.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p1.12.5.m5.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.12.5.m5.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.12.5.m5.1d">1</annotation></semantics></math> are visited by <math alttext="\bar{h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.13.6.m6.1"><semantics id="S4.Thmtheorem5.p1.13.6.m6.1a"><mover accent="true" id="S4.Thmtheorem5.p1.13.6.m6.1.1" xref="S4.Thmtheorem5.p1.13.6.m6.1.1.cmml"><mi id="S4.Thmtheorem5.p1.13.6.m6.1.1.2" xref="S4.Thmtheorem5.p1.13.6.m6.1.1.2.cmml">h</mi><mo id="S4.Thmtheorem5.p1.13.6.m6.1.1.1" xref="S4.Thmtheorem5.p1.13.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.13.6.m6.1b"><apply id="S4.Thmtheorem5.p1.13.6.m6.1.1.cmml" xref="S4.Thmtheorem5.p1.13.6.m6.1.1"><ci id="S4.Thmtheorem5.p1.13.6.m6.1.1.1.cmml" xref="S4.Thmtheorem5.p1.13.6.m6.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p1.13.6.m6.1.1.2.cmml" xref="S4.Thmtheorem5.p1.13.6.m6.1.1.2">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.13.6.m6.1c">\bar{h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.13.6.m6.1d">over¯ start_ARG italic_h end_ARG</annotation></semantics></math> such that <math alttext="h=\bar{h}\Last{h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.14.7.m7.1"><semantics id="S4.Thmtheorem5.p1.14.7.m7.1a"><mrow id="S4.Thmtheorem5.p1.14.7.m7.1.1" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.cmml"><mi id="S4.Thmtheorem5.p1.14.7.m7.1.1.2" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.2.cmml">h</mi><mo id="S4.Thmtheorem5.p1.14.7.m7.1.1.1" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.1.cmml">=</mo><mrow id="S4.Thmtheorem5.p1.14.7.m7.1.1.3" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.cmml"><mi id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.2" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.2.cmml">h</mi><mo id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.1" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.1" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.1.cmml">⁢</mo><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3b.cmml"><mtext id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3a" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3b.cmml">\Last</mtext></merror><mo id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.1a" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.4" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.4.cmml">h</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.14.7.m7.1b"><apply id="S4.Thmtheorem5.p1.14.7.m7.1.1.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1"><eq id="S4.Thmtheorem5.p1.14.7.m7.1.1.1.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.1"></eq><ci id="S4.Thmtheorem5.p1.14.7.m7.1.1.2.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.2">ℎ</ci><apply id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3"><times id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.1.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.1"></times><apply id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2"><ci id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.1.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.2.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.2.2">ℎ</ci></apply><ci id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3b.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3"><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3"><mtext id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3a.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.3">\Last</mtext></merror></ci><ci id="S4.Thmtheorem5.p1.14.7.m7.1.1.3.4.cmml" xref="S4.Thmtheorem5.p1.14.7.m7.1.1.3.4">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.14.7.m7.1c">h=\bar{h}\Last{h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.14.7.m7.1d">italic_h = over¯ start_ARG italic_h end_ARG italic_h</annotation></semantics></math>. Therefore the subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S4.Thmtheorem5.p1.15.8.m8.1"><semantics id="S4.Thmtheorem5.p1.15.8.m8.1a"><msub id="S4.Thmtheorem5.p1.15.8.m8.1.1"><mi id="S4.Thmtheorem5.p1.15.8.m8.1.1.2">G</mi><mrow id="S4.Thmtheorem5.p1.15.8.m8.1.1.3"><mo fence="false" id="S4.Thmtheorem5.p1.15.8.m8.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.Thmtheorem5.p1.15.8.m8.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.15.8.m8.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.15.8.m8.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> has dimension <math alttext="k\leq t" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.16.9.m9.1"><semantics id="S4.Thmtheorem5.p1.16.9.m9.1a"><mrow id="S4.Thmtheorem5.p1.16.9.m9.1.1" xref="S4.Thmtheorem5.p1.16.9.m9.1.1.cmml"><mi id="S4.Thmtheorem5.p1.16.9.m9.1.1.2" xref="S4.Thmtheorem5.p1.16.9.m9.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem5.p1.16.9.m9.1.1.1" xref="S4.Thmtheorem5.p1.16.9.m9.1.1.1.cmml">≤</mo><mi id="S4.Thmtheorem5.p1.16.9.m9.1.1.3" xref="S4.Thmtheorem5.p1.16.9.m9.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.16.9.m9.1b"><apply id="S4.Thmtheorem5.p1.16.9.m9.1.1.cmml" xref="S4.Thmtheorem5.p1.16.9.m9.1.1"><leq id="S4.Thmtheorem5.p1.16.9.m9.1.1.1.cmml" xref="S4.Thmtheorem5.p1.16.9.m9.1.1.1"></leq><ci id="S4.Thmtheorem5.p1.16.9.m9.1.1.2.cmml" xref="S4.Thmtheorem5.p1.16.9.m9.1.1.2">𝑘</ci><ci id="S4.Thmtheorem5.p1.16.9.m9.1.1.3.cmml" xref="S4.Thmtheorem5.p1.16.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.16.9.m9.1c">k\leq t</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.16.9.m9.1d">italic_k ≤ italic_t</annotation></semantics></math>. Notice that <math alttext="k>0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.17.10.m10.1"><semantics id="S4.Thmtheorem5.p1.17.10.m10.1a"><mrow id="S4.Thmtheorem5.p1.17.10.m10.1.1" xref="S4.Thmtheorem5.p1.17.10.m10.1.1.cmml"><mi id="S4.Thmtheorem5.p1.17.10.m10.1.1.2" xref="S4.Thmtheorem5.p1.17.10.m10.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem5.p1.17.10.m10.1.1.1" xref="S4.Thmtheorem5.p1.17.10.m10.1.1.1.cmml">></mo><mn id="S4.Thmtheorem5.p1.17.10.m10.1.1.3" xref="S4.Thmtheorem5.p1.17.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.17.10.m10.1b"><apply id="S4.Thmtheorem5.p1.17.10.m10.1.1.cmml" xref="S4.Thmtheorem5.p1.17.10.m10.1.1"><gt id="S4.Thmtheorem5.p1.17.10.m10.1.1.1.cmml" xref="S4.Thmtheorem5.p1.17.10.m10.1.1.1"></gt><ci id="S4.Thmtheorem5.p1.17.10.m10.1.1.2.cmml" xref="S4.Thmtheorem5.p1.17.10.m10.1.1.2">𝑘</ci><cn id="S4.Thmtheorem5.p1.17.10.m10.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.17.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.17.10.m10.1c">k>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.17.10.m10.1d">italic_k > 0</annotation></semantics></math> by definition of <math alttext="c_{i}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.18.11.m11.1"><semantics id="S4.Thmtheorem5.p1.18.11.m11.1a"><msub id="S4.Thmtheorem5.p1.18.11.m11.1.1" xref="S4.Thmtheorem5.p1.18.11.m11.1.1.cmml"><mi id="S4.Thmtheorem5.p1.18.11.m11.1.1.2" xref="S4.Thmtheorem5.p1.18.11.m11.1.1.2.cmml">c</mi><mi id="S4.Thmtheorem5.p1.18.11.m11.1.1.3" xref="S4.Thmtheorem5.p1.18.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.18.11.m11.1b"><apply id="S4.Thmtheorem5.p1.18.11.m11.1.1.cmml" xref="S4.Thmtheorem5.p1.18.11.m11.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.18.11.m11.1.1.1.cmml" xref="S4.Thmtheorem5.p1.18.11.m11.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p1.18.11.m11.1.1.2.cmml" xref="S4.Thmtheorem5.p1.18.11.m11.1.1.2">𝑐</ci><ci id="S4.Thmtheorem5.p1.18.11.m11.1.1.3.cmml" xref="S4.Thmtheorem5.p1.18.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.18.11.m11.1c">c_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.18.11.m11.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, see (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E8" title="In Proof 4.5 (Proof of Lemma 4.4) ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">8</span></a>).</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem5.p2"> <p class="ltx_p" id="S4.Thmtheorem5.p2.1"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p2.1.1">Let us consider the SP game <math alttext="\bar{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.1.1.m1.1"><semantics id="S4.Thmtheorem5.p2.1.1.m1.1a"><mover accent="true" id="S4.Thmtheorem5.p2.1.1.m1.1.1" xref="S4.Thmtheorem5.p2.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem5.p2.1.1.m1.1.1.2" xref="S4.Thmtheorem5.p2.1.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem5.p2.1.1.m1.1.1.1" xref="S4.Thmtheorem5.p2.1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.1.1.m1.1b"><apply id="S4.Thmtheorem5.p2.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p2.1.1.m1.1.1"><ci id="S4.Thmtheorem5.p2.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p2.1.1.m1.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p2.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p2.1.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.1.1.m1.1c">\bar{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.1.1.m1.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math></span></p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.1">with the same arena as </span><math alttext="G" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.1"><semantics id="S4.I1.i1.p1.1.m1.1a"><mi id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.1b"><ci id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.2"> but with the initial vertex </span><math alttext="\Last{h}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.1"><semantics id="S4.I1.i1.p1.2.m2.1a"><mrow id="S4.I1.i1.p1.2.m2.1.1" xref="S4.I1.i1.p1.2.m2.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="S4.I1.i1.p1.2.m2.1.1.2" xref="S4.I1.i1.p1.2.m2.1.1.2b.cmml"><mtext id="S4.I1.i1.p1.2.m2.1.1.2a" xref="S4.I1.i1.p1.2.m2.1.1.2b.cmml">\Last</mtext></merror><mo id="S4.I1.i1.p1.2.m2.1.1.1" xref="S4.I1.i1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.I1.i1.p1.2.m2.1.1.3" xref="S4.I1.i1.p1.2.m2.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.2.m2.1b"><apply id="S4.I1.i1.p1.2.m2.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1"><times id="S4.I1.i1.p1.2.m2.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1"></times><ci id="S4.I1.i1.p1.2.m2.1.1.2b.cmml" xref="S4.I1.i1.p1.2.m2.1.1.2"><merror class="ltx_ERROR undefined undefined" id="S4.I1.i1.p1.2.m2.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.1.1.2"><mtext id="S4.I1.i1.p1.2.m2.1.1.2a.cmml" xref="S4.I1.i1.p1.2.m2.1.1.2">\Last</mtext></merror></ci><ci id="S4.I1.i1.p1.2.m2.1.1.3.cmml" xref="S4.I1.i1.p1.2.m2.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.2.m2.1c">\Last{h}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.2.m2.1d">italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.3"> (instead of </span><math alttext="v_{0}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.3.m3.1"><semantics id="S4.I1.i1.p1.3.m3.1a"><msub id="S4.I1.i1.p1.3.m3.1.1" xref="S4.I1.i1.p1.3.m3.1.1.cmml"><mi id="S4.I1.i1.p1.3.m3.1.1.2" xref="S4.I1.i1.p1.3.m3.1.1.2.cmml">v</mi><mn id="S4.I1.i1.p1.3.m3.1.1.3" xref="S4.I1.i1.p1.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.3.m3.1b"><apply id="S4.I1.i1.p1.3.m3.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.3.m3.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.3.m3.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.1.1.2">𝑣</ci><cn id="S4.I1.i1.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.I1.i1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.3.m3.1c">v_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.3.m3.1d">italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.4">),</span></p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.6"><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.1">with Player </span><math alttext="0" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.1"><semantics id="S4.I1.i2.p1.1.m1.1a"><mn id="S4.I1.i2.p1.1.m1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.1b"><cn id="S4.I1.i2.p1.1.m1.1.1.cmml" type="integer" xref="S4.I1.i2.p1.1.m1.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.2">’s target equal to </span><math alttext="\{\Last{h}\}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.2.m2.1"><semantics id="S4.I1.i2.p1.2.m2.1a"><mrow id="S4.I1.i2.p1.2.m2.1.1.1" xref="S4.I1.i2.p1.2.m2.1.1.2.cmml"><mo id="S4.I1.i2.p1.2.m2.1.1.1.2" stretchy="false" xref="S4.I1.i2.p1.2.m2.1.1.2.cmml">{</mo><mrow id="S4.I1.i2.p1.2.m2.1.1.1.1" xref="S4.I1.i2.p1.2.m2.1.1.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="S4.I1.i2.p1.2.m2.1.1.1.1.2" xref="S4.I1.i2.p1.2.m2.1.1.1.1.2b.cmml"><mtext id="S4.I1.i2.p1.2.m2.1.1.1.1.2a" xref="S4.I1.i2.p1.2.m2.1.1.1.1.2b.cmml">\Last</mtext></merror><mo id="S4.I1.i2.p1.2.m2.1.1.1.1.1" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S4.I1.i2.p1.2.m2.1.1.1.1.3" xref="S4.I1.i2.p1.2.m2.1.1.1.1.3.cmml">h</mi></mrow><mo id="S4.I1.i2.p1.2.m2.1.1.1.3" stretchy="false" xref="S4.I1.i2.p1.2.m2.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.2.m2.1b"><set id="S4.I1.i2.p1.2.m2.1.1.2.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1"><apply id="S4.I1.i2.p1.2.m2.1.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1"><times id="S4.I1.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1"></times><ci id="S4.I1.i2.p1.2.m2.1.1.1.1.2b.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.2"><merror class="ltx_ERROR undefined undefined" id="S4.I1.i2.p1.2.m2.1.1.1.1.2.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.2"><mtext id="S4.I1.i2.p1.2.m2.1.1.1.1.2a.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.2">\Last</mtext></merror></ci><ci id="S4.I1.i2.p1.2.m2.1.1.1.1.3.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.3">ℎ</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.2.m2.1c">\{\Last{h}\}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.2.m2.1d">{ italic_h }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.3"> (since </span><math alttext="\bar{h}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.3.m3.1"><semantics id="S4.I1.i2.p1.3.m3.1a"><mover accent="true" id="S4.I1.i2.p1.3.m3.1.1" xref="S4.I1.i2.p1.3.m3.1.1.cmml"><mi id="S4.I1.i2.p1.3.m3.1.1.2" xref="S4.I1.i2.p1.3.m3.1.1.2.cmml">h</mi><mo id="S4.I1.i2.p1.3.m3.1.1.1" xref="S4.I1.i2.p1.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.3.m3.1b"><apply id="S4.I1.i2.p1.3.m3.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1"><ci id="S4.I1.i2.p1.3.m3.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1">¯</ci><ci id="S4.I1.i2.p1.3.m3.1.1.2.cmml" xref="S4.I1.i2.p1.3.m3.1.1.2">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.3.m3.1c">\bar{h}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.3.m3.1d">over¯ start_ARG italic_h end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.4"> has visited the target of Player </span><math alttext="0" class="ltx_Math" display="inline" id="S4.I1.i2.p1.4.m4.1"><semantics id="S4.I1.i2.p1.4.m4.1a"><mn id="S4.I1.i2.p1.4.m4.1.1" xref="S4.I1.i2.p1.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.4.m4.1b"><cn id="S4.I1.i2.p1.4.m4.1.1.cmml" type="integer" xref="S4.I1.i2.p1.4.m4.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.5"> in </span><math alttext="G" class="ltx_Math" display="inline" id="S4.I1.i2.p1.5.m5.1"><semantics id="S4.I1.i2.p1.5.m5.1a"><mi id="S4.I1.i2.p1.5.m5.1.1" xref="S4.I1.i2.p1.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.5.m5.1b"><ci id="S4.I1.i2.p1.5.m5.1.1.cmml" xref="S4.I1.i2.p1.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.5.m5.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.6">, we replace it by the initial vertex of </span><math alttext="\bar{G}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.6.m6.1"><semantics id="S4.I1.i2.p1.6.m6.1a"><mover accent="true" id="S4.I1.i2.p1.6.m6.1.1" xref="S4.I1.i2.p1.6.m6.1.1.cmml"><mi id="S4.I1.i2.p1.6.m6.1.1.2" xref="S4.I1.i2.p1.6.m6.1.1.2.cmml">G</mi><mo id="S4.I1.i2.p1.6.m6.1.1.1" xref="S4.I1.i2.p1.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.6.m6.1b"><apply id="S4.I1.i2.p1.6.m6.1.1.cmml" xref="S4.I1.i2.p1.6.m6.1.1"><ci id="S4.I1.i2.p1.6.m6.1.1.1.cmml" xref="S4.I1.i2.p1.6.m6.1.1.1">¯</ci><ci id="S4.I1.i2.p1.6.m6.1.1.2.cmml" xref="S4.I1.i2.p1.6.m6.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.6.m6.1c">\bar{G}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.6.m6.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.7">),</span><span class="ltx_note ltx_role_footnote" id="footnote8"><sup class="ltx_note_mark">8</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">8</sup><span class="ltx_tag ltx_tag_note">8</span>We recall that a target must be non-empty by definition.</span></span></span><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.6.8"></span></p> </div> </li> <li class="ltx_item" id="S4.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i3.p1"> <p class="ltx_p" id="S4.I1.i3.p1.6"><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.6.1">and with the targets visited by </span><math alttext="\bar{h}" class="ltx_Math" display="inline" id="S4.I1.i3.p1.1.m1.1"><semantics id="S4.I1.i3.p1.1.m1.1a"><mover accent="true" id="S4.I1.i3.p1.1.m1.1.1" xref="S4.I1.i3.p1.1.m1.1.1.cmml"><mi id="S4.I1.i3.p1.1.m1.1.1.2" xref="S4.I1.i3.p1.1.m1.1.1.2.cmml">h</mi><mo id="S4.I1.i3.p1.1.m1.1.1.1" xref="S4.I1.i3.p1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.1.m1.1b"><apply id="S4.I1.i3.p1.1.m1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.1.1"><ci id="S4.I1.i3.p1.1.m1.1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.1.1.1">¯</ci><ci id="S4.I1.i3.p1.1.m1.1.1.2.cmml" xref="S4.I1.i3.p1.1.m1.1.1.2">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.1.m1.1c">\bar{h}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.1.m1.1d">over¯ start_ARG italic_h end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.6.2"> removed from Player </span><math alttext="1" class="ltx_Math" display="inline" id="S4.I1.i3.p1.2.m2.1"><semantics id="S4.I1.i3.p1.2.m2.1a"><mn id="S4.I1.i3.p1.2.m2.1.1" xref="S4.I1.i3.p1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.2.m2.1b"><cn id="S4.I1.i3.p1.2.m2.1.1.cmml" type="integer" xref="S4.I1.i3.p1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.2.m2.1d">1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.6.3">’s set of targets, and if </span><math alttext="k<t" class="ltx_Math" display="inline" id="S4.I1.i3.p1.3.m3.1"><semantics id="S4.I1.i3.p1.3.m3.1a"><mrow id="S4.I1.i3.p1.3.m3.1.1" xref="S4.I1.i3.p1.3.m3.1.1.cmml"><mi id="S4.I1.i3.p1.3.m3.1.1.2" xref="S4.I1.i3.p1.3.m3.1.1.2.cmml">k</mi><mo id="S4.I1.i3.p1.3.m3.1.1.1" xref="S4.I1.i3.p1.3.m3.1.1.1.cmml"><</mo><mi id="S4.I1.i3.p1.3.m3.1.1.3" xref="S4.I1.i3.p1.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.3.m3.1b"><apply id="S4.I1.i3.p1.3.m3.1.1.cmml" xref="S4.I1.i3.p1.3.m3.1.1"><lt id="S4.I1.i3.p1.3.m3.1.1.1.cmml" xref="S4.I1.i3.p1.3.m3.1.1.1"></lt><ci id="S4.I1.i3.p1.3.m3.1.1.2.cmml" xref="S4.I1.i3.p1.3.m3.1.1.2">𝑘</ci><ci id="S4.I1.i3.p1.3.m3.1.1.3.cmml" xref="S4.I1.i3.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.3.m3.1c">k<t</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.3.m3.1d">italic_k < italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.6.4"> with </span><math alttext="t-k" class="ltx_Math" display="inline" id="S4.I1.i3.p1.4.m4.1"><semantics id="S4.I1.i3.p1.4.m4.1a"><mrow id="S4.I1.i3.p1.4.m4.1.1" xref="S4.I1.i3.p1.4.m4.1.1.cmml"><mi id="S4.I1.i3.p1.4.m4.1.1.2" xref="S4.I1.i3.p1.4.m4.1.1.2.cmml">t</mi><mo id="S4.I1.i3.p1.4.m4.1.1.1" xref="S4.I1.i3.p1.4.m4.1.1.1.cmml">−</mo><mi id="S4.I1.i3.p1.4.m4.1.1.3" xref="S4.I1.i3.p1.4.m4.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.4.m4.1b"><apply id="S4.I1.i3.p1.4.m4.1.1.cmml" xref="S4.I1.i3.p1.4.m4.1.1"><minus id="S4.I1.i3.p1.4.m4.1.1.1.cmml" xref="S4.I1.i3.p1.4.m4.1.1.1"></minus><ci id="S4.I1.i3.p1.4.m4.1.1.2.cmml" xref="S4.I1.i3.p1.4.m4.1.1.2">𝑡</ci><ci id="S4.I1.i3.p1.4.m4.1.1.3.cmml" xref="S4.I1.i3.p1.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.4.m4.1c">t-k</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.4.m4.1d">italic_t - italic_k</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.6.5"> additional targets that are copies of some remaining (non-empty) targets in a way to have exactly </span><math alttext="t" class="ltx_Math" display="inline" id="S4.I1.i3.p1.5.m5.1"><semantics id="S4.I1.i3.p1.5.m5.1a"><mi id="S4.I1.i3.p1.5.m5.1.1" xref="S4.I1.i3.p1.5.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.5.m5.1b"><ci id="S4.I1.i3.p1.5.m5.1.1.cmml" xref="S4.I1.i3.p1.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.5.m5.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.6.6"> targets for Player </span><math alttext="1" class="ltx_Math" display="inline" id="S4.I1.i3.p1.6.m6.1"><semantics id="S4.I1.i3.p1.6.m6.1a"><mn id="S4.I1.i3.p1.6.m6.1.1" xref="S4.I1.i3.p1.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.6.m6.1b"><cn id="S4.I1.i3.p1.6.m6.1.1.cmml" type="integer" xref="S4.I1.i3.p1.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.6.m6.1d">1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.6.7">.</span></p> </div> </li> </ul> <p class="ltx_p" id="S4.Thmtheorem5.p2.19"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p2.19.18">This game <math alttext="\bar{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.2.1.m1.1"><semantics id="S4.Thmtheorem5.p2.2.1.m1.1a"><mover accent="true" id="S4.Thmtheorem5.p2.2.1.m1.1.1" xref="S4.Thmtheorem5.p2.2.1.m1.1.1.cmml"><mi id="S4.Thmtheorem5.p2.2.1.m1.1.1.2" xref="S4.Thmtheorem5.p2.2.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem5.p2.2.1.m1.1.1.1" xref="S4.Thmtheorem5.p2.2.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.2.1.m1.1b"><apply id="S4.Thmtheorem5.p2.2.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p2.2.1.m1.1.1"><ci id="S4.Thmtheorem5.p2.2.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p2.2.1.m1.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p2.2.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p2.2.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.2.1.m1.1c">\bar{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.2.1.m1.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math> has dimension <math alttext="t" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.3.2.m2.1"><semantics id="S4.Thmtheorem5.p2.3.2.m2.1a"><mi id="S4.Thmtheorem5.p2.3.2.m2.1.1" xref="S4.Thmtheorem5.p2.3.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.3.2.m2.1b"><ci id="S4.Thmtheorem5.p2.3.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p2.3.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.3.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.3.2.m2.1d">italic_t</annotation></semantics></math>. We also consider the strategy <math alttext="\bar{\sigma}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.4.3.m3.1"><semantics id="S4.Thmtheorem5.p2.4.3.m3.1a"><msub id="S4.Thmtheorem5.p2.4.3.m3.1.1" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p2.4.3.m3.1.1.2" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.2.cmml"><mi id="S4.Thmtheorem5.p2.4.3.m3.1.1.2.2" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p2.4.3.m3.1.1.2.1" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p2.4.3.m3.1.1.3" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.4.3.m3.1b"><apply id="S4.Thmtheorem5.p2.4.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p2.4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p2.4.3.m3.1.1.1.cmml" xref="S4.Thmtheorem5.p2.4.3.m3.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p2.4.3.m3.1.1.2.cmml" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.2"><ci id="S4.Thmtheorem5.p2.4.3.m3.1.1.2.1.cmml" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p2.4.3.m3.1.1.2.2.cmml" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p2.4.3.m3.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p2.4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.4.3.m3.1c">\bar{\sigma}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.4.3.m3.1d">over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> for Player <math alttext="0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.5.4.m4.1"><semantics id="S4.Thmtheorem5.p2.5.4.m4.1a"><mn id="S4.Thmtheorem5.p2.5.4.m4.1.1" xref="S4.Thmtheorem5.p2.5.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.5.4.m4.1b"><cn id="S4.Thmtheorem5.p2.5.4.m4.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p2.5.4.m4.1.1">0</cn></annotation-xml></semantics></math> in <math alttext="\bar{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.6.5.m5.1"><semantics id="S4.Thmtheorem5.p2.6.5.m5.1a"><mover accent="true" id="S4.Thmtheorem5.p2.6.5.m5.1.1" xref="S4.Thmtheorem5.p2.6.5.m5.1.1.cmml"><mi id="S4.Thmtheorem5.p2.6.5.m5.1.1.2" xref="S4.Thmtheorem5.p2.6.5.m5.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem5.p2.6.5.m5.1.1.1" xref="S4.Thmtheorem5.p2.6.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.6.5.m5.1b"><apply id="S4.Thmtheorem5.p2.6.5.m5.1.1.cmml" xref="S4.Thmtheorem5.p2.6.5.m5.1.1"><ci id="S4.Thmtheorem5.p2.6.5.m5.1.1.1.cmml" xref="S4.Thmtheorem5.p2.6.5.m5.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p2.6.5.m5.1.1.2.cmml" xref="S4.Thmtheorem5.p2.6.5.m5.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.6.5.m5.1c">\bar{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.6.5.m5.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math> constructed from the strategy <math alttext="\sigma_{0|h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.7.6.m6.1"><semantics id="S4.Thmtheorem5.p2.7.6.m6.1a"><msub id="S4.Thmtheorem5.p2.7.6.m6.1.1" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.cmml"><mi id="S4.Thmtheorem5.p2.7.6.m6.1.1.2" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.2.cmml">σ</mi><mrow id="S4.Thmtheorem5.p2.7.6.m6.1.1.3" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3.cmml"><mn id="S4.Thmtheorem5.p2.7.6.m6.1.1.3.2" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p2.7.6.m6.1.1.3.1" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p2.7.6.m6.1.1.3.3" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3.3.cmml">h</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.7.6.m6.1b"><apply id="S4.Thmtheorem5.p2.7.6.m6.1.1.cmml" xref="S4.Thmtheorem5.p2.7.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p2.7.6.m6.1.1.1.cmml" xref="S4.Thmtheorem5.p2.7.6.m6.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p2.7.6.m6.1.1.2.cmml" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.2">𝜎</ci><apply id="S4.Thmtheorem5.p2.7.6.m6.1.1.3.cmml" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3"><csymbol cd="latexml" id="S4.Thmtheorem5.p2.7.6.m6.1.1.3.1.cmml" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3.1">conditional</csymbol><cn id="S4.Thmtheorem5.p2.7.6.m6.1.1.3.2.cmml" type="integer" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3.2">0</cn><ci id="S4.Thmtheorem5.p2.7.6.m6.1.1.3.3.cmml" xref="S4.Thmtheorem5.p2.7.6.m6.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.7.6.m6.1c">\sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.7.6.m6.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S4.Thmtheorem5.p2.8.7.m7.1"><semantics id="S4.Thmtheorem5.p2.8.7.m7.1a"><msub id="S4.Thmtheorem5.p2.8.7.m7.1.1"><mi id="S4.Thmtheorem5.p2.8.7.m7.1.1.2">G</mi><mrow id="S4.Thmtheorem5.p2.8.7.m7.1.1.3"><mo fence="false" id="S4.Thmtheorem5.p2.8.7.m7.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.Thmtheorem5.p2.8.7.m7.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.8.7.m7.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.8.7.m7.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> as follows: <math alttext="\bar{\sigma}_{0}(g)=\sigma_{0|h}(\bar{h}g)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.9.8.m8.2"><semantics id="S4.Thmtheorem5.p2.9.8.m8.2a"><mrow id="S4.Thmtheorem5.p2.9.8.m8.2.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.cmml"><mrow id="S4.Thmtheorem5.p2.9.8.m8.2.2.3" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.cmml"><msub id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.cmml"><mi id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.3" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.3.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.cmml"><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.cmml">(</mo><mi id="S4.Thmtheorem5.p2.9.8.m8.1.1" xref="S4.Thmtheorem5.p2.9.8.m8.1.1.cmml">g</mi><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.2.cmml">=</mo><mrow id="S4.Thmtheorem5.p2.9.8.m8.2.2.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.cmml"><msub id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.cmml"><mi id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.2.cmml">σ</mi><mrow id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.cmml"><mn id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.3" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.3.cmml">h</mi></mrow></msub><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.cmml"><mi id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.2" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.2.cmml">h</mi><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.1" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.3" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.3.cmml">g</mi></mrow><mo id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.9.8.m8.2b"><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2"><eq id="S4.Thmtheorem5.p2.9.8.m8.2.2.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.2"></eq><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3"><times id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.1"></times><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2">subscript</csymbol><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2"><ci id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.3.2.3">0</cn></apply><ci id="S4.Thmtheorem5.p2.9.8.m8.1.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.1.1">𝑔</ci></apply><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1"><times id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.2"></times><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.2">𝜎</ci><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3"><csymbol cd="latexml" id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.1">conditional</csymbol><cn id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.2">0</cn><ci id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.3.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.3.3.3">ℎ</ci></apply></apply><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1"><times id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.1"></times><apply id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2"><ci id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.2.2">ℎ</ci></apply><ci id="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p2.9.8.m8.2.2.1.1.1.1.3">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.9.8.m8.2c">\bar{\sigma}_{0}(g)=\sigma_{0|h}(\bar{h}g)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.9.8.m8.2d">over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_g ) = italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT ( over¯ start_ARG italic_h end_ARG italic_g )</annotation></semantics></math> for all histories <math alttext="g\in\textsf{Hist}_{\bar{G}}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.10.9.m9.1"><semantics id="S4.Thmtheorem5.p2.10.9.m9.1a"><mrow id="S4.Thmtheorem5.p2.10.9.m9.1.1" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.cmml"><mi id="S4.Thmtheorem5.p2.10.9.m9.1.1.2" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.2.cmml">g</mi><mo id="S4.Thmtheorem5.p2.10.9.m9.1.1.1" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p2.10.9.m9.1.1.3" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.2" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.2a.cmml">Hist</mtext><mover accent="true" id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.cmml"><mi id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.2" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.2.cmml">G</mi><mo id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.1" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.1.cmml">¯</mo></mover></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.10.9.m9.1b"><apply id="S4.Thmtheorem5.p2.10.9.m9.1.1.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1"><in id="S4.Thmtheorem5.p2.10.9.m9.1.1.1.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.1"></in><ci id="S4.Thmtheorem5.p2.10.9.m9.1.1.2.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.2">𝑔</ci><apply id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.1.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.2a.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.2.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.2">Hist</mtext></ci><apply id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3"><ci id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.1">¯</ci><ci id="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p2.10.9.m9.1.1.3.3.2">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.10.9.m9.1c">g\in\textsf{Hist}_{\bar{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.10.9.m9.1d">italic_g ∈ Hist start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> (Player <math alttext="0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.11.10.m10.1"><semantics id="S4.Thmtheorem5.p2.11.10.m10.1a"><mn id="S4.Thmtheorem5.p2.11.10.m10.1.1" xref="S4.Thmtheorem5.p2.11.10.m10.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.11.10.m10.1b"><cn id="S4.Thmtheorem5.p2.11.10.m10.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p2.11.10.m10.1.1">0</cn></annotation-xml></semantics></math> plays in <math alttext="\bar{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.12.11.m11.1"><semantics id="S4.Thmtheorem5.p2.12.11.m11.1a"><mover accent="true" id="S4.Thmtheorem5.p2.12.11.m11.1.1" xref="S4.Thmtheorem5.p2.12.11.m11.1.1.cmml"><mi id="S4.Thmtheorem5.p2.12.11.m11.1.1.2" xref="S4.Thmtheorem5.p2.12.11.m11.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem5.p2.12.11.m11.1.1.1" xref="S4.Thmtheorem5.p2.12.11.m11.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.12.11.m11.1b"><apply id="S4.Thmtheorem5.p2.12.11.m11.1.1.cmml" xref="S4.Thmtheorem5.p2.12.11.m11.1.1"><ci id="S4.Thmtheorem5.p2.12.11.m11.1.1.1.cmml" xref="S4.Thmtheorem5.p2.12.11.m11.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p2.12.11.m11.1.1.2.cmml" xref="S4.Thmtheorem5.p2.12.11.m11.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.12.11.m11.1c">\bar{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.12.11.m11.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math> from <math alttext="\Last{h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.13.12.m12.1"><semantics id="S4.Thmtheorem5.p2.13.12.m12.1a"><mrow id="S4.Thmtheorem5.p2.13.12.m12.1.1" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem5.p2.13.12.m12.1.1.2" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.2b.cmml"><mtext id="S4.Thmtheorem5.p2.13.12.m12.1.1.2a" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.2b.cmml">\Last</mtext></merror><mo id="S4.Thmtheorem5.p2.13.12.m12.1.1.1" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem5.p2.13.12.m12.1.1.3" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.13.12.m12.1b"><apply id="S4.Thmtheorem5.p2.13.12.m12.1.1.cmml" xref="S4.Thmtheorem5.p2.13.12.m12.1.1"><times id="S4.Thmtheorem5.p2.13.12.m12.1.1.1.cmml" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.1"></times><ci id="S4.Thmtheorem5.p2.13.12.m12.1.1.2b.cmml" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.2"><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem5.p2.13.12.m12.1.1.2.cmml" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.2"><mtext id="S4.Thmtheorem5.p2.13.12.m12.1.1.2a.cmml" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.2">\Last</mtext></merror></ci><ci id="S4.Thmtheorem5.p2.13.12.m12.1.1.3.cmml" xref="S4.Thmtheorem5.p2.13.12.m12.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.13.12.m12.1c">\Last{h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.13.12.m12.1d">italic_h</annotation></semantics></math> as he plays in <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S4.Thmtheorem5.p2.14.13.m13.1"><semantics id="S4.Thmtheorem5.p2.14.13.m13.1a"><msub id="S4.Thmtheorem5.p2.14.13.m13.1.1"><mi id="S4.Thmtheorem5.p2.14.13.m13.1.1.2">G</mi><mrow id="S4.Thmtheorem5.p2.14.13.m13.1.1.3"><mo fence="false" id="S4.Thmtheorem5.p2.14.13.m13.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.Thmtheorem5.p2.14.13.m13.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.14.13.m13.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.14.13.m13.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.15.14.m14.1"><semantics id="S4.Thmtheorem5.p2.15.14.m14.1a"><mi id="S4.Thmtheorem5.p2.15.14.m14.1.1" xref="S4.Thmtheorem5.p2.15.14.m14.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.15.14.m14.1b"><ci id="S4.Thmtheorem5.p2.15.14.m14.1.1.cmml" xref="S4.Thmtheorem5.p2.15.14.m14.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.15.14.m14.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.15.14.m14.1d">italic_h</annotation></semantics></math>). We have that <math alttext="\bar{\sigma}_{0}\in\mbox{SPS}({\bar{G}},{0})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.16.15.m15.2"><semantics id="S4.Thmtheorem5.p2.16.15.m15.2a"><mrow id="S4.Thmtheorem5.p2.16.15.m15.2.3" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.cmml"><msub id="S4.Thmtheorem5.p2.16.15.m15.2.3.2" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.cmml"><mi id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.2" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.1" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.3" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p2.16.15.m15.2.3.1" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem5.p2.16.15.m15.2.3.3" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.2" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.1" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.2" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem5.p2.16.15.m15.1.1" xref="S4.Thmtheorem5.p2.16.15.m15.1.1.cmml"><mi id="S4.Thmtheorem5.p2.16.15.m15.1.1.2" xref="S4.Thmtheorem5.p2.16.15.m15.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem5.p2.16.15.m15.1.1.1" xref="S4.Thmtheorem5.p2.16.15.m15.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.2.2" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.1.cmml">,</mo><mn id="S4.Thmtheorem5.p2.16.15.m15.2.2" xref="S4.Thmtheorem5.p2.16.15.m15.2.2.cmml">0</mn><mo id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.16.15.m15.2b"><apply id="S4.Thmtheorem5.p2.16.15.m15.2.3.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3"><in id="S4.Thmtheorem5.p2.16.15.m15.2.3.1.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.1"></in><apply id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.1.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2">subscript</csymbol><apply id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2"><ci id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.1.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.2.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p2.16.15.m15.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3"><times id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.1.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.1"></times><ci id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.2a.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.2.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.1.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.2.3.3.3.2"><apply id="S4.Thmtheorem5.p2.16.15.m15.1.1.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.1.1"><ci id="S4.Thmtheorem5.p2.16.15.m15.1.1.1.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p2.16.15.m15.1.1.2.cmml" xref="S4.Thmtheorem5.p2.16.15.m15.1.1.2">𝐺</ci></apply><cn id="S4.Thmtheorem5.p2.16.15.m15.2.2.cmml" type="integer" xref="S4.Thmtheorem5.p2.16.15.m15.2.2">0</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.16.15.m15.2c">\bar{\sigma}_{0}\in\mbox{SPS}({\bar{G}},{0})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.16.15.m15.2d">over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( over¯ start_ARG italic_G end_ARG , 0 )</annotation></semantics></math> (the bound is equal to <math alttext="0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.17.16.m16.1"><semantics id="S4.Thmtheorem5.p2.17.16.m16.1a"><mn id="S4.Thmtheorem5.p2.17.16.m16.1.1" xref="S4.Thmtheorem5.p2.17.16.m16.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.17.16.m16.1b"><cn id="S4.Thmtheorem5.p2.17.16.m16.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p2.17.16.m16.1.1">0</cn></annotation-xml></semantics></math> as Player <math alttext="0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.18.17.m17.1"><semantics id="S4.Thmtheorem5.p2.18.17.m17.1a"><mn id="S4.Thmtheorem5.p2.18.17.m17.1.1" xref="S4.Thmtheorem5.p2.18.17.m17.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.18.17.m17.1b"><cn id="S4.Thmtheorem5.p2.18.17.m17.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p2.18.17.m17.1.1">0</cn></annotation-xml></semantics></math>’s target is equal to <math alttext="\Last{h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p2.19.18.m18.1"><semantics id="S4.Thmtheorem5.p2.19.18.m18.1a"><mrow id="S4.Thmtheorem5.p2.19.18.m18.1.1" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem5.p2.19.18.m18.1.1.2" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.2b.cmml"><mtext id="S4.Thmtheorem5.p2.19.18.m18.1.1.2a" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.2b.cmml">\Last</mtext></merror><mo id="S4.Thmtheorem5.p2.19.18.m18.1.1.1" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem5.p2.19.18.m18.1.1.3" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p2.19.18.m18.1b"><apply id="S4.Thmtheorem5.p2.19.18.m18.1.1.cmml" xref="S4.Thmtheorem5.p2.19.18.m18.1.1"><times id="S4.Thmtheorem5.p2.19.18.m18.1.1.1.cmml" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.1"></times><ci id="S4.Thmtheorem5.p2.19.18.m18.1.1.2b.cmml" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.2"><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem5.p2.19.18.m18.1.1.2.cmml" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.2"><mtext id="S4.Thmtheorem5.p2.19.18.m18.1.1.2a.cmml" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.2">\Last</mtext></merror></ci><ci id="S4.Thmtheorem5.p2.19.18.m18.1.1.3.cmml" xref="S4.Thmtheorem5.p2.19.18.m18.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p2.19.18.m18.1c">\Last{h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p2.19.18.m18.1d">italic_h</annotation></semantics></math>).</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem5.p3"> <p class="ltx_p" id="S4.Thmtheorem5.p3.17"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p3.17.17">We can thus apply the induction hypothesis: by Equation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E3" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3</span></a>), there exists <math alttext="\bar{\tau}_{0}\in\mbox{SPS}({\bar{G}},{0})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.1.1.m1.2"><semantics id="S4.Thmtheorem5.p3.1.1.m1.2a"><mrow id="S4.Thmtheorem5.p3.1.1.m1.2.3" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.cmml"><msub id="S4.Thmtheorem5.p3.1.1.m1.2.3.2" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.cmml"><mi id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.2" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.1" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.3" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p3.1.1.m1.2.3.1" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem5.p3.1.1.m1.2.3.3" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.2" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.1" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.2" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem5.p3.1.1.m1.1.1" xref="S4.Thmtheorem5.p3.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem5.p3.1.1.m1.1.1.2" xref="S4.Thmtheorem5.p3.1.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem5.p3.1.1.m1.1.1.1" xref="S4.Thmtheorem5.p3.1.1.m1.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.2.2" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.1.cmml">,</mo><mn id="S4.Thmtheorem5.p3.1.1.m1.2.2" xref="S4.Thmtheorem5.p3.1.1.m1.2.2.cmml">0</mn><mo id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.1.1.m1.2b"><apply id="S4.Thmtheorem5.p3.1.1.m1.2.3.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3"><in id="S4.Thmtheorem5.p3.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.1"></in><apply id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2">subscript</csymbol><apply id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2"><ci id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.1.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.2.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p3.1.1.m1.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3"><times id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.1"></times><ci id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.2a.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.1.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.2.3.3.3.2"><apply id="S4.Thmtheorem5.p3.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.1.1"><ci id="S4.Thmtheorem5.p3.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p3.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p3.1.1.m1.1.1.2">𝐺</ci></apply><cn id="S4.Thmtheorem5.p3.1.1.m1.2.2.cmml" type="integer" xref="S4.Thmtheorem5.p3.1.1.m1.2.2">0</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.1.1.m1.2c">\bar{\tau}_{0}\in\mbox{SPS}({\bar{G}},{0})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.1.1.m1.2d">over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( over¯ start_ARG italic_G end_ARG , 0 )</annotation></semantics></math> without cycles such that <math alttext="\bar{\tau}_{0}\preceq\bar{\sigma}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.2.2.m2.1"><semantics id="S4.Thmtheorem5.p3.2.2.m2.1a"><mrow id="S4.Thmtheorem5.p3.2.2.m2.1.1" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.cmml"><msub id="S4.Thmtheorem5.p3.2.2.m2.1.1.2" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.cmml"><mi id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.2" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.1" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.3" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p3.2.2.m2.1.1.1" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.1.cmml">⪯</mo><msub id="S4.Thmtheorem5.p3.2.2.m2.1.1.3" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.cmml"><mi id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.2" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.1" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.3" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.2.2.m2.1b"><apply id="S4.Thmtheorem5.p3.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1"><csymbol cd="latexml" id="S4.Thmtheorem5.p3.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.1">precedes-or-equals</csymbol><apply id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.1.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2">subscript</csymbol><apply id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2"><ci id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.1.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.2.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p3.2.2.m2.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3">subscript</csymbol><apply id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2"><ci id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.1.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.2.cmml" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p3.2.2.m2.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.2.2.m2.1c">\bar{\tau}_{0}\preceq\bar{\sigma}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.2.2.m2.1d">over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and for all <math alttext="\bar{c}\in C_{\bar{\tau}_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.3.3.m3.1"><semantics id="S4.Thmtheorem5.p3.3.3.m3.1a"><mrow id="S4.Thmtheorem5.p3.3.3.m3.1.1" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.3.3.m3.1.1.2" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.2.cmml"><mi id="S4.Thmtheorem5.p3.3.3.m3.1.1.2.2" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.2.2.cmml">c</mi><mo id="S4.Thmtheorem5.p3.3.3.m3.1.1.2.1" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.3.3.m3.1.1.1" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p3.3.3.m3.1.1.3" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.cmml"><mi id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.2" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.2.cmml">C</mi><msub id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.cmml"><mi id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.2" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.1" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.3" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.3.3.m3.1b"><apply id="S4.Thmtheorem5.p3.3.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1"><in id="S4.Thmtheorem5.p3.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.1"></in><apply id="S4.Thmtheorem5.p3.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.2"><ci id="S4.Thmtheorem5.p3.3.3.m3.1.1.2.1.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.3.3.m3.1.1.2.2.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.2.2">𝑐</ci></apply><apply id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.2">𝐶</ci><apply id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3">subscript</csymbol><apply id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2"><ci id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.1.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.2.cmml" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.3.3.m3.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.3.3.m3.1c">\bar{c}\in C_{\bar{\tau}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.3.3.m3.1d">over¯ start_ARG italic_c end_ARG ∈ italic_C start_POSTSUBSCRIPT over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and all <math alttext="j\in\{1,\ldots,k\},\bar{c}_{j}\leq f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.4.4.m4.7"><semantics id="S4.Thmtheorem5.p3.4.4.m4.7a"><mrow id="S4.Thmtheorem5.p3.4.4.m4.7.7.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.3.cmml"><mrow id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.cmml"><mi id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.2" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.2.cmml">j</mi><mo id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.1" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.1.cmml">∈</mo><mrow id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.2" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.1.cmml"><mo id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.1.cmml">{</mo><mn id="S4.Thmtheorem5.p3.4.4.m4.1.1" xref="S4.Thmtheorem5.p3.4.4.m4.1.1.cmml">1</mn><mo id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.2.2" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p3.4.4.m4.2.2" mathvariant="normal" xref="S4.Thmtheorem5.p3.4.4.m4.2.2.cmml">…</mi><mo id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.2.3" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p3.4.4.m4.3.3" xref="S4.Thmtheorem5.p3.4.4.m4.3.3.cmml">k</mi><mo id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.2.4" stretchy="false" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.1.cmml">}</mo></mrow></mrow><mo id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.3" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.3a.cmml">,</mo><mrow id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.cmml"><msub id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.cmml"><mi id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.2.cmml">c</mi><mo id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.1" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.1.cmml">¯</mo></mover><mi id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.3" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.3.cmml">j</mi></msub><mo id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.1" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.1.cmml">≤</mo><mrow id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.cmml"><mi id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.2.cmml">f</mi><mo id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.1" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.1.cmml"><mo id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem5.p3.4.4.m4.4.4" xref="S4.Thmtheorem5.p3.4.4.m4.4.4.cmml">0</mn><mo id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.2.2" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p3.4.4.m4.5.5" xref="S4.Thmtheorem5.p3.4.4.m4.5.5.cmml">t</mi><mo id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.4.4.m4.7b"><apply id="S4.Thmtheorem5.p3.4.4.m4.7.7.3.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.4.4.m4.7.7.3a.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.3">formulae-sequence</csymbol><apply id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1"><in id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.1"></in><ci id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.2.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.2">𝑗</ci><set id="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.6.6.1.1.3.2"><cn id="S4.Thmtheorem5.p3.4.4.m4.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p3.4.4.m4.1.1">1</cn><ci id="S4.Thmtheorem5.p3.4.4.m4.2.2.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.2.2">…</ci><ci id="S4.Thmtheorem5.p3.4.4.m4.3.3.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.3.3">𝑘</ci></set></apply><apply id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2"><leq id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.1"></leq><apply id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2">subscript</csymbol><apply id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2"><ci id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.2.2">𝑐</ci></apply><ci id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.3.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.2.3">𝑗</ci></apply><apply id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3"><times id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.1"></times><ci id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.2.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.1.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.7.7.2.2.3.3.2"><cn id="S4.Thmtheorem5.p3.4.4.m4.4.4.cmml" type="integer" xref="S4.Thmtheorem5.p3.4.4.m4.4.4">0</cn><ci id="S4.Thmtheorem5.p3.4.4.m4.5.5.cmml" xref="S4.Thmtheorem5.p3.4.4.m4.5.5">𝑡</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.4.4.m4.7c">j\in\{1,\ldots,k\},\bar{c}_{j}\leq f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.4.4.m4.7d">italic_j ∈ { 1 , … , italic_k } , over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≤ italic_f ( 0 , italic_t )</annotation></semantics></math> or <math alttext="\bar{c}_{j}=\infty" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.5.5.m5.1"><semantics id="S4.Thmtheorem5.p3.5.5.m5.1a"><mrow id="S4.Thmtheorem5.p3.5.5.m5.1.1" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.cmml"><msub id="S4.Thmtheorem5.p3.5.5.m5.1.1.2" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.cmml"><mi id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.2" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.2.cmml">c</mi><mo id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.1" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.1.cmml">¯</mo></mover><mi id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.3" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.3.cmml">j</mi></msub><mo id="S4.Thmtheorem5.p3.5.5.m5.1.1.1" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.1.cmml">=</mo><mi id="S4.Thmtheorem5.p3.5.5.m5.1.1.3" mathvariant="normal" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.5.5.m5.1b"><apply id="S4.Thmtheorem5.p3.5.5.m5.1.1.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1"><eq id="S4.Thmtheorem5.p3.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.1"></eq><apply id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.1.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2">subscript</csymbol><apply id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2"><ci id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.1.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.2.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.2.2">𝑐</ci></apply><ci id="S4.Thmtheorem5.p3.5.5.m5.1.1.2.3.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.2.3">𝑗</ci></apply><infinity id="S4.Thmtheorem5.p3.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem5.p3.5.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.5.5.m5.1c">\bar{c}_{j}=\infty</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.5.5.m5.1d">over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∞</annotation></semantics></math>. Notice that one can choose for <math alttext="\bar{\sigma}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.6.6.m6.1"><semantics id="S4.Thmtheorem5.p3.6.6.m6.1a"><msub id="S4.Thmtheorem5.p3.6.6.m6.1.1" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.6.6.m6.1.1.2" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.2.cmml"><mi id="S4.Thmtheorem5.p3.6.6.m6.1.1.2.2" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p3.6.6.m6.1.1.2.1" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.6.6.m6.1.1.3" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.6.6.m6.1b"><apply id="S4.Thmtheorem5.p3.6.6.m6.1.1.cmml" xref="S4.Thmtheorem5.p3.6.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem5.p3.6.6.m6.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p3.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.2"><ci id="S4.Thmtheorem5.p3.6.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.6.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p3.6.6.m6.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.6.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.6.6.m6.1c">\bar{\sigma}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.6.6.m6.1d">over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> a set of witnesses derived from those of <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.7.7.m7.1"><semantics id="S4.Thmtheorem5.p3.7.7.m7.1a"><msub id="S4.Thmtheorem5.p3.7.7.m7.1.1" xref="S4.Thmtheorem5.p3.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem5.p3.7.7.m7.1.1.2" xref="S4.Thmtheorem5.p3.7.7.m7.1.1.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p3.7.7.m7.1.1.3" xref="S4.Thmtheorem5.p3.7.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.7.7.m7.1b"><apply id="S4.Thmtheorem5.p3.7.7.m7.1.1.cmml" xref="S4.Thmtheorem5.p3.7.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem5.p3.7.7.m7.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p3.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem5.p3.7.7.m7.1.1.2">𝜎</ci><cn id="S4.Thmtheorem5.p3.7.7.m7.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.7.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.7.7.m7.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.7.7.m7.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> having <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.8.8.m8.1"><semantics id="S4.Thmtheorem5.p3.8.8.m8.1a"><mi id="S4.Thmtheorem5.p3.8.8.m8.1.1" xref="S4.Thmtheorem5.p3.8.8.m8.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.8.8.m8.1b"><ci id="S4.Thmtheorem5.p3.8.8.m8.1.1.cmml" xref="S4.Thmtheorem5.p3.8.8.m8.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.8.8.m8.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.8.8.m8.1d">italic_h</annotation></semantics></math> as prefix: <math alttext="\textsf{Wit}_{\bar{\sigma}_{0}}=\{\bar{\pi}\in\textsf{Play}_{\bar{G}}\mid\bar{% h}\bar{\pi}\in\textsf{Wit}_{\sigma_{0}}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.9.9.m9.2"><semantics id="S4.Thmtheorem5.p3.9.9.m9.2a"><mrow id="S4.Thmtheorem5.p3.9.9.m9.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.cmml"><msub id="S4.Thmtheorem5.p3.9.9.m9.2.2.4" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.cmml"><mi id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.1" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.3" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.3" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.3.cmml">=</mo><mrow id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.3.cmml"><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.3.1.cmml">{</mo><mrow id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.cmml"><mi id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.2.cmml">π</mi><mo id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.1" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.1" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.2" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.2a.cmml">Play</mtext><mover accent="true" id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.cmml"><mi id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.2" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.2.cmml">G</mi><mo id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.1" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.1.cmml">¯</mo></mover></msub></mrow><mo fence="true" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.4" lspace="0em" rspace="0em" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.3.1.cmml">∣</mo><mrow id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.cmml"><mrow id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.2.cmml">h</mi><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.1" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.1" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.1.cmml">⁢</mo><mover accent="true" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.cmml"><mi id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.2.cmml">π</mi><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.1" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.1.cmml">¯</mo></mover></mrow><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.1" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.cmml"><mi id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.2" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.3" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.3.cmml">0</mn></msub></msub></mrow><mo id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.9.9.m9.2b"><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2"><eq id="S4.Thmtheorem5.p3.9.9.m9.2.2.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.3"></eq><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4">subscript</csymbol><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.2a.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.2">Wit</mtext></ci><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3">subscript</csymbol><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2"><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.4.3.3">0</cn></apply></apply><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2"><csymbol cd="latexml" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.3.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.3">conditional-set</csymbol><apply id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1"><in id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.1"></in><apply id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2"><ci id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.2.2">𝜋</ci></apply><apply id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.2a.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.2">Play</mtext></ci><apply id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3"><ci id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.1">¯</ci><ci id="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.1.1.1.1.1.3.3.2">𝐺</ci></apply></apply></apply><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2"><in id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.1"></in><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2"><times id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.1"></times><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2"><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.2.2">ℎ</ci></apply><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3"><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.1">¯</ci><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.2.3.2">𝜋</ci></apply></apply><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.2a.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.2">Wit</mtext></ci><apply id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.1.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.2.cmml" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.2">𝜎</ci><cn id="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.9.9.m9.2.2.2.2.2.3.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.9.9.m9.2c">\textsf{Wit}_{\bar{\sigma}_{0}}=\{\bar{\pi}\in\textsf{Play}_{\bar{G}}\mid\bar{% h}\bar{\pi}\in\textsf{Wit}_{\sigma_{0}}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.9.9.m9.2d">Wit start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { over¯ start_ARG italic_π end_ARG ∈ Play start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ∣ over¯ start_ARG italic_h end_ARG over¯ start_ARG italic_π end_ARG ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math>. In particular, there exists <math alttext="\bar{\rho}\in\textsf{Wit}_{\bar{\sigma}_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.10.10.m10.1"><semantics id="S4.Thmtheorem5.p3.10.10.m10.1a"><mrow id="S4.Thmtheorem5.p3.10.10.m10.1.1" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.10.10.m10.1.1.2" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.2.cmml"><mi id="S4.Thmtheorem5.p3.10.10.m10.1.1.2.2" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.2.2.cmml">ρ</mi><mo id="S4.Thmtheorem5.p3.10.10.m10.1.1.2.1" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.10.10.m10.1.1.1" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p3.10.10.m10.1.1.3" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.2" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.cmml"><mi id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.2" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.1" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.3" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.10.10.m10.1b"><apply id="S4.Thmtheorem5.p3.10.10.m10.1.1.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1"><in id="S4.Thmtheorem5.p3.10.10.m10.1.1.1.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.1"></in><apply id="S4.Thmtheorem5.p3.10.10.m10.1.1.2.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.2"><ci id="S4.Thmtheorem5.p3.10.10.m10.1.1.2.1.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.10.10.m10.1.1.2.2.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.2.2">𝜌</ci></apply><apply id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.1.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.2a.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.2.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.2">Wit</mtext></ci><apply id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3">subscript</csymbol><apply id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2"><ci id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.1.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.2.cmml" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.10.10.m10.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.10.10.m10.1c">\bar{\rho}\in\textsf{Wit}_{\bar{\sigma}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.10.10.m10.1d">over¯ start_ARG italic_ρ end_ARG ∈ Wit start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\rho=\bar{h}\bar{\rho}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.11.11.m11.1"><semantics id="S4.Thmtheorem5.p3.11.11.m11.1a"><mrow id="S4.Thmtheorem5.p3.11.11.m11.1.1" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.cmml"><mi id="S4.Thmtheorem5.p3.11.11.m11.1.1.2" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.2.cmml">ρ</mi><mo id="S4.Thmtheorem5.p3.11.11.m11.1.1.1" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.1.cmml">=</mo><mrow id="S4.Thmtheorem5.p3.11.11.m11.1.1.3" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.cmml"><mi id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.2" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.2.cmml">h</mi><mo id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.1" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.1" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.1.cmml">⁢</mo><mover accent="true" id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.cmml"><mi id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.2" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.2.cmml">ρ</mi><mo id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.1" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.11.11.m11.1b"><apply id="S4.Thmtheorem5.p3.11.11.m11.1.1.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1"><eq id="S4.Thmtheorem5.p3.11.11.m11.1.1.1.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.1"></eq><ci id="S4.Thmtheorem5.p3.11.11.m11.1.1.2.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.2">𝜌</ci><apply id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3"><times id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.1.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.1"></times><apply id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2"><ci id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.1.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.2.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.2.2">ℎ</ci></apply><apply id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3"><ci id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.1">¯</ci><ci id="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p3.11.11.m11.1.1.3.3.2">𝜌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.11.11.m11.1c">\rho=\bar{h}\bar{\rho}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.11.11.m11.1d">italic_ρ = over¯ start_ARG italic_h end_ARG over¯ start_ARG italic_ρ end_ARG</annotation></semantics></math>. By (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E8" title="In Proof 4.5 (Proof of Lemma 4.4) ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E9" title="In Proof 4.5 (Proof of Lemma 4.4) ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">9</span></a>), we have that <math alttext="\textsf{cost}_{T_{i}}(\bar{\rho})>f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.12.12.m12.3"><semantics id="S4.Thmtheorem5.p3.12.12.m12.3a"><mrow id="S4.Thmtheorem5.p3.12.12.m12.3.4" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.cmml"><mrow id="S4.Thmtheorem5.p3.12.12.m12.3.4.2" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.cmml"><msub id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.2" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.2a.cmml">cost</mtext><msub id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.cmml"><mi id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.2" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.2.cmml">T</mi><mi id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.3" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.3.cmml">i</mi></msub></msub><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.1" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.3.2" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.cmml"><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem5.p3.12.12.m12.1.1" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.cmml"><mi id="S4.Thmtheorem5.p3.12.12.m12.1.1.2" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.2.cmml">ρ</mi><mo id="S4.Thmtheorem5.p3.12.12.m12.1.1.1" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.1" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.1.cmml">></mo><mrow id="S4.Thmtheorem5.p3.12.12.m12.3.4.3" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.cmml"><mi id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.2" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.2.cmml">f</mi><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.1" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.2" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.1.cmml"><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem5.p3.12.12.m12.2.2" xref="S4.Thmtheorem5.p3.12.12.m12.2.2.cmml">0</mn><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.2.2" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p3.12.12.m12.3.3" xref="S4.Thmtheorem5.p3.12.12.m12.3.3.cmml">t</mi><mo id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.12.12.m12.3b"><apply id="S4.Thmtheorem5.p3.12.12.m12.3.4.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4"><gt id="S4.Thmtheorem5.p3.12.12.m12.3.4.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.1"></gt><apply id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2"><times id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.1"></times><apply id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2">subscript</csymbol><ci id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.2a.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.2.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.2">cost</mtext></ci><apply id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.2.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.2">𝑇</ci><ci id="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.3.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.2.3.3">𝑖</ci></apply></apply><apply id="S4.Thmtheorem5.p3.12.12.m12.1.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.2.3.2"><ci id="S4.Thmtheorem5.p3.12.12.m12.1.1.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p3.12.12.m12.1.1.2.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.1.1.2">𝜌</ci></apply></apply><apply id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3"><times id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.1"></times><ci id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.2.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.1.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.4.3.3.2"><cn id="S4.Thmtheorem5.p3.12.12.m12.2.2.cmml" type="integer" xref="S4.Thmtheorem5.p3.12.12.m12.2.2">0</cn><ci id="S4.Thmtheorem5.p3.12.12.m12.3.3.cmml" xref="S4.Thmtheorem5.p3.12.12.m12.3.3">𝑡</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.12.12.m12.3c">\textsf{cost}_{T_{i}}(\bar{\rho})>f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.12.12.m12.3d">cost start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over¯ start_ARG italic_ρ end_ARG ) > italic_f ( 0 , italic_t )</annotation></semantics></math> (<math alttext="\textsf{cost}_{T_{i}}(\bar{\rho})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.13.13.m13.1"><semantics id="S4.Thmtheorem5.p3.13.13.m13.1a"><mrow id="S4.Thmtheorem5.p3.13.13.m13.1.2" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.cmml"><msub id="S4.Thmtheorem5.p3.13.13.m13.1.2.2" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.2" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.2a.cmml">cost</mtext><msub id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.cmml"><mi id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.2" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.2.cmml">T</mi><mi id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.3" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.3.cmml">i</mi></msub></msub><mo id="S4.Thmtheorem5.p3.13.13.m13.1.2.1" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.13.13.m13.1.2.3.2" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.cmml"><mo id="S4.Thmtheorem5.p3.13.13.m13.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem5.p3.13.13.m13.1.1" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.cmml"><mi id="S4.Thmtheorem5.p3.13.13.m13.1.1.2" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.2.cmml">ρ</mi><mo id="S4.Thmtheorem5.p3.13.13.m13.1.1.1" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p3.13.13.m13.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.13.13.m13.1b"><apply id="S4.Thmtheorem5.p3.13.13.m13.1.2.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2"><times id="S4.Thmtheorem5.p3.13.13.m13.1.2.1.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.1"></times><apply id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.1.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.2a.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.2.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.2">cost</mtext></ci><apply id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.1.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.2.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.2">𝑇</ci><ci id="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.3.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.2.3.3">𝑖</ci></apply></apply><apply id="S4.Thmtheorem5.p3.13.13.m13.1.1.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.2.3.2"><ci id="S4.Thmtheorem5.p3.13.13.m13.1.1.1.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p3.13.13.m13.1.1.2.cmml" xref="S4.Thmtheorem5.p3.13.13.m13.1.1.2">𝜌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.13.13.m13.1c">\textsf{cost}_{T_{i}}(\bar{\rho})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.13.13.m13.1d">cost start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over¯ start_ARG italic_ρ end_ARG )</annotation></semantics></math> and <math alttext="\textsf{cost}_{T_{i}}(\rho)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.14.14.m14.1"><semantics id="S4.Thmtheorem5.p3.14.14.m14.1a"><mrow id="S4.Thmtheorem5.p3.14.14.m14.1.2" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.cmml"><msub id="S4.Thmtheorem5.p3.14.14.m14.1.2.2" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.2" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.2a.cmml">cost</mtext><msub id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.cmml"><mi id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.2" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.2.cmml">T</mi><mi id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.3" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.3.cmml">i</mi></msub></msub><mo id="S4.Thmtheorem5.p3.14.14.m14.1.2.1" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.14.14.m14.1.2.3.2" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.cmml"><mo id="S4.Thmtheorem5.p3.14.14.m14.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.cmml">(</mo><mi id="S4.Thmtheorem5.p3.14.14.m14.1.1" xref="S4.Thmtheorem5.p3.14.14.m14.1.1.cmml">ρ</mi><mo id="S4.Thmtheorem5.p3.14.14.m14.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.14.14.m14.1b"><apply id="S4.Thmtheorem5.p3.14.14.m14.1.2.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2"><times id="S4.Thmtheorem5.p3.14.14.m14.1.2.1.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.1"></times><apply id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.1.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.2a.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.2.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.2">cost</mtext></ci><apply id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.1.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.2.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.2">𝑇</ci><ci id="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.3.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.2.2.3.3">𝑖</ci></apply></apply><ci id="S4.Thmtheorem5.p3.14.14.m14.1.1.cmml" xref="S4.Thmtheorem5.p3.14.14.m14.1.1">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.14.14.m14.1c">\textsf{cost}_{T_{i}}(\rho)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.14.14.m14.1d">cost start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ )</annotation></semantics></math> differ by <math alttext="w(h)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.15.15.m15.1"><semantics id="S4.Thmtheorem5.p3.15.15.m15.1a"><mrow id="S4.Thmtheorem5.p3.15.15.m15.1.2" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.cmml"><mi id="S4.Thmtheorem5.p3.15.15.m15.1.2.2" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.2.cmml">w</mi><mo id="S4.Thmtheorem5.p3.15.15.m15.1.2.1" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.15.15.m15.1.2.3.2" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.cmml"><mo id="S4.Thmtheorem5.p3.15.15.m15.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.cmml">(</mo><mi id="S4.Thmtheorem5.p3.15.15.m15.1.1" xref="S4.Thmtheorem5.p3.15.15.m15.1.1.cmml">h</mi><mo id="S4.Thmtheorem5.p3.15.15.m15.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.15.15.m15.1b"><apply id="S4.Thmtheorem5.p3.15.15.m15.1.2.cmml" xref="S4.Thmtheorem5.p3.15.15.m15.1.2"><times id="S4.Thmtheorem5.p3.15.15.m15.1.2.1.cmml" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.1"></times><ci id="S4.Thmtheorem5.p3.15.15.m15.1.2.2.cmml" xref="S4.Thmtheorem5.p3.15.15.m15.1.2.2">𝑤</ci><ci id="S4.Thmtheorem5.p3.15.15.m15.1.1.cmml" xref="S4.Thmtheorem5.p3.15.15.m15.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.15.15.m15.1c">w(h)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.15.15.m15.1d">italic_w ( italic_h )</annotation></semantics></math>). As <math alttext="\bar{c}_{i}\leq f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.16.16.m16.2"><semantics id="S4.Thmtheorem5.p3.16.16.m16.2a"><mrow id="S4.Thmtheorem5.p3.16.16.m16.2.3" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.cmml"><msub id="S4.Thmtheorem5.p3.16.16.m16.2.3.2" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.cmml"><mi id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.2" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.2.cmml">c</mi><mo id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.1" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.1.cmml">¯</mo></mover><mi id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.3" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.3.cmml">i</mi></msub><mo id="S4.Thmtheorem5.p3.16.16.m16.2.3.1" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.1.cmml">≤</mo><mrow id="S4.Thmtheorem5.p3.16.16.m16.2.3.3" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.cmml"><mi id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.2" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.2.cmml">f</mi><mo id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.1" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.2" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem5.p3.16.16.m16.1.1" xref="S4.Thmtheorem5.p3.16.16.m16.1.1.cmml">0</mn><mo id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.2.2" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p3.16.16.m16.2.2" xref="S4.Thmtheorem5.p3.16.16.m16.2.2.cmml">t</mi><mo id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.16.16.m16.2b"><apply id="S4.Thmtheorem5.p3.16.16.m16.2.3.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3"><leq id="S4.Thmtheorem5.p3.16.16.m16.2.3.1.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.1"></leq><apply id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.1.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2">subscript</csymbol><apply id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2"><ci id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.1.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.2.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.2.2">𝑐</ci></apply><ci id="S4.Thmtheorem5.p3.16.16.m16.2.3.2.3.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.2.3">𝑖</ci></apply><apply id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3"><times id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.1.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.1"></times><ci id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.2.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.1.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.3.3.3.2"><cn id="S4.Thmtheorem5.p3.16.16.m16.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p3.16.16.m16.1.1">0</cn><ci id="S4.Thmtheorem5.p3.16.16.m16.2.2.cmml" xref="S4.Thmtheorem5.p3.16.16.m16.2.2">𝑡</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.16.16.m16.2c">\bar{c}_{i}\leq f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.16.16.m16.2d">over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_f ( 0 , italic_t )</annotation></semantics></math>, it follows that <math alttext="\bar{\tau}_{0}\prec\bar{\sigma}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p3.17.17.m17.1"><semantics id="S4.Thmtheorem5.p3.17.17.m17.1a"><mrow id="S4.Thmtheorem5.p3.17.17.m17.1.1" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.cmml"><msub id="S4.Thmtheorem5.p3.17.17.m17.1.1.2" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.cmml"><mi id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.2" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.1" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.3" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p3.17.17.m17.1.1.1" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.1.cmml">≺</mo><msub id="S4.Thmtheorem5.p3.17.17.m17.1.1.3" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.cmml"><mi id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.2" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.1" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.3" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p3.17.17.m17.1b"><apply id="S4.Thmtheorem5.p3.17.17.m17.1.1.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1"><csymbol cd="latexml" id="S4.Thmtheorem5.p3.17.17.m17.1.1.1.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.1">precedes</csymbol><apply id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.1.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2">subscript</csymbol><apply id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2"><ci id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.1.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.2.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p3.17.17.m17.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.1.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3">subscript</csymbol><apply id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2"><ci id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.1.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.2.cmml" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p3.17.17.m17.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p3.17.17.m17.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p3.17.17.m17.1c">\bar{\tau}_{0}\prec\bar{\sigma}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p3.17.17.m17.1d">over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem5.p4"> <p class="ltx_p" id="S4.Thmtheorem5.p4.28"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p4.28.28">We now want to transfer the previous solution <math alttext="\bar{\tau}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.1.1.m1.1"><semantics id="S4.Thmtheorem5.p4.1.1.m1.1a"><msub id="S4.Thmtheorem5.p4.1.1.m1.1.1" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.1.1.m1.1.1.2" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.1.1.m1.1.1.2.2" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p4.1.1.m1.1.1.2.1" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.1.1.m1.1.1.3" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.1.1.m1.1b"><apply id="S4.Thmtheorem5.p4.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p4.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p4.1.1.m1.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p4.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.2"><ci id="S4.Thmtheorem5.p4.1.1.m1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.1.1.m1.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p4.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.1.1.m1.1c">\bar{\tau}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.1.1.m1.1d">over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\bar{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.2.2.m2.1"><semantics id="S4.Thmtheorem5.p4.2.2.m2.1a"><mover accent="true" id="S4.Thmtheorem5.p4.2.2.m2.1.1" xref="S4.Thmtheorem5.p4.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem5.p4.2.2.m2.1.1.2" xref="S4.Thmtheorem5.p4.2.2.m2.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem5.p4.2.2.m2.1.1.1" xref="S4.Thmtheorem5.p4.2.2.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.2.2.m2.1b"><apply id="S4.Thmtheorem5.p4.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p4.2.2.m2.1.1"><ci id="S4.Thmtheorem5.p4.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p4.2.2.m2.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p4.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p4.2.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.2.2.m2.1c">\bar{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.2.2.m2.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math> to the subgame <math alttext="G_{|h}" class="ltx_math_unparsed" display="inline" id="S4.Thmtheorem5.p4.3.3.m3.1"><semantics id="S4.Thmtheorem5.p4.3.3.m3.1a"><msub id="S4.Thmtheorem5.p4.3.3.m3.1.1"><mi id="S4.Thmtheorem5.p4.3.3.m3.1.1.2">G</mi><mrow id="S4.Thmtheorem5.p4.3.3.m3.1.1.3"><mo fence="false" id="S4.Thmtheorem5.p4.3.3.m3.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.Thmtheorem5.p4.3.3.m3.1.1.3.2">h</mi></mrow></msub><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.3.3.m3.1b">G_{|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.3.3.m3.1c">italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> in a way to apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem1" title="Lemma 3.1 ‣ 3.2 Improving a Solution in a Subgame ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.1</span></a> and thus obtain the desired strategy <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.4.4.m4.1"><semantics id="S4.Thmtheorem5.p4.4.4.m4.1a"><msubsup id="S4.Thmtheorem5.p4.4.4.m4.1.1" xref="S4.Thmtheorem5.p4.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem5.p4.4.4.m4.1.1.2.2" xref="S4.Thmtheorem5.p4.4.4.m4.1.1.2.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p4.4.4.m4.1.1.3" xref="S4.Thmtheorem5.p4.4.4.m4.1.1.3.cmml">0</mn><mo id="S4.Thmtheorem5.p4.4.4.m4.1.1.2.3" xref="S4.Thmtheorem5.p4.4.4.m4.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.4.4.m4.1b"><apply id="S4.Thmtheorem5.p4.4.4.m4.1.1.cmml" xref="S4.Thmtheorem5.p4.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem5.p4.4.4.m4.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p4.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem5.p4.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.4.4.m4.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.4.4.m4.1.1">superscript</csymbol><ci id="S4.Thmtheorem5.p4.4.4.m4.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.4.4.m4.1.1.2.2">𝜎</ci><ci id="S4.Thmtheorem5.p4.4.4.m4.1.1.2.3.cmml" xref="S4.Thmtheorem5.p4.4.4.m4.1.1.2.3">′</ci></apply><cn id="S4.Thmtheorem5.p4.4.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.4.4.m4.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.4.4.m4.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.5.5.m5.1"><semantics id="S4.Thmtheorem5.p4.5.5.m5.1a"><mi id="S4.Thmtheorem5.p4.5.5.m5.1.1" xref="S4.Thmtheorem5.p4.5.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.5.5.m5.1b"><ci id="S4.Thmtheorem5.p4.5.5.m5.1.1.cmml" xref="S4.Thmtheorem5.p4.5.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.5.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.5.5.m5.1d">italic_G</annotation></semantics></math>. Recall that <math alttext="\bar{\sigma}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.6.6.m6.1"><semantics id="S4.Thmtheorem5.p4.6.6.m6.1a"><msub id="S4.Thmtheorem5.p4.6.6.m6.1.1" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.6.6.m6.1.1.2" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.6.6.m6.1.1.2.2" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p4.6.6.m6.1.1.2.1" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.6.6.m6.1.1.3" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.6.6.m6.1b"><apply id="S4.Thmtheorem5.p4.6.6.m6.1.1.cmml" xref="S4.Thmtheorem5.p4.6.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem5.p4.6.6.m6.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p4.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.2"><ci id="S4.Thmtheorem5.p4.6.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.6.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p4.6.6.m6.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.6.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.6.6.m6.1c">\bar{\sigma}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.6.6.m6.1d">over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> was constructed from <math alttext="\sigma_{0|h}\in\Sigma_{0|h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.7.7.m7.1"><semantics id="S4.Thmtheorem5.p4.7.7.m7.1a"><mrow id="S4.Thmtheorem5.p4.7.7.m7.1.1" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.cmml"><msub id="S4.Thmtheorem5.p4.7.7.m7.1.1.2" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.2" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.2.cmml">σ</mi><mrow id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.cmml"><mn id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.2" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.1" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.3" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.3.cmml">h</mi></mrow></msub><mo id="S4.Thmtheorem5.p4.7.7.m7.1.1.1" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p4.7.7.m7.1.1.3" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.2" mathvariant="normal" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.2.cmml">Σ</mi><mrow id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.cmml"><mn id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.2" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.1" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.3" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.7.7.m7.1b"><apply id="S4.Thmtheorem5.p4.7.7.m7.1.1.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1"><in id="S4.Thmtheorem5.p4.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.1"></in><apply id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.2">𝜎</ci><apply id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.1.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.1">conditional</csymbol><cn id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.2.cmml" type="integer" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.2">0</cn><ci id="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.3.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.2.3.3">ℎ</ci></apply></apply><apply id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.2">Σ</ci><apply id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.1">conditional</csymbol><cn id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.2">0</cn><ci id="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.3.cmml" xref="S4.Thmtheorem5.p4.7.7.m7.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.7.7.m7.1c">\sigma_{0|h}\in\Sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.7.7.m7.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math>. Let us conversely define the strategy <math alttext="\tau_{0}\in\Sigma_{0|h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.8.8.m8.1"><semantics id="S4.Thmtheorem5.p4.8.8.m8.1a"><mrow id="S4.Thmtheorem5.p4.8.8.m8.1.1" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.cmml"><msub id="S4.Thmtheorem5.p4.8.8.m8.1.1.2" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.8.8.m8.1.1.2.2" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.2.2.cmml">τ</mi><mn id="S4.Thmtheorem5.p4.8.8.m8.1.1.2.3" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p4.8.8.m8.1.1.1" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p4.8.8.m8.1.1.3" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.2" mathvariant="normal" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.2.cmml">Σ</mi><mrow id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.cmml"><mn id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.2" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.1" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.3" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.8.8.m8.1b"><apply id="S4.Thmtheorem5.p4.8.8.m8.1.1.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1"><in id="S4.Thmtheorem5.p4.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.1"></in><apply id="S4.Thmtheorem5.p4.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.8.8.m8.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem5.p4.8.8.m8.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.2.2">𝜏</ci><cn id="S4.Thmtheorem5.p4.8.8.m8.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.2">Σ</ci><apply id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.1">conditional</csymbol><cn id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.2">0</cn><ci id="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.3.cmml" xref="S4.Thmtheorem5.p4.8.8.m8.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.8.8.m8.1c">\tau_{0}\in\Sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.8.8.m8.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="\bar{\tau}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.9.9.m9.1"><semantics id="S4.Thmtheorem5.p4.9.9.m9.1a"><msub id="S4.Thmtheorem5.p4.9.9.m9.1.1" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.9.9.m9.1.1.2" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.9.9.m9.1.1.2.2" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p4.9.9.m9.1.1.2.1" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.9.9.m9.1.1.3" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.9.9.m9.1b"><apply id="S4.Thmtheorem5.p4.9.9.m9.1.1.cmml" xref="S4.Thmtheorem5.p4.9.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.9.9.m9.1.1.1.cmml" xref="S4.Thmtheorem5.p4.9.9.m9.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p4.9.9.m9.1.1.2.cmml" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.2"><ci id="S4.Thmtheorem5.p4.9.9.m9.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.9.9.m9.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p4.9.9.m9.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.9.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.9.9.m9.1c">\bar{\tau}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.9.9.m9.1d">over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>: <math alttext="\tau_{0}(\bar{h}g)=\bar{\tau}_{0}(g)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.10.10.m10.2"><semantics id="S4.Thmtheorem5.p4.10.10.m10.2a"><mrow id="S4.Thmtheorem5.p4.10.10.m10.2.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.cmml"><mrow id="S4.Thmtheorem5.p4.10.10.m10.2.2.1" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.cmml"><msub id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.cmml"><mi id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.2.cmml">τ</mi><mn id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.3" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.2.cmml">h</mi><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.1" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.1" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.3" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.3.cmml">g</mi></mrow><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.2.cmml">=</mo><mrow id="S4.Thmtheorem5.p4.10.10.m10.2.2.3" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.cmml"><msub id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.cmml"><mi id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.1" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.3" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.1" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.3.2" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.cmml"><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.cmml">(</mo><mi id="S4.Thmtheorem5.p4.10.10.m10.1.1" xref="S4.Thmtheorem5.p4.10.10.m10.1.1.cmml">g</mi><mo id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.10.10.m10.2b"><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2"><eq id="S4.Thmtheorem5.p4.10.10.m10.2.2.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.2"></eq><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1"><times id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.2"></times><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.2">𝜏</ci><cn id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.3.3">0</cn></apply><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1"><times id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.1"></times><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2"><ci id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.2.2">ℎ</ci></apply><ci id="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.1.1.1.1.3">𝑔</ci></apply></apply><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3"><times id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.1"></times><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2">subscript</csymbol><apply id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2"><ci id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.2.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.10.10.m10.2.2.3.2.3">0</cn></apply><ci id="S4.Thmtheorem5.p4.10.10.m10.1.1.cmml" xref="S4.Thmtheorem5.p4.10.10.m10.1.1">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.10.10.m10.2c">\tau_{0}(\bar{h}g)=\bar{\tau}_{0}(g)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.10.10.m10.2d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( over¯ start_ARG italic_h end_ARG italic_g ) = over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_g )</annotation></semantics></math> for all histories <math alttext="\bar{h}g\in\textsf{Hist}_{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.11.11.m11.1"><semantics id="S4.Thmtheorem5.p4.11.11.m11.1a"><mrow id="S4.Thmtheorem5.p4.11.11.m11.1.1" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.cmml"><mrow id="S4.Thmtheorem5.p4.11.11.m11.1.1.2" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.cmml"><mi id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.2" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.2.cmml">h</mi><mo id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.1" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.1" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.3" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.3.cmml">g</mi></mrow><mo id="S4.Thmtheorem5.p4.11.11.m11.1.1.1" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p4.11.11.m11.1.1.3" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p4.11.11.m11.1.1.3.2" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3.2a.cmml">Hist</mtext><mi id="S4.Thmtheorem5.p4.11.11.m11.1.1.3.3" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3.3.cmml">G</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.11.11.m11.1b"><apply id="S4.Thmtheorem5.p4.11.11.m11.1.1.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1"><in id="S4.Thmtheorem5.p4.11.11.m11.1.1.1.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.1"></in><apply id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2"><times id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.1"></times><apply id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2"><ci id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.1.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.2.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.2.2">ℎ</ci></apply><ci id="S4.Thmtheorem5.p4.11.11.m11.1.1.2.3.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.2.3">𝑔</ci></apply><apply id="S4.Thmtheorem5.p4.11.11.m11.1.1.3.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.11.11.m11.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.11.11.m11.1.1.3.2a.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p4.11.11.m11.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3.2">Hist</mtext></ci><ci id="S4.Thmtheorem5.p4.11.11.m11.1.1.3.3.cmml" xref="S4.Thmtheorem5.p4.11.11.m11.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.11.11.m11.1c">\bar{h}g\in\textsf{Hist}_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.11.11.m11.1d">over¯ start_ARG italic_h end_ARG italic_g ∈ Hist start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, we can choose the following sets of witnesses for <math alttext="\sigma_{0|h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.12.12.m12.1"><semantics id="S4.Thmtheorem5.p4.12.12.m12.1a"><msub id="S4.Thmtheorem5.p4.12.12.m12.1.1" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.cmml"><mi id="S4.Thmtheorem5.p4.12.12.m12.1.1.2" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.2.cmml">σ</mi><mrow id="S4.Thmtheorem5.p4.12.12.m12.1.1.3" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3.cmml"><mn id="S4.Thmtheorem5.p4.12.12.m12.1.1.3.2" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p4.12.12.m12.1.1.3.1" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p4.12.12.m12.1.1.3.3" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3.3.cmml">h</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.12.12.m12.1b"><apply id="S4.Thmtheorem5.p4.12.12.m12.1.1.cmml" xref="S4.Thmtheorem5.p4.12.12.m12.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.12.12.m12.1.1.1.cmml" xref="S4.Thmtheorem5.p4.12.12.m12.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p4.12.12.m12.1.1.2.cmml" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.2">𝜎</ci><apply id="S4.Thmtheorem5.p4.12.12.m12.1.1.3.cmml" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.12.12.m12.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3.1">conditional</csymbol><cn id="S4.Thmtheorem5.p4.12.12.m12.1.1.3.2.cmml" type="integer" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3.2">0</cn><ci id="S4.Thmtheorem5.p4.12.12.m12.1.1.3.3.cmml" xref="S4.Thmtheorem5.p4.12.12.m12.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.12.12.m12.1c">\sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.12.12.m12.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tau_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.13.13.m13.1"><semantics id="S4.Thmtheorem5.p4.13.13.m13.1a"><msub id="S4.Thmtheorem5.p4.13.13.m13.1.1" xref="S4.Thmtheorem5.p4.13.13.m13.1.1.cmml"><mi id="S4.Thmtheorem5.p4.13.13.m13.1.1.2" xref="S4.Thmtheorem5.p4.13.13.m13.1.1.2.cmml">τ</mi><mn id="S4.Thmtheorem5.p4.13.13.m13.1.1.3" xref="S4.Thmtheorem5.p4.13.13.m13.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.13.13.m13.1b"><apply id="S4.Thmtheorem5.p4.13.13.m13.1.1.cmml" xref="S4.Thmtheorem5.p4.13.13.m13.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.13.13.m13.1.1.1.cmml" xref="S4.Thmtheorem5.p4.13.13.m13.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p4.13.13.m13.1.1.2.cmml" xref="S4.Thmtheorem5.p4.13.13.m13.1.1.2">𝜏</ci><cn id="S4.Thmtheorem5.p4.13.13.m13.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.13.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.13.13.m13.1c">\tau_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.13.13.m13.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>: <math alttext="\textsf{Wit}_{\sigma_{0|h}}=\{\bar{h}\bar{\pi}\mid\bar{\pi}\in\textsf{Wit}_{% \bar{\sigma}_{0}}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.14.14.m14.2"><semantics id="S4.Thmtheorem5.p4.14.14.m14.2a"><mrow id="S4.Thmtheorem5.p4.14.14.m14.2.2" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.cmml"><msub id="S4.Thmtheorem5.p4.14.14.m14.2.2.4" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p4.14.14.m14.2.2.4.2" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.cmml"><mi id="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.2" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.2.cmml">σ</mi><mrow id="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3.cmml"><mn id="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3.2" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3.1" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3.3" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.4.3.3.3.cmml">h</mi></mrow></msub></msub><mo id="S4.Thmtheorem5.p4.14.14.m14.2.2.3" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.3.cmml">=</mo><mrow id="S4.Thmtheorem5.p4.14.14.m14.2.2.2.2" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.2.3.cmml"><mo id="S4.Thmtheorem5.p4.14.14.m14.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.2.3.1.cmml">{</mo><mrow id="S4.Thmtheorem5.p4.14.14.m14.1.1.1.1.1" xref="S4.Thmtheorem5.p4.14.14.m14.1.1.1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.14.14.m14.1.1.1.1.1.2" xref="S4.Thmtheorem5.p4.14.14.m14.1.1.1.1.1.2.cmml"><mi 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type="integer" xref="S4.Thmtheorem5.p4.14.14.m14.2.2.2.2.2.3.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.14.14.m14.2c">\textsf{Wit}_{\sigma_{0|h}}=\{\bar{h}\bar{\pi}\mid\bar{\pi}\in\textsf{Wit}_{% \bar{\sigma}_{0}}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.14.14.m14.2d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { over¯ start_ARG italic_h end_ARG over¯ start_ARG italic_π end_ARG ∣ over¯ start_ARG italic_π end_ARG ∈ Wit start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math> and <math alttext="\textsf{Wit}_{\tau_{0}}=\{\bar{h}\bar{\pi}\mid\bar{\pi}\in\textsf{Wit}_{\bar{% \tau}_{0}}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.15.15.m15.2"><semantics id="S4.Thmtheorem5.p4.15.15.m15.2a"><mrow 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xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.2" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.2.cmml">h</mi><mo id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.1" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.1" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.1.cmml">⁢</mo><mover accent="true" id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.2" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.2.cmml">π</mi><mo id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.1" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.1.cmml">¯</mo></mover></mrow><mo fence="true" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.4" lspace="0em" rspace="0em" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.3.1.cmml">∣</mo><mrow id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.2" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.2.cmml">π</mi><mo id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.1" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.1" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.2" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.cmml"><mi id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.2" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.1" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.3" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.3.cmml">0</mn></msub></msub></mrow><mo id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.15.15.m15.2b"><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2"><eq id="S4.Thmtheorem5.p4.15.15.m15.2.2.3.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.3"></eq><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4">subscript</csymbol><ci id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.2a.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4.2">Wit</mtext></ci><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3.2">𝜏</ci><cn id="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.4.3.3">0</cn></apply></apply><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.3.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.3.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.3">conditional-set</csymbol><apply id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1"><times id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.1"></times><apply id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2"><ci id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.2.2">ℎ</ci></apply><apply id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3"><ci id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.1">¯</ci><ci id="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.1.1.1.1.1.3.2">𝜋</ci></apply></apply><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2"><in id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.1"></in><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2"><ci id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.2.2">𝜋</ci></apply><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.2a.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.2">Wit</mtext></ci><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3">subscript</csymbol><apply id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2"><ci id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.1.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.2.cmml" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.15.15.m15.2.2.2.2.2.3.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.15.15.m15.2c">\textsf{Wit}_{\tau_{0}}=\{\bar{h}\bar{\pi}\mid\bar{\pi}\in\textsf{Wit}_{\bar{% \tau}_{0}}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.15.15.m15.2d">Wit start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { over¯ start_ARG italic_h end_ARG over¯ start_ARG italic_π end_ARG ∣ over¯ start_ARG italic_π end_ARG ∈ Wit start_POSTSUBSCRIPT over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math>. From <math alttext="\bar{\tau}_{0}\prec\bar{\sigma}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.16.16.m16.1"><semantics id="S4.Thmtheorem5.p4.16.16.m16.1a"><mrow id="S4.Thmtheorem5.p4.16.16.m16.1.1" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.cmml"><msub id="S4.Thmtheorem5.p4.16.16.m16.1.1.2" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.cmml"><mi id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.2" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.1" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.3" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p4.16.16.m16.1.1.1" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.1.cmml">≺</mo><msub id="S4.Thmtheorem5.p4.16.16.m16.1.1.3" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.cmml"><mi id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.2" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.1" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.3" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.16.16.m16.1b"><apply id="S4.Thmtheorem5.p4.16.16.m16.1.1.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.16.16.m16.1.1.1.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.1">precedes</csymbol><apply id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2">subscript</csymbol><apply id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2"><ci id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.1.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.2.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.2.2">𝜏</ci></apply><cn id="S4.Thmtheorem5.p4.16.16.m16.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3">subscript</csymbol><apply id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2"><ci id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.1.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.2.cmml" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p4.16.16.m16.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.16.16.m16.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.16.16.m16.1c">\bar{\tau}_{0}\prec\bar{\sigma}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.16.16.m16.1d">over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, it follows that <math alttext="\tau_{0}\prec\sigma_{0|h}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.17.17.m17.1"><semantics id="S4.Thmtheorem5.p4.17.17.m17.1a"><mrow id="S4.Thmtheorem5.p4.17.17.m17.1.1" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.cmml"><msub id="S4.Thmtheorem5.p4.17.17.m17.1.1.2" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.17.17.m17.1.1.2.2" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.2.2.cmml">τ</mi><mn id="S4.Thmtheorem5.p4.17.17.m17.1.1.2.3" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p4.17.17.m17.1.1.1" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.1.cmml">≺</mo><msub id="S4.Thmtheorem5.p4.17.17.m17.1.1.3" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.2" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.2.cmml">σ</mi><mrow id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.cmml"><mn id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.2" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.2.cmml">0</mn><mo fence="false" id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.1" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.1.cmml">|</mo><mi id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.3" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.3.cmml">h</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.17.17.m17.1b"><apply id="S4.Thmtheorem5.p4.17.17.m17.1.1.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.17.17.m17.1.1.1.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.1">precedes</csymbol><apply id="S4.Thmtheorem5.p4.17.17.m17.1.1.2.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.17.17.m17.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem5.p4.17.17.m17.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.2.2">𝜏</ci><cn id="S4.Thmtheorem5.p4.17.17.m17.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.2.3">0</cn></apply><apply id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.2">𝜎</ci><apply id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.1">conditional</csymbol><cn id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.2">0</cn><ci id="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.3.cmml" xref="S4.Thmtheorem5.p4.17.17.m17.1.1.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.17.17.m17.1c">\tau_{0}\prec\sigma_{0|h}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.17.17.m17.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 | italic_h end_POSTSUBSCRIPT</annotation></semantics></math> (again, the Pareto-optimal costs of <math alttext="\bar{\tau}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.18.18.m18.1"><semantics id="S4.Thmtheorem5.p4.18.18.m18.1a"><mover accent="true" id="S4.Thmtheorem5.p4.18.18.m18.1.1" xref="S4.Thmtheorem5.p4.18.18.m18.1.1.cmml"><mi id="S4.Thmtheorem5.p4.18.18.m18.1.1.2" xref="S4.Thmtheorem5.p4.18.18.m18.1.1.2.cmml">τ</mi><mo id="S4.Thmtheorem5.p4.18.18.m18.1.1.1" xref="S4.Thmtheorem5.p4.18.18.m18.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.18.18.m18.1b"><apply id="S4.Thmtheorem5.p4.18.18.m18.1.1.cmml" xref="S4.Thmtheorem5.p4.18.18.m18.1.1"><ci id="S4.Thmtheorem5.p4.18.18.m18.1.1.1.cmml" xref="S4.Thmtheorem5.p4.18.18.m18.1.1.1">¯</ci><ci id="S4.Thmtheorem5.p4.18.18.m18.1.1.2.cmml" xref="S4.Thmtheorem5.p4.18.18.m18.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.18.18.m18.1c">\bar{\tau}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.18.18.m18.1d">over¯ start_ARG italic_τ end_ARG</annotation></semantics></math> and <math alttext="\tau" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.19.19.m19.1"><semantics id="S4.Thmtheorem5.p4.19.19.m19.1a"><mi id="S4.Thmtheorem5.p4.19.19.m19.1.1" xref="S4.Thmtheorem5.p4.19.19.m19.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.19.19.m19.1b"><ci id="S4.Thmtheorem5.p4.19.19.m19.1.1.cmml" xref="S4.Thmtheorem5.p4.19.19.m19.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.19.19.m19.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.19.19.m19.1d">italic_τ</annotation></semantics></math> differ by <math alttext="w(h)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.20.20.m20.1"><semantics id="S4.Thmtheorem5.p4.20.20.m20.1a"><mrow id="S4.Thmtheorem5.p4.20.20.m20.1.2" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.cmml"><mi id="S4.Thmtheorem5.p4.20.20.m20.1.2.2" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.2.cmml">w</mi><mo id="S4.Thmtheorem5.p4.20.20.m20.1.2.1" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p4.20.20.m20.1.2.3.2" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.cmml"><mo id="S4.Thmtheorem5.p4.20.20.m20.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.cmml">(</mo><mi id="S4.Thmtheorem5.p4.20.20.m20.1.1" xref="S4.Thmtheorem5.p4.20.20.m20.1.1.cmml">h</mi><mo id="S4.Thmtheorem5.p4.20.20.m20.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.20.20.m20.1b"><apply id="S4.Thmtheorem5.p4.20.20.m20.1.2.cmml" xref="S4.Thmtheorem5.p4.20.20.m20.1.2"><times id="S4.Thmtheorem5.p4.20.20.m20.1.2.1.cmml" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.1"></times><ci id="S4.Thmtheorem5.p4.20.20.m20.1.2.2.cmml" xref="S4.Thmtheorem5.p4.20.20.m20.1.2.2">𝑤</ci><ci id="S4.Thmtheorem5.p4.20.20.m20.1.1.cmml" xref="S4.Thmtheorem5.p4.20.20.m20.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.20.20.m20.1c">w(h)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.20.20.m20.1d">italic_w ( italic_h )</annotation></semantics></math>, and so do the ones of <math alttext="\bar{\sigma}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.21.21.m21.1"><semantics id="S4.Thmtheorem5.p4.21.21.m21.1a"><msub id="S4.Thmtheorem5.p4.21.21.m21.1.1" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p4.21.21.m21.1.1.2" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.21.21.m21.1.1.2.2" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.2.2.cmml">σ</mi><mo id="S4.Thmtheorem5.p4.21.21.m21.1.1.2.1" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.2.1.cmml">¯</mo></mover><mn id="S4.Thmtheorem5.p4.21.21.m21.1.1.3" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.21.21.m21.1b"><apply id="S4.Thmtheorem5.p4.21.21.m21.1.1.cmml" xref="S4.Thmtheorem5.p4.21.21.m21.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.21.21.m21.1.1.1.cmml" xref="S4.Thmtheorem5.p4.21.21.m21.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p4.21.21.m21.1.1.2.cmml" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.2"><ci id="S4.Thmtheorem5.p4.21.21.m21.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p4.21.21.m21.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.2.2">𝜎</ci></apply><cn id="S4.Thmtheorem5.p4.21.21.m21.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.21.21.m21.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.21.21.m21.1c">\bar{\sigma}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.21.21.m21.1d">over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.22.22.m22.1"><semantics id="S4.Thmtheorem5.p4.22.22.m22.1a"><msub id="S4.Thmtheorem5.p4.22.22.m22.1.1" xref="S4.Thmtheorem5.p4.22.22.m22.1.1.cmml"><mi id="S4.Thmtheorem5.p4.22.22.m22.1.1.2" xref="S4.Thmtheorem5.p4.22.22.m22.1.1.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p4.22.22.m22.1.1.3" xref="S4.Thmtheorem5.p4.22.22.m22.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.22.22.m22.1b"><apply id="S4.Thmtheorem5.p4.22.22.m22.1.1.cmml" xref="S4.Thmtheorem5.p4.22.22.m22.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.22.22.m22.1.1.1.cmml" xref="S4.Thmtheorem5.p4.22.22.m22.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p4.22.22.m22.1.1.2.cmml" xref="S4.Thmtheorem5.p4.22.22.m22.1.1.2">𝜎</ci><cn id="S4.Thmtheorem5.p4.22.22.m22.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.22.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.22.22.m22.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.22.22.m22.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>). Moreover <math alttext="\tau_{0}\in\mbox{SPS}({G_{|h}},{B})" class="ltx_math_unparsed" display="inline" id="S4.Thmtheorem5.p4.23.23.m23.2"><semantics id="S4.Thmtheorem5.p4.23.23.m23.2a"><mrow id="S4.Thmtheorem5.p4.23.23.m23.2.2"><msub id="S4.Thmtheorem5.p4.23.23.m23.2.2.3"><mi id="S4.Thmtheorem5.p4.23.23.m23.2.2.3.2">τ</mi><mn id="S4.Thmtheorem5.p4.23.23.m23.2.2.3.3">0</mn></msub><mo id="S4.Thmtheorem5.p4.23.23.m23.2.2.2">∈</mo><mrow id="S4.Thmtheorem5.p4.23.23.m23.2.2.1"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.3">SPS</mtext><mo id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.2">⁢</mo><mrow id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1"><mo id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.2" stretchy="false">(</mo><msub id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.1"><mi id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.1.2">G</mi><mrow id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.1.3"><mo fence="false" id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.1.3.2">h</mi></mrow></msub><mo id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.3">,</mo><mi id="S4.Thmtheorem5.p4.23.23.m23.1.1">B</mi><mo id="S4.Thmtheorem5.p4.23.23.m23.2.2.1.1.1.4" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.23.23.m23.2b">\tau_{0}\in\mbox{SPS}({G_{|h}},{B})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.23.23.m23.2c">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G start_POSTSUBSCRIPT | italic_h end_POSTSUBSCRIPT , italic_B )</annotation></semantics></math> since <math alttext="h" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.24.24.m24.1"><semantics id="S4.Thmtheorem5.p4.24.24.m24.1a"><mi id="S4.Thmtheorem5.p4.24.24.m24.1.1" xref="S4.Thmtheorem5.p4.24.24.m24.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.24.24.m24.1b"><ci id="S4.Thmtheorem5.p4.24.24.m24.1.1.cmml" xref="S4.Thmtheorem5.p4.24.24.m24.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.24.24.m24.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.24.24.m24.1d">italic_h</annotation></semantics></math> visits Player <math alttext="0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.25.25.m25.1"><semantics id="S4.Thmtheorem5.p4.25.25.m25.1a"><mn id="S4.Thmtheorem5.p4.25.25.m25.1.1" xref="S4.Thmtheorem5.p4.25.25.m25.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.25.25.m25.1b"><cn id="S4.Thmtheorem5.p4.25.25.m25.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p4.25.25.m25.1.1">0</cn></annotation-xml></semantics></math>’s target. Hence, by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem1" title="Lemma 3.1 ‣ 3.2 Improving a Solution in a Subgame ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.1</span></a>, the strategy <math alttext="\sigma^{\prime}_{0}=\sigma_{0}[h\rightarrow\tau_{0}]" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.26.26.m26.1"><semantics id="S4.Thmtheorem5.p4.26.26.m26.1a"><mrow id="S4.Thmtheorem5.p4.26.26.m26.1.1" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.cmml"><msubsup id="S4.Thmtheorem5.p4.26.26.m26.1.1.3" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.26.26.m26.1.1.3.2.2" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p4.26.26.m26.1.1.3.3" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.3.3.cmml">0</mn><mo id="S4.Thmtheorem5.p4.26.26.m26.1.1.3.2.3" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.3.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem5.p4.26.26.m26.1.1.2" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.2.cmml">=</mo><mrow id="S4.Thmtheorem5.p4.26.26.m26.1.1.1" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.cmml"><msub id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.2" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.3" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.3.cmml">0</mn></msub><mo id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.2" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.2.cmml"><mo id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.2" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.2.cmml">h</mi><mo id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.1" stretchy="false" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.1.cmml">→</mo><msub id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.2" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.2.cmml">τ</mi><mn id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.3" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.26.26.m26.1b"><apply id="S4.Thmtheorem5.p4.26.26.m26.1.1.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1"><eq id="S4.Thmtheorem5.p4.26.26.m26.1.1.2.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.2"></eq><apply id="S4.Thmtheorem5.p4.26.26.m26.1.1.3.cmml" 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xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.3.3">0</cn></apply><apply id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1"><ci id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.2">ℎ</ci><apply id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.2">𝜏</ci><cn id="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.26.26.m26.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.26.26.m26.1c">\sigma^{\prime}_{0}=\sigma_{0}[h\rightarrow\tau_{0}]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.26.26.m26.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_h → italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math> is a solution in <math alttext="\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.27.27.m27.2"><semantics id="S4.Thmtheorem5.p4.27.27.m27.2a"><mrow id="S4.Thmtheorem5.p4.27.27.m27.2.3" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p4.27.27.m27.2.3.2" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem5.p4.27.27.m27.2.3.1" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p4.27.27.m27.2.3.3.2" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.3.1.cmml"><mo id="S4.Thmtheorem5.p4.27.27.m27.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem5.p4.27.27.m27.1.1" xref="S4.Thmtheorem5.p4.27.27.m27.1.1.cmml">G</mi><mo id="S4.Thmtheorem5.p4.27.27.m27.2.3.3.2.2" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p4.27.27.m27.2.2" xref="S4.Thmtheorem5.p4.27.27.m27.2.2.cmml">B</mi><mo id="S4.Thmtheorem5.p4.27.27.m27.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.27.27.m27.2b"><apply id="S4.Thmtheorem5.p4.27.27.m27.2.3.cmml" xref="S4.Thmtheorem5.p4.27.27.m27.2.3"><times id="S4.Thmtheorem5.p4.27.27.m27.2.3.1.cmml" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.1"></times><ci id="S4.Thmtheorem5.p4.27.27.m27.2.3.2a.cmml" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem5.p4.27.27.m27.2.3.2.cmml" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem5.p4.27.27.m27.2.3.3.1.cmml" xref="S4.Thmtheorem5.p4.27.27.m27.2.3.3.2"><ci id="S4.Thmtheorem5.p4.27.27.m27.1.1.cmml" xref="S4.Thmtheorem5.p4.27.27.m27.1.1">𝐺</ci><ci id="S4.Thmtheorem5.p4.27.27.m27.2.2.cmml" xref="S4.Thmtheorem5.p4.27.27.m27.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.27.27.m27.2c">\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.27.27.m27.2d">SPS ( italic_G , italic_B )</annotation></semantics></math> and <math alttext="\sigma_{0}^{\prime}\prec\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p4.28.28.m28.1"><semantics id="S4.Thmtheorem5.p4.28.28.m28.1a"><mrow id="S4.Thmtheorem5.p4.28.28.m28.1.1" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.cmml"><msubsup id="S4.Thmtheorem5.p4.28.28.m28.1.1.2" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2.cmml"><mi id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.2" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.3" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.3.cmml">0</mn><mo id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.3" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem5.p4.28.28.m28.1.1.1" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.1.cmml">≺</mo><msub id="S4.Thmtheorem5.p4.28.28.m28.1.1.3" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.3.cmml"><mi id="S4.Thmtheorem5.p4.28.28.m28.1.1.3.2" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p4.28.28.m28.1.1.3.3" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p4.28.28.m28.1b"><apply id="S4.Thmtheorem5.p4.28.28.m28.1.1.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1"><csymbol cd="latexml" id="S4.Thmtheorem5.p4.28.28.m28.1.1.1.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.1">precedes</csymbol><apply id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.1.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2">superscript</csymbol><apply id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.1.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.2.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.2">𝜎</ci><cn id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2.2.3">0</cn></apply><ci id="S4.Thmtheorem5.p4.28.28.m28.1.1.2.3.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.2.3">′</ci></apply><apply id="S4.Thmtheorem5.p4.28.28.m28.1.1.3.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p4.28.28.m28.1.1.3.1.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p4.28.28.m28.1.1.3.2.cmml" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem5.p4.28.28.m28.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p4.28.28.m28.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p4.28.28.m28.1c">\sigma_{0}^{\prime}\prec\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p4.28.28.m28.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≺ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem5.p5"> <p class="ltx_p" id="S4.Thmtheorem5.p5.1"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p5.1.1">We repeat the process described above as long as there remain costs <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p5.1.1.m1.1"><semantics id="S4.Thmtheorem5.p5.1.1.m1.1a"><mrow id="S4.Thmtheorem5.p5.1.1.m1.1.1" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem5.p5.1.1.m1.1.1.2" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem5.p5.1.1.m1.1.1.1" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p5.1.1.m1.1.1.3" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.2" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.2.cmml">C</mi><msub id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.cmml"><mi id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.2" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.3" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p5.1.1.m1.1b"><apply id="S4.Thmtheorem5.p5.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1"><in id="S4.Thmtheorem5.p5.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem5.p5.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.2">𝑐</ci><apply id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.2">𝐶</ci><apply id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.1.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.2.cmml" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p5.1.1.m1.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p5.1.1.m1.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p5.1.1.m1.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> that are too large. The process terminates as we are always building strictly better strategies. If the resulting strategy is not without cycles, we can eliminate them, one by one, to get a better strategy by applying Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>. This second process also terminates by the strict reduction of the length of the witness set (by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>).</span></p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Bounding the Minimum Component of Pareto-Optimal Costs</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.10">To prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>, in view of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem4" title="Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.4</span></a>, our last step is to provide a bound on <math alttext="c_{min}" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><msub id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml">c</mi><mrow id="S4.SS3.p1.1.m1.1.1.3" xref="S4.SS3.p1.1.m1.1.1.3.cmml"><mi id="S4.SS3.p1.1.m1.1.1.3.2" xref="S4.SS3.p1.1.m1.1.1.3.2.cmml">m</mi><mo id="S4.SS3.p1.1.m1.1.1.3.1" xref="S4.SS3.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.1.m1.1.1.3.3" xref="S4.SS3.p1.1.m1.1.1.3.3.cmml">i</mi><mo id="S4.SS3.p1.1.m1.1.1.3.1a" xref="S4.SS3.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.1.m1.1.1.3.4" xref="S4.SS3.p1.1.m1.1.1.3.4.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><apply id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2">𝑐</ci><apply id="S4.SS3.p1.1.m1.1.1.3.cmml" xref="S4.SS3.p1.1.m1.1.1.3"><times id="S4.SS3.p1.1.m1.1.1.3.1.cmml" xref="S4.SS3.p1.1.m1.1.1.3.1"></times><ci id="S4.SS3.p1.1.m1.1.1.3.2.cmml" xref="S4.SS3.p1.1.m1.1.1.3.2">𝑚</ci><ci id="S4.SS3.p1.1.m1.1.1.3.3.cmml" xref="S4.SS3.p1.1.m1.1.1.3.3">𝑖</ci><ci id="S4.SS3.p1.1.m1.1.1.3.4.cmml" xref="S4.SS3.p1.1.m1.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">c_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, the minimum component of each Pareto-optimal cost <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mrow id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml"><mi id="S4.SS3.p1.2.m2.1.1.2" xref="S4.SS3.p1.2.m2.1.1.2.cmml">c</mi><mo id="S4.SS3.p1.2.m2.1.1.1" xref="S4.SS3.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S4.SS3.p1.2.m2.1.1.3" xref="S4.SS3.p1.2.m2.1.1.3.cmml"><mi id="S4.SS3.p1.2.m2.1.1.3.2" xref="S4.SS3.p1.2.m2.1.1.3.2.cmml">C</mi><msub id="S4.SS3.p1.2.m2.1.1.3.3" xref="S4.SS3.p1.2.m2.1.1.3.3.cmml"><mi id="S4.SS3.p1.2.m2.1.1.3.3.2" xref="S4.SS3.p1.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S4.SS3.p1.2.m2.1.1.3.3.3" xref="S4.SS3.p1.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><apply id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1"><in id="S4.SS3.p1.2.m2.1.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1.1"></in><ci id="S4.SS3.p1.2.m2.1.1.2.cmml" xref="S4.SS3.p1.2.m2.1.1.2">𝑐</ci><apply id="S4.SS3.p1.2.m2.1.1.3.cmml" xref="S4.SS3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.2.m2.1.1.3.1.cmml" xref="S4.SS3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS3.p1.2.m2.1.1.3.2.cmml" xref="S4.SS3.p1.2.m2.1.1.3.2">𝐶</ci><apply id="S4.SS3.p1.2.m2.1.1.3.3.cmml" xref="S4.SS3.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.2.m2.1.1.3.3.1.cmml" xref="S4.SS3.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S4.SS3.p1.2.m2.1.1.3.3.2.cmml" xref="S4.SS3.p1.2.m2.1.1.3.3.2">𝜎</ci><cn id="S4.SS3.p1.2.m2.1.1.3.3.3.cmml" type="integer" xref="S4.SS3.p1.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Notice that if <math alttext="c_{min}=\infty" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><mrow id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml"><msub id="S4.SS3.p1.3.m3.1.1.2" xref="S4.SS3.p1.3.m3.1.1.2.cmml"><mi id="S4.SS3.p1.3.m3.1.1.2.2" xref="S4.SS3.p1.3.m3.1.1.2.2.cmml">c</mi><mrow id="S4.SS3.p1.3.m3.1.1.2.3" xref="S4.SS3.p1.3.m3.1.1.2.3.cmml"><mi id="S4.SS3.p1.3.m3.1.1.2.3.2" xref="S4.SS3.p1.3.m3.1.1.2.3.2.cmml">m</mi><mo id="S4.SS3.p1.3.m3.1.1.2.3.1" xref="S4.SS3.p1.3.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.3.m3.1.1.2.3.3" xref="S4.SS3.p1.3.m3.1.1.2.3.3.cmml">i</mi><mo id="S4.SS3.p1.3.m3.1.1.2.3.1a" xref="S4.SS3.p1.3.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.3.m3.1.1.2.3.4" xref="S4.SS3.p1.3.m3.1.1.2.3.4.cmml">n</mi></mrow></msub><mo id="S4.SS3.p1.3.m3.1.1.1" xref="S4.SS3.p1.3.m3.1.1.1.cmml">=</mo><mi id="S4.SS3.p1.3.m3.1.1.3" mathvariant="normal" xref="S4.SS3.p1.3.m3.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><apply id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1"><eq id="S4.SS3.p1.3.m3.1.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1.1"></eq><apply id="S4.SS3.p1.3.m3.1.1.2.cmml" xref="S4.SS3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m3.1.1.2.1.cmml" xref="S4.SS3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.3.m3.1.1.2.2.cmml" xref="S4.SS3.p1.3.m3.1.1.2.2">𝑐</ci><apply id="S4.SS3.p1.3.m3.1.1.2.3.cmml" xref="S4.SS3.p1.3.m3.1.1.2.3"><times id="S4.SS3.p1.3.m3.1.1.2.3.1.cmml" xref="S4.SS3.p1.3.m3.1.1.2.3.1"></times><ci id="S4.SS3.p1.3.m3.1.1.2.3.2.cmml" xref="S4.SS3.p1.3.m3.1.1.2.3.2">𝑚</ci><ci id="S4.SS3.p1.3.m3.1.1.2.3.3.cmml" xref="S4.SS3.p1.3.m3.1.1.2.3.3">𝑖</ci><ci id="S4.SS3.p1.3.m3.1.1.2.3.4.cmml" xref="S4.SS3.p1.3.m3.1.1.2.3.4">𝑛</ci></apply></apply><infinity id="S4.SS3.p1.3.m3.1.1.3.cmml" xref="S4.SS3.p1.3.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">c_{min}=\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT = ∞</annotation></semantics></math>, then all the components of <math alttext="c" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><mi id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><ci id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">italic_c</annotation></semantics></math> are equal to <math alttext="\infty" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m5.1"><semantics id="S4.SS3.p1.5.m5.1a"><mi id="S4.SS3.p1.5.m5.1.1" mathvariant="normal" xref="S4.SS3.p1.5.m5.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m5.1b"><infinity id="S4.SS3.p1.5.m5.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m5.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m5.1d">∞</annotation></semantics></math>. In this case, <math alttext="C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m6.4"><semantics id="S4.SS3.p1.6.m6.4a"><mrow id="S4.SS3.p1.6.m6.4.4" xref="S4.SS3.p1.6.m6.4.4.cmml"><msub id="S4.SS3.p1.6.m6.4.4.3" xref="S4.SS3.p1.6.m6.4.4.3.cmml"><mi id="S4.SS3.p1.6.m6.4.4.3.2" xref="S4.SS3.p1.6.m6.4.4.3.2.cmml">C</mi><msub id="S4.SS3.p1.6.m6.4.4.3.3" xref="S4.SS3.p1.6.m6.4.4.3.3.cmml"><mi id="S4.SS3.p1.6.m6.4.4.3.3.2" xref="S4.SS3.p1.6.m6.4.4.3.3.2.cmml">σ</mi><mn id="S4.SS3.p1.6.m6.4.4.3.3.3" xref="S4.SS3.p1.6.m6.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S4.SS3.p1.6.m6.4.4.2" xref="S4.SS3.p1.6.m6.4.4.2.cmml">=</mo><mrow id="S4.SS3.p1.6.m6.4.4.1.1" xref="S4.SS3.p1.6.m6.4.4.1.2.cmml"><mo id="S4.SS3.p1.6.m6.4.4.1.1.2" stretchy="false" xref="S4.SS3.p1.6.m6.4.4.1.2.cmml">{</mo><mrow id="S4.SS3.p1.6.m6.4.4.1.1.1.2" xref="S4.SS3.p1.6.m6.4.4.1.1.1.1.cmml"><mo id="S4.SS3.p1.6.m6.4.4.1.1.1.2.1" stretchy="false" xref="S4.SS3.p1.6.m6.4.4.1.1.1.1.cmml">(</mo><mi id="S4.SS3.p1.6.m6.1.1" mathvariant="normal" xref="S4.SS3.p1.6.m6.1.1.cmml">∞</mi><mo id="S4.SS3.p1.6.m6.4.4.1.1.1.2.2" xref="S4.SS3.p1.6.m6.4.4.1.1.1.1.cmml">,</mo><mi id="S4.SS3.p1.6.m6.2.2" mathvariant="normal" xref="S4.SS3.p1.6.m6.2.2.cmml">…</mi><mo id="S4.SS3.p1.6.m6.4.4.1.1.1.2.3" xref="S4.SS3.p1.6.m6.4.4.1.1.1.1.cmml">,</mo><mi id="S4.SS3.p1.6.m6.3.3" mathvariant="normal" xref="S4.SS3.p1.6.m6.3.3.cmml">∞</mi><mo id="S4.SS3.p1.6.m6.4.4.1.1.1.2.4" stretchy="false" xref="S4.SS3.p1.6.m6.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS3.p1.6.m6.4.4.1.1.3" stretchy="false" xref="S4.SS3.p1.6.m6.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.6.m6.4b"><apply id="S4.SS3.p1.6.m6.4.4.cmml" xref="S4.SS3.p1.6.m6.4.4"><eq id="S4.SS3.p1.6.m6.4.4.2.cmml" xref="S4.SS3.p1.6.m6.4.4.2"></eq><apply id="S4.SS3.p1.6.m6.4.4.3.cmml" xref="S4.SS3.p1.6.m6.4.4.3"><csymbol cd="ambiguous" id="S4.SS3.p1.6.m6.4.4.3.1.cmml" xref="S4.SS3.p1.6.m6.4.4.3">subscript</csymbol><ci id="S4.SS3.p1.6.m6.4.4.3.2.cmml" xref="S4.SS3.p1.6.m6.4.4.3.2">𝐶</ci><apply id="S4.SS3.p1.6.m6.4.4.3.3.cmml" xref="S4.SS3.p1.6.m6.4.4.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.6.m6.4.4.3.3.1.cmml" xref="S4.SS3.p1.6.m6.4.4.3.3">subscript</csymbol><ci id="S4.SS3.p1.6.m6.4.4.3.3.2.cmml" xref="S4.SS3.p1.6.m6.4.4.3.3.2">𝜎</ci><cn id="S4.SS3.p1.6.m6.4.4.3.3.3.cmml" type="integer" xref="S4.SS3.p1.6.m6.4.4.3.3.3">0</cn></apply></apply><set id="S4.SS3.p1.6.m6.4.4.1.2.cmml" xref="S4.SS3.p1.6.m6.4.4.1.1"><vector id="S4.SS3.p1.6.m6.4.4.1.1.1.1.cmml" xref="S4.SS3.p1.6.m6.4.4.1.1.1.2"><infinity id="S4.SS3.p1.6.m6.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1"></infinity><ci id="S4.SS3.p1.6.m6.2.2.cmml" xref="S4.SS3.p1.6.m6.2.2">…</ci><infinity id="S4.SS3.p1.6.m6.3.3.cmml" xref="S4.SS3.p1.6.m6.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m6.4c">C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m6.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { ( ∞ , … , ∞ ) }</annotation></semantics></math>, i.e., there is no play in <math alttext="\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m7.1"><semantics id="S4.SS3.p1.7.m7.1a"><msub id="S4.SS3.p1.7.m7.1.1" xref="S4.SS3.p1.7.m7.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.SS3.p1.7.m7.1.1.2" xref="S4.SS3.p1.7.m7.1.1.2a.cmml">Play</mtext><msub id="S4.SS3.p1.7.m7.1.1.3" xref="S4.SS3.p1.7.m7.1.1.3.cmml"><mi id="S4.SS3.p1.7.m7.1.1.3.2" xref="S4.SS3.p1.7.m7.1.1.3.2.cmml">σ</mi><mn id="S4.SS3.p1.7.m7.1.1.3.3" xref="S4.SS3.p1.7.m7.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m7.1b"><apply id="S4.SS3.p1.7.m7.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.7.m7.1.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS3.p1.7.m7.1.1.2a.cmml" xref="S4.SS3.p1.7.m7.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.SS3.p1.7.m7.1.1.2.cmml" xref="S4.SS3.p1.7.m7.1.1.2">Play</mtext></ci><apply id="S4.SS3.p1.7.m7.1.1.3.cmml" xref="S4.SS3.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.7.m7.1.1.3.1.cmml" xref="S4.SS3.p1.7.m7.1.1.3">subscript</csymbol><ci id="S4.SS3.p1.7.m7.1.1.3.2.cmml" xref="S4.SS3.p1.7.m7.1.1.3.2">𝜎</ci><cn id="S4.SS3.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S4.SS3.p1.7.m7.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.7.m7.1c">\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.7.m7.1d">Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> visiting Player <math alttext="1" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m8.1"><semantics id="S4.SS3.p1.8.m8.1a"><mn id="S4.SS3.p1.8.m8.1.1" xref="S4.SS3.p1.8.m8.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m8.1b"><cn id="S4.SS3.p1.8.m8.1.1.cmml" type="integer" xref="S4.SS3.p1.8.m8.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m8.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m8.1d">1</annotation></semantics></math>’s targets. Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a> provides a bound on <math alttext="c_{min}" class="ltx_Math" display="inline" id="S4.SS3.p1.9.m9.1"><semantics id="S4.SS3.p1.9.m9.1a"><msub id="S4.SS3.p1.9.m9.1.1" xref="S4.SS3.p1.9.m9.1.1.cmml"><mi id="S4.SS3.p1.9.m9.1.1.2" xref="S4.SS3.p1.9.m9.1.1.2.cmml">c</mi><mrow id="S4.SS3.p1.9.m9.1.1.3" xref="S4.SS3.p1.9.m9.1.1.3.cmml"><mi id="S4.SS3.p1.9.m9.1.1.3.2" xref="S4.SS3.p1.9.m9.1.1.3.2.cmml">m</mi><mo id="S4.SS3.p1.9.m9.1.1.3.1" xref="S4.SS3.p1.9.m9.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.9.m9.1.1.3.3" xref="S4.SS3.p1.9.m9.1.1.3.3.cmml">i</mi><mo id="S4.SS3.p1.9.m9.1.1.3.1a" xref="S4.SS3.p1.9.m9.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.9.m9.1.1.3.4" xref="S4.SS3.p1.9.m9.1.1.3.4.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.9.m9.1b"><apply id="S4.SS3.p1.9.m9.1.1.cmml" xref="S4.SS3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.9.m9.1.1.1.cmml" xref="S4.SS3.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p1.9.m9.1.1.2.cmml" xref="S4.SS3.p1.9.m9.1.1.2">𝑐</ci><apply id="S4.SS3.p1.9.m9.1.1.3.cmml" xref="S4.SS3.p1.9.m9.1.1.3"><times id="S4.SS3.p1.9.m9.1.1.3.1.cmml" xref="S4.SS3.p1.9.m9.1.1.3.1"></times><ci id="S4.SS3.p1.9.m9.1.1.3.2.cmml" xref="S4.SS3.p1.9.m9.1.1.3.2">𝑚</ci><ci id="S4.SS3.p1.9.m9.1.1.3.3.cmml" xref="S4.SS3.p1.9.m9.1.1.3.3">𝑖</ci><ci id="S4.SS3.p1.9.m9.1.1.3.4.cmml" xref="S4.SS3.p1.9.m9.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.9.m9.1c">c_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.9.m9.1d">italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math> when <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S4.SS3.p1.10.m10.4"><semantics id="S4.SS3.p1.10.m10.4a"><mrow id="S4.SS3.p1.10.m10.4.4" xref="S4.SS3.p1.10.m10.4.4.cmml"><msub id="S4.SS3.p1.10.m10.4.4.3" xref="S4.SS3.p1.10.m10.4.4.3.cmml"><mi id="S4.SS3.p1.10.m10.4.4.3.2" xref="S4.SS3.p1.10.m10.4.4.3.2.cmml">C</mi><msub id="S4.SS3.p1.10.m10.4.4.3.3" xref="S4.SS3.p1.10.m10.4.4.3.3.cmml"><mi id="S4.SS3.p1.10.m10.4.4.3.3.2" xref="S4.SS3.p1.10.m10.4.4.3.3.2.cmml">σ</mi><mn id="S4.SS3.p1.10.m10.4.4.3.3.3" xref="S4.SS3.p1.10.m10.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S4.SS3.p1.10.m10.4.4.2" xref="S4.SS3.p1.10.m10.4.4.2.cmml">≠</mo><mrow id="S4.SS3.p1.10.m10.4.4.1.1" xref="S4.SS3.p1.10.m10.4.4.1.2.cmml"><mo id="S4.SS3.p1.10.m10.4.4.1.1.2" stretchy="false" xref="S4.SS3.p1.10.m10.4.4.1.2.cmml">{</mo><mrow id="S4.SS3.p1.10.m10.4.4.1.1.1.2" xref="S4.SS3.p1.10.m10.4.4.1.1.1.1.cmml"><mo id="S4.SS3.p1.10.m10.4.4.1.1.1.2.1" stretchy="false" xref="S4.SS3.p1.10.m10.4.4.1.1.1.1.cmml">(</mo><mi id="S4.SS3.p1.10.m10.1.1" mathvariant="normal" xref="S4.SS3.p1.10.m10.1.1.cmml">∞</mi><mo id="S4.SS3.p1.10.m10.4.4.1.1.1.2.2" xref="S4.SS3.p1.10.m10.4.4.1.1.1.1.cmml">,</mo><mi id="S4.SS3.p1.10.m10.2.2" mathvariant="normal" xref="S4.SS3.p1.10.m10.2.2.cmml">…</mi><mo id="S4.SS3.p1.10.m10.4.4.1.1.1.2.3" xref="S4.SS3.p1.10.m10.4.4.1.1.1.1.cmml">,</mo><mi id="S4.SS3.p1.10.m10.3.3" mathvariant="normal" xref="S4.SS3.p1.10.m10.3.3.cmml">∞</mi><mo id="S4.SS3.p1.10.m10.4.4.1.1.1.2.4" stretchy="false" xref="S4.SS3.p1.10.m10.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS3.p1.10.m10.4.4.1.1.3" stretchy="false" xref="S4.SS3.p1.10.m10.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.10.m10.4b"><apply id="S4.SS3.p1.10.m10.4.4.cmml" xref="S4.SS3.p1.10.m10.4.4"><neq id="S4.SS3.p1.10.m10.4.4.2.cmml" xref="S4.SS3.p1.10.m10.4.4.2"></neq><apply id="S4.SS3.p1.10.m10.4.4.3.cmml" xref="S4.SS3.p1.10.m10.4.4.3"><csymbol cd="ambiguous" id="S4.SS3.p1.10.m10.4.4.3.1.cmml" xref="S4.SS3.p1.10.m10.4.4.3">subscript</csymbol><ci id="S4.SS3.p1.10.m10.4.4.3.2.cmml" xref="S4.SS3.p1.10.m10.4.4.3.2">𝐶</ci><apply id="S4.SS3.p1.10.m10.4.4.3.3.cmml" xref="S4.SS3.p1.10.m10.4.4.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.10.m10.4.4.3.3.1.cmml" xref="S4.SS3.p1.10.m10.4.4.3.3">subscript</csymbol><ci id="S4.SS3.p1.10.m10.4.4.3.3.2.cmml" xref="S4.SS3.p1.10.m10.4.4.3.3.2">𝜎</ci><cn id="S4.SS3.p1.10.m10.4.4.3.3.3.cmml" type="integer" xref="S4.SS3.p1.10.m10.4.4.3.3.3">0</cn></apply></apply><set id="S4.SS3.p1.10.m10.4.4.1.2.cmml" xref="S4.SS3.p1.10.m10.4.4.1.1"><vector id="S4.SS3.p1.10.m10.4.4.1.1.1.1.cmml" xref="S4.SS3.p1.10.m10.4.4.1.1.1.2"><infinity id="S4.SS3.p1.10.m10.1.1.cmml" xref="S4.SS3.p1.10.m10.1.1"></infinity><ci id="S4.SS3.p1.10.m10.2.2.cmml" xref="S4.SS3.p1.10.m10.2.2">…</ci><infinity id="S4.SS3.p1.10.m10.3.3.cmml" xref="S4.SS3.p1.10.m10.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.10.m10.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.10.m10.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.1.1.1">Lemma 4.6</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem6.p1.5.5">Let <math alttext="G\in\textsf{BinGames}_{t+1}" 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">G</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem6.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2a.cmml">BinGames</mtext><mrow id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml"><mi id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.2" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.2.cmml">t</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.1.cmml">+</mo><mn id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.3" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.cmml">1</mn></mrow></msub></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"><in id="S4.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.2">𝐺</ci><apply id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2a.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2">BinGames</mtext></ci><apply id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3"><plus id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.1"></plus><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.2">𝑡</ci><cn id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.1.1.m1.1c">G\in\textsf{BinGames}_{t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.1.1.m1.1d">italic_G ∈ BinGames start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a binary SP game with dimension <math alttext="t+1" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.2.m2.1"><semantics id="S4.Thmtheorem6.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem6.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.2.2.m2.1b"><apply id="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1"><plus id="S4.Thmtheorem6.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.1"></plus><ci id="S4.Thmtheorem6.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.2">𝑡</ci><cn id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.2.m2.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.2.m2.1d">italic_t + 1</annotation></semantics></math>, <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.3.3.m3.1"><semantics id="S4.Thmtheorem6.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem6.p1.3.3.m3.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.2.cmml">B</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.3.3.m3.1b"><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1"><in id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1"></in><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.2">𝐵</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.3.3.m3.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.3.3.m3.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.4.4.m4.2"><semantics id="S4.Thmtheorem6.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem6.p1.4.4.m4.2.3" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.cmml"><msub id="S4.Thmtheorem6.p1.4.4.m4.2.3.2" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.2.cmml"><mi id="S4.Thmtheorem6.p1.4.4.m4.2.3.2.2" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem6.p1.4.4.m4.2.3.2.3" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem6.p1.4.4.m4.2.3.1" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem6.p1.4.4.m4.2.3.3" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.2" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.1" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.2" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem6.p1.4.4.m4.1.1" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.cmml">G</mi><mo id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.2.2" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem6.p1.4.4.m4.2.2" xref="S4.Thmtheorem6.p1.4.4.m4.2.2.cmml">B</mi><mo id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.4.4.m4.2b"><apply id="S4.Thmtheorem6.p1.4.4.m4.2.3.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3"><in id="S4.Thmtheorem6.p1.4.4.m4.2.3.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.1"></in><apply id="S4.Thmtheorem6.p1.4.4.m4.2.3.2.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.4.4.m4.2.3.2.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem6.p1.4.4.m4.2.3.2.2.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem6.p1.4.4.m4.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3"><times id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.1"></times><ci id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.2a.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.2.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.3.3.3.2"><ci id="S4.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1">𝐺</ci><ci id="S4.Thmtheorem6.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.4.4.m4.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.4.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution without cycles and satisfying (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E7" title="In Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">7</span></a>). Suppose that <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.5.5.m5.4"><semantics id="S4.Thmtheorem6.p1.5.5.m5.4a"><mrow id="S4.Thmtheorem6.p1.5.5.m5.4.4" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.cmml"><msub id="S4.Thmtheorem6.p1.5.5.m5.4.4.3" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.cmml"><mi id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.2" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.2.cmml">C</mi><msub id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.cmml"><mi id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.2" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.3" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem6.p1.5.5.m5.4.4.2" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.2.cmml">≠</mo><mrow id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.2.cmml"><mo id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.2" stretchy="false" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.2.cmml">{</mo><mrow id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.2" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem6.p1.5.5.m5.1.1" mathvariant="normal" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.cmml">∞</mi><mo id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.2.2" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem6.p1.5.5.m5.2.2" mathvariant="normal" xref="S4.Thmtheorem6.p1.5.5.m5.2.2.cmml">…</mi><mo id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.2.3" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem6.p1.5.5.m5.3.3" mathvariant="normal" xref="S4.Thmtheorem6.p1.5.5.m5.3.3.cmml">∞</mi><mo id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.2.4" stretchy="false" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.3" stretchy="false" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.5.5.m5.4b"><apply id="S4.Thmtheorem6.p1.5.5.m5.4.4.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4"><neq id="S4.Thmtheorem6.p1.5.5.m5.4.4.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.2"></neq><apply id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3">subscript</csymbol><ci id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.2">𝐶</ci><apply id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3">subscript</csymbol><ci id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.2">𝜎</ci><cn id="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.3.3.3">0</cn></apply></apply><set id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1"><vector id="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.4.4.1.1.1.2"><infinity id="S4.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1"></infinity><ci id="S4.Thmtheorem6.p1.5.5.m5.2.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.2.2">…</ci><infinity id="S4.Thmtheorem6.p1.5.5.m5.3.3.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.5.5.m5.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.5.5.m5.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>. Then,</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx10"> <tbody id="S4.E10"><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 c\in C_{\sigma_{0}}:~{}c_{min}" class="ltx_Math" display="inline" id="S4.E10.m1.1"><semantics id="S4.E10.m1.1a"><mrow id="S4.E10.m1.1.1" xref="S4.E10.m1.1.1.cmml"><mrow id="S4.E10.m1.1.1.2" xref="S4.E10.m1.1.1.2.cmml"><mrow id="S4.E10.m1.1.1.2.2" xref="S4.E10.m1.1.1.2.2.cmml"><mo id="S4.E10.m1.1.1.2.2.1" rspace="0.167em" xref="S4.E10.m1.1.1.2.2.1.cmml">∀</mo><mi id="S4.E10.m1.1.1.2.2.2" xref="S4.E10.m1.1.1.2.2.2.cmml">c</mi></mrow><mo id="S4.E10.m1.1.1.2.1" xref="S4.E10.m1.1.1.2.1.cmml">∈</mo><msub id="S4.E10.m1.1.1.2.3" xref="S4.E10.m1.1.1.2.3.cmml"><mi id="S4.E10.m1.1.1.2.3.2" xref="S4.E10.m1.1.1.2.3.2.cmml">C</mi><msub id="S4.E10.m1.1.1.2.3.3" xref="S4.E10.m1.1.1.2.3.3.cmml"><mi id="S4.E10.m1.1.1.2.3.3.2" xref="S4.E10.m1.1.1.2.3.3.2.cmml">σ</mi><mn id="S4.E10.m1.1.1.2.3.3.3" xref="S4.E10.m1.1.1.2.3.3.3.cmml">0</mn></msub></msub></mrow><mo id="S4.E10.m1.1.1.1" lspace="0.278em" rspace="0.608em" xref="S4.E10.m1.1.1.1.cmml">:</mo><msub id="S4.E10.m1.1.1.3" xref="S4.E10.m1.1.1.3.cmml"><mi id="S4.E10.m1.1.1.3.2" xref="S4.E10.m1.1.1.3.2.cmml">c</mi><mrow id="S4.E10.m1.1.1.3.3" xref="S4.E10.m1.1.1.3.3.cmml"><mi id="S4.E10.m1.1.1.3.3.2" xref="S4.E10.m1.1.1.3.3.2.cmml">m</mi><mo id="S4.E10.m1.1.1.3.3.1" xref="S4.E10.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S4.E10.m1.1.1.3.3.3" xref="S4.E10.m1.1.1.3.3.3.cmml">i</mi><mo id="S4.E10.m1.1.1.3.3.1a" xref="S4.E10.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S4.E10.m1.1.1.3.3.4" xref="S4.E10.m1.1.1.3.3.4.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.E10.m1.1b"><apply id="S4.E10.m1.1.1.cmml" xref="S4.E10.m1.1.1"><ci id="S4.E10.m1.1.1.1.cmml" xref="S4.E10.m1.1.1.1">:</ci><apply id="S4.E10.m1.1.1.2.cmml" xref="S4.E10.m1.1.1.2"><in id="S4.E10.m1.1.1.2.1.cmml" xref="S4.E10.m1.1.1.2.1"></in><apply id="S4.E10.m1.1.1.2.2.cmml" xref="S4.E10.m1.1.1.2.2"><csymbol cd="latexml" id="S4.E10.m1.1.1.2.2.1.cmml" xref="S4.E10.m1.1.1.2.2.1">for-all</csymbol><ci id="S4.E10.m1.1.1.2.2.2.cmml" xref="S4.E10.m1.1.1.2.2.2">𝑐</ci></apply><apply id="S4.E10.m1.1.1.2.3.cmml" xref="S4.E10.m1.1.1.2.3"><csymbol cd="ambiguous" id="S4.E10.m1.1.1.2.3.1.cmml" xref="S4.E10.m1.1.1.2.3">subscript</csymbol><ci id="S4.E10.m1.1.1.2.3.2.cmml" xref="S4.E10.m1.1.1.2.3.2">𝐶</ci><apply id="S4.E10.m1.1.1.2.3.3.cmml" xref="S4.E10.m1.1.1.2.3.3"><csymbol cd="ambiguous" id="S4.E10.m1.1.1.2.3.3.1.cmml" xref="S4.E10.m1.1.1.2.3.3">subscript</csymbol><ci id="S4.E10.m1.1.1.2.3.3.2.cmml" xref="S4.E10.m1.1.1.2.3.3.2">𝜎</ci><cn id="S4.E10.m1.1.1.2.3.3.3.cmml" type="integer" xref="S4.E10.m1.1.1.2.3.3.3">0</cn></apply></apply></apply><apply id="S4.E10.m1.1.1.3.cmml" xref="S4.E10.m1.1.1.3"><csymbol cd="ambiguous" id="S4.E10.m1.1.1.3.1.cmml" xref="S4.E10.m1.1.1.3">subscript</csymbol><ci id="S4.E10.m1.1.1.3.2.cmml" xref="S4.E10.m1.1.1.3.2">𝑐</ci><apply id="S4.E10.m1.1.1.3.3.cmml" xref="S4.E10.m1.1.1.3.3"><times id="S4.E10.m1.1.1.3.3.1.cmml" xref="S4.E10.m1.1.1.3.3.1"></times><ci id="S4.E10.m1.1.1.3.3.2.cmml" xref="S4.E10.m1.1.1.3.3.2">𝑚</ci><ci id="S4.E10.m1.1.1.3.3.3.cmml" xref="S4.E10.m1.1.1.3.3.3">𝑖</ci><ci id="S4.E10.m1.1.1.3.3.4.cmml" xref="S4.E10.m1.1.1.3.3.4">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10.m1.1c">\displaystyle\forall c\in C_{\sigma_{0}}:~{}c_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.E10.m1.1d">∀ italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" 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end_POSTSUBSCRIPT ( | italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ) + 1 ) + 1 + italic_f ( 0 , italic_t ) ) ,</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">(10)</span></td> </tr></tbody> <tbody id="S4.E11"><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\mbox{ with }\quad|C_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S4.E11.m1.2"><semantics id="S4.E11.m1.2a"><mrow id="S4.E11.m1.2.2.1" xref="S4.E11.m1.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.E11.m1.1.1" xref="S4.E11.m1.1.1a.cmml">with </mtext><mspace id="S4.E11.m1.2.2.1.2" width="1em" xref="S4.E11.m1.2.2.2.cmml"></mspace><mrow id="S4.E11.m1.2.2.1.1.1" xref="S4.E11.m1.2.2.1.1.2.cmml"><mo id="S4.E11.m1.2.2.1.1.1.2" stretchy="false" xref="S4.E11.m1.2.2.1.1.2.1.cmml">|</mo><msub id="S4.E11.m1.2.2.1.1.1.1" xref="S4.E11.m1.2.2.1.1.1.1.cmml"><mi id="S4.E11.m1.2.2.1.1.1.1.2" xref="S4.E11.m1.2.2.1.1.1.1.2.cmml">C</mi><msub id="S4.E11.m1.2.2.1.1.1.1.3" xref="S4.E11.m1.2.2.1.1.1.1.3.cmml"><mi id="S4.E11.m1.2.2.1.1.1.1.3.2" xref="S4.E11.m1.2.2.1.1.1.1.3.2.cmml">σ</mi><mn id="S4.E11.m1.2.2.1.1.1.1.3.3" xref="S4.E11.m1.2.2.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.E11.m1.2.2.1.1.1.3" stretchy="false" xref="S4.E11.m1.2.2.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E11.m1.2b"><list id="S4.E11.m1.2.2.2.cmml" xref="S4.E11.m1.2.2.1"><ci id="S4.E11.m1.1.1a.cmml" xref="S4.E11.m1.1.1"><mtext class="ltx_mathvariant_italic" id="S4.E11.m1.1.1.cmml" xref="S4.E11.m1.1.1">with </mtext></ci><apply id="S4.E11.m1.2.2.1.1.2.cmml" xref="S4.E11.m1.2.2.1.1.1"><abs id="S4.E11.m1.2.2.1.1.2.1.cmml" xref="S4.E11.m1.2.2.1.1.1.2"></abs><apply id="S4.E11.m1.2.2.1.1.1.1.cmml" xref="S4.E11.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.E11.m1.2.2.1.1.1.1.1.cmml" xref="S4.E11.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S4.E11.m1.2.2.1.1.1.1.2.cmml" xref="S4.E11.m1.2.2.1.1.1.1.2">𝐶</ci><apply id="S4.E11.m1.2.2.1.1.1.1.3.cmml" xref="S4.E11.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E11.m1.2.2.1.1.1.1.3.1.cmml" xref="S4.E11.m1.2.2.1.1.1.1.3">subscript</csymbol><ci id="S4.E11.m1.2.2.1.1.1.1.3.2.cmml" xref="S4.E11.m1.2.2.1.1.1.1.3.2">𝜎</ci><cn id="S4.E11.m1.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S4.E11.m1.2.2.1.1.1.1.3.3">0</cn></apply></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.E11.m1.2c">\displaystyle\mbox{ with }\quad|C_{\sigma_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m1.2d">with | italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.E11.m2.1"><semantics id="S4.E11.m2.1a"><mo id="S4.E11.m2.1.1" xref="S4.E11.m2.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.E11.m2.1b"><leq id="S4.E11.m2.1.1.cmml" xref="S4.E11.m2.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.E11.m2.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m2.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\big{(}f({0},{t})+B+3\big{)}^{t+1}." class="ltx_Math" display="inline" id="S4.E11.m3.3"><semantics id="S4.E11.m3.3a"><mrow id="S4.E11.m3.3.3.1" xref="S4.E11.m3.3.3.1.1.cmml"><msup id="S4.E11.m3.3.3.1.1" xref="S4.E11.m3.3.3.1.1.cmml"><mrow id="S4.E11.m3.3.3.1.1.1.1" xref="S4.E11.m3.3.3.1.1.1.1.1.cmml"><mo id="S4.E11.m3.3.3.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S4.E11.m3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S4.E11.m3.3.3.1.1.1.1.1" xref="S4.E11.m3.3.3.1.1.1.1.1.cmml"><mrow id="S4.E11.m3.3.3.1.1.1.1.1.2" xref="S4.E11.m3.3.3.1.1.1.1.1.2.cmml"><mi id="S4.E11.m3.3.3.1.1.1.1.1.2.2" xref="S4.E11.m3.3.3.1.1.1.1.1.2.2.cmml">f</mi><mo id="S4.E11.m3.3.3.1.1.1.1.1.2.1" xref="S4.E11.m3.3.3.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.E11.m3.3.3.1.1.1.1.1.2.3.2" xref="S4.E11.m3.3.3.1.1.1.1.1.2.3.1.cmml"><mo id="S4.E11.m3.3.3.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.E11.m3.3.3.1.1.1.1.1.2.3.1.cmml">(</mo><mn id="S4.E11.m3.1.1" xref="S4.E11.m3.1.1.cmml">0</mn><mo id="S4.E11.m3.3.3.1.1.1.1.1.2.3.2.2" xref="S4.E11.m3.3.3.1.1.1.1.1.2.3.1.cmml">,</mo><mi id="S4.E11.m3.2.2" xref="S4.E11.m3.2.2.cmml">t</mi><mo id="S4.E11.m3.3.3.1.1.1.1.1.2.3.2.3" stretchy="false" xref="S4.E11.m3.3.3.1.1.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.E11.m3.3.3.1.1.1.1.1.1" xref="S4.E11.m3.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="S4.E11.m3.3.3.1.1.1.1.1.3" xref="S4.E11.m3.3.3.1.1.1.1.1.3.cmml">B</mi><mo id="S4.E11.m3.3.3.1.1.1.1.1.1a" xref="S4.E11.m3.3.3.1.1.1.1.1.1.cmml">+</mo><mn id="S4.E11.m3.3.3.1.1.1.1.1.4" xref="S4.E11.m3.3.3.1.1.1.1.1.4.cmml">3</mn></mrow><mo id="S4.E11.m3.3.3.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S4.E11.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S4.E11.m3.3.3.1.1.3" xref="S4.E11.m3.3.3.1.1.3.cmml"><mi id="S4.E11.m3.3.3.1.1.3.2" xref="S4.E11.m3.3.3.1.1.3.2.cmml">t</mi><mo id="S4.E11.m3.3.3.1.1.3.1" xref="S4.E11.m3.3.3.1.1.3.1.cmml">+</mo><mn id="S4.E11.m3.3.3.1.1.3.3" xref="S4.E11.m3.3.3.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.E11.m3.3.3.1.2" lspace="0em" xref="S4.E11.m3.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E11.m3.3b"><apply id="S4.E11.m3.3.3.1.1.cmml" xref="S4.E11.m3.3.3.1"><csymbol cd="ambiguous" id="S4.E11.m3.3.3.1.1.2.cmml" xref="S4.E11.m3.3.3.1">superscript</csymbol><apply id="S4.E11.m3.3.3.1.1.1.1.1.cmml" 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id="S4.E11.m3.3.3.1.1.3.3.cmml" type="integer" xref="S4.E11.m3.3.3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E11.m3.3c">\displaystyle\big{(}f({0},{t})+B+3\big{)}^{t+1}.</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m3.3d">( italic_f ( 0 , italic_t ) + italic_B + 3 ) start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT .</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">(11)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.3">The proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a> requires the next technical property about trees. We recall the notion of <em class="ltx_emph ltx_font_italic" id="S4.SS3.p2.3.1">depth</em> of a node in a tree: the root has depth <math alttext="0" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mn id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><cn id="S4.SS3.p2.1.m1.1.1.cmml" type="integer" xref="S4.SS3.p2.1.m1.1.1">0</cn></annotation-xml></semantics></math>, and if a node has depth <math alttext="d" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><mi id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><ci id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">italic_d</annotation></semantics></math>, then its sons have depth <math alttext="d+1" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.1"><semantics id="S4.SS3.p2.3.m3.1a"><mrow id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml"><mi id="S4.SS3.p2.3.m3.1.1.2" xref="S4.SS3.p2.3.m3.1.1.2.cmml">d</mi><mo id="S4.SS3.p2.3.m3.1.1.1" xref="S4.SS3.p2.3.m3.1.1.1.cmml">+</mo><mn id="S4.SS3.p2.3.m3.1.1.3" xref="S4.SS3.p2.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.1b"><apply id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1"><plus id="S4.SS3.p2.3.m3.1.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1.1"></plus><ci id="S4.SS3.p2.3.m3.1.1.2.cmml" xref="S4.SS3.p2.3.m3.1.1.2">𝑑</ci><cn id="S4.SS3.p2.3.m3.1.1.3.cmml" type="integer" xref="S4.SS3.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.1c">d+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.1d">italic_d + 1</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.1.1.1">Lemma 4.7</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem7.p1"> <p class="ltx_p" id="S4.Thmtheorem7.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem7.p1.4.4">Let <math alttext="n,\ell\in{\rm Nature}\setminus\{0\}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.1.1.m1.3"><semantics id="S4.Thmtheorem7.p1.1.1.m1.3a"><mrow id="S4.Thmtheorem7.p1.1.1.m1.3.4" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.cmml"><mrow id="S4.Thmtheorem7.p1.1.1.m1.3.4.2.2" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.2.1.cmml"><mi id="S4.Thmtheorem7.p1.1.1.m1.2.2" xref="S4.Thmtheorem7.p1.1.1.m1.2.2.cmml">n</mi><mo id="S4.Thmtheorem7.p1.1.1.m1.3.4.2.2.1" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.2.1.cmml">,</mo><mi id="S4.Thmtheorem7.p1.1.1.m1.3.3" mathvariant="normal" xref="S4.Thmtheorem7.p1.1.1.m1.3.3.cmml">ℓ</mi></mrow><mo id="S4.Thmtheorem7.p1.1.1.m1.3.4.1" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S4.Thmtheorem7.p1.1.1.m1.3.4.3" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.cmml"><mi id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.2" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.2.cmml">Nature</mi><mo id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.1" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml">∖</mo><mrow id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.2" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mn id="S4.Thmtheorem7.p1.1.1.m1.1.1" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml">0</mn><mo id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.1.1.m1.3b"><apply id="S4.Thmtheorem7.p1.1.1.m1.3.4.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.4"><in id="S4.Thmtheorem7.p1.1.1.m1.3.4.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.1"></in><list id="S4.Thmtheorem7.p1.1.1.m1.3.4.2.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.2.2"><ci id="S4.Thmtheorem7.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.2.2">𝑛</ci><ci id="S4.Thmtheorem7.p1.1.1.m1.3.3.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.3">ℓ</ci></list><apply id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3"><setdiff id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.1"></setdiff><ci id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.2.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.2">Nature</ci><set id="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.3.4.3.3.2"><cn id="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem7.p1.1.1.m1.1.1">0</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.1.1.m1.3c">n,\ell\in{\rm Nature}\setminus\{0\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.1.1.m1.3d">italic_n , roman_ℓ ∈ roman_Nature ∖ { 0 }</annotation></semantics></math>. Consider a finite tree with at most <math alttext="n" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.2.2.m2.1"><semantics id="S4.Thmtheorem7.p1.2.2.m2.1a"><mi id="S4.Thmtheorem7.p1.2.2.m2.1.1" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.2.2.m2.1b"><ci id="S4.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.2.2.m2.1d">italic_n</annotation></semantics></math> leaves such that there are at most <math alttext="\ell-1" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.3.3.m3.1"><semantics id="S4.Thmtheorem7.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem7.p1.3.3.m3.1.1" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem7.p1.3.3.m3.1.1.2" mathvariant="normal" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.2.cmml">ℓ</mi><mo id="S4.Thmtheorem7.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem7.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.3.3.m3.1b"><apply id="S4.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1"><minus id="S4.Thmtheorem7.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.1"></minus><ci id="S4.Thmtheorem7.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.2">ℓ</ci><cn id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.3.3.m3.1c">\ell-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.3.3.m3.1d">roman_ℓ - 1</annotation></semantics></math> consecutive nodes with degree one along any branch of the tree. Then the leaves with the smallest depth have a depth bounded by <math alttext="\ell\cdot(\log_{2}(n)+1)" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.4.4.m4.2"><semantics id="S4.Thmtheorem7.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m4.2.2.3" mathvariant="normal" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.3.cmml">ℓ</mi><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.2" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.cmml">⋅</mo><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.cmml"><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.2.cmml"><msub id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.2.cmml">log</mi><mn id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.3" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1a" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.2.cmml">⁡</mo><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.2.cmml"><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.2.cmml">(</mo><mi id="S4.Thmtheorem7.p1.4.4.m4.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.cmml">n</mi><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.2.2" stretchy="false" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.2.cmml">+</mo><mn id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.3" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.4.4.m4.2b"><apply id="S4.Thmtheorem7.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2"><ci id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2">⋅</ci><ci id="S4.Thmtheorem7.p1.4.4.m4.2.2.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.3">ℓ</ci><apply id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1"><plus id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.2"></plus><apply id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1"><apply id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1">subscript</csymbol><log id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.2"></log><cn id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.1.1.1.3">2</cn></apply><ci id="S4.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.1.1">𝑛</ci></apply><cn id="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.4.4.m4.2c">\ell\cdot(\log_{2}(n)+1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.4.4.m4.2d">roman_ℓ ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_n ) + 1 )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.1.1.1">Proof 4.8</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem8.p1"> <p class="ltx_p" id="S4.Thmtheorem8.p1.9"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem8.p1.9.9">When <math alttext="n=1" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.1.1.m1.1"><semantics id="S4.Thmtheorem8.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem8.p1.1.1.m1.1.1" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem8.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem8.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.Thmtheorem8.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.1.1.m1.1b"><apply id="S4.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.1.1"><eq id="S4.Thmtheorem8.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.1"></eq><ci id="S4.Thmtheorem8.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.2">𝑛</ci><cn id="S4.Thmtheorem8.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.1.1.m1.1c">n=1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.1.1.m1.1d">italic_n = 1</annotation></semantics></math>, the worst case is the tree composed of one branch with <math alttext="\ell-1" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.2.2.m2.1"><semantics id="S4.Thmtheorem8.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem8.p1.2.2.m2.1.1" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem8.p1.2.2.m2.1.1.2" mathvariant="normal" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.2.2.m2.1b"><apply id="S4.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.1.1"><minus id="S4.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.1"></minus><ci id="S4.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.2">ℓ</ci><cn id="S4.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.2.2.m2.1c">\ell-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.2.2.m2.1d">roman_ℓ - 1</annotation></semantics></math> nodes. When <math alttext="n\geq 2" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.3.3.m3.1"><semantics id="S4.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem8.p1.3.3.m3.1.1" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.3.3.m3.1b"><apply id="S4.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1"><geq id="S4.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.1"></geq><ci id="S4.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.2">𝑛</ci><cn id="S4.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.3.3.m3.1c">n\geq 2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.3.3.m3.1d">italic_n ≥ 2</annotation></semantics></math>, the worst case happens when the tree is binary, all its leaves have the same depth, and along any branch, there are <math alttext="\ell-1" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.4.4.m4.1"><semantics id="S4.Thmtheorem8.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem8.p1.4.4.m4.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem8.p1.4.4.m4.1.1.2" mathvariant="normal" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.2.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p1.4.4.m4.1.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem8.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.4.4.m4.1b"><apply id="S4.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.1.1"><minus id="S4.Thmtheorem8.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.1"></minus><ci id="S4.Thmtheorem8.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.2">ℓ</ci><cn id="S4.Thmtheorem8.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.4.4.m4.1c">\ell-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.4.4.m4.1d">roman_ℓ - 1</annotation></semantics></math> consecutive nodes with degree 1 before and after each node with degree 2. In both cases <math alttext="n=1" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.5.5.m5.1"><semantics id="S4.Thmtheorem8.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem8.p1.5.5.m5.1.1" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem8.p1.5.5.m5.1.1.2" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem8.p1.5.5.m5.1.1.1" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.1.cmml">=</mo><mn id="S4.Thmtheorem8.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.5.5.m5.1b"><apply id="S4.Thmtheorem8.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem8.p1.5.5.m5.1.1"><eq id="S4.Thmtheorem8.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.1"></eq><ci id="S4.Thmtheorem8.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.2">𝑛</ci><cn id="S4.Thmtheorem8.p1.5.5.m5.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.5.5.m5.1c">n=1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.5.5.m5.1d">italic_n = 1</annotation></semantics></math> and <math alttext="n\geq 2" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.6.6.m6.1"><semantics id="S4.Thmtheorem8.p1.6.6.m6.1a"><mrow id="S4.Thmtheorem8.p1.6.6.m6.1.1" xref="S4.Thmtheorem8.p1.6.6.m6.1.1.cmml"><mi id="S4.Thmtheorem8.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem8.p1.6.6.m6.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem8.p1.6.6.m6.1.1.1" xref="S4.Thmtheorem8.p1.6.6.m6.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem8.p1.6.6.m6.1.1.3" xref="S4.Thmtheorem8.p1.6.6.m6.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.6.6.m6.1b"><apply id="S4.Thmtheorem8.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem8.p1.6.6.m6.1.1"><geq id="S4.Thmtheorem8.p1.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem8.p1.6.6.m6.1.1.1"></geq><ci id="S4.Thmtheorem8.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem8.p1.6.6.m6.1.1.2">𝑛</ci><cn id="S4.Thmtheorem8.p1.6.6.m6.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.6.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.6.6.m6.1c">n\geq 2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.6.6.m6.1d">italic_n ≥ 2</annotation></semantics></math>, the worse tree has <math alttext="n=2^{k}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.7.7.m7.1"><semantics id="S4.Thmtheorem8.p1.7.7.m7.1a"><mrow id="S4.Thmtheorem8.p1.7.7.m7.1.1" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem8.p1.7.7.m7.1.1.2" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem8.p1.7.7.m7.1.1.1" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.1.cmml">=</mo><msup id="S4.Thmtheorem8.p1.7.7.m7.1.1.3" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.3.cmml"><mn id="S4.Thmtheorem8.p1.7.7.m7.1.1.3.2" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.3.2.cmml">2</mn><mi id="S4.Thmtheorem8.p1.7.7.m7.1.1.3.3" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.7.7.m7.1b"><apply id="S4.Thmtheorem8.p1.7.7.m7.1.1.cmml" xref="S4.Thmtheorem8.p1.7.7.m7.1.1"><eq id="S4.Thmtheorem8.p1.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.1"></eq><ci id="S4.Thmtheorem8.p1.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.2">𝑛</ci><apply id="S4.Thmtheorem8.p1.7.7.m7.1.1.3.cmml" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.7.7.m7.1.1.3.1.cmml" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.3">superscript</csymbol><cn id="S4.Thmtheorem8.p1.7.7.m7.1.1.3.2.cmml" type="integer" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.3.2">2</cn><ci id="S4.Thmtheorem8.p1.7.7.m7.1.1.3.3.cmml" xref="S4.Thmtheorem8.p1.7.7.m7.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.7.7.m7.1c">n=2^{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.7.7.m7.1d">italic_n = 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> leaves for some <math alttext="k\geq 0" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.8.8.m8.1"><semantics id="S4.Thmtheorem8.p1.8.8.m8.1a"><mrow id="S4.Thmtheorem8.p1.8.8.m8.1.1" xref="S4.Thmtheorem8.p1.8.8.m8.1.1.cmml"><mi id="S4.Thmtheorem8.p1.8.8.m8.1.1.2" xref="S4.Thmtheorem8.p1.8.8.m8.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem8.p1.8.8.m8.1.1.1" xref="S4.Thmtheorem8.p1.8.8.m8.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem8.p1.8.8.m8.1.1.3" xref="S4.Thmtheorem8.p1.8.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.8.8.m8.1b"><apply id="S4.Thmtheorem8.p1.8.8.m8.1.1.cmml" xref="S4.Thmtheorem8.p1.8.8.m8.1.1"><geq id="S4.Thmtheorem8.p1.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem8.p1.8.8.m8.1.1.1"></geq><ci id="S4.Thmtheorem8.p1.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem8.p1.8.8.m8.1.1.2">𝑘</ci><cn id="S4.Thmtheorem8.p1.8.8.m8.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.8.8.m8.1c">k\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.8.8.m8.1d">italic_k ≥ 0</annotation></semantics></math>. We denote this tree by <math alttext="\mathcal{T}_{k}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.9.9.m9.1"><semantics id="S4.Thmtheorem8.p1.9.9.m9.1a"><msub id="S4.Thmtheorem8.p1.9.9.m9.1.1" xref="S4.Thmtheorem8.p1.9.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem8.p1.9.9.m9.1.1.2" xref="S4.Thmtheorem8.p1.9.9.m9.1.1.2.cmml">𝒯</mi><mi id="S4.Thmtheorem8.p1.9.9.m9.1.1.3" xref="S4.Thmtheorem8.p1.9.9.m9.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.9.9.m9.1b"><apply id="S4.Thmtheorem8.p1.9.9.m9.1.1.cmml" xref="S4.Thmtheorem8.p1.9.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.9.9.m9.1.1.1.cmml" xref="S4.Thmtheorem8.p1.9.9.m9.1.1">subscript</csymbol><ci id="S4.Thmtheorem8.p1.9.9.m9.1.1.2.cmml" xref="S4.Thmtheorem8.p1.9.9.m9.1.1.2">𝒯</ci><ci id="S4.Thmtheorem8.p1.9.9.m9.1.1.3.cmml" xref="S4.Thmtheorem8.p1.9.9.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.9.9.m9.1c">\mathcal{T}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.9.9.m9.1d">caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.F4" title="Figure 4 ‣ Proof 4.8 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>).</span></p> </div> <figure class="ltx_figure" id="S4.F4"><svg class="ltx_picture ltx_centering" height="150.71" id="S4.F4.pic1" overflow="visible" version="1.1" width="170.4"><g fill="#000000" stroke="#000000" transform="translate(0,150.71) matrix(1 0 0 -1 0 0) translate(85.2,0) translate(0,6.46)"><g stroke-width="0.8pt"><path d="M 5.91 137.8 C 5.91 141.06 3.26 143.7 0 143.7 C -3.26 143.7 -5.91 141.06 -5.91 137.8 C -5.91 134.53 -3.26 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124.57" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 0 111.65 L 0 104.88" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M -4.03 93.38 L -11.71 83.78" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M -19.78 73.7 L -27.46 64.1" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M -35.53 54.01 L -43.21 44.41" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 4.03 93.38 L 11.71 83.78" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 19.78 73.7 L 27.46 64.1" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 35.53 54.01 L 43.21 44.41" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M -51.28 34.33 L -58.96 24.73" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M -67.02 14.64 L -74.7 5.04" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M -43.21 34.33 L -35.53 24.73" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M -27.46 14.64 L -19.78 5.04" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 43.21 34.33 L 35.53 24.73" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 27.46 14.64 L 19.78 5.04" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 51.28 34.33 L 58.96 24.73" style="fill:none"></path></g><g stroke-width="1.1381pt"><path d="M 67.02 14.64 L 74.7 5.04" style="fill:none"></path></g></g></svg> <figcaption class="ltx_caption ltx_centering ltx_font_italic"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text ltx_font_upright" id="S4.F4.11.1.1">Figure 4</span>: </span>Tree <math alttext="\mathcal{T}_{k}" class="ltx_Math" display="inline" id="S4.F4.4.m1.1"><semantics id="S4.F4.4.m1.1b"><msub id="S4.F4.4.m1.1.1" xref="S4.F4.4.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.F4.4.m1.1.1.2" xref="S4.F4.4.m1.1.1.2.cmml">𝒯</mi><mi id="S4.F4.4.m1.1.1.3" xref="S4.F4.4.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.F4.4.m1.1c"><apply id="S4.F4.4.m1.1.1.cmml" xref="S4.F4.4.m1.1.1"><csymbol cd="ambiguous" id="S4.F4.4.m1.1.1.1.cmml" xref="S4.F4.4.m1.1.1">subscript</csymbol><ci id="S4.F4.4.m1.1.1.2.cmml" xref="S4.F4.4.m1.1.1.2">𝒯</ci><ci id="S4.F4.4.m1.1.1.3.cmml" xref="S4.F4.4.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.4.m1.1d">\mathcal{T}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.4.m1.1e">caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="k=2" class="ltx_Math" display="inline" id="S4.F4.5.m2.1"><semantics id="S4.F4.5.m2.1b"><mrow id="S4.F4.5.m2.1.1" xref="S4.F4.5.m2.1.1.cmml"><mi id="S4.F4.5.m2.1.1.2" xref="S4.F4.5.m2.1.1.2.cmml">k</mi><mo id="S4.F4.5.m2.1.1.1" xref="S4.F4.5.m2.1.1.1.cmml">=</mo><mn id="S4.F4.5.m2.1.1.3" xref="S4.F4.5.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.5.m2.1c"><apply id="S4.F4.5.m2.1.1.cmml" xref="S4.F4.5.m2.1.1"><eq id="S4.F4.5.m2.1.1.1.cmml" xref="S4.F4.5.m2.1.1.1"></eq><ci id="S4.F4.5.m2.1.1.2.cmml" xref="S4.F4.5.m2.1.1.2">𝑘</ci><cn id="S4.F4.5.m2.1.1.3.cmml" type="integer" xref="S4.F4.5.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.5.m2.1d">k=2</annotation><annotation encoding="application/x-llamapun" id="S4.F4.5.m2.1e">italic_k = 2</annotation></semantics></math> and <math alttext="\ell=3" class="ltx_Math" display="inline" id="S4.F4.6.m3.1"><semantics id="S4.F4.6.m3.1b"><mrow id="S4.F4.6.m3.1.1" xref="S4.F4.6.m3.1.1.cmml"><mi id="S4.F4.6.m3.1.1.2" mathvariant="normal" xref="S4.F4.6.m3.1.1.2.cmml">ℓ</mi><mo id="S4.F4.6.m3.1.1.1" xref="S4.F4.6.m3.1.1.1.cmml">=</mo><mn id="S4.F4.6.m3.1.1.3" xref="S4.F4.6.m3.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.6.m3.1c"><apply id="S4.F4.6.m3.1.1.cmml" xref="S4.F4.6.m3.1.1"><eq id="S4.F4.6.m3.1.1.1.cmml" xref="S4.F4.6.m3.1.1.1"></eq><ci id="S4.F4.6.m3.1.1.2.cmml" xref="S4.F4.6.m3.1.1.2">ℓ</ci><cn id="S4.F4.6.m3.1.1.3.cmml" type="integer" xref="S4.F4.6.m3.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.6.m3.1d">\ell=3</annotation><annotation encoding="application/x-llamapun" id="S4.F4.6.m3.1e">roman_ℓ = 3</annotation></semantics></math></figcaption> </figure> <div class="ltx_para" id="S4.Thmtheorem8.p2"> <p class="ltx_p" id="S4.Thmtheorem8.p2.18"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem8.p2.18.18">Let us prove by induction on <math alttext="k" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.1.1.m1.1"><semantics id="S4.Thmtheorem8.p2.1.1.m1.1a"><mi id="S4.Thmtheorem8.p2.1.1.m1.1.1" xref="S4.Thmtheorem8.p2.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.1.1.m1.1b"><ci id="S4.Thmtheorem8.p2.1.1.m1.1.1.cmml" xref="S4.Thmtheorem8.p2.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.1.1.m1.1d">italic_k</annotation></semantics></math> that the depth of the leaves of <math alttext="\mathcal{T}_{k}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.2.2.m2.1"><semantics id="S4.Thmtheorem8.p2.2.2.m2.1a"><msub id="S4.Thmtheorem8.p2.2.2.m2.1.1" xref="S4.Thmtheorem8.p2.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem8.p2.2.2.m2.1.1.2" xref="S4.Thmtheorem8.p2.2.2.m2.1.1.2.cmml">𝒯</mi><mi id="S4.Thmtheorem8.p2.2.2.m2.1.1.3" xref="S4.Thmtheorem8.p2.2.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.2.2.m2.1b"><apply id="S4.Thmtheorem8.p2.2.2.m2.1.1.cmml" xref="S4.Thmtheorem8.p2.2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p2.2.2.m2.1.1.1.cmml" 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xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.2" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.2.cmml"><mo id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.2.cmml">(</mo><mi id="S4.Thmtheorem8.p2.3.3.m3.1.1" xref="S4.Thmtheorem8.p2.3.3.m3.1.1.cmml">n</mi><mo id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.2.2" stretchy="false" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.2" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.2.cmml">+</mo><mn id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.3" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem8.p2.3.3.m3.3.3.3" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.3.cmml">=</mo><mrow id="S4.Thmtheorem8.p2.3.3.m3.3.3.2" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.cmml"><mi id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.3" mathvariant="normal" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.3.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.2" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.2.cmml">⋅</mo><mrow id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.cmml"><mo id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.2" stretchy="false" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.cmml"><mi id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.2" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.1" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.3" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.3" stretchy="false" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.3.3.m3.3b"><apply id="S4.Thmtheorem8.p2.3.3.m3.3.3.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3"><eq id="S4.Thmtheorem8.p2.3.3.m3.3.3.3.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.3"></eq><apply id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1"><ci id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.2.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.2">⋅</ci><ci id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.3.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.3">ℓ</ci><apply id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1"><plus id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.2"></plus><apply id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1"><apply id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1">subscript</csymbol><log id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1.2"></log><cn id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.1.1.1.3">2</cn></apply><ci id="S4.Thmtheorem8.p2.3.3.m3.1.1.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.1.1">𝑛</ci></apply><cn id="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.3.3.m3.2.2.1.1.1.1.3">1</cn></apply></apply><apply id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2"><ci id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.2.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.2">⋅</ci><ci id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.3.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.3">ℓ</ci><apply id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1"><plus id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.1.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.1"></plus><ci id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.2.cmml" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.2">𝑘</ci><cn id="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.3.3.m3.3.3.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.3.3.m3.3c">\ell\cdot(\log_{2}(n)+1)=\ell\cdot(k+1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.3.3.m3.3d">roman_ℓ ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_n ) + 1 ) = roman_ℓ ⋅ ( italic_k + 1 )</annotation></semantics></math>. When <math alttext="k=0" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.4.4.m4.1"><semantics id="S4.Thmtheorem8.p2.4.4.m4.1a"><mrow id="S4.Thmtheorem8.p2.4.4.m4.1.1" xref="S4.Thmtheorem8.p2.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem8.p2.4.4.m4.1.1.2" xref="S4.Thmtheorem8.p2.4.4.m4.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem8.p2.4.4.m4.1.1.1" xref="S4.Thmtheorem8.p2.4.4.m4.1.1.1.cmml">=</mo><mn id="S4.Thmtheorem8.p2.4.4.m4.1.1.3" xref="S4.Thmtheorem8.p2.4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.4.4.m4.1b"><apply id="S4.Thmtheorem8.p2.4.4.m4.1.1.cmml" xref="S4.Thmtheorem8.p2.4.4.m4.1.1"><eq id="S4.Thmtheorem8.p2.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem8.p2.4.4.m4.1.1.1"></eq><ci id="S4.Thmtheorem8.p2.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem8.p2.4.4.m4.1.1.2">𝑘</ci><cn id="S4.Thmtheorem8.p2.4.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.4.4.m4.1c">k=0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.4.4.m4.1d">italic_k = 0</annotation></semantics></math>, the unique leaf of <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.5.5.m5.1"><semantics id="S4.Thmtheorem8.p2.5.5.m5.1a"><msub id="S4.Thmtheorem8.p2.5.5.m5.1.1" xref="S4.Thmtheorem8.p2.5.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem8.p2.5.5.m5.1.1.2" xref="S4.Thmtheorem8.p2.5.5.m5.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem8.p2.5.5.m5.1.1.3" xref="S4.Thmtheorem8.p2.5.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.5.5.m5.1b"><apply id="S4.Thmtheorem8.p2.5.5.m5.1.1.cmml" xref="S4.Thmtheorem8.p2.5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p2.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem8.p2.5.5.m5.1.1">subscript</csymbol><ci id="S4.Thmtheorem8.p2.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem8.p2.5.5.m5.1.1.2">𝒯</ci><cn id="S4.Thmtheorem8.p2.5.5.m5.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.5.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.5.5.m5.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.5.5.m5.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> has depth <math alttext="\ell-2\leq\ell" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.6.6.m6.1"><semantics id="S4.Thmtheorem8.p2.6.6.m6.1a"><mrow id="S4.Thmtheorem8.p2.6.6.m6.1.1" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.cmml"><mrow id="S4.Thmtheorem8.p2.6.6.m6.1.1.2" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2.cmml"><mi id="S4.Thmtheorem8.p2.6.6.m6.1.1.2.2" mathvariant="normal" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2.2.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p2.6.6.m6.1.1.2.1" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2.1.cmml">−</mo><mn id="S4.Thmtheorem8.p2.6.6.m6.1.1.2.3" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2.3.cmml">2</mn></mrow><mo id="S4.Thmtheorem8.p2.6.6.m6.1.1.1" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.1.cmml">≤</mo><mi id="S4.Thmtheorem8.p2.6.6.m6.1.1.3" mathvariant="normal" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.3.cmml">ℓ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.6.6.m6.1b"><apply id="S4.Thmtheorem8.p2.6.6.m6.1.1.cmml" xref="S4.Thmtheorem8.p2.6.6.m6.1.1"><leq id="S4.Thmtheorem8.p2.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.1"></leq><apply id="S4.Thmtheorem8.p2.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2"><minus id="S4.Thmtheorem8.p2.6.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2.1"></minus><ci id="S4.Thmtheorem8.p2.6.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2.2">ℓ</ci><cn id="S4.Thmtheorem8.p2.6.6.m6.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.2.3">2</cn></apply><ci id="S4.Thmtheorem8.p2.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem8.p2.6.6.m6.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.6.6.m6.1c">\ell-2\leq\ell</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.6.6.m6.1d">roman_ℓ - 2 ≤ roman_ℓ</annotation></semantics></math>. When <math alttext="k\geq 1" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.7.7.m7.1"><semantics id="S4.Thmtheorem8.p2.7.7.m7.1a"><mrow id="S4.Thmtheorem8.p2.7.7.m7.1.1" xref="S4.Thmtheorem8.p2.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem8.p2.7.7.m7.1.1.2" xref="S4.Thmtheorem8.p2.7.7.m7.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem8.p2.7.7.m7.1.1.1" xref="S4.Thmtheorem8.p2.7.7.m7.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem8.p2.7.7.m7.1.1.3" xref="S4.Thmtheorem8.p2.7.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.7.7.m7.1b"><apply id="S4.Thmtheorem8.p2.7.7.m7.1.1.cmml" xref="S4.Thmtheorem8.p2.7.7.m7.1.1"><geq id="S4.Thmtheorem8.p2.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem8.p2.7.7.m7.1.1.1"></geq><ci id="S4.Thmtheorem8.p2.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem8.p2.7.7.m7.1.1.2">𝑘</ci><cn id="S4.Thmtheorem8.p2.7.7.m7.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.7.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.7.7.m7.1c">k\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.7.7.m7.1d">italic_k ≥ 1</annotation></semantics></math>, consider the node <math alttext="v" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.8.8.m8.1"><semantics id="S4.Thmtheorem8.p2.8.8.m8.1a"><mi id="S4.Thmtheorem8.p2.8.8.m8.1.1" xref="S4.Thmtheorem8.p2.8.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.8.8.m8.1b"><ci id="S4.Thmtheorem8.p2.8.8.m8.1.1.cmml" xref="S4.Thmtheorem8.p2.8.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.8.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.8.8.m8.1d">italic_v</annotation></semantics></math> of <math alttext="\mathcal{T}_{k}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.9.9.m9.1"><semantics id="S4.Thmtheorem8.p2.9.9.m9.1a"><msub id="S4.Thmtheorem8.p2.9.9.m9.1.1" xref="S4.Thmtheorem8.p2.9.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem8.p2.9.9.m9.1.1.2" xref="S4.Thmtheorem8.p2.9.9.m9.1.1.2.cmml">𝒯</mi><mi id="S4.Thmtheorem8.p2.9.9.m9.1.1.3" xref="S4.Thmtheorem8.p2.9.9.m9.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.9.9.m9.1b"><apply id="S4.Thmtheorem8.p2.9.9.m9.1.1.cmml" xref="S4.Thmtheorem8.p2.9.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p2.9.9.m9.1.1.1.cmml" xref="S4.Thmtheorem8.p2.9.9.m9.1.1">subscript</csymbol><ci id="S4.Thmtheorem8.p2.9.9.m9.1.1.2.cmml" xref="S4.Thmtheorem8.p2.9.9.m9.1.1.2">𝒯</ci><ci id="S4.Thmtheorem8.p2.9.9.m9.1.1.3.cmml" xref="S4.Thmtheorem8.p2.9.9.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.9.9.m9.1c">\mathcal{T}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.9.9.m9.1d">caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with degree 2 and with the smallest depth <math alttext="d" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.10.10.m10.1"><semantics id="S4.Thmtheorem8.p2.10.10.m10.1a"><mi id="S4.Thmtheorem8.p2.10.10.m10.1.1" xref="S4.Thmtheorem8.p2.10.10.m10.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.10.10.m10.1b"><ci id="S4.Thmtheorem8.p2.10.10.m10.1.1.cmml" xref="S4.Thmtheorem8.p2.10.10.m10.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.10.10.m10.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.10.10.m10.1d">italic_d</annotation></semantics></math>. The nodes with depth <math alttext="<d" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.11.11.m11.1"><semantics id="S4.Thmtheorem8.p2.11.11.m11.1a"><mrow id="S4.Thmtheorem8.p2.11.11.m11.1.1" xref="S4.Thmtheorem8.p2.11.11.m11.1.1.cmml"><mi id="S4.Thmtheorem8.p2.11.11.m11.1.1.2" xref="S4.Thmtheorem8.p2.11.11.m11.1.1.2.cmml"></mi><mo id="S4.Thmtheorem8.p2.11.11.m11.1.1.1" xref="S4.Thmtheorem8.p2.11.11.m11.1.1.1.cmml"><</mo><mi id="S4.Thmtheorem8.p2.11.11.m11.1.1.3" xref="S4.Thmtheorem8.p2.11.11.m11.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.11.11.m11.1b"><apply id="S4.Thmtheorem8.p2.11.11.m11.1.1.cmml" xref="S4.Thmtheorem8.p2.11.11.m11.1.1"><lt id="S4.Thmtheorem8.p2.11.11.m11.1.1.1.cmml" xref="S4.Thmtheorem8.p2.11.11.m11.1.1.1"></lt><csymbol cd="latexml" id="S4.Thmtheorem8.p2.11.11.m11.1.1.2.cmml" xref="S4.Thmtheorem8.p2.11.11.m11.1.1.2">absent</csymbol><ci id="S4.Thmtheorem8.p2.11.11.m11.1.1.3.cmml" xref="S4.Thmtheorem8.p2.11.11.m11.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.11.11.m11.1c"><d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.11.11.m11.1d">< italic_d</annotation></semantics></math> are ancestors of <math alttext="v" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.12.12.m12.1"><semantics id="S4.Thmtheorem8.p2.12.12.m12.1a"><mi id="S4.Thmtheorem8.p2.12.12.m12.1.1" xref="S4.Thmtheorem8.p2.12.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.12.12.m12.1b"><ci id="S4.Thmtheorem8.p2.12.12.m12.1.1.cmml" xref="S4.Thmtheorem8.p2.12.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.12.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.12.12.m12.1d">italic_v</annotation></semantics></math> and there are <math alttext="\ell-1" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.13.13.m13.1"><semantics id="S4.Thmtheorem8.p2.13.13.m13.1a"><mrow id="S4.Thmtheorem8.p2.13.13.m13.1.1" xref="S4.Thmtheorem8.p2.13.13.m13.1.1.cmml"><mi id="S4.Thmtheorem8.p2.13.13.m13.1.1.2" mathvariant="normal" xref="S4.Thmtheorem8.p2.13.13.m13.1.1.2.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p2.13.13.m13.1.1.1" xref="S4.Thmtheorem8.p2.13.13.m13.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem8.p2.13.13.m13.1.1.3" xref="S4.Thmtheorem8.p2.13.13.m13.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.13.13.m13.1b"><apply id="S4.Thmtheorem8.p2.13.13.m13.1.1.cmml" xref="S4.Thmtheorem8.p2.13.13.m13.1.1"><minus id="S4.Thmtheorem8.p2.13.13.m13.1.1.1.cmml" xref="S4.Thmtheorem8.p2.13.13.m13.1.1.1"></minus><ci id="S4.Thmtheorem8.p2.13.13.m13.1.1.2.cmml" xref="S4.Thmtheorem8.p2.13.13.m13.1.1.2">ℓ</ci><cn id="S4.Thmtheorem8.p2.13.13.m13.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.13.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.13.13.m13.1c">\ell-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.13.13.m13.1d">roman_ℓ - 1</annotation></semantics></math> of them. Hence <math alttext="d=\ell-1" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.14.14.m14.1"><semantics id="S4.Thmtheorem8.p2.14.14.m14.1a"><mrow id="S4.Thmtheorem8.p2.14.14.m14.1.1" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.cmml"><mi id="S4.Thmtheorem8.p2.14.14.m14.1.1.2" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.2.cmml">d</mi><mo id="S4.Thmtheorem8.p2.14.14.m14.1.1.1" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.1.cmml">=</mo><mrow id="S4.Thmtheorem8.p2.14.14.m14.1.1.3" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3.cmml"><mi id="S4.Thmtheorem8.p2.14.14.m14.1.1.3.2" mathvariant="normal" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3.2.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p2.14.14.m14.1.1.3.1" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3.1.cmml">−</mo><mn id="S4.Thmtheorem8.p2.14.14.m14.1.1.3.3" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.14.14.m14.1b"><apply id="S4.Thmtheorem8.p2.14.14.m14.1.1.cmml" xref="S4.Thmtheorem8.p2.14.14.m14.1.1"><eq id="S4.Thmtheorem8.p2.14.14.m14.1.1.1.cmml" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.1"></eq><ci id="S4.Thmtheorem8.p2.14.14.m14.1.1.2.cmml" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.2">𝑑</ci><apply id="S4.Thmtheorem8.p2.14.14.m14.1.1.3.cmml" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3"><minus id="S4.Thmtheorem8.p2.14.14.m14.1.1.3.1.cmml" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3.1"></minus><ci id="S4.Thmtheorem8.p2.14.14.m14.1.1.3.2.cmml" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3.2">ℓ</ci><cn id="S4.Thmtheorem8.p2.14.14.m14.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.14.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.14.14.m14.1c">d=\ell-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.14.14.m14.1d">italic_d = roman_ℓ - 1</annotation></semantics></math>. Moreover, the two subtrees of <math alttext="v" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.15.15.m15.1"><semantics id="S4.Thmtheorem8.p2.15.15.m15.1a"><mi id="S4.Thmtheorem8.p2.15.15.m15.1.1" xref="S4.Thmtheorem8.p2.15.15.m15.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.15.15.m15.1b"><ci id="S4.Thmtheorem8.p2.15.15.m15.1.1.cmml" xref="S4.Thmtheorem8.p2.15.15.m15.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.15.15.m15.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.15.15.m15.1d">italic_v</annotation></semantics></math> are isomorphic to <math alttext="\mathcal{T}_{k-1}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.16.16.m16.1"><semantics id="S4.Thmtheorem8.p2.16.16.m16.1a"><msub id="S4.Thmtheorem8.p2.16.16.m16.1.1" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem8.p2.16.16.m16.1.1.2" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.2.cmml">𝒯</mi><mrow id="S4.Thmtheorem8.p2.16.16.m16.1.1.3" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3.cmml"><mi id="S4.Thmtheorem8.p2.16.16.m16.1.1.3.2" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3.2.cmml">k</mi><mo id="S4.Thmtheorem8.p2.16.16.m16.1.1.3.1" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3.1.cmml">−</mo><mn id="S4.Thmtheorem8.p2.16.16.m16.1.1.3.3" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.16.16.m16.1b"><apply id="S4.Thmtheorem8.p2.16.16.m16.1.1.cmml" xref="S4.Thmtheorem8.p2.16.16.m16.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p2.16.16.m16.1.1.1.cmml" xref="S4.Thmtheorem8.p2.16.16.m16.1.1">subscript</csymbol><ci id="S4.Thmtheorem8.p2.16.16.m16.1.1.2.cmml" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.2">𝒯</ci><apply id="S4.Thmtheorem8.p2.16.16.m16.1.1.3.cmml" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3"><minus id="S4.Thmtheorem8.p2.16.16.m16.1.1.3.1.cmml" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3.1"></minus><ci id="S4.Thmtheorem8.p2.16.16.m16.1.1.3.2.cmml" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3.2">𝑘</ci><cn id="S4.Thmtheorem8.p2.16.16.m16.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.16.16.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.16.16.m16.1c">\mathcal{T}_{k-1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.16.16.m16.1d">caligraphic_T start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, by induction hypothesis, the leaves of <math alttext="\mathcal{T}_{k}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.17.17.m17.1"><semantics id="S4.Thmtheorem8.p2.17.17.m17.1a"><msub id="S4.Thmtheorem8.p2.17.17.m17.1.1" xref="S4.Thmtheorem8.p2.17.17.m17.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem8.p2.17.17.m17.1.1.2" xref="S4.Thmtheorem8.p2.17.17.m17.1.1.2.cmml">𝒯</mi><mi id="S4.Thmtheorem8.p2.17.17.m17.1.1.3" xref="S4.Thmtheorem8.p2.17.17.m17.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.17.17.m17.1b"><apply id="S4.Thmtheorem8.p2.17.17.m17.1.1.cmml" xref="S4.Thmtheorem8.p2.17.17.m17.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p2.17.17.m17.1.1.1.cmml" xref="S4.Thmtheorem8.p2.17.17.m17.1.1">subscript</csymbol><ci id="S4.Thmtheorem8.p2.17.17.m17.1.1.2.cmml" xref="S4.Thmtheorem8.p2.17.17.m17.1.1.2">𝒯</ci><ci id="S4.Thmtheorem8.p2.17.17.m17.1.1.3.cmml" xref="S4.Thmtheorem8.p2.17.17.m17.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.17.17.m17.1c">\mathcal{T}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.17.17.m17.1d">caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> have depth bounded by <math alttext="(\ell-1)+1+\ell\cdot k\leq\ell\cdot(k+1)" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p2.18.18.m18.2"><semantics id="S4.Thmtheorem8.p2.18.18.m18.2a"><mrow id="S4.Thmtheorem8.p2.18.18.m18.2.2" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.cmml"><mrow id="S4.Thmtheorem8.p2.18.18.m18.1.1.1" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.cmml"><mrow id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.2" mathvariant="normal" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.2.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.1" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.3" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.2" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.2.cmml">+</mo><mn id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.3" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.3.cmml">1</mn><mo id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.2a" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.2.cmml">+</mo><mrow id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.cmml"><mi id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.2" mathvariant="normal" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.2.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.1" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.1.cmml">⋅</mo><mi id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.3" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.3.cmml">k</mi></mrow></mrow><mo id="S4.Thmtheorem8.p2.18.18.m18.2.2.3" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.3.cmml">≤</mo><mrow id="S4.Thmtheorem8.p2.18.18.m18.2.2.2" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.cmml"><mi id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.3" mathvariant="normal" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.3.cmml">ℓ</mi><mo id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.2" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.2.cmml">⋅</mo><mrow id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.2" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.1" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.3" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p2.18.18.m18.2b"><apply id="S4.Thmtheorem8.p2.18.18.m18.2.2.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2"><leq id="S4.Thmtheorem8.p2.18.18.m18.2.2.3.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.3"></leq><apply id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1"><plus id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.2.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.2"></plus><apply id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1"><minus id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.1"></minus><ci id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.2">ℓ</ci><cn id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.1.1.1.3">1</cn></apply><cn id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.3">1</cn><apply id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4"><ci id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.1.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.1">⋅</ci><ci id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.2.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.2">ℓ</ci><ci id="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.3.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.1.1.1.4.3">𝑘</ci></apply></apply><apply id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2"><ci id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.2.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.2">⋅</ci><ci id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.3.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.3">ℓ</ci><apply id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1"><plus id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.1"></plus><ci id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.2">𝑘</ci><cn id="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p2.18.18.m18.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p2.18.18.m18.2c">(\ell-1)+1+\ell\cdot k\leq\ell\cdot(k+1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p2.18.18.m18.2d">( roman_ℓ - 1 ) + 1 + roman_ℓ ⋅ italic_k ≤ roman_ℓ ⋅ ( italic_k + 1 )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.1">We are now able to establish Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.1.1.1">Proof 4.9</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.2.2"> </span>(Proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a>)</h6> <div class="ltx_para" id="S4.Thmtheorem9.p1"> <p class="ltx_p" id="S4.Thmtheorem9.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p1.5.5">Let <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.1.1.m1.2"><semantics id="S4.Thmtheorem9.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem9.p1.1.1.m1.2.3" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.cmml"><msub id="S4.Thmtheorem9.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2.cmml"><mi id="S4.Thmtheorem9.p1.1.1.m1.2.3.2.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p1.1.1.m1.2.3.2.3" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem9.p1.1.1.m1.2.3.3" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem9.p1.1.1.m1.1.1" xref="S4.Thmtheorem9.p1.1.1.m1.1.1.cmml">G</mi><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.2.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.1.1.m1.2.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.2.cmml">B</mi><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.1.1.m1.2b"><apply id="S4.Thmtheorem9.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3"><in id="S4.Thmtheorem9.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.1"></in><apply id="S4.Thmtheorem9.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.1.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem9.p1.1.1.m1.2.3.2.2.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.1.1.m1.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3"><times id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1"></times><ci id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2a.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.3.2"><ci id="S4.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.1.1">𝐺</ci><ci id="S4.Thmtheorem9.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.1.1.m1.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.1.1.m1.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution without cycles and satisfying (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E7" title="In Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">7</span></a>). Suppose that <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.2.2.m2.4"><semantics id="S4.Thmtheorem9.p1.2.2.m2.4a"><mrow id="S4.Thmtheorem9.p1.2.2.m2.4.4" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.cmml"><msub id="S4.Thmtheorem9.p1.2.2.m2.4.4.3" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.cmml"><mi id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.2" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.2.cmml">C</mi><msub id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.cmml"><mi id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.2" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.3" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem9.p1.2.2.m2.4.4.2" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.2.cmml">≠</mo><mrow id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.2.cmml"><mo id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.2.cmml">{</mo><mrow id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.2" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem9.p1.2.2.m2.1.1" mathvariant="normal" xref="S4.Thmtheorem9.p1.2.2.m2.1.1.cmml">∞</mi><mo id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.2.2" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.2.2.m2.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p1.2.2.m2.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.2.3" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.2.2.m2.3.3" mathvariant="normal" xref="S4.Thmtheorem9.p1.2.2.m2.3.3.cmml">∞</mi><mo id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.2.4" stretchy="false" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.2.2.m2.4b"><apply id="S4.Thmtheorem9.p1.2.2.m2.4.4.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4"><neq id="S4.Thmtheorem9.p1.2.2.m2.4.4.2.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.2"></neq><apply id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.2.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.2">𝐶</ci><apply id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.2.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.3.3.3">0</cn></apply></apply><set id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.2.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1"><vector id="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.4.4.1.1.1.2"><infinity id="S4.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.1.1"></infinity><ci id="S4.Thmtheorem9.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.2">…</ci><infinity id="S4.Thmtheorem9.p1.2.2.m2.3.3.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.2.2.m2.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.2.2.m2.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math> and let <math alttext="\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.3.3.m3.1"><semantics id="S4.Thmtheorem9.p1.3.3.m3.1a"><msub id="S4.Thmtheorem9.p1.3.3.m3.1.1" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem9.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.3.3.m3.1b"><apply id="S4.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.3.3.m3.1.1.2a.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.2">Wit</mtext></ci><apply id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.3.3.m3.1c">\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.3.3.m3.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be a set of witnesses for <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.4.4.m4.1"><semantics id="S4.Thmtheorem9.p1.4.4.m4.1a"><msub id="S4.Thmtheorem9.p1.4.4.m4.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem9.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.4.4.m4.1.1.3.2" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p1.4.4.m4.1.1.3.3" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.4.4.m4.1b"><apply id="S4.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.4.4.m4.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.4.4.m4.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.4.4.m4.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.4.4.m4.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.4.4.m4.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Assume by contradiction that Inequality (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E10" title="In Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">10</span></a>) does not hold, that is, there exists <math alttext="d\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.5.5.m5.1"><semantics id="S4.Thmtheorem9.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem9.p1.5.5.m5.1.1" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem9.p1.5.5.m5.1.1.2" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.2.cmml">d</mi><mo id="S4.Thmtheorem9.p1.5.5.m5.1.1.1" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem9.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.2" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.2.cmml">C</mi><msub id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.cmml"><mi id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.2" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.3" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.5.5.m5.1b"><apply id="S4.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1"><in id="S4.Thmtheorem9.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.1"></in><ci id="S4.Thmtheorem9.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.2">𝑑</ci><apply id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.2">𝐶</ci><apply id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.1.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.5.5.m5.1c">d\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.5.5.m5.1d">italic_d ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx11"> <tbody id="S4.E12"><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 B+2^{t+1}\big{(}\delta+1+f({0},{t})\big{)}<d_{min}<\infty." class="ltx_Math" display="inline" id="S4.E12.m1.3"><semantics id="S4.E12.m1.3a"><mrow id="S4.E12.m1.3.3.1" xref="S4.E12.m1.3.3.1.1.cmml"><mrow id="S4.E12.m1.3.3.1.1" xref="S4.E12.m1.3.3.1.1.cmml"><mrow id="S4.E12.m1.3.3.1.1.1" xref="S4.E12.m1.3.3.1.1.1.cmml"><mi id="S4.E12.m1.3.3.1.1.1.3" xref="S4.E12.m1.3.3.1.1.1.3.cmml">B</mi><mo id="S4.E12.m1.3.3.1.1.1.2" xref="S4.E12.m1.3.3.1.1.1.2.cmml">+</mo><mrow id="S4.E12.m1.3.3.1.1.1.1" xref="S4.E12.m1.3.3.1.1.1.1.cmml"><msup id="S4.E12.m1.3.3.1.1.1.1.3" xref="S4.E12.m1.3.3.1.1.1.1.3.cmml"><mn id="S4.E12.m1.3.3.1.1.1.1.3.2" xref="S4.E12.m1.3.3.1.1.1.1.3.2.cmml">2</mn><mrow id="S4.E12.m1.3.3.1.1.1.1.3.3" xref="S4.E12.m1.3.3.1.1.1.1.3.3.cmml"><mi id="S4.E12.m1.3.3.1.1.1.1.3.3.2" xref="S4.E12.m1.3.3.1.1.1.1.3.3.2.cmml">t</mi><mo id="S4.E12.m1.3.3.1.1.1.1.3.3.1" xref="S4.E12.m1.3.3.1.1.1.1.3.3.1.cmml">+</mo><mn id="S4.E12.m1.3.3.1.1.1.1.3.3.3" xref="S4.E12.m1.3.3.1.1.1.1.3.3.3.cmml">1</mn></mrow></msup><mo id="S4.E12.m1.3.3.1.1.1.1.2" xref="S4.E12.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E12.m1.3.3.1.1.1.1.1.1" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E12.m1.3.3.1.1.1.1.1.1.1" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S4.E12.m1.3.3.1.1.1.1.1.1.1.2" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.2.cmml">δ</mi><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.1.1" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.E12.m1.3.3.1.1.1.1.1.1.1.3" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.1.1a" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.E12.m1.3.3.1.1.1.1.1.1.1.4" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.cmml"><mi id="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.2" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.2.cmml">f</mi><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.1" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.2" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.1.cmml"><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.2.1" stretchy="false" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.1.cmml">(</mo><mn id="S4.E12.m1.1.1" xref="S4.E12.m1.1.1.cmml">0</mn><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.2.2" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.1.cmml">,</mo><mi id="S4.E12.m1.2.2" xref="S4.E12.m1.2.2.cmml">t</mi><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.2.3" stretchy="false" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.4.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E12.m1.3.3.1.1.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S4.E12.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E12.m1.3.3.1.1.3" xref="S4.E12.m1.3.3.1.1.3.cmml"><</mo><msub id="S4.E12.m1.3.3.1.1.4" xref="S4.E12.m1.3.3.1.1.4.cmml"><mi id="S4.E12.m1.3.3.1.1.4.2" xref="S4.E12.m1.3.3.1.1.4.2.cmml">d</mi><mrow id="S4.E12.m1.3.3.1.1.4.3" xref="S4.E12.m1.3.3.1.1.4.3.cmml"><mi id="S4.E12.m1.3.3.1.1.4.3.2" xref="S4.E12.m1.3.3.1.1.4.3.2.cmml">m</mi><mo id="S4.E12.m1.3.3.1.1.4.3.1" xref="S4.E12.m1.3.3.1.1.4.3.1.cmml">⁢</mo><mi id="S4.E12.m1.3.3.1.1.4.3.3" xref="S4.E12.m1.3.3.1.1.4.3.3.cmml">i</mi><mo id="S4.E12.m1.3.3.1.1.4.3.1a" xref="S4.E12.m1.3.3.1.1.4.3.1.cmml">⁢</mo><mi id="S4.E12.m1.3.3.1.1.4.3.4" xref="S4.E12.m1.3.3.1.1.4.3.4.cmml">n</mi></mrow></msub><mo id="S4.E12.m1.3.3.1.1.5" xref="S4.E12.m1.3.3.1.1.5.cmml"><</mo><mi id="S4.E12.m1.3.3.1.1.6" mathvariant="normal" xref="S4.E12.m1.3.3.1.1.6.cmml">∞</mi></mrow><mo id="S4.E12.m1.3.3.1.2" lspace="0em" xref="S4.E12.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E12.m1.3b"><apply id="S4.E12.m1.3.3.1.1.cmml" xref="S4.E12.m1.3.3.1"><and id="S4.E12.m1.3.3.1.1a.cmml" xref="S4.E12.m1.3.3.1"></and><apply id="S4.E12.m1.3.3.1.1b.cmml" xref="S4.E12.m1.3.3.1"><lt id="S4.E12.m1.3.3.1.1.3.cmml" xref="S4.E12.m1.3.3.1.1.3"></lt><apply 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id="S4.E12.m1.3.3.1.1.4.1.cmml" xref="S4.E12.m1.3.3.1.1.4">subscript</csymbol><ci id="S4.E12.m1.3.3.1.1.4.2.cmml" xref="S4.E12.m1.3.3.1.1.4.2">𝑑</ci><apply id="S4.E12.m1.3.3.1.1.4.3.cmml" xref="S4.E12.m1.3.3.1.1.4.3"><times id="S4.E12.m1.3.3.1.1.4.3.1.cmml" xref="S4.E12.m1.3.3.1.1.4.3.1"></times><ci id="S4.E12.m1.3.3.1.1.4.3.2.cmml" xref="S4.E12.m1.3.3.1.1.4.3.2">𝑚</ci><ci id="S4.E12.m1.3.3.1.1.4.3.3.cmml" xref="S4.E12.m1.3.3.1.1.4.3.3">𝑖</ci><ci id="S4.E12.m1.3.3.1.1.4.3.4.cmml" xref="S4.E12.m1.3.3.1.1.4.3.4">𝑛</ci></apply></apply></apply><apply id="S4.E12.m1.3.3.1.1c.cmml" xref="S4.E12.m1.3.3.1"><lt id="S4.E12.m1.3.3.1.1.5.cmml" xref="S4.E12.m1.3.3.1.1.5"></lt><share href="https://arxiv.org/html/2308.09443v2#S4.E12.m1.3.3.1.1.4.cmml" id="S4.E12.m1.3.3.1.1d.cmml" xref="S4.E12.m1.3.3.1"></share><infinity id="S4.E12.m1.3.3.1.1.6.cmml" xref="S4.E12.m1.3.3.1.1.6"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E12.m1.3c">\displaystyle B+2^{t+1}\big{(}\delta+1+f({0},{t})\big{)}<d_{min}<\infty.</annotation><annotation encoding="application/x-llamapun" id="S4.E12.m1.3d">italic_B + 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ( italic_δ + 1 + italic_f ( 0 , italic_t ) ) < italic_d start_POSTSUBSCRIPT italic_m italic_i italic_n 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">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem9.p1.10"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p1.10.5">with <math alttext="\delta=|V|\cdot(\log_{2}(|C_{\sigma_{0}}|)+1)" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.6.1.m1.2"><semantics id="S4.Thmtheorem9.p1.6.1.m1.2a"><mrow id="S4.Thmtheorem9.p1.6.1.m1.2.2" xref="S4.Thmtheorem9.p1.6.1.m1.2.2.cmml"><mi 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xref="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.2.2.2.1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.2.2.2.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.2.2.2.1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.2.2.2.1.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.2.2.2.1.1.1.3.3">0</cn></apply></apply></apply></apply><cn id="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.4.cmml" type="integer" xref="S4.Thmtheorem9.p1.6.1.m1.2.2.1.1.1.1.4">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.6.1.m1.2c">\delta=|V|\cdot(\log_{2}(|C_{\sigma_{0}}|)+1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.6.1.m1.2d">italic_δ = | italic_V | ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( | italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ) + 1 )</annotation></semantics></math>. Let <math alttext="\rho\in\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.7.2.m2.1"><semantics id="S4.Thmtheorem9.p1.7.2.m2.1a"><mrow id="S4.Thmtheorem9.p1.7.2.m2.1.1" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.cmml"><mi id="S4.Thmtheorem9.p1.7.2.m2.1.1.2" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.2.cmml">ρ</mi><mo id="S4.Thmtheorem9.p1.7.2.m2.1.1.1" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem9.p1.7.2.m2.1.1.3" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.2" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.cmml"><mi id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.2" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.3" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.7.2.m2.1b"><apply id="S4.Thmtheorem9.p1.7.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1"><in id="S4.Thmtheorem9.p1.7.2.m2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.1"></in><ci id="S4.Thmtheorem9.p1.7.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.2">𝜌</ci><apply id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.2a.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.2">Wit</mtext></ci><apply id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.1.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.2.cmml" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.7.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.7.2.m2.1c">\rho\in\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.7.2.m2.1d">italic_ρ ∈ Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be a witness such that <math alttext="\textsf{cost}({\rho})=d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.8.3.m3.1"><semantics id="S4.Thmtheorem9.p1.8.3.m3.1a"><mrow id="S4.Thmtheorem9.p1.8.3.m3.1.2" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.cmml"><mrow id="S4.Thmtheorem9.p1.8.3.m3.1.2.2" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.2" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.2a.cmml">cost</mtext><mo id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.1" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.3.2" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.cmml"><mo id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p1.8.3.m3.1.1" xref="S4.Thmtheorem9.p1.8.3.m3.1.1.cmml">ρ</mi><mo id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p1.8.3.m3.1.2.1" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.1.cmml">=</mo><mi id="S4.Thmtheorem9.p1.8.3.m3.1.2.3" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.8.3.m3.1b"><apply id="S4.Thmtheorem9.p1.8.3.m3.1.2.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.2"><eq id="S4.Thmtheorem9.p1.8.3.m3.1.2.1.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.1"></eq><apply id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2"><times id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.1.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.1"></times><ci id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.2a.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p1.8.3.m3.1.2.2.2.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.2.2">cost</mtext></ci><ci id="S4.Thmtheorem9.p1.8.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.1">𝜌</ci></apply><ci id="S4.Thmtheorem9.p1.8.3.m3.1.2.3.cmml" xref="S4.Thmtheorem9.p1.8.3.m3.1.2.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.8.3.m3.1c">\textsf{cost}({\rho})=d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.8.3.m3.1d">cost ( italic_ρ ) = italic_d</annotation></semantics></math>. We are going to build a finite sequence of Pareto-optimal costs <math alttext="(c^{(k)})_{k\in\{0,\ldots,2^{t+1}\}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.9.4.m4.5"><semantics id="S4.Thmtheorem9.p1.9.4.m4.5a"><msub id="S4.Thmtheorem9.p1.9.4.m4.5.5" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.cmml"><mrow id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.2" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p1.9.4.m4.1.1.1.3" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.9.4.m4.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem9.p1.9.4.m4.1.1.1.1" xref="S4.Thmtheorem9.p1.9.4.m4.1.1.1.1.cmml">k</mi><mo id="S4.Thmtheorem9.p1.9.4.m4.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml">)</mo></mrow><mrow id="S4.Thmtheorem9.p1.9.4.m4.4.4.3" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.cmml"><mi id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.5" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.5.cmml">k</mi><mo id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.4" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.4.cmml">∈</mo><mrow id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.2.cmml"><mo id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.2.cmml">{</mo><mn id="S4.Thmtheorem9.p1.9.4.m4.2.2.1.1" xref="S4.Thmtheorem9.p1.9.4.m4.2.2.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.3" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.2.cmml">,</mo><mi id="S4.Thmtheorem9.p1.9.4.m4.3.3.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p1.9.4.m4.3.3.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.4" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.2.cmml">,</mo><msup id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.cmml"><mn id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.2" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.2.cmml">2</mn><mrow id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.2" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.1" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.3" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.5" stretchy="false" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.2.cmml">}</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.9.4.m4.5b"><apply id="S4.Thmtheorem9.p1.9.4.m4.5.5.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.5.5"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.9.4.m4.5.5.2.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.5.5">subscript</csymbol><apply id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.5.5.1.1.1.2">𝑐</ci><ci id="S4.Thmtheorem9.p1.9.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.1.1.1.1">𝑘</ci></apply><apply id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3"><in id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.4.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.4"></in><ci id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.5.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.5">𝑘</ci><set id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.2.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1"><cn id="S4.Thmtheorem9.p1.9.4.m4.2.2.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p1.9.4.m4.2.2.1.1">0</cn><ci id="S4.Thmtheorem9.p1.9.4.m4.3.3.2.2.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.3.3.2.2">…</ci><apply id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.1.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1">superscript</csymbol><cn id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.2.cmml" type="integer" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.2">2</cn><apply id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3"><plus id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.1"></plus><ci id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.2">𝑡</ci><cn id="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.9.4.m4.4.4.3.3.1.1.3.3">1</cn></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.9.4.m4.5c">(c^{(k)})_{k\in\{0,\ldots,2^{t+1}\}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.9.4.m4.5d">( italic_c start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ { 0 , … , 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT } end_POSTSUBSCRIPT</annotation></semantics></math> such that, for all <math alttext="k\in\{0,\ldots,2^{t+1}-1\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.10.5.m5.3"><semantics id="S4.Thmtheorem9.p1.10.5.m5.3a"><mrow id="S4.Thmtheorem9.p1.10.5.m5.3.3" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.cmml"><mi id="S4.Thmtheorem9.p1.10.5.m5.3.3.3" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.3.cmml">k</mi><mo id="S4.Thmtheorem9.p1.10.5.m5.3.3.2" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.2.cmml">∈</mo><mrow id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.2.cmml"><mo id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.2.cmml">{</mo><mn id="S4.Thmtheorem9.p1.10.5.m5.1.1" xref="S4.Thmtheorem9.p1.10.5.m5.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.3" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.2.cmml">,</mo><mi id="S4.Thmtheorem9.p1.10.5.m5.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p1.10.5.m5.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.4" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.2.cmml">,</mo><mrow id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.cmml"><msup id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.cmml"><mn 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id="S4.Thmtheorem9.p1.10.5.m5.3.3.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3"><in id="S4.Thmtheorem9.p1.10.5.m5.3.3.2.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.2"></in><ci id="S4.Thmtheorem9.p1.10.5.m5.3.3.3.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.3">𝑘</ci><set id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.2.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1"><cn id="S4.Thmtheorem9.p1.10.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p1.10.5.m5.1.1">0</cn><ci id="S4.Thmtheorem9.p1.10.5.m5.2.2.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.2.2">…</ci><apply id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1"><minus id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.1"></minus><apply id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.1.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2">superscript</csymbol><cn id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.2.cmml" type="integer" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.2">2</cn><apply id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3"><plus id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3.1.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3.1"></plus><ci id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3.2.cmml" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3.2">𝑡</ci><cn id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.2.3.3">1</cn></apply></apply><cn id="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.10.5.m5.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.10.5.m5.3c">k\in\{0,\ldots,2^{t+1}-1\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.10.5.m5.3d">italic_k ∈ { 0 , … , 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT - 1 }</annotation></semantics></math>,</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx12"> <tbody id="S4.E13"><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\{c^{(k)}_{i}\mid c^{(k)}_{i}\neq\infty\}<c^{(k+1)}_{min}." class="ltx_Math" display="inline" id="S4.E13.m1.5"><semantics id="S4.E13.m1.5a"><mrow id="S4.E13.m1.5.5.1" xref="S4.E13.m1.5.5.1.1.cmml"><mrow id="S4.E13.m1.5.5.1.1" xref="S4.E13.m1.5.5.1.1.cmml"><mrow id="S4.E13.m1.5.5.1.1.1.1" xref="S4.E13.m1.5.5.1.1.1.2.cmml"><mi id="S4.E13.m1.4.4" xref="S4.E13.m1.4.4.cmml">max</mi><mo id="S4.E13.m1.5.5.1.1.1.1a" xref="S4.E13.m1.5.5.1.1.1.2.cmml">⁡</mo><mrow id="S4.E13.m1.5.5.1.1.1.1.1" xref="S4.E13.m1.5.5.1.1.1.2.cmml"><mo id="S4.E13.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S4.E13.m1.5.5.1.1.1.2.cmml">{</mo><mrow 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xref="S4.E13.m1.5.5.1.1.3.2.2.cmml">c</mi><mrow id="S4.E13.m1.5.5.1.1.3.3" xref="S4.E13.m1.5.5.1.1.3.3.cmml"><mi id="S4.E13.m1.5.5.1.1.3.3.2" xref="S4.E13.m1.5.5.1.1.3.3.2.cmml">m</mi><mo id="S4.E13.m1.5.5.1.1.3.3.1" xref="S4.E13.m1.5.5.1.1.3.3.1.cmml">⁢</mo><mi id="S4.E13.m1.5.5.1.1.3.3.3" xref="S4.E13.m1.5.5.1.1.3.3.3.cmml">i</mi><mo id="S4.E13.m1.5.5.1.1.3.3.1a" xref="S4.E13.m1.5.5.1.1.3.3.1.cmml">⁢</mo><mi id="S4.E13.m1.5.5.1.1.3.3.4" xref="S4.E13.m1.5.5.1.1.3.3.4.cmml">n</mi></mrow><mrow id="S4.E13.m1.3.3.1.1" xref="S4.E13.m1.3.3.1.1.1.cmml"><mo id="S4.E13.m1.3.3.1.1.2" stretchy="false" xref="S4.E13.m1.3.3.1.1.1.cmml">(</mo><mrow id="S4.E13.m1.3.3.1.1.1" xref="S4.E13.m1.3.3.1.1.1.cmml"><mi id="S4.E13.m1.3.3.1.1.1.2" xref="S4.E13.m1.3.3.1.1.1.2.cmml">k</mi><mo id="S4.E13.m1.3.3.1.1.1.1" xref="S4.E13.m1.3.3.1.1.1.1.cmml">+</mo><mn id="S4.E13.m1.3.3.1.1.1.3" xref="S4.E13.m1.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S4.E13.m1.3.3.1.1.3" stretchy="false" 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type="integer" xref="S4.E13.m1.3.3.1.1.1.3">1</cn></apply></apply><apply id="S4.E13.m1.5.5.1.1.3.3.cmml" xref="S4.E13.m1.5.5.1.1.3.3"><times id="S4.E13.m1.5.5.1.1.3.3.1.cmml" xref="S4.E13.m1.5.5.1.1.3.3.1"></times><ci id="S4.E13.m1.5.5.1.1.3.3.2.cmml" xref="S4.E13.m1.5.5.1.1.3.3.2">𝑚</ci><ci id="S4.E13.m1.5.5.1.1.3.3.3.cmml" xref="S4.E13.m1.5.5.1.1.3.3.3">𝑖</ci><ci id="S4.E13.m1.5.5.1.1.3.3.4.cmml" xref="S4.E13.m1.5.5.1.1.3.3.4">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13.m1.5c">\displaystyle\max\{c^{(k)}_{i}\mid c^{(k)}_{i}\neq\infty\}<c^{(k+1)}_{min}.</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m1.5d">roman_max { italic_c start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_c start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∞ } < italic_c start_POSTSUPERSCRIPT ( italic_k + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n 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">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem9.p1.12"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p1.12.2">Given the size of this sequence, by the pigeonhole principle, there must exist <math alttext="k,k^{\prime}\in\{0,\ldots,2^{t+1}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.11.1.m1.5"><semantics id="S4.Thmtheorem9.p1.11.1.m1.5a"><mrow id="S4.Thmtheorem9.p1.11.1.m1.5.5" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.cmml"><mrow id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.2.cmml"><mi id="S4.Thmtheorem9.p1.11.1.m1.3.3" xref="S4.Thmtheorem9.p1.11.1.m1.3.3.cmml">k</mi><mo id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.2" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.2.cmml">,</mo><msup id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.2" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.3" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S4.Thmtheorem9.p1.11.1.m1.5.5.3" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.3.cmml">∈</mo><mrow id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.2.cmml"><mo id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.2.cmml">{</mo><mn id="S4.Thmtheorem9.p1.11.1.m1.1.1" xref="S4.Thmtheorem9.p1.11.1.m1.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.3" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.2.cmml">,</mo><mi id="S4.Thmtheorem9.p1.11.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p1.11.1.m1.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.4" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.2.cmml">,</mo><msup id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.cmml"><mn id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.2" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.2.cmml">2</mn><mrow id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.2" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.1" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.3" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.5" stretchy="false" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.11.1.m1.5b"><apply id="S4.Thmtheorem9.p1.11.1.m1.5.5.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5"><in id="S4.Thmtheorem9.p1.11.1.m1.5.5.3.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.3"></in><list id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.2.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1"><ci id="S4.Thmtheorem9.p1.11.1.m1.3.3.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.3.3">𝑘</ci><apply id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.2">𝑘</ci><ci id="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.4.4.1.1.1.3">′</ci></apply></list><set id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.2.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1"><cn id="S4.Thmtheorem9.p1.11.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p1.11.1.m1.1.1">0</cn><ci id="S4.Thmtheorem9.p1.11.1.m1.2.2.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.2.2">…</ci><apply id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1">superscript</csymbol><cn id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.2.cmml" type="integer" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.2">2</cn><apply id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3"><plus id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.1"></plus><ci id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.2">𝑡</ci><cn id="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.11.1.m1.5.5.2.1.1.3.3">1</cn></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.11.1.m1.5c">k,k^{\prime}\in\{0,\ldots,2^{t+1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.11.1.m1.5d">italic_k , italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ { 0 , … , 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT }</annotation></semantics></math> such that <math alttext="k<k^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.12.2.m2.1"><semantics id="S4.Thmtheorem9.p1.12.2.m2.1a"><mrow id="S4.Thmtheorem9.p1.12.2.m2.1.1" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.cmml"><mi id="S4.Thmtheorem9.p1.12.2.m2.1.1.2" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem9.p1.12.2.m2.1.1.1" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.1.cmml"><</mo><msup id="S4.Thmtheorem9.p1.12.2.m2.1.1.3" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.12.2.m2.1.1.3.2" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.3.2.cmml">k</mi><mo id="S4.Thmtheorem9.p1.12.2.m2.1.1.3.3" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.12.2.m2.1b"><apply id="S4.Thmtheorem9.p1.12.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.12.2.m2.1.1"><lt id="S4.Thmtheorem9.p1.12.2.m2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.1"></lt><ci id="S4.Thmtheorem9.p1.12.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.2">𝑘</ci><apply id="S4.Thmtheorem9.p1.12.2.m2.1.1.3.cmml" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.12.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem9.p1.12.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.3.2">𝑘</ci><ci id="S4.Thmtheorem9.p1.12.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem9.p1.12.2.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.12.2.m2.1c">k<k^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.12.2.m2.1d">italic_k < italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and</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="\{i\in\{1,\ldots,t+1\}\mid c_{i}^{(k)}=\infty\}=\{i\in\{1,\ldots,t+1\}\mid c_{% i}^{(k^{\prime})}=\infty\}" class="ltx_Math" display="block" id="S4.Ex6.m1.10"><semantics id="S4.Ex6.m1.10a"><mrow id="S4.Ex6.m1.10.10" xref="S4.Ex6.m1.10.10.cmml"><mrow id="S4.Ex6.m1.8.8.2.2" xref="S4.Ex6.m1.8.8.2.3.cmml"><mo id="S4.Ex6.m1.8.8.2.2.3" stretchy="false" xref="S4.Ex6.m1.8.8.2.3.1.cmml">{</mo><mrow id="S4.Ex6.m1.7.7.1.1.1" xref="S4.Ex6.m1.7.7.1.1.1.cmml"><mi id="S4.Ex6.m1.7.7.1.1.1.3" xref="S4.Ex6.m1.7.7.1.1.1.3.cmml">i</mi><mo id="S4.Ex6.m1.7.7.1.1.1.2" xref="S4.Ex6.m1.7.7.1.1.1.2.cmml">∈</mo><mrow 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xref="S4.Ex6.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.Ex6.m1.2.2.1.1.1.1.cmml" xref="S4.Ex6.m1.2.2.1.1">superscript</csymbol><ci id="S4.Ex6.m1.2.2.1.1.1.2.cmml" xref="S4.Ex6.m1.2.2.1.1.1.2">𝑘</ci><ci id="S4.Ex6.m1.2.2.1.1.1.3.cmml" xref="S4.Ex6.m1.2.2.1.1.1.3">′</ci></apply></apply><infinity id="S4.Ex6.m1.10.10.4.2.2.3.cmml" xref="S4.Ex6.m1.10.10.4.2.2.3"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex6.m1.10c">\{i\in\{1,\ldots,t+1\}\mid c_{i}^{(k)}=\infty\}=\{i\in\{1,\ldots,t+1\}\mid c_{% i}^{(k^{\prime})}=\infty\}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex6.m1.10d">{ italic_i ∈ { 1 , … , italic_t + 1 } ∣ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = ∞ } = { italic_i ∈ { 1 , … , italic_t + 1 } ∣ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT = ∞ }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem9.p1.14"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p1.14.2">(a cost component is either finite or infinite). It follows from (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E13" title="In Proof 4.9 (Proof of Lemma 4.6) ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">13</span></a>) that <math alttext="c^{(k)}<c^{(k^{\prime})}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.13.1.m1.2"><semantics id="S4.Thmtheorem9.p1.13.1.m1.2a"><mrow id="S4.Thmtheorem9.p1.13.1.m1.2.3" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.cmml"><msup id="S4.Thmtheorem9.p1.13.1.m1.2.3.2" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2.cmml"><mi id="S4.Thmtheorem9.p1.13.1.m1.2.3.2.2" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p1.13.1.m1.1.1.1.3" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2.cmml"><mo id="S4.Thmtheorem9.p1.13.1.m1.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem9.p1.13.1.m1.1.1.1.1" xref="S4.Thmtheorem9.p1.13.1.m1.1.1.1.1.cmml">k</mi><mo id="S4.Thmtheorem9.p1.13.1.m1.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2.cmml">)</mo></mrow></msup><mo id="S4.Thmtheorem9.p1.13.1.m1.2.3.1" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.1.cmml"><</mo><msup id="S4.Thmtheorem9.p1.13.1.m1.2.3.3" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.3.cmml"><mi id="S4.Thmtheorem9.p1.13.1.m1.2.3.3.2" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.3.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.2" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.3" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.cmml">)</mo></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.13.1.m1.2b"><apply id="S4.Thmtheorem9.p1.13.1.m1.2.3.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3"><lt id="S4.Thmtheorem9.p1.13.1.m1.2.3.1.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.1"></lt><apply id="S4.Thmtheorem9.p1.13.1.m1.2.3.2.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.13.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2">superscript</csymbol><ci id="S4.Thmtheorem9.p1.13.1.m1.2.3.2.2.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.2.2">𝑐</ci><ci id="S4.Thmtheorem9.p1.13.1.m1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.1.1.1.1">𝑘</ci></apply><apply id="S4.Thmtheorem9.p1.13.1.m1.2.3.3.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.13.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.3">superscript</csymbol><ci id="S4.Thmtheorem9.p1.13.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.3.3.2">𝑐</ci><apply id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.2">𝑘</ci><ci id="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.13.1.m1.2.2.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.13.1.m1.2c">c^{(k)}<c^{(k^{\prime})}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.13.1.m1.2d">italic_c start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT < italic_c start_POSTSUPERSCRIPT ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math>. This is impossible as <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.14.2.m2.1"><semantics id="S4.Thmtheorem9.p1.14.2.m2.1a"><msub id="S4.Thmtheorem9.p1.14.2.m2.1.1" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.cmml"><mi id="S4.Thmtheorem9.p1.14.2.m2.1.1.2" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p1.14.2.m2.1.1.3" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.14.2.m2.1.1.3.2" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p1.14.2.m2.1.1.3.3" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.14.2.m2.1b"><apply id="S4.Thmtheorem9.p1.14.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.14.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.14.2.m2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.14.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.14.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p1.14.2.m2.1.1.3.cmml" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.14.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.14.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p1.14.2.m2.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.14.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.14.2.m2.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.14.2.m2.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is an antichain.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem9.p2"> <p class="ltx_p" id="S4.Thmtheorem9.p2.16"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p2.16.16">To build the sequence <math alttext="(c^{(k)})_{k\in\{0,\ldots,2^{t+1}\}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.1.1.m1.5"><semantics id="S4.Thmtheorem9.p2.1.1.m1.5a"><msub id="S4.Thmtheorem9.p2.1.1.m1.5.5" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.cmml"><mrow id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml"><mo id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml"><mi id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.2" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p2.1.1.m1.1.1.1.3" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml"><mo id="S4.Thmtheorem9.p2.1.1.m1.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem9.p2.1.1.m1.1.1.1.1" xref="S4.Thmtheorem9.p2.1.1.m1.1.1.1.1.cmml">k</mi><mo id="S4.Thmtheorem9.p2.1.1.m1.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml">)</mo></mrow><mrow id="S4.Thmtheorem9.p2.1.1.m1.4.4.3" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.cmml"><mi id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.5" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.5.cmml">k</mi><mo id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.4" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.4.cmml">∈</mo><mrow id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.2.cmml"><mo id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.2.cmml">{</mo><mn id="S4.Thmtheorem9.p2.1.1.m1.2.2.1.1" xref="S4.Thmtheorem9.p2.1.1.m1.2.2.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.3" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.2.cmml">,</mo><mi id="S4.Thmtheorem9.p2.1.1.m1.3.3.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p2.1.1.m1.3.3.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.4" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.2.cmml">,</mo><msup id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.cmml"><mn id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.2" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.2.cmml">2</mn><mrow id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.cmml"><mi id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.2" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.1" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.3" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.5" stretchy="false" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.2.cmml">}</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.1.1.m1.5b"><apply id="S4.Thmtheorem9.p2.1.1.m1.5.5.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.5.5"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.1.1.m1.5.5.2.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.5.5">subscript</csymbol><apply id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.2.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.5.5.1.1.1.2">𝑐</ci><ci id="S4.Thmtheorem9.p2.1.1.m1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.1.1.1.1">𝑘</ci></apply><apply id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3"><in id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.4.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.4"></in><ci id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.5.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.5">𝑘</ci><set id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.2.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1"><cn id="S4.Thmtheorem9.p2.1.1.m1.2.2.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p2.1.1.m1.2.2.1.1">0</cn><ci id="S4.Thmtheorem9.p2.1.1.m1.3.3.2.2.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.3.3.2.2">…</ci><apply id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.1.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1">superscript</csymbol><cn id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.2.cmml" type="integer" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.2">2</cn><apply id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3"><plus id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.1.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.1"></plus><ci id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.2.cmml" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.2">𝑡</ci><cn id="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p2.1.1.m1.4.4.3.3.1.1.3.3">1</cn></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.1.1.m1.5c">(c^{(k)})_{k\in\{0,\ldots,2^{t+1}\}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.1.1.m1.5d">( italic_c start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ { 0 , … , 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT } end_POSTSUBSCRIPT</annotation></semantics></math>, we consider the tree <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.2.2.m2.1"><semantics id="S4.Thmtheorem9.p2.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p2.2.2.m2.1.1" xref="S4.Thmtheorem9.p2.2.2.m2.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.2.2.m2.1b"><ci id="S4.Thmtheorem9.p2.2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p2.2.2.m2.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.2.2.m2.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.2.2.m2.1d">caligraphic_T</annotation></semantics></math> of witnesses of <math alttext="\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.3.3.m3.1"><semantics id="S4.Thmtheorem9.p2.3.3.m3.1a"><msub id="S4.Thmtheorem9.p2.3.3.m3.1.1" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p2.3.3.m3.1.1.2" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.2a.cmml">Wit</mtext><msub id="S4.Thmtheorem9.p2.3.3.m3.1.1.3" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.3.cmml"><mi id="S4.Thmtheorem9.p2.3.3.m3.1.1.3.2" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p2.3.3.m3.1.1.3.3" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.3.3.m3.1b"><apply id="S4.Thmtheorem9.p2.3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem9.p2.3.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p2.3.3.m3.1.1.2a.cmml" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p2.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.2">Wit</mtext></ci><apply id="S4.Thmtheorem9.p2.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p2.3.3.m3.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p2.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.3.3.m3.1c">\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.3.3.m3.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, obtained by truncating each witness at its first visit to a target of Player 1. Notice that each witness visits at least one target of Player <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.4.4.m4.1"><semantics id="S4.Thmtheorem9.p2.4.4.m4.1a"><mn id="S4.Thmtheorem9.p2.4.4.m4.1.1" xref="S4.Thmtheorem9.p2.4.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.4.4.m4.1b"><cn id="S4.Thmtheorem9.p2.4.4.m4.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p2.4.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.4.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.4.4.m4.1d">1</annotation></semantics></math> as <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.5.5.m5.4"><semantics id="S4.Thmtheorem9.p2.5.5.m5.4a"><mrow id="S4.Thmtheorem9.p2.5.5.m5.4.4" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.cmml"><msub id="S4.Thmtheorem9.p2.5.5.m5.4.4.3" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.cmml"><mi id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.2" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.2.cmml">C</mi><msub id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.cmml"><mi id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.2" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.3" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem9.p2.5.5.m5.4.4.2" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.2.cmml">≠</mo><mrow id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.2.cmml"><mo id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.2.cmml">{</mo><mrow id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.2" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem9.p2.5.5.m5.1.1" mathvariant="normal" xref="S4.Thmtheorem9.p2.5.5.m5.1.1.cmml">∞</mi><mo id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.2.2" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem9.p2.5.5.m5.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p2.5.5.m5.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.2.3" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem9.p2.5.5.m5.3.3" mathvariant="normal" xref="S4.Thmtheorem9.p2.5.5.m5.3.3.cmml">∞</mi><mo id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.2.4" stretchy="false" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.5.5.m5.4b"><apply id="S4.Thmtheorem9.p2.5.5.m5.4.4.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4"><neq id="S4.Thmtheorem9.p2.5.5.m5.4.4.2.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.2"></neq><apply id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.1.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.2.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.2">𝐶</ci><apply id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.1.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.2.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.3.3.3">0</cn></apply></apply><set id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.2.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1"><vector id="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.4.4.1.1.1.2"><infinity id="S4.Thmtheorem9.p2.5.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.1.1"></infinity><ci id="S4.Thmtheorem9.p2.5.5.m5.2.2.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.2.2">…</ci><infinity id="S4.Thmtheorem9.p2.5.5.m5.3.3.cmml" xref="S4.Thmtheorem9.p2.5.5.m5.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.5.5.m5.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.5.5.m5.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>, which explains that this truncation always exists. This truncated tree is finite: <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.6.6.m6.1"><semantics id="S4.Thmtheorem9.p2.6.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p2.6.6.m6.1.1" xref="S4.Thmtheorem9.p2.6.6.m6.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.6.6.m6.1b"><ci id="S4.Thmtheorem9.p2.6.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p2.6.6.m6.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.6.6.m6.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.6.6.m6.1d">caligraphic_T</annotation></semantics></math> has at most <math alttext="|C_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.7.7.m7.1"><semantics id="S4.Thmtheorem9.p2.7.7.m7.1a"><mrow id="S4.Thmtheorem9.p2.7.7.m7.1.1.1" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.2.cmml"><mo id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.2.1.cmml">|</mo><msub id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.2" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.2" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.3" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.7.7.m7.1b"><apply id="S4.Thmtheorem9.p2.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1"><abs id="S4.Thmtheorem9.p2.7.7.m7.1.1.2.1.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.2"></abs><apply id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p2.7.7.m7.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.7.7.m7.1c">|C_{\sigma_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.7.7.m7.1d">| italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math> leaves that correspond to the histories <math alttext="g" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.8.8.m8.1"><semantics id="S4.Thmtheorem9.p2.8.8.m8.1a"><mi id="S4.Thmtheorem9.p2.8.8.m8.1.1" xref="S4.Thmtheorem9.p2.8.8.m8.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.8.8.m8.1b"><ci id="S4.Thmtheorem9.p2.8.8.m8.1.1.cmml" xref="S4.Thmtheorem9.p2.8.8.m8.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.8.8.m8.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.8.8.m8.1d">italic_g</annotation></semantics></math>, prefixes of witnesses, such that <math alttext="w(g)=c_{min}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.9.9.m9.1"><semantics id="S4.Thmtheorem9.p2.9.9.m9.1a"><mrow id="S4.Thmtheorem9.p2.9.9.m9.1.2" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.cmml"><mrow id="S4.Thmtheorem9.p2.9.9.m9.1.2.2" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.cmml"><mi id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.2" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.2.cmml">w</mi><mo id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.1" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.3.2" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.cmml"><mo id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.9.9.m9.1.1" xref="S4.Thmtheorem9.p2.9.9.m9.1.1.cmml">g</mi><mo id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.9.9.m9.1.2.1" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.1.cmml">=</mo><msub id="S4.Thmtheorem9.p2.9.9.m9.1.2.3" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.cmml"><mi id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.2" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.cmml"><mi id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.2" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.1" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.3" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.1a" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.4" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.4.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.9.9.m9.1b"><apply id="S4.Thmtheorem9.p2.9.9.m9.1.2.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2"><eq id="S4.Thmtheorem9.p2.9.9.m9.1.2.1.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.1"></eq><apply id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2"><times id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.1.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.1"></times><ci id="S4.Thmtheorem9.p2.9.9.m9.1.2.2.2.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.2.2">𝑤</ci><ci id="S4.Thmtheorem9.p2.9.9.m9.1.1.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.1">𝑔</ci></apply><apply id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.1.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.2.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.2">𝑐</ci><apply id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3"><times id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.1.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.1"></times><ci id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.2.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.3.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.4.cmml" xref="S4.Thmtheorem9.p2.9.9.m9.1.2.3.3.4">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.9.9.m9.1c">w(g)=c_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.9.9.m9.1d">italic_w ( italic_g ) = italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, with <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.10.10.m10.1"><semantics id="S4.Thmtheorem9.p2.10.10.m10.1a"><mrow id="S4.Thmtheorem9.p2.10.10.m10.1.1" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.cmml"><mi id="S4.Thmtheorem9.p2.10.10.m10.1.1.2" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem9.p2.10.10.m10.1.1.1" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem9.p2.10.10.m10.1.1.3" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.cmml"><mi id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.2" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.2.cmml">C</mi><msub id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.cmml"><mi id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.2" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.3" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.10.10.m10.1b"><apply id="S4.Thmtheorem9.p2.10.10.m10.1.1.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1"><in id="S4.Thmtheorem9.p2.10.10.m10.1.1.1.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.1"></in><ci id="S4.Thmtheorem9.p2.10.10.m10.1.1.2.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.2">𝑐</ci><apply id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.1.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.2.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.2">𝐶</ci><apply id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.1.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.2.cmml" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p2.10.10.m10.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.10.10.m10.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.10.10.m10.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and such that the first visit of <math alttext="g" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.11.11.m11.1"><semantics id="S4.Thmtheorem9.p2.11.11.m11.1a"><mi id="S4.Thmtheorem9.p2.11.11.m11.1.1" xref="S4.Thmtheorem9.p2.11.11.m11.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.11.11.m11.1b"><ci id="S4.Thmtheorem9.p2.11.11.m11.1.1.cmml" xref="S4.Thmtheorem9.p2.11.11.m11.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.11.11.m11.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.11.11.m11.1d">italic_g</annotation></semantics></math> of some target of Player <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.12.12.m12.1"><semantics id="S4.Thmtheorem9.p2.12.12.m12.1a"><mn id="S4.Thmtheorem9.p2.12.12.m12.1.1" xref="S4.Thmtheorem9.p2.12.12.m12.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.12.12.m12.1b"><cn id="S4.Thmtheorem9.p2.12.12.m12.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p2.12.12.m12.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.12.12.m12.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.12.12.m12.1d">1</annotation></semantics></math> is in its last vertex <math alttext="\Last{g}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.13.13.m13.1"><semantics id="S4.Thmtheorem9.p2.13.13.m13.1a"><mrow id="S4.Thmtheorem9.p2.13.13.m13.1.1" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.cmml"><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem9.p2.13.13.m13.1.1.2" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.2b.cmml"><mtext id="S4.Thmtheorem9.p2.13.13.m13.1.1.2a" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.2b.cmml">\Last</mtext></merror><mo id="S4.Thmtheorem9.p2.13.13.m13.1.1.1" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p2.13.13.m13.1.1.3" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.13.13.m13.1b"><apply id="S4.Thmtheorem9.p2.13.13.m13.1.1.cmml" xref="S4.Thmtheorem9.p2.13.13.m13.1.1"><times id="S4.Thmtheorem9.p2.13.13.m13.1.1.1.cmml" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.1"></times><ci id="S4.Thmtheorem9.p2.13.13.m13.1.1.2b.cmml" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.2"><merror class="ltx_ERROR undefined undefined" id="S4.Thmtheorem9.p2.13.13.m13.1.1.2.cmml" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.2"><mtext id="S4.Thmtheorem9.p2.13.13.m13.1.1.2a.cmml" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.2">\Last</mtext></merror></ci><ci id="S4.Thmtheorem9.p2.13.13.m13.1.1.3.cmml" xref="S4.Thmtheorem9.p2.13.13.m13.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.13.13.m13.1c">\Last{g}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.13.13.m13.1d">italic_g</annotation></semantics></math>. Notice that one leaf of <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.14.14.m14.1"><semantics id="S4.Thmtheorem9.p2.14.14.m14.1a"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p2.14.14.m14.1.1" xref="S4.Thmtheorem9.p2.14.14.m14.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.14.14.m14.1b"><ci id="S4.Thmtheorem9.p2.14.14.m14.1.1.cmml" xref="S4.Thmtheorem9.p2.14.14.m14.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.14.14.m14.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.14.14.m14.1d">caligraphic_T</annotation></semantics></math> corresponds to some <math alttext="g" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.15.15.m15.1"><semantics id="S4.Thmtheorem9.p2.15.15.m15.1a"><mi id="S4.Thmtheorem9.p2.15.15.m15.1.1" xref="S4.Thmtheorem9.p2.15.15.m15.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.15.15.m15.1b"><ci id="S4.Thmtheorem9.p2.15.15.m15.1.1.cmml" xref="S4.Thmtheorem9.p2.15.15.m15.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.15.15.m15.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.15.15.m15.1d">italic_g</annotation></semantics></math> such that <math alttext="w(g)=d_{min}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.16.16.m16.1"><semantics id="S4.Thmtheorem9.p2.16.16.m16.1a"><mrow id="S4.Thmtheorem9.p2.16.16.m16.1.2" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.cmml"><mrow id="S4.Thmtheorem9.p2.16.16.m16.1.2.2" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.cmml"><mi id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.2" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.2.cmml">w</mi><mo id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.1" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.3.2" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.cmml"><mo id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.16.16.m16.1.1" xref="S4.Thmtheorem9.p2.16.16.m16.1.1.cmml">g</mi><mo id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.16.16.m16.1.2.1" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.1.cmml">=</mo><msub id="S4.Thmtheorem9.p2.16.16.m16.1.2.3" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.cmml"><mi id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.2" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.2.cmml">d</mi><mrow id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.cmml"><mi id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.2" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.1" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.3" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.1a" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.4" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.4.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.16.16.m16.1b"><apply id="S4.Thmtheorem9.p2.16.16.m16.1.2.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2"><eq id="S4.Thmtheorem9.p2.16.16.m16.1.2.1.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.1"></eq><apply id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2"><times id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.1.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.1"></times><ci id="S4.Thmtheorem9.p2.16.16.m16.1.2.2.2.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.2.2">𝑤</ci><ci id="S4.Thmtheorem9.p2.16.16.m16.1.1.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.1">𝑔</ci></apply><apply id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.1.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3">subscript</csymbol><ci id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.2.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.2">𝑑</ci><apply id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3"><times id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.1.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.1"></times><ci id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.2.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.3.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.4.cmml" xref="S4.Thmtheorem9.p2.16.16.m16.1.2.3.3.4">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.16.16.m16.1c">w(g)=d_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.16.16.m16.1d">italic_w ( italic_g ) = italic_d start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem9.p3"> <p class="ltx_p" id="S4.Thmtheorem9.p3.24"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p3.24.24">Let us construct the first Pareto-optimal cost <math alttext="c^{(0)}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.1.1.m1.1"><semantics id="S4.Thmtheorem9.p3.1.1.m1.1a"><msup id="S4.Thmtheorem9.p3.1.1.m1.1.2" xref="S4.Thmtheorem9.p3.1.1.m1.1.2.cmml"><mi id="S4.Thmtheorem9.p3.1.1.m1.1.2.2" xref="S4.Thmtheorem9.p3.1.1.m1.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p3.1.1.m1.1.1.1.3" xref="S4.Thmtheorem9.p3.1.1.m1.1.2.cmml"><mo id="S4.Thmtheorem9.p3.1.1.m1.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p3.1.1.m1.1.2.cmml">(</mo><mn id="S4.Thmtheorem9.p3.1.1.m1.1.1.1.1" xref="S4.Thmtheorem9.p3.1.1.m1.1.1.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p3.1.1.m1.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p3.1.1.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.1.1.m1.1b"><apply id="S4.Thmtheorem9.p3.1.1.m1.1.2.cmml" xref="S4.Thmtheorem9.p3.1.1.m1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.1.1.m1.1.2.1.cmml" xref="S4.Thmtheorem9.p3.1.1.m1.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p3.1.1.m1.1.2.2.cmml" xref="S4.Thmtheorem9.p3.1.1.m1.1.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p3.1.1.m1.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p3.1.1.m1.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.1.1.m1.1c">c^{(0)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.1.1.m1.1d">italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="h_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.2.2.m2.1"><semantics id="S4.Thmtheorem9.p3.2.2.m2.1a"><msub id="S4.Thmtheorem9.p3.2.2.m2.1.1" xref="S4.Thmtheorem9.p3.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem9.p3.2.2.m2.1.1.2" xref="S4.Thmtheorem9.p3.2.2.m2.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p3.2.2.m2.1.1.3" xref="S4.Thmtheorem9.p3.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.2.2.m2.1b"><apply id="S4.Thmtheorem9.p3.2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p3.2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem9.p3.2.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p3.2.2.m2.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p3.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.2.2.m2.1c">h_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.2.2.m2.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be the history of maximal length such that <math alttext="h_{0}\rho" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.3.3.m3.1"><semantics id="S4.Thmtheorem9.p3.3.3.m3.1a"><mrow id="S4.Thmtheorem9.p3.3.3.m3.1.1" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.cmml"><msub id="S4.Thmtheorem9.p3.3.3.m3.1.1.2" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.2.cmml"><mi id="S4.Thmtheorem9.p3.3.3.m3.1.1.2.2" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.2.2.cmml">h</mi><mn id="S4.Thmtheorem9.p3.3.3.m3.1.1.2.3" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem9.p3.3.3.m3.1.1.1" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p3.3.3.m3.1.1.3" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.3.3.m3.1b"><apply id="S4.Thmtheorem9.p3.3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p3.3.3.m3.1.1"><times id="S4.Thmtheorem9.p3.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.1"></times><apply id="S4.Thmtheorem9.p3.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.3.3.m3.1.1.2.1.cmml" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem9.p3.3.3.m3.1.1.2.2.cmml" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.2.2">ℎ</ci><cn id="S4.Thmtheorem9.p3.3.3.m3.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.2.3">0</cn></apply><ci id="S4.Thmtheorem9.p3.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem9.p3.3.3.m3.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.3.3.m3.1c">h_{0}\rho</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.3.3.m3.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ρ</annotation></semantics></math> and <math alttext="w(h_{0})=B" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.4.4.m4.1"><semantics id="S4.Thmtheorem9.p3.4.4.m4.1a"><mrow id="S4.Thmtheorem9.p3.4.4.m4.1.1" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.cmml"><mrow id="S4.Thmtheorem9.p3.4.4.m4.1.1.1" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.cmml"><mi id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.3" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.3.cmml">w</mi><mo id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.2" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.2" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.3" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p3.4.4.m4.1.1.2" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.2.cmml">=</mo><mi id="S4.Thmtheorem9.p3.4.4.m4.1.1.3" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.4.4.m4.1b"><apply id="S4.Thmtheorem9.p3.4.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1"><eq id="S4.Thmtheorem9.p3.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.2"></eq><apply id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1"><times id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.2.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.2"></times><ci id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.3.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.3">𝑤</ci><apply id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.1.1.1.1.3">0</cn></apply></apply><ci id="S4.Thmtheorem9.p3.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem9.p3.4.4.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.4.4.m4.1c">w(h_{0})=B</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.4.4.m4.1d">italic_w ( italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_B</annotation></semantics></math>. This history <math alttext="h_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.5.5.m5.1"><semantics id="S4.Thmtheorem9.p3.5.5.m5.1a"><msub id="S4.Thmtheorem9.p3.5.5.m5.1.1" xref="S4.Thmtheorem9.p3.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem9.p3.5.5.m5.1.1.2" xref="S4.Thmtheorem9.p3.5.5.m5.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p3.5.5.m5.1.1.3" xref="S4.Thmtheorem9.p3.5.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.5.5.m5.1b"><apply id="S4.Thmtheorem9.p3.5.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p3.5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem9.p3.5.5.m5.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem9.p3.5.5.m5.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p3.5.5.m5.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.5.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.5.5.m5.1c">h_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.5.5.m5.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> exists because the arena is binary and <math alttext="\textsf{cost}({\rho})=d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.6.6.m6.1"><semantics id="S4.Thmtheorem9.p3.6.6.m6.1a"><mrow id="S4.Thmtheorem9.p3.6.6.m6.1.2" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.cmml"><mrow id="S4.Thmtheorem9.p3.6.6.m6.1.2.2" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.2" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.2a.cmml">cost</mtext><mo id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.1" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.3.2" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.cmml"><mo id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p3.6.6.m6.1.1" xref="S4.Thmtheorem9.p3.6.6.m6.1.1.cmml">ρ</mi><mo id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p3.6.6.m6.1.2.1" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.1.cmml">=</mo><mi id="S4.Thmtheorem9.p3.6.6.m6.1.2.3" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.6.6.m6.1b"><apply id="S4.Thmtheorem9.p3.6.6.m6.1.2.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.2"><eq id="S4.Thmtheorem9.p3.6.6.m6.1.2.1.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.1"></eq><apply id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2"><times id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.1.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.1"></times><ci id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.2a.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem9.p3.6.6.m6.1.2.2.2.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.2.2">cost</mtext></ci><ci id="S4.Thmtheorem9.p3.6.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.1">𝜌</ci></apply><ci id="S4.Thmtheorem9.p3.6.6.m6.1.2.3.cmml" xref="S4.Thmtheorem9.p3.6.6.m6.1.2.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.6.6.m6.1c">\textsf{cost}({\rho})=d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.6.6.m6.1d">cost ( italic_ρ ) = italic_d</annotation></semantics></math> with <math alttext="d_{min}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.7.7.m7.1"><semantics id="S4.Thmtheorem9.p3.7.7.m7.1a"><msub id="S4.Thmtheorem9.p3.7.7.m7.1.1" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem9.p3.7.7.m7.1.1.2" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.2.cmml">d</mi><mrow id="S4.Thmtheorem9.p3.7.7.m7.1.1.3" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.cmml"><mi id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.2" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.1" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.3" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.1a" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.4" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.4.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.7.7.m7.1b"><apply id="S4.Thmtheorem9.p3.7.7.m7.1.1.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.2">𝑑</ci><apply id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3"><times id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.1.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.1"></times><ci id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.2.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.3.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p3.7.7.m7.1.1.3.4.cmml" xref="S4.Thmtheorem9.p3.7.7.m7.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.7.7.m7.1c">d_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.7.7.m7.1d">italic_d start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math> satisfying (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E12" title="In Proof 4.9 (Proof of Lemma 4.6) ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">12</span></a>). Moreover, as <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.8.8.m8.1"><semantics id="S4.Thmtheorem9.p3.8.8.m8.1a"><msub id="S4.Thmtheorem9.p3.8.8.m8.1.1" xref="S4.Thmtheorem9.p3.8.8.m8.1.1.cmml"><mi id="S4.Thmtheorem9.p3.8.8.m8.1.1.2" xref="S4.Thmtheorem9.p3.8.8.m8.1.1.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p3.8.8.m8.1.1.3" xref="S4.Thmtheorem9.p3.8.8.m8.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.8.8.m8.1b"><apply id="S4.Thmtheorem9.p3.8.8.m8.1.1.cmml" xref="S4.Thmtheorem9.p3.8.8.m8.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem9.p3.8.8.m8.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem9.p3.8.8.m8.1.1.2">𝜎</ci><cn id="S4.Thmtheorem9.p3.8.8.m8.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.8.8.m8.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.8.8.m8.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution, <math alttext="h_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.9.9.m9.1"><semantics id="S4.Thmtheorem9.p3.9.9.m9.1a"><msub id="S4.Thmtheorem9.p3.9.9.m9.1.1" xref="S4.Thmtheorem9.p3.9.9.m9.1.1.cmml"><mi id="S4.Thmtheorem9.p3.9.9.m9.1.1.2" xref="S4.Thmtheorem9.p3.9.9.m9.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p3.9.9.m9.1.1.3" xref="S4.Thmtheorem9.p3.9.9.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.9.9.m9.1b"><apply id="S4.Thmtheorem9.p3.9.9.m9.1.1.cmml" xref="S4.Thmtheorem9.p3.9.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.9.9.m9.1.1.1.cmml" xref="S4.Thmtheorem9.p3.9.9.m9.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.9.9.m9.1.1.2.cmml" xref="S4.Thmtheorem9.p3.9.9.m9.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p3.9.9.m9.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.9.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.9.9.m9.1c">h_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.9.9.m9.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> visits Player <math alttext="0" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.10.10.m10.1"><semantics id="S4.Thmtheorem9.p3.10.10.m10.1a"><mn id="S4.Thmtheorem9.p3.10.10.m10.1.1" xref="S4.Thmtheorem9.p3.10.10.m10.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.10.10.m10.1b"><cn id="S4.Thmtheorem9.p3.10.10.m10.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p3.10.10.m10.1.1">0</cn></annotation-xml></semantics></math>’s target. We consider the subtree <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.11.11.m11.1"><semantics id="S4.Thmtheorem9.p3.11.11.m11.1a"><msub id="S4.Thmtheorem9.p3.11.11.m11.1.1" xref="S4.Thmtheorem9.p3.11.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p3.11.11.m11.1.1.2" xref="S4.Thmtheorem9.p3.11.11.m11.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p3.11.11.m11.1.1.3" xref="S4.Thmtheorem9.p3.11.11.m11.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.11.11.m11.1b"><apply id="S4.Thmtheorem9.p3.11.11.m11.1.1.cmml" xref="S4.Thmtheorem9.p3.11.11.m11.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.11.11.m11.1.1.1.cmml" xref="S4.Thmtheorem9.p3.11.11.m11.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.11.11.m11.1.1.2.cmml" xref="S4.Thmtheorem9.p3.11.11.m11.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p3.11.11.m11.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.11.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.11.11.m11.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.11.11.m11.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.12.12.m12.1"><semantics id="S4.Thmtheorem9.p3.12.12.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p3.12.12.m12.1.1" xref="S4.Thmtheorem9.p3.12.12.m12.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.12.12.m12.1b"><ci id="S4.Thmtheorem9.p3.12.12.m12.1.1.cmml" xref="S4.Thmtheorem9.p3.12.12.m12.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.12.12.m12.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.12.12.m12.1d">caligraphic_T</annotation></semantics></math> rooted in the last vertex of <math alttext="h_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.13.13.m13.1"><semantics id="S4.Thmtheorem9.p3.13.13.m13.1a"><msub id="S4.Thmtheorem9.p3.13.13.m13.1.1" xref="S4.Thmtheorem9.p3.13.13.m13.1.1.cmml"><mi id="S4.Thmtheorem9.p3.13.13.m13.1.1.2" xref="S4.Thmtheorem9.p3.13.13.m13.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p3.13.13.m13.1.1.3" xref="S4.Thmtheorem9.p3.13.13.m13.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.13.13.m13.1b"><apply id="S4.Thmtheorem9.p3.13.13.m13.1.1.cmml" xref="S4.Thmtheorem9.p3.13.13.m13.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.13.13.m13.1.1.1.cmml" xref="S4.Thmtheorem9.p3.13.13.m13.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.13.13.m13.1.1.2.cmml" xref="S4.Thmtheorem9.p3.13.13.m13.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p3.13.13.m13.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.13.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.13.13.m13.1c">h_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.13.13.m13.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. This subtree has at most <math alttext="|C_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.14.14.m14.1"><semantics id="S4.Thmtheorem9.p3.14.14.m14.1a"><mrow id="S4.Thmtheorem9.p3.14.14.m14.1.1.1" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.2.cmml"><mo id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.2.1.cmml">|</mo><msub id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.2" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.2" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.3" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.14.14.m14.1b"><apply id="S4.Thmtheorem9.p3.14.14.m14.1.1.2.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1"><abs id="S4.Thmtheorem9.p3.14.14.m14.1.1.2.1.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.2"></abs><apply id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.14.14.m14.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.14.14.m14.1c">|C_{\sigma_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.14.14.m14.1d">| italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math> leaves as it is the case for <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.15.15.m15.1"><semantics id="S4.Thmtheorem9.p3.15.15.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p3.15.15.m15.1.1" xref="S4.Thmtheorem9.p3.15.15.m15.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.15.15.m15.1b"><ci id="S4.Thmtheorem9.p3.15.15.m15.1.1.cmml" xref="S4.Thmtheorem9.p3.15.15.m15.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.15.15.m15.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.15.15.m15.1d">caligraphic_T</annotation></semantics></math>. Its nodes with degree <math alttext="\geq 2" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.16.16.m16.1"><semantics id="S4.Thmtheorem9.p3.16.16.m16.1a"><mrow id="S4.Thmtheorem9.p3.16.16.m16.1.1" xref="S4.Thmtheorem9.p3.16.16.m16.1.1.cmml"><mi id="S4.Thmtheorem9.p3.16.16.m16.1.1.2" xref="S4.Thmtheorem9.p3.16.16.m16.1.1.2.cmml"></mi><mo id="S4.Thmtheorem9.p3.16.16.m16.1.1.1" xref="S4.Thmtheorem9.p3.16.16.m16.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem9.p3.16.16.m16.1.1.3" xref="S4.Thmtheorem9.p3.16.16.m16.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.16.16.m16.1b"><apply id="S4.Thmtheorem9.p3.16.16.m16.1.1.cmml" xref="S4.Thmtheorem9.p3.16.16.m16.1.1"><geq id="S4.Thmtheorem9.p3.16.16.m16.1.1.1.cmml" xref="S4.Thmtheorem9.p3.16.16.m16.1.1.1"></geq><csymbol cd="latexml" id="S4.Thmtheorem9.p3.16.16.m16.1.1.2.cmml" xref="S4.Thmtheorem9.p3.16.16.m16.1.1.2">absent</csymbol><cn id="S4.Thmtheorem9.p3.16.16.m16.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.16.16.m16.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.16.16.m16.1c">\geq 2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.16.16.m16.1d">≥ 2</annotation></semantics></math> correspond to histories that are branching points<span class="ltx_note ltx_role_footnote" id="footnote9"><sup class="ltx_note_mark">9</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">9</sup><span class="ltx_tag ltx_tag_note"><span class="ltx_text ltx_font_upright" id="footnote9.1.1.1">9</span></span><span class="ltx_text ltx_font_upright" id="footnote9.5">We recall that a branching point is a history which is the greatest common prefix of two witnesses.</span></span></span></span>. Moreover, any two consecutive nodes with degree one along any branch of <math alttext="{\mathcal{T}}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.17.17.m17.1"><semantics id="S4.Thmtheorem9.p3.17.17.m17.1a"><msub id="S4.Thmtheorem9.p3.17.17.m17.1.1" xref="S4.Thmtheorem9.p3.17.17.m17.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p3.17.17.m17.1.1.2" xref="S4.Thmtheorem9.p3.17.17.m17.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p3.17.17.m17.1.1.3" xref="S4.Thmtheorem9.p3.17.17.m17.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.17.17.m17.1b"><apply id="S4.Thmtheorem9.p3.17.17.m17.1.1.cmml" xref="S4.Thmtheorem9.p3.17.17.m17.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.17.17.m17.1.1.1.cmml" xref="S4.Thmtheorem9.p3.17.17.m17.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.17.17.m17.1.1.2.cmml" xref="S4.Thmtheorem9.p3.17.17.m17.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p3.17.17.m17.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.17.17.m17.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.17.17.m17.1c">{\mathcal{T}}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.17.17.m17.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> are in the same region, as Player <math alttext="0" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.18.18.m18.1"><semantics id="S4.Thmtheorem9.p3.18.18.m18.1a"><mn id="S4.Thmtheorem9.p3.18.18.m18.1.1" xref="S4.Thmtheorem9.p3.18.18.m18.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.18.18.m18.1b"><cn id="S4.Thmtheorem9.p3.18.18.m18.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p3.18.18.m18.1.1">0</cn></annotation-xml></semantics></math>’s target is visited by <math alttext="h_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.19.19.m19.1"><semantics id="S4.Thmtheorem9.p3.19.19.m19.1a"><msub id="S4.Thmtheorem9.p3.19.19.m19.1.1" xref="S4.Thmtheorem9.p3.19.19.m19.1.1.cmml"><mi id="S4.Thmtheorem9.p3.19.19.m19.1.1.2" xref="S4.Thmtheorem9.p3.19.19.m19.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p3.19.19.m19.1.1.3" xref="S4.Thmtheorem9.p3.19.19.m19.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.19.19.m19.1b"><apply id="S4.Thmtheorem9.p3.19.19.m19.1.1.cmml" xref="S4.Thmtheorem9.p3.19.19.m19.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.19.19.m19.1.1.1.cmml" xref="S4.Thmtheorem9.p3.19.19.m19.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.19.19.m19.1.1.2.cmml" xref="S4.Thmtheorem9.p3.19.19.m19.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p3.19.19.m19.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.19.19.m19.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.19.19.m19.1c">h_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.19.19.m19.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and the first visit to a target of Player <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.20.20.m20.1"><semantics id="S4.Thmtheorem9.p3.20.20.m20.1a"><mn id="S4.Thmtheorem9.p3.20.20.m20.1.1" xref="S4.Thmtheorem9.p3.20.20.m20.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.20.20.m20.1b"><cn id="S4.Thmtheorem9.p3.20.20.m20.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p3.20.20.m20.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.20.20.m20.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.20.20.m20.1d">1</annotation></semantics></math> is in a leaf of <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.21.21.m21.1"><semantics id="S4.Thmtheorem9.p3.21.21.m21.1a"><msub id="S4.Thmtheorem9.p3.21.21.m21.1.1" xref="S4.Thmtheorem9.p3.21.21.m21.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p3.21.21.m21.1.1.2" xref="S4.Thmtheorem9.p3.21.21.m21.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p3.21.21.m21.1.1.3" xref="S4.Thmtheorem9.p3.21.21.m21.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.21.21.m21.1b"><apply id="S4.Thmtheorem9.p3.21.21.m21.1.1.cmml" xref="S4.Thmtheorem9.p3.21.21.m21.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.21.21.m21.1.1.1.cmml" xref="S4.Thmtheorem9.p3.21.21.m21.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.21.21.m21.1.1.2.cmml" xref="S4.Thmtheorem9.p3.21.21.m21.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p3.21.21.m21.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.21.21.m21.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.21.21.m21.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.21.21.m21.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. As <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.22.22.m22.1"><semantics id="S4.Thmtheorem9.p3.22.22.m22.1a"><msub id="S4.Thmtheorem9.p3.22.22.m22.1.1" xref="S4.Thmtheorem9.p3.22.22.m22.1.1.cmml"><mi id="S4.Thmtheorem9.p3.22.22.m22.1.1.2" xref="S4.Thmtheorem9.p3.22.22.m22.1.1.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p3.22.22.m22.1.1.3" xref="S4.Thmtheorem9.p3.22.22.m22.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.22.22.m22.1b"><apply id="S4.Thmtheorem9.p3.22.22.m22.1.1.cmml" xref="S4.Thmtheorem9.p3.22.22.m22.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.22.22.m22.1.1.1.cmml" xref="S4.Thmtheorem9.p3.22.22.m22.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.22.22.m22.1.1.2.cmml" xref="S4.Thmtheorem9.p3.22.22.m22.1.1.2">𝜎</ci><cn id="S4.Thmtheorem9.p3.22.22.m22.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.22.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.22.22.m22.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.22.22.m22.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is without cycle in the sense of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>, it follows that there are at most <math alttext="|V|-1" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.23.23.m23.1"><semantics id="S4.Thmtheorem9.p3.23.23.m23.1a"><mrow id="S4.Thmtheorem9.p3.23.23.m23.1.2" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.cmml"><mrow id="S4.Thmtheorem9.p3.23.23.m23.1.2.2.2" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.2.1.cmml"><mo id="S4.Thmtheorem9.p3.23.23.m23.1.2.2.2.1" stretchy="false" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.2.1.1.cmml">|</mo><mi id="S4.Thmtheorem9.p3.23.23.m23.1.1" xref="S4.Thmtheorem9.p3.23.23.m23.1.1.cmml">V</mi><mo id="S4.Thmtheorem9.p3.23.23.m23.1.2.2.2.2" stretchy="false" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem9.p3.23.23.m23.1.2.1" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.1.cmml">−</mo><mn id="S4.Thmtheorem9.p3.23.23.m23.1.2.3" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.23.23.m23.1b"><apply id="S4.Thmtheorem9.p3.23.23.m23.1.2.cmml" xref="S4.Thmtheorem9.p3.23.23.m23.1.2"><minus id="S4.Thmtheorem9.p3.23.23.m23.1.2.1.cmml" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.1"></minus><apply id="S4.Thmtheorem9.p3.23.23.m23.1.2.2.1.cmml" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.2.2"><abs id="S4.Thmtheorem9.p3.23.23.m23.1.2.2.1.1.cmml" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.2.2.1"></abs><ci id="S4.Thmtheorem9.p3.23.23.m23.1.1.cmml" xref="S4.Thmtheorem9.p3.23.23.m23.1.1">𝑉</ci></apply><cn id="S4.Thmtheorem9.p3.23.23.m23.1.2.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.23.23.m23.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.23.23.m23.1c">|V|-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.23.23.m23.1d">| italic_V | - 1</annotation></semantics></math> consecutive nodes with degree one along any branch of <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.24.24.m24.1"><semantics id="S4.Thmtheorem9.p3.24.24.m24.1a"><msub id="S4.Thmtheorem9.p3.24.24.m24.1.1" xref="S4.Thmtheorem9.p3.24.24.m24.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p3.24.24.m24.1.1.2" xref="S4.Thmtheorem9.p3.24.24.m24.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p3.24.24.m24.1.1.3" xref="S4.Thmtheorem9.p3.24.24.m24.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.24.24.m24.1b"><apply id="S4.Thmtheorem9.p3.24.24.m24.1.1.cmml" xref="S4.Thmtheorem9.p3.24.24.m24.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.24.24.m24.1.1.1.cmml" xref="S4.Thmtheorem9.p3.24.24.m24.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.24.24.m24.1.1.2.cmml" xref="S4.Thmtheorem9.p3.24.24.m24.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p3.24.24.m24.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.24.24.m24.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.24.24.m24.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.24.24.m24.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem9.p4"> <p class="ltx_p" id="S4.Thmtheorem9.p4.12"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p4.12.12">Therefore, we can apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem7" title="Lemma 4.7 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.7</span></a> to <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.1.1.m1.1"><semantics id="S4.Thmtheorem9.p4.1.1.m1.1a"><msub id="S4.Thmtheorem9.p4.1.1.m1.1.1" xref="S4.Thmtheorem9.p4.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p4.1.1.m1.1.1.2" xref="S4.Thmtheorem9.p4.1.1.m1.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p4.1.1.m1.1.1.3" xref="S4.Thmtheorem9.p4.1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.1.1.m1.1b"><apply id="S4.Thmtheorem9.p4.1.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p4.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.1.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem9.p4.1.1.m1.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p4.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.1.1.m1.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.1.1.m1.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with the parameters <math alttext="n=|C_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.2.2.m2.1"><semantics id="S4.Thmtheorem9.p4.2.2.m2.1a"><mrow id="S4.Thmtheorem9.p4.2.2.m2.1.1" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem9.p4.2.2.m2.1.1.3" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.3.cmml">n</mi><mo id="S4.Thmtheorem9.p4.2.2.m2.1.1.2" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.2.cmml">=</mo><mrow id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.2.cmml"><mo id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.2.1.cmml">|</mo><msub id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.2" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.2" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.3" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.2.2.m2.1b"><apply id="S4.Thmtheorem9.p4.2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1"><eq id="S4.Thmtheorem9.p4.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.2"></eq><ci id="S4.Thmtheorem9.p4.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.3">𝑛</ci><apply id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.2.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1"><abs id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.2.1.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.2"></abs><apply id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.2.2.m2.1.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.2.2.m2.1c">n=|C_{\sigma_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.2.2.m2.1d">italic_n = | italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math> and <math alttext="\ell=|V|" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.3.3.m3.1"><semantics id="S4.Thmtheorem9.p4.3.3.m3.1a"><mrow id="S4.Thmtheorem9.p4.3.3.m3.1.2" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.cmml"><mi id="S4.Thmtheorem9.p4.3.3.m3.1.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.2.cmml">ℓ</mi><mo id="S4.Thmtheorem9.p4.3.3.m3.1.2.1" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p4.3.3.m3.1.2.3.2" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.3.1.cmml"><mo id="S4.Thmtheorem9.p4.3.3.m3.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem9.p4.3.3.m3.1.1" xref="S4.Thmtheorem9.p4.3.3.m3.1.1.cmml">V</mi><mo id="S4.Thmtheorem9.p4.3.3.m3.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.3.3.m3.1b"><apply id="S4.Thmtheorem9.p4.3.3.m3.1.2.cmml" xref="S4.Thmtheorem9.p4.3.3.m3.1.2"><eq id="S4.Thmtheorem9.p4.3.3.m3.1.2.1.cmml" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.1"></eq><ci id="S4.Thmtheorem9.p4.3.3.m3.1.2.2.cmml" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.2">ℓ</ci><apply id="S4.Thmtheorem9.p4.3.3.m3.1.2.3.1.cmml" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.3.2"><abs id="S4.Thmtheorem9.p4.3.3.m3.1.2.3.1.1.cmml" xref="S4.Thmtheorem9.p4.3.3.m3.1.2.3.2.1"></abs><ci id="S4.Thmtheorem9.p4.3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p4.3.3.m3.1.1">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.3.3.m3.1c">\ell=|V|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.3.3.m3.1d">roman_ℓ = | italic_V |</annotation></semantics></math>. It follows that the leaves of <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.4.4.m4.1"><semantics id="S4.Thmtheorem9.p4.4.4.m4.1a"><msub id="S4.Thmtheorem9.p4.4.4.m4.1.1" xref="S4.Thmtheorem9.p4.4.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p4.4.4.m4.1.1.2" xref="S4.Thmtheorem9.p4.4.4.m4.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p4.4.4.m4.1.1.3" xref="S4.Thmtheorem9.p4.4.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.4.4.m4.1b"><apply id="S4.Thmtheorem9.p4.4.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p4.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem9.p4.4.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem9.p4.4.4.m4.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p4.4.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.4.4.m4.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.4.4.m4.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with the smallest depth have a depth <math alttext="\leq\delta=|V|\cdot(\log_{2}(|C_{\sigma_{0}}|)+1)" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.5.5.m5.2"><semantics id="S4.Thmtheorem9.p4.5.5.m5.2a"><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.cmml"><mi id="S4.Thmtheorem9.p4.5.5.m5.2.2.3" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.3.cmml"></mi><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.4" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.4.cmml">≤</mo><mi id="S4.Thmtheorem9.p4.5.5.m5.2.2.5" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.5.cmml">δ</mi><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.6" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.6.cmml">=</mo><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2.1" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.cmml"><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.2" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.1.cmml"><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem9.p4.5.5.m5.1.1" xref="S4.Thmtheorem9.p4.5.5.m5.1.1.cmml">V</mi><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.2.2" rspace="0.055em" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.2" rspace="0.222em" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.2.cmml">⋅</mo><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.cmml"><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.3.cmml"><msub id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.2" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.2.cmml">log</mi><mn id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2a" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.3.cmml">⁡</mo><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.3.cmml"><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.2" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.3.cmml">(</mo><mrow id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.2.cmml"><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.2.1.cmml">|</mo><msub id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.2" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.2" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.3" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.3" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.3" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.3.cmml">+</mo><mn id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.4" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.4.cmml">1</mn></mrow><mo id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.5.5.m5.2b"><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2"><and id="S4.Thmtheorem9.p4.5.5.m5.2.2a.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2"></and><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2b.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2"><leq id="S4.Thmtheorem9.p4.5.5.m5.2.2.4.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.4"></leq><csymbol cd="latexml" id="S4.Thmtheorem9.p4.5.5.m5.2.2.3.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.3">absent</csymbol><ci id="S4.Thmtheorem9.p4.5.5.m5.2.2.5.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.5">𝛿</ci></apply><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2c.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2"><eq id="S4.Thmtheorem9.p4.5.5.m5.2.2.6.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.6"></eq><share href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem9.p4.5.5.m5.2.2.5.cmml" id="S4.Thmtheorem9.p4.5.5.m5.2.2d.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2"></share><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1"><ci id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.2.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.2">⋅</ci><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.2"><abs id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.1.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.3.2.1"></abs><ci id="S4.Thmtheorem9.p4.5.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.1.1">𝑉</ci></apply><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1"><plus id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.3"></plus><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.3.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2"><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1">subscript</csymbol><log id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.2"></log><cn id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.2.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1"><abs id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.2.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.2"></abs><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.2.2.2.1.1.1.3.3">0</cn></apply></apply></apply></apply><cn id="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.4.cmml" type="integer" xref="S4.Thmtheorem9.p4.5.5.m5.2.2.1.1.1.1.4">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.5.5.m5.2c">\leq\delta=|V|\cdot(\log_{2}(|C_{\sigma_{0}}|)+1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.5.5.m5.2d">≤ italic_δ = | italic_V | ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( | italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ) + 1 )</annotation></semantics></math>. We set <math alttext="c^{(0)}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.6.6.m6.1"><semantics id="S4.Thmtheorem9.p4.6.6.m6.1a"><msup id="S4.Thmtheorem9.p4.6.6.m6.1.2" xref="S4.Thmtheorem9.p4.6.6.m6.1.2.cmml"><mi id="S4.Thmtheorem9.p4.6.6.m6.1.2.2" xref="S4.Thmtheorem9.p4.6.6.m6.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p4.6.6.m6.1.1.1.3" xref="S4.Thmtheorem9.p4.6.6.m6.1.2.cmml"><mo id="S4.Thmtheorem9.p4.6.6.m6.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p4.6.6.m6.1.2.cmml">(</mo><mn id="S4.Thmtheorem9.p4.6.6.m6.1.1.1.1" xref="S4.Thmtheorem9.p4.6.6.m6.1.1.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p4.6.6.m6.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p4.6.6.m6.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.6.6.m6.1b"><apply id="S4.Thmtheorem9.p4.6.6.m6.1.2.cmml" xref="S4.Thmtheorem9.p4.6.6.m6.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.6.6.m6.1.2.1.cmml" xref="S4.Thmtheorem9.p4.6.6.m6.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p4.6.6.m6.1.2.2.cmml" xref="S4.Thmtheorem9.p4.6.6.m6.1.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p4.6.6.m6.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p4.6.6.m6.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.6.6.m6.1c">c^{(0)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.6.6.m6.1d">italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math> as the cost of a witness associated with one of those leaves. We get that <math alttext="c^{(0)}_{min}\leq B+\delta" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.7.7.m7.1"><semantics id="S4.Thmtheorem9.p4.7.7.m7.1a"><mrow id="S4.Thmtheorem9.p4.7.7.m7.1.2" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.cmml"><msubsup id="S4.Thmtheorem9.p4.7.7.m7.1.2.2" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.cmml"><mi id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.2.2" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.cmml"><mi id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.2" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.1" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.3" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.1a" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.4" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.4.cmml">n</mi></mrow><mrow id="S4.Thmtheorem9.p4.7.7.m7.1.1.1.3" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.cmml"><mo id="S4.Thmtheorem9.p4.7.7.m7.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.cmml">(</mo><mn id="S4.Thmtheorem9.p4.7.7.m7.1.1.1.1" xref="S4.Thmtheorem9.p4.7.7.m7.1.1.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p4.7.7.m7.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.cmml">)</mo></mrow></msubsup><mo id="S4.Thmtheorem9.p4.7.7.m7.1.2.1" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.1.cmml">≤</mo><mrow id="S4.Thmtheorem9.p4.7.7.m7.1.2.3" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3.cmml"><mi id="S4.Thmtheorem9.p4.7.7.m7.1.2.3.2" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3.2.cmml">B</mi><mo id="S4.Thmtheorem9.p4.7.7.m7.1.2.3.1" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3.1.cmml">+</mo><mi id="S4.Thmtheorem9.p4.7.7.m7.1.2.3.3" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3.3.cmml">δ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.7.7.m7.1b"><apply id="S4.Thmtheorem9.p4.7.7.m7.1.2.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2"><leq id="S4.Thmtheorem9.p4.7.7.m7.1.2.1.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.1"></leq><apply id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.1.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2">subscript</csymbol><apply id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.2.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.2.1.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2">superscript</csymbol><ci id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.2.2.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p4.7.7.m7.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p4.7.7.m7.1.1.1.1">0</cn></apply><apply id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3"><times id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.1.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.1"></times><ci id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.2.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.3.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.4.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.2.3.4">𝑛</ci></apply></apply><apply id="S4.Thmtheorem9.p4.7.7.m7.1.2.3.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3"><plus id="S4.Thmtheorem9.p4.7.7.m7.1.2.3.1.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3.1"></plus><ci id="S4.Thmtheorem9.p4.7.7.m7.1.2.3.2.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3.2">𝐵</ci><ci id="S4.Thmtheorem9.p4.7.7.m7.1.2.3.3.cmml" xref="S4.Thmtheorem9.p4.7.7.m7.1.2.3.3">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.7.7.m7.1c">c^{(0)}_{min}\leq B+\delta</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.7.7.m7.1d">italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ≤ italic_B + italic_δ</annotation></semantics></math> because <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.8.8.m8.1"><semantics id="S4.Thmtheorem9.p4.8.8.m8.1a"><msub id="S4.Thmtheorem9.p4.8.8.m8.1.1" xref="S4.Thmtheorem9.p4.8.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p4.8.8.m8.1.1.2" xref="S4.Thmtheorem9.p4.8.8.m8.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p4.8.8.m8.1.1.3" xref="S4.Thmtheorem9.p4.8.8.m8.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.8.8.m8.1b"><apply id="S4.Thmtheorem9.p4.8.8.m8.1.1.cmml" xref="S4.Thmtheorem9.p4.8.8.m8.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem9.p4.8.8.m8.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem9.p4.8.8.m8.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p4.8.8.m8.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.8.8.m8.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.8.8.m8.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the subtree rooted at <math alttext="h_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.9.9.m9.1"><semantics id="S4.Thmtheorem9.p4.9.9.m9.1a"><msub id="S4.Thmtheorem9.p4.9.9.m9.1.1" xref="S4.Thmtheorem9.p4.9.9.m9.1.1.cmml"><mi id="S4.Thmtheorem9.p4.9.9.m9.1.1.2" xref="S4.Thmtheorem9.p4.9.9.m9.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p4.9.9.m9.1.1.3" xref="S4.Thmtheorem9.p4.9.9.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.9.9.m9.1b"><apply id="S4.Thmtheorem9.p4.9.9.m9.1.1.cmml" xref="S4.Thmtheorem9.p4.9.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.9.9.m9.1.1.1.cmml" xref="S4.Thmtheorem9.p4.9.9.m9.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.9.9.m9.1.1.2.cmml" xref="S4.Thmtheorem9.p4.9.9.m9.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p4.9.9.m9.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.9.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.9.9.m9.1c">h_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.9.9.m9.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="w(h_{0})=B" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.10.10.m10.1"><semantics id="S4.Thmtheorem9.p4.10.10.m10.1a"><mrow id="S4.Thmtheorem9.p4.10.10.m10.1.1" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.cmml"><mrow id="S4.Thmtheorem9.p4.10.10.m10.1.1.1" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.cmml"><mi id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.3" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.3.cmml">w</mi><mo id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.2" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.2" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.3" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p4.10.10.m10.1.1.2" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.2.cmml">=</mo><mi id="S4.Thmtheorem9.p4.10.10.m10.1.1.3" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.10.10.m10.1b"><apply id="S4.Thmtheorem9.p4.10.10.m10.1.1.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1"><eq id="S4.Thmtheorem9.p4.10.10.m10.1.1.2.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.2"></eq><apply id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1"><times id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.2.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.2"></times><ci id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.3.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.3">𝑤</ci><apply id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.1.1.1.1.3">0</cn></apply></apply><ci id="S4.Thmtheorem9.p4.10.10.m10.1.1.3.cmml" xref="S4.Thmtheorem9.p4.10.10.m10.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.10.10.m10.1c">w(h_{0})=B</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.10.10.m10.1d">italic_w ( italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_B</annotation></semantics></math>. As <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.11.11.m11.1"><semantics id="S4.Thmtheorem9.p4.11.11.m11.1a"><msub id="S4.Thmtheorem9.p4.11.11.m11.1.1" xref="S4.Thmtheorem9.p4.11.11.m11.1.1.cmml"><mi id="S4.Thmtheorem9.p4.11.11.m11.1.1.2" xref="S4.Thmtheorem9.p4.11.11.m11.1.1.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p4.11.11.m11.1.1.3" xref="S4.Thmtheorem9.p4.11.11.m11.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p4.11.11.m11.1b"><apply id="S4.Thmtheorem9.p4.11.11.m11.1.1.cmml" xref="S4.Thmtheorem9.p4.11.11.m11.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p4.11.11.m11.1.1.1.cmml" xref="S4.Thmtheorem9.p4.11.11.m11.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p4.11.11.m11.1.1.2.cmml" xref="S4.Thmtheorem9.p4.11.11.m11.1.1.2">𝜎</ci><cn id="S4.Thmtheorem9.p4.11.11.m11.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p4.11.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.11.11.m11.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.11.11.m11.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E7" title="In Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">7</span></a>) by hypothesis, we get that <math alttext="\max\{c^{(0)}_{i}\mid c^{(0)}_{i}\neq\infty\}\leq\max\{{c^{(0)}_{min},B}\}+1+f% ({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p4.12.12.m12.10"><semantics id="S4.Thmtheorem9.p4.12.12.m12.10a"><mrow id="S4.Thmtheorem9.p4.12.12.m12.10.10" xref="S4.Thmtheorem9.p4.12.12.m12.10.10.cmml"><mrow id="S4.Thmtheorem9.p4.12.12.m12.9.9.1.1" xref="S4.Thmtheorem9.p4.12.12.m12.9.9.1.2.cmml"><mi 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id="S4.Thmtheorem9.p4.12.12.m12.10.10.2.4.2.cmml" xref="S4.Thmtheorem9.p4.12.12.m12.10.10.2.4.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem9.p4.12.12.m12.10.10.2.4.3.1.cmml" xref="S4.Thmtheorem9.p4.12.12.m12.10.10.2.4.3.2"><cn id="S4.Thmtheorem9.p4.12.12.m12.7.7.cmml" type="integer" xref="S4.Thmtheorem9.p4.12.12.m12.7.7">0</cn><ci id="S4.Thmtheorem9.p4.12.12.m12.8.8.cmml" xref="S4.Thmtheorem9.p4.12.12.m12.8.8">𝑡</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p4.12.12.m12.10c">\max\{c^{(0)}_{i}\mid c^{(0)}_{i}\neq\infty\}\leq\max\{{c^{(0)}_{min},B}\}+1+f% ({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p4.12.12.m12.10d">roman_max { italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∞ } ≤ roman_max { italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_B } + 1 + italic_f ( 0 , italic_t )</annotation></semantics></math>, that is,</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx13"> <tbody id="S4.E14"><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\{c^{(0)}_{i}\mid c^{(0)}_{i}\neq\infty\}\leq B+\big{(}\delta% +1+f({0},{t})\big{)}." class="ltx_Math" display="inline" id="S4.E14.m1.6"><semantics id="S4.E14.m1.6a"><mrow id="S4.E14.m1.6.6.1" xref="S4.E14.m1.6.6.1.1.cmml"><mrow id="S4.E14.m1.6.6.1.1" xref="S4.E14.m1.6.6.1.1.cmml"><mrow id="S4.E14.m1.6.6.1.1.1.1" xref="S4.E14.m1.6.6.1.1.1.2.cmml"><mi id="S4.E14.m1.3.3" xref="S4.E14.m1.3.3.cmml">max</mi><mo id="S4.E14.m1.6.6.1.1.1.1a" xref="S4.E14.m1.6.6.1.1.1.2.cmml">⁡</mo><mrow id="S4.E14.m1.6.6.1.1.1.1.1" xref="S4.E14.m1.6.6.1.1.1.2.cmml"><mo id="S4.E14.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S4.E14.m1.6.6.1.1.1.2.cmml">{</mo><mrow id="S4.E14.m1.6.6.1.1.1.1.1.1" xref="S4.E14.m1.6.6.1.1.1.1.1.1.cmml"><mrow id="S4.E14.m1.6.6.1.1.1.1.1.1.2" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.cmml"><msubsup id="S4.E14.m1.6.6.1.1.1.1.1.1.2.2" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.cmml"><mi id="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.2.2" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.2.2.cmml">c</mi><mi id="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.3" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.3.cmml">i</mi><mrow id="S4.E14.m1.1.1.1.3" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.cmml"><mo id="S4.E14.m1.1.1.1.3.1" stretchy="false" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.cmml">(</mo><mn id="S4.E14.m1.1.1.1.1" xref="S4.E14.m1.1.1.1.1.cmml">0</mn><mo id="S4.E14.m1.1.1.1.3.2" stretchy="false" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.cmml">)</mo></mrow></msubsup><mo id="S4.E14.m1.6.6.1.1.1.1.1.1.2.1" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.1.cmml">∣</mo><msubsup id="S4.E14.m1.6.6.1.1.1.1.1.1.2.3" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.cmml"><mi id="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.2.2" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.2.2.cmml">c</mi><mi id="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.3" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.3.cmml">i</mi><mrow id="S4.E14.m1.2.2.1.3" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.cmml"><mo id="S4.E14.m1.2.2.1.3.1" stretchy="false" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.cmml">(</mo><mn id="S4.E14.m1.2.2.1.1" xref="S4.E14.m1.2.2.1.1.cmml">0</mn><mo id="S4.E14.m1.2.2.1.3.2" stretchy="false" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.3.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.E14.m1.6.6.1.1.1.1.1.1.1" xref="S4.E14.m1.6.6.1.1.1.1.1.1.1.cmml">≠</mo><mi id="S4.E14.m1.6.6.1.1.1.1.1.1.3" mathvariant="normal" xref="S4.E14.m1.6.6.1.1.1.1.1.1.3.cmml">∞</mi></mrow><mo id="S4.E14.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="S4.E14.m1.6.6.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S4.E14.m1.6.6.1.1.3" xref="S4.E14.m1.6.6.1.1.3.cmml">≤</mo><mrow id="S4.E14.m1.6.6.1.1.2" xref="S4.E14.m1.6.6.1.1.2.cmml"><mi id="S4.E14.m1.6.6.1.1.2.3" xref="S4.E14.m1.6.6.1.1.2.3.cmml">B</mi><mo id="S4.E14.m1.6.6.1.1.2.2" xref="S4.E14.m1.6.6.1.1.2.2.cmml">+</mo><mrow id="S4.E14.m1.6.6.1.1.2.1.1" xref="S4.E14.m1.6.6.1.1.2.1.1.1.cmml"><mo id="S4.E14.m1.6.6.1.1.2.1.1.2" maxsize="120%" minsize="120%" xref="S4.E14.m1.6.6.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.E14.m1.6.6.1.1.2.1.1.1" xref="S4.E14.m1.6.6.1.1.2.1.1.1.cmml"><mi id="S4.E14.m1.6.6.1.1.2.1.1.1.2" xref="S4.E14.m1.6.6.1.1.2.1.1.1.2.cmml">δ</mi><mo id="S4.E14.m1.6.6.1.1.2.1.1.1.1" xref="S4.E14.m1.6.6.1.1.2.1.1.1.1.cmml">+</mo><mn id="S4.E14.m1.6.6.1.1.2.1.1.1.3" xref="S4.E14.m1.6.6.1.1.2.1.1.1.3.cmml">1</mn><mo id="S4.E14.m1.6.6.1.1.2.1.1.1.1a" xref="S4.E14.m1.6.6.1.1.2.1.1.1.1.cmml">+</mo><mrow id="S4.E14.m1.6.6.1.1.2.1.1.1.4" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.cmml"><mi id="S4.E14.m1.6.6.1.1.2.1.1.1.4.2" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.2.cmml">f</mi><mo id="S4.E14.m1.6.6.1.1.2.1.1.1.4.1" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.2" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.1.cmml"><mo id="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.2.1" stretchy="false" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.1.cmml">(</mo><mn id="S4.E14.m1.4.4" xref="S4.E14.m1.4.4.cmml">0</mn><mo id="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.2.2" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.1.cmml">,</mo><mi id="S4.E14.m1.5.5" xref="S4.E14.m1.5.5.cmml">t</mi><mo id="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.2.3" stretchy="false" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E14.m1.6.6.1.1.2.1.1.3" maxsize="120%" minsize="120%" xref="S4.E14.m1.6.6.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E14.m1.6.6.1.2" lspace="0em" xref="S4.E14.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E14.m1.6b"><apply id="S4.E14.m1.6.6.1.1.cmml" xref="S4.E14.m1.6.6.1"><leq id="S4.E14.m1.6.6.1.1.3.cmml" xref="S4.E14.m1.6.6.1.1.3"></leq><apply id="S4.E14.m1.6.6.1.1.1.2.cmml" xref="S4.E14.m1.6.6.1.1.1.1"><max id="S4.E14.m1.3.3.cmml" xref="S4.E14.m1.3.3"></max><apply id="S4.E14.m1.6.6.1.1.1.1.1.1.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1"><neq id="S4.E14.m1.6.6.1.1.1.1.1.1.1.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.1"></neq><apply id="S4.E14.m1.6.6.1.1.1.1.1.1.2.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2"><csymbol cd="latexml" id="S4.E14.m1.6.6.1.1.1.1.1.1.2.1.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.1">conditional</csymbol><apply id="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2">subscript</csymbol><apply id="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.E14.m1.6.6.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.2.2">superscript</csymbol><ci 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id="S4.E14.m1.6.6.1.1.1.1.1.1.3.cmml" xref="S4.E14.m1.6.6.1.1.1.1.1.1.3"></infinity></apply></apply><apply id="S4.E14.m1.6.6.1.1.2.cmml" xref="S4.E14.m1.6.6.1.1.2"><plus id="S4.E14.m1.6.6.1.1.2.2.cmml" xref="S4.E14.m1.6.6.1.1.2.2"></plus><ci id="S4.E14.m1.6.6.1.1.2.3.cmml" xref="S4.E14.m1.6.6.1.1.2.3">𝐵</ci><apply id="S4.E14.m1.6.6.1.1.2.1.1.1.cmml" xref="S4.E14.m1.6.6.1.1.2.1.1"><plus id="S4.E14.m1.6.6.1.1.2.1.1.1.1.cmml" xref="S4.E14.m1.6.6.1.1.2.1.1.1.1"></plus><ci id="S4.E14.m1.6.6.1.1.2.1.1.1.2.cmml" xref="S4.E14.m1.6.6.1.1.2.1.1.1.2">𝛿</ci><cn id="S4.E14.m1.6.6.1.1.2.1.1.1.3.cmml" type="integer" xref="S4.E14.m1.6.6.1.1.2.1.1.1.3">1</cn><apply id="S4.E14.m1.6.6.1.1.2.1.1.1.4.cmml" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4"><times id="S4.E14.m1.6.6.1.1.2.1.1.1.4.1.cmml" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.1"></times><ci id="S4.E14.m1.6.6.1.1.2.1.1.1.4.2.cmml" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.2">𝑓</ci><interval closure="open" id="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.1.cmml" xref="S4.E14.m1.6.6.1.1.2.1.1.1.4.3.2"><cn id="S4.E14.m1.4.4.cmml" type="integer" xref="S4.E14.m1.4.4">0</cn><ci id="S4.E14.m1.5.5.cmml" xref="S4.E14.m1.5.5">𝑡</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E14.m1.6c">\displaystyle\max\{c^{(0)}_{i}\mid c^{(0)}_{i}\neq\infty\}\leq B+\big{(}\delta% +1+f({0},{t})\big{)}.</annotation><annotation encoding="application/x-llamapun" id="S4.E14.m1.6d">roman_max { italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∞ } ≤ italic_B + ( italic_δ + 1 + italic_f ( 0 , italic_t ) ) .</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">(14)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.Thmtheorem9.p5"> <p class="ltx_p" id="S4.Thmtheorem9.p5.16"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p5.16.16">Let us construct the second Pareto-optimal cost <math alttext="c^{(1)}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.1.1.m1.1"><semantics id="S4.Thmtheorem9.p5.1.1.m1.1a"><msup id="S4.Thmtheorem9.p5.1.1.m1.1.2" xref="S4.Thmtheorem9.p5.1.1.m1.1.2.cmml"><mi id="S4.Thmtheorem9.p5.1.1.m1.1.2.2" xref="S4.Thmtheorem9.p5.1.1.m1.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.1.1.m1.1.1.1.3" xref="S4.Thmtheorem9.p5.1.1.m1.1.2.cmml"><mo id="S4.Thmtheorem9.p5.1.1.m1.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.1.1.m1.1.2.cmml">(</mo><mn id="S4.Thmtheorem9.p5.1.1.m1.1.1.1.1" xref="S4.Thmtheorem9.p5.1.1.m1.1.1.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p5.1.1.m1.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.1.1.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.1.1.m1.1b"><apply id="S4.Thmtheorem9.p5.1.1.m1.1.2.cmml" xref="S4.Thmtheorem9.p5.1.1.m1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.1.1.m1.1.2.1.cmml" xref="S4.Thmtheorem9.p5.1.1.m1.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p5.1.1.m1.1.2.2.cmml" xref="S4.Thmtheorem9.p5.1.1.m1.1.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.1.1.m1.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.1.1.m1.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.1.1.m1.1c">c^{(1)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.1.1.m1.1d">italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> as we did for <math alttext="c^{(0)}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.2.2.m2.1"><semantics id="S4.Thmtheorem9.p5.2.2.m2.1a"><msup id="S4.Thmtheorem9.p5.2.2.m2.1.2" xref="S4.Thmtheorem9.p5.2.2.m2.1.2.cmml"><mi id="S4.Thmtheorem9.p5.2.2.m2.1.2.2" xref="S4.Thmtheorem9.p5.2.2.m2.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.2.2.m2.1.1.1.3" xref="S4.Thmtheorem9.p5.2.2.m2.1.2.cmml"><mo id="S4.Thmtheorem9.p5.2.2.m2.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.2.2.m2.1.2.cmml">(</mo><mn id="S4.Thmtheorem9.p5.2.2.m2.1.1.1.1" xref="S4.Thmtheorem9.p5.2.2.m2.1.1.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p5.2.2.m2.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.2.2.m2.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.2.2.m2.1b"><apply id="S4.Thmtheorem9.p5.2.2.m2.1.2.cmml" xref="S4.Thmtheorem9.p5.2.2.m2.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.2.2.m2.1.2.1.cmml" xref="S4.Thmtheorem9.p5.2.2.m2.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p5.2.2.m2.1.2.2.cmml" xref="S4.Thmtheorem9.p5.2.2.m2.1.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.2.2.m2.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.2.2.m2.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.2.2.m2.1c">c^{(0)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.2.2.m2.1d">italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="h_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.3.3.m3.1"><semantics id="S4.Thmtheorem9.p5.3.3.m3.1a"><msub id="S4.Thmtheorem9.p5.3.3.m3.1.1" xref="S4.Thmtheorem9.p5.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem9.p5.3.3.m3.1.1.2" xref="S4.Thmtheorem9.p5.3.3.m3.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.3.3.m3.1.1.3" xref="S4.Thmtheorem9.p5.3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.3.3.m3.1b"><apply id="S4.Thmtheorem9.p5.3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p5.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem9.p5.3.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem9.p5.3.3.m3.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.3.3.m3.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.3.3.m3.1c">h_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.3.3.m3.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be the history of maximal length such that <math alttext="h_{1}\rho" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.4.4.m4.1"><semantics id="S4.Thmtheorem9.p5.4.4.m4.1a"><mrow id="S4.Thmtheorem9.p5.4.4.m4.1.1" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.cmml"><msub id="S4.Thmtheorem9.p5.4.4.m4.1.1.2" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.2.cmml"><mi id="S4.Thmtheorem9.p5.4.4.m4.1.1.2.2" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.2.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.4.4.m4.1.1.2.3" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p5.4.4.m4.1.1.1" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p5.4.4.m4.1.1.3" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.4.4.m4.1b"><apply id="S4.Thmtheorem9.p5.4.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p5.4.4.m4.1.1"><times id="S4.Thmtheorem9.p5.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.1"></times><apply id="S4.Thmtheorem9.p5.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.4.4.m4.1.1.2.1.cmml" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem9.p5.4.4.m4.1.1.2.2.cmml" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.2.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.4.4.m4.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.2.3">1</cn></apply><ci id="S4.Thmtheorem9.p5.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem9.p5.4.4.m4.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.4.4.m4.1c">h_{1}\rho</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.4.4.m4.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ρ</annotation></semantics></math> and <math alttext="w(h_{1})=B+\delta+1+f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.5.5.m5.3"><semantics id="S4.Thmtheorem9.p5.5.5.m5.3a"><mrow id="S4.Thmtheorem9.p5.5.5.m5.3.3" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.cmml"><mrow id="S4.Thmtheorem9.p5.5.5.m5.3.3.1" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.cmml"><mi id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.3" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.3.cmml">w</mi><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.2" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.2" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.3" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.2" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.2.cmml">=</mo><mrow id="S4.Thmtheorem9.p5.5.5.m5.3.3.3" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.cmml"><mi id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.2" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.2.cmml">B</mi><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1.cmml">+</mo><mi id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.3" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.3.cmml">δ</mi><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1a" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.4" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.4.cmml">1</mn><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1b" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1.cmml">+</mo><mrow id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.cmml"><mi id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.2" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.2.cmml">f</mi><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.1" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.2" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.1.cmml"><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p5.5.5.m5.1.1" xref="S4.Thmtheorem9.p5.5.5.m5.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.2.2" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p5.5.5.m5.2.2" xref="S4.Thmtheorem9.p5.5.5.m5.2.2.cmml">t</mi><mo id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.5.5.m5.3b"><apply id="S4.Thmtheorem9.p5.5.5.m5.3.3.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3"><eq id="S4.Thmtheorem9.p5.5.5.m5.3.3.2.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.2"></eq><apply id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1"><times id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.2.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.2"></times><ci id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.3.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.3">𝑤</ci><apply id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.1.1.1.1.3">1</cn></apply></apply><apply id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3"><plus id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.1"></plus><ci id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.2.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.2">𝐵</ci><ci id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.3.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.3">𝛿</ci><cn id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.4.cmml" type="integer" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.4">1</cn><apply id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5"><times id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.1.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.1"></times><ci id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.2.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.1.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.3.3.3.5.3.2"><cn id="S4.Thmtheorem9.p5.5.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.5.5.m5.1.1">0</cn><ci id="S4.Thmtheorem9.p5.5.5.m5.2.2.cmml" xref="S4.Thmtheorem9.p5.5.5.m5.2.2">𝑡</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.5.5.m5.3c">w(h_{1})=B+\delta+1+f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.5.5.m5.3d">italic_w ( italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_B + italic_δ + 1 + italic_f ( 0 , italic_t )</annotation></semantics></math> (notice that we use the bound of (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E14" title="In Proof 4.9 (Proof of Lemma 4.6) ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">14</span></a>)). This history <math alttext="h_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.6.6.m6.1"><semantics id="S4.Thmtheorem9.p5.6.6.m6.1a"><msub id="S4.Thmtheorem9.p5.6.6.m6.1.1" xref="S4.Thmtheorem9.p5.6.6.m6.1.1.cmml"><mi id="S4.Thmtheorem9.p5.6.6.m6.1.1.2" xref="S4.Thmtheorem9.p5.6.6.m6.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.6.6.m6.1.1.3" xref="S4.Thmtheorem9.p5.6.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.6.6.m6.1b"><apply id="S4.Thmtheorem9.p5.6.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p5.6.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem9.p5.6.6.m6.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem9.p5.6.6.m6.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.6.6.m6.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.6.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.6.6.m6.1c">h_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.6.6.m6.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> exists for the same reasons as for <math alttext="h_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.7.7.m7.1"><semantics id="S4.Thmtheorem9.p5.7.7.m7.1a"><msub id="S4.Thmtheorem9.p5.7.7.m7.1.1" xref="S4.Thmtheorem9.p5.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem9.p5.7.7.m7.1.1.2" xref="S4.Thmtheorem9.p5.7.7.m7.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.7.7.m7.1.1.3" xref="S4.Thmtheorem9.p5.7.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.7.7.m7.1b"><apply id="S4.Thmtheorem9.p5.7.7.m7.1.1.cmml" xref="S4.Thmtheorem9.p5.7.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem9.p5.7.7.m7.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem9.p5.7.7.m7.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.7.7.m7.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.7.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.7.7.m7.1c">h_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.7.7.m7.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="\mathcal{T}_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.8.8.m8.1"><semantics id="S4.Thmtheorem9.p5.8.8.m8.1a"><msub id="S4.Thmtheorem9.p5.8.8.m8.1.1" xref="S4.Thmtheorem9.p5.8.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p5.8.8.m8.1.1.2" xref="S4.Thmtheorem9.p5.8.8.m8.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p5.8.8.m8.1.1.3" xref="S4.Thmtheorem9.p5.8.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.8.8.m8.1b"><apply id="S4.Thmtheorem9.p5.8.8.m8.1.1.cmml" xref="S4.Thmtheorem9.p5.8.8.m8.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem9.p5.8.8.m8.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem9.p5.8.8.m8.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p5.8.8.m8.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.8.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.8.8.m8.1c">\mathcal{T}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.8.8.m8.1d">caligraphic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be the subtree rooted in the last vertex of <math alttext="h_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.9.9.m9.1"><semantics id="S4.Thmtheorem9.p5.9.9.m9.1a"><msub id="S4.Thmtheorem9.p5.9.9.m9.1.1" xref="S4.Thmtheorem9.p5.9.9.m9.1.1.cmml"><mi id="S4.Thmtheorem9.p5.9.9.m9.1.1.2" xref="S4.Thmtheorem9.p5.9.9.m9.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.9.9.m9.1.1.3" xref="S4.Thmtheorem9.p5.9.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.9.9.m9.1b"><apply id="S4.Thmtheorem9.p5.9.9.m9.1.1.cmml" xref="S4.Thmtheorem9.p5.9.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.9.9.m9.1.1.1.cmml" xref="S4.Thmtheorem9.p5.9.9.m9.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.9.9.m9.1.1.2.cmml" xref="S4.Thmtheorem9.p5.9.9.m9.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.9.9.m9.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.9.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.9.9.m9.1c">h_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.9.9.m9.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> (it is a subtree of <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.10.10.m10.1"><semantics id="S4.Thmtheorem9.p5.10.10.m10.1a"><msub id="S4.Thmtheorem9.p5.10.10.m10.1.1" xref="S4.Thmtheorem9.p5.10.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p5.10.10.m10.1.1.2" xref="S4.Thmtheorem9.p5.10.10.m10.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p5.10.10.m10.1.1.3" xref="S4.Thmtheorem9.p5.10.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.10.10.m10.1b"><apply id="S4.Thmtheorem9.p5.10.10.m10.1.1.cmml" xref="S4.Thmtheorem9.p5.10.10.m10.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.10.10.m10.1.1.1.cmml" xref="S4.Thmtheorem9.p5.10.10.m10.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.10.10.m10.1.1.2.cmml" xref="S4.Thmtheorem9.p5.10.10.m10.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p5.10.10.m10.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.10.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.10.10.m10.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.10.10.m10.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>). We can apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem7" title="Lemma 4.7 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.7</span></a> as for <math alttext="\mathcal{T}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.11.11.m11.1"><semantics id="S4.Thmtheorem9.p5.11.11.m11.1a"><msub id="S4.Thmtheorem9.p5.11.11.m11.1.1" xref="S4.Thmtheorem9.p5.11.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p5.11.11.m11.1.1.2" xref="S4.Thmtheorem9.p5.11.11.m11.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p5.11.11.m11.1.1.3" xref="S4.Thmtheorem9.p5.11.11.m11.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.11.11.m11.1b"><apply id="S4.Thmtheorem9.p5.11.11.m11.1.1.cmml" xref="S4.Thmtheorem9.p5.11.11.m11.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.11.11.m11.1.1.1.cmml" xref="S4.Thmtheorem9.p5.11.11.m11.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.11.11.m11.1.1.2.cmml" xref="S4.Thmtheorem9.p5.11.11.m11.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p5.11.11.m11.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.11.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.11.11.m11.1c">\mathcal{T}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.11.11.m11.1d">caligraphic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>: the leaves of <math alttext="\mathcal{T}_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.12.12.m12.1"><semantics id="S4.Thmtheorem9.p5.12.12.m12.1a"><msub id="S4.Thmtheorem9.p5.12.12.m12.1.1" xref="S4.Thmtheorem9.p5.12.12.m12.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p5.12.12.m12.1.1.2" xref="S4.Thmtheorem9.p5.12.12.m12.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p5.12.12.m12.1.1.3" xref="S4.Thmtheorem9.p5.12.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.12.12.m12.1b"><apply id="S4.Thmtheorem9.p5.12.12.m12.1.1.cmml" xref="S4.Thmtheorem9.p5.12.12.m12.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.12.12.m12.1.1.1.cmml" xref="S4.Thmtheorem9.p5.12.12.m12.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.12.12.m12.1.1.2.cmml" xref="S4.Thmtheorem9.p5.12.12.m12.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p5.12.12.m12.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.12.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.12.12.m12.1c">\mathcal{T}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.12.12.m12.1d">caligraphic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> with the smallest depth have a depth <math alttext="\leq\delta" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.13.13.m13.1"><semantics id="S4.Thmtheorem9.p5.13.13.m13.1a"><mrow id="S4.Thmtheorem9.p5.13.13.m13.1.1" xref="S4.Thmtheorem9.p5.13.13.m13.1.1.cmml"><mi id="S4.Thmtheorem9.p5.13.13.m13.1.1.2" xref="S4.Thmtheorem9.p5.13.13.m13.1.1.2.cmml"></mi><mo id="S4.Thmtheorem9.p5.13.13.m13.1.1.1" xref="S4.Thmtheorem9.p5.13.13.m13.1.1.1.cmml">≤</mo><mi id="S4.Thmtheorem9.p5.13.13.m13.1.1.3" xref="S4.Thmtheorem9.p5.13.13.m13.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.13.13.m13.1b"><apply id="S4.Thmtheorem9.p5.13.13.m13.1.1.cmml" xref="S4.Thmtheorem9.p5.13.13.m13.1.1"><leq id="S4.Thmtheorem9.p5.13.13.m13.1.1.1.cmml" xref="S4.Thmtheorem9.p5.13.13.m13.1.1.1"></leq><csymbol cd="latexml" id="S4.Thmtheorem9.p5.13.13.m13.1.1.2.cmml" xref="S4.Thmtheorem9.p5.13.13.m13.1.1.2">absent</csymbol><ci id="S4.Thmtheorem9.p5.13.13.m13.1.1.3.cmml" xref="S4.Thmtheorem9.p5.13.13.m13.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.13.13.m13.1c">\leq\delta</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.13.13.m13.1d">≤ italic_δ</annotation></semantics></math> . Thus, we set <math alttext="c^{(1)}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.14.14.m14.1"><semantics id="S4.Thmtheorem9.p5.14.14.m14.1a"><msup id="S4.Thmtheorem9.p5.14.14.m14.1.2" xref="S4.Thmtheorem9.p5.14.14.m14.1.2.cmml"><mi id="S4.Thmtheorem9.p5.14.14.m14.1.2.2" xref="S4.Thmtheorem9.p5.14.14.m14.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.14.14.m14.1.1.1.3" xref="S4.Thmtheorem9.p5.14.14.m14.1.2.cmml"><mo id="S4.Thmtheorem9.p5.14.14.m14.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.14.14.m14.1.2.cmml">(</mo><mn id="S4.Thmtheorem9.p5.14.14.m14.1.1.1.1" xref="S4.Thmtheorem9.p5.14.14.m14.1.1.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p5.14.14.m14.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.14.14.m14.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.14.14.m14.1b"><apply id="S4.Thmtheorem9.p5.14.14.m14.1.2.cmml" xref="S4.Thmtheorem9.p5.14.14.m14.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.14.14.m14.1.2.1.cmml" xref="S4.Thmtheorem9.p5.14.14.m14.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p5.14.14.m14.1.2.2.cmml" xref="S4.Thmtheorem9.p5.14.14.m14.1.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.14.14.m14.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.14.14.m14.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.14.14.m14.1c">c^{(1)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.14.14.m14.1d">italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> as the cost of a witness associated with one of these leaves. We get that <math alttext="c^{(1)}_{min}\leq B+\delta+1+f({0},{t})+\delta" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.15.15.m15.3"><semantics id="S4.Thmtheorem9.p5.15.15.m15.3a"><mrow id="S4.Thmtheorem9.p5.15.15.m15.3.4" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.cmml"><msubsup id="S4.Thmtheorem9.p5.15.15.m15.3.4.2" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.cmml"><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.2.2" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.cmml"><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.2" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.1" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.3" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.1a" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.4" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.4.cmml">n</mi></mrow><mrow id="S4.Thmtheorem9.p5.15.15.m15.1.1.1.3" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.cmml"><mo id="S4.Thmtheorem9.p5.15.15.m15.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.cmml">(</mo><mn id="S4.Thmtheorem9.p5.15.15.m15.1.1.1.1" xref="S4.Thmtheorem9.p5.15.15.m15.1.1.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p5.15.15.m15.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.cmml">)</mo></mrow></msubsup><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.1" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.1.cmml">≤</mo><mrow id="S4.Thmtheorem9.p5.15.15.m15.3.4.3" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.cmml"><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.2" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.2.cmml">B</mi><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1.cmml">+</mo><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.3" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.3.cmml">δ</mi><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1a" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.4" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.4.cmml">1</mn><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1b" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1.cmml">+</mo><mrow id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.cmml"><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.2" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.2.cmml">f</mi><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.1" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.2" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.1.cmml"><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p5.15.15.m15.2.2" xref="S4.Thmtheorem9.p5.15.15.m15.2.2.cmml">0</mn><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.2.2" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p5.15.15.m15.3.3" xref="S4.Thmtheorem9.p5.15.15.m15.3.3.cmml">t</mi><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1c" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1.cmml">+</mo><mi id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.6" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.6.cmml">δ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.15.15.m15.3b"><apply id="S4.Thmtheorem9.p5.15.15.m15.3.4.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4"><leq id="S4.Thmtheorem9.p5.15.15.m15.3.4.1.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.1"></leq><apply id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.1.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2">subscript</csymbol><apply id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.2.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.2.1.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2">superscript</csymbol><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.2.2.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.15.15.m15.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.15.15.m15.1.1.1.1">1</cn></apply><apply id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3"><times id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.1.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.1"></times><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.2.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.3.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.4.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.2.3.4">𝑛</ci></apply></apply><apply id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3"><plus id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.1"></plus><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.2.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.2">𝐵</ci><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.3.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.3">𝛿</ci><cn id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.4.cmml" type="integer" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.4">1</cn><apply id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5"><times id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.1.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.1"></times><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.2.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.1.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.5.3.2"><cn id="S4.Thmtheorem9.p5.15.15.m15.2.2.cmml" type="integer" xref="S4.Thmtheorem9.p5.15.15.m15.2.2">0</cn><ci id="S4.Thmtheorem9.p5.15.15.m15.3.3.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.3">𝑡</ci></interval></apply><ci id="S4.Thmtheorem9.p5.15.15.m15.3.4.3.6.cmml" xref="S4.Thmtheorem9.p5.15.15.m15.3.4.3.6">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.15.15.m15.3c">c^{(1)}_{min}\leq B+\delta+1+f({0},{t})+\delta</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.15.15.m15.3d">italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ≤ italic_B + italic_δ + 1 + italic_f ( 0 , italic_t ) + italic_δ</annotation></semantics></math> by definition of <math alttext="w(h_{1})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.16.16.m16.1"><semantics id="S4.Thmtheorem9.p5.16.16.m16.1a"><mrow id="S4.Thmtheorem9.p5.16.16.m16.1.1" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.cmml"><mi id="S4.Thmtheorem9.p5.16.16.m16.1.1.3" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.3.cmml">w</mi><mo id="S4.Thmtheorem9.p5.16.16.m16.1.1.2" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.2" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.3" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.16.16.m16.1b"><apply id="S4.Thmtheorem9.p5.16.16.m16.1.1.cmml" xref="S4.Thmtheorem9.p5.16.16.m16.1.1"><times id="S4.Thmtheorem9.p5.16.16.m16.1.1.2.cmml" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.2"></times><ci id="S4.Thmtheorem9.p5.16.16.m16.1.1.3.cmml" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.3">𝑤</ci><apply id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.16.16.m16.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.16.16.m16.1c">w(h_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.16.16.m16.1d">italic_w ( italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. By (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E7" title="In Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">7</span></a>), we get</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx14"> <tbody id="S4.E15"><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\{c^{(1)}_{i}\mid c^{(1)}_{i}\neq\infty\}\leq B+2\big{(}% \delta+1+f({0},{t})\big{)}." class="ltx_Math" display="inline" id="S4.E15.m1.6"><semantics id="S4.E15.m1.6a"><mrow id="S4.E15.m1.6.6.1" xref="S4.E15.m1.6.6.1.1.cmml"><mrow id="S4.E15.m1.6.6.1.1" xref="S4.E15.m1.6.6.1.1.cmml"><mrow id="S4.E15.m1.6.6.1.1.1.1" xref="S4.E15.m1.6.6.1.1.1.2.cmml"><mi id="S4.E15.m1.3.3" xref="S4.E15.m1.3.3.cmml">max</mi><mo id="S4.E15.m1.6.6.1.1.1.1a" xref="S4.E15.m1.6.6.1.1.1.2.cmml">⁡</mo><mrow id="S4.E15.m1.6.6.1.1.1.1.1" xref="S4.E15.m1.6.6.1.1.1.2.cmml"><mo id="S4.E15.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S4.E15.m1.6.6.1.1.1.2.cmml">{</mo><mrow id="S4.E15.m1.6.6.1.1.1.1.1.1" xref="S4.E15.m1.6.6.1.1.1.1.1.1.cmml"><mrow id="S4.E15.m1.6.6.1.1.1.1.1.1.2" xref="S4.E15.m1.6.6.1.1.1.1.1.1.2.cmml"><msubsup id="S4.E15.m1.6.6.1.1.1.1.1.1.2.2" xref="S4.E15.m1.6.6.1.1.1.1.1.1.2.2.cmml"><mi id="S4.E15.m1.6.6.1.1.1.1.1.1.2.2.2.2" xref="S4.E15.m1.6.6.1.1.1.1.1.1.2.2.2.2.cmml">c</mi><mi id="S4.E15.m1.6.6.1.1.1.1.1.1.2.2.3" xref="S4.E15.m1.6.6.1.1.1.1.1.1.2.2.3.cmml">i</mi><mrow id="S4.E15.m1.1.1.1.3" xref="S4.E15.m1.6.6.1.1.1.1.1.1.2.2.cmml"><mo id="S4.E15.m1.1.1.1.3.1" stretchy="false" xref="S4.E15.m1.6.6.1.1.1.1.1.1.2.2.cmml">(</mo><mn id="S4.E15.m1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.cmml">1</mn><mo id="S4.E15.m1.1.1.1.3.2" stretchy="false" 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id="S4.E15.m1.6c">\displaystyle\max\{c^{(1)}_{i}\mid c^{(1)}_{i}\neq\infty\}\leq B+2\big{(}% \delta+1+f({0},{t})\big{)}.</annotation><annotation encoding="application/x-llamapun" id="S4.E15.m1.6d">roman_max { italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∞ } ≤ italic_B + 2 ( italic_δ + 1 + italic_f ( 0 , italic_t ) ) .</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">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem9.p5.25"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p5.25.9">We also have that <math alttext="\max\{c^{(0)}_{i}\mid c^{(0)}_{i}\neq\infty\}<c^{(1)}_{min}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.17.1.m1.5"><semantics id="S4.Thmtheorem9.p5.17.1.m1.5a"><mrow id="S4.Thmtheorem9.p5.17.1.m1.5.5" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.cmml"><mrow id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.2.cmml"><mi id="S4.Thmtheorem9.p5.17.1.m1.4.4" xref="S4.Thmtheorem9.p5.17.1.m1.4.4.cmml">max</mi><mo id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1a" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.2.cmml">⁡</mo><mrow id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.2.cmml"><mo id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.2.cmml">{</mo><mrow id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.cmml"><mrow id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.cmml"><msubsup id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.cmml"><mi 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xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.2.2.cmml">c</mi><mi id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.3" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.3.cmml">i</mi><mrow id="S4.Thmtheorem9.p5.17.1.m1.2.2.1.3" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.cmml"><mo id="S4.Thmtheorem9.p5.17.1.m1.2.2.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.cmml">(</mo><mn id="S4.Thmtheorem9.p5.17.1.m1.2.2.1.1" xref="S4.Thmtheorem9.p5.17.1.m1.2.2.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p5.17.1.m1.2.2.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.1" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.1.cmml">≠</mo><mi id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.3" mathvariant="normal" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.3.cmml">∞</mi></mrow><mo id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.2.cmml">}</mo></mrow></mrow><mo id="S4.Thmtheorem9.p5.17.1.m1.5.5.2" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.2.cmml"><</mo><msubsup id="S4.Thmtheorem9.p5.17.1.m1.5.5.3" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.cmml"><mi id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.2.2" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.cmml"><mi id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.2" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.1" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.3" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.1a" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.4" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.4.cmml">n</mi></mrow><mrow id="S4.Thmtheorem9.p5.17.1.m1.3.3.1.3" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.cmml"><mo id="S4.Thmtheorem9.p5.17.1.m1.3.3.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.cmml">(</mo><mn id="S4.Thmtheorem9.p5.17.1.m1.3.3.1.1" xref="S4.Thmtheorem9.p5.17.1.m1.3.3.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p5.17.1.m1.3.3.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.17.1.m1.5b"><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5"><lt id="S4.Thmtheorem9.p5.17.1.m1.5.5.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.2"></lt><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1"><max id="S4.Thmtheorem9.p5.17.1.m1.4.4.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.4.4"></max><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1"><neq id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.1"></neq><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2"><csymbol cd="latexml" id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.1">conditional</csymbol><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2">subscript</csymbol><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.2.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2">superscript</csymbol><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.2.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.17.1.m1.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.17.1.m1.1.1.1.1">0</cn></apply><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.3.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.2.3">𝑖</ci></apply><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3">subscript</csymbol><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.2.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3">superscript</csymbol><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.2.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.17.1.m1.2.2.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.17.1.m1.2.2.1.1">0</cn></apply><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.3.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.2.3.3">𝑖</ci></apply></apply><infinity id="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.1.1.1.1.3"></infinity></apply></apply><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3">subscript</csymbol><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.2.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3">superscript</csymbol><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.2.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.17.1.m1.3.3.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.17.1.m1.3.3.1.1">1</cn></apply><apply id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3"><times id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.1.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.1"></times><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.2.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.3.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.4.cmml" xref="S4.Thmtheorem9.p5.17.1.m1.5.5.3.3.4">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.17.1.m1.5c">\max\{c^{(0)}_{i}\mid c^{(0)}_{i}\neq\infty\}<c^{(1)}_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.17.1.m1.5d">roman_max { italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∞ } < italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as required in (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E13" title="In Proof 4.9 (Proof of Lemma 4.6) ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">13</span></a>). Indeed, <math alttext="c^{(1)}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.18.2.m2.1"><semantics id="S4.Thmtheorem9.p5.18.2.m2.1a"><msup id="S4.Thmtheorem9.p5.18.2.m2.1.2" xref="S4.Thmtheorem9.p5.18.2.m2.1.2.cmml"><mi id="S4.Thmtheorem9.p5.18.2.m2.1.2.2" xref="S4.Thmtheorem9.p5.18.2.m2.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.18.2.m2.1.1.1.3" xref="S4.Thmtheorem9.p5.18.2.m2.1.2.cmml"><mo id="S4.Thmtheorem9.p5.18.2.m2.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.18.2.m2.1.2.cmml">(</mo><mn id="S4.Thmtheorem9.p5.18.2.m2.1.1.1.1" xref="S4.Thmtheorem9.p5.18.2.m2.1.1.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p5.18.2.m2.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.18.2.m2.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.18.2.m2.1b"><apply id="S4.Thmtheorem9.p5.18.2.m2.1.2.cmml" xref="S4.Thmtheorem9.p5.18.2.m2.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.18.2.m2.1.2.1.cmml" xref="S4.Thmtheorem9.p5.18.2.m2.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p5.18.2.m2.1.2.2.cmml" xref="S4.Thmtheorem9.p5.18.2.m2.1.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.18.2.m2.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.18.2.m2.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.18.2.m2.1c">c^{(1)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.18.2.m2.1d">italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.19.3.m3.1"><semantics id="S4.Thmtheorem9.p5.19.3.m3.1a"><mi id="S4.Thmtheorem9.p5.19.3.m3.1.1" xref="S4.Thmtheorem9.p5.19.3.m3.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.19.3.m3.1b"><ci id="S4.Thmtheorem9.p5.19.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p5.19.3.m3.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.19.3.m3.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.19.3.m3.1d">italic_d</annotation></semantics></math> correspond to two different leaves of <math alttext="\mathcal{T}_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.20.4.m4.1"><semantics id="S4.Thmtheorem9.p5.20.4.m4.1a"><msub id="S4.Thmtheorem9.p5.20.4.m4.1.1" xref="S4.Thmtheorem9.p5.20.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p5.20.4.m4.1.1.2" xref="S4.Thmtheorem9.p5.20.4.m4.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p5.20.4.m4.1.1.3" xref="S4.Thmtheorem9.p5.20.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.20.4.m4.1b"><apply id="S4.Thmtheorem9.p5.20.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p5.20.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.20.4.m4.1.1.1.cmml" xref="S4.Thmtheorem9.p5.20.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.20.4.m4.1.1.2.cmml" xref="S4.Thmtheorem9.p5.20.4.m4.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p5.20.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.20.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.20.4.m4.1c">\mathcal{T}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.20.4.m4.1d">caligraphic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and thus <math alttext="c^{(1)}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.21.5.m5.1"><semantics id="S4.Thmtheorem9.p5.21.5.m5.1a"><msup id="S4.Thmtheorem9.p5.21.5.m5.1.2" xref="S4.Thmtheorem9.p5.21.5.m5.1.2.cmml"><mi id="S4.Thmtheorem9.p5.21.5.m5.1.2.2" xref="S4.Thmtheorem9.p5.21.5.m5.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.21.5.m5.1.1.1.3" xref="S4.Thmtheorem9.p5.21.5.m5.1.2.cmml"><mo id="S4.Thmtheorem9.p5.21.5.m5.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.21.5.m5.1.2.cmml">(</mo><mn id="S4.Thmtheorem9.p5.21.5.m5.1.1.1.1" xref="S4.Thmtheorem9.p5.21.5.m5.1.1.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p5.21.5.m5.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.21.5.m5.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.21.5.m5.1b"><apply id="S4.Thmtheorem9.p5.21.5.m5.1.2.cmml" xref="S4.Thmtheorem9.p5.21.5.m5.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.21.5.m5.1.2.1.cmml" xref="S4.Thmtheorem9.p5.21.5.m5.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p5.21.5.m5.1.2.2.cmml" xref="S4.Thmtheorem9.p5.21.5.m5.1.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.21.5.m5.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.21.5.m5.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.21.5.m5.1c">c^{(1)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.21.5.m5.1d">italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> does not correspond to the root of <math alttext="\mathcal{T}_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.22.6.m6.1"><semantics id="S4.Thmtheorem9.p5.22.6.m6.1a"><msub id="S4.Thmtheorem9.p5.22.6.m6.1.1" xref="S4.Thmtheorem9.p5.22.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem9.p5.22.6.m6.1.1.2" xref="S4.Thmtheorem9.p5.22.6.m6.1.1.2.cmml">𝒯</mi><mn id="S4.Thmtheorem9.p5.22.6.m6.1.1.3" xref="S4.Thmtheorem9.p5.22.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.22.6.m6.1b"><apply id="S4.Thmtheorem9.p5.22.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p5.22.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.22.6.m6.1.1.1.cmml" xref="S4.Thmtheorem9.p5.22.6.m6.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.22.6.m6.1.1.2.cmml" xref="S4.Thmtheorem9.p5.22.6.m6.1.1.2">𝒯</ci><cn id="S4.Thmtheorem9.p5.22.6.m6.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.22.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.22.6.m6.1c">\mathcal{T}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.22.6.m6.1d">caligraphic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. By definition of <math alttext="h_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.23.7.m7.1"><semantics id="S4.Thmtheorem9.p5.23.7.m7.1a"><msub id="S4.Thmtheorem9.p5.23.7.m7.1.1" xref="S4.Thmtheorem9.p5.23.7.m7.1.1.cmml"><mi id="S4.Thmtheorem9.p5.23.7.m7.1.1.2" xref="S4.Thmtheorem9.p5.23.7.m7.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.23.7.m7.1.1.3" xref="S4.Thmtheorem9.p5.23.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.23.7.m7.1b"><apply id="S4.Thmtheorem9.p5.23.7.m7.1.1.cmml" xref="S4.Thmtheorem9.p5.23.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.23.7.m7.1.1.1.cmml" xref="S4.Thmtheorem9.p5.23.7.m7.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.23.7.m7.1.1.2.cmml" xref="S4.Thmtheorem9.p5.23.7.m7.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.23.7.m7.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.23.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.23.7.m7.1c">h_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.23.7.m7.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, we get that <math alttext="c^{(1)}_{min}>w(h_{1})=B+\delta+1+f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p5.24.8.m8.4"><semantics id="S4.Thmtheorem9.p5.24.8.m8.4a"><mrow id="S4.Thmtheorem9.p5.24.8.m8.4.4" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.cmml"><msubsup id="S4.Thmtheorem9.p5.24.8.m8.4.4.3" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.cmml"><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.2.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.cmml"><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.1" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.3" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.1a" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.4" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.4.cmml">n</mi></mrow><mrow id="S4.Thmtheorem9.p5.24.8.m8.1.1.1.3" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.cmml"><mo id="S4.Thmtheorem9.p5.24.8.m8.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.cmml">(</mo><mn id="S4.Thmtheorem9.p5.24.8.m8.1.1.1.1" xref="S4.Thmtheorem9.p5.24.8.m8.1.1.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p5.24.8.m8.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.cmml">)</mo></mrow></msubsup><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.4" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.4.cmml">></mo><mrow id="S4.Thmtheorem9.p5.24.8.m8.4.4.1" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.cmml"><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.3" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.3.cmml">w</mi><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.2.cmml">h</mi><mn id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.3" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.5" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.5.cmml">=</mo><mrow id="S4.Thmtheorem9.p5.24.8.m8.4.4.6" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.cmml"><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.2.cmml">B</mi><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1.cmml">+</mo><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.3" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.3.cmml">δ</mi><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1a" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1.cmml">+</mo><mn id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.4" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.4.cmml">1</mn><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1b" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1.cmml">+</mo><mrow id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.cmml"><mi id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.2.cmml">f</mi><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.1" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.1.cmml"><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p5.24.8.m8.2.2" xref="S4.Thmtheorem9.p5.24.8.m8.2.2.cmml">0</mn><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.2.2" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p5.24.8.m8.3.3" xref="S4.Thmtheorem9.p5.24.8.m8.3.3.cmml">t</mi><mo id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p5.24.8.m8.4b"><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4"><and id="S4.Thmtheorem9.p5.24.8.m8.4.4a.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4"></and><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4b.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4"><gt id="S4.Thmtheorem9.p5.24.8.m8.4.4.4.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.4"></gt><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3">subscript</csymbol><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.2.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.2.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3">superscript</csymbol><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.2.2.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.24.8.m8.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.24.8.m8.1.1.1.1">1</cn></apply><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3"><times id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.1"></times><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.2.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.3.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.4.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.3.3.4">𝑛</ci></apply></apply><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1"><times id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.2.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.2"></times><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.3.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.3">𝑤</ci><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.2">ℎ</ci><cn id="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.1.1.1.1.3">1</cn></apply></apply></apply><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4c.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4"><eq id="S4.Thmtheorem9.p5.24.8.m8.4.4.5.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.5"></eq><share href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem9.p5.24.8.m8.4.4.1.cmml" id="S4.Thmtheorem9.p5.24.8.m8.4.4d.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4"></share><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6"><plus id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.1"></plus><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.2.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.2">𝐵</ci><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.3.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.3">𝛿</ci><cn id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.4.cmml" type="integer" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.4">1</cn><apply id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5"><times id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.1"></times><ci id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.2.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.1.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.4.4.6.5.3.2"><cn id="S4.Thmtheorem9.p5.24.8.m8.2.2.cmml" type="integer" xref="S4.Thmtheorem9.p5.24.8.m8.2.2">0</cn><ci id="S4.Thmtheorem9.p5.24.8.m8.3.3.cmml" xref="S4.Thmtheorem9.p5.24.8.m8.3.3">𝑡</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.24.8.m8.4c">c^{(1)}_{min}>w(h_{1})=B+\delta+1+f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.24.8.m8.4d">italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT > italic_w ( italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_B + italic_δ + 1 + italic_f ( 0 , italic_t )</annotation></semantics></math>, and thus <math 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xref="S4.Thmtheorem9.p5.25.9.m9.5.5.3">superscript</csymbol><ci id="S4.Thmtheorem9.p5.25.9.m9.5.5.3.2.2.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.3.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.25.9.m9.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.25.9.m9.1.1.1.1">1</cn></apply><apply id="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3"><times id="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.1.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.1"></times><ci id="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.2.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.3.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.4.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.3.3.4">𝑛</ci></apply></apply><apply id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.2.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1"><max id="S4.Thmtheorem9.p5.25.9.m9.4.4.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.4.4"></max><apply 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id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.2.2.2.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.2.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.25.9.m9.2.2.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.25.9.m9.2.2.1.1">0</cn></apply><ci id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.2.3.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.2.3">𝑖</ci></apply><apply id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.1.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3">subscript</csymbol><apply id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.2.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.2.1.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3">superscript</csymbol><ci id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.2.2.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p5.25.9.m9.3.3.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p5.25.9.m9.3.3.1.1">0</cn></apply><ci id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.3.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.2.3.3">𝑖</ci></apply></apply><infinity id="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p5.25.9.m9.5.5.1.1.1.1.3"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p5.25.9.m9.5c">c^{(1)}_{min}>\max\{c^{(0)}_{i}\mid c^{(0)}_{i}\neq\infty\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p5.25.9.m9.5d">italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT > roman_max { italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∞ }</annotation></semantics></math> by (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E14" title="In Proof 4.9 (Proof of Lemma 4.6) ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">14</span></a>).</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem9.p6"> <p class="ltx_p" id="S4.Thmtheorem9.p6.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p6.4.4">As <math alttext="d_{min}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p6.1.1.m1.1"><semantics id="S4.Thmtheorem9.p6.1.1.m1.1a"><msub id="S4.Thmtheorem9.p6.1.1.m1.1.1" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem9.p6.1.1.m1.1.1.2" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.2.cmml">d</mi><mrow id="S4.Thmtheorem9.p6.1.1.m1.1.1.3" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.2" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.1" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.3" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.1a" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.4" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.4.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p6.1.1.m1.1b"><apply id="S4.Thmtheorem9.p6.1.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p6.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p6.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.2">𝑑</ci><apply id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3"><times id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.1"></times><ci id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p6.1.1.m1.1.1.3.4.cmml" xref="S4.Thmtheorem9.p6.1.1.m1.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p6.1.1.m1.1c">d_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p6.1.1.m1.1d">italic_d start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math> satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E12" title="In Proof 4.9 (Proof of Lemma 4.6) ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">12</span></a>), we can repeat this process to construct the next costs <math alttext="c^{(2)},c^{(3)},\ldots," class="ltx_Math" display="inline" id="S4.Thmtheorem9.p6.2.2.m2.4"><semantics id="S4.Thmtheorem9.p6.2.2.m2.4a"><mrow id="S4.Thmtheorem9.p6.2.2.m2.4.4.1"><mrow id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.3.cmml"><msup id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.2" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p6.2.2.m2.1.1.1.3" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p6.2.2.m2.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.cmml">(</mo><mn id="S4.Thmtheorem9.p6.2.2.m2.1.1.1.1" xref="S4.Thmtheorem9.p6.2.2.m2.1.1.1.1.cmml">2</mn><mo id="S4.Thmtheorem9.p6.2.2.m2.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.3" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.3.cmml">,</mo><msup id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.cmml"><mi id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.2" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p6.2.2.m2.2.2.1.3" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.cmml"><mo id="S4.Thmtheorem9.p6.2.2.m2.2.2.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.cmml">(</mo><mn id="S4.Thmtheorem9.p6.2.2.m2.2.2.1.1" xref="S4.Thmtheorem9.p6.2.2.m2.2.2.1.1.cmml">3</mn><mo id="S4.Thmtheorem9.p6.2.2.m2.2.2.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.cmml">)</mo></mrow></msup><mo id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.4" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.3.cmml">,</mo><mi id="S4.Thmtheorem9.p6.2.2.m2.3.3" mathvariant="normal" xref="S4.Thmtheorem9.p6.2.2.m2.3.3.cmml">…</mi></mrow><mo id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p6.2.2.m2.4b"><list id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.3.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2"><apply id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.1.1.2">𝑐</ci><cn id="S4.Thmtheorem9.p6.2.2.m2.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p6.2.2.m2.1.1.1.1">2</cn></apply><apply id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.1.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2">superscript</csymbol><ci id="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.2.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.4.4.1.1.2.2.2">𝑐</ci><cn id="S4.Thmtheorem9.p6.2.2.m2.2.2.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p6.2.2.m2.2.2.1.1">3</cn></apply><ci id="S4.Thmtheorem9.p6.2.2.m2.3.3.cmml" xref="S4.Thmtheorem9.p6.2.2.m2.3.3">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p6.2.2.m2.4c">c^{(2)},c^{(3)},\ldots,</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p6.2.2.m2.4d">italic_c start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_c start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT , … ,</annotation></semantics></math> until the last cost <math alttext="c^{(2^{t+1})}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p6.3.3.m3.1"><semantics id="S4.Thmtheorem9.p6.3.3.m3.1a"><msup id="S4.Thmtheorem9.p6.3.3.m3.1.2" xref="S4.Thmtheorem9.p6.3.3.m3.1.2.cmml"><mi id="S4.Thmtheorem9.p6.3.3.m3.1.2.2" xref="S4.Thmtheorem9.p6.3.3.m3.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.cmml"><mn id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.2.cmml">2</mn><mrow id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.2" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.1" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.3" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p6.3.3.m3.1b"><apply id="S4.Thmtheorem9.p6.3.3.m3.1.2.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p6.3.3.m3.1.2.1.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.2">superscript</csymbol><ci id="S4.Thmtheorem9.p6.3.3.m3.1.2.2.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.2.2">𝑐</ci><apply id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1">superscript</csymbol><cn id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.2">2</cn><apply id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3"><plus id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.1"></plus><ci id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.2">𝑡</ci><cn id="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p6.3.3.m3.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p6.3.3.m3.1c">c^{(2^{t+1})}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p6.3.3.m3.1d">italic_c start_POSTSUPERSCRIPT ( 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math>. This completes the construction of the announced sequence <math alttext="(c^{(k)})_{k\in\{0,\ldots,2^{t+1}\}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p6.4.4.m4.5"><semantics id="S4.Thmtheorem9.p6.4.4.m4.5a"><msub id="S4.Thmtheorem9.p6.4.4.m4.5.5" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.cmml"><mrow id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml"><mo id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml"><mi id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.2" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p6.4.4.m4.1.1.1.3" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml"><mo id="S4.Thmtheorem9.p6.4.4.m4.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem9.p6.4.4.m4.1.1.1.1" xref="S4.Thmtheorem9.p6.4.4.m4.1.1.1.1.cmml">k</mi><mo id="S4.Thmtheorem9.p6.4.4.m4.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml">)</mo></mrow><mrow id="S4.Thmtheorem9.p6.4.4.m4.4.4.3" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.cmml"><mi id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.5" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.5.cmml">k</mi><mo id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.4" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.4.cmml">∈</mo><mrow id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.2.cmml"><mo id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.2" stretchy="false" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.2.cmml">{</mo><mn id="S4.Thmtheorem9.p6.4.4.m4.2.2.1.1" xref="S4.Thmtheorem9.p6.4.4.m4.2.2.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.3" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.2.cmml">,</mo><mi id="S4.Thmtheorem9.p6.4.4.m4.3.3.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p6.4.4.m4.3.3.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.4" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.2.cmml">,</mo><msup id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.cmml"><mn id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.2" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.2.cmml">2</mn><mrow id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.cmml"><mi id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.2" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.1" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.3" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.5" stretchy="false" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.2.cmml">}</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p6.4.4.m4.5b"><apply id="S4.Thmtheorem9.p6.4.4.m4.5.5.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.5.5"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p6.4.4.m4.5.5.2.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.5.5">subscript</csymbol><apply id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.1.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.2.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.5.5.1.1.1.2">𝑐</ci><ci id="S4.Thmtheorem9.p6.4.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.1.1.1.1">𝑘</ci></apply><apply id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3"><in id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.4.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.4"></in><ci 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id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.2.cmml" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.2">𝑡</ci><cn id="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p6.4.4.m4.4.4.3.3.1.1.3.3">1</cn></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p6.4.4.m4.5c">(c^{(k)})_{k\in\{0,\ldots,2^{t+1}\}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p6.4.4.m4.5d">( italic_c start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ { 0 , … , 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT } end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem9.p7"> <p class="ltx_p" id="S4.Thmtheorem9.p7.17"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p7.17.17">It remains to prove Inequality (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E11" title="In Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">11</span></a>). By hypothesis, the solution <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.1.1.m1.1"><semantics id="S4.Thmtheorem9.p7.1.1.m1.1a"><msub id="S4.Thmtheorem9.p7.1.1.m1.1.1" xref="S4.Thmtheorem9.p7.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem9.p7.1.1.m1.1.1.2" xref="S4.Thmtheorem9.p7.1.1.m1.1.1.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p7.1.1.m1.1.1.3" xref="S4.Thmtheorem9.p7.1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.1.1.m1.1b"><apply id="S4.Thmtheorem9.p7.1.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p7.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.1.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p7.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem9.p7.1.1.m1.1.1.2">𝜎</ci><cn id="S4.Thmtheorem9.p7.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.1.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E7" title="In Lemma 4.4 ‣ 4.2 Bounding the Pareto-Optimal Costs by Induction ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">7</span></a>), that is, for all <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.2.2.m2.1"><semantics id="S4.Thmtheorem9.p7.2.2.m2.1a"><mrow id="S4.Thmtheorem9.p7.2.2.m2.1.1" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem9.p7.2.2.m2.1.1.2" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem9.p7.2.2.m2.1.1.1" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem9.p7.2.2.m2.1.1.3" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.2" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.2.cmml">C</mi><msub id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.cmml"><mi id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.2" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.3" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.2.2.m2.1b"><apply id="S4.Thmtheorem9.p7.2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1"><in id="S4.Thmtheorem9.p7.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem9.p7.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.2">𝑐</ci><apply id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.2">𝐶</ci><apply id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.1.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.2.cmml" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.2.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.2.2.m2.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.2.2.m2.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, for all <math alttext="i\in\{1,\ldots,t+1\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.3.3.m3.3"><semantics id="S4.Thmtheorem9.p7.3.3.m3.3a"><mrow id="S4.Thmtheorem9.p7.3.3.m3.3.3" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.cmml"><mi id="S4.Thmtheorem9.p7.3.3.m3.3.3.3" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p7.3.3.m3.3.3.2" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.2.cmml">∈</mo><mrow id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.2.cmml"><mo id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.2.cmml">{</mo><mn id="S4.Thmtheorem9.p7.3.3.m3.1.1" xref="S4.Thmtheorem9.p7.3.3.m3.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.3" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.2.cmml">,</mo><mi id="S4.Thmtheorem9.p7.3.3.m3.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p7.3.3.m3.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.4" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.2.cmml">,</mo><mrow id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.cmml"><mi id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.2" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.1" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.3" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.5" stretchy="false" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.3.3.m3.3b"><apply id="S4.Thmtheorem9.p7.3.3.m3.3.3.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.3.3"><in id="S4.Thmtheorem9.p7.3.3.m3.3.3.2.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.2"></in><ci id="S4.Thmtheorem9.p7.3.3.m3.3.3.3.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.3">𝑖</ci><set id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.2.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1"><cn id="S4.Thmtheorem9.p7.3.3.m3.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p7.3.3.m3.1.1">1</cn><ci id="S4.Thmtheorem9.p7.3.3.m3.2.2.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.2.2">…</ci><apply id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1"><plus id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.1"></plus><ci id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.2">𝑡</ci><cn id="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.3.3.m3.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.3.3.m3.3c">i\in\{1,\ldots,t+1\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.3.3.m3.3d">italic_i ∈ { 1 , … , italic_t + 1 }</annotation></semantics></math>, <math alttext="c_{i}\leq\max\{c_{min},B\}+1+f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.4.4.m4.5"><semantics id="S4.Thmtheorem9.p7.4.4.m4.5a"><mrow id="S4.Thmtheorem9.p7.4.4.m4.5.5" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.cmml"><msub id="S4.Thmtheorem9.p7.4.4.m4.5.5.3" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.3.cmml"><mi id="S4.Thmtheorem9.p7.4.4.m4.5.5.3.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.3.2.cmml">c</mi><mi id="S4.Thmtheorem9.p7.4.4.m4.5.5.3.3" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.3.3.cmml">i</mi></msub><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.2.cmml">≤</mo><mrow id="S4.Thmtheorem9.p7.4.4.m4.5.5.1" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.cmml"><mrow id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.2.cmml"><mi id="S4.Thmtheorem9.p7.4.4.m4.1.1" xref="S4.Thmtheorem9.p7.4.4.m4.1.1.cmml">max</mi><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1a" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.2.cmml">⁡</mo><mrow id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.2.cmml"><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.2.cmml">{</mo><msub id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.1" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.3" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.1a" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.4" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.4.cmml">n</mi></mrow></msub><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.3" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.2.cmml">,</mo><mi id="S4.Thmtheorem9.p7.4.4.m4.2.2" xref="S4.Thmtheorem9.p7.4.4.m4.2.2.cmml">B</mi><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.4" stretchy="false" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.2.cmml">}</mo></mrow></mrow><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.2.cmml">+</mo><mn id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.3" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.3.cmml">1</mn><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.2a" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.2.cmml">+</mo><mrow id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.cmml"><mi id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.2.cmml">f</mi><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.1" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.1.cmml"><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p7.4.4.m4.3.3" xref="S4.Thmtheorem9.p7.4.4.m4.3.3.cmml">0</mn><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.2.2" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p7.4.4.m4.4.4" xref="S4.Thmtheorem9.p7.4.4.m4.4.4.cmml">t</mi><mo id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.4.4.m4.5b"><apply id="S4.Thmtheorem9.p7.4.4.m4.5.5.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5"><leq id="S4.Thmtheorem9.p7.4.4.m4.5.5.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.2"></leq><apply id="S4.Thmtheorem9.p7.4.4.m4.5.5.3.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.4.4.m4.5.5.3.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.4.4.m4.5.5.3.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.3.2">𝑐</ci><ci id="S4.Thmtheorem9.p7.4.4.m4.5.5.3.3.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.3.3">𝑖</ci></apply><apply id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1"><plus id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.2"></plus><apply id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1"><max id="S4.Thmtheorem9.p7.4.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.1.1"></max><apply id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.2">𝑐</ci><apply id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3"><times id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.1"></times><ci id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.3.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.4.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.1.1.1.1.3.4">𝑛</ci></apply></apply><ci id="S4.Thmtheorem9.p7.4.4.m4.2.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.2.2">𝐵</ci></apply><cn id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.3">1</cn><apply id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4"><times id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.1"></times><ci id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.2.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.1.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.5.5.1.4.3.2"><cn id="S4.Thmtheorem9.p7.4.4.m4.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.4.4.m4.3.3">0</cn><ci id="S4.Thmtheorem9.p7.4.4.m4.4.4.cmml" xref="S4.Thmtheorem9.p7.4.4.m4.4.4">𝑡</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.4.4.m4.5c">c_{i}\leq\max\{c_{min},B\}+1+f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.4.4.m4.5d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ roman_max { italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_B } + 1 + italic_f ( 0 , italic_t )</annotation></semantics></math> or <math alttext="c_{i}=\infty" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.5.5.m5.1"><semantics id="S4.Thmtheorem9.p7.5.5.m5.1a"><mrow id="S4.Thmtheorem9.p7.5.5.m5.1.1" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.cmml"><msub id="S4.Thmtheorem9.p7.5.5.m5.1.1.2" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.2.cmml"><mi id="S4.Thmtheorem9.p7.5.5.m5.1.1.2.2" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.2.2.cmml">c</mi><mi id="S4.Thmtheorem9.p7.5.5.m5.1.1.2.3" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S4.Thmtheorem9.p7.5.5.m5.1.1.1" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.1.cmml">=</mo><mi id="S4.Thmtheorem9.p7.5.5.m5.1.1.3" mathvariant="normal" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.5.5.m5.1b"><apply id="S4.Thmtheorem9.p7.5.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p7.5.5.m5.1.1"><eq id="S4.Thmtheorem9.p7.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.1"></eq><apply id="S4.Thmtheorem9.p7.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.5.5.m5.1.1.2.1.cmml" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem9.p7.5.5.m5.1.1.2.2.cmml" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.2.2">𝑐</ci><ci id="S4.Thmtheorem9.p7.5.5.m5.1.1.2.3.cmml" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.2.3">𝑖</ci></apply><infinity id="S4.Thmtheorem9.p7.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem9.p7.5.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.5.5.m5.1c">c_{i}=\infty</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.5.5.m5.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∞</annotation></semantics></math>. Therefore, we can write each <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.6.6.m6.1"><semantics id="S4.Thmtheorem9.p7.6.6.m6.1a"><mrow id="S4.Thmtheorem9.p7.6.6.m6.1.1" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.cmml"><mi id="S4.Thmtheorem9.p7.6.6.m6.1.1.2" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem9.p7.6.6.m6.1.1.1" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem9.p7.6.6.m6.1.1.3" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.cmml"><mi id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.2" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.2.cmml">C</mi><msub id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.cmml"><mi id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.2" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.3" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.6.6.m6.1b"><apply id="S4.Thmtheorem9.p7.6.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1"><in id="S4.Thmtheorem9.p7.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.1"></in><ci id="S4.Thmtheorem9.p7.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.2">𝑐</ci><apply id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.1.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.2.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.2">𝐶</ci><apply id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.1.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.2.cmml" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.6.6.m6.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.6.6.m6.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.6.6.m6.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> as <math alttext="c=c_{min}(1,\ldots,1)+d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.7.7.m7.3"><semantics id="S4.Thmtheorem9.p7.7.7.m7.3a"><mrow id="S4.Thmtheorem9.p7.7.7.m7.3.4" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.cmml"><mi id="S4.Thmtheorem9.p7.7.7.m7.3.4.2" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.2.cmml">c</mi><mo id="S4.Thmtheorem9.p7.7.7.m7.3.4.1" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p7.7.7.m7.3.4.3" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.cmml"><mrow id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.cmml"><msub id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.cmml"><mi id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.2" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.cmml"><mi id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.2" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.1" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.3" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.1a" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.4" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.4.cmml">n</mi></mrow></msub><mo id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.1" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.1.cmml">⁢</mo><mrow 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xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.7.7.m7.3b"><apply id="S4.Thmtheorem9.p7.7.7.m7.3.4.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4"><eq id="S4.Thmtheorem9.p7.7.7.m7.3.4.1.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.1"></eq><ci id="S4.Thmtheorem9.p7.7.7.m7.3.4.2.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.2">𝑐</ci><apply id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3"><plus id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.1.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.1"></plus><apply id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2"><times id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.1.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.1"></times><apply id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.1.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2">subscript</csymbol><ci id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.2.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.2">𝑐</ci><apply id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3"><times id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.1.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.1"></times><ci id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.2.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.3.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.4.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.2.3.4">𝑛</ci></apply></apply><vector id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.3.1.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.2.3.2"><cn id="S4.Thmtheorem9.p7.7.7.m7.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p7.7.7.m7.1.1">1</cn><ci id="S4.Thmtheorem9.p7.7.7.m7.2.2.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.2.2">…</ci><cn id="S4.Thmtheorem9.p7.7.7.m7.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.7.7.m7.3.3">1</cn></vector></apply><ci id="S4.Thmtheorem9.p7.7.7.m7.3.4.3.3.cmml" xref="S4.Thmtheorem9.p7.7.7.m7.3.4.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.7.7.m7.3c">c=c_{min}(1,\ldots,1)+d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.7.7.m7.3d">italic_c = italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ( 1 , … , 1 ) + italic_d</annotation></semantics></math> with <math alttext="d_{i}\in\{0,\ldots,B+1+f({0},{t})\}\cup\{\infty\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.8.8.m8.6"><semantics id="S4.Thmtheorem9.p7.8.8.m8.6a"><mrow id="S4.Thmtheorem9.p7.8.8.m8.6.6" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.cmml"><msub id="S4.Thmtheorem9.p7.8.8.m8.6.6.3" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.3.cmml"><mi id="S4.Thmtheorem9.p7.8.8.m8.6.6.3.2" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.3.2.cmml">d</mi><mi id="S4.Thmtheorem9.p7.8.8.m8.6.6.3.3" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.3.3.cmml">i</mi></msub><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.2" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.2.cmml">∈</mo><mrow id="S4.Thmtheorem9.p7.8.8.m8.6.6.1" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.cmml"><mrow id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.2.cmml"><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.2.cmml">{</mo><mn id="S4.Thmtheorem9.p7.8.8.m8.3.3" xref="S4.Thmtheorem9.p7.8.8.m8.3.3.cmml">0</mn><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.3" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.2.cmml">,</mo><mi id="S4.Thmtheorem9.p7.8.8.m8.4.4" mathvariant="normal" xref="S4.Thmtheorem9.p7.8.8.m8.4.4.cmml">…</mi><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.4" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.2.cmml">,</mo><mrow id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.2" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.2.cmml">B</mi><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.1" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.3" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.3.cmml">1</mn><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.1a" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.1.cmml">+</mo><mrow id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.cmml"><mi id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.2" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.2.cmml">f</mi><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.1" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.3.2" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.3.1.cmml"><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.3.2.1" stretchy="false" 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xref="S4.Thmtheorem9.p7.8.8.m8.5.5.cmml">∞</mi><mo id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.8.8.m8.6b"><apply id="S4.Thmtheorem9.p7.8.8.m8.6.6.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6"><in id="S4.Thmtheorem9.p7.8.8.m8.6.6.2.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.2"></in><apply id="S4.Thmtheorem9.p7.8.8.m8.6.6.3.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.8.8.m8.6.6.3.1.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.8.8.m8.6.6.3.2.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.3.2">𝑑</ci><ci id="S4.Thmtheorem9.p7.8.8.m8.6.6.3.3.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.3.3">𝑖</ci></apply><apply id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1"><union id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.2.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.2"></union><set id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.2.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1"><cn id="S4.Thmtheorem9.p7.8.8.m8.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.8.8.m8.3.3">0</cn><ci id="S4.Thmtheorem9.p7.8.8.m8.4.4.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.4.4">…</ci><apply id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1"><plus id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.1"></plus><ci id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.2">𝐵</ci><cn id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.3">1</cn><apply id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4"><times id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.1.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.1"></times><ci id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.2.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.3.1.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.1.1.1.4.3.2"><cn id="S4.Thmtheorem9.p7.8.8.m8.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p7.8.8.m8.1.1">0</cn><ci id="S4.Thmtheorem9.p7.8.8.m8.2.2.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.2.2">𝑡</ci></interval></apply></apply></set><set id="S4.Thmtheorem9.p7.8.8.m8.6.6.1.3.1.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.6.6.1.3.2"><infinity id="S4.Thmtheorem9.p7.8.8.m8.5.5.cmml" xref="S4.Thmtheorem9.p7.8.8.m8.5.5"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.8.8.m8.6c">d_{i}\in\{0,\ldots,B+1+f({0},{t})\}\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.8.8.m8.6d">italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { 0 , … , italic_B + 1 + italic_f ( 0 , italic_t ) } ∪ { ∞ }</annotation></semantics></math> for all <math alttext="i" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.9.9.m9.1"><semantics id="S4.Thmtheorem9.p7.9.9.m9.1a"><mi id="S4.Thmtheorem9.p7.9.9.m9.1.1" xref="S4.Thmtheorem9.p7.9.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.9.9.m9.1b"><ci id="S4.Thmtheorem9.p7.9.9.m9.1.1.cmml" xref="S4.Thmtheorem9.p7.9.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.9.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.9.9.m9.1d">italic_i</annotation></semantics></math>. If two costs <math alttext="c,c^{\prime}\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.10.10.m10.2"><semantics id="S4.Thmtheorem9.p7.10.10.m10.2a"><mrow id="S4.Thmtheorem9.p7.10.10.m10.2.2" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.cmml"><mrow id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.2.cmml"><mi id="S4.Thmtheorem9.p7.10.10.m10.1.1" xref="S4.Thmtheorem9.p7.10.10.m10.1.1.cmml">c</mi><mo id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.2" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.2.cmml">,</mo><msup id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.2" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.3" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S4.Thmtheorem9.p7.10.10.m10.2.2.2" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.2.cmml">∈</mo><msub id="S4.Thmtheorem9.p7.10.10.m10.2.2.3" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.cmml"><mi id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.2" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.2.cmml">C</mi><msub id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.cmml"><mi id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.2" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.3" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.10.10.m10.2b"><apply id="S4.Thmtheorem9.p7.10.10.m10.2.2.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2"><in id="S4.Thmtheorem9.p7.10.10.m10.2.2.2.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.2"></in><list id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.2.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1"><ci id="S4.Thmtheorem9.p7.10.10.m10.1.1.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.1.1">𝑐</ci><apply id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.2">𝑐</ci><ci id="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.1.1.1.3">′</ci></apply></list><apply id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.1.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.2.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.2">𝐶</ci><apply id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.1.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.2.cmml" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.10.10.m10.2.2.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.10.10.m10.2c">c,c^{\prime}\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.10.10.m10.2d">italic_c , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> are such that <math alttext="c=c_{min}(1,\ldots,1)+d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.11.11.m11.3"><semantics id="S4.Thmtheorem9.p7.11.11.m11.3a"><mrow id="S4.Thmtheorem9.p7.11.11.m11.3.4" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.cmml"><mi id="S4.Thmtheorem9.p7.11.11.m11.3.4.2" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.2.cmml">c</mi><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.1" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p7.11.11.m11.3.4.3" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.cmml"><mrow id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.cmml"><msub id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.cmml"><mi id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.2" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.cmml"><mi id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.2" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.1" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.3" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.1a" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.4" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.4.cmml">n</mi></mrow></msub><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.1" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.2" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.1.cmml"><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p7.11.11.m11.1.1" xref="S4.Thmtheorem9.p7.11.11.m11.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.2.2" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p7.11.11.m11.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p7.11.11.m11.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.2.3" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.1.cmml">,</mo><mn id="S4.Thmtheorem9.p7.11.11.m11.3.3" xref="S4.Thmtheorem9.p7.11.11.m11.3.3.cmml">1</mn><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.2.4" stretchy="false" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.1" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.1.cmml">+</mo><mi id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.3" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.11.11.m11.3b"><apply id="S4.Thmtheorem9.p7.11.11.m11.3.4.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4"><eq id="S4.Thmtheorem9.p7.11.11.m11.3.4.1.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.1"></eq><ci id="S4.Thmtheorem9.p7.11.11.m11.3.4.2.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.2">𝑐</ci><apply id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3"><plus id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.1.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.1"></plus><apply id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2"><times id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.1.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.1"></times><apply id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.1.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2">subscript</csymbol><ci id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.2.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.2">𝑐</ci><apply id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3"><times id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.1.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.1"></times><ci id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.2.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.3.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.4.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.2.3.4">𝑛</ci></apply></apply><vector id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.1.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.2.3.2"><cn id="S4.Thmtheorem9.p7.11.11.m11.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p7.11.11.m11.1.1">1</cn><ci id="S4.Thmtheorem9.p7.11.11.m11.2.2.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.2.2">…</ci><cn id="S4.Thmtheorem9.p7.11.11.m11.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.11.11.m11.3.3">1</cn></vector></apply><ci id="S4.Thmtheorem9.p7.11.11.m11.3.4.3.3.cmml" xref="S4.Thmtheorem9.p7.11.11.m11.3.4.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.11.11.m11.3c">c=c_{min}(1,\ldots,1)+d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.11.11.m11.3d">italic_c = italic_c start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ( 1 , … , 1 ) + italic_d</annotation></semantics></math> and <math alttext="c^{\prime}=c^{\prime}_{min}(1,\ldots,1)+d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.12.12.m12.3"><semantics id="S4.Thmtheorem9.p7.12.12.m12.3a"><mrow id="S4.Thmtheorem9.p7.12.12.m12.3.4" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.cmml"><msup id="S4.Thmtheorem9.p7.12.12.m12.3.4.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.2.cmml"><mi id="S4.Thmtheorem9.p7.12.12.m12.3.4.2.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.2.2.cmml">c</mi><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.2.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.1" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p7.12.12.m12.3.4.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.cmml"><mrow id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.cmml"><msubsup id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.cmml"><mi id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.2.cmml">c</mi><mrow id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.cmml"><mi id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.1" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.3.cmml">i</mi><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.1a" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.4" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.4.cmml">n</mi></mrow><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.1" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.1.cmml"><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p7.12.12.m12.1.1" xref="S4.Thmtheorem9.p7.12.12.m12.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.2.2" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p7.12.12.m12.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p7.12.12.m12.2.2.cmml">…</mi><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.2.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.1.cmml">,</mo><mn id="S4.Thmtheorem9.p7.12.12.m12.3.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.3.cmml">1</mn><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.2.4" stretchy="false" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.1" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.1.cmml">+</mo><mi id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.3" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.12.12.m12.3b"><apply id="S4.Thmtheorem9.p7.12.12.m12.3.4.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4"><eq id="S4.Thmtheorem9.p7.12.12.m12.3.4.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.1"></eq><apply id="S4.Thmtheorem9.p7.12.12.m12.3.4.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.12.12.m12.3.4.2.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.2">superscript</csymbol><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.2.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.2.2">𝑐</ci><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.2.3.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.2.3">′</ci></apply><apply id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3"><plus id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.1"></plus><apply id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2"><times id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.1"></times><apply id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2">subscript</csymbol><apply id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2">superscript</csymbol><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.2">𝑐</ci><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.3.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.2.3">′</ci></apply><apply id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3"><times id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.1"></times><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.2">𝑚</ci><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.3.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.3">𝑖</ci><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.4.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.2.3.4">𝑛</ci></apply></apply><vector id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.1.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.2.3.2"><cn id="S4.Thmtheorem9.p7.12.12.m12.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p7.12.12.m12.1.1">1</cn><ci id="S4.Thmtheorem9.p7.12.12.m12.2.2.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.2.2">…</ci><cn id="S4.Thmtheorem9.p7.12.12.m12.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.12.12.m12.3.3">1</cn></vector></apply><ci id="S4.Thmtheorem9.p7.12.12.m12.3.4.3.3.cmml" xref="S4.Thmtheorem9.p7.12.12.m12.3.4.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.12.12.m12.3c">c^{\prime}=c^{\prime}_{min}(1,\ldots,1)+d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.12.12.m12.3d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ( 1 , … , 1 ) + italic_d</annotation></semantics></math>, with the same vector <math alttext="d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.13.13.m13.1"><semantics id="S4.Thmtheorem9.p7.13.13.m13.1a"><mi id="S4.Thmtheorem9.p7.13.13.m13.1.1" xref="S4.Thmtheorem9.p7.13.13.m13.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.13.13.m13.1b"><ci id="S4.Thmtheorem9.p7.13.13.m13.1.1.cmml" xref="S4.Thmtheorem9.p7.13.13.m13.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.13.13.m13.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.13.13.m13.1d">italic_d</annotation></semantics></math>, then they are comparable. This is impossible as <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.14.14.m14.1"><semantics id="S4.Thmtheorem9.p7.14.14.m14.1a"><msub id="S4.Thmtheorem9.p7.14.14.m14.1.1" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.cmml"><mi id="S4.Thmtheorem9.p7.14.14.m14.1.1.2" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p7.14.14.m14.1.1.3" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.3.cmml"><mi id="S4.Thmtheorem9.p7.14.14.m14.1.1.3.2" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p7.14.14.m14.1.1.3.3" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.14.14.m14.1b"><apply id="S4.Thmtheorem9.p7.14.14.m14.1.1.cmml" xref="S4.Thmtheorem9.p7.14.14.m14.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.14.14.m14.1.1.1.cmml" xref="S4.Thmtheorem9.p7.14.14.m14.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p7.14.14.m14.1.1.2.cmml" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p7.14.14.m14.1.1.3.cmml" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.14.14.m14.1.1.3.1.cmml" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.14.14.m14.1.1.3.2.cmml" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p7.14.14.m14.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.14.14.m14.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.14.14.m14.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.14.14.m14.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is an antichain. Hence, the size of <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.15.15.m15.1"><semantics id="S4.Thmtheorem9.p7.15.15.m15.1a"><msub id="S4.Thmtheorem9.p7.15.15.m15.1.1" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.cmml"><mi id="S4.Thmtheorem9.p7.15.15.m15.1.1.2" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.2.cmml">C</mi><msub id="S4.Thmtheorem9.p7.15.15.m15.1.1.3" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.3.cmml"><mi id="S4.Thmtheorem9.p7.15.15.m15.1.1.3.2" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem9.p7.15.15.m15.1.1.3.3" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.15.15.m15.1b"><apply id="S4.Thmtheorem9.p7.15.15.m15.1.1.cmml" xref="S4.Thmtheorem9.p7.15.15.m15.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.15.15.m15.1.1.1.cmml" xref="S4.Thmtheorem9.p7.15.15.m15.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p7.15.15.m15.1.1.2.cmml" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.2">𝐶</ci><apply id="S4.Thmtheorem9.p7.15.15.m15.1.1.3.cmml" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.15.15.m15.1.1.3.1.cmml" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p7.15.15.m15.1.1.3.2.cmml" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem9.p7.15.15.m15.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.15.15.m15.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.15.15.m15.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.15.15.m15.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is bounded by the number of vectors <math alttext="d" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.16.16.m16.1"><semantics id="S4.Thmtheorem9.p7.16.16.m16.1a"><mi id="S4.Thmtheorem9.p7.16.16.m16.1.1" xref="S4.Thmtheorem9.p7.16.16.m16.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.16.16.m16.1b"><ci id="S4.Thmtheorem9.p7.16.16.m16.1.1.cmml" xref="S4.Thmtheorem9.p7.16.16.m16.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.16.16.m16.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.16.16.m16.1d">italic_d</annotation></semantics></math>, that is, by <math alttext="\big{(}f({0},{t})+B+3\big{)}^{t+1}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p7.17.17.m17.3"><semantics id="S4.Thmtheorem9.p7.17.17.m17.3a"><msup id="S4.Thmtheorem9.p7.17.17.m17.3.3" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.cmml"><mrow id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.cmml"><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.2" maxsize="120%" minsize="120%" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.cmml"><mrow id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.cmml"><mi id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.2" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.2.cmml">f</mi><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.1" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.2" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.1.cmml"><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p7.17.17.m17.1.1" xref="S4.Thmtheorem9.p7.17.17.m17.1.1.cmml">0</mn><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.2.2" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p7.17.17.m17.2.2" xref="S4.Thmtheorem9.p7.17.17.m17.2.2.cmml">t</mi><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.1" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.3" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.3.cmml">B</mi><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.1a" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.4" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.4.cmml">3</mn></mrow><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.3" maxsize="120%" minsize="120%" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.cmml">)</mo></mrow><mrow id="S4.Thmtheorem9.p7.17.17.m17.3.3.3" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3.cmml"><mi id="S4.Thmtheorem9.p7.17.17.m17.3.3.3.2" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p7.17.17.m17.3.3.3.1" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p7.17.17.m17.3.3.3.3" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p7.17.17.m17.3b"><apply id="S4.Thmtheorem9.p7.17.17.m17.3.3.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p7.17.17.m17.3.3.2.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3">superscript</csymbol><apply id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1"><plus id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.1"></plus><apply id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2"><times id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.1.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.1"></times><ci id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.2.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.1.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.2.3.2"><cn id="S4.Thmtheorem9.p7.17.17.m17.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p7.17.17.m17.1.1">0</cn><ci id="S4.Thmtheorem9.p7.17.17.m17.2.2.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.2.2">𝑡</ci></interval></apply><ci id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.3.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.3">𝐵</ci><cn id="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.4.cmml" type="integer" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.1.1.1.4">3</cn></apply><apply id="S4.Thmtheorem9.p7.17.17.m17.3.3.3.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3"><plus id="S4.Thmtheorem9.p7.17.17.m17.3.3.3.1.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3.1"></plus><ci id="S4.Thmtheorem9.p7.17.17.m17.3.3.3.2.cmml" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3.2">𝑡</ci><cn id="S4.Thmtheorem9.p7.17.17.m17.3.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p7.17.17.m17.3.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p7.17.17.m17.3c">\big{(}f({0},{t})+B+3\big{)}^{t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p7.17.17.m17.3d">( italic_f ( 0 , italic_t ) + italic_B + 3 ) start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, and (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E11" title="In Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">11</span></a>) is then established.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.4 </span>Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> </h3> <div class="ltx_para" id="S4.SS4.p1"> <p class="ltx_p" id="S4.SS4.p1.1">Finally, we gather all our intermediate results and combine them, in a way to complete the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>. Thanks to Lemmas <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem2" title="Lemma 4.2 ‣ 4.1 Dimension One ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.2</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a>, calculations can be done in a way to have an explicit formula for <math alttext="f({B},{t})" class="ltx_Math" display="inline" id="S4.SS4.p1.1.m1.2"><semantics id="S4.SS4.p1.1.m1.2a"><mrow id="S4.SS4.p1.1.m1.2.3" xref="S4.SS4.p1.1.m1.2.3.cmml"><mi id="S4.SS4.p1.1.m1.2.3.2" xref="S4.SS4.p1.1.m1.2.3.2.cmml">f</mi><mo id="S4.SS4.p1.1.m1.2.3.1" xref="S4.SS4.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS4.p1.1.m1.2.3.3.2" xref="S4.SS4.p1.1.m1.2.3.3.1.cmml"><mo id="S4.SS4.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS4.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS4.p1.1.m1.1.1" xref="S4.SS4.p1.1.m1.1.1.cmml">B</mi><mo id="S4.SS4.p1.1.m1.2.3.3.2.2" xref="S4.SS4.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS4.p1.1.m1.2.2" xref="S4.SS4.p1.1.m1.2.2.cmml">t</mi><mo id="S4.SS4.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS4.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.1.m1.2b"><apply id="S4.SS4.p1.1.m1.2.3.cmml" xref="S4.SS4.p1.1.m1.2.3"><times id="S4.SS4.p1.1.m1.2.3.1.cmml" xref="S4.SS4.p1.1.m1.2.3.1"></times><ci id="S4.SS4.p1.1.m1.2.3.2.cmml" xref="S4.SS4.p1.1.m1.2.3.2">𝑓</ci><interval closure="open" id="S4.SS4.p1.1.m1.2.3.3.1.cmml" xref="S4.SS4.p1.1.m1.2.3.3.2"><ci id="S4.SS4.p1.1.m1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1">𝐵</ci><ci id="S4.SS4.p1.1.m1.2.2.cmml" xref="S4.SS4.p1.1.m1.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.1.m1.2c">f({B},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.1.m1.2d">italic_f ( italic_B , italic_t )</annotation></semantics></math> and a bound on its value.</p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem10.1.1.1">Proof 4.10</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem10.2.2"> </span>(Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>)</h6> <div class="ltx_para" id="S4.Thmtheorem10.p1"> <p class="ltx_p" id="S4.Thmtheorem10.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem10.p1.8.8">Let <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.1.1.m1.1"><semantics id="S4.Thmtheorem10.p1.1.1.m1.1a"><mi id="S4.Thmtheorem10.p1.1.1.m1.1.1" xref="S4.Thmtheorem10.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.1.1.m1.1b"><ci id="S4.Thmtheorem10.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem10.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a binary SP game with dimension <math alttext="t" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.2.2.m2.1"><semantics id="S4.Thmtheorem10.p1.2.2.m2.1a"><mi id="S4.Thmtheorem10.p1.2.2.m2.1.1" xref="S4.Thmtheorem10.p1.2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.2.2.m2.1b"><ci id="S4.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem10.p1.2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.2.2.m2.1d">italic_t</annotation></semantics></math>, <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.3.3.m3.1"><semantics id="S4.Thmtheorem10.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem10.p1.3.3.m3.1.1" xref="S4.Thmtheorem10.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem10.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem10.p1.3.3.m3.1.1.2.cmml">B</mi><mo id="S4.Thmtheorem10.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem10.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem10.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem10.p1.3.3.m3.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.3.3.m3.1b"><apply id="S4.Thmtheorem10.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem10.p1.3.3.m3.1.1"><in id="S4.Thmtheorem10.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem10.p1.3.3.m3.1.1.1"></in><ci id="S4.Thmtheorem10.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem10.p1.3.3.m3.1.1.2">𝐵</ci><ci id="S4.Thmtheorem10.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem10.p1.3.3.m3.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.3.3.m3.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.3.3.m3.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.4.4.m4.2"><semantics id="S4.Thmtheorem10.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem10.p1.4.4.m4.2.3" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.cmml"><msub id="S4.Thmtheorem10.p1.4.4.m4.2.3.2" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.2.cmml"><mi id="S4.Thmtheorem10.p1.4.4.m4.2.3.2.2" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem10.p1.4.4.m4.2.3.2.3" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem10.p1.4.4.m4.2.3.1" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem10.p1.4.4.m4.2.3.3" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.2" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.1" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.2" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem10.p1.4.4.m4.1.1" xref="S4.Thmtheorem10.p1.4.4.m4.1.1.cmml">G</mi><mo id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.2.2" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem10.p1.4.4.m4.2.2" xref="S4.Thmtheorem10.p1.4.4.m4.2.2.cmml">B</mi><mo id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.4.4.m4.2b"><apply id="S4.Thmtheorem10.p1.4.4.m4.2.3.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3"><in id="S4.Thmtheorem10.p1.4.4.m4.2.3.1.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.1"></in><apply id="S4.Thmtheorem10.p1.4.4.m4.2.3.2.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.4.4.m4.2.3.2.1.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem10.p1.4.4.m4.2.3.2.2.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.2.2">𝜎</ci><cn id="S4.Thmtheorem10.p1.4.4.m4.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3"><times id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.1.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.1"></times><ci id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.2a.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.2.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.1.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.3.3.3.2"><ci id="S4.Thmtheorem10.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.1.1">𝐺</ci><ci id="S4.Thmtheorem10.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem10.p1.4.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.4.4.m4.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.4.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. We assume that <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.5.5.m5.4"><semantics id="S4.Thmtheorem10.p1.5.5.m5.4a"><mrow id="S4.Thmtheorem10.p1.5.5.m5.4.4" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.cmml"><msub id="S4.Thmtheorem10.p1.5.5.m5.4.4.3" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.cmml"><mi id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.2" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.2.cmml">C</mi><msub id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.cmml"><mi id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.2" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.3" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem10.p1.5.5.m5.4.4.2" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.2.cmml">≠</mo><mrow id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.2.cmml"><mo id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.2" stretchy="false" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.2.cmml">{</mo><mrow id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.2" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem10.p1.5.5.m5.1.1" mathvariant="normal" xref="S4.Thmtheorem10.p1.5.5.m5.1.1.cmml">∞</mi><mo id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.2.2" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem10.p1.5.5.m5.2.2" mathvariant="normal" xref="S4.Thmtheorem10.p1.5.5.m5.2.2.cmml">…</mi><mo id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.2.3" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem10.p1.5.5.m5.3.3" mathvariant="normal" xref="S4.Thmtheorem10.p1.5.5.m5.3.3.cmml">∞</mi><mo id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.2.4" stretchy="false" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.3" stretchy="false" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.5.5.m5.4b"><apply id="S4.Thmtheorem10.p1.5.5.m5.4.4.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4"><neq id="S4.Thmtheorem10.p1.5.5.m5.4.4.2.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.2"></neq><apply id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.1.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3">subscript</csymbol><ci id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.2.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.2">𝐶</ci><apply id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.1.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3">subscript</csymbol><ci id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.2.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.2">𝜎</ci><cn id="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.3.cmml" type="integer" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.3.3.3">0</cn></apply></apply><set id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.2.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1"><vector id="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.4.4.1.1.1.2"><infinity id="S4.Thmtheorem10.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.1.1"></infinity><ci id="S4.Thmtheorem10.p1.5.5.m5.2.2.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.2.2">…</ci><infinity id="S4.Thmtheorem10.p1.5.5.m5.3.3.cmml" xref="S4.Thmtheorem10.p1.5.5.m5.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.5.5.m5.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.5.5.m5.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>, as Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> is trivially true in case <math alttext="C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.6.6.m6.4"><semantics id="S4.Thmtheorem10.p1.6.6.m6.4a"><mrow id="S4.Thmtheorem10.p1.6.6.m6.4.4" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.cmml"><msub id="S4.Thmtheorem10.p1.6.6.m6.4.4.3" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.cmml"><mi id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.2" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.2.cmml">C</mi><msub id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.cmml"><mi id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.2" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.2.cmml">σ</mi><mn id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.3" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S4.Thmtheorem10.p1.6.6.m6.4.4.2" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.2.cmml">=</mo><mrow id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.2.cmml"><mo id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.2" stretchy="false" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.2.cmml">{</mo><mrow id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.2" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem10.p1.6.6.m6.1.1" mathvariant="normal" xref="S4.Thmtheorem10.p1.6.6.m6.1.1.cmml">∞</mi><mo id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.2.2" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem10.p1.6.6.m6.2.2" mathvariant="normal" xref="S4.Thmtheorem10.p1.6.6.m6.2.2.cmml">…</mi><mo id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.2.3" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.1.cmml">,</mo><mi id="S4.Thmtheorem10.p1.6.6.m6.3.3" mathvariant="normal" xref="S4.Thmtheorem10.p1.6.6.m6.3.3.cmml">∞</mi><mo id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.2.4" stretchy="false" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.3" stretchy="false" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.6.6.m6.4b"><apply id="S4.Thmtheorem10.p1.6.6.m6.4.4.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4"><eq id="S4.Thmtheorem10.p1.6.6.m6.4.4.2.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.2"></eq><apply id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.1.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3">subscript</csymbol><ci id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.2.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.2">𝐶</ci><apply id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.1.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3">subscript</csymbol><ci id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.2.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.2">𝜎</ci><cn id="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.3.cmml" type="integer" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.3.3.3">0</cn></apply></apply><set id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.2.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1"><vector id="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.4.4.1.1.1.2"><infinity id="S4.Thmtheorem10.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.1.1"></infinity><ci id="S4.Thmtheorem10.p1.6.6.m6.2.2.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.2.2">…</ci><infinity id="S4.Thmtheorem10.p1.6.6.m6.3.3.cmml" xref="S4.Thmtheorem10.p1.6.6.m6.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.6.6.m6.4c">C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.6.6.m6.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { ( ∞ , … , ∞ ) }</annotation></semantics></math>. By Lemmas <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem2" title="Lemma 4.2 ‣ 4.1 Dimension One ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.2</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a>, there exists <math alttext="\sigma^{\prime}_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.7.7.m7.2"><semantics id="S4.Thmtheorem10.p1.7.7.m7.2a"><mrow id="S4.Thmtheorem10.p1.7.7.m7.2.3" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.cmml"><msubsup id="S4.Thmtheorem10.p1.7.7.m7.2.3.2" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2.cmml"><mi id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.2" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.3" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2.3.cmml">0</mn><mo id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.3" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem10.p1.7.7.m7.2.3.1" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem10.p1.7.7.m7.2.3.3" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.2" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.1" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.2" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem10.p1.7.7.m7.1.1" xref="S4.Thmtheorem10.p1.7.7.m7.1.1.cmml">G</mi><mo id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.2.2" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem10.p1.7.7.m7.2.2" xref="S4.Thmtheorem10.p1.7.7.m7.2.2.cmml">B</mi><mo id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.7.7.m7.2b"><apply id="S4.Thmtheorem10.p1.7.7.m7.2.3.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3"><in id="S4.Thmtheorem10.p1.7.7.m7.2.3.1.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.1"></in><apply id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.1.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2">subscript</csymbol><apply id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.1.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2">superscript</csymbol><ci id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.2.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.2">𝜎</ci><ci id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.3.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2.2.3">′</ci></apply><cn id="S4.Thmtheorem10.p1.7.7.m7.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.2.3">0</cn></apply><apply id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3"><times id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.1.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.1"></times><ci id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.2a.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.2.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.1.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.3.3.3.2"><ci id="S4.Thmtheorem10.p1.7.7.m7.1.1.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.1.1">𝐺</ci><ci id="S4.Thmtheorem10.p1.7.7.m7.2.2.cmml" xref="S4.Thmtheorem10.p1.7.7.m7.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.7.7.m7.2c">\sigma^{\prime}_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.7.7.m7.2d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> without cycles such that <math alttext="\sigma^{\prime}_{0}\preceq\sigma_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.8.8.m8.1"><semantics id="S4.Thmtheorem10.p1.8.8.m8.1a"><mrow id="S4.Thmtheorem10.p1.8.8.m8.1.1" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.cmml"><msubsup id="S4.Thmtheorem10.p1.8.8.m8.1.1.2" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2.cmml"><mi id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.2" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.3" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2.3.cmml">0</mn><mo id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.3" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem10.p1.8.8.m8.1.1.1" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.1.cmml">⪯</mo><msub id="S4.Thmtheorem10.p1.8.8.m8.1.1.3" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.3.cmml"><mi id="S4.Thmtheorem10.p1.8.8.m8.1.1.3.2" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.3.2.cmml">σ</mi><mn id="S4.Thmtheorem10.p1.8.8.m8.1.1.3.3" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.8.8.m8.1b"><apply id="S4.Thmtheorem10.p1.8.8.m8.1.1.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1"><csymbol cd="latexml" id="S4.Thmtheorem10.p1.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.1">precedes-or-equals</csymbol><apply id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.1.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2">subscript</csymbol><apply id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.1.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2">superscript</csymbol><ci id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.2.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.2">𝜎</ci><ci id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.3.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2.2.3">′</ci></apply><cn id="S4.Thmtheorem10.p1.8.8.m8.1.1.2.3.cmml" type="integer" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.2.3">0</cn></apply><apply id="S4.Thmtheorem10.p1.8.8.m8.1.1.3.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.8.8.m8.1.1.3.1.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem10.p1.8.8.m8.1.1.3.2.cmml" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.3.2">𝜎</ci><cn id="S4.Thmtheorem10.p1.8.8.m8.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem10.p1.8.8.m8.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.8.8.m8.1c">\sigma^{\prime}_{0}\preceq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.8.8.m8.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⪯ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and</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="\forall c^{\prime}\in C_{\sigma^{\prime}_{0}},\forall i\in\{1,\ldots,t\}:~{}c^% {\prime}_{i}\leq f({B},{t})~{}\vee~{}c^{\prime}_{i}=\infty," class="ltx_Math" display="block" id="S4.Ex7.m1.6"><semantics id="S4.Ex7.m1.6a"><mrow id="S4.Ex7.m1.6.6.1" xref="S4.Ex7.m1.6.6.1.1.cmml"><mrow id="S4.Ex7.m1.6.6.1.1" xref="S4.Ex7.m1.6.6.1.1.cmml"><mrow id="S4.Ex7.m1.6.6.1.1.2.2" xref="S4.Ex7.m1.6.6.1.1.2.3.cmml"><mrow id="S4.Ex7.m1.6.6.1.1.1.1.1" xref="S4.Ex7.m1.6.6.1.1.1.1.1.cmml"><mrow id="S4.Ex7.m1.6.6.1.1.1.1.1.2" xref="S4.Ex7.m1.6.6.1.1.1.1.1.2.cmml"><mo id="S4.Ex7.m1.6.6.1.1.1.1.1.2.1" rspace="0.167em" xref="S4.Ex7.m1.6.6.1.1.1.1.1.2.1.cmml">∀</mo><msup id="S4.Ex7.m1.6.6.1.1.1.1.1.2.2" xref="S4.Ex7.m1.6.6.1.1.1.1.1.2.2.cmml"><mi id="S4.Ex7.m1.6.6.1.1.1.1.1.2.2.2" xref="S4.Ex7.m1.6.6.1.1.1.1.1.2.2.2.cmml">c</mi><mo id="S4.Ex7.m1.6.6.1.1.1.1.1.2.2.3" xref="S4.Ex7.m1.6.6.1.1.1.1.1.2.2.3.cmml">′</mo></msup></mrow><mo id="S4.Ex7.m1.6.6.1.1.1.1.1.1" xref="S4.Ex7.m1.6.6.1.1.1.1.1.1.cmml">∈</mo><msub id="S4.Ex7.m1.6.6.1.1.1.1.1.3" xref="S4.Ex7.m1.6.6.1.1.1.1.1.3.cmml"><mi id="S4.Ex7.m1.6.6.1.1.1.1.1.3.2" xref="S4.Ex7.m1.6.6.1.1.1.1.1.3.2.cmml">C</mi><msubsup id="S4.Ex7.m1.6.6.1.1.1.1.1.3.3" xref="S4.Ex7.m1.6.6.1.1.1.1.1.3.3.cmml"><mi id="S4.Ex7.m1.6.6.1.1.1.1.1.3.3.2.2" xref="S4.Ex7.m1.6.6.1.1.1.1.1.3.3.2.2.cmml">σ</mi><mn id="S4.Ex7.m1.6.6.1.1.1.1.1.3.3.3" xref="S4.Ex7.m1.6.6.1.1.1.1.1.3.3.3.cmml">0</mn><mo id="S4.Ex7.m1.6.6.1.1.1.1.1.3.3.2.3" xref="S4.Ex7.m1.6.6.1.1.1.1.1.3.3.2.3.cmml">′</mo></msubsup></msub></mrow><mo id="S4.Ex7.m1.6.6.1.1.2.2.3" xref="S4.Ex7.m1.6.6.1.1.2.3a.cmml">,</mo><mrow id="S4.Ex7.m1.6.6.1.1.2.2.2" xref="S4.Ex7.m1.6.6.1.1.2.2.2.cmml"><mrow id="S4.Ex7.m1.6.6.1.1.2.2.2.2" xref="S4.Ex7.m1.6.6.1.1.2.2.2.2.cmml"><mo id="S4.Ex7.m1.6.6.1.1.2.2.2.2.1" rspace="0.167em" xref="S4.Ex7.m1.6.6.1.1.2.2.2.2.1.cmml">∀</mo><mi id="S4.Ex7.m1.6.6.1.1.2.2.2.2.2" xref="S4.Ex7.m1.6.6.1.1.2.2.2.2.2.cmml">i</mi></mrow><mo id="S4.Ex7.m1.6.6.1.1.2.2.2.1" xref="S4.Ex7.m1.6.6.1.1.2.2.2.1.cmml">∈</mo><mrow id="S4.Ex7.m1.6.6.1.1.2.2.2.3.2" xref="S4.Ex7.m1.6.6.1.1.2.2.2.3.1.cmml"><mo id="S4.Ex7.m1.6.6.1.1.2.2.2.3.2.1" stretchy="false" xref="S4.Ex7.m1.6.6.1.1.2.2.2.3.1.cmml">{</mo><mn id="S4.Ex7.m1.1.1" xref="S4.Ex7.m1.1.1.cmml">1</mn><mo id="S4.Ex7.m1.6.6.1.1.2.2.2.3.2.2" xref="S4.Ex7.m1.6.6.1.1.2.2.2.3.1.cmml">,</mo><mi id="S4.Ex7.m1.2.2" mathvariant="normal" xref="S4.Ex7.m1.2.2.cmml">…</mi><mo id="S4.Ex7.m1.6.6.1.1.2.2.2.3.2.3" xref="S4.Ex7.m1.6.6.1.1.2.2.2.3.1.cmml">,</mo><mi id="S4.Ex7.m1.3.3" xref="S4.Ex7.m1.3.3.cmml">t</mi><mo id="S4.Ex7.m1.6.6.1.1.2.2.2.3.2.4" rspace="0.278em" stretchy="false" xref="S4.Ex7.m1.6.6.1.1.2.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S4.Ex7.m1.6.6.1.1.3" rspace="0.608em" xref="S4.Ex7.m1.6.6.1.1.3.cmml">:</mo><mrow id="S4.Ex7.m1.6.6.1.1.4" xref="S4.Ex7.m1.6.6.1.1.4.cmml"><msubsup id="S4.Ex7.m1.6.6.1.1.4.2" xref="S4.Ex7.m1.6.6.1.1.4.2.cmml"><mi id="S4.Ex7.m1.6.6.1.1.4.2.2.2" xref="S4.Ex7.m1.6.6.1.1.4.2.2.2.cmml">c</mi><mi id="S4.Ex7.m1.6.6.1.1.4.2.3" xref="S4.Ex7.m1.6.6.1.1.4.2.3.cmml">i</mi><mo id="S4.Ex7.m1.6.6.1.1.4.2.2.3" xref="S4.Ex7.m1.6.6.1.1.4.2.2.3.cmml">′</mo></msubsup><mo id="S4.Ex7.m1.6.6.1.1.4.3" xref="S4.Ex7.m1.6.6.1.1.4.3.cmml">≤</mo><mrow id="S4.Ex7.m1.6.6.1.1.4.4" xref="S4.Ex7.m1.6.6.1.1.4.4.cmml"><mrow id="S4.Ex7.m1.6.6.1.1.4.4.2" xref="S4.Ex7.m1.6.6.1.1.4.4.2.cmml"><mi id="S4.Ex7.m1.6.6.1.1.4.4.2.2" xref="S4.Ex7.m1.6.6.1.1.4.4.2.2.cmml">f</mi><mo id="S4.Ex7.m1.6.6.1.1.4.4.2.1" xref="S4.Ex7.m1.6.6.1.1.4.4.2.1.cmml">⁢</mo><mrow id="S4.Ex7.m1.6.6.1.1.4.4.2.3.2" xref="S4.Ex7.m1.6.6.1.1.4.4.2.3.1.cmml"><mo id="S4.Ex7.m1.6.6.1.1.4.4.2.3.2.1" stretchy="false" xref="S4.Ex7.m1.6.6.1.1.4.4.2.3.1.cmml">(</mo><mi id="S4.Ex7.m1.4.4" xref="S4.Ex7.m1.4.4.cmml">B</mi><mo id="S4.Ex7.m1.6.6.1.1.4.4.2.3.2.2" xref="S4.Ex7.m1.6.6.1.1.4.4.2.3.1.cmml">,</mo><mi id="S4.Ex7.m1.5.5" xref="S4.Ex7.m1.5.5.cmml">t</mi><mo id="S4.Ex7.m1.6.6.1.1.4.4.2.3.2.3" rspace="0.330em" stretchy="false" xref="S4.Ex7.m1.6.6.1.1.4.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.6.6.1.1.4.4.1" rspace="0.552em" 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id="S4.Ex7.m1.6c">\forall c^{\prime}\in C_{\sigma^{\prime}_{0}},\forall i\in\{1,\ldots,t\}:~{}c^% {\prime}_{i}\leq f({B},{t})~{}\vee~{}c^{\prime}_{i}=\infty,</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.6d">∀ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ∀ italic_i ∈ { 1 , … , italic_t } : italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_f ( italic_B , italic_t ) ∨ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT 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="S4.Thmtheorem10.p1.9"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem10.p1.9.1">where</span></p> <ul class="ltx_itemize" id="S4.I2"> <li class="ltx_item" id="S4.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I2.i1.p1"> <p class="ltx_p" id="S4.I2.i1.p1.2"><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.2.1">in dimension </span><math alttext="t=1" class="ltx_Math" display="inline" id="S4.I2.i1.p1.1.m1.1"><semantics id="S4.I2.i1.p1.1.m1.1a"><mrow id="S4.I2.i1.p1.1.m1.1.1" xref="S4.I2.i1.p1.1.m1.1.1.cmml"><mi id="S4.I2.i1.p1.1.m1.1.1.2" xref="S4.I2.i1.p1.1.m1.1.1.2.cmml">t</mi><mo id="S4.I2.i1.p1.1.m1.1.1.1" xref="S4.I2.i1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.I2.i1.p1.1.m1.1.1.3" xref="S4.I2.i1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.1.m1.1b"><apply id="S4.I2.i1.p1.1.m1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.1.1"><eq id="S4.I2.i1.p1.1.m1.1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.1.1.1"></eq><ci id="S4.I2.i1.p1.1.m1.1.1.2.cmml" xref="S4.I2.i1.p1.1.m1.1.1.2">𝑡</ci><cn id="S4.I2.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I2.i1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.1.m1.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.1.m1.1d">italic_t = 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.2.2">: </span><math alttext="f({B},{1})=B+|V|," class="ltx_Math" display="inline" id="S4.I2.i1.p1.2.m2.4"><semantics id="S4.I2.i1.p1.2.m2.4a"><mrow id="S4.I2.i1.p1.2.m2.4.4.1" xref="S4.I2.i1.p1.2.m2.4.4.1.1.cmml"><mrow id="S4.I2.i1.p1.2.m2.4.4.1.1" xref="S4.I2.i1.p1.2.m2.4.4.1.1.cmml"><mrow id="S4.I2.i1.p1.2.m2.4.4.1.1.2" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2.cmml"><mi id="S4.I2.i1.p1.2.m2.4.4.1.1.2.2" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2.2.cmml">f</mi><mo id="S4.I2.i1.p1.2.m2.4.4.1.1.2.1" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S4.I2.i1.p1.2.m2.4.4.1.1.2.3.2" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2.3.1.cmml"><mo id="S4.I2.i1.p1.2.m2.4.4.1.1.2.3.2.1" stretchy="false" 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id="S4.I2.i1.p1.2.m2.4.4.1.1.3.3.2.2" stretchy="false" xref="S4.I2.i1.p1.2.m2.4.4.1.1.3.3.1.1.cmml">|</mo></mrow></mrow></mrow><mo id="S4.I2.i1.p1.2.m2.4.4.1.2" xref="S4.I2.i1.p1.2.m2.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.2.m2.4b"><apply id="S4.I2.i1.p1.2.m2.4.4.1.1.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1"><eq id="S4.I2.i1.p1.2.m2.4.4.1.1.1.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.1"></eq><apply id="S4.I2.i1.p1.2.m2.4.4.1.1.2.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2"><times id="S4.I2.i1.p1.2.m2.4.4.1.1.2.1.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2.1"></times><ci id="S4.I2.i1.p1.2.m2.4.4.1.1.2.2.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2.2">𝑓</ci><interval closure="open" id="S4.I2.i1.p1.2.m2.4.4.1.1.2.3.1.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.2.3.2"><ci id="S4.I2.i1.p1.2.m2.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1">𝐵</ci><cn id="S4.I2.i1.p1.2.m2.2.2.cmml" type="integer" xref="S4.I2.i1.p1.2.m2.2.2">1</cn></interval></apply><apply id="S4.I2.i1.p1.2.m2.4.4.1.1.3.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.3"><plus id="S4.I2.i1.p1.2.m2.4.4.1.1.3.1.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.3.1"></plus><ci id="S4.I2.i1.p1.2.m2.4.4.1.1.3.2.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.3.2">𝐵</ci><apply id="S4.I2.i1.p1.2.m2.4.4.1.1.3.3.1.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.3.3.2"><abs id="S4.I2.i1.p1.2.m2.4.4.1.1.3.3.1.1.cmml" xref="S4.I2.i1.p1.2.m2.4.4.1.1.3.3.2.1"></abs><ci id="S4.I2.i1.p1.2.m2.3.3.cmml" xref="S4.I2.i1.p1.2.m2.3.3">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.2.m2.4c">f({B},{1})=B+|V|,</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.2.m2.4d">italic_f ( italic_B , 1 ) = italic_B + | italic_V | ,</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.2.3"></span></p> </div> </li> <li class="ltx_item" id="S4.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I2.i2.p1"> <p class="ltx_p" id="S4.I2.i2.p1.2"><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.2.1">in dimension </span><math alttext="t+1" class="ltx_Math" display="inline" id="S4.I2.i2.p1.1.m1.1"><semantics id="S4.I2.i2.p1.1.m1.1a"><mrow id="S4.I2.i2.p1.1.m1.1.1" xref="S4.I2.i2.p1.1.m1.1.1.cmml"><mi id="S4.I2.i2.p1.1.m1.1.1.2" xref="S4.I2.i2.p1.1.m1.1.1.2.cmml">t</mi><mo id="S4.I2.i2.p1.1.m1.1.1.1" xref="S4.I2.i2.p1.1.m1.1.1.1.cmml">+</mo><mn id="S4.I2.i2.p1.1.m1.1.1.3" xref="S4.I2.i2.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.1.m1.1b"><apply id="S4.I2.i2.p1.1.m1.1.1.cmml" xref="S4.I2.i2.p1.1.m1.1.1"><plus id="S4.I2.i2.p1.1.m1.1.1.1.cmml" xref="S4.I2.i2.p1.1.m1.1.1.1"></plus><ci id="S4.I2.i2.p1.1.m1.1.1.2.cmml" xref="S4.I2.i2.p1.1.m1.1.1.2">𝑡</ci><cn id="S4.I2.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I2.i2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.1.m1.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.1.m1.1d">italic_t + 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.2.2"> (under the induction hypothesis for </span><math alttext="t" class="ltx_Math" display="inline" id="S4.I2.i2.p1.2.m2.1"><semantics id="S4.I2.i2.p1.2.m2.1a"><mi id="S4.I2.i2.p1.2.m2.1.1" xref="S4.I2.i2.p1.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.2.m2.1b"><ci id="S4.I2.i2.p1.2.m2.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.2.m2.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.2.3">):</span></p> <dl class="ltx_description" id="S4.I2.i2.I1"> <dt class="ltx_item" id="S4.I2.i2.I1.ix1"><span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I2.i2.I1.ix1.1.1.1">(a)</span></span></dt> <dd class="ltx_item"> <div class="ltx_para" id="S4.I2.i2.I1.ix1.p1"> <p class="ltx_p" id="S4.I2.i2.I1.ix1.p1.1"><math alttext="c^{\prime}_{i}\leq\max\{c^{\prime}_{min},B\}+1+f({0},{t})~{}\vee~{}c^{\prime}_% {i}=\infty" class="ltx_Math" display="inline" id="S4.I2.i2.I1.ix1.p1.1.m1.5"><semantics id="S4.I2.i2.I1.ix1.p1.1.m1.5a"><mrow id="S4.I2.i2.I1.ix1.p1.1.m1.5.5" xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5.cmml"><msubsup id="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3" xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3.cmml"><mi id="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3.2.2" xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3.2.2.cmml">c</mi><mi id="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3.3" xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3.3.cmml">i</mi><mo id="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3.2.3" xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5.3.2.3.cmml">′</mo></msubsup><mo id="S4.I2.i2.I1.ix1.p1.1.m1.5.5.4" xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5.4.cmml">≤</mo><mrow id="S4.I2.i2.I1.ix1.p1.1.m1.5.5.1" 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xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5"></share><infinity id="S4.I2.i2.I1.ix1.p1.1.m1.5.5.6.cmml" xref="S4.I2.i2.I1.ix1.p1.1.m1.5.5.6"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.I1.ix1.p1.1.m1.5c">c^{\prime}_{i}\leq\max\{c^{\prime}_{min},B\}+1+f({0},{t})~{}\vee~{}c^{\prime}_% {i}=\infty</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.I1.ix1.p1.1.m1.5d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ roman_max { italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_B } + 1 + italic_f ( 0 , italic_t ) ∨ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∞</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.I1.ix1.p1.1.1">,</span></p> </div> </dd> <dt class="ltx_item" id="S4.I2.i2.I1.ix2"><span class="ltx_tag 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id="S4.I2.i2.I1.ix2.p1.1.m1.4c">c^{\prime}_{min}\leq B+2^{t+1}\big{(}|V|\cdot(\log_{2}(|C^{\prime}_{\sigma_{0}% }|)+1)+1+f({0},{t})\big{)}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.I1.ix2.p1.1.m1.4d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ≤ italic_B + 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ( | italic_V | ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( | italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ) + 1 ) + 1 + italic_f ( 0 , italic_t ) )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.I1.ix2.p1.1.1">,</span></p> </div> </dd> <dt class="ltx_item" id="S4.I2.i2.I1.ix3"><span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I2.i2.I1.ix3.1.1.1">(c)</span></span></dt> <dd class="ltx_item"> <div 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id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.2.2.cmml" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.2.2">𝑓</ci><interval closure="open" id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.2.3.1.cmml" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.2.3.2"><cn id="S4.I2.i2.I1.ix3.p1.1.m1.1.1.cmml" type="integer" xref="S4.I2.i2.I1.ix3.p1.1.m1.1.1">0</cn><ci id="S4.I2.i2.I1.ix3.p1.1.m1.2.2.cmml" xref="S4.I2.i2.I1.ix3.p1.1.m1.2.2">𝑡</ci></interval></apply><ci id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.3.cmml" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.3">𝐵</ci><cn id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.4.cmml" type="integer" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.1.1.1.4">3</cn></apply><apply id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3.cmml" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3"><plus id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3.1.cmml" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3.1"></plus><ci id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3.2.cmml" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3.2">𝑡</ci><cn id="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3.3.cmml" type="integer" xref="S4.I2.i2.I1.ix3.p1.1.m1.4.4.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.I1.ix3.p1.1.m1.4c">|C^{\prime}_{\sigma_{0}}|\leq\big{(}f({0},{t})+B+3\big{)}^{t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.I1.ix3.p1.1.m1.4d">| italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ≤ ( italic_f ( 0 , italic_t ) + italic_B + 3 ) start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.I1.ix3.p1.1.1">,</span></p> </div> </dd> </dl> <p class="ltx_p" id="S4.I2.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.3.1">that is,</span></p> </div> <div class="ltx_para" id="S4.I2.i2.p2"> <p class="ltx_p ltx_align_center" id="S4.I2.i2.p2.1"><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p2.1.1"><math alttext="\begin{array}[]{ll}f({B},{t+1})=\big{(}B+2^{t+1}\big{(}|V|\cdot(\log_{2}(% \alpha)+1)+1+f({0},{t})\big{)}\big{)}+1+f({0},{t})&(d)\end{array}" class="ltx_Math" display="inline" id="S4.I2.i2.p2.1.1.m1.10"><semantics id="S4.I2.i2.p2.1.1.m1.10a"><mtable columnspacing="5pt" id="S4.I2.i2.p2.1.1.m1.10.10" xref="S4.I2.i2.p2.1.1.m1.10.10.cmml"><mtr id="S4.I2.i2.p2.1.1.m1.10.10a" xref="S4.I2.i2.p2.1.1.m1.10.10.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.I2.i2.p2.1.1.m1.10.10b" xref="S4.I2.i2.p2.1.1.m1.10.10.cmml"><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.cmml"><mrow id="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8" xref="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8.cmml"><mi id="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8.3" xref="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8.3.cmml">f</mi><mo id="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8.2" xref="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8.2.cmml">⁢</mo><mrow id="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8.1.1" xref="S4.I2.i2.p2.1.1.m1.8.8.8.8.8.8.1.2.cmml"><mo 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xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.cmml"><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.cmml"><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.2" maxsize="120%" minsize="120%" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.cmml">(</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.cmml"><mi id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.3" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.3.cmml">B</mi><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.2.cmml">+</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.cmml"><msup id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.cmml"><mn id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.2.cmml">2</mn><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3.cmml"><mi id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3.2.cmml">t</mi><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3.1.cmml">+</mo><mn id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3.3" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.3.3.3.cmml">1</mn></mrow></msup><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.cmml"><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.cmml"><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.cmml"><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.3.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.3.1.1.cmml">|</mo><mi id="S4.I2.i2.p2.1.1.m1.2.2.2.2.2.2" xref="S4.I2.i2.p2.1.1.m1.2.2.2.2.2.2.cmml">V</mi><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.3.2.2" rspace="0.055em" stretchy="false" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.2" rspace="0.222em" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.2.cmml">⋅</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><msub id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">log</mi><mn id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1a" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1" stretchy="false" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">(</mo><mi id="S4.I2.i2.p2.1.1.m1.3.3.3.3.3.3" xref="S4.I2.i2.p2.1.1.m1.3.3.3.3.3.3.cmml">α</mi><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2" stretchy="false" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.2.cmml">+</mo><mn id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.2.cmml">+</mo><mn id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.3" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.2a" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.2.cmml">+</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.cmml"><mi id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.2.cmml">f</mi><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.1" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.3.2" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.3.1.cmml"><mo id="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.3.2.1" stretchy="false" xref="S4.I2.i2.p2.1.1.m1.9.9.9.9.9.9.1.1.1.1.1.1.1.4.3.1.cmml">(</mo><mn id="S4.I2.i2.p2.1.1.m1.4.4.4.4.4.4" 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end_POSTSUPERSCRIPT ( | italic_V | ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_α ) + 1 ) + 1 + italic_f ( 0 , italic_t ) ) ) + 1 + italic_f ( 0 , italic_t ) end_CELL start_CELL ( italic_d ) end_CELL end_ROW end_ARRAY</annotation></semantics></math></span></p> </div> <div class="ltx_para" id="S4.I2.i2.p3"> <p class="ltx_p" id="S4.I2.i2.p3.1"><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p3.1.1">with </span><math alttext="\alpha=\big{(}f({0},{t})+B+3\big{)}^{t+1}" class="ltx_Math" display="inline" id="S4.I2.i2.p3.1.m1.3"><semantics id="S4.I2.i2.p3.1.m1.3a"><mrow id="S4.I2.i2.p3.1.m1.3.3" xref="S4.I2.i2.p3.1.m1.3.3.cmml"><mi id="S4.I2.i2.p3.1.m1.3.3.3" xref="S4.I2.i2.p3.1.m1.3.3.3.cmml">α</mi><mo id="S4.I2.i2.p3.1.m1.3.3.2" xref="S4.I2.i2.p3.1.m1.3.3.2.cmml">=</mo><msup id="S4.I2.i2.p3.1.m1.3.3.1" xref="S4.I2.i2.p3.1.m1.3.3.1.cmml"><mrow id="S4.I2.i2.p3.1.m1.3.3.1.1.1" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.cmml"><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.2" maxsize="120%" minsize="120%" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.cmml"><mrow id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.cmml"><mi id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.2" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.2.cmml">f</mi><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.1" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.2" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.1.cmml"><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.1.cmml">(</mo><mn id="S4.I2.i2.p3.1.m1.1.1" xref="S4.I2.i2.p3.1.m1.1.1.cmml">0</mn><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.2.2" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.1.cmml">,</mo><mi id="S4.I2.i2.p3.1.m1.2.2" xref="S4.I2.i2.p3.1.m1.2.2.cmml">t</mi><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.2.3" stretchy="false" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.1" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.1.cmml">+</mo><mi id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.3" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.3.cmml">B</mi><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.1a" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.1.cmml">+</mo><mn id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.4" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.4.cmml">3</mn></mrow><mo id="S4.I2.i2.p3.1.m1.3.3.1.1.1.3" maxsize="120%" minsize="120%" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow><mrow id="S4.I2.i2.p3.1.m1.3.3.1.3" xref="S4.I2.i2.p3.1.m1.3.3.1.3.cmml"><mi id="S4.I2.i2.p3.1.m1.3.3.1.3.2" xref="S4.I2.i2.p3.1.m1.3.3.1.3.2.cmml">t</mi><mo id="S4.I2.i2.p3.1.m1.3.3.1.3.1" xref="S4.I2.i2.p3.1.m1.3.3.1.3.1.cmml">+</mo><mn id="S4.I2.i2.p3.1.m1.3.3.1.3.3" xref="S4.I2.i2.p3.1.m1.3.3.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p3.1.m1.3b"><apply id="S4.I2.i2.p3.1.m1.3.3.cmml" xref="S4.I2.i2.p3.1.m1.3.3"><eq id="S4.I2.i2.p3.1.m1.3.3.2.cmml" xref="S4.I2.i2.p3.1.m1.3.3.2"></eq><ci id="S4.I2.i2.p3.1.m1.3.3.3.cmml" xref="S4.I2.i2.p3.1.m1.3.3.3">𝛼</ci><apply id="S4.I2.i2.p3.1.m1.3.3.1.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1"><csymbol cd="ambiguous" id="S4.I2.i2.p3.1.m1.3.3.1.2.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1">superscript</csymbol><apply id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1"><plus id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.1.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.1"></plus><apply id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2"><times id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.1.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.1"></times><ci id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.2.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.2">𝑓</ci><interval closure="open" id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.1.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.2.3.2"><cn id="S4.I2.i2.p3.1.m1.1.1.cmml" type="integer" xref="S4.I2.i2.p3.1.m1.1.1">0</cn><ci id="S4.I2.i2.p3.1.m1.2.2.cmml" xref="S4.I2.i2.p3.1.m1.2.2">𝑡</ci></interval></apply><ci id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.3.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.3">𝐵</ci><cn id="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.4.cmml" type="integer" xref="S4.I2.i2.p3.1.m1.3.3.1.1.1.1.4">3</cn></apply><apply id="S4.I2.i2.p3.1.m1.3.3.1.3.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.3"><plus id="S4.I2.i2.p3.1.m1.3.3.1.3.1.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.3.1"></plus><ci id="S4.I2.i2.p3.1.m1.3.3.1.3.2.cmml" xref="S4.I2.i2.p3.1.m1.3.3.1.3.2">𝑡</ci><cn id="S4.I2.i2.p3.1.m1.3.3.1.3.3.cmml" type="integer" xref="S4.I2.i2.p3.1.m1.3.3.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p3.1.m1.3c">\alpha=\big{(}f({0},{t})+B+3\big{)}^{t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p3.1.m1.3d">italic_α = ( italic_f ( 0 , italic_t ) + italic_B + 3 ) start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p3.1.2">.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S4.Thmtheorem10.p2"> <p class="ltx_p" id="S4.Thmtheorem10.p2.12"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem10.p2.12.1">We have to prove Inequation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E4" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>) holds, that is,</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx15"> <tbody id="S4.Ex8"><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({B},{t})\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot(B+3)" class="ltx_Math" display="inline" id="S4.Ex8.m1.6"><semantics id="S4.Ex8.m1.6a"><mrow id="S4.Ex8.m1.6.6" xref="S4.Ex8.m1.6.6.cmml"><mrow id="S4.Ex8.m1.6.6.3" xref="S4.Ex8.m1.6.6.3.cmml"><mi id="S4.Ex8.m1.6.6.3.2" xref="S4.Ex8.m1.6.6.3.2.cmml">f</mi><mo id="S4.Ex8.m1.6.6.3.1" xref="S4.Ex8.m1.6.6.3.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.3.3.2" xref="S4.Ex8.m1.6.6.3.3.1.cmml"><mo id="S4.Ex8.m1.6.6.3.3.2.1" stretchy="false" xref="S4.Ex8.m1.6.6.3.3.1.cmml">(</mo><mi id="S4.Ex8.m1.3.3" xref="S4.Ex8.m1.3.3.cmml">B</mi><mo id="S4.Ex8.m1.6.6.3.3.2.2" xref="S4.Ex8.m1.6.6.3.3.1.cmml">,</mo><mi id="S4.Ex8.m1.4.4" xref="S4.Ex8.m1.4.4.cmml">t</mi><mo id="S4.Ex8.m1.6.6.3.3.2.3" stretchy="false" xref="S4.Ex8.m1.6.6.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex8.m1.6.6.2" xref="S4.Ex8.m1.6.6.2.cmml">≤</mo><mrow id="S4.Ex8.m1.6.6.1" xref="S4.Ex8.m1.6.6.1.cmml"><msup id="S4.Ex8.m1.6.6.1.3" xref="S4.Ex8.m1.6.6.1.3.cmml"><mn id="S4.Ex8.m1.6.6.1.3.2" xref="S4.Ex8.m1.6.6.1.3.2.cmml">2</mn><mrow id="S4.Ex8.m1.1.1.1" xref="S4.Ex8.m1.1.1.1.cmml"><mi id="S4.Ex8.m1.1.1.1.3" mathvariant="normal" xref="S4.Ex8.m1.1.1.1.3.cmml">Θ</mi><mo id="S4.Ex8.m1.1.1.1.2" xref="S4.Ex8.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex8.m1.1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.1.1.cmml"><mo id="S4.Ex8.m1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex8.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex8.m1.1.1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex8.m1.1.1.1.1.1.1.2" xref="S4.Ex8.m1.1.1.1.1.1.1.2.cmml">t</mi><mn id="S4.Ex8.m1.1.1.1.1.1.1.3" xref="S4.Ex8.m1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S4.Ex8.m1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex8.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex8.m1.6.6.1.2" lspace="0.222em" rspace="0.222em" xref="S4.Ex8.m1.6.6.1.2.cmml">⋅</mo><msup id="S4.Ex8.m1.6.6.1.4" xref="S4.Ex8.m1.6.6.1.4.cmml"><mrow id="S4.Ex8.m1.6.6.1.4.2.2" xref="S4.Ex8.m1.6.6.1.4.2.1.cmml"><mo id="S4.Ex8.m1.6.6.1.4.2.2.1" stretchy="false" xref="S4.Ex8.m1.6.6.1.4.2.1.1.cmml">|</mo><mi id="S4.Ex8.m1.5.5" xref="S4.Ex8.m1.5.5.cmml">V</mi><mo id="S4.Ex8.m1.6.6.1.4.2.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex8.m1.6.6.1.4.2.1.1.cmml">|</mo></mrow><mrow id="S4.Ex8.m1.2.2.1" xref="S4.Ex8.m1.2.2.1.cmml"><mi id="S4.Ex8.m1.2.2.1.3" mathvariant="normal" xref="S4.Ex8.m1.2.2.1.3.cmml">Θ</mi><mo id="S4.Ex8.m1.2.2.1.2" xref="S4.Ex8.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex8.m1.2.2.1.4.2" xref="S4.Ex8.m1.2.2.1.cmml"><mo id="S4.Ex8.m1.2.2.1.4.2.1" stretchy="false" xref="S4.Ex8.m1.2.2.1.cmml">(</mo><mi id="S4.Ex8.m1.2.2.1.1" xref="S4.Ex8.m1.2.2.1.1.cmml">t</mi><mo id="S4.Ex8.m1.2.2.1.4.2.2" stretchy="false" xref="S4.Ex8.m1.2.2.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex8.m1.6.6.1.2a" rspace="0.222em" xref="S4.Ex8.m1.6.6.1.2.cmml">⋅</mo><mrow id="S4.Ex8.m1.6.6.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.1.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.2" stretchy="false" xref="S4.Ex8.m1.6.6.1.1.1.1.cmml">(</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.1.cmml"><mi id="S4.Ex8.m1.6.6.1.1.1.1.2" xref="S4.Ex8.m1.6.6.1.1.1.1.2.cmml">B</mi><mo id="S4.Ex8.m1.6.6.1.1.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.cmml">+</mo><mn id="S4.Ex8.m1.6.6.1.1.1.1.3" xref="S4.Ex8.m1.6.6.1.1.1.1.3.cmml">3</mn></mrow><mo id="S4.Ex8.m1.6.6.1.1.1.3" stretchy="false" xref="S4.Ex8.m1.6.6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex8.m1.6b"><apply id="S4.Ex8.m1.6.6.cmml" xref="S4.Ex8.m1.6.6"><leq id="S4.Ex8.m1.6.6.2.cmml" xref="S4.Ex8.m1.6.6.2"></leq><apply id="S4.Ex8.m1.6.6.3.cmml" xref="S4.Ex8.m1.6.6.3"><times id="S4.Ex8.m1.6.6.3.1.cmml" xref="S4.Ex8.m1.6.6.3.1"></times><ci id="S4.Ex8.m1.6.6.3.2.cmml" xref="S4.Ex8.m1.6.6.3.2">𝑓</ci><interval closure="open" id="S4.Ex8.m1.6.6.3.3.1.cmml" xref="S4.Ex8.m1.6.6.3.3.2"><ci id="S4.Ex8.m1.3.3.cmml" xref="S4.Ex8.m1.3.3">𝐵</ci><ci id="S4.Ex8.m1.4.4.cmml" xref="S4.Ex8.m1.4.4">𝑡</ci></interval></apply><apply id="S4.Ex8.m1.6.6.1.cmml" xref="S4.Ex8.m1.6.6.1"><ci id="S4.Ex8.m1.6.6.1.2.cmml" xref="S4.Ex8.m1.6.6.1.2">⋅</ci><apply id="S4.Ex8.m1.6.6.1.3.cmml" xref="S4.Ex8.m1.6.6.1.3"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.3.1.cmml" xref="S4.Ex8.m1.6.6.1.3">superscript</csymbol><cn id="S4.Ex8.m1.6.6.1.3.2.cmml" type="integer" xref="S4.Ex8.m1.6.6.1.3.2">2</cn><apply id="S4.Ex8.m1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1"><times id="S4.Ex8.m1.1.1.1.2.cmml" xref="S4.Ex8.m1.1.1.1.2"></times><ci id="S4.Ex8.m1.1.1.1.3.cmml" xref="S4.Ex8.m1.1.1.1.3">Θ</ci><apply id="S4.Ex8.m1.1.1.1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex8.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex8.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex8.m1.1.1.1.1.1.1.2">𝑡</ci><cn id="S4.Ex8.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Ex8.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply><apply id="S4.Ex8.m1.6.6.1.4.cmml" xref="S4.Ex8.m1.6.6.1.4"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.4.1.cmml" xref="S4.Ex8.m1.6.6.1.4">superscript</csymbol><apply id="S4.Ex8.m1.6.6.1.4.2.1.cmml" xref="S4.Ex8.m1.6.6.1.4.2.2"><abs id="S4.Ex8.m1.6.6.1.4.2.1.1.cmml" xref="S4.Ex8.m1.6.6.1.4.2.2.1"></abs><ci id="S4.Ex8.m1.5.5.cmml" xref="S4.Ex8.m1.5.5">𝑉</ci></apply><apply id="S4.Ex8.m1.2.2.1.cmml" xref="S4.Ex8.m1.2.2.1"><times id="S4.Ex8.m1.2.2.1.2.cmml" xref="S4.Ex8.m1.2.2.1.2"></times><ci id="S4.Ex8.m1.2.2.1.3.cmml" xref="S4.Ex8.m1.2.2.1.3">Θ</ci><ci id="S4.Ex8.m1.2.2.1.1.cmml" xref="S4.Ex8.m1.2.2.1.1">𝑡</ci></apply></apply><apply id="S4.Ex8.m1.6.6.1.1.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1"><plus id="S4.Ex8.m1.6.6.1.1.1.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1"></plus><ci id="S4.Ex8.m1.6.6.1.1.1.1.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.2">𝐵</ci><cn id="S4.Ex8.m1.6.6.1.1.1.1.3.cmml" type="integer" xref="S4.Ex8.m1.6.6.1.1.1.1.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.6c">\displaystyle f({B},{t})\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot(B+3)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.6d">italic_f ( italic_B , italic_t ) ≤ 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem10.p2.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem10.p2.6.6">for all <math alttext="t\geq 1" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.1.1.m1.1"><semantics id="S4.Thmtheorem10.p2.1.1.m1.1a"><mrow id="S4.Thmtheorem10.p2.1.1.m1.1.1" xref="S4.Thmtheorem10.p2.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem10.p2.1.1.m1.1.1.2" xref="S4.Thmtheorem10.p2.1.1.m1.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem10.p2.1.1.m1.1.1.1" xref="S4.Thmtheorem10.p2.1.1.m1.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem10.p2.1.1.m1.1.1.3" xref="S4.Thmtheorem10.p2.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.1.1.m1.1b"><apply id="S4.Thmtheorem10.p2.1.1.m1.1.1.cmml" xref="S4.Thmtheorem10.p2.1.1.m1.1.1"><geq id="S4.Thmtheorem10.p2.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem10.p2.1.1.m1.1.1.1"></geq><ci id="S4.Thmtheorem10.p2.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem10.p2.1.1.m1.1.1.2">𝑡</ci><cn id="S4.Thmtheorem10.p2.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem10.p2.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.1.1.m1.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.1.1.m1.1d">italic_t ≥ 1</annotation></semantics></math>. This is true when <math alttext="t=1" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.2.2.m2.1"><semantics id="S4.Thmtheorem10.p2.2.2.m2.1a"><mrow id="S4.Thmtheorem10.p2.2.2.m2.1.1" xref="S4.Thmtheorem10.p2.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem10.p2.2.2.m2.1.1.2" xref="S4.Thmtheorem10.p2.2.2.m2.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem10.p2.2.2.m2.1.1.1" xref="S4.Thmtheorem10.p2.2.2.m2.1.1.1.cmml">=</mo><mn id="S4.Thmtheorem10.p2.2.2.m2.1.1.3" xref="S4.Thmtheorem10.p2.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.2.2.m2.1b"><apply id="S4.Thmtheorem10.p2.2.2.m2.1.1.cmml" xref="S4.Thmtheorem10.p2.2.2.m2.1.1"><eq id="S4.Thmtheorem10.p2.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem10.p2.2.2.m2.1.1.1"></eq><ci id="S4.Thmtheorem10.p2.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem10.p2.2.2.m2.1.1.2">𝑡</ci><cn id="S4.Thmtheorem10.p2.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem10.p2.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.2.2.m2.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.2.2.m2.1d">italic_t = 1</annotation></semantics></math> by (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E6" title="In 4.1 Dimension One ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">6</span></a>). Let us suppose that it is true for <math alttext="t" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.3.3.m3.1"><semantics id="S4.Thmtheorem10.p2.3.3.m3.1a"><mi id="S4.Thmtheorem10.p2.3.3.m3.1.1" xref="S4.Thmtheorem10.p2.3.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.3.3.m3.1b"><ci id="S4.Thmtheorem10.p2.3.3.m3.1.1.cmml" xref="S4.Thmtheorem10.p2.3.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.3.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.3.3.m3.1d">italic_t</annotation></semantics></math>, and let us prove that it remains true for <math alttext="t+1" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.4.4.m4.1"><semantics id="S4.Thmtheorem10.p2.4.4.m4.1a"><mrow id="S4.Thmtheorem10.p2.4.4.m4.1.1" xref="S4.Thmtheorem10.p2.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem10.p2.4.4.m4.1.1.2" xref="S4.Thmtheorem10.p2.4.4.m4.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem10.p2.4.4.m4.1.1.1" xref="S4.Thmtheorem10.p2.4.4.m4.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem10.p2.4.4.m4.1.1.3" xref="S4.Thmtheorem10.p2.4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.4.4.m4.1b"><apply id="S4.Thmtheorem10.p2.4.4.m4.1.1.cmml" xref="S4.Thmtheorem10.p2.4.4.m4.1.1"><plus id="S4.Thmtheorem10.p2.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem10.p2.4.4.m4.1.1.1"></plus><ci id="S4.Thmtheorem10.p2.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem10.p2.4.4.m4.1.1.2">𝑡</ci><cn id="S4.Thmtheorem10.p2.4.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem10.p2.4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.4.4.m4.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.4.4.m4.1d">italic_t + 1</annotation></semantics></math>. Let us begin with the factor <math alttext="|V|\cdot(\log_{2}(\alpha)+1)+1+f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.5.5.m5.5"><semantics id="S4.Thmtheorem10.p2.5.5.m5.5a"><mrow id="S4.Thmtheorem10.p2.5.5.m5.5.5" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.cmml"><mrow id="S4.Thmtheorem10.p2.5.5.m5.5.5.1" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.cmml"><mrow id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.3.2" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.3.1.cmml"><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.3.2.1" stretchy="false" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem10.p2.5.5.m5.1.1" xref="S4.Thmtheorem10.p2.5.5.m5.1.1.cmml">V</mi><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.3.2.2" rspace="0.055em" stretchy="false" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.3.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.2" rspace="0.222em" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.2.cmml">⋅</mo><mrow id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1" 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id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.1.1.2.1" stretchy="false" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.1.2.cmml">(</mo><mi id="S4.Thmtheorem10.p2.5.5.m5.2.2" xref="S4.Thmtheorem10.p2.5.5.m5.2.2.cmml">α</mi><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.1.1.2.2" stretchy="false" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.2" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.2.cmml">+</mo><mn id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.3" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.3" stretchy="false" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.2" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.2.cmml">+</mo><mn id="S4.Thmtheorem10.p2.5.5.m5.5.5.3" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.3.cmml">1</mn><mo id="S4.Thmtheorem10.p2.5.5.m5.5.5.2a" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.2.cmml">+</mo><mrow 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xref="S4.Thmtheorem10.p2.5.5.m5.5.5.4.1"></times><ci id="S4.Thmtheorem10.p2.5.5.m5.5.5.4.2.cmml" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.4.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem10.p2.5.5.m5.5.5.4.3.1.cmml" xref="S4.Thmtheorem10.p2.5.5.m5.5.5.4.3.2"><cn id="S4.Thmtheorem10.p2.5.5.m5.3.3.cmml" type="integer" xref="S4.Thmtheorem10.p2.5.5.m5.3.3">0</cn><ci id="S4.Thmtheorem10.p2.5.5.m5.4.4.cmml" xref="S4.Thmtheorem10.p2.5.5.m5.4.4">𝑡</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.5.5.m5.5c">|V|\cdot(\log_{2}(\alpha)+1)+1+f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.5.5.m5.5d">| italic_V | ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_α ) + 1 ) + 1 + italic_f ( 0 , italic_t )</annotation></semantics></math> appearing in the bound on <math alttext="c^{\prime}_{min}" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.6.6.m6.1"><semantics id="S4.Thmtheorem10.p2.6.6.m6.1a"><msubsup id="S4.Thmtheorem10.p2.6.6.m6.1.1" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.cmml"><mi id="S4.Thmtheorem10.p2.6.6.m6.1.1.2.2" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.2.2.cmml">c</mi><mrow id="S4.Thmtheorem10.p2.6.6.m6.1.1.3" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.3.cmml"><mi id="S4.Thmtheorem10.p2.6.6.m6.1.1.3.2" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.3.2.cmml">m</mi><mo id="S4.Thmtheorem10.p2.6.6.m6.1.1.3.1" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem10.p2.6.6.m6.1.1.3.3" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.3.3.cmml">i</mi><mo id="S4.Thmtheorem10.p2.6.6.m6.1.1.3.1a" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem10.p2.6.6.m6.1.1.3.4" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.3.4.cmml">n</mi></mrow><mo id="S4.Thmtheorem10.p2.6.6.m6.1.1.2.3" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.6.6.m6.1b"><apply 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id="S4.Thmtheorem10.p2.6.6.m6.1.1.3.4.cmml" xref="S4.Thmtheorem10.p2.6.6.m6.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.6.6.m6.1c">c^{\prime}_{min}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.6.6.m6.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT</annotation></semantics></math>:</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex9"> <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="\begin{array}[]{lll}&|V|\cdot(\log_{2}(\alpha)+1)+1+f({0},{t})\\ &=|V|\cdot\big{(}(t+1)\cdot\log_{2}(f({0},{t})+B+3)+1\big{)}+1+f({0},{t})&% \mbox{as $\log_{2}(x^{k})=k\log_{2}(x)$}\\ &\leq|V|\cdot\big{(}(t+1)\cdot(f({0},{t})+B+3)+1\big{)}+1+f({0},{t})&\mbox{as % }\log_{2}(x)\leq x\\ &\leq(f({0},{t})+B+3)\cdot|V|\cdot(t+2)&\mbox{(e).}\end{array}" class="ltx_Math" display="block" id="S4.Ex9.m1.25"><semantics id="S4.Ex9.m1.25a"><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex9.m1.25.25" rowspacing="0pt" xref="S4.Ex9.m1.25.25.cmml"><mtr id="S4.Ex9.m1.25.25a" xref="S4.Ex9.m1.25.25.cmml"><mtd id="S4.Ex9.m1.25.25b" xref="S4.Ex9.m1.25.25.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex9.m1.25.25c" xref="S4.Ex9.m1.25.25.cmml"><mrow id="S4.Ex9.m1.6.6.6.5.5" xref="S4.Ex9.m1.6.6.6.5.5.cmml"><mrow id="S4.Ex9.m1.6.6.6.5.5.5" xref="S4.Ex9.m1.6.6.6.5.5.5.cmml"><mrow id="S4.Ex9.m1.6.6.6.5.5.5.3.2" xref="S4.Ex9.m1.6.6.6.5.5.5.3.1.cmml"><mo id="S4.Ex9.m1.6.6.6.5.5.5.3.2.1" stretchy="false" xref="S4.Ex9.m1.6.6.6.5.5.5.3.1.1.cmml">|</mo><mi id="S4.Ex9.m1.2.2.2.1.1.1" xref="S4.Ex9.m1.2.2.2.1.1.1.cmml">V</mi><mo id="S4.Ex9.m1.6.6.6.5.5.5.3.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex9.m1.6.6.6.5.5.5.3.1.1.cmml">|</mo></mrow><mo 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xref="S4.Ex9.m1.12.12.12.7.6.7.cmml">=</mo><mrow id="S4.Ex9.m1.12.12.12.7.6.6" xref="S4.Ex9.m1.12.12.12.7.6.6.cmml"><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1" xref="S4.Ex9.m1.12.12.12.7.6.6.1.cmml"><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.3.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.3.1.cmml"><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.3.2.1" stretchy="false" xref="S4.Ex9.m1.12.12.12.7.6.6.1.3.1.1.cmml">|</mo><mi id="S4.Ex9.m1.7.7.7.2.1.1" xref="S4.Ex9.m1.7.7.7.2.1.1.cmml">V</mi><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.3.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex9.m1.12.12.12.7.6.6.1.3.1.1.cmml">|</mo></mrow><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.2" rspace="0.222em" xref="S4.Ex9.m1.12.12.12.7.6.6.1.2.cmml">⋅</mo><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.cmml"><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.2" maxsize="120%" minsize="120%" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.cmml">(</mo><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.cmml"><mrow 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xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.4.cmml">⋅</mo><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.3.cmml"><msub id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.2.2.1.1" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.2.2.1.1.cmml"><mi id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.2.2.1.1.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.2.2.1.1.2.cmml">log</mi><mn id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.2.2.1.1.3" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.2.2.1.1.3.cmml">2</mn></msub><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2a" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.3.cmml">⁡</mo><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.3.cmml"><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.2" stretchy="false" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.3.cmml">(</mo><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.cmml"><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.cmml"><mi id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.2.cmml">f</mi><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.1" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.1.cmml">⁢</mo><mrow id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.1.cmml"><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.2.1" stretchy="false" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.1.cmml">(</mo><mn id="S4.Ex9.m1.8.8.8.3.2.2" xref="S4.Ex9.m1.8.8.8.3.2.2.cmml">0</mn><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.2.2" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.1.cmml">,</mo><mi id="S4.Ex9.m1.9.9.9.4.3.3" xref="S4.Ex9.m1.9.9.9.4.3.3.cmml">t</mi><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.2.3" stretchy="false" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.1" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.1.cmml">+</mo><mi id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.3" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.3.cmml">B</mi><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.1a" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.1.cmml">+</mo><mn id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.4" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.1.4.cmml">3</mn></mrow><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.2.2.3" stretchy="false" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.3.3.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.4" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.4.cmml">+</mo><mn id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.5" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.5.cmml">1</mn></mrow><mo id="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.3" maxsize="120%" minsize="120%" xref="S4.Ex9.m1.12.12.12.7.6.6.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.12.12.12.7.6.6.2" 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xref="S4.Ex9.m1.25.25.25.7.1"><mtext class="ltx_mathvariant_italic" id="S4.Ex9.m1.25.25.25.7.1.cmml" xref="S4.Ex9.m1.25.25.25.7.1">(e).</mtext></ci></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.25c">\begin{array}[]{lll}&|V|\cdot(\log_{2}(\alpha)+1)+1+f({0},{t})\\ &=|V|\cdot\big{(}(t+1)\cdot\log_{2}(f({0},{t})+B+3)+1\big{)}+1+f({0},{t})&% \mbox{as $\log_{2}(x^{k})=k\log_{2}(x)$}\\ &\leq|V|\cdot\big{(}(t+1)\cdot(f({0},{t})+B+3)+1\big{)}+1+f({0},{t})&\mbox{as % }\log_{2}(x)\leq x\\ &\leq(f({0},{t})+B+3)\cdot|V|\cdot(t+2)&\mbox{(e).}\end{array}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.25d">start_ARRAY start_ROW start_CELL end_CELL start_CELL | italic_V | ⋅ ( roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_α ) + 1 ) + 1 + italic_f ( 0 , italic_t ) end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = | italic_V | ⋅ ( ( italic_t + 1 ) ⋅ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_f ( 0 , italic_t ) + italic_B + 3 ) + 1 ) + 1 + italic_f ( 0 , italic_t ) end_CELL start_CELL as roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = italic_k roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ | italic_V | ⋅ ( ( italic_t + 1 ) ⋅ ( italic_f ( 0 , italic_t ) + italic_B + 3 ) + 1 ) + 1 + italic_f ( 0 , italic_t ) end_CELL start_CELL as roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) ≤ italic_x end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ ( italic_f ( 0 , italic_t ) + italic_B + 3 ) ⋅ | italic_V | ⋅ ( italic_t + 2 ) end_CELL start_CELL (e). end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem10.p2.7"><span class="ltx_text ltx_font_italic" 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B+2^{t+1}\Big{(}(f({0},{t})+B+3)\cdot|V|\cdot(t+2)+1+f({0},{t})\Big{)}&% \mbox{by (d) and (e)}\\ &\leq 2^{t+1}\big{(}f({0},{t})+B+3\big{)}\cdot|V|\cdot(t+3)&\mbox{(f).}\end{array}" class="ltx_Math" display="block" id="S4.Ex10.m1.13"><semantics id="S4.Ex10.m1.13a"><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex10.m1.13.13" rowspacing="0pt" xref="S4.Ex10.m1.13.13.cmml"><mtr id="S4.Ex10.m1.13.13a" xref="S4.Ex10.m1.13.13.cmml"><mtd id="S4.Ex10.m1.13.13b" xref="S4.Ex10.m1.13.13.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex10.m1.13.13c" xref="S4.Ex10.m1.13.13.cmml"><mrow id="S4.Ex10.m1.2.2.2.2.2" xref="S4.Ex10.m1.2.2.2.2.2.cmml"><mi id="S4.Ex10.m1.2.2.2.2.2.4" xref="S4.Ex10.m1.2.2.2.2.2.4.cmml">f</mi><mo id="S4.Ex10.m1.2.2.2.2.2.3" xref="S4.Ex10.m1.2.2.2.2.2.3.cmml">⁢</mo><mrow id="S4.Ex10.m1.2.2.2.2.2.2.1" xref="S4.Ex10.m1.2.2.2.2.2.2.2.cmml"><mo id="S4.Ex10.m1.2.2.2.2.2.2.1.2" stretchy="false" xref="S4.Ex10.m1.2.2.2.2.2.2.2.cmml">(</mo><mi 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xref="S4.Ex10.m1.8.8.8.6.6.8.cmml"></mi><mo id="S4.Ex10.m1.8.8.8.6.6.7" xref="S4.Ex10.m1.8.8.8.6.6.7.cmml">≤</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6" xref="S4.Ex10.m1.8.8.8.6.6.6.cmml"><mi id="S4.Ex10.m1.8.8.8.6.6.6.3" xref="S4.Ex10.m1.8.8.8.6.6.6.3.cmml">B</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.2" xref="S4.Ex10.m1.8.8.8.6.6.6.2.cmml">+</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.cmml"><msup id="S4.Ex10.m1.8.8.8.6.6.6.1.3" xref="S4.Ex10.m1.8.8.8.6.6.6.1.3.cmml"><mn id="S4.Ex10.m1.8.8.8.6.6.6.1.3.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.3.2.cmml">2</mn><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.3.3" xref="S4.Ex10.m1.8.8.8.6.6.6.1.3.3.cmml"><mi id="S4.Ex10.m1.8.8.8.6.6.6.1.3.3.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.3.3.2.cmml">t</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.3.3.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.3.3.1.cmml">+</mo><mn id="S4.Ex10.m1.8.8.8.6.6.6.1.3.3.3" xref="S4.Ex10.m1.8.8.8.6.6.6.1.3.3.3.cmml">1</mn></mrow></msup><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.2.cmml">⁢</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.cmml"><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.2" maxsize="160%" minsize="160%" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.cmml">(</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.cmml"><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.cmml"><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.2.cmml">f</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.1.cmml"><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.1.cmml">(</mo><mn id="S4.Ex10.m1.3.3.3.1.1.1" xref="S4.Ex10.m1.3.3.3.1.1.1.cmml">0</mn><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.2.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.1.cmml">,</mo><mi id="S4.Ex10.m1.4.4.4.2.2.2" xref="S4.Ex10.m1.4.4.4.2.2.2.cmml">t</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.2.3" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.3" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.3.cmml">B</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.1a" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.4" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.4.cmml">3</mn></mrow><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.3" rspace="0.222em" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.3.cmml">⋅</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.4.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.4.1.cmml"><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.4.2.1" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.4.1.1.cmml">|</mo><mi id="S4.Ex10.m1.5.5.5.3.3.3" xref="S4.Ex10.m1.5.5.5.3.3.3.cmml">V</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.4.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.4.1.1.cmml">|</mo></mrow><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.3a" rspace="0.222em" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.3.cmml">⋅</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.cmml"><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.2" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.cmml">(</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.cmml"><mi id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.2.cmml">t</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.1.cmml">+</mo><mn id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.3" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.3.cmml">2</mn></mrow><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.3" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.3" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.3.cmml">+</mo><mn id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.4" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.4.cmml">1</mn><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.3a" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.3.cmml">+</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.cmml"><mi id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.2.cmml">f</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.1" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.1.cmml">⁢</mo><mrow id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.1.cmml"><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.2.1" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.1.cmml">(</mo><mn id="S4.Ex10.m1.6.6.6.4.4.4" xref="S4.Ex10.m1.6.6.6.4.4.4.cmml">0</mn><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.2.2" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.1.cmml">,</mo><mi id="S4.Ex10.m1.7.7.7.5.5.5" xref="S4.Ex10.m1.7.7.7.5.5.5.cmml">t</mi><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.2.3" stretchy="false" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.5.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.3" maxsize="160%" minsize="160%" xref="S4.Ex10.m1.8.8.8.6.6.6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex10.m1.13.13h" xref="S4.Ex10.m1.13.13.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex10.m1.8.8.8.8.1" xref="S4.Ex10.m1.8.8.8.8.1a.cmml">by (d) and (e)</mtext></mtd></mtr><mtr id="S4.Ex10.m1.13.13i" xref="S4.Ex10.m1.13.13.cmml"><mtd id="S4.Ex10.m1.13.13j" xref="S4.Ex10.m1.13.13.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex10.m1.13.13k" xref="S4.Ex10.m1.13.13.cmml"><mrow id="S4.Ex10.m1.13.13.13.5.5" xref="S4.Ex10.m1.13.13.13.5.5.cmml"><mi id="S4.Ex10.m1.13.13.13.5.5.7" xref="S4.Ex10.m1.13.13.13.5.5.7.cmml"></mi><mo id="S4.Ex10.m1.13.13.13.5.5.6" xref="S4.Ex10.m1.13.13.13.5.5.6.cmml">≤</mo><mrow id="S4.Ex10.m1.13.13.13.5.5.5" xref="S4.Ex10.m1.13.13.13.5.5.5.cmml"><mrow 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xref="S4.Ex10.m1.13.13.13.5.5.5.4.2.1"></abs><ci id="S4.Ex10.m1.11.11.11.3.3.3.cmml" xref="S4.Ex10.m1.11.11.11.3.3.3">𝑉</ci></apply><apply id="S4.Ex10.m1.13.13.13.5.5.5.2.1.1.cmml" xref="S4.Ex10.m1.13.13.13.5.5.5.2.1"><plus id="S4.Ex10.m1.13.13.13.5.5.5.2.1.1.1.cmml" xref="S4.Ex10.m1.13.13.13.5.5.5.2.1.1.1"></plus><ci id="S4.Ex10.m1.13.13.13.5.5.5.2.1.1.2.cmml" xref="S4.Ex10.m1.13.13.13.5.5.5.2.1.1.2">𝑡</ci><cn id="S4.Ex10.m1.13.13.13.5.5.5.2.1.1.3.cmml" type="integer" xref="S4.Ex10.m1.13.13.13.5.5.5.2.1.1.3">3</cn></apply></apply></apply><ci id="S4.Ex10.m1.13.13.13.7.1a.cmml" xref="S4.Ex10.m1.13.13.13.7.1"><mtext class="ltx_mathvariant_italic" id="S4.Ex10.m1.13.13.13.7.1.cmml" xref="S4.Ex10.m1.13.13.13.7.1">(f).</mtext></ci></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex10.m1.13c">\begin{array}[]{lll}&f({B},{t+1})\\ &\leq B+2^{t+1}\Big{(}(f({0},{t})+B+3)\cdot|V|\cdot(t+2)+1+f({0},{t})\Big{)}&% \mbox{by (d) and (e)}\\ &\leq 2^{t+1}\big{(}f({0},{t})+B+3\big{)}\cdot|V|\cdot(t+3)&\mbox{(f).}\end{array}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m1.13d">start_ARRAY start_ROW start_CELL end_CELL start_CELL italic_f ( italic_B , italic_t + 1 ) end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ italic_B + 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ( ( italic_f ( 0 , italic_t ) + italic_B + 3 ) ⋅ | italic_V | ⋅ ( italic_t + 2 ) + 1 + italic_f ( 0 , italic_t ) ) end_CELL start_CELL by (d) and (e) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ( italic_f ( 0 , italic_t ) + italic_B + 3 ) ⋅ | italic_V | ⋅ ( italic_t + 3 ) end_CELL start_CELL (f). end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem10.p2.9"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem10.p2.9.2">It follows that <math alttext="f({B},{t+1})" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.8.1.m1.2"><semantics id="S4.Thmtheorem10.p2.8.1.m1.2a"><mrow id="S4.Thmtheorem10.p2.8.1.m1.2.2" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.cmml"><mi id="S4.Thmtheorem10.p2.8.1.m1.2.2.3" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.3.cmml">f</mi><mo id="S4.Thmtheorem10.p2.8.1.m1.2.2.2" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.2.cmml"><mo id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.2.cmml">(</mo><mi id="S4.Thmtheorem10.p2.8.1.m1.1.1" xref="S4.Thmtheorem10.p2.8.1.m1.1.1.cmml">B</mi><mo id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.3" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.2.cmml">,</mo><mrow id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.2" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.1" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.3" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.4" stretchy="false" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.8.1.m1.2b"><apply id="S4.Thmtheorem10.p2.8.1.m1.2.2.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.2.2"><times id="S4.Thmtheorem10.p2.8.1.m1.2.2.2.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.2"></times><ci id="S4.Thmtheorem10.p2.8.1.m1.2.2.3.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.3">𝑓</ci><interval closure="open" id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.2.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1"><ci id="S4.Thmtheorem10.p2.8.1.m1.1.1.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.1.1">𝐵</ci><apply id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1"><plus id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.1"></plus><ci id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.2">𝑡</ci><cn id="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem10.p2.8.1.m1.2.2.1.1.1.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.8.1.m1.2c">f({B},{t+1})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.8.1.m1.2d">italic_f ( italic_B , italic_t + 1 )</annotation></semantics></math> can be computed thanks to <math alttext="f({0},{t})" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.9.2.m2.2"><semantics id="S4.Thmtheorem10.p2.9.2.m2.2a"><mrow id="S4.Thmtheorem10.p2.9.2.m2.2.3" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.cmml"><mi id="S4.Thmtheorem10.p2.9.2.m2.2.3.2" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.2.cmml">f</mi><mo id="S4.Thmtheorem10.p2.9.2.m2.2.3.1" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem10.p2.9.2.m2.2.3.3.2" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.3.1.cmml"><mo id="S4.Thmtheorem10.p2.9.2.m2.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem10.p2.9.2.m2.1.1" xref="S4.Thmtheorem10.p2.9.2.m2.1.1.cmml">0</mn><mo id="S4.Thmtheorem10.p2.9.2.m2.2.3.3.2.2" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem10.p2.9.2.m2.2.2" xref="S4.Thmtheorem10.p2.9.2.m2.2.2.cmml">t</mi><mo id="S4.Thmtheorem10.p2.9.2.m2.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.9.2.m2.2b"><apply id="S4.Thmtheorem10.p2.9.2.m2.2.3.cmml" xref="S4.Thmtheorem10.p2.9.2.m2.2.3"><times id="S4.Thmtheorem10.p2.9.2.m2.2.3.1.cmml" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.1"></times><ci id="S4.Thmtheorem10.p2.9.2.m2.2.3.2.cmml" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.2">𝑓</ci><interval closure="open" id="S4.Thmtheorem10.p2.9.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem10.p2.9.2.m2.2.3.3.2"><cn id="S4.Thmtheorem10.p2.9.2.m2.1.1.cmml" type="integer" xref="S4.Thmtheorem10.p2.9.2.m2.1.1">0</cn><ci id="S4.Thmtheorem10.p2.9.2.m2.2.2.cmml" xref="S4.Thmtheorem10.p2.9.2.m2.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.9.2.m2.2c">f({0},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.9.2.m2.2d">italic_f ( 0 , italic_t )</annotation></semantics></math> bounded by induction:</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex11"> <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="\begin{array}[]{lll}&f({B},{t+1})\leq\\ &\leq 2^{t+1}\big{(}3\cdot 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}+B+3\big{)}% \cdot|V|\cdot(t+3)&\mbox{by (f) and (\ref{eq:O()})}\\ &\leq 2^{\Theta((t+1)^{2})}\cdot|V|^{\Theta(t+1)}\cdot(B+3)&\mbox{as }3\cdot t% +3\leq 2^{t+3}.\end{array}" class="ltx_Math" display="block" id="S4.Ex11.m1.13"><semantics id="S4.Ex11.m1.13a"><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex11.m1.13.13" rowspacing="0pt" xref="S4.Ex11.m1.13.13.cmml"><mtr id="S4.Ex11.m1.13.13a" xref="S4.Ex11.m1.13.13.cmml"><mtd id="S4.Ex11.m1.13.13b" xref="S4.Ex11.m1.13.13.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex11.m1.13.13c" xref="S4.Ex11.m1.13.13.cmml"><mrow id="S4.Ex11.m1.2.2.2.2.2" xref="S4.Ex11.m1.2.2.2.2.2.cmml"><mrow id="S4.Ex11.m1.2.2.2.2.2.2" xref="S4.Ex11.m1.2.2.2.2.2.2.cmml"><mi id="S4.Ex11.m1.2.2.2.2.2.2.3" xref="S4.Ex11.m1.2.2.2.2.2.2.3.cmml">f</mi><mo id="S4.Ex11.m1.2.2.2.2.2.2.2" xref="S4.Ex11.m1.2.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.2.2.2.2.2.2.1.1" xref="S4.Ex11.m1.2.2.2.2.2.2.1.2.cmml"><mo id="S4.Ex11.m1.2.2.2.2.2.2.1.1.2" stretchy="false" xref="S4.Ex11.m1.2.2.2.2.2.2.1.2.cmml">(</mo><mi id="S4.Ex11.m1.1.1.1.1.1.1" xref="S4.Ex11.m1.1.1.1.1.1.1.cmml">B</mi><mo id="S4.Ex11.m1.2.2.2.2.2.2.1.1.3" xref="S4.Ex11.m1.2.2.2.2.2.2.1.2.cmml">,</mo><mrow id="S4.Ex11.m1.2.2.2.2.2.2.1.1.1" xref="S4.Ex11.m1.2.2.2.2.2.2.1.1.1.cmml"><mi id="S4.Ex11.m1.2.2.2.2.2.2.1.1.1.2" xref="S4.Ex11.m1.2.2.2.2.2.2.1.1.1.2.cmml">t</mi><mo id="S4.Ex11.m1.2.2.2.2.2.2.1.1.1.1" xref="S4.Ex11.m1.2.2.2.2.2.2.1.1.1.1.cmml">+</mo><mn id="S4.Ex11.m1.2.2.2.2.2.2.1.1.1.3" xref="S4.Ex11.m1.2.2.2.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Ex11.m1.2.2.2.2.2.2.1.1.4" stretchy="false" xref="S4.Ex11.m1.2.2.2.2.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.2.2.2.2.2.3" xref="S4.Ex11.m1.2.2.2.2.2.3.cmml">≤</mo><mi id="S4.Ex11.m1.2.2.2.2.2.4" xref="S4.Ex11.m1.2.2.2.2.2.4.cmml"></mi></mrow></mtd><mtd id="S4.Ex11.m1.13.13d" xref="S4.Ex11.m1.13.13.cmml"></mtd></mtr><mtr id="S4.Ex11.m1.13.13e" xref="S4.Ex11.m1.13.13.cmml"><mtd id="S4.Ex11.m1.13.13f" xref="S4.Ex11.m1.13.13.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex11.m1.13.13g" xref="S4.Ex11.m1.13.13.cmml"><mrow id="S4.Ex11.m1.8.8.8.6.6" xref="S4.Ex11.m1.8.8.8.6.6.cmml"><mi id="S4.Ex11.m1.8.8.8.6.6.8" xref="S4.Ex11.m1.8.8.8.6.6.8.cmml"></mi><mo id="S4.Ex11.m1.8.8.8.6.6.7" xref="S4.Ex11.m1.8.8.8.6.6.7.cmml">≤</mo><mrow id="S4.Ex11.m1.8.8.8.6.6.6" xref="S4.Ex11.m1.8.8.8.6.6.6.cmml"><mrow id="S4.Ex11.m1.7.7.7.5.5.5.1" xref="S4.Ex11.m1.7.7.7.5.5.5.1.cmml"><msup id="S4.Ex11.m1.7.7.7.5.5.5.1.3" xref="S4.Ex11.m1.7.7.7.5.5.5.1.3.cmml"><mn id="S4.Ex11.m1.7.7.7.5.5.5.1.3.2" xref="S4.Ex11.m1.7.7.7.5.5.5.1.3.2.cmml">2</mn><mrow id="S4.Ex11.m1.7.7.7.5.5.5.1.3.3" xref="S4.Ex11.m1.7.7.7.5.5.5.1.3.3.cmml"><mi id="S4.Ex11.m1.7.7.7.5.5.5.1.3.3.2" xref="S4.Ex11.m1.7.7.7.5.5.5.1.3.3.2.cmml">t</mi><mo id="S4.Ex11.m1.7.7.7.5.5.5.1.3.3.1" xref="S4.Ex11.m1.7.7.7.5.5.5.1.3.3.1.cmml">+</mo><mn id="S4.Ex11.m1.7.7.7.5.5.5.1.3.3.3" xref="S4.Ex11.m1.7.7.7.5.5.5.1.3.3.3.cmml">1</mn></mrow></msup><mo id="S4.Ex11.m1.7.7.7.5.5.5.1.2" xref="S4.Ex11.m1.7.7.7.5.5.5.1.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1" xref="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.cmml"><mo id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.2" maxsize="120%" minsize="120%" xref="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1" xref="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.cmml"><mrow id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2" xref="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.cmml"><mn id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.2" xref="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.2.cmml">3</mn><mo id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.1.cmml">⋅</mo><msup id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.3" xref="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.3.cmml"><mn id="S4.Ex11.m1.7.7.7.5.5.5.1.1.1.1.2.3.2" 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stretchy="false" xref="S4.Ex11.m1.8.8.8.6.6.6.4.1.1.cmml">|</mo><mi id="S4.Ex11.m1.6.6.6.4.4.4" xref="S4.Ex11.m1.6.6.6.4.4.4.cmml">V</mi><mo id="S4.Ex11.m1.8.8.8.6.6.6.4.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex11.m1.8.8.8.6.6.6.4.1.1.cmml">|</mo></mrow><mo id="S4.Ex11.m1.8.8.8.6.6.6.3a" rspace="0.222em" xref="S4.Ex11.m1.8.8.8.6.6.6.3.cmml">⋅</mo><mrow id="S4.Ex11.m1.8.8.8.6.6.6.2.1" xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.cmml"><mo id="S4.Ex11.m1.8.8.8.6.6.6.2.1.2" stretchy="false" xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.8.8.8.6.6.6.2.1.1" xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.cmml"><mi id="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.2" xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.2.cmml">t</mi><mo id="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.1" xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.1.cmml">+</mo><mn id="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.3" xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.3.cmml">3</mn></mrow><mo id="S4.Ex11.m1.8.8.8.6.6.6.2.1.3" stretchy="false" 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xref="S4.Ex11.m1.10.10.10.2.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.10.10.10.2.2.2.1.1.1" xref="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.cmml"><mo id="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.2" stretchy="false" xref="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1" xref="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.cmml"><mi id="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.2" xref="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.2.cmml">t</mi><mo id="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.1" xref="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.1.cmml">+</mo><mn id="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.3" xref="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.3" stretchy="false" xref="S4.Ex11.m1.10.10.10.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex11.m1.12.12.12.4.4.4.2a" rspace="0.222em" xref="S4.Ex11.m1.12.12.12.4.4.4.2.cmml">⋅</mo><mrow id="S4.Ex11.m1.12.12.12.4.4.4.1.1" xref="S4.Ex11.m1.12.12.12.4.4.4.1.1.1.cmml"><mo 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xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.2">𝑡</ci><cn id="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.3.cmml" type="integer" xref="S4.Ex11.m1.8.8.8.6.6.6.2.1.1.3">3</cn></apply></apply></apply><ci id="S4.Ex11.m1.8.8.8.8.1f.cmml" xref="S4.Ex11.m1.8.8.8.8.1"><mrow id="S4.Ex11.m1.8.8.8.8.1.cmml" xref="S4.Ex11.m1.8.8.8.8.1"><mtext class="ltx_mathvariant_italic" id="S4.Ex11.m1.8.8.8.8.1a.cmml" xref="S4.Ex11.m1.8.8.8.8.1">by (f) and (</mtext><mtext class="ltx_mathvariant_italic" id="S4.Ex11.m1.8.8.8.8.1b.cmml" xref="S4.Ex11.m1.8.8.8.8.1"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2308.09443v2#S4.E2" title="In Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2</span></a></mtext><mtext class="ltx_mathvariant_italic" id="S4.Ex11.m1.8.8.8.8.1e.cmml" xref="S4.Ex11.m1.8.8.8.8.1">)</mtext></mrow></ci></matrixrow><matrixrow id="S4.Ex11.m1.13.13i.cmml" xref="S4.Ex11.m1.13.13"><cerror 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id="S4.Ex11.m1.13.13.13.5.1.1.1.3.1.cmml" xref="S4.Ex11.m1.13.13.13.5.1.1.1.3">superscript</csymbol><cn id="S4.Ex11.m1.13.13.13.5.1.1.1.3.2.cmml" type="integer" xref="S4.Ex11.m1.13.13.13.5.1.1.1.3.2">2</cn><apply id="S4.Ex11.m1.13.13.13.5.1.1.1.3.3.cmml" xref="S4.Ex11.m1.13.13.13.5.1.1.1.3.3"><plus id="S4.Ex11.m1.13.13.13.5.1.1.1.3.3.1.cmml" xref="S4.Ex11.m1.13.13.13.5.1.1.1.3.3.1"></plus><ci id="S4.Ex11.m1.13.13.13.5.1.1.1.3.3.2.cmml" xref="S4.Ex11.m1.13.13.13.5.1.1.1.3.3.2">𝑡</ci><cn id="S4.Ex11.m1.13.13.13.5.1.1.1.3.3.3.cmml" type="integer" xref="S4.Ex11.m1.13.13.13.5.1.1.1.3.3.3">3</cn></apply></apply></apply></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.13c">\begin{array}[]{lll}&f({B},{t+1})\leq\\ &\leq 2^{t+1}\big{(}3\cdot 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}+B+3\big{)}% \cdot|V|\cdot(t+3)&\mbox{by (f) and (\ref{eq:O()})}\\ &\leq 2^{\Theta((t+1)^{2})}\cdot|V|^{\Theta(t+1)}\cdot(B+3)&\mbox{as }3\cdot t% +3\leq 2^{t+3}.\end{array}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.13d">start_ARRAY start_ROW start_CELL end_CELL start_CELL italic_f ( italic_B , italic_t + 1 ) ≤ end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ 2 start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ( 3 ⋅ 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT + italic_B + 3 ) ⋅ | italic_V | ⋅ ( italic_t + 3 ) end_CELL start_CELL by (f) and ( ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ 2 start_POSTSUPERSCRIPT roman_Θ ( ( italic_t + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t + 1 ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 ) end_CELL start_CELL as 3 ⋅ italic_t + 3 ≤ 2 start_POSTSUPERSCRIPT italic_t + 3 end_POSTSUPERSCRIPT . end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem10.p2.11"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem10.p2.11.2">Therefore, Inequation (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E4" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>) holds for <math alttext="f({B},{t+1})" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.10.1.m1.2"><semantics id="S4.Thmtheorem10.p2.10.1.m1.2a"><mrow id="S4.Thmtheorem10.p2.10.1.m1.2.2" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.cmml"><mi id="S4.Thmtheorem10.p2.10.1.m1.2.2.3" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.3.cmml">f</mi><mo id="S4.Thmtheorem10.p2.10.1.m1.2.2.2" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.2.cmml"><mo id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.2.cmml">(</mo><mi id="S4.Thmtheorem10.p2.10.1.m1.1.1" xref="S4.Thmtheorem10.p2.10.1.m1.1.1.cmml">B</mi><mo id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.3" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.2.cmml">,</mo><mrow id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.2" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.1" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.1.cmml">+</mo><mn id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.3" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.4" stretchy="false" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.10.1.m1.2b"><apply id="S4.Thmtheorem10.p2.10.1.m1.2.2.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.2.2"><times id="S4.Thmtheorem10.p2.10.1.m1.2.2.2.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.2"></times><ci id="S4.Thmtheorem10.p2.10.1.m1.2.2.3.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.3">𝑓</ci><interval closure="open" id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.2.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1"><ci id="S4.Thmtheorem10.p2.10.1.m1.1.1.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.1.1">𝐵</ci><apply id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1"><plus id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.1"></plus><ci id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.2">𝑡</ci><cn id="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem10.p2.10.1.m1.2.2.1.1.1.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.10.1.m1.2c">f({B},{t+1})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.10.1.m1.2d">italic_f ( italic_B , italic_t + 1 )</annotation></semantics></math>. This completes the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> by induction on <math alttext="t" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p2.11.2.m2.1"><semantics id="S4.Thmtheorem10.p2.11.2.m2.1a"><mi id="S4.Thmtheorem10.p2.11.2.m2.1.1" xref="S4.Thmtheorem10.p2.11.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p2.11.2.m2.1b"><ci id="S4.Thmtheorem10.p2.11.2.m2.1.1.cmml" xref="S4.Thmtheorem10.p2.11.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p2.11.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p2.11.2.m2.1d">italic_t</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS4.p2"> <p class="ltx_p" id="S4.SS4.p2.3">Thanks to Lemmas <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem2" title="Lemma 2.2 ‣ Reduction to binary arenas. ‣ 2.3 Tools ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.2</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a> and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>, we easily get a bound for <math alttext="|C_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S4.SS4.p2.1.m1.1"><semantics id="S4.SS4.p2.1.m1.1a"><mrow id="S4.SS4.p2.1.m1.1.1.1" xref="S4.SS4.p2.1.m1.1.1.2.cmml"><mo id="S4.SS4.p2.1.m1.1.1.1.2" stretchy="false" xref="S4.SS4.p2.1.m1.1.1.2.1.cmml">|</mo><msub id="S4.SS4.p2.1.m1.1.1.1.1" xref="S4.SS4.p2.1.m1.1.1.1.1.cmml"><mi id="S4.SS4.p2.1.m1.1.1.1.1.2" xref="S4.SS4.p2.1.m1.1.1.1.1.2.cmml">C</mi><msub id="S4.SS4.p2.1.m1.1.1.1.1.3" xref="S4.SS4.p2.1.m1.1.1.1.1.3.cmml"><mi id="S4.SS4.p2.1.m1.1.1.1.1.3.2" xref="S4.SS4.p2.1.m1.1.1.1.1.3.2.cmml">σ</mi><mn id="S4.SS4.p2.1.m1.1.1.1.1.3.3" xref="S4.SS4.p2.1.m1.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.SS4.p2.1.m1.1.1.1.3" stretchy="false" xref="S4.SS4.p2.1.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.1.m1.1b"><apply id="S4.SS4.p2.1.m1.1.1.2.cmml" xref="S4.SS4.p2.1.m1.1.1.1"><abs id="S4.SS4.p2.1.m1.1.1.2.1.cmml" xref="S4.SS4.p2.1.m1.1.1.1.2"></abs><apply id="S4.SS4.p2.1.m1.1.1.1.1.cmml" xref="S4.SS4.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.1.m1.1.1.1.1.1.cmml" xref="S4.SS4.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.SS4.p2.1.m1.1.1.1.1.2.cmml" xref="S4.SS4.p2.1.m1.1.1.1.1.2">𝐶</ci><apply id="S4.SS4.p2.1.m1.1.1.1.1.3.cmml" xref="S4.SS4.p2.1.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS4.p2.1.m1.1.1.1.1.3.1.cmml" xref="S4.SS4.p2.1.m1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS4.p2.1.m1.1.1.1.1.3.2.cmml" xref="S4.SS4.p2.1.m1.1.1.1.1.3.2">𝜎</ci><cn id="S4.SS4.p2.1.m1.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS4.p2.1.m1.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.1.m1.1c">|C_{\sigma_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.1.m1.1d">| italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math> depending on <math alttext="G" class="ltx_Math" display="inline" id="S4.SS4.p2.2.m2.1"><semantics id="S4.SS4.p2.2.m2.1a"><mi id="S4.SS4.p2.2.m2.1.1" xref="S4.SS4.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.2.m2.1b"><ci id="S4.SS4.p2.2.m2.1.1.cmml" xref="S4.SS4.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.2.m2.1d">italic_G</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S4.SS4.p2.3.m3.1"><semantics id="S4.SS4.p2.3.m3.1a"><mi id="S4.SS4.p2.3.m3.1.1" xref="S4.SS4.p2.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.3.m3.1b"><ci id="S4.SS4.p2.3.m3.1.1.cmml" xref="S4.SS4.p2.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.3.m3.1d">italic_B</annotation></semantics></math>, as stated in the next corollary.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S4.Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem11.1.1.1">Corollary 4.11</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem11.p1"> <p class="ltx_p" id="S4.Thmtheorem11.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem11.p1.4.4">For all games <math alttext="G\in\textsf{Games}_{t}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.1.1.m1.1"><semantics id="S4.Thmtheorem11.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem11.p1.1.1.m1.1.1" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem11.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem11.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem11.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2a.cmml">Games</mtext><mi id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.1.1.m1.1b"><apply id="S4.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1"><in id="S4.Thmtheorem11.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem11.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.2">𝐺</ci><apply id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2a.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2">Games</mtext></ci><ci id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.1.1.m1.1c">G\in\textsf{Games}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.1.1.m1.1d">italic_G ∈ Games start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and for all Pareto-bounded<span class="ltx_note ltx_role_footnote" id="footnote10"><sup class="ltx_note_mark">10</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">10</sup><span class="ltx_tag ltx_tag_note"><span class="ltx_text ltx_font_upright" id="footnote10.1.1.1">10</span></span><span class="ltx_text ltx_font_upright" id="footnote10.5">The notion of Pareto-bounded solution has been defined below Theorem </span><a class="ltx_ref ltx_font_upright" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a><span class="ltx_text ltx_font_upright" id="footnote10.6">.</span></span></span></span> solutions <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.2.2.m2.2"><semantics id="S4.Thmtheorem11.p1.2.2.m2.2a"><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.cmml"><msub id="S4.Thmtheorem11.p1.2.2.m2.2.3.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.2.cmml"><mi id="S4.Thmtheorem11.p1.2.2.m2.2.3.2.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.2.2.cmml">σ</mi><mn id="S4.Thmtheorem11.p1.2.2.m2.2.3.2.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem11.p1.2.2.m2.2.3.1" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.3.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem11.p1.2.2.m2.2.3.3.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.2a.cmml">SPS</mtext><mo id="S4.Thmtheorem11.p1.2.2.m2.2.3.3.1" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem11.p1.2.2.m2.1.1" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.cmml">G</mi><mo id="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.2.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.cmml">B</mi><mo id="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.2.2.m2.2b"><apply 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xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.3.3.3.2"><ci id="S4.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1">𝐺</ci><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.2.2.m2.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.2.2.m2.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math>, the size <math alttext="|C_{\sigma_{0}}|" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.3.3.m3.1"><semantics id="S4.Thmtheorem11.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem11.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem11.p1.3.3.m3.1.1.2.cmml"><mo id="S4.Thmtheorem11.p1.3.3.m3.1.1.1.2" stretchy="false" 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xref="S4.Thmtheorem11.p1.4.4.m4.5.5.1.1.1.1.3">𝑊</ci></apply><apply id="S4.Thmtheorem11.p1.4.4.m4.2.2.1.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.2.2.1"><times id="S4.Thmtheorem11.p1.4.4.m4.2.2.1.2.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.2.2.1.2"></times><ci id="S4.Thmtheorem11.p1.4.4.m4.2.2.1.3.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.2.2.1.3">Θ</ci><apply id="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1.1.2">𝑡</ci><cn id="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem11.p1.4.4.m4.2.2.1.1.1.1.3">2</cn></apply></apply></apply><apply id="S4.Thmtheorem11.p1.4.4.m4.6.6.2.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.6.6.2"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.4.4.m4.6.6.2.2.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.6.6.2">superscript</csymbol><apply id="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1.1.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1"><plus id="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1.1.1"></plus><ci id="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1.1.2">𝐵</ci><cn id="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem11.p1.4.4.m4.6.6.2.1.1.1.3">3</cn></apply><apply id="S4.Thmtheorem11.p1.4.4.m4.3.3.1.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.3.3.1"><times id="S4.Thmtheorem11.p1.4.4.m4.3.3.1.2.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.3.3.1.2"></times><ci id="S4.Thmtheorem11.p1.4.4.m4.3.3.1.3.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.3.3.1.3">Θ</ci><ci id="S4.Thmtheorem11.p1.4.4.m4.3.3.1.1.cmml" xref="S4.Thmtheorem11.p1.4.4.m4.3.3.1.1">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.4.4.m4.6c">2^{\Theta(t^{3})}\cdot(|V|\cdot W)^{\Theta(t^{2})}\cdot(B+3)^{\Theta(t)}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.4.4.m4.6d">2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ ( | italic_V | ⋅ italic_W ) start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 ) start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem12.1.1.1">Proof 4.12</span></span></h6> <div class="ltx_para" id="S4.Thmtheorem12.p1"> <p class="ltx_p" id="S4.Thmtheorem12.p1.1"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem12.p1.1.1">Suppose that <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem12.p1.1.1.m1.1"><semantics id="S4.Thmtheorem12.p1.1.1.m1.1a"><mi id="S4.Thmtheorem12.p1.1.1.m1.1.1" xref="S4.Thmtheorem12.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem12.p1.1.1.m1.1b"><ci id="S4.Thmtheorem12.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem12.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem12.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem12.p1.1.1.m1.1d">italic_G</annotation></semantics></math> is binary. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex12"> <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="|C_{\sigma_{0}}|\leq\big{(}f({0},{t})+B+3\big{)}^{t+1}" class="ltx_Math" display="block" id="S4.Ex12.m1.4"><semantics id="S4.Ex12.m1.4a"><mrow id="S4.Ex12.m1.4.4" xref="S4.Ex12.m1.4.4.cmml"><mrow id="S4.Ex12.m1.3.3.1.1" xref="S4.Ex12.m1.3.3.1.2.cmml"><mo id="S4.Ex12.m1.3.3.1.1.2" stretchy="false" xref="S4.Ex12.m1.3.3.1.2.1.cmml">|</mo><msub id="S4.Ex12.m1.3.3.1.1.1" xref="S4.Ex12.m1.3.3.1.1.1.cmml"><mi id="S4.Ex12.m1.3.3.1.1.1.2" xref="S4.Ex12.m1.3.3.1.1.1.2.cmml">C</mi><msub id="S4.Ex12.m1.3.3.1.1.1.3" xref="S4.Ex12.m1.3.3.1.1.1.3.cmml"><mi id="S4.Ex12.m1.3.3.1.1.1.3.2" xref="S4.Ex12.m1.3.3.1.1.1.3.2.cmml">σ</mi><mn id="S4.Ex12.m1.3.3.1.1.1.3.3" xref="S4.Ex12.m1.3.3.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Ex12.m1.3.3.1.1.3" stretchy="false" xref="S4.Ex12.m1.3.3.1.2.1.cmml">|</mo></mrow><mo id="S4.Ex12.m1.4.4.3" xref="S4.Ex12.m1.4.4.3.cmml">≤</mo><msup id="S4.Ex12.m1.4.4.2" xref="S4.Ex12.m1.4.4.2.cmml"><mrow id="S4.Ex12.m1.4.4.2.1.1" xref="S4.Ex12.m1.4.4.2.1.1.1.cmml"><mo id="S4.Ex12.m1.4.4.2.1.1.2" maxsize="120%" minsize="120%" xref="S4.Ex12.m1.4.4.2.1.1.1.cmml">(</mo><mrow id="S4.Ex12.m1.4.4.2.1.1.1" xref="S4.Ex12.m1.4.4.2.1.1.1.cmml"><mrow id="S4.Ex12.m1.4.4.2.1.1.1.2" xref="S4.Ex12.m1.4.4.2.1.1.1.2.cmml"><mi id="S4.Ex12.m1.4.4.2.1.1.1.2.2" xref="S4.Ex12.m1.4.4.2.1.1.1.2.2.cmml">f</mi><mo id="S4.Ex12.m1.4.4.2.1.1.1.2.1" xref="S4.Ex12.m1.4.4.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex12.m1.4.4.2.1.1.1.2.3.2" xref="S4.Ex12.m1.4.4.2.1.1.1.2.3.1.cmml"><mo id="S4.Ex12.m1.4.4.2.1.1.1.2.3.2.1" stretchy="false" xref="S4.Ex12.m1.4.4.2.1.1.1.2.3.1.cmml">(</mo><mn id="S4.Ex12.m1.1.1" xref="S4.Ex12.m1.1.1.cmml">0</mn><mo id="S4.Ex12.m1.4.4.2.1.1.1.2.3.2.2" xref="S4.Ex12.m1.4.4.2.1.1.1.2.3.1.cmml">,</mo><mi id="S4.Ex12.m1.2.2" xref="S4.Ex12.m1.2.2.cmml">t</mi><mo id="S4.Ex12.m1.4.4.2.1.1.1.2.3.2.3" stretchy="false" xref="S4.Ex12.m1.4.4.2.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex12.m1.4.4.2.1.1.1.1" xref="S4.Ex12.m1.4.4.2.1.1.1.1.cmml">+</mo><mi id="S4.Ex12.m1.4.4.2.1.1.1.3" xref="S4.Ex12.m1.4.4.2.1.1.1.3.cmml">B</mi><mo id="S4.Ex12.m1.4.4.2.1.1.1.1a" xref="S4.Ex12.m1.4.4.2.1.1.1.1.cmml">+</mo><mn id="S4.Ex12.m1.4.4.2.1.1.1.4" xref="S4.Ex12.m1.4.4.2.1.1.1.4.cmml">3</mn></mrow><mo id="S4.Ex12.m1.4.4.2.1.1.3" maxsize="120%" minsize="120%" xref="S4.Ex12.m1.4.4.2.1.1.1.cmml">)</mo></mrow><mrow id="S4.Ex12.m1.4.4.2.3" xref="S4.Ex12.m1.4.4.2.3.cmml"><mi id="S4.Ex12.m1.4.4.2.3.2" xref="S4.Ex12.m1.4.4.2.3.2.cmml">t</mi><mo id="S4.Ex12.m1.4.4.2.3.1" xref="S4.Ex12.m1.4.4.2.3.1.cmml">+</mo><mn id="S4.Ex12.m1.4.4.2.3.3" xref="S4.Ex12.m1.4.4.2.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.4b"><apply id="S4.Ex12.m1.4.4.cmml" xref="S4.Ex12.m1.4.4"><leq id="S4.Ex12.m1.4.4.3.cmml" xref="S4.Ex12.m1.4.4.3"></leq><apply id="S4.Ex12.m1.3.3.1.2.cmml" xref="S4.Ex12.m1.3.3.1.1"><abs id="S4.Ex12.m1.3.3.1.2.1.cmml" xref="S4.Ex12.m1.3.3.1.1.2"></abs><apply id="S4.Ex12.m1.3.3.1.1.1.cmml" xref="S4.Ex12.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Ex12.m1.3.3.1.1.1.1.cmml" xref="S4.Ex12.m1.3.3.1.1.1">subscript</csymbol><ci id="S4.Ex12.m1.3.3.1.1.1.2.cmml" xref="S4.Ex12.m1.3.3.1.1.1.2">𝐶</ci><apply id="S4.Ex12.m1.3.3.1.1.1.3.cmml" xref="S4.Ex12.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex12.m1.3.3.1.1.1.3.1.cmml" xref="S4.Ex12.m1.3.3.1.1.1.3">subscript</csymbol><ci id="S4.Ex12.m1.3.3.1.1.1.3.2.cmml" xref="S4.Ex12.m1.3.3.1.1.1.3.2">𝜎</ci><cn id="S4.Ex12.m1.3.3.1.1.1.3.3.cmml" type="integer" xref="S4.Ex12.m1.3.3.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S4.Ex12.m1.4.4.2.cmml" xref="S4.Ex12.m1.4.4.2"><csymbol cd="ambiguous" id="S4.Ex12.m1.4.4.2.2.cmml" xref="S4.Ex12.m1.4.4.2">superscript</csymbol><apply id="S4.Ex12.m1.4.4.2.1.1.1.cmml" xref="S4.Ex12.m1.4.4.2.1.1"><plus id="S4.Ex12.m1.4.4.2.1.1.1.1.cmml" xref="S4.Ex12.m1.4.4.2.1.1.1.1"></plus><apply id="S4.Ex12.m1.4.4.2.1.1.1.2.cmml" xref="S4.Ex12.m1.4.4.2.1.1.1.2"><times id="S4.Ex12.m1.4.4.2.1.1.1.2.1.cmml" xref="S4.Ex12.m1.4.4.2.1.1.1.2.1"></times><ci id="S4.Ex12.m1.4.4.2.1.1.1.2.2.cmml" xref="S4.Ex12.m1.4.4.2.1.1.1.2.2">𝑓</ci><interval closure="open" id="S4.Ex12.m1.4.4.2.1.1.1.2.3.1.cmml" xref="S4.Ex12.m1.4.4.2.1.1.1.2.3.2"><cn id="S4.Ex12.m1.1.1.cmml" type="integer" xref="S4.Ex12.m1.1.1">0</cn><ci id="S4.Ex12.m1.2.2.cmml" xref="S4.Ex12.m1.2.2">𝑡</ci></interval></apply><ci id="S4.Ex12.m1.4.4.2.1.1.1.3.cmml" xref="S4.Ex12.m1.4.4.2.1.1.1.3">𝐵</ci><cn id="S4.Ex12.m1.4.4.2.1.1.1.4.cmml" type="integer" xref="S4.Ex12.m1.4.4.2.1.1.1.4">3</cn></apply><apply id="S4.Ex12.m1.4.4.2.3.cmml" xref="S4.Ex12.m1.4.4.2.3"><plus id="S4.Ex12.m1.4.4.2.3.1.cmml" xref="S4.Ex12.m1.4.4.2.3.1"></plus><ci id="S4.Ex12.m1.4.4.2.3.2.cmml" xref="S4.Ex12.m1.4.4.2.3.2">𝑡</ci><cn id="S4.Ex12.m1.4.4.2.3.3.cmml" type="integer" xref="S4.Ex12.m1.4.4.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.4c">|C_{\sigma_{0}}|\leq\big{(}f({0},{t})+B+3\big{)}^{t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.4d">| italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ≤ ( italic_f ( 0 , italic_t ) + italic_B + 3 ) start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem12.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem12.p1.4.1">by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem6" title="Lemma 4.6 ‣ 4.3 Bounding the Minimum Component of Pareto-Optimal Costs ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.6</span></a> and</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex13"> <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({B},{t})\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot(B+3)" class="ltx_Math" display="block" id="S4.Ex13.m1.6"><semantics id="S4.Ex13.m1.6a"><mrow id="S4.Ex13.m1.6.6" xref="S4.Ex13.m1.6.6.cmml"><mrow id="S4.Ex13.m1.6.6.3" xref="S4.Ex13.m1.6.6.3.cmml"><mi id="S4.Ex13.m1.6.6.3.2" xref="S4.Ex13.m1.6.6.3.2.cmml">f</mi><mo id="S4.Ex13.m1.6.6.3.1" xref="S4.Ex13.m1.6.6.3.1.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.3.3.2" xref="S4.Ex13.m1.6.6.3.3.1.cmml"><mo id="S4.Ex13.m1.6.6.3.3.2.1" stretchy="false" xref="S4.Ex13.m1.6.6.3.3.1.cmml">(</mo><mi id="S4.Ex13.m1.3.3" xref="S4.Ex13.m1.3.3.cmml">B</mi><mo id="S4.Ex13.m1.6.6.3.3.2.2" xref="S4.Ex13.m1.6.6.3.3.1.cmml">,</mo><mi id="S4.Ex13.m1.4.4" xref="S4.Ex13.m1.4.4.cmml">t</mi><mo id="S4.Ex13.m1.6.6.3.3.2.3" stretchy="false" xref="S4.Ex13.m1.6.6.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.6.6.2" xref="S4.Ex13.m1.6.6.2.cmml">≤</mo><mrow id="S4.Ex13.m1.6.6.1" xref="S4.Ex13.m1.6.6.1.cmml"><msup id="S4.Ex13.m1.6.6.1.3" xref="S4.Ex13.m1.6.6.1.3.cmml"><mn id="S4.Ex13.m1.6.6.1.3.2" xref="S4.Ex13.m1.6.6.1.3.2.cmml">2</mn><mrow id="S4.Ex13.m1.1.1.1" xref="S4.Ex13.m1.1.1.1.cmml"><mi id="S4.Ex13.m1.1.1.1.3" mathvariant="normal" xref="S4.Ex13.m1.1.1.1.3.cmml">Θ</mi><mo id="S4.Ex13.m1.1.1.1.2" xref="S4.Ex13.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.1.1.cmml"><mo id="S4.Ex13.m1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex13.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex13.m1.1.1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex13.m1.1.1.1.1.1.1.2" xref="S4.Ex13.m1.1.1.1.1.1.1.2.cmml">t</mi><mn id="S4.Ex13.m1.1.1.1.1.1.1.3" xref="S4.Ex13.m1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S4.Ex13.m1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex13.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex13.m1.6.6.1.2" lspace="0.222em" rspace="0.222em" xref="S4.Ex13.m1.6.6.1.2.cmml">⋅</mo><msup id="S4.Ex13.m1.6.6.1.4" xref="S4.Ex13.m1.6.6.1.4.cmml"><mrow id="S4.Ex13.m1.6.6.1.4.2.2" xref="S4.Ex13.m1.6.6.1.4.2.1.cmml"><mo id="S4.Ex13.m1.6.6.1.4.2.2.1" stretchy="false" xref="S4.Ex13.m1.6.6.1.4.2.1.1.cmml">|</mo><mi id="S4.Ex13.m1.5.5" xref="S4.Ex13.m1.5.5.cmml">V</mi><mo id="S4.Ex13.m1.6.6.1.4.2.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex13.m1.6.6.1.4.2.1.1.cmml">|</mo></mrow><mrow id="S4.Ex13.m1.2.2.1" xref="S4.Ex13.m1.2.2.1.cmml"><mi id="S4.Ex13.m1.2.2.1.3" mathvariant="normal" xref="S4.Ex13.m1.2.2.1.3.cmml">Θ</mi><mo id="S4.Ex13.m1.2.2.1.2" xref="S4.Ex13.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.2.2.1.4.2" xref="S4.Ex13.m1.2.2.1.cmml"><mo id="S4.Ex13.m1.2.2.1.4.2.1" stretchy="false" xref="S4.Ex13.m1.2.2.1.cmml">(</mo><mi id="S4.Ex13.m1.2.2.1.1" xref="S4.Ex13.m1.2.2.1.1.cmml">t</mi><mo id="S4.Ex13.m1.2.2.1.4.2.2" stretchy="false" xref="S4.Ex13.m1.2.2.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex13.m1.6.6.1.2a" rspace="0.222em" xref="S4.Ex13.m1.6.6.1.2.cmml">⋅</mo><mrow id="S4.Ex13.m1.6.6.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.cmml">(</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.cmml"><mi id="S4.Ex13.m1.6.6.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.2.cmml">B</mi><mo id="S4.Ex13.m1.6.6.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.cmml">+</mo><mn id="S4.Ex13.m1.6.6.1.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.1.1.3.cmml">3</mn></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.3" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m1.6b"><apply id="S4.Ex13.m1.6.6.cmml" xref="S4.Ex13.m1.6.6"><leq 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xref="S4.Ex13.m1.1.1.1.3">Θ</ci><apply id="S4.Ex13.m1.1.1.1.1.1.1.cmml" xref="S4.Ex13.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex13.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex13.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex13.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex13.m1.1.1.1.1.1.1.2">𝑡</ci><cn id="S4.Ex13.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Ex13.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply><apply id="S4.Ex13.m1.6.6.1.4.cmml" xref="S4.Ex13.m1.6.6.1.4"><csymbol cd="ambiguous" id="S4.Ex13.m1.6.6.1.4.1.cmml" xref="S4.Ex13.m1.6.6.1.4">superscript</csymbol><apply id="S4.Ex13.m1.6.6.1.4.2.1.cmml" xref="S4.Ex13.m1.6.6.1.4.2.2"><abs id="S4.Ex13.m1.6.6.1.4.2.1.1.cmml" xref="S4.Ex13.m1.6.6.1.4.2.2.1"></abs><ci id="S4.Ex13.m1.5.5.cmml" xref="S4.Ex13.m1.5.5">𝑉</ci></apply><apply id="S4.Ex13.m1.2.2.1.cmml" xref="S4.Ex13.m1.2.2.1"><times id="S4.Ex13.m1.2.2.1.2.cmml" xref="S4.Ex13.m1.2.2.1.2"></times><ci id="S4.Ex13.m1.2.2.1.3.cmml" xref="S4.Ex13.m1.2.2.1.3">Θ</ci><ci id="S4.Ex13.m1.2.2.1.1.cmml" xref="S4.Ex13.m1.2.2.1.1">𝑡</ci></apply></apply><apply id="S4.Ex13.m1.6.6.1.1.1.1.cmml" xref="S4.Ex13.m1.6.6.1.1.1"><plus id="S4.Ex13.m1.6.6.1.1.1.1.1.cmml" xref="S4.Ex13.m1.6.6.1.1.1.1.1"></plus><ci id="S4.Ex13.m1.6.6.1.1.1.1.2.cmml" xref="S4.Ex13.m1.6.6.1.1.1.1.2">𝐵</ci><cn id="S4.Ex13.m1.6.6.1.1.1.1.3.cmml" type="integer" xref="S4.Ex13.m1.6.6.1.1.1.1.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.6c">f({B},{t})\leq 2^{\Theta(t^{2})}\cdot|V|^{\Theta(t)}\cdot(B+3)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.6d">italic_f ( italic_B , italic_t ) ≤ 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem12.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem12.p1.5.1">by (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E4" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>). Therefore, we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex14"> <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="\begin{array}[]{ll}|C_{\sigma_{0}}|&\leq\big{(}2^{\Theta(t^{2})}\cdot|V|^{% \Theta(t)}\cdot 3+B+3\big{)}^{t+1}\\ &\leq\big{(}2^{\Theta(t^{2})+2}\cdot|V|^{\Theta(t)}\cdot(B+3)\big{)}^{t+1}\\ &\leq 2^{\Theta(t^{3})}\cdot|V|^{\Theta(t^{2})}\cdot(B+3)^{\Theta(t)}.\end{array}" class="ltx_Math" display="block" id="S4.Ex14.m1.14"><semantics id="S4.Ex14.m1.14a"><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex14.m1.14.14" rowspacing="0pt" xref="S4.Ex14.m1.14.14.cmml"><mtr id="S4.Ex14.m1.14.14a" xref="S4.Ex14.m1.14.14.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex14.m1.14.14b" xref="S4.Ex14.m1.14.14.cmml"><mrow id="S4.Ex14.m1.1.1.1.1.1.1" xref="S4.Ex14.m1.1.1.1.1.1.2.cmml"><mo id="S4.Ex14.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.1.1.1.1.1.2.1.cmml">|</mo><msub id="S4.Ex14.m1.1.1.1.1.1.1.1" xref="S4.Ex14.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.1.2" xref="S4.Ex14.m1.1.1.1.1.1.1.1.2.cmml">C</mi><msub id="S4.Ex14.m1.1.1.1.1.1.1.1.3" xref="S4.Ex14.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.1.3.2" xref="S4.Ex14.m1.1.1.1.1.1.1.1.3.2.cmml">σ</mi><mn id="S4.Ex14.m1.1.1.1.1.1.1.1.3.3" xref="S4.Ex14.m1.1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Ex14.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex14.m1.1.1.1.1.1.2.1.cmml">|</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex14.m1.14.14c" xref="S4.Ex14.m1.14.14.cmml"><mrow id="S4.Ex14.m1.5.5.5.5.4" xref="S4.Ex14.m1.5.5.5.5.4.cmml"><mi id="S4.Ex14.m1.5.5.5.5.4.6" xref="S4.Ex14.m1.5.5.5.5.4.6.cmml"></mi><mo id="S4.Ex14.m1.5.5.5.5.4.5" xref="S4.Ex14.m1.5.5.5.5.4.5.cmml">≤</mo><msup id="S4.Ex14.m1.5.5.5.5.4.4" xref="S4.Ex14.m1.5.5.5.5.4.4.cmml"><mrow id="S4.Ex14.m1.5.5.5.5.4.4.1.1" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.cmml"><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.2" maxsize="120%" minsize="120%" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.cmml">(</mo><mrow id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.cmml"><mrow id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.cmml"><msup id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.2" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.2.cmml"><mn id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.2.2" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.2.2.cmml">2</mn><mrow id="S4.Ex14.m1.2.2.2.2.1.1.1" xref="S4.Ex14.m1.2.2.2.2.1.1.1.cmml"><mi id="S4.Ex14.m1.2.2.2.2.1.1.1.3" mathvariant="normal" xref="S4.Ex14.m1.2.2.2.2.1.1.1.3.cmml">Θ</mi><mo id="S4.Ex14.m1.2.2.2.2.1.1.1.2" xref="S4.Ex14.m1.2.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex14.m1.2.2.2.2.1.1.1.1.1" xref="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1" xref="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.2" xref="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.2.cmml">t</mi><mn id="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.3" xref="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.Ex14.m1.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.1.cmml">⋅</mo><msup id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3.cmml"><mrow id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3.2.2" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3.2.1.cmml"><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3.2.2.1" stretchy="false" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3.2.1.1.cmml">|</mo><mi id="S4.Ex14.m1.4.4.4.4.3.3" xref="S4.Ex14.m1.4.4.4.4.3.3.cmml">V</mi><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.3.2.1.1.cmml">|</mo></mrow><mrow id="S4.Ex14.m1.3.3.3.3.2.2.1" xref="S4.Ex14.m1.3.3.3.3.2.2.1.cmml"><mi id="S4.Ex14.m1.3.3.3.3.2.2.1.3" mathvariant="normal" xref="S4.Ex14.m1.3.3.3.3.2.2.1.3.cmml">Θ</mi><mo id="S4.Ex14.m1.3.3.3.3.2.2.1.2" xref="S4.Ex14.m1.3.3.3.3.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex14.m1.3.3.3.3.2.2.1.4.2" xref="S4.Ex14.m1.3.3.3.3.2.2.1.cmml"><mo id="S4.Ex14.m1.3.3.3.3.2.2.1.4.2.1" stretchy="false" xref="S4.Ex14.m1.3.3.3.3.2.2.1.cmml">(</mo><mi id="S4.Ex14.m1.3.3.3.3.2.2.1.1" xref="S4.Ex14.m1.3.3.3.3.2.2.1.1.cmml">t</mi><mo id="S4.Ex14.m1.3.3.3.3.2.2.1.4.2.2" stretchy="false" xref="S4.Ex14.m1.3.3.3.3.2.2.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.1a" rspace="0.222em" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.1.cmml">⋅</mo><mn id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.4" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.2.4.cmml">3</mn></mrow><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.1" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.1.cmml">+</mo><mi id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.3" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.3.cmml">B</mi><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.1a" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.1.cmml">+</mo><mn id="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.4" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.4.cmml">3</mn></mrow><mo id="S4.Ex14.m1.5.5.5.5.4.4.1.1.3" maxsize="120%" minsize="120%" xref="S4.Ex14.m1.5.5.5.5.4.4.1.1.1.cmml">)</mo></mrow><mrow id="S4.Ex14.m1.5.5.5.5.4.4.3" xref="S4.Ex14.m1.5.5.5.5.4.4.3.cmml"><mi id="S4.Ex14.m1.5.5.5.5.4.4.3.2" xref="S4.Ex14.m1.5.5.5.5.4.4.3.2.cmml">t</mi><mo id="S4.Ex14.m1.5.5.5.5.4.4.3.1" xref="S4.Ex14.m1.5.5.5.5.4.4.3.1.cmml">+</mo><mn id="S4.Ex14.m1.5.5.5.5.4.4.3.3" xref="S4.Ex14.m1.5.5.5.5.4.4.3.3.cmml">1</mn></mrow></msup></mrow></mtd></mtr><mtr id="S4.Ex14.m1.14.14d" xref="S4.Ex14.m1.14.14.cmml"><mtd id="S4.Ex14.m1.14.14e" xref="S4.Ex14.m1.14.14.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex14.m1.14.14f" xref="S4.Ex14.m1.14.14.cmml"><mrow id="S4.Ex14.m1.9.9.9.4.4" xref="S4.Ex14.m1.9.9.9.4.4.cmml"><mi id="S4.Ex14.m1.9.9.9.4.4.6" xref="S4.Ex14.m1.9.9.9.4.4.6.cmml"></mi><mo id="S4.Ex14.m1.9.9.9.4.4.5" xref="S4.Ex14.m1.9.9.9.4.4.5.cmml">≤</mo><msup id="S4.Ex14.m1.9.9.9.4.4.4" xref="S4.Ex14.m1.9.9.9.4.4.4.cmml"><mrow id="S4.Ex14.m1.9.9.9.4.4.4.1.1" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.cmml"><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.2" maxsize="120%" minsize="120%" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.cmml">(</mo><mrow id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.cmml"><msup id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.3" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.3.cmml"><mn id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.3.2" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.3.2.cmml">2</mn><mrow id="S4.Ex14.m1.6.6.6.1.1.1.1" xref="S4.Ex14.m1.6.6.6.1.1.1.1.cmml"><mrow id="S4.Ex14.m1.6.6.6.1.1.1.1.1" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.6.6.6.1.1.1.1.1.3" mathvariant="normal" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.3.cmml">Θ</mi><mo id="S4.Ex14.m1.6.6.6.1.1.1.1.1.2" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.2" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.2.cmml">t</mi><mn id="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.3" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex14.m1.6.6.6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex14.m1.6.6.6.1.1.1.1.2" xref="S4.Ex14.m1.6.6.6.1.1.1.1.2.cmml">+</mo><mn id="S4.Ex14.m1.6.6.6.1.1.1.1.3" xref="S4.Ex14.m1.6.6.6.1.1.1.1.3.cmml">2</mn></mrow></msup><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.2.cmml">⋅</mo><msup id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4.cmml"><mrow id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4.2.2" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4.2.1.cmml"><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4.2.2.1" stretchy="false" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4.2.1.1.cmml">|</mo><mi id="S4.Ex14.m1.8.8.8.3.3.3" xref="S4.Ex14.m1.8.8.8.3.3.3.cmml">V</mi><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4.2.2.2" rspace="0.055em" stretchy="false" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.4.2.1.1.cmml">|</mo></mrow><mrow id="S4.Ex14.m1.7.7.7.2.2.2.1" xref="S4.Ex14.m1.7.7.7.2.2.2.1.cmml"><mi id="S4.Ex14.m1.7.7.7.2.2.2.1.3" mathvariant="normal" xref="S4.Ex14.m1.7.7.7.2.2.2.1.3.cmml">Θ</mi><mo id="S4.Ex14.m1.7.7.7.2.2.2.1.2" xref="S4.Ex14.m1.7.7.7.2.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex14.m1.7.7.7.2.2.2.1.4.2" xref="S4.Ex14.m1.7.7.7.2.2.2.1.cmml"><mo id="S4.Ex14.m1.7.7.7.2.2.2.1.4.2.1" stretchy="false" xref="S4.Ex14.m1.7.7.7.2.2.2.1.cmml">(</mo><mi id="S4.Ex14.m1.7.7.7.2.2.2.1.1" xref="S4.Ex14.m1.7.7.7.2.2.2.1.1.cmml">t</mi><mo id="S4.Ex14.m1.7.7.7.2.2.2.1.4.2.2" stretchy="false" xref="S4.Ex14.m1.7.7.7.2.2.2.1.cmml">)</mo></mrow></mrow></msup><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.2a" rspace="0.222em" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.2.cmml">⋅</mo><mrow id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.cmml"><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.2" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.2.cmml">B</mi><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.1" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.3" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.3.cmml">3</mn></mrow><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.3" stretchy="false" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex14.m1.9.9.9.4.4.4.1.1.3" maxsize="120%" minsize="120%" xref="S4.Ex14.m1.9.9.9.4.4.4.1.1.1.cmml">)</mo></mrow><mrow id="S4.Ex14.m1.9.9.9.4.4.4.3" xref="S4.Ex14.m1.9.9.9.4.4.4.3.cmml"><mi id="S4.Ex14.m1.9.9.9.4.4.4.3.2" xref="S4.Ex14.m1.9.9.9.4.4.4.3.2.cmml">t</mi><mo id="S4.Ex14.m1.9.9.9.4.4.4.3.1" xref="S4.Ex14.m1.9.9.9.4.4.4.3.1.cmml">+</mo><mn id="S4.Ex14.m1.9.9.9.4.4.4.3.3" xref="S4.Ex14.m1.9.9.9.4.4.4.3.3.cmml">1</mn></mrow></msup></mrow></mtd></mtr><mtr id="S4.Ex14.m1.14.14g" xref="S4.Ex14.m1.14.14.cmml"><mtd id="S4.Ex14.m1.14.14h" xref="S4.Ex14.m1.14.14.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex14.m1.14.14i" xref="S4.Ex14.m1.14.14.cmml"><mrow id="S4.Ex14.m1.14.14.14.5.5.5" xref="S4.Ex14.m1.14.14.14.5.5.5.1.cmml"><mrow id="S4.Ex14.m1.14.14.14.5.5.5.1" xref="S4.Ex14.m1.14.14.14.5.5.5.1.cmml"><mi id="S4.Ex14.m1.14.14.14.5.5.5.1.3" xref="S4.Ex14.m1.14.14.14.5.5.5.1.3.cmml"></mi><mo id="S4.Ex14.m1.14.14.14.5.5.5.1.2" xref="S4.Ex14.m1.14.14.14.5.5.5.1.2.cmml">≤</mo><mrow id="S4.Ex14.m1.14.14.14.5.5.5.1.1" xref="S4.Ex14.m1.14.14.14.5.5.5.1.1.cmml"><msup id="S4.Ex14.m1.14.14.14.5.5.5.1.1.3" xref="S4.Ex14.m1.14.14.14.5.5.5.1.1.3.cmml"><mn id="S4.Ex14.m1.14.14.14.5.5.5.1.1.3.2" xref="S4.Ex14.m1.14.14.14.5.5.5.1.1.3.2.cmml">2</mn><mrow id="S4.Ex14.m1.10.10.10.1.1.1.1" xref="S4.Ex14.m1.10.10.10.1.1.1.1.cmml"><mi id="S4.Ex14.m1.10.10.10.1.1.1.1.3" mathvariant="normal" xref="S4.Ex14.m1.10.10.10.1.1.1.1.3.cmml">Θ</mi><mo id="S4.Ex14.m1.10.10.10.1.1.1.1.2" xref="S4.Ex14.m1.10.10.10.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex14.m1.10.10.10.1.1.1.1.1.1" xref="S4.Ex14.m1.10.10.10.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex14.m1.10.10.10.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.10.10.10.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex14.m1.10.10.10.1.1.1.1.1.1.1" xref="S4.Ex14.m1.10.10.10.1.1.1.1.1.1.1.cmml"><mi 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xref="S4.Ex14.m1.12.12.12.3.3.3.1.1">𝑡</ci></apply></apply></apply></apply></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m1.14c">\begin{array}[]{ll}|C_{\sigma_{0}}|&\leq\big{(}2^{\Theta(t^{2})}\cdot|V|^{% \Theta(t)}\cdot 3+B+3\big{)}^{t+1}\\ &\leq\big{(}2^{\Theta(t^{2})+2}\cdot|V|^{\Theta(t)}\cdot(B+3)\big{)}^{t+1}\\ &\leq 2^{\Theta(t^{3})}\cdot|V|^{\Theta(t^{2})}\cdot(B+3)^{\Theta(t)}.\end{array}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.14d">start_ARRAY start_ROW start_CELL | italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | end_CELL start_CELL ≤ ( 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT ⋅ 3 + italic_B + 3 ) start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ ( 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + 2 end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 ) ) start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ 2 start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ | italic_V | start_POSTSUPERSCRIPT roman_Θ ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ⋅ ( italic_B + 3 ) start_POSTSUPERSCRIPT roman_Θ ( italic_t ) end_POSTSUPERSCRIPT . end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem12.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem12.p1.3.2">In the general case, by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem2" title="Lemma 2.2 ‣ Reduction to binary arenas. ‣ 2.3 Tools ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.2</span></a>, <math alttext="|V|" class="ltx_Math" display="inline" id="S4.Thmtheorem12.p1.2.1.m1.1"><semantics id="S4.Thmtheorem12.p1.2.1.m1.1a"><mrow id="S4.Thmtheorem12.p1.2.1.m1.1.2.2" xref="S4.Thmtheorem12.p1.2.1.m1.1.2.1.cmml"><mo id="S4.Thmtheorem12.p1.2.1.m1.1.2.2.1" stretchy="false" xref="S4.Thmtheorem12.p1.2.1.m1.1.2.1.1.cmml">|</mo><mi id="S4.Thmtheorem12.p1.2.1.m1.1.1" xref="S4.Thmtheorem12.p1.2.1.m1.1.1.cmml">V</mi><mo id="S4.Thmtheorem12.p1.2.1.m1.1.2.2.2" stretchy="false" xref="S4.Thmtheorem12.p1.2.1.m1.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem12.p1.2.1.m1.1b"><apply id="S4.Thmtheorem12.p1.2.1.m1.1.2.1.cmml" xref="S4.Thmtheorem12.p1.2.1.m1.1.2.2"><abs id="S4.Thmtheorem12.p1.2.1.m1.1.2.1.1.cmml" xref="S4.Thmtheorem12.p1.2.1.m1.1.2.2.1"></abs><ci id="S4.Thmtheorem12.p1.2.1.m1.1.1.cmml" xref="S4.Thmtheorem12.p1.2.1.m1.1.1">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem12.p1.2.1.m1.1c">|V|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem12.p1.2.1.m1.1d">| italic_V |</annotation></semantics></math> is multiplied by <math alttext="W" class="ltx_Math" display="inline" id="S4.Thmtheorem12.p1.3.2.m2.1"><semantics id="S4.Thmtheorem12.p1.3.2.m2.1a"><mi id="S4.Thmtheorem12.p1.3.2.m2.1.1" xref="S4.Thmtheorem12.p1.3.2.m2.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem12.p1.3.2.m2.1b"><ci id="S4.Thmtheorem12.p1.3.2.m2.1.1.cmml" xref="S4.Thmtheorem12.p1.3.2.m2.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem12.p1.3.2.m2.1c">W</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem12.p1.3.2.m2.1d">italic_W</annotation></semantics></math> in the last inequality.</span></p> </div> </div> <div class="ltx_para" id="S4.SS4.p3"> <p class="ltx_p" id="S4.SS4.p3.3">In the sequel, we use the same notation <math alttext="f({B},{t})" class="ltx_Math" display="inline" id="S4.SS4.p3.1.m1.2"><semantics id="S4.SS4.p3.1.m1.2a"><mrow id="S4.SS4.p3.1.m1.2.3" xref="S4.SS4.p3.1.m1.2.3.cmml"><mi id="S4.SS4.p3.1.m1.2.3.2" xref="S4.SS4.p3.1.m1.2.3.2.cmml">f</mi><mo id="S4.SS4.p3.1.m1.2.3.1" xref="S4.SS4.p3.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS4.p3.1.m1.2.3.3.2" xref="S4.SS4.p3.1.m1.2.3.3.1.cmml"><mo id="S4.SS4.p3.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS4.p3.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS4.p3.1.m1.1.1" xref="S4.SS4.p3.1.m1.1.1.cmml">B</mi><mo id="S4.SS4.p3.1.m1.2.3.3.2.2" xref="S4.SS4.p3.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS4.p3.1.m1.2.2" xref="S4.SS4.p3.1.m1.2.2.cmml">t</mi><mo id="S4.SS4.p3.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS4.p3.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.1.m1.2b"><apply id="S4.SS4.p3.1.m1.2.3.cmml" xref="S4.SS4.p3.1.m1.2.3"><times id="S4.SS4.p3.1.m1.2.3.1.cmml" xref="S4.SS4.p3.1.m1.2.3.1"></times><ci id="S4.SS4.p3.1.m1.2.3.2.cmml" xref="S4.SS4.p3.1.m1.2.3.2">𝑓</ci><interval closure="open" id="S4.SS4.p3.1.m1.2.3.3.1.cmml" xref="S4.SS4.p3.1.m1.2.3.3.2"><ci id="S4.SS4.p3.1.m1.1.1.cmml" xref="S4.SS4.p3.1.m1.1.1">𝐵</ci><ci id="S4.SS4.p3.1.m1.2.2.cmml" xref="S4.SS4.p3.1.m1.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.1.m1.2c">f({B},{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.1.m1.2d">italic_f ( italic_B , italic_t )</annotation></semantics></math> for any SP games, having in mind that <math alttext="|V|" class="ltx_Math" display="inline" id="S4.SS4.p3.2.m2.1"><semantics id="S4.SS4.p3.2.m2.1a"><mrow id="S4.SS4.p3.2.m2.1.2.2" xref="S4.SS4.p3.2.m2.1.2.1.cmml"><mo id="S4.SS4.p3.2.m2.1.2.2.1" stretchy="false" xref="S4.SS4.p3.2.m2.1.2.1.1.cmml">|</mo><mi id="S4.SS4.p3.2.m2.1.1" xref="S4.SS4.p3.2.m2.1.1.cmml">V</mi><mo id="S4.SS4.p3.2.m2.1.2.2.2" stretchy="false" xref="S4.SS4.p3.2.m2.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.2.m2.1b"><apply id="S4.SS4.p3.2.m2.1.2.1.cmml" xref="S4.SS4.p3.2.m2.1.2.2"><abs id="S4.SS4.p3.2.m2.1.2.1.1.cmml" xref="S4.SS4.p3.2.m2.1.2.2.1"></abs><ci id="S4.SS4.p3.2.m2.1.1.cmml" xref="S4.SS4.p3.2.m2.1.1">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.2.m2.1c">|V|</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.2.m2.1d">| italic_V |</annotation></semantics></math> has to be multiplied by <math alttext="W" class="ltx_Math" display="inline" id="S4.SS4.p3.3.m3.1"><semantics id="S4.SS4.p3.3.m3.1a"><mi id="S4.SS4.p3.3.m3.1.1" xref="S4.SS4.p3.3.m3.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.3.m3.1b"><ci id="S4.SS4.p3.3.m3.1.1.cmml" xref="S4.SS4.p3.3.m3.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.3.m3.1c">W</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.3.m3.1d">italic_W</annotation></semantics></math> when the game arena is not binary.</p> </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>Complexity of the SPS Problem</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this section, we prove our main result: the SPS problem is <span class="ltx_text ltx_font_sansserif" id="S5.p1.1.1">NEXPTIME</span>-complete (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem1" title="Theorem 2.1 ‣ Stackelberg-Pareto synthesis problem. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.1</span></a>). It follows the same pattern as for Boolean reachability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite>, however it requires the results of Section <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4" title="4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a> (which is meaningless in the Boolean case) and some modifications to handle quantitative reachability.</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>Finite-Memory Solutions</h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.1">We first show that if there exists a solution to the SPS problem, then there is one that is finite-memory and whose memory size is bounded exponentially. This intermediate step is necessary to prove that the SPS problem is in <span class="ltx_text ltx_font_sansserif" id="S5.SS1.p1.1.1">NEXPTIME</span>: we will guess such a strategy and check that it is a solution in exponential time.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.1.1.1">Proposition 5.1</span></span></h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p1.5.5">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 SP game, <math alttext="B\in{\rm Nature}" 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">B</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">Nature</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"><in id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1"></in><ci id="S5.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.2">𝐵</ci><ci id="S5.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" 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.m3.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.cmml"><msub id="S5.Thmtheorem1.p1.3.3.m3.2.3.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.2.cmml"><mi id="S5.Thmtheorem1.p1.3.3.m3.2.3.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.2.2.cmml">σ</mi><mn id="S5.Thmtheorem1.p1.3.3.m3.2.3.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.2.3.cmml">0</mn></msub><mo id="S5.Thmtheorem1.p1.3.3.m3.2.3.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.3.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml"><mo id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.cmml">G</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem1.p1.3.3.m3.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.cmml">B</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.2.3" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.3.m3.2b"><apply id="S5.Thmtheorem1.p1.3.3.m3.2.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3"><in id="S5.Thmtheorem1.p1.3.3.m3.2.3.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.1"></in><apply id="S5.Thmtheorem1.p1.3.3.m3.2.3.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.3.3.m3.2.3.2.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem1.p1.3.3.m3.2.3.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.2.2">𝜎</ci><cn id="S5.Thmtheorem1.p1.3.3.m3.2.3.2.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.2.3">0</cn></apply><apply id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3"><times id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.1"></times><ci id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.2a.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.3.3.3.2"><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1">𝐺</ci><ci id="S5.Thmtheorem1.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.3.m3.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.3.m3.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a solution. Then there exists a Pareto-bounded solution <math alttext="\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.4.4.m4.2"><semantics id="S5.Thmtheorem1.p1.4.4.m4.2a"><mrow id="S5.Thmtheorem1.p1.4.4.m4.2.3" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.cmml"><msubsup id="S5.Thmtheorem1.p1.4.4.m4.2.3.2" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2.cmml"><mi id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.2" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.2.cmml">σ</mi><mn id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.3" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.3.cmml">0</mn><mo id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.3" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2.3.cmml">′</mo></msubsup><mo id="S5.Thmtheorem1.p1.4.4.m4.2.3.1" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem1.p1.4.4.m4.2.3.3" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.2" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.1" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.2" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml"><mo id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem1.p1.4.4.m4.1.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.cmml">G</mi><mo id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.2.2" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem1.p1.4.4.m4.2.2" xref="S5.Thmtheorem1.p1.4.4.m4.2.2.cmml">B</mi><mo id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.2.3" stretchy="false" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.4.4.m4.2b"><apply id="S5.Thmtheorem1.p1.4.4.m4.2.3.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3"><in id="S5.Thmtheorem1.p1.4.4.m4.2.3.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.1"></in><apply id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2">superscript</csymbol><apply id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.2">𝜎</ci><cn id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2.2.3">0</cn></apply><ci id="S5.Thmtheorem1.p1.4.4.m4.2.3.2.3.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.2.3">′</ci></apply><apply id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3"><times id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.1"></times><ci id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.2a.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.3.3.3.2"><ci id="S5.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1">𝐺</ci><ci id="S5.Thmtheorem1.p1.4.4.m4.2.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.4.4.m4.2c">\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.4.4.m4.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> such that <math alttext="\sigma_{0}^{\prime}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.5.5.m5.1"><semantics id="S5.Thmtheorem1.p1.5.5.m5.1a"><msubsup id="S5.Thmtheorem1.p1.5.5.m5.1.1" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.cmml"><mi id="S5.Thmtheorem1.p1.5.5.m5.1.1.2.2" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.2.2.cmml">σ</mi><mn id="S5.Thmtheorem1.p1.5.5.m5.1.1.2.3" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.2.3.cmml">0</mn><mo id="S5.Thmtheorem1.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.5.5.m5.1b"><apply id="S5.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.1.1">superscript</csymbol><apply id="S5.Thmtheorem1.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.5.5.m5.1.1.2.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.5.5.m5.1.1.2.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.2.2">𝜎</ci><cn id="S5.Thmtheorem1.p1.5.5.m5.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.2.3">0</cn></apply><ci id="S5.Thmtheorem1.p1.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.5.5.m5.1c">\sigma_{0}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.5.5.m5.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is finite-memory and its memory size is bounded exponentially.</span></p> </div> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.2">When <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.4"><semantics id="S5.SS1.p2.1.m1.4a"><mrow id="S5.SS1.p2.1.m1.4.4" xref="S5.SS1.p2.1.m1.4.4.cmml"><msub id="S5.SS1.p2.1.m1.4.4.3" xref="S5.SS1.p2.1.m1.4.4.3.cmml"><mi id="S5.SS1.p2.1.m1.4.4.3.2" xref="S5.SS1.p2.1.m1.4.4.3.2.cmml">C</mi><msub id="S5.SS1.p2.1.m1.4.4.3.3" xref="S5.SS1.p2.1.m1.4.4.3.3.cmml"><mi id="S5.SS1.p2.1.m1.4.4.3.3.2" xref="S5.SS1.p2.1.m1.4.4.3.3.2.cmml">σ</mi><mn id="S5.SS1.p2.1.m1.4.4.3.3.3" xref="S5.SS1.p2.1.m1.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S5.SS1.p2.1.m1.4.4.2" xref="S5.SS1.p2.1.m1.4.4.2.cmml">≠</mo><mrow id="S5.SS1.p2.1.m1.4.4.1.1" xref="S5.SS1.p2.1.m1.4.4.1.2.cmml"><mo id="S5.SS1.p2.1.m1.4.4.1.1.2" stretchy="false" xref="S5.SS1.p2.1.m1.4.4.1.2.cmml">{</mo><mrow id="S5.SS1.p2.1.m1.4.4.1.1.1.2" xref="S5.SS1.p2.1.m1.4.4.1.1.1.1.cmml"><mo id="S5.SS1.p2.1.m1.4.4.1.1.1.2.1" stretchy="false" xref="S5.SS1.p2.1.m1.4.4.1.1.1.1.cmml">(</mo><mi id="S5.SS1.p2.1.m1.1.1" mathvariant="normal" xref="S5.SS1.p2.1.m1.1.1.cmml">∞</mi><mo id="S5.SS1.p2.1.m1.4.4.1.1.1.2.2" xref="S5.SS1.p2.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.SS1.p2.1.m1.2.2" mathvariant="normal" xref="S5.SS1.p2.1.m1.2.2.cmml">…</mi><mo id="S5.SS1.p2.1.m1.4.4.1.1.1.2.3" xref="S5.SS1.p2.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.SS1.p2.1.m1.3.3" mathvariant="normal" xref="S5.SS1.p2.1.m1.3.3.cmml">∞</mi><mo id="S5.SS1.p2.1.m1.4.4.1.1.1.2.4" stretchy="false" xref="S5.SS1.p2.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS1.p2.1.m1.4.4.1.1.3" stretchy="false" xref="S5.SS1.p2.1.m1.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.4b"><apply id="S5.SS1.p2.1.m1.4.4.cmml" xref="S5.SS1.p2.1.m1.4.4"><neq id="S5.SS1.p2.1.m1.4.4.2.cmml" xref="S5.SS1.p2.1.m1.4.4.2"></neq><apply id="S5.SS1.p2.1.m1.4.4.3.cmml" xref="S5.SS1.p2.1.m1.4.4.3"><csymbol cd="ambiguous" id="S5.SS1.p2.1.m1.4.4.3.1.cmml" xref="S5.SS1.p2.1.m1.4.4.3">subscript</csymbol><ci id="S5.SS1.p2.1.m1.4.4.3.2.cmml" xref="S5.SS1.p2.1.m1.4.4.3.2">𝐶</ci><apply id="S5.SS1.p2.1.m1.4.4.3.3.cmml" xref="S5.SS1.p2.1.m1.4.4.3.3"><csymbol cd="ambiguous" id="S5.SS1.p2.1.m1.4.4.3.3.1.cmml" xref="S5.SS1.p2.1.m1.4.4.3.3">subscript</csymbol><ci id="S5.SS1.p2.1.m1.4.4.3.3.2.cmml" xref="S5.SS1.p2.1.m1.4.4.3.3.2">𝜎</ci><cn id="S5.SS1.p2.1.m1.4.4.3.3.3.cmml" type="integer" xref="S5.SS1.p2.1.m1.4.4.3.3.3">0</cn></apply></apply><set id="S5.SS1.p2.1.m1.4.4.1.2.cmml" xref="S5.SS1.p2.1.m1.4.4.1.1"><vector id="S5.SS1.p2.1.m1.4.4.1.1.1.1.cmml" xref="S5.SS1.p2.1.m1.4.4.1.1.1.2"><infinity id="S5.SS1.p2.1.m1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1"></infinity><ci id="S5.SS1.p2.1.m1.2.2.cmml" xref="S5.SS1.p2.1.m1.2.2">…</ci><infinity id="S5.SS1.p2.1.m1.3.3.cmml" xref="S5.SS1.p2.1.m1.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>, the proof of this proposition is based on the following principles, that are detailed below (the case <math alttext="C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m2.4"><semantics id="S5.SS1.p2.2.m2.4a"><mrow id="S5.SS1.p2.2.m2.4.4" xref="S5.SS1.p2.2.m2.4.4.cmml"><msub id="S5.SS1.p2.2.m2.4.4.3" xref="S5.SS1.p2.2.m2.4.4.3.cmml"><mi id="S5.SS1.p2.2.m2.4.4.3.2" xref="S5.SS1.p2.2.m2.4.4.3.2.cmml">C</mi><msub id="S5.SS1.p2.2.m2.4.4.3.3" xref="S5.SS1.p2.2.m2.4.4.3.3.cmml"><mi id="S5.SS1.p2.2.m2.4.4.3.3.2" xref="S5.SS1.p2.2.m2.4.4.3.3.2.cmml">σ</mi><mn id="S5.SS1.p2.2.m2.4.4.3.3.3" xref="S5.SS1.p2.2.m2.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S5.SS1.p2.2.m2.4.4.2" xref="S5.SS1.p2.2.m2.4.4.2.cmml">=</mo><mrow id="S5.SS1.p2.2.m2.4.4.1.1" xref="S5.SS1.p2.2.m2.4.4.1.2.cmml"><mo id="S5.SS1.p2.2.m2.4.4.1.1.2" stretchy="false" xref="S5.SS1.p2.2.m2.4.4.1.2.cmml">{</mo><mrow id="S5.SS1.p2.2.m2.4.4.1.1.1.2" xref="S5.SS1.p2.2.m2.4.4.1.1.1.1.cmml"><mo id="S5.SS1.p2.2.m2.4.4.1.1.1.2.1" stretchy="false" xref="S5.SS1.p2.2.m2.4.4.1.1.1.1.cmml">(</mo><mi id="S5.SS1.p2.2.m2.1.1" mathvariant="normal" xref="S5.SS1.p2.2.m2.1.1.cmml">∞</mi><mo id="S5.SS1.p2.2.m2.4.4.1.1.1.2.2" xref="S5.SS1.p2.2.m2.4.4.1.1.1.1.cmml">,</mo><mi id="S5.SS1.p2.2.m2.2.2" mathvariant="normal" xref="S5.SS1.p2.2.m2.2.2.cmml">…</mi><mo id="S5.SS1.p2.2.m2.4.4.1.1.1.2.3" xref="S5.SS1.p2.2.m2.4.4.1.1.1.1.cmml">,</mo><mi id="S5.SS1.p2.2.m2.3.3" mathvariant="normal" xref="S5.SS1.p2.2.m2.3.3.cmml">∞</mi><mo id="S5.SS1.p2.2.m2.4.4.1.1.1.2.4" stretchy="false" xref="S5.SS1.p2.2.m2.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS1.p2.2.m2.4.4.1.1.3" stretchy="false" xref="S5.SS1.p2.2.m2.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m2.4b"><apply id="S5.SS1.p2.2.m2.4.4.cmml" xref="S5.SS1.p2.2.m2.4.4"><eq id="S5.SS1.p2.2.m2.4.4.2.cmml" xref="S5.SS1.p2.2.m2.4.4.2"></eq><apply id="S5.SS1.p2.2.m2.4.4.3.cmml" xref="S5.SS1.p2.2.m2.4.4.3"><csymbol cd="ambiguous" id="S5.SS1.p2.2.m2.4.4.3.1.cmml" xref="S5.SS1.p2.2.m2.4.4.3">subscript</csymbol><ci id="S5.SS1.p2.2.m2.4.4.3.2.cmml" xref="S5.SS1.p2.2.m2.4.4.3.2">𝐶</ci><apply id="S5.SS1.p2.2.m2.4.4.3.3.cmml" xref="S5.SS1.p2.2.m2.4.4.3.3"><csymbol cd="ambiguous" id="S5.SS1.p2.2.m2.4.4.3.3.1.cmml" xref="S5.SS1.p2.2.m2.4.4.3.3">subscript</csymbol><ci id="S5.SS1.p2.2.m2.4.4.3.3.2.cmml" xref="S5.SS1.p2.2.m2.4.4.3.3.2">𝜎</ci><cn id="S5.SS1.p2.2.m2.4.4.3.3.3.cmml" type="integer" xref="S5.SS1.p2.2.m2.4.4.3.3.3">0</cn></apply></apply><set id="S5.SS1.p2.2.m2.4.4.1.2.cmml" xref="S5.SS1.p2.2.m2.4.4.1.1"><vector id="S5.SS1.p2.2.m2.4.4.1.1.1.1.cmml" xref="S5.SS1.p2.2.m2.4.4.1.1.1.2"><infinity id="S5.SS1.p2.2.m2.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1"></infinity><ci id="S5.SS1.p2.2.m2.2.2.cmml" xref="S5.SS1.p2.2.m2.2.2">…</ci><infinity id="S5.SS1.p2.2.m2.3.3.cmml" xref="S5.SS1.p2.2.m2.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m2.4c">C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m2.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { ( ∞ , … , ∞ ) }</annotation></semantics></math> is treated separately):</p> </div> <div class="ltx_para" id="S5.SS1.p3"> <ul class="ltx_itemize" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.6">We first transform the arena of <math alttext="G" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><mi id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><ci id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_G</annotation></semantics></math> into a binary arena and adapt the given solution <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S5.I1.i1.p1.2.m2.2"><semantics id="S5.I1.i1.p1.2.m2.2a"><mrow id="S5.I1.i1.p1.2.m2.2.3" xref="S5.I1.i1.p1.2.m2.2.3.cmml"><msub id="S5.I1.i1.p1.2.m2.2.3.2" xref="S5.I1.i1.p1.2.m2.2.3.2.cmml"><mi id="S5.I1.i1.p1.2.m2.2.3.2.2" xref="S5.I1.i1.p1.2.m2.2.3.2.2.cmml">σ</mi><mn id="S5.I1.i1.p1.2.m2.2.3.2.3" xref="S5.I1.i1.p1.2.m2.2.3.2.3.cmml">0</mn></msub><mo id="S5.I1.i1.p1.2.m2.2.3.1" xref="S5.I1.i1.p1.2.m2.2.3.1.cmml">∈</mo><mrow id="S5.I1.i1.p1.2.m2.2.3.3" xref="S5.I1.i1.p1.2.m2.2.3.3.cmml"><mtext id="S5.I1.i1.p1.2.m2.2.3.3.2" xref="S5.I1.i1.p1.2.m2.2.3.3.2a.cmml">SPS</mtext><mo id="S5.I1.i1.p1.2.m2.2.3.3.1" xref="S5.I1.i1.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S5.I1.i1.p1.2.m2.2.3.3.3.2" xref="S5.I1.i1.p1.2.m2.2.3.3.3.1.cmml"><mo id="S5.I1.i1.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S5.I1.i1.p1.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S5.I1.i1.p1.2.m2.1.1" xref="S5.I1.i1.p1.2.m2.1.1.cmml">G</mi><mo id="S5.I1.i1.p1.2.m2.2.3.3.3.2.2" xref="S5.I1.i1.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S5.I1.i1.p1.2.m2.2.2" xref="S5.I1.i1.p1.2.m2.2.2.cmml">B</mi><mo id="S5.I1.i1.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="S5.I1.i1.p1.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.2.m2.2b"><apply id="S5.I1.i1.p1.2.m2.2.3.cmml" xref="S5.I1.i1.p1.2.m2.2.3"><in id="S5.I1.i1.p1.2.m2.2.3.1.cmml" xref="S5.I1.i1.p1.2.m2.2.3.1"></in><apply id="S5.I1.i1.p1.2.m2.2.3.2.cmml" xref="S5.I1.i1.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.2.m2.2.3.2.1.cmml" xref="S5.I1.i1.p1.2.m2.2.3.2">subscript</csymbol><ci id="S5.I1.i1.p1.2.m2.2.3.2.2.cmml" xref="S5.I1.i1.p1.2.m2.2.3.2.2">𝜎</ci><cn id="S5.I1.i1.p1.2.m2.2.3.2.3.cmml" type="integer" xref="S5.I1.i1.p1.2.m2.2.3.2.3">0</cn></apply><apply id="S5.I1.i1.p1.2.m2.2.3.3.cmml" xref="S5.I1.i1.p1.2.m2.2.3.3"><times id="S5.I1.i1.p1.2.m2.2.3.3.1.cmml" xref="S5.I1.i1.p1.2.m2.2.3.3.1"></times><ci id="S5.I1.i1.p1.2.m2.2.3.3.2a.cmml" xref="S5.I1.i1.p1.2.m2.2.3.3.2"><mtext id="S5.I1.i1.p1.2.m2.2.3.3.2.cmml" xref="S5.I1.i1.p1.2.m2.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S5.I1.i1.p1.2.m2.2.3.3.3.1.cmml" xref="S5.I1.i1.p1.2.m2.2.3.3.3.2"><ci id="S5.I1.i1.p1.2.m2.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1">𝐺</ci><ci id="S5.I1.i1.p1.2.m2.2.2.cmml" xref="S5.I1.i1.p1.2.m2.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.2.m2.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.2.m2.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> to the new game. We keep the same notations <math alttext="G" class="ltx_Math" display="inline" id="S5.I1.i1.p1.3.m3.1"><semantics id="S5.I1.i1.p1.3.m3.1a"><mi id="S5.I1.i1.p1.3.m3.1.1" xref="S5.I1.i1.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.3.m3.1b"><ci id="S5.I1.i1.p1.3.m3.1.1.cmml" xref="S5.I1.i1.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.3.m3.1d">italic_G</annotation></semantics></math> and <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.4.m4.1"><semantics id="S5.I1.i1.p1.4.m4.1a"><msub id="S5.I1.i1.p1.4.m4.1.1" xref="S5.I1.i1.p1.4.m4.1.1.cmml"><mi id="S5.I1.i1.p1.4.m4.1.1.2" xref="S5.I1.i1.p1.4.m4.1.1.2.cmml">σ</mi><mn id="S5.I1.i1.p1.4.m4.1.1.3" xref="S5.I1.i1.p1.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.4.m4.1b"><apply id="S5.I1.i1.p1.4.m4.1.1.cmml" xref="S5.I1.i1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.4.m4.1.1.1.cmml" xref="S5.I1.i1.p1.4.m4.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.4.m4.1.1.2.cmml" xref="S5.I1.i1.p1.4.m4.1.1.2">𝜎</ci><cn id="S5.I1.i1.p1.4.m4.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.4.m4.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. We can suppose that <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.5.m5.1"><semantics id="S5.I1.i1.p1.5.m5.1a"><msub id="S5.I1.i1.p1.5.m5.1.1" xref="S5.I1.i1.p1.5.m5.1.1.cmml"><mi id="S5.I1.i1.p1.5.m5.1.1.2" xref="S5.I1.i1.p1.5.m5.1.1.2.cmml">σ</mi><mn id="S5.I1.i1.p1.5.m5.1.1.3" xref="S5.I1.i1.p1.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.5.m5.1b"><apply id="S5.I1.i1.p1.5.m5.1.1.cmml" xref="S5.I1.i1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.5.m5.1.1.1.cmml" xref="S5.I1.i1.p1.5.m5.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.5.m5.1.1.2.cmml" xref="S5.I1.i1.p1.5.m5.1.1.2">𝜎</ci><cn id="S5.I1.i1.p1.5.m5.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.5.m5.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.5.m5.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is Pareto-bounded by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>. We consider a set of witnesses <math alttext="\textsf{Wit}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.6.m6.1"><semantics id="S5.I1.i1.p1.6.m6.1a"><msub id="S5.I1.i1.p1.6.m6.1.1" xref="S5.I1.i1.p1.6.m6.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.I1.i1.p1.6.m6.1.1.2" xref="S5.I1.i1.p1.6.m6.1.1.2a.cmml">Wit</mtext><msub id="S5.I1.i1.p1.6.m6.1.1.3" xref="S5.I1.i1.p1.6.m6.1.1.3.cmml"><mi id="S5.I1.i1.p1.6.m6.1.1.3.2" xref="S5.I1.i1.p1.6.m6.1.1.3.2.cmml">σ</mi><mn id="S5.I1.i1.p1.6.m6.1.1.3.3" xref="S5.I1.i1.p1.6.m6.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.6.m6.1b"><apply id="S5.I1.i1.p1.6.m6.1.1.cmml" xref="S5.I1.i1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.6.m6.1.1.1.cmml" xref="S5.I1.i1.p1.6.m6.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.6.m6.1.1.2a.cmml" xref="S5.I1.i1.p1.6.m6.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.I1.i1.p1.6.m6.1.1.2.cmml" xref="S5.I1.i1.p1.6.m6.1.1.2">Wit</mtext></ci><apply id="S5.I1.i1.p1.6.m6.1.1.3.cmml" xref="S5.I1.i1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i1.p1.6.m6.1.1.3.1.cmml" xref="S5.I1.i1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S5.I1.i1.p1.6.m6.1.1.3.2.cmml" xref="S5.I1.i1.p1.6.m6.1.1.3.2">𝜎</ci><cn id="S5.I1.i1.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S5.I1.i1.p1.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.6.m6.1c">\textsf{Wit}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.6.m6.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.4">We show that at any deviation<span class="ltx_note ltx_role_footnote" id="footnote11"><sup class="ltx_note_mark">11</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">11</sup><span class="ltx_tag ltx_tag_note">11</span>We recall that a deviation is a history <math alttext="hv" class="ltx_Math" display="inline" id="footnote11.m1.1"><semantics id="footnote11.m1.1b"><mrow id="footnote11.m1.1.1" xref="footnote11.m1.1.1.cmml"><mi id="footnote11.m1.1.1.2" xref="footnote11.m1.1.1.2.cmml">h</mi><mo id="footnote11.m1.1.1.1" xref="footnote11.m1.1.1.1.cmml">⁢</mo><mi id="footnote11.m1.1.1.3" xref="footnote11.m1.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote11.m1.1c"><apply id="footnote11.m1.1.1.cmml" xref="footnote11.m1.1.1"><times id="footnote11.m1.1.1.1.cmml" xref="footnote11.m1.1.1.1"></times><ci id="footnote11.m1.1.1.2.cmml" xref="footnote11.m1.1.1.2">ℎ</ci><ci id="footnote11.m1.1.1.3.cmml" xref="footnote11.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m1.1d">hv</annotation><annotation encoding="application/x-llamapun" id="footnote11.m1.1e">italic_h italic_v</annotation></semantics></math> with <math alttext="h\in\textsf{Hist}^{1}" class="ltx_Math" display="inline" id="footnote11.m2.1"><semantics id="footnote11.m2.1b"><mrow id="footnote11.m2.1.1" xref="footnote11.m2.1.1.cmml"><mi id="footnote11.m2.1.1.2" xref="footnote11.m2.1.1.2.cmml">h</mi><mo id="footnote11.m2.1.1.1" xref="footnote11.m2.1.1.1.cmml">∈</mo><msup id="footnote11.m2.1.1.3" xref="footnote11.m2.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="footnote11.m2.1.1.3.2" xref="footnote11.m2.1.1.3.2a.cmml">Hist</mtext><mn id="footnote11.m2.1.1.3.3" xref="footnote11.m2.1.1.3.3.cmml">1</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="footnote11.m2.1c"><apply id="footnote11.m2.1.1.cmml" xref="footnote11.m2.1.1"><in id="footnote11.m2.1.1.1.cmml" xref="footnote11.m2.1.1.1"></in><ci id="footnote11.m2.1.1.2.cmml" xref="footnote11.m2.1.1.2">ℎ</ci><apply id="footnote11.m2.1.1.3.cmml" xref="footnote11.m2.1.1.3"><csymbol cd="ambiguous" id="footnote11.m2.1.1.3.1.cmml" xref="footnote11.m2.1.1.3">superscript</csymbol><ci id="footnote11.m2.1.1.3.2a.cmml" xref="footnote11.m2.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="footnote11.m2.1.1.3.2.cmml" xref="footnote11.m2.1.1.3.2">Hist</mtext></ci><cn id="footnote11.m2.1.1.3.3.cmml" type="integer" xref="footnote11.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m2.1d">h\in\textsf{Hist}^{1}</annotation><annotation encoding="application/x-llamapun" id="footnote11.m2.1e">italic_h ∈ Hist start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="v\in V" class="ltx_Math" display="inline" id="footnote11.m3.1"><semantics id="footnote11.m3.1b"><mrow id="footnote11.m3.1.1" xref="footnote11.m3.1.1.cmml"><mi id="footnote11.m3.1.1.2" xref="footnote11.m3.1.1.2.cmml">v</mi><mo id="footnote11.m3.1.1.1" xref="footnote11.m3.1.1.1.cmml">∈</mo><mi id="footnote11.m3.1.1.3" xref="footnote11.m3.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote11.m3.1c"><apply id="footnote11.m3.1.1.cmml" xref="footnote11.m3.1.1"><in id="footnote11.m3.1.1.1.cmml" xref="footnote11.m3.1.1.1"></in><ci id="footnote11.m3.1.1.2.cmml" xref="footnote11.m3.1.1.2">𝑣</ci><ci id="footnote11.m3.1.1.3.cmml" xref="footnote11.m3.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m3.1d">v\in V</annotation><annotation encoding="application/x-llamapun" id="footnote11.m3.1e">italic_v ∈ italic_V</annotation></semantics></math>, such that <math alttext="h" class="ltx_Math" display="inline" id="footnote11.m4.1"><semantics id="footnote11.m4.1b"><mi id="footnote11.m4.1.1" xref="footnote11.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="footnote11.m4.1c"><ci id="footnote11.m4.1.1.cmml" xref="footnote11.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m4.1d">h</annotation><annotation encoding="application/x-llamapun" id="footnote11.m4.1e">italic_h</annotation></semantics></math> is prefix of some witness, but <math alttext="hv" class="ltx_Math" display="inline" id="footnote11.m5.1"><semantics id="footnote11.m5.1b"><mrow id="footnote11.m5.1.1" xref="footnote11.m5.1.1.cmml"><mi id="footnote11.m5.1.1.2" xref="footnote11.m5.1.1.2.cmml">h</mi><mo id="footnote11.m5.1.1.1" xref="footnote11.m5.1.1.1.cmml">⁢</mo><mi id="footnote11.m5.1.1.3" xref="footnote11.m5.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote11.m5.1c"><apply id="footnote11.m5.1.1.cmml" xref="footnote11.m5.1.1"><times id="footnote11.m5.1.1.1.cmml" xref="footnote11.m5.1.1.1"></times><ci id="footnote11.m5.1.1.2.cmml" xref="footnote11.m5.1.1.2">ℎ</ci><ci id="footnote11.m5.1.1.3.cmml" xref="footnote11.m5.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m5.1d">hv</annotation><annotation encoding="application/x-llamapun" id="footnote11.m5.1e">italic_h italic_v</annotation></semantics></math> is prefix of no witness.</span></span></span>, Player <math alttext="0" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.1"><semantics id="S5.I1.i2.p1.1.m1.1a"><mn id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.1b"><cn id="S5.I1.i2.p1.1.m1.1.1.cmml" type="integer" xref="S5.I1.i2.p1.1.m1.1.1">0</cn></annotation-xml></semantics></math> can switch to a <em class="ltx_emph ltx_font_italic" id="S5.I1.i2.p1.4.1">punishing strategy</em> that imposes that the consistent plays <math alttext="\pi" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.1"><semantics id="S5.I1.i2.p1.2.m2.1a"><mi id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.1b"><ci id="S5.I1.i2.p1.2.m2.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.1d">italic_π</annotation></semantics></math> either satisfy <math alttext="\textsf{val}(\pi)\leq B" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m3.1"><semantics id="S5.I1.i2.p1.3.m3.1a"><mrow id="S5.I1.i2.p1.3.m3.1.2" xref="S5.I1.i2.p1.3.m3.1.2.cmml"><mrow id="S5.I1.i2.p1.3.m3.1.2.2" xref="S5.I1.i2.p1.3.m3.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.I1.i2.p1.3.m3.1.2.2.2" xref="S5.I1.i2.p1.3.m3.1.2.2.2a.cmml">val</mtext><mo id="S5.I1.i2.p1.3.m3.1.2.2.1" xref="S5.I1.i2.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.I1.i2.p1.3.m3.1.2.2.3.2" xref="S5.I1.i2.p1.3.m3.1.2.2.cmml"><mo id="S5.I1.i2.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S5.I1.i2.p1.3.m3.1.2.2.cmml">(</mo><mi id="S5.I1.i2.p1.3.m3.1.1" xref="S5.I1.i2.p1.3.m3.1.1.cmml">π</mi><mo id="S5.I1.i2.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="S5.I1.i2.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.I1.i2.p1.3.m3.1.2.1" xref="S5.I1.i2.p1.3.m3.1.2.1.cmml">≤</mo><mi id="S5.I1.i2.p1.3.m3.1.2.3" xref="S5.I1.i2.p1.3.m3.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.3.m3.1b"><apply id="S5.I1.i2.p1.3.m3.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.2"><leq id="S5.I1.i2.p1.3.m3.1.2.1.cmml" xref="S5.I1.i2.p1.3.m3.1.2.1"></leq><apply id="S5.I1.i2.p1.3.m3.1.2.2.cmml" xref="S5.I1.i2.p1.3.m3.1.2.2"><times id="S5.I1.i2.p1.3.m3.1.2.2.1.cmml" xref="S5.I1.i2.p1.3.m3.1.2.2.1"></times><ci id="S5.I1.i2.p1.3.m3.1.2.2.2a.cmml" xref="S5.I1.i2.p1.3.m3.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.I1.i2.p1.3.m3.1.2.2.2.cmml" xref="S5.I1.i2.p1.3.m3.1.2.2.2">val</mtext></ci><ci id="S5.I1.i2.p1.3.m3.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1">𝜋</ci></apply><ci id="S5.I1.i2.p1.3.m3.1.2.3.cmml" xref="S5.I1.i2.p1.3.m3.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m3.1c">\textsf{val}(\pi)\leq B</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m3.1d">val ( italic_π ) ≤ italic_B</annotation></semantics></math> or <math alttext="\textsf{cost}({\pi})" class="ltx_Math" display="inline" id="S5.I1.i2.p1.4.m4.1"><semantics id="S5.I1.i2.p1.4.m4.1a"><mrow id="S5.I1.i2.p1.4.m4.1.2" xref="S5.I1.i2.p1.4.m4.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.I1.i2.p1.4.m4.1.2.2" xref="S5.I1.i2.p1.4.m4.1.2.2a.cmml">cost</mtext><mo id="S5.I1.i2.p1.4.m4.1.2.1" xref="S5.I1.i2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S5.I1.i2.p1.4.m4.1.2.3.2" xref="S5.I1.i2.p1.4.m4.1.2.cmml"><mo id="S5.I1.i2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S5.I1.i2.p1.4.m4.1.2.cmml">(</mo><mi id="S5.I1.i2.p1.4.m4.1.1" xref="S5.I1.i2.p1.4.m4.1.1.cmml">π</mi><mo id="S5.I1.i2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S5.I1.i2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.4.m4.1b"><apply id="S5.I1.i2.p1.4.m4.1.2.cmml" xref="S5.I1.i2.p1.4.m4.1.2"><times id="S5.I1.i2.p1.4.m4.1.2.1.cmml" xref="S5.I1.i2.p1.4.m4.1.2.1"></times><ci id="S5.I1.i2.p1.4.m4.1.2.2a.cmml" xref="S5.I1.i2.p1.4.m4.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.I1.i2.p1.4.m4.1.2.2.cmml" xref="S5.I1.i2.p1.4.m4.1.2.2">cost</mtext></ci><ci id="S5.I1.i2.p1.4.m4.1.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.4.m4.1c">\textsf{cost}({\pi})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.4.m4.1d">cost ( italic_π )</annotation></semantics></math> is not Pareto-optimal. Moreover, this punishing strategy is finite-memory with an exponential memory.</p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.1">We then show how to transform the witnesses into lassos, and how they can be produced by a finite-memory strategy with exponential memory. We also show that we need at most exponentially many different punishing strategies.</p> </div> </li> </ul> <p class="ltx_p" id="S5.SS1.p3.1">In this way, we get a strategy solution to the SPS problem whose memory size is exponential.</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>Punishing Strategies</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.13">Let <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><msub id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml"><mi id="S5.SS2.p1.1.m1.1.1.2" xref="S5.SS2.p1.1.m1.1.1.2.cmml">σ</mi><mn id="S5.SS2.p1.1.m1.1.1.3" xref="S5.SS2.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><apply id="S5.SS2.p1.1.m1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p1.1.m1.1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS2.p1.1.m1.1.1.2.cmml" xref="S5.SS2.p1.1.m1.1.1.2">𝜎</ci><cn id="S5.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.SS2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be a Pareto-bounded solution to the SPS Problem. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E3" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3</span></a>) (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E4" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>), we get that <math alttext="c_{i}\leq f({B},{t})" class="ltx_Math" display="inline" id="S5.SS2.p1.2.m2.2"><semantics id="S5.SS2.p1.2.m2.2a"><mrow id="S5.SS2.p1.2.m2.2.3" xref="S5.SS2.p1.2.m2.2.3.cmml"><msub id="S5.SS2.p1.2.m2.2.3.2" xref="S5.SS2.p1.2.m2.2.3.2.cmml"><mi id="S5.SS2.p1.2.m2.2.3.2.2" xref="S5.SS2.p1.2.m2.2.3.2.2.cmml">c</mi><mi id="S5.SS2.p1.2.m2.2.3.2.3" xref="S5.SS2.p1.2.m2.2.3.2.3.cmml">i</mi></msub><mo id="S5.SS2.p1.2.m2.2.3.1" xref="S5.SS2.p1.2.m2.2.3.1.cmml">≤</mo><mrow id="S5.SS2.p1.2.m2.2.3.3" xref="S5.SS2.p1.2.m2.2.3.3.cmml"><mi id="S5.SS2.p1.2.m2.2.3.3.2" xref="S5.SS2.p1.2.m2.2.3.3.2.cmml">f</mi><mo id="S5.SS2.p1.2.m2.2.3.3.1" xref="S5.SS2.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S5.SS2.p1.2.m2.2.3.3.3.2" xref="S5.SS2.p1.2.m2.2.3.3.3.1.cmml"><mo id="S5.SS2.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S5.SS2.p1.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S5.SS2.p1.2.m2.1.1" xref="S5.SS2.p1.2.m2.1.1.cmml">B</mi><mo id="S5.SS2.p1.2.m2.2.3.3.3.2.2" xref="S5.SS2.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S5.SS2.p1.2.m2.2.2" xref="S5.SS2.p1.2.m2.2.2.cmml">t</mi><mo id="S5.SS2.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="S5.SS2.p1.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.2.m2.2b"><apply id="S5.SS2.p1.2.m2.2.3.cmml" xref="S5.SS2.p1.2.m2.2.3"><leq id="S5.SS2.p1.2.m2.2.3.1.cmml" xref="S5.SS2.p1.2.m2.2.3.1"></leq><apply id="S5.SS2.p1.2.m2.2.3.2.cmml" xref="S5.SS2.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S5.SS2.p1.2.m2.2.3.2.1.cmml" xref="S5.SS2.p1.2.m2.2.3.2">subscript</csymbol><ci id="S5.SS2.p1.2.m2.2.3.2.2.cmml" xref="S5.SS2.p1.2.m2.2.3.2.2">𝑐</ci><ci id="S5.SS2.p1.2.m2.2.3.2.3.cmml" xref="S5.SS2.p1.2.m2.2.3.2.3">𝑖</ci></apply><apply id="S5.SS2.p1.2.m2.2.3.3.cmml" xref="S5.SS2.p1.2.m2.2.3.3"><times id="S5.SS2.p1.2.m2.2.3.3.1.cmml" xref="S5.SS2.p1.2.m2.2.3.3.1"></times><ci id="S5.SS2.p1.2.m2.2.3.3.2.cmml" xref="S5.SS2.p1.2.m2.2.3.3.2">𝑓</ci><interval closure="open" id="S5.SS2.p1.2.m2.2.3.3.3.1.cmml" xref="S5.SS2.p1.2.m2.2.3.3.3.2"><ci id="S5.SS2.p1.2.m2.1.1.cmml" xref="S5.SS2.p1.2.m2.1.1">𝐵</ci><ci id="S5.SS2.p1.2.m2.2.2.cmml" xref="S5.SS2.p1.2.m2.2.2">𝑡</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.2.m2.2c">c_{i}\leq f({B},{t})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.2.m2.2d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_f ( italic_B , italic_t )</annotation></semantics></math> or <math alttext="c_{i}=\infty" class="ltx_Math" display="inline" id="S5.SS2.p1.3.m3.1"><semantics id="S5.SS2.p1.3.m3.1a"><mrow id="S5.SS2.p1.3.m3.1.1" xref="S5.SS2.p1.3.m3.1.1.cmml"><msub id="S5.SS2.p1.3.m3.1.1.2" xref="S5.SS2.p1.3.m3.1.1.2.cmml"><mi id="S5.SS2.p1.3.m3.1.1.2.2" xref="S5.SS2.p1.3.m3.1.1.2.2.cmml">c</mi><mi id="S5.SS2.p1.3.m3.1.1.2.3" xref="S5.SS2.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS2.p1.3.m3.1.1.1" xref="S5.SS2.p1.3.m3.1.1.1.cmml">=</mo><mi id="S5.SS2.p1.3.m3.1.1.3" mathvariant="normal" xref="S5.SS2.p1.3.m3.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.3.m3.1b"><apply id="S5.SS2.p1.3.m3.1.1.cmml" xref="S5.SS2.p1.3.m3.1.1"><eq id="S5.SS2.p1.3.m3.1.1.1.cmml" xref="S5.SS2.p1.3.m3.1.1.1"></eq><apply id="S5.SS2.p1.3.m3.1.1.2.cmml" xref="S5.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.p1.3.m3.1.1.2.1.cmml" xref="S5.SS2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S5.SS2.p1.3.m3.1.1.2.2.cmml" xref="S5.SS2.p1.3.m3.1.1.2.2">𝑐</ci><ci id="S5.SS2.p1.3.m3.1.1.2.3.cmml" xref="S5.SS2.p1.3.m3.1.1.2.3">𝑖</ci></apply><infinity id="S5.SS2.p1.3.m3.1.1.3.cmml" xref="S5.SS2.p1.3.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.3.m3.1c">c_{i}=\infty</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.3.m3.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∞</annotation></semantics></math> for all <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.SS2.p1.4.m4.1"><semantics id="S5.SS2.p1.4.m4.1a"><mrow id="S5.SS2.p1.4.m4.1.1" xref="S5.SS2.p1.4.m4.1.1.cmml"><mi id="S5.SS2.p1.4.m4.1.1.2" xref="S5.SS2.p1.4.m4.1.1.2.cmml">c</mi><mo id="S5.SS2.p1.4.m4.1.1.1" xref="S5.SS2.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S5.SS2.p1.4.m4.1.1.3" xref="S5.SS2.p1.4.m4.1.1.3.cmml"><mi id="S5.SS2.p1.4.m4.1.1.3.2" xref="S5.SS2.p1.4.m4.1.1.3.2.cmml">C</mi><msub id="S5.SS2.p1.4.m4.1.1.3.3" xref="S5.SS2.p1.4.m4.1.1.3.3.cmml"><mi id="S5.SS2.p1.4.m4.1.1.3.3.2" xref="S5.SS2.p1.4.m4.1.1.3.3.2.cmml">σ</mi><mn id="S5.SS2.p1.4.m4.1.1.3.3.3" xref="S5.SS2.p1.4.m4.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.4.m4.1b"><apply id="S5.SS2.p1.4.m4.1.1.cmml" xref="S5.SS2.p1.4.m4.1.1"><in id="S5.SS2.p1.4.m4.1.1.1.cmml" xref="S5.SS2.p1.4.m4.1.1.1"></in><ci id="S5.SS2.p1.4.m4.1.1.2.cmml" xref="S5.SS2.p1.4.m4.1.1.2">𝑐</ci><apply id="S5.SS2.p1.4.m4.1.1.3.cmml" xref="S5.SS2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p1.4.m4.1.1.3.1.cmml" xref="S5.SS2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.SS2.p1.4.m4.1.1.3.2.cmml" xref="S5.SS2.p1.4.m4.1.1.3.2">𝐶</ci><apply id="S5.SS2.p1.4.m4.1.1.3.3.cmml" xref="S5.SS2.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS2.p1.4.m4.1.1.3.3.1.cmml" xref="S5.SS2.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S5.SS2.p1.4.m4.1.1.3.3.2.cmml" xref="S5.SS2.p1.4.m4.1.1.3.3.2">𝜎</ci><cn id="S5.SS2.p1.4.m4.1.1.3.3.3.cmml" type="integer" xref="S5.SS2.p1.4.m4.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.4.m4.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.4.m4.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and all <math alttext="i\in\{1,\ldots,t\}" class="ltx_Math" display="inline" id="S5.SS2.p1.5.m5.3"><semantics id="S5.SS2.p1.5.m5.3a"><mrow id="S5.SS2.p1.5.m5.3.4" xref="S5.SS2.p1.5.m5.3.4.cmml"><mi id="S5.SS2.p1.5.m5.3.4.2" xref="S5.SS2.p1.5.m5.3.4.2.cmml">i</mi><mo id="S5.SS2.p1.5.m5.3.4.1" xref="S5.SS2.p1.5.m5.3.4.1.cmml">∈</mo><mrow id="S5.SS2.p1.5.m5.3.4.3.2" xref="S5.SS2.p1.5.m5.3.4.3.1.cmml"><mo id="S5.SS2.p1.5.m5.3.4.3.2.1" stretchy="false" xref="S5.SS2.p1.5.m5.3.4.3.1.cmml">{</mo><mn id="S5.SS2.p1.5.m5.1.1" xref="S5.SS2.p1.5.m5.1.1.cmml">1</mn><mo id="S5.SS2.p1.5.m5.3.4.3.2.2" xref="S5.SS2.p1.5.m5.3.4.3.1.cmml">,</mo><mi id="S5.SS2.p1.5.m5.2.2" mathvariant="normal" xref="S5.SS2.p1.5.m5.2.2.cmml">…</mi><mo id="S5.SS2.p1.5.m5.3.4.3.2.3" xref="S5.SS2.p1.5.m5.3.4.3.1.cmml">,</mo><mi id="S5.SS2.p1.5.m5.3.3" xref="S5.SS2.p1.5.m5.3.3.cmml">t</mi><mo id="S5.SS2.p1.5.m5.3.4.3.2.4" stretchy="false" xref="S5.SS2.p1.5.m5.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.5.m5.3b"><apply id="S5.SS2.p1.5.m5.3.4.cmml" xref="S5.SS2.p1.5.m5.3.4"><in id="S5.SS2.p1.5.m5.3.4.1.cmml" xref="S5.SS2.p1.5.m5.3.4.1"></in><ci id="S5.SS2.p1.5.m5.3.4.2.cmml" xref="S5.SS2.p1.5.m5.3.4.2">𝑖</ci><set id="S5.SS2.p1.5.m5.3.4.3.1.cmml" xref="S5.SS2.p1.5.m5.3.4.3.2"><cn id="S5.SS2.p1.5.m5.1.1.cmml" type="integer" xref="S5.SS2.p1.5.m5.1.1">1</cn><ci id="S5.SS2.p1.5.m5.2.2.cmml" xref="S5.SS2.p1.5.m5.2.2">…</ci><ci id="S5.SS2.p1.5.m5.3.3.cmml" xref="S5.SS2.p1.5.m5.3.3">𝑡</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.5.m5.3c">i\in\{1,\ldots,t\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.5.m5.3d">italic_i ∈ { 1 , … , italic_t }</annotation></semantics></math>, such that <math alttext="f({B},{t})" class="ltx_Math" display="inline" id="S5.SS2.p1.6.m6.2"><semantics id="S5.SS2.p1.6.m6.2a"><mrow id="S5.SS2.p1.6.m6.2.3" xref="S5.SS2.p1.6.m6.2.3.cmml"><mi id="S5.SS2.p1.6.m6.2.3.2" xref="S5.SS2.p1.6.m6.2.3.2.cmml">f</mi><mo id="S5.SS2.p1.6.m6.2.3.1" xref="S5.SS2.p1.6.m6.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.p1.6.m6.2.3.3.2" xref="S5.SS2.p1.6.m6.2.3.3.1.cmml"><mo id="S5.SS2.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S5.SS2.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="S5.SS2.p1.6.m6.1.1" xref="S5.SS2.p1.6.m6.1.1.cmml">B</mi><mo id="S5.SS2.p1.6.m6.2.3.3.2.2" xref="S5.SS2.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="S5.SS2.p1.6.m6.2.2" xref="S5.SS2.p1.6.m6.2.2.cmml">t</mi><mo id="S5.SS2.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S5.SS2.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.6.m6.2b"><apply id="S5.SS2.p1.6.m6.2.3.cmml" xref="S5.SS2.p1.6.m6.2.3"><times id="S5.SS2.p1.6.m6.2.3.1.cmml" xref="S5.SS2.p1.6.m6.2.3.1"></times><ci id="S5.SS2.p1.6.m6.2.3.2.cmml" xref="S5.SS2.p1.6.m6.2.3.2">𝑓</ci><interval closure="open" id="S5.SS2.p1.6.m6.2.3.3.1.cmml" xref="S5.SS2.p1.6.m6.2.3.3.2"><ci id="S5.SS2.p1.6.m6.1.1.cmml" xref="S5.SS2.p1.6.m6.1.1">𝐵</ci><ci id="S5.SS2.p1.6.m6.2.2.cmml" xref="S5.SS2.p1.6.m6.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.6.m6.2c">f({B},{t})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.6.m6.2d">italic_f ( italic_B , italic_t )</annotation></semantics></math> is exponentially bounded. Moreover, if a play <math alttext="\rho" class="ltx_Math" display="inline" id="S5.SS2.p1.7.m7.1"><semantics id="S5.SS2.p1.7.m7.1a"><mi id="S5.SS2.p1.7.m7.1.1" xref="S5.SS2.p1.7.m7.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.7.m7.1b"><ci id="S5.SS2.p1.7.m7.1.1.cmml" xref="S5.SS2.p1.7.m7.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.7.m7.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.7.m7.1d">italic_ρ</annotation></semantics></math> is Pareto-optimal, then <math alttext="\textsf{val}(\rho)\leq B" class="ltx_Math" display="inline" id="S5.SS2.p1.8.m8.1"><semantics id="S5.SS2.p1.8.m8.1a"><mrow id="S5.SS2.p1.8.m8.1.2" xref="S5.SS2.p1.8.m8.1.2.cmml"><mrow id="S5.SS2.p1.8.m8.1.2.2" xref="S5.SS2.p1.8.m8.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.8.m8.1.2.2.2" xref="S5.SS2.p1.8.m8.1.2.2.2a.cmml">val</mtext><mo id="S5.SS2.p1.8.m8.1.2.2.1" xref="S5.SS2.p1.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p1.8.m8.1.2.2.3.2" xref="S5.SS2.p1.8.m8.1.2.2.cmml"><mo id="S5.SS2.p1.8.m8.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.p1.8.m8.1.2.2.cmml">(</mo><mi id="S5.SS2.p1.8.m8.1.1" xref="S5.SS2.p1.8.m8.1.1.cmml">ρ</mi><mo id="S5.SS2.p1.8.m8.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.p1.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p1.8.m8.1.2.1" xref="S5.SS2.p1.8.m8.1.2.1.cmml">≤</mo><mi id="S5.SS2.p1.8.m8.1.2.3" xref="S5.SS2.p1.8.m8.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.8.m8.1b"><apply id="S5.SS2.p1.8.m8.1.2.cmml" xref="S5.SS2.p1.8.m8.1.2"><leq id="S5.SS2.p1.8.m8.1.2.1.cmml" xref="S5.SS2.p1.8.m8.1.2.1"></leq><apply id="S5.SS2.p1.8.m8.1.2.2.cmml" xref="S5.SS2.p1.8.m8.1.2.2"><times id="S5.SS2.p1.8.m8.1.2.2.1.cmml" xref="S5.SS2.p1.8.m8.1.2.2.1"></times><ci id="S5.SS2.p1.8.m8.1.2.2.2a.cmml" xref="S5.SS2.p1.8.m8.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.8.m8.1.2.2.2.cmml" xref="S5.SS2.p1.8.m8.1.2.2.2">val</mtext></ci><ci id="S5.SS2.p1.8.m8.1.1.cmml" xref="S5.SS2.p1.8.m8.1.1">𝜌</ci></apply><ci id="S5.SS2.p1.8.m8.1.2.3.cmml" xref="S5.SS2.p1.8.m8.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.8.m8.1c">\textsf{val}(\rho)\leq B</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.8.m8.1d">val ( italic_ρ ) ≤ italic_B</annotation></semantics></math>. We define for each history <math alttext="g\in\textsf{Hist}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.SS2.p1.9.m9.1"><semantics id="S5.SS2.p1.9.m9.1a"><mrow id="S5.SS2.p1.9.m9.1.1" xref="S5.SS2.p1.9.m9.1.1.cmml"><mi id="S5.SS2.p1.9.m9.1.1.2" xref="S5.SS2.p1.9.m9.1.1.2.cmml">g</mi><mo id="S5.SS2.p1.9.m9.1.1.1" xref="S5.SS2.p1.9.m9.1.1.1.cmml">∈</mo><msub id="S5.SS2.p1.9.m9.1.1.3" xref="S5.SS2.p1.9.m9.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.9.m9.1.1.3.2" xref="S5.SS2.p1.9.m9.1.1.3.2a.cmml">Hist</mtext><msub id="S5.SS2.p1.9.m9.1.1.3.3" xref="S5.SS2.p1.9.m9.1.1.3.3.cmml"><mi id="S5.SS2.p1.9.m9.1.1.3.3.2" xref="S5.SS2.p1.9.m9.1.1.3.3.2.cmml">σ</mi><mn id="S5.SS2.p1.9.m9.1.1.3.3.3" xref="S5.SS2.p1.9.m9.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.9.m9.1b"><apply id="S5.SS2.p1.9.m9.1.1.cmml" xref="S5.SS2.p1.9.m9.1.1"><in id="S5.SS2.p1.9.m9.1.1.1.cmml" xref="S5.SS2.p1.9.m9.1.1.1"></in><ci id="S5.SS2.p1.9.m9.1.1.2.cmml" xref="S5.SS2.p1.9.m9.1.1.2">𝑔</ci><apply id="S5.SS2.p1.9.m9.1.1.3.cmml" xref="S5.SS2.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p1.9.m9.1.1.3.1.cmml" xref="S5.SS2.p1.9.m9.1.1.3">subscript</csymbol><ci id="S5.SS2.p1.9.m9.1.1.3.2a.cmml" xref="S5.SS2.p1.9.m9.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.9.m9.1.1.3.2.cmml" xref="S5.SS2.p1.9.m9.1.1.3.2">Hist</mtext></ci><apply id="S5.SS2.p1.9.m9.1.1.3.3.cmml" xref="S5.SS2.p1.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS2.p1.9.m9.1.1.3.3.1.cmml" xref="S5.SS2.p1.9.m9.1.1.3.3">subscript</csymbol><ci id="S5.SS2.p1.9.m9.1.1.3.3.2.cmml" xref="S5.SS2.p1.9.m9.1.1.3.3.2">𝜎</ci><cn id="S5.SS2.p1.9.m9.1.1.3.3.3.cmml" type="integer" xref="S5.SS2.p1.9.m9.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.9.m9.1c">g\in\textsf{Hist}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.9.m9.1d">italic_g ∈ Hist start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> its <em class="ltx_emph ltx_font_italic" id="S5.SS2.p1.13.1">record</em> <math alttext="\textsf{rec}({h})=" class="ltx_Math" display="inline" id="S5.SS2.p1.10.m10.1"><semantics id="S5.SS2.p1.10.m10.1a"><mrow id="S5.SS2.p1.10.m10.1.2" xref="S5.SS2.p1.10.m10.1.2.cmml"><mrow id="S5.SS2.p1.10.m10.1.2.2" xref="S5.SS2.p1.10.m10.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.10.m10.1.2.2.2" xref="S5.SS2.p1.10.m10.1.2.2.2a.cmml">rec</mtext><mo id="S5.SS2.p1.10.m10.1.2.2.1" xref="S5.SS2.p1.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p1.10.m10.1.2.2.3.2" xref="S5.SS2.p1.10.m10.1.2.2.cmml"><mo id="S5.SS2.p1.10.m10.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.p1.10.m10.1.2.2.cmml">(</mo><mi id="S5.SS2.p1.10.m10.1.1" xref="S5.SS2.p1.10.m10.1.1.cmml">h</mi><mo id="S5.SS2.p1.10.m10.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.p1.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p1.10.m10.1.2.1" xref="S5.SS2.p1.10.m10.1.2.1.cmml">=</mo><mi id="S5.SS2.p1.10.m10.1.2.3" xref="S5.SS2.p1.10.m10.1.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.10.m10.1b"><apply id="S5.SS2.p1.10.m10.1.2.cmml" xref="S5.SS2.p1.10.m10.1.2"><eq id="S5.SS2.p1.10.m10.1.2.1.cmml" xref="S5.SS2.p1.10.m10.1.2.1"></eq><apply id="S5.SS2.p1.10.m10.1.2.2.cmml" xref="S5.SS2.p1.10.m10.1.2.2"><times id="S5.SS2.p1.10.m10.1.2.2.1.cmml" xref="S5.SS2.p1.10.m10.1.2.2.1"></times><ci id="S5.SS2.p1.10.m10.1.2.2.2a.cmml" xref="S5.SS2.p1.10.m10.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.10.m10.1.2.2.2.cmml" xref="S5.SS2.p1.10.m10.1.2.2.2">rec</mtext></ci><ci id="S5.SS2.p1.10.m10.1.1.cmml" xref="S5.SS2.p1.10.m10.1.1">ℎ</ci></apply><csymbol cd="latexml" id="S5.SS2.p1.10.m10.1.2.3.cmml" xref="S5.SS2.p1.10.m10.1.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.10.m10.1c">\textsf{rec}({h})=</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.10.m10.1d">rec ( italic_h ) =</annotation></semantics></math> <math alttext="(w(h),\textsf{val}(h),\textsf{cost}({h}))" class="ltx_Math" display="inline" id="S5.SS2.p1.11.m11.6"><semantics id="S5.SS2.p1.11.m11.6a"><mrow id="S5.SS2.p1.11.m11.6.6.3" xref="S5.SS2.p1.11.m11.6.6.4.cmml"><mo id="S5.SS2.p1.11.m11.6.6.3.4" stretchy="false" xref="S5.SS2.p1.11.m11.6.6.4.cmml">(</mo><mrow id="S5.SS2.p1.11.m11.4.4.1.1" xref="S5.SS2.p1.11.m11.4.4.1.1.cmml"><mi id="S5.SS2.p1.11.m11.4.4.1.1.2" xref="S5.SS2.p1.11.m11.4.4.1.1.2.cmml">w</mi><mo id="S5.SS2.p1.11.m11.4.4.1.1.1" xref="S5.SS2.p1.11.m11.4.4.1.1.1.cmml">⁢</mo><mrow id="S5.SS2.p1.11.m11.4.4.1.1.3.2" xref="S5.SS2.p1.11.m11.4.4.1.1.cmml"><mo id="S5.SS2.p1.11.m11.4.4.1.1.3.2.1" stretchy="false" xref="S5.SS2.p1.11.m11.4.4.1.1.cmml">(</mo><mi id="S5.SS2.p1.11.m11.1.1" xref="S5.SS2.p1.11.m11.1.1.cmml">h</mi><mo id="S5.SS2.p1.11.m11.4.4.1.1.3.2.2" stretchy="false" xref="S5.SS2.p1.11.m11.4.4.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p1.11.m11.6.6.3.5" xref="S5.SS2.p1.11.m11.6.6.4.cmml">,</mo><mrow id="S5.SS2.p1.11.m11.5.5.2.2" xref="S5.SS2.p1.11.m11.5.5.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.11.m11.5.5.2.2.2" xref="S5.SS2.p1.11.m11.5.5.2.2.2a.cmml">val</mtext><mo id="S5.SS2.p1.11.m11.5.5.2.2.1" xref="S5.SS2.p1.11.m11.5.5.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p1.11.m11.5.5.2.2.3.2" xref="S5.SS2.p1.11.m11.5.5.2.2.cmml"><mo id="S5.SS2.p1.11.m11.5.5.2.2.3.2.1" stretchy="false" xref="S5.SS2.p1.11.m11.5.5.2.2.cmml">(</mo><mi id="S5.SS2.p1.11.m11.2.2" xref="S5.SS2.p1.11.m11.2.2.cmml">h</mi><mo id="S5.SS2.p1.11.m11.5.5.2.2.3.2.2" stretchy="false" xref="S5.SS2.p1.11.m11.5.5.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p1.11.m11.6.6.3.6" xref="S5.SS2.p1.11.m11.6.6.4.cmml">,</mo><mrow id="S5.SS2.p1.11.m11.6.6.3.3" xref="S5.SS2.p1.11.m11.6.6.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.11.m11.6.6.3.3.2" xref="S5.SS2.p1.11.m11.6.6.3.3.2a.cmml">cost</mtext><mo id="S5.SS2.p1.11.m11.6.6.3.3.1" xref="S5.SS2.p1.11.m11.6.6.3.3.1.cmml">⁢</mo><mrow id="S5.SS2.p1.11.m11.6.6.3.3.3.2" xref="S5.SS2.p1.11.m11.6.6.3.3.cmml"><mo id="S5.SS2.p1.11.m11.6.6.3.3.3.2.1" stretchy="false" xref="S5.SS2.p1.11.m11.6.6.3.3.cmml">(</mo><mi id="S5.SS2.p1.11.m11.3.3" xref="S5.SS2.p1.11.m11.3.3.cmml">h</mi><mo id="S5.SS2.p1.11.m11.6.6.3.3.3.2.2" stretchy="false" xref="S5.SS2.p1.11.m11.6.6.3.3.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p1.11.m11.6.6.3.7" stretchy="false" xref="S5.SS2.p1.11.m11.6.6.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.11.m11.6b"><vector id="S5.SS2.p1.11.m11.6.6.4.cmml" xref="S5.SS2.p1.11.m11.6.6.3"><apply id="S5.SS2.p1.11.m11.4.4.1.1.cmml" xref="S5.SS2.p1.11.m11.4.4.1.1"><times id="S5.SS2.p1.11.m11.4.4.1.1.1.cmml" xref="S5.SS2.p1.11.m11.4.4.1.1.1"></times><ci id="S5.SS2.p1.11.m11.4.4.1.1.2.cmml" xref="S5.SS2.p1.11.m11.4.4.1.1.2">𝑤</ci><ci id="S5.SS2.p1.11.m11.1.1.cmml" xref="S5.SS2.p1.11.m11.1.1">ℎ</ci></apply><apply id="S5.SS2.p1.11.m11.5.5.2.2.cmml" xref="S5.SS2.p1.11.m11.5.5.2.2"><times id="S5.SS2.p1.11.m11.5.5.2.2.1.cmml" xref="S5.SS2.p1.11.m11.5.5.2.2.1"></times><ci id="S5.SS2.p1.11.m11.5.5.2.2.2a.cmml" xref="S5.SS2.p1.11.m11.5.5.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.11.m11.5.5.2.2.2.cmml" xref="S5.SS2.p1.11.m11.5.5.2.2.2">val</mtext></ci><ci id="S5.SS2.p1.11.m11.2.2.cmml" xref="S5.SS2.p1.11.m11.2.2">ℎ</ci></apply><apply id="S5.SS2.p1.11.m11.6.6.3.3.cmml" xref="S5.SS2.p1.11.m11.6.6.3.3"><times id="S5.SS2.p1.11.m11.6.6.3.3.1.cmml" xref="S5.SS2.p1.11.m11.6.6.3.3.1"></times><ci id="S5.SS2.p1.11.m11.6.6.3.3.2a.cmml" xref="S5.SS2.p1.11.m11.6.6.3.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p1.11.m11.6.6.3.3.2.cmml" xref="S5.SS2.p1.11.m11.6.6.3.3.2">cost</mtext></ci><ci id="S5.SS2.p1.11.m11.3.3.cmml" xref="S5.SS2.p1.11.m11.3.3">ℎ</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.11.m11.6c">(w(h),\textsf{val}(h),\textsf{cost}({h}))</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.11.m11.6d">( italic_w ( italic_h ) , val ( italic_h ) , cost ( italic_h ) )</annotation></semantics></math> whose values are <em class="ltx_emph ltx_font_italic" id="S5.SS2.p1.13.2">truncated</em> to <math alttext="\infty" class="ltx_Math" display="inline" id="S5.SS2.p1.12.m12.1"><semantics id="S5.SS2.p1.12.m12.1a"><mi id="S5.SS2.p1.12.m12.1.1" mathvariant="normal" xref="S5.SS2.p1.12.m12.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.12.m12.1b"><infinity id="S5.SS2.p1.12.m12.1.1.cmml" xref="S5.SS2.p1.12.m12.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.12.m12.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.12.m12.1d">∞</annotation></semantics></math> if they are greater than <math alttext="f({B},{t})" class="ltx_Math" display="inline" id="S5.SS2.p1.13.m13.2"><semantics id="S5.SS2.p1.13.m13.2a"><mrow id="S5.SS2.p1.13.m13.2.3" xref="S5.SS2.p1.13.m13.2.3.cmml"><mi id="S5.SS2.p1.13.m13.2.3.2" xref="S5.SS2.p1.13.m13.2.3.2.cmml">f</mi><mo id="S5.SS2.p1.13.m13.2.3.1" xref="S5.SS2.p1.13.m13.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.p1.13.m13.2.3.3.2" xref="S5.SS2.p1.13.m13.2.3.3.1.cmml"><mo id="S5.SS2.p1.13.m13.2.3.3.2.1" stretchy="false" xref="S5.SS2.p1.13.m13.2.3.3.1.cmml">(</mo><mi id="S5.SS2.p1.13.m13.1.1" xref="S5.SS2.p1.13.m13.1.1.cmml">B</mi><mo id="S5.SS2.p1.13.m13.2.3.3.2.2" xref="S5.SS2.p1.13.m13.2.3.3.1.cmml">,</mo><mi id="S5.SS2.p1.13.m13.2.2" xref="S5.SS2.p1.13.m13.2.2.cmml">t</mi><mo id="S5.SS2.p1.13.m13.2.3.3.2.3" stretchy="false" xref="S5.SS2.p1.13.m13.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.13.m13.2b"><apply id="S5.SS2.p1.13.m13.2.3.cmml" xref="S5.SS2.p1.13.m13.2.3"><times id="S5.SS2.p1.13.m13.2.3.1.cmml" xref="S5.SS2.p1.13.m13.2.3.1"></times><ci id="S5.SS2.p1.13.m13.2.3.2.cmml" xref="S5.SS2.p1.13.m13.2.3.2">𝑓</ci><interval closure="open" id="S5.SS2.p1.13.m13.2.3.3.1.cmml" xref="S5.SS2.p1.13.m13.2.3.3.2"><ci id="S5.SS2.p1.13.m13.1.1.cmml" xref="S5.SS2.p1.13.m13.1.1">𝐵</ci><ci id="S5.SS2.p1.13.m13.2.2.cmml" xref="S5.SS2.p1.13.m13.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.13.m13.2c">f({B},{t})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.13.m13.2d">italic_f ( italic_B , italic_t )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.2">We define for each deviation <math alttext="hv" class="ltx_Math" display="inline" id="S5.SS2.p2.1.m1.1"><semantics id="S5.SS2.p2.1.m1.1a"><mrow id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml"><mi id="S5.SS2.p2.1.m1.1.1.2" xref="S5.SS2.p2.1.m1.1.1.2.cmml">h</mi><mo id="S5.SS2.p2.1.m1.1.1.1" xref="S5.SS2.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S5.SS2.p2.1.m1.1.1.3" xref="S5.SS2.p2.1.m1.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.1.m1.1b"><apply id="S5.SS2.p2.1.m1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1"><times id="S5.SS2.p2.1.m1.1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1.1"></times><ci id="S5.SS2.p2.1.m1.1.1.2.cmml" xref="S5.SS2.p2.1.m1.1.1.2">ℎ</ci><ci id="S5.SS2.p2.1.m1.1.1.3.cmml" xref="S5.SS2.p2.1.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.1.m1.1d">italic_h italic_v</annotation></semantics></math> a punishing strategy <math alttext="\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}" class="ltx_Math" display="inline" id="S5.SS2.p2.2.m2.2"><semantics id="S5.SS2.p2.2.m2.2a"><msubsup id="S5.SS2.p2.2.m2.2.3" xref="S5.SS2.p2.2.m2.2.3.cmml"><mi id="S5.SS2.p2.2.m2.2.3.2.2" xref="S5.SS2.p2.2.m2.2.3.2.2.cmml">τ</mi><mrow id="S5.SS2.p2.2.m2.2.2.2.2" xref="S5.SS2.p2.2.m2.2.2.2.3.cmml"><mi id="S5.SS2.p2.2.m2.1.1.1.1" xref="S5.SS2.p2.2.m2.1.1.1.1.cmml">v</mi><mo id="S5.SS2.p2.2.m2.2.2.2.2.2" xref="S5.SS2.p2.2.m2.2.2.2.3.cmml">,</mo><mrow id="S5.SS2.p2.2.m2.2.2.2.2.1" xref="S5.SS2.p2.2.m2.2.2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p2.2.m2.2.2.2.2.1.3" xref="S5.SS2.p2.2.m2.2.2.2.2.1.3a.cmml">rec</mtext><mo id="S5.SS2.p2.2.m2.2.2.2.2.1.2" xref="S5.SS2.p2.2.m2.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.cmml"><mo id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.cmml"><mi id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.2" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.2.cmml">h</mi><mo id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.1" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.3" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p2.2.m2.2.3.2.3" xref="S5.SS2.p2.2.m2.2.3.2.3a.cmml">Pun</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.2.m2.2b"><apply id="S5.SS2.p2.2.m2.2.3.cmml" xref="S5.SS2.p2.2.m2.2.3"><csymbol cd="ambiguous" id="S5.SS2.p2.2.m2.2.3.1.cmml" xref="S5.SS2.p2.2.m2.2.3">subscript</csymbol><apply id="S5.SS2.p2.2.m2.2.3.2.cmml" xref="S5.SS2.p2.2.m2.2.3"><csymbol cd="ambiguous" id="S5.SS2.p2.2.m2.2.3.2.1.cmml" xref="S5.SS2.p2.2.m2.2.3">superscript</csymbol><ci id="S5.SS2.p2.2.m2.2.3.2.2.cmml" xref="S5.SS2.p2.2.m2.2.3.2.2">𝜏</ci><ci id="S5.SS2.p2.2.m2.2.3.2.3a.cmml" xref="S5.SS2.p2.2.m2.2.3.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p2.2.m2.2.3.2.3.cmml" mathsize="70%" xref="S5.SS2.p2.2.m2.2.3.2.3">Pun</mtext></ci></apply><list id="S5.SS2.p2.2.m2.2.2.2.3.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2"><ci id="S5.SS2.p2.2.m2.1.1.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1.1.1">𝑣</ci><apply id="S5.SS2.p2.2.m2.2.2.2.2.1.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2.1"><times id="S5.SS2.p2.2.m2.2.2.2.2.1.2.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2.1.2"></times><ci id="S5.SS2.p2.2.m2.2.2.2.2.1.3a.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.SS2.p2.2.m2.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.SS2.p2.2.m2.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1"><times id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.1.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.1"></times><ci id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.2.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.3.cmml" xref="S5.SS2.p2.2.m2.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.2.m2.2c">\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.2.m2.2d">italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT</annotation></semantics></math> as stated in the next lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.1.1.1">Lemma 5.2</span></span></h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.11"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem2.p1.11.11">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> be an SP game, <math alttext="B\in{\rm Nature}" 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">B</mi><mo id="S5.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S5.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">Nature</mi></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"><in id="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1"></in><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2">𝐵</ci><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>, and <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" 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.m3.2.3" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.cmml"><msub id="S5.Thmtheorem2.p1.3.3.m3.2.3.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2.cmml"><mi id="S5.Thmtheorem2.p1.3.3.m3.2.3.2.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2.2.cmml">σ</mi><mn id="S5.Thmtheorem2.p1.3.3.m3.2.3.2.3" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2.3.cmml">0</mn></msub><mo id="S5.Thmtheorem2.p1.3.3.m3.2.3.1" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem2.p1.3.3.m3.2.3.3" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.1" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.1.cmml"><mo id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem2.p1.3.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml">G</mi><mo id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.2.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.3.3.m3.2.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.cmml">B</mi><mo id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.2.3" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.3.m3.2b"><apply id="S5.Thmtheorem2.p1.3.3.m3.2.3.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3"><in id="S5.Thmtheorem2.p1.3.3.m3.2.3.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.1"></in><apply id="S5.Thmtheorem2.p1.3.3.m3.2.3.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.3.m3.2.3.2.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem2.p1.3.3.m3.2.3.2.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2.2">𝜎</ci><cn id="S5.Thmtheorem2.p1.3.3.m3.2.3.2.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2.3">0</cn></apply><apply id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3"><times id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.1"></times><ci id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.2a.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.3.3.2"><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1">𝐺</ci><ci id="S5.Thmtheorem2.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.3.m3.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.3.m3.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> be a Pareto-bounded solution. Suppose <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.4.4.m4.4"><semantics id="S5.Thmtheorem2.p1.4.4.m4.4a"><mrow id="S5.Thmtheorem2.p1.4.4.m4.4.4" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.cmml"><msub id="S5.Thmtheorem2.p1.4.4.m4.4.4.3" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.cmml"><mi id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.2" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.2.cmml">C</mi><msub id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.cmml"><mi id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.2" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.2.cmml">σ</mi><mn id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.3" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S5.Thmtheorem2.p1.4.4.m4.4.4.2" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.2.cmml">≠</mo><mrow id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.2.cmml"><mo id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.2.cmml">{</mo><mrow id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.2" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.1.cmml"><mo id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.1.cmml">(</mo><mi id="S5.Thmtheorem2.p1.4.4.m4.1.1" mathvariant="normal" xref="S5.Thmtheorem2.p1.4.4.m4.1.1.cmml">∞</mi><mo id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.2.2" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.4.4.m4.2.2" mathvariant="normal" xref="S5.Thmtheorem2.p1.4.4.m4.2.2.cmml">…</mi><mo id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.2.3" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.4.4.m4.3.3" mathvariant="normal" xref="S5.Thmtheorem2.p1.4.4.m4.3.3.cmml">∞</mi><mo id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.2.4" stretchy="false" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.4.4.m4.4b"><apply id="S5.Thmtheorem2.p1.4.4.m4.4.4.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4"><neq id="S5.Thmtheorem2.p1.4.4.m4.4.4.2.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.2"></neq><apply id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.1.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3">subscript</csymbol><ci id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.2.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.2">𝐶</ci><apply id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.1.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3">subscript</csymbol><ci id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.2.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.2">𝜎</ci><cn id="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.3.3.3">0</cn></apply></apply><set id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.2.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1"><vector id="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.4.4.1.1.1.2"><infinity id="S5.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.1.1"></infinity><ci id="S5.Thmtheorem2.p1.4.4.m4.2.2.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.2.2">…</ci><infinity id="S5.Thmtheorem2.p1.4.4.m4.3.3.cmml" xref="S5.Thmtheorem2.p1.4.4.m4.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.4.4.m4.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.4.4.m4.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>. Let <math alttext="hv" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.5.5.m5.1"><semantics id="S5.Thmtheorem2.p1.5.5.m5.1a"><mrow id="S5.Thmtheorem2.p1.5.5.m5.1.1" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.cmml"><mi id="S5.Thmtheorem2.p1.5.5.m5.1.1.2" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem2.p1.5.5.m5.1.1.1" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem2.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.5.5.m5.1b"><apply id="S5.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1"><times id="S5.Thmtheorem2.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.1"></times><ci id="S5.Thmtheorem2.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.2">ℎ</ci><ci id="S5.Thmtheorem2.p1.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem2.p1.5.5.m5.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.5.5.m5.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.5.5.m5.1d">italic_h italic_v</annotation></semantics></math> be a deviation such that <math alttext="\textsf{val}(h)=\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.6.6.m6.1"><semantics id="S5.Thmtheorem2.p1.6.6.m6.1a"><mrow id="S5.Thmtheorem2.p1.6.6.m6.1.2" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.cmml"><mrow id="S5.Thmtheorem2.p1.6.6.m6.1.2.2" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.2" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.1" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.3.2" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.cmml"><mo id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem2.p1.6.6.m6.1.1" xref="S5.Thmtheorem2.p1.6.6.m6.1.1.cmml">h</mi><mo id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem2.p1.6.6.m6.1.2.1" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.1.cmml">=</mo><mi id="S5.Thmtheorem2.p1.6.6.m6.1.2.3" mathvariant="normal" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.6.6.m6.1b"><apply id="S5.Thmtheorem2.p1.6.6.m6.1.2.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.2"><eq id="S5.Thmtheorem2.p1.6.6.m6.1.2.1.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.1"></eq><apply id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2"><times id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.1.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.1"></times><ci id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.2a.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.6.6.m6.1.2.2.2.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.1">ℎ</ci></apply><infinity id="S5.Thmtheorem2.p1.6.6.m6.1.2.3.cmml" xref="S5.Thmtheorem2.p1.6.6.m6.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.6.6.m6.1c">\textsf{val}(h)=\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.6.6.m6.1d">val ( italic_h ) = ∞</annotation></semantics></math> (resp. <math alttext="\textsf{val}(h)<\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.7.7.m7.1"><semantics id="S5.Thmtheorem2.p1.7.7.m7.1a"><mrow id="S5.Thmtheorem2.p1.7.7.m7.1.2" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.cmml"><mrow id="S5.Thmtheorem2.p1.7.7.m7.1.2.2" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.2" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.1" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.3.2" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.cmml"><mo id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem2.p1.7.7.m7.1.1" xref="S5.Thmtheorem2.p1.7.7.m7.1.1.cmml">h</mi><mo id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem2.p1.7.7.m7.1.2.1" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.1.cmml"><</mo><mi id="S5.Thmtheorem2.p1.7.7.m7.1.2.3" mathvariant="normal" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.7.7.m7.1b"><apply id="S5.Thmtheorem2.p1.7.7.m7.1.2.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.2"><lt id="S5.Thmtheorem2.p1.7.7.m7.1.2.1.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.1"></lt><apply id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2"><times id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.1.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.1"></times><ci id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.2a.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.7.7.m7.1.2.2.2.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.1">ℎ</ci></apply><infinity id="S5.Thmtheorem2.p1.7.7.m7.1.2.3.cmml" xref="S5.Thmtheorem2.p1.7.7.m7.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.7.7.m7.1c">\textsf{val}(h)<\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.7.7.m7.1d">val ( italic_h ) < ∞</annotation></semantics></math>). Then there exists a finite-memory strategy <math alttext="\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.8.8.m8.2"><semantics id="S5.Thmtheorem2.p1.8.8.m8.2a"><msubsup id="S5.Thmtheorem2.p1.8.8.m8.2.3" xref="S5.Thmtheorem2.p1.8.8.m8.2.3.cmml"><mi id="S5.Thmtheorem2.p1.8.8.m8.2.3.2.2" xref="S5.Thmtheorem2.p1.8.8.m8.2.3.2.2.cmml">τ</mi><mrow id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.3.cmml"><mi id="S5.Thmtheorem2.p1.8.8.m8.1.1.1.1" xref="S5.Thmtheorem2.p1.8.8.m8.1.1.1.1.cmml">v</mi><mo id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.2" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.3.cmml">,</mo><mrow id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.3" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.3a.cmml">rec</mtext><mo id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.2" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.cmml"><mo id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.2" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.1" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.3" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.8.8.m8.2.3.2.3" xref="S5.Thmtheorem2.p1.8.8.m8.2.3.2.3a.cmml">Pun</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.8.8.m8.2b"><apply id="S5.Thmtheorem2.p1.8.8.m8.2.3.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.8.8.m8.2.3.1.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.3">subscript</csymbol><apply id="S5.Thmtheorem2.p1.8.8.m8.2.3.2.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.8.8.m8.2.3.2.1.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.3">superscript</csymbol><ci id="S5.Thmtheorem2.p1.8.8.m8.2.3.2.2.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.3.2.2">𝜏</ci><ci id="S5.Thmtheorem2.p1.8.8.m8.2.3.2.3a.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.3.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.8.8.m8.2.3.2.3.cmml" mathsize="70%" xref="S5.Thmtheorem2.p1.8.8.m8.2.3.2.3">Pun</mtext></ci></apply><list id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.3.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2"><ci id="S5.Thmtheorem2.p1.8.8.m8.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.1.1.1.1">𝑣</ci><apply id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1"><times id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.2"></times><ci id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.3a.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1"><times id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.1"></times><ci id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.8.8.m8.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.8.8.m8.2c">\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.8.8.m8.2d">italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT</annotation></semantics></math> with an exponential memory size (resp. with size <math alttext="1" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.9.9.m9.1"><semantics id="S5.Thmtheorem2.p1.9.9.m9.1a"><mn id="S5.Thmtheorem2.p1.9.9.m9.1.1" xref="S5.Thmtheorem2.p1.9.9.m9.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.9.9.m9.1b"><cn id="S5.Thmtheorem2.p1.9.9.m9.1.1.cmml" type="integer" xref="S5.Thmtheorem2.p1.9.9.m9.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.9.9.m9.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.9.9.m9.1d">1</annotation></semantics></math>) such that <math alttext="\sigma_{0}^{\prime}=\sigma_{0}[hv\rightarrow\tau^{\textsf{Pun}}_{v,\textsf{rec% }({hv})}]" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.10.10.m10.3"><semantics id="S5.Thmtheorem2.p1.10.10.m10.3a"><mrow id="S5.Thmtheorem2.p1.10.10.m10.3.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.cmml"><msubsup id="S5.Thmtheorem2.p1.10.10.m10.3.3.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3.cmml"><mi id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.2" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.2.cmml">σ</mi><mn id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.3.cmml">0</mn><mo id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3.3.cmml">′</mo></msubsup><mo id="S5.Thmtheorem2.p1.10.10.m10.3.3.2" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.2.cmml">=</mo><mrow id="S5.Thmtheorem2.p1.10.10.m10.3.3.1" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.cmml"><msub id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.cmml"><mi id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.2" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.2.cmml">σ</mi><mn id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.3.cmml">0</mn></msub><mo id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.2" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.2.cmml"><mo id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.2.1.cmml">[</mo><mrow id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.cmml"><mrow id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.cmml"><mi id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.2" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.2.cmml">h</mi><mo id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.1" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.1" stretchy="false" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.1.cmml">→</mo><msubsup id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.2" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.2.cmml">τ</mi><mrow id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.3.cmml"><mi id="S5.Thmtheorem2.p1.10.10.m10.1.1.1.1" xref="S5.Thmtheorem2.p1.10.10.m10.1.1.1.1.cmml">v</mi><mo id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.2" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.3.cmml">,</mo><mrow id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1" 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xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.3" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.3a.cmml">Pun</mtext></msubsup></mrow><mo id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.10.10.m10.3b"><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3"><eq id="S5.Thmtheorem2.p1.10.10.m10.3.3.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.2"></eq><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3">superscript</csymbol><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3">subscript</csymbol><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.2">𝜎</ci><cn id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3.2.3">0</cn></apply><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.3.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.3.3">′</ci></apply><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1"><times id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.2"></times><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3">subscript</csymbol><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.2">𝜎</ci><cn id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.3.3">0</cn></apply><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1"><csymbol cd="latexml" id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.2.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.2">delimited-[]</csymbol><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1"><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.1">→</ci><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2"><times id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.1"></times><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.2">ℎ</ci><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.2.3">𝑣</ci></apply><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3">subscript</csymbol><apply id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.2">𝜏</ci><ci id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.3a.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.3.cmml" mathsize="70%" xref="S5.Thmtheorem2.p1.10.10.m10.3.3.1.1.1.1.3.2.3">Pun</mtext></ci></apply><list id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2"><ci id="S5.Thmtheorem2.p1.10.10.m10.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.1.1.1.1">𝑣</ci><apply id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1"><times id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.2"></times><ci id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.3a.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1"><times id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.1"></times><ci id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.10.10.m10.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.10.10.m10.3c">\sigma_{0}^{\prime}=\sigma_{0}[hv\rightarrow\tau^{\textsf{Pun}}_{v,\textsf{rec% }({hv})}]</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.10.10.m10.3d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_h italic_v → italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT ]</annotation></semantics></math> is also a solution in <math alttext="\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.11.11.m11.2"><semantics id="S5.Thmtheorem2.p1.11.11.m11.2a"><mrow id="S5.Thmtheorem2.p1.11.11.m11.2.3" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem2.p1.11.11.m11.2.3.2" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem2.p1.11.11.m11.2.3.1" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.11.11.m11.2.3.3.2" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.3.1.cmml"><mo id="S5.Thmtheorem2.p1.11.11.m11.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem2.p1.11.11.m11.1.1" xref="S5.Thmtheorem2.p1.11.11.m11.1.1.cmml">G</mi><mo id="S5.Thmtheorem2.p1.11.11.m11.2.3.3.2.2" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.11.11.m11.2.2" xref="S5.Thmtheorem2.p1.11.11.m11.2.2.cmml">B</mi><mo id="S5.Thmtheorem2.p1.11.11.m11.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.11.11.m11.2b"><apply id="S5.Thmtheorem2.p1.11.11.m11.2.3.cmml" xref="S5.Thmtheorem2.p1.11.11.m11.2.3"><times id="S5.Thmtheorem2.p1.11.11.m11.2.3.1.cmml" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.1"></times><ci id="S5.Thmtheorem2.p1.11.11.m11.2.3.2a.cmml" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem2.p1.11.11.m11.2.3.2.cmml" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem2.p1.11.11.m11.2.3.3.1.cmml" xref="S5.Thmtheorem2.p1.11.11.m11.2.3.3.2"><ci id="S5.Thmtheorem2.p1.11.11.m11.1.1.cmml" xref="S5.Thmtheorem2.p1.11.11.m11.1.1">𝐺</ci><ci id="S5.Thmtheorem2.p1.11.11.m11.2.2.cmml" xref="S5.Thmtheorem2.p1.11.11.m11.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.11.11.m11.2c">\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.11.11.m11.2d">SPS ( italic_G , italic_B )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.1.1.1">Proof 5.3</span></span></h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.32"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.32.32">Let <math alttext="hv" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S5.Thmtheorem3.p1.1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.1.1.m1.1.1.1" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.1.m1.1b"><apply id="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1"><times id="S5.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.1"></times><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.1.m1.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.1.m1.1d">italic_h italic_v</annotation></semantics></math> be a deviation such that <math alttext="\textsf{val}(h)=\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem3.p1.2.2.m2.1.2" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.cmml"><mrow id="S5.Thmtheorem3.p1.2.2.m2.1.2.2" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.2" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.3.2" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.cmml"><mo id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml">h</mi><mo id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.2.2.m2.1.2.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.1.cmml">=</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.1.2.3" mathvariant="normal" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.2.m2.1b"><apply id="S5.Thmtheorem3.p1.2.2.m2.1.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.2"><eq id="S5.Thmtheorem3.p1.2.2.m2.1.2.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.1"></eq><apply id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2"><times id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.1"></times><ci id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.2a.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.2.2.m2.1.2.2.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1">ℎ</ci></apply><infinity id="S5.Thmtheorem3.p1.2.2.m2.1.2.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.2.m2.1c">\textsf{val}(h)=\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.2.m2.1d">val ( italic_h ) = ∞</annotation></semantics></math>, that is, <math alttext="h" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.3.m3.1"><semantics id="S5.Thmtheorem3.p1.3.3.m3.1a"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.3.m3.1b"><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.3.m3.1d">italic_h</annotation></semantics></math> does not visit Player <math alttext="0" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.4.4.m4.1"><semantics id="S5.Thmtheorem3.p1.4.4.m4.1a"><mn id="S5.Thmtheorem3.p1.4.4.m4.1.1" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.4.4.m4.1b"><cn id="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml" type="integer" xref="S5.Thmtheorem3.p1.4.4.m4.1.1">0</cn></annotation-xml></semantics></math>’s target. As <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.5.5.m5.1"><semantics id="S5.Thmtheorem3.p1.5.5.m5.1a"><msub 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">σ</mi><mn id="S5.Thmtheorem3.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.3.cmml">0</mn></msub><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"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2">𝜎</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">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.5.5.m5.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution, notice that <math alttext="\sigma_{0|hv}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.6.6.m6.1"><semantics id="S5.Thmtheorem3.p1.6.6.m6.1a"><msub id="S5.Thmtheorem3.p1.6.6.m6.1.1" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.cmml"><mi id="S5.Thmtheorem3.p1.6.6.m6.1.1.2" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.2.cmml">σ</mi><mrow id="S5.Thmtheorem3.p1.6.6.m6.1.1.3" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.cmml"><mn id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.2" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.2.cmml">0</mn><mo fence="false" id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.1" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.1.cmml">|</mo><mrow id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.2" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.1" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.3" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.3.cmml">v</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.6.6.m6.1b"><apply id="S5.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.6.6.m6.1.1.1.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.6.6.m6.1.1.2.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.2">𝜎</ci><apply id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3"><csymbol cd="latexml" id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.1">conditional</csymbol><cn id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.2">0</cn><apply id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3"><times id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.1"></times><ci id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.3.cmml" xref="S5.Thmtheorem3.p1.6.6.m6.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.6.6.m6.1c">\sigma_{0|hv}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.6.6.m6.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h italic_v end_POSTSUBSCRIPT</annotation></semantics></math> imposes to each consistent play <math alttext="\pi" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.7.7.m7.1"><semantics id="S5.Thmtheorem3.p1.7.7.m7.1a"><mi id="S5.Thmtheorem3.p1.7.7.m7.1.1" xref="S5.Thmtheorem3.p1.7.7.m7.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.7.7.m7.1b"><ci id="S5.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="S5.Thmtheorem3.p1.7.7.m7.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.7.7.m7.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.7.7.m7.1d">italic_π</annotation></semantics></math> in <math alttext="G_{|hv}" class="ltx_math_unparsed" display="inline" id="S5.Thmtheorem3.p1.8.8.m8.1"><semantics id="S5.Thmtheorem3.p1.8.8.m8.1a"><msub id="S5.Thmtheorem3.p1.8.8.m8.1.1"><mi id="S5.Thmtheorem3.p1.8.8.m8.1.1.2">G</mi><mrow id="S5.Thmtheorem3.p1.8.8.m8.1.1.3"><mo fence="false" id="S5.Thmtheorem3.p1.8.8.m8.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S5.Thmtheorem3.p1.8.8.m8.1.1.3.2">h</mi><mi id="S5.Thmtheorem3.p1.8.8.m8.1.1.3.3">v</mi></mrow></msub><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.8.8.m8.1b">G_{|hv}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.8.8.m8.1c">italic_G start_POSTSUBSCRIPT | italic_h italic_v end_POSTSUBSCRIPT</annotation></semantics></math> to satisfy <math alttext="\textsf{val}(\pi)\leq B" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.9.9.m9.1"><semantics id="S5.Thmtheorem3.p1.9.9.m9.1a"><mrow id="S5.Thmtheorem3.p1.9.9.m9.1.2" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.cmml"><mrow id="S5.Thmtheorem3.p1.9.9.m9.1.2.2" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.2" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.1" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.3.2" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.cmml"><mo id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem3.p1.9.9.m9.1.1" xref="S5.Thmtheorem3.p1.9.9.m9.1.1.cmml">π</mi><mo id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.9.9.m9.1.2.1" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.1.cmml">≤</mo><mi id="S5.Thmtheorem3.p1.9.9.m9.1.2.3" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.9.9.m9.1b"><apply id="S5.Thmtheorem3.p1.9.9.m9.1.2.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.2"><leq id="S5.Thmtheorem3.p1.9.9.m9.1.2.1.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.1"></leq><apply id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2"><times id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.1.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.1"></times><ci id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.2a.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.9.9.m9.1.2.2.2.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem3.p1.9.9.m9.1.1.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.1">𝜋</ci></apply><ci id="S5.Thmtheorem3.p1.9.9.m9.1.2.3.cmml" xref="S5.Thmtheorem3.p1.9.9.m9.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.9.9.m9.1c">\textsf{val}(\pi)\leq B</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.9.9.m9.1d">val ( italic_π ) ≤ italic_B</annotation></semantics></math> or <math alttext="\textsf{cost}({\pi})>c" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.10.10.m10.1"><semantics id="S5.Thmtheorem3.p1.10.10.m10.1a"><mrow id="S5.Thmtheorem3.p1.10.10.m10.1.2" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.cmml"><mrow id="S5.Thmtheorem3.p1.10.10.m10.1.2.2" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.2" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.2a.cmml">cost</mtext><mo id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.1" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.3.2" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.cmml"><mo id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem3.p1.10.10.m10.1.1" xref="S5.Thmtheorem3.p1.10.10.m10.1.1.cmml">π</mi><mo id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.10.10.m10.1.2.1" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.1.cmml">></mo><mi id="S5.Thmtheorem3.p1.10.10.m10.1.2.3" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.10.10.m10.1b"><apply id="S5.Thmtheorem3.p1.10.10.m10.1.2.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.2"><gt id="S5.Thmtheorem3.p1.10.10.m10.1.2.1.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.1"></gt><apply id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2"><times id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.1.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.1"></times><ci id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.2a.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.10.10.m10.1.2.2.2.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.2.2">cost</mtext></ci><ci id="S5.Thmtheorem3.p1.10.10.m10.1.1.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.1">𝜋</ci></apply><ci id="S5.Thmtheorem3.p1.10.10.m10.1.2.3.cmml" xref="S5.Thmtheorem3.p1.10.10.m10.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.10.10.m10.1c">\textsf{cost}({\pi})>c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.10.10.m10.1d">cost ( italic_π ) > italic_c</annotation></semantics></math> for some <math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.11.11.m11.1"><semantics id="S5.Thmtheorem3.p1.11.11.m11.1a"><mrow id="S5.Thmtheorem3.p1.11.11.m11.1.1" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.cmml"><mi id="S5.Thmtheorem3.p1.11.11.m11.1.1.2" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.2.cmml">c</mi><mo id="S5.Thmtheorem3.p1.11.11.m11.1.1.1" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem3.p1.11.11.m11.1.1.3" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.cmml"><mi id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.2" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.2.cmml">C</mi><msub id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.2" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.2.cmml">σ</mi><mn id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.3" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.11.11.m11.1b"><apply id="S5.Thmtheorem3.p1.11.11.m11.1.1.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1"><in id="S5.Thmtheorem3.p1.11.11.m11.1.1.1.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.1"></in><ci id="S5.Thmtheorem3.p1.11.11.m11.1.1.2.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.2">𝑐</ci><apply id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.2">𝐶</ci><apply id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.2">𝜎</ci><cn id="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.11.11.m11.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.11.11.m11.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.11.11.m11.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. We are going to replace <math alttext="\sigma_{0|hv}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.12.12.m12.1"><semantics id="S5.Thmtheorem3.p1.12.12.m12.1a"><msub id="S5.Thmtheorem3.p1.12.12.m12.1.1" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.cmml"><mi id="S5.Thmtheorem3.p1.12.12.m12.1.1.2" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.2.cmml">σ</mi><mrow id="S5.Thmtheorem3.p1.12.12.m12.1.1.3" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.cmml"><mn id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.2" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.2.cmml">0</mn><mo fence="false" id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.1" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.1.cmml">|</mo><mrow id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.2" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.1" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.3" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.3.cmml">v</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.12.12.m12.1b"><apply id="S5.Thmtheorem3.p1.12.12.m12.1.1.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.12.12.m12.1.1.1.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.12.12.m12.1.1.2.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.2">𝜎</ci><apply id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3"><csymbol cd="latexml" id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.1">conditional</csymbol><cn id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.2">0</cn><apply id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3"><times id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.1"></times><ci id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.3.cmml" xref="S5.Thmtheorem3.p1.12.12.m12.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.12.12.m12.1c">\sigma_{0|hv}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.12.12.m12.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h italic_v end_POSTSUBSCRIPT</annotation></semantics></math> by a winning strategy in a zero-sum game <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.13.13.m13.1"><semantics id="S5.Thmtheorem3.p1.13.13.m13.1a"><mi id="S5.Thmtheorem3.p1.13.13.m13.1.1" xref="S5.Thmtheorem3.p1.13.13.m13.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.13.13.m13.1b"><ci id="S5.Thmtheorem3.p1.13.13.m13.1.1.cmml" xref="S5.Thmtheorem3.p1.13.13.m13.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.13.13.m13.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.13.13.m13.1d">italic_H</annotation></semantics></math> with an exponential arena and an omega-regular objective<span class="ltx_note ltx_role_footnote" id="footnote12"><sup class="ltx_note_mark">12</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">12</sup><span class="ltx_tag ltx_tag_note"><span class="ltx_text ltx_font_upright" id="footnote12.1.1.1">12</span></span><span class="ltx_text ltx_font_upright" id="footnote12.5">We assume that the reader is familiar with the concept of zero-sum game with an omega-regular objective, see e.g. </span><cite class="ltx_cite ltx_citemacro_cite"><span class="ltx_text ltx_font_upright" id="footnote12.6.1">[</span><a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib25" title="">25</a><span class="ltx_text ltx_font_upright" id="footnote12.7.2">]</span></cite><span class="ltx_text ltx_font_upright" id="footnote12.8">.</span></span></span></span> that is equivalent to what <math alttext="\sigma_{0|hv}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.14.14.m14.1"><semantics id="S5.Thmtheorem3.p1.14.14.m14.1a"><msub id="S5.Thmtheorem3.p1.14.14.m14.1.1" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.cmml"><mi id="S5.Thmtheorem3.p1.14.14.m14.1.1.2" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.2.cmml">σ</mi><mrow id="S5.Thmtheorem3.p1.14.14.m14.1.1.3" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.cmml"><mn id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.2" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.2.cmml">0</mn><mo fence="false" id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.1" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.1.cmml">|</mo><mrow id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.2" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.1" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.3" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.3.cmml">v</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.14.14.m14.1b"><apply id="S5.Thmtheorem3.p1.14.14.m14.1.1.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.14.14.m14.1.1.1.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.14.14.m14.1.1.2.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.2">𝜎</ci><apply id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3"><csymbol cd="latexml" id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.1">conditional</csymbol><cn id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.2">0</cn><apply id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3"><times id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.1"></times><ci id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.3.cmml" xref="S5.Thmtheorem3.p1.14.14.m14.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.14.14.m14.1c">\sigma_{0|hv}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.14.14.m14.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h italic_v end_POSTSUBSCRIPT</annotation></semantics></math> imposes to plays. The arena of <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.15.15.m15.1"><semantics id="S5.Thmtheorem3.p1.15.15.m15.1a"><mi id="S5.Thmtheorem3.p1.15.15.m15.1.1" xref="S5.Thmtheorem3.p1.15.15.m15.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.15.15.m15.1b"><ci id="S5.Thmtheorem3.p1.15.15.m15.1.1.cmml" xref="S5.Thmtheorem3.p1.15.15.m15.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.15.15.m15.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.15.15.m15.1d">italic_H</annotation></semantics></math> is the arena of <math alttext="G" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.16.16.m16.1"><semantics id="S5.Thmtheorem3.p1.16.16.m16.1a"><mi id="S5.Thmtheorem3.p1.16.16.m16.1.1" xref="S5.Thmtheorem3.p1.16.16.m16.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.16.16.m16.1b"><ci id="S5.Thmtheorem3.p1.16.16.m16.1.1.cmml" xref="S5.Thmtheorem3.p1.16.16.m16.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.16.16.m16.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.16.16.m16.1d">italic_G</annotation></semantics></math> extended with the record <math alttext="\textsf{rec}({g})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.17.17.m17.1"><semantics id="S5.Thmtheorem3.p1.17.17.m17.1a"><mrow id="S5.Thmtheorem3.p1.17.17.m17.1.2" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.17.17.m17.1.2.2" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.2a.cmml">rec</mtext><mo id="S5.Thmtheorem3.p1.17.17.m17.1.2.1" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.17.17.m17.1.2.3.2" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.cmml"><mo id="S5.Thmtheorem3.p1.17.17.m17.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.cmml">(</mo><mi id="S5.Thmtheorem3.p1.17.17.m17.1.1" xref="S5.Thmtheorem3.p1.17.17.m17.1.1.cmml">g</mi><mo id="S5.Thmtheorem3.p1.17.17.m17.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.17.17.m17.1b"><apply id="S5.Thmtheorem3.p1.17.17.m17.1.2.cmml" xref="S5.Thmtheorem3.p1.17.17.m17.1.2"><times id="S5.Thmtheorem3.p1.17.17.m17.1.2.1.cmml" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.1"></times><ci id="S5.Thmtheorem3.p1.17.17.m17.1.2.2a.cmml" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.17.17.m17.1.2.2.cmml" xref="S5.Thmtheorem3.p1.17.17.m17.1.2.2">rec</mtext></ci><ci id="S5.Thmtheorem3.p1.17.17.m17.1.1.cmml" xref="S5.Thmtheorem3.p1.17.17.m17.1.1">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.17.17.m17.1c">\textsf{rec}({g})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.17.17.m17.1d">rec ( italic_g )</annotation></semantics></math> of the current history <math alttext="g" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.18.18.m18.1"><semantics id="S5.Thmtheorem3.p1.18.18.m18.1a"><mi id="S5.Thmtheorem3.p1.18.18.m18.1.1" xref="S5.Thmtheorem3.p1.18.18.m18.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.18.18.m18.1b"><ci id="S5.Thmtheorem3.p1.18.18.m18.1.1.cmml" xref="S5.Thmtheorem3.p1.18.18.m18.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.18.18.m18.1c">g</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.18.18.m18.1d">italic_g</annotation></semantics></math>. More precisely, vertices are of the form <math alttext="(v,(m_{1},m_{2},m_{3}))" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.19.19.m19.2"><semantics id="S5.Thmtheorem3.p1.19.19.m19.2a"><mrow id="S5.Thmtheorem3.p1.19.19.m19.2.2.1" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.2.cmml"><mo id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.2.cmml">(</mo><mi id="S5.Thmtheorem3.p1.19.19.m19.1.1" xref="S5.Thmtheorem3.p1.19.19.m19.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.3" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.2.cmml">,</mo><mrow id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.4.cmml"><mo id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.4" stretchy="false" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.4.cmml">(</mo><msub id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.2" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.3" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.5" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.4.cmml">,</mo><msub id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.cmml"><mi id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.2" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.3" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.3.cmml">2</mn></msub><mo id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.6" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.4.cmml">,</mo><msub id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.2" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.3" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.3.cmml">3</mn></msub><mo id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.7" stretchy="false" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.4.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.4" stretchy="false" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.19.19.m19.2b"><interval closure="open" id="S5.Thmtheorem3.p1.19.19.m19.2.2.2.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1"><ci id="S5.Thmtheorem3.p1.19.19.m19.1.1.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.1.1">𝑣</ci><vector id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.4.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3"><apply id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.1.1.3">1</cn></apply><apply id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.1.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.2.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.2.2.3">2</cn></apply><apply id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.19.19.m19.2.2.1.1.3.3.3">3</cn></apply></vector></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.19.19.m19.2c">(v,(m_{1},m_{2},m_{3}))</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.19.19.m19.2d">( italic_v , ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) )</annotation></semantics></math> with <math alttext="v\in V" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.20.20.m20.1"><semantics id="S5.Thmtheorem3.p1.20.20.m20.1a"><mrow id="S5.Thmtheorem3.p1.20.20.m20.1.1" xref="S5.Thmtheorem3.p1.20.20.m20.1.1.cmml"><mi id="S5.Thmtheorem3.p1.20.20.m20.1.1.2" xref="S5.Thmtheorem3.p1.20.20.m20.1.1.2.cmml">v</mi><mo id="S5.Thmtheorem3.p1.20.20.m20.1.1.1" 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xref="S5.Thmtheorem3.p1.22.22.m22.6.6.2"></in><apply id="S5.Thmtheorem3.p1.22.22.m22.6.6.3.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.22.22.m22.6.6.3.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.22.22.m22.6.6.3.2.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.3.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.22.22.m22.6.6.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.3.3">3</cn></apply><apply id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.2.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1">superscript</csymbol><apply id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1"><union id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.2"></union><set id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1"><cn id="S5.Thmtheorem3.p1.22.22.m22.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.22.22.m22.3.3">0</cn><ci id="S5.Thmtheorem3.p1.22.22.m22.4.4.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.4.4">…</ci><apply id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1"><times id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1.1"></times><ci id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1.2">𝑓</ci><interval closure="open" id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.1.1.1.3.2"><ci id="S5.Thmtheorem3.p1.22.22.m22.1.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.1.1">𝐵</ci><ci id="S5.Thmtheorem3.p1.22.22.m22.2.2.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.2.2">𝑡</ci></interval></apply></set><set id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.1.1.1.3.2"><infinity id="S5.Thmtheorem3.p1.22.22.m22.5.5.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.5.5"></infinity></set></apply><ci id="S5.Thmtheorem3.p1.22.22.m22.6.6.1.3.cmml" xref="S5.Thmtheorem3.p1.22.22.m22.6.6.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.22.22.m22.6c">m_{3}\in\big{(}\{0,\ldots,f({B},{t})\}\cup\{\infty\}\big{)}^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.22.22.m22.6d">italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ ( { 0 , … , italic_f ( italic_B , italic_t ) } ∪ { ∞ } ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, such that, whenever <math alttext="v" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.23.23.m23.1"><semantics id="S5.Thmtheorem3.p1.23.23.m23.1a"><mi id="S5.Thmtheorem3.p1.23.23.m23.1.1" xref="S5.Thmtheorem3.p1.23.23.m23.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.23.23.m23.1b"><ci id="S5.Thmtheorem3.p1.23.23.m23.1.1.cmml" xref="S5.Thmtheorem3.p1.23.23.m23.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.23.23.m23.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.23.23.m23.1d">italic_v</annotation></semantics></math> belongs to some target, the weight component <math alttext="m_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.24.24.m24.1"><semantics id="S5.Thmtheorem3.p1.24.24.m24.1a"><msub id="S5.Thmtheorem3.p1.24.24.m24.1.1" xref="S5.Thmtheorem3.p1.24.24.m24.1.1.cmml"><mi id="S5.Thmtheorem3.p1.24.24.m24.1.1.2" xref="S5.Thmtheorem3.p1.24.24.m24.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.24.24.m24.1.1.3" xref="S5.Thmtheorem3.p1.24.24.m24.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.24.24.m24.1b"><apply id="S5.Thmtheorem3.p1.24.24.m24.1.1.cmml" xref="S5.Thmtheorem3.p1.24.24.m24.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.24.24.m24.1.1.1.cmml" xref="S5.Thmtheorem3.p1.24.24.m24.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.24.24.m24.1.1.2.cmml" xref="S5.Thmtheorem3.p1.24.24.m24.1.1.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.24.24.m24.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.24.24.m24.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.24.24.m24.1c">m_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.24.24.m24.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> allows to update the (truncated) <span class="ltx_text ltx_markedasmath ltx_font_sansserif" id="S5.Thmtheorem3.p1.32.32.1">val</span> component <math alttext="m_{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.26.26.m26.1"><semantics id="S5.Thmtheorem3.p1.26.26.m26.1a"><msub id="S5.Thmtheorem3.p1.26.26.m26.1.1" xref="S5.Thmtheorem3.p1.26.26.m26.1.1.cmml"><mi id="S5.Thmtheorem3.p1.26.26.m26.1.1.2" xref="S5.Thmtheorem3.p1.26.26.m26.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.26.26.m26.1.1.3" xref="S5.Thmtheorem3.p1.26.26.m26.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.26.26.m26.1b"><apply id="S5.Thmtheorem3.p1.26.26.m26.1.1.cmml" xref="S5.Thmtheorem3.p1.26.26.m26.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.26.26.m26.1.1.1.cmml" xref="S5.Thmtheorem3.p1.26.26.m26.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.26.26.m26.1.1.2.cmml" xref="S5.Thmtheorem3.p1.26.26.m26.1.1.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.26.26.m26.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.26.26.m26.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.26.26.m26.1c">m_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.26.26.m26.1d">italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and the (truncated) <span class="ltx_text ltx_markedasmath ltx_font_sansserif" id="S5.Thmtheorem3.p1.32.32.2">cost</span> component <math alttext="m_{3}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.28.28.m28.1"><semantics id="S5.Thmtheorem3.p1.28.28.m28.1a"><msub id="S5.Thmtheorem3.p1.28.28.m28.1.1" xref="S5.Thmtheorem3.p1.28.28.m28.1.1.cmml"><mi id="S5.Thmtheorem3.p1.28.28.m28.1.1.2" xref="S5.Thmtheorem3.p1.28.28.m28.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.28.28.m28.1.1.3" xref="S5.Thmtheorem3.p1.28.28.m28.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.28.28.m28.1b"><apply id="S5.Thmtheorem3.p1.28.28.m28.1.1.cmml" xref="S5.Thmtheorem3.p1.28.28.m28.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.28.28.m28.1.1.1.cmml" xref="S5.Thmtheorem3.p1.28.28.m28.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.28.28.m28.1.1.2.cmml" xref="S5.Thmtheorem3.p1.28.28.m28.1.1.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.28.28.m28.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.28.28.m28.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.28.28.m28.1c">m_{3}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.28.28.m28.1d">italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>. The initial vertex of <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.29.29.m29.1"><semantics id="S5.Thmtheorem3.p1.29.29.m29.1a"><mi id="S5.Thmtheorem3.p1.29.29.m29.1.1" xref="S5.Thmtheorem3.p1.29.29.m29.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.29.29.m29.1b"><ci id="S5.Thmtheorem3.p1.29.29.m29.1.1.cmml" xref="S5.Thmtheorem3.p1.29.29.m29.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.29.29.m29.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.29.29.m29.1d">italic_H</annotation></semantics></math> is equal to <math alttext="(v,\textsf{rec}({hv}))" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.30.30.m30.2"><semantics id="S5.Thmtheorem3.p1.30.30.m30.2a"><mrow id="S5.Thmtheorem3.p1.30.30.m30.2.2.1" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.2.cmml"><mo id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.2.cmml">(</mo><mi id="S5.Thmtheorem3.p1.30.30.m30.1.1" xref="S5.Thmtheorem3.p1.30.30.m30.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.3" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.2.cmml">,</mo><mrow id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.3" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.3a.cmml">rec</mtext><mo id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.2" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.4" stretchy="false" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.30.30.m30.2b"><interval closure="open" id="S5.Thmtheorem3.p1.30.30.m30.2.2.2.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1"><ci id="S5.Thmtheorem3.p1.30.30.m30.1.1.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.1.1">𝑣</ci><apply id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1"><times id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.2.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.2"></times><ci id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.3a.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.3.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.3">rec</mtext></ci><apply id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1"><times id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.1"></times><ci id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.30.30.m30.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.30.30.m30.2c">(v,\textsf{rec}({hv}))</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.30.30.m30.2d">( italic_v , rec ( italic_h italic_v ) )</annotation></semantics></math>. This arena is finite and of exponential size by the way the values of <math alttext="(m_{1},m_{2},m_{3})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.31.31.m31.3"><semantics id="S5.Thmtheorem3.p1.31.31.m31.3a"><mrow id="S5.Thmtheorem3.p1.31.31.m31.3.3.3" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.4.cmml"><mo id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.4" stretchy="false" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.4.cmml">(</mo><msub id="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1" xref="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.2" xref="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.3" xref="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.5" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.4.cmml">,</mo><msub id="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2" xref="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.cmml"><mi id="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.2" xref="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.3" xref="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.3.cmml">2</mn></msub><mo id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.6" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.4.cmml">,</mo><msub id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.cmml"><mi id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.2" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.2.cmml">m</mi><mn id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.3" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.3.cmml">3</mn></msub><mo id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.7" stretchy="false" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.31.31.m31.3b"><vector id="S5.Thmtheorem3.p1.31.31.m31.3.3.4.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3"><apply id="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.31.31.m31.1.1.1.1.3">1</cn></apply><apply id="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.1.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.2.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.31.31.m31.2.2.2.2.3">2</cn></apply><apply id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.1.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.2.cmml" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.2">𝑚</ci><cn id="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.31.31.m31.3.3.3.3.3">3</cn></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.31.31.m31.3c">(m_{1},m_{2},m_{3})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.31.31.m31.3d">( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math> are truncated and by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a> (<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.E4" title="In 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4</span></a>). The omega regular objective of <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.32.32.m32.1"><semantics id="S5.Thmtheorem3.p1.32.32.m32.1a"><mi id="S5.Thmtheorem3.p1.32.32.m32.1.1" xref="S5.Thmtheorem3.p1.32.32.m32.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.32.32.m32.1b"><ci id="S5.Thmtheorem3.p1.32.32.m32.1.1.cmml" xref="S5.Thmtheorem3.p1.32.32.m32.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.32.32.m32.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.32.32.m32.1d">italic_H</annotation></semantics></math> is the disjunction between</span></p> <ul class="ltx_itemize" id="S5.I2"> <li class="ltx_item" id="S5.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I2.i1.p1"> <p class="ltx_p" id="S5.I2.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.1.1">the reachability objective </span><math alttext="\{(v,(m_{1},m_{2},m_{3}))\mid m_{2}\leq B\}" class="ltx_Math" display="inline" id="S5.I2.i1.p1.1.m1.3"><semantics id="S5.I2.i1.p1.1.m1.3a"><mrow id="S5.I2.i1.p1.1.m1.3.3.2" xref="S5.I2.i1.p1.1.m1.3.3.3.cmml"><mo id="S5.I2.i1.p1.1.m1.3.3.2.3" stretchy="false" xref="S5.I2.i1.p1.1.m1.3.3.3.1.cmml">{</mo><mrow id="S5.I2.i1.p1.1.m1.2.2.1.1.1" xref="S5.I2.i1.p1.1.m1.2.2.1.1.2.cmml"><mo id="S5.I2.i1.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S5.I2.i1.p1.1.m1.2.2.1.1.2.cmml">(</mo><mi id="S5.I2.i1.p1.1.m1.1.1" xref="S5.I2.i1.p1.1.m1.1.1.cmml">v</mi><mo id="S5.I2.i1.p1.1.m1.2.2.1.1.1.3" xref="S5.I2.i1.p1.1.m1.2.2.1.1.2.cmml">,</mo><mrow id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.4.cmml"><mo id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.4" stretchy="false" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.4.cmml">(</mo><msub id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml">m</mi><mn id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.5" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.4.cmml">,</mo><msub id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.2.2" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.2.2.cmml"><mi id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.2.2.2" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.2.2.2.cmml">m</mi><mn id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.2.2.3" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.2.2.3.cmml">2</mn></msub><mo id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.6" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.4.cmml">,</mo><msub id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.3" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.3.cmml"><mi id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.3.2" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.3.2.cmml">m</mi><mn id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.3.3" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.3.3.cmml">3</mn></msub><mo id="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.3.7" stretchy="false" xref="S5.I2.i1.p1.1.m1.2.2.1.1.1.1.4.cmml">)</mo></mrow><mo id="S5.I2.i1.p1.1.m1.2.2.1.1.1.4" stretchy="false" xref="S5.I2.i1.p1.1.m1.2.2.1.1.2.cmml">)</mo></mrow><mo fence="true" id="S5.I2.i1.p1.1.m1.3.3.2.4" lspace="0em" rspace="0em" xref="S5.I2.i1.p1.1.m1.3.3.3.1.cmml">∣</mo><mrow id="S5.I2.i1.p1.1.m1.3.3.2.2" xref="S5.I2.i1.p1.1.m1.3.3.2.2.cmml"><msub id="S5.I2.i1.p1.1.m1.3.3.2.2.2" xref="S5.I2.i1.p1.1.m1.3.3.2.2.2.cmml"><mi id="S5.I2.i1.p1.1.m1.3.3.2.2.2.2" xref="S5.I2.i1.p1.1.m1.3.3.2.2.2.2.cmml">m</mi><mn id="S5.I2.i1.p1.1.m1.3.3.2.2.2.3" xref="S5.I2.i1.p1.1.m1.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S5.I2.i1.p1.1.m1.3.3.2.2.1" xref="S5.I2.i1.p1.1.m1.3.3.2.2.1.cmml">≤</mo><mi id="S5.I2.i1.p1.1.m1.3.3.2.2.3" xref="S5.I2.i1.p1.1.m1.3.3.2.2.3.cmml">B</mi></mrow><mo id="S5.I2.i1.p1.1.m1.3.3.2.5" stretchy="false" xref="S5.I2.i1.p1.1.m1.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.1.m1.3b"><apply id="S5.I2.i1.p1.1.m1.3.3.3.cmml" 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id="S5.I2.i1.p1.1.m1.3.3.2.2.2.1.cmml" xref="S5.I2.i1.p1.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S5.I2.i1.p1.1.m1.3.3.2.2.2.2.cmml" xref="S5.I2.i1.p1.1.m1.3.3.2.2.2.2">𝑚</ci><cn id="S5.I2.i1.p1.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S5.I2.i1.p1.1.m1.3.3.2.2.2.3">2</cn></apply><ci id="S5.I2.i1.p1.1.m1.3.3.2.2.3.cmml" xref="S5.I2.i1.p1.1.m1.3.3.2.2.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.1.m1.3c">\{(v,(m_{1},m_{2},m_{3}))\mid m_{2}\leq B\}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.1.m1.3d">{ ( italic_v , ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ) ∣ italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ italic_B }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.1.2">, and</span></p> </div> </li> <li class="ltx_item" id="S5.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I2.i2.p1"> <p class="ltx_p" id="S5.I2.i2.p1.2"><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.2.1">the safety objective </span><math alttext="\{(v,(m_{1},m_{2},m_{3}))\mid m_{3}>c" class="ltx_math_unparsed" display="inline" id="S5.I2.i2.p1.1.m1.1"><semantics id="S5.I2.i2.p1.1.m1.1a"><mrow id="S5.I2.i2.p1.1.m1.1b"><mo id="S5.I2.i2.p1.1.m1.1.2" stretchy="false">{</mo><mrow id="S5.I2.i2.p1.1.m1.1.3"><mo id="S5.I2.i2.p1.1.m1.1.3.1" stretchy="false">(</mo><mi id="S5.I2.i2.p1.1.m1.1.1">v</mi><mo id="S5.I2.i2.p1.1.m1.1.3.2">,</mo><mrow id="S5.I2.i2.p1.1.m1.1.3.3"><mo id="S5.I2.i2.p1.1.m1.1.3.3.1" stretchy="false">(</mo><msub id="S5.I2.i2.p1.1.m1.1.3.3.2"><mi id="S5.I2.i2.p1.1.m1.1.3.3.2.2">m</mi><mn id="S5.I2.i2.p1.1.m1.1.3.3.2.3">1</mn></msub><mo id="S5.I2.i2.p1.1.m1.1.3.3.3">,</mo><msub id="S5.I2.i2.p1.1.m1.1.3.3.4"><mi id="S5.I2.i2.p1.1.m1.1.3.3.4.2">m</mi><mn id="S5.I2.i2.p1.1.m1.1.3.3.4.3">2</mn></msub><mo id="S5.I2.i2.p1.1.m1.1.3.3.5">,</mo><msub id="S5.I2.i2.p1.1.m1.1.3.3.6"><mi id="S5.I2.i2.p1.1.m1.1.3.3.6.2">m</mi><mn id="S5.I2.i2.p1.1.m1.1.3.3.6.3">3</mn></msub><mo id="S5.I2.i2.p1.1.m1.1.3.3.7" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p1.1.m1.1.3.4" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p1.1.m1.1.4" lspace="0em" rspace="0.167em">∣</mo><msub id="S5.I2.i2.p1.1.m1.1.5"><mi id="S5.I2.i2.p1.1.m1.1.5.2">m</mi><mn id="S5.I2.i2.p1.1.m1.1.5.3">3</mn></msub><mo id="S5.I2.i2.p1.1.m1.1.6">></mo><mi id="S5.I2.i2.p1.1.m1.1.7">c</mi></mrow><annotation encoding="application/x-tex" id="S5.I2.i2.p1.1.m1.1c">\{(v,(m_{1},m_{2},m_{3}))\mid m_{3}>c</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.1.m1.1d">{ ( italic_v , ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ) ∣ italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT > italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.2.2"> for some </span><math alttext="c\in C_{\sigma_{0}}\}" class="ltx_math_unparsed" display="inline" id="S5.I2.i2.p1.2.m2.1"><semantics id="S5.I2.i2.p1.2.m2.1a"><mrow id="S5.I2.i2.p1.2.m2.1b"><mi id="S5.I2.i2.p1.2.m2.1.1">c</mi><mo id="S5.I2.i2.p1.2.m2.1.2">∈</mo><msub id="S5.I2.i2.p1.2.m2.1.3"><mi id="S5.I2.i2.p1.2.m2.1.3.2">C</mi><msub id="S5.I2.i2.p1.2.m2.1.3.3"><mi id="S5.I2.i2.p1.2.m2.1.3.3.2">σ</mi><mn id="S5.I2.i2.p1.2.m2.1.3.3.3">0</mn></msub></msub><mo id="S5.I2.i2.p1.2.m2.1.4" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S5.I2.i2.p1.2.m2.1c">c\in C_{\sigma_{0}}\}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.2.m2.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.2.3">.</span></p> </div> </li> </ul> <p class="ltx_p" id="S5.Thmtheorem3.p1.39"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.39.7">It is known, see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib15" title="">15</a>]</cite>, that if there exists a winning strategy for zero-sum games with an objective which is the disjunction of a reachability objective and a safety objective, then there exists one that is memoryless. This is the case here for the extended game <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.33.1.m1.1"><semantics id="S5.Thmtheorem3.p1.33.1.m1.1a"><mi id="S5.Thmtheorem3.p1.33.1.m1.1.1" xref="S5.Thmtheorem3.p1.33.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.33.1.m1.1b"><ci id="S5.Thmtheorem3.p1.33.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.33.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.33.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.33.1.m1.1d">italic_H</annotation></semantics></math>: as <math alttext="\sigma_{0|hv}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.34.2.m2.1"><semantics id="S5.Thmtheorem3.p1.34.2.m2.1a"><msub id="S5.Thmtheorem3.p1.34.2.m2.1.1" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.cmml"><mi id="S5.Thmtheorem3.p1.34.2.m2.1.1.2" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.2.cmml">σ</mi><mrow id="S5.Thmtheorem3.p1.34.2.m2.1.1.3" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.cmml"><mn id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.2" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.2.cmml">0</mn><mo fence="false" id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.1" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.1.cmml">|</mo><mrow id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.2" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.1" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.3" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.3.cmml">v</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.34.2.m2.1b"><apply id="S5.Thmtheorem3.p1.34.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.34.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.34.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.2">𝜎</ci><apply id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3"><csymbol cd="latexml" id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.1">conditional</csymbol><cn id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.2">0</cn><apply id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3"><times id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.1"></times><ci id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.3.cmml" xref="S5.Thmtheorem3.p1.34.2.m2.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.34.2.m2.1c">\sigma_{0|hv}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.34.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is winning in <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.35.3.m3.1"><semantics id="S5.Thmtheorem3.p1.35.3.m3.1a"><mi id="S5.Thmtheorem3.p1.35.3.m3.1.1" xref="S5.Thmtheorem3.p1.35.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.35.3.m3.1b"><ci id="S5.Thmtheorem3.p1.35.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.35.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.35.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.35.3.m3.1d">italic_H</annotation></semantics></math>, there exists a winning memoryless strategy in <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.36.4.m4.1"><semantics id="S5.Thmtheorem3.p1.36.4.m4.1a"><mi id="S5.Thmtheorem3.p1.36.4.m4.1.1" xref="S5.Thmtheorem3.p1.36.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.36.4.m4.1b"><ci id="S5.Thmtheorem3.p1.36.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p1.36.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.36.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.36.4.m4.1d">italic_H</annotation></semantics></math>, and thus a winning finite-memory strategy <math alttext="\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.37.5.m5.2"><semantics id="S5.Thmtheorem3.p1.37.5.m5.2a"><msubsup id="S5.Thmtheorem3.p1.37.5.m5.2.3" xref="S5.Thmtheorem3.p1.37.5.m5.2.3.cmml"><mi id="S5.Thmtheorem3.p1.37.5.m5.2.3.2.2" xref="S5.Thmtheorem3.p1.37.5.m5.2.3.2.2.cmml">τ</mi><mrow id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.3.cmml"><mi id="S5.Thmtheorem3.p1.37.5.m5.1.1.1.1" xref="S5.Thmtheorem3.p1.37.5.m5.1.1.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.2" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.3.cmml">,</mo><mrow id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.3" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.3a.cmml">rec</mtext><mo id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.2" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.2" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.1" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.3" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.37.5.m5.2.3.2.3" xref="S5.Thmtheorem3.p1.37.5.m5.2.3.2.3a.cmml">Pun</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.37.5.m5.2b"><apply id="S5.Thmtheorem3.p1.37.5.m5.2.3.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.37.5.m5.2.3.1.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.3">subscript</csymbol><apply id="S5.Thmtheorem3.p1.37.5.m5.2.3.2.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.37.5.m5.2.3.2.1.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.3">superscript</csymbol><ci id="S5.Thmtheorem3.p1.37.5.m5.2.3.2.2.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.3.2.2">𝜏</ci><ci id="S5.Thmtheorem3.p1.37.5.m5.2.3.2.3a.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.3.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.37.5.m5.2.3.2.3.cmml" mathsize="70%" xref="S5.Thmtheorem3.p1.37.5.m5.2.3.2.3">Pun</mtext></ci></apply><list id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.3.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2"><ci id="S5.Thmtheorem3.p1.37.5.m5.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.1.1.1.1">𝑣</ci><apply id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1"><times id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.2"></times><ci id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.3a.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1"><times id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.1"></times><ci id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.37.5.m5.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.37.5.m5.2c">\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.37.5.m5.2d">italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT</annotation></semantics></math> with exponential size in the original game. Therefore, the strategy <math alttext="\sigma_{0}[hv\rightarrow\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}]" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.38.6.m6.3"><semantics id="S5.Thmtheorem3.p1.38.6.m6.3a"><mrow id="S5.Thmtheorem3.p1.38.6.m6.3.3" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.cmml"><msub id="S5.Thmtheorem3.p1.38.6.m6.3.3.3" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.3.cmml"><mi id="S5.Thmtheorem3.p1.38.6.m6.3.3.3.2" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.3.2.cmml">σ</mi><mn id="S5.Thmtheorem3.p1.38.6.m6.3.3.3.3" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.3.3.cmml">0</mn></msub><mo id="S5.Thmtheorem3.p1.38.6.m6.3.3.2" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.2.cmml"><mo id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.2.1.cmml">[</mo><mrow id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.cmml"><mrow id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.cmml"><mi id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.2" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.1" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.3" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.1" stretchy="false" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.1.cmml">→</mo><msubsup id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.2" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.2.cmml">τ</mi><mrow id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.3.cmml"><mi id="S5.Thmtheorem3.p1.38.6.m6.1.1.1.1" xref="S5.Thmtheorem3.p1.38.6.m6.1.1.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.2" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.3.cmml">,</mo><mrow id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.3" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.3a.cmml">rec</mtext><mo id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.2" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.2" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.1" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.3" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.3" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.3a.cmml">Pun</mtext></msubsup></mrow><mo id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.38.6.m6.3b"><apply id="S5.Thmtheorem3.p1.38.6.m6.3.3.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3"><times id="S5.Thmtheorem3.p1.38.6.m6.3.3.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.2"></times><apply id="S5.Thmtheorem3.p1.38.6.m6.3.3.3.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.38.6.m6.3.3.3.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.38.6.m6.3.3.3.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.3.2">𝜎</ci><cn id="S5.Thmtheorem3.p1.38.6.m6.3.3.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.3.3">0</cn></apply><apply id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1"><csymbol cd="latexml" id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.2.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.2">delimited-[]</csymbol><apply id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1"><ci id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.1">→</ci><apply id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2"><times id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.1"></times><ci id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.3.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.2.3">𝑣</ci></apply><apply id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3">subscript</csymbol><apply id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.2">𝜏</ci><ci id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.3a.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.3.cmml" mathsize="70%" xref="S5.Thmtheorem3.p1.38.6.m6.3.3.1.1.1.3.2.3">Pun</mtext></ci></apply><list id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.3.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2"><ci id="S5.Thmtheorem3.p1.38.6.m6.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.1.1.1.1">𝑣</ci><apply id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1"><times id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.2"></times><ci id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.3a.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1"><times id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.1"></times><ci id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.38.6.m6.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.38.6.m6.3c">\sigma_{0}[hv\rightarrow\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}]</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.38.6.m6.3d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_h italic_v → italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT ]</annotation></semantics></math> is again a solution in <math alttext="\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.39.7.m7.2"><semantics id="S5.Thmtheorem3.p1.39.7.m7.2a"><mrow id="S5.Thmtheorem3.p1.39.7.m7.2.3" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p1.39.7.m7.2.3.2" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem3.p1.39.7.m7.2.3.1" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.39.7.m7.2.3.3.2" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p1.39.7.m7.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.39.7.m7.1.1" xref="S5.Thmtheorem3.p1.39.7.m7.1.1.cmml">G</mi><mo id="S5.Thmtheorem3.p1.39.7.m7.2.3.3.2.2" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.39.7.m7.2.2" xref="S5.Thmtheorem3.p1.39.7.m7.2.2.cmml">B</mi><mo id="S5.Thmtheorem3.p1.39.7.m7.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.39.7.m7.2b"><apply id="S5.Thmtheorem3.p1.39.7.m7.2.3.cmml" xref="S5.Thmtheorem3.p1.39.7.m7.2.3"><times id="S5.Thmtheorem3.p1.39.7.m7.2.3.1.cmml" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.1"></times><ci id="S5.Thmtheorem3.p1.39.7.m7.2.3.2a.cmml" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p1.39.7.m7.2.3.2.cmml" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem3.p1.39.7.m7.2.3.3.1.cmml" xref="S5.Thmtheorem3.p1.39.7.m7.2.3.3.2"><ci id="S5.Thmtheorem3.p1.39.7.m7.1.1.cmml" xref="S5.Thmtheorem3.p1.39.7.m7.1.1">𝐺</ci><ci id="S5.Thmtheorem3.p1.39.7.m7.2.2.cmml" xref="S5.Thmtheorem3.p1.39.7.m7.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.39.7.m7.2c">\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.39.7.m7.2d">SPS ( italic_G , italic_B )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p2"> <p class="ltx_p" id="S5.Thmtheorem3.p2.6"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p2.6.6">Suppose now that <math alttext="hv" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.1.1.m1.1"><semantics id="S5.Thmtheorem3.p2.1.1.m1.1a"><mrow id="S5.Thmtheorem3.p2.1.1.m1.1.1" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p2.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem3.p2.1.1.m1.1.1.1" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p2.1.1.m1.1.1.3" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.1.1.m1.1b"><apply id="S5.Thmtheorem3.p2.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p2.1.1.m1.1.1"><times id="S5.Thmtheorem3.p2.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.1"></times><ci id="S5.Thmtheorem3.p2.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem3.p2.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.1.1.m1.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.1.1.m1.1d">italic_h italic_v</annotation></semantics></math> is a deviation with <math alttext="\textsf{val}(h)<\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.2.2.m2.1"><semantics id="S5.Thmtheorem3.p2.2.2.m2.1a"><mrow id="S5.Thmtheorem3.p2.2.2.m2.1.2" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.cmml"><mrow id="S5.Thmtheorem3.p2.2.2.m2.1.2.2" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.2" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.1" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.3.2" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.cmml"><mo id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem3.p2.2.2.m2.1.1" xref="S5.Thmtheorem3.p2.2.2.m2.1.1.cmml">h</mi><mo id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p2.2.2.m2.1.2.1" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.1.cmml"><</mo><mi id="S5.Thmtheorem3.p2.2.2.m2.1.2.3" mathvariant="normal" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.2.2.m2.1b"><apply id="S5.Thmtheorem3.p2.2.2.m2.1.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.2"><lt id="S5.Thmtheorem3.p2.2.2.m2.1.2.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.1"></lt><apply id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2"><times id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.1"></times><ci id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.2a.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p2.2.2.m2.1.2.2.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem3.p2.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.1">ℎ</ci></apply><infinity id="S5.Thmtheorem3.p2.2.2.m2.1.2.3.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.2.2.m2.1c">\textsf{val}(h)<\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.2.2.m2.1d">val ( italic_h ) < ∞</annotation></semantics></math>. As <math alttext="h" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.3.3.m3.1"><semantics id="S5.Thmtheorem3.p2.3.3.m3.1a"><mi id="S5.Thmtheorem3.p2.3.3.m3.1.1" xref="S5.Thmtheorem3.p2.3.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.3.3.m3.1b"><ci id="S5.Thmtheorem3.p2.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.3.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.3.3.m3.1d">italic_h</annotation></semantics></math> already visits Player <math alttext="0" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.4.4.m4.1"><semantics id="S5.Thmtheorem3.p2.4.4.m4.1a"><mn id="S5.Thmtheorem3.p2.4.4.m4.1.1" xref="S5.Thmtheorem3.p2.4.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.4.4.m4.1b"><cn id="S5.Thmtheorem3.p2.4.4.m4.1.1.cmml" type="integer" xref="S5.Thmtheorem3.p2.4.4.m4.1.1">0</cn></annotation-xml></semantics></math>’s target, we can use, in place of <math alttext="\sigma_{0|hv}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.5.5.m5.1"><semantics id="S5.Thmtheorem3.p2.5.5.m5.1a"><msub id="S5.Thmtheorem3.p2.5.5.m5.1.1" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.cmml"><mi id="S5.Thmtheorem3.p2.5.5.m5.1.1.2" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2.cmml">σ</mi><mrow id="S5.Thmtheorem3.p2.5.5.m5.1.1.3" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.cmml"><mn id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.2" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.2.cmml">0</mn><mo fence="false" id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.1" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.1.cmml">|</mo><mrow id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.2" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.2.cmml">h</mi><mo id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.1" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.3" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.3.cmml">v</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.5.5.m5.1b"><apply id="S5.Thmtheorem3.p2.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p2.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2">𝜎</ci><apply id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3"><csymbol cd="latexml" id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.1.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.1">conditional</csymbol><cn id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.2">0</cn><apply id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3"><times id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.1"></times><ci id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.2">ℎ</ci><ci id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.3.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.5.5.m5.1c">\sigma_{0|hv}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.5.5.m5.1d">italic_σ start_POSTSUBSCRIPT 0 | italic_h italic_v end_POSTSUBSCRIPT</annotation></semantics></math>, any memoryless strategy <math alttext="\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.6.6.m6.2"><semantics id="S5.Thmtheorem3.p2.6.6.m6.2a"><msubsup id="S5.Thmtheorem3.p2.6.6.m6.2.3" xref="S5.Thmtheorem3.p2.6.6.m6.2.3.cmml"><mi id="S5.Thmtheorem3.p2.6.6.m6.2.3.2.2" xref="S5.Thmtheorem3.p2.6.6.m6.2.3.2.2.cmml">τ</mi><mrow id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.3.cmml"><mi 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xref="S5.Thmtheorem3.p2.6.6.m6.2.3">superscript</csymbol><ci id="S5.Thmtheorem3.p2.6.6.m6.2.3.2.2.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.3.2.2">𝜏</ci><ci id="S5.Thmtheorem3.p2.6.6.m6.2.3.2.3a.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.3.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p2.6.6.m6.2.3.2.3.cmml" mathsize="70%" xref="S5.Thmtheorem3.p2.6.6.m6.2.3.2.3">Pun</mtext></ci></apply><list id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.3.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2"><ci id="S5.Thmtheorem3.p2.6.6.m6.1.1.1.1.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.1.1.1.1">𝑣</ci><apply id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1"><times id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.2"></times><ci id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.3a.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1"><times id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1.1.1"></times><ci id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.6.6.m6.2c">\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.6.6.m6.2d">italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT</annotation></semantics></math> as deviating strategy.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S5.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.3 </span>Lasso Witnesses</h3> <div class="ltx_para" id="S5.SS3.p1"> <p class="ltx_p" id="S5.SS3.p1.4">To get Proposition <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem1" title="Proposition 5.1 ‣ 5.1 Finite-Memory Solutions ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.1</span></a>, we show that one can play with an exponential finite-memory strategy over the witness tree and an exponential number of deviating strategies. The main idea of the proof is to transform each witness <math alttext="\rho" class="ltx_Math" display="inline" id="S5.SS3.p1.1.m1.1"><semantics id="S5.SS3.p1.1.m1.1a"><mi id="S5.SS3.p1.1.m1.1.1" xref="S5.SS3.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.1.m1.1b"><ci id="S5.SS3.p1.1.m1.1.1.cmml" xref="S5.SS3.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.1.m1.1d">italic_ρ</annotation></semantics></math> into a <em class="ltx_emph ltx_font_italic" id="S5.SS3.p1.4.1">lasso</em>: as the solution <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.SS3.p1.2.m2.1"><semantics id="S5.SS3.p1.2.m2.1a"><msub id="S5.SS3.p1.2.m2.1.1" xref="S5.SS3.p1.2.m2.1.1.cmml"><mi id="S5.SS3.p1.2.m2.1.1.2" xref="S5.SS3.p1.2.m2.1.1.2.cmml">σ</mi><mn id="S5.SS3.p1.2.m2.1.1.3" xref="S5.SS3.p1.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.2.m2.1b"><apply id="S5.SS3.p1.2.m2.1.1.cmml" xref="S5.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.2.m2.1.1.1.cmml" xref="S5.SS3.p1.2.m2.1.1">subscript</csymbol><ci id="S5.SS3.p1.2.m2.1.1.2.cmml" xref="S5.SS3.p1.2.m2.1.1.2">𝜎</ci><cn id="S5.SS3.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.SS3.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.2.m2.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is Pareto-bounded, in the region decomposition of <math alttext="\rho" class="ltx_Math" display="inline" id="S5.SS3.p1.3.m3.1"><semantics id="S5.SS3.p1.3.m3.1a"><mi id="S5.SS3.p1.3.m3.1.1" xref="S5.SS3.p1.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.3.m3.1b"><ci id="S5.SS3.p1.3.m3.1.1.cmml" xref="S5.SS3.p1.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.3.m3.1d">italic_ρ</annotation></semantics></math>, each region contains no cycle, except the last one; in the last region, as soon as a vertex is repeated, we replace the suffix of <math alttext="\rho" class="ltx_Math" display="inline" id="S5.SS3.p1.4.m4.1"><semantics id="S5.SS3.p1.4.m4.1a"><mi id="S5.SS3.p1.4.m4.1.1" xref="S5.SS3.p1.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.4.m4.1b"><ci id="S5.SS3.p1.4.m4.1.1.cmml" xref="S5.SS3.p1.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.4.m4.1d">italic_ρ</annotation></semantics></math> by the infinite repetition of this cycle. We will show that the number of regions traversed by these lasso witnesses is exponential.</p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S5.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.1.1.1">Proof 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.2.2"> </span>(Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem1" title="Proposition 5.1 ‣ 5.1 Finite-Memory Solutions ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.1</span></a>)</h6> <div class="ltx_para" id="S5.Thmtheorem4.p1"> <p class="ltx_p" id="S5.Thmtheorem4.p1.9"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p1.9.9">(1) We first suppose that <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.1.1.m1.4"><semantics id="S5.Thmtheorem4.p1.1.1.m1.4a"><mrow id="S5.Thmtheorem4.p1.1.1.m1.4.4" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.cmml"><msub id="S5.Thmtheorem4.p1.1.1.m1.4.4.3" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.cmml"><mi id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.2" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.2.cmml">C</mi><msub id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.cmml"><mi id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.2" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.3" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S5.Thmtheorem4.p1.1.1.m1.4.4.2" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.2.cmml">≠</mo><mrow id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.2.cmml"><mo id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.2.cmml">{</mo><mrow id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.2" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.1.cmml"><mo id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.2.1" stretchy="false" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.1.cmml">(</mo><mi id="S5.Thmtheorem4.p1.1.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml">∞</mi><mo id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.2.2" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem4.p1.1.1.m1.2.2" mathvariant="normal" xref="S5.Thmtheorem4.p1.1.1.m1.2.2.cmml">…</mi><mo id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.2.3" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem4.p1.1.1.m1.3.3" mathvariant="normal" xref="S5.Thmtheorem4.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.2.4" stretchy="false" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.3" stretchy="false" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.1.1.m1.4b"><apply id="S5.Thmtheorem4.p1.1.1.m1.4.4.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4"><neq id="S5.Thmtheorem4.p1.1.1.m1.4.4.2.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.2"></neq><apply id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3">subscript</csymbol><ci id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.2.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.2">𝐶</ci><apply id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3">subscript</csymbol><ci id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.2.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.2">𝜎</ci><cn id="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.3.3.3">0</cn></apply></apply><set id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.2.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1"><vector id="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.4.4.1.1.1.2"><infinity id="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.1.1"></infinity><ci id="S5.Thmtheorem4.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.2.2">…</ci><infinity id="S5.Thmtheorem4.p1.1.1.m1.3.3.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.1.1.m1.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.1.1.m1.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>. According to Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>, from <math alttext="\sigma_{0}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.2.2.m2.2"><semantics id="S5.Thmtheorem4.p1.2.2.m2.2a"><mrow id="S5.Thmtheorem4.p1.2.2.m2.2.3" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.cmml"><msub id="S5.Thmtheorem4.p1.2.2.m2.2.3.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2.cmml"><mi id="S5.Thmtheorem4.p1.2.2.m2.2.3.2.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p1.2.2.m2.2.3.2.3" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2.3.cmml">0</mn></msub><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.1" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem4.p1.2.2.m2.2.3.3" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.1.cmml"><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem4.p1.2.2.m2.1.1" xref="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml">G</mi><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.2.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem4.p1.2.2.m2.2.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.2.cmml">B</mi><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.2.3" stretchy="false" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.2.2.m2.2b"><apply id="S5.Thmtheorem4.p1.2.2.m2.2.3.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3"><in id="S5.Thmtheorem4.p1.2.2.m2.2.3.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.1"></in><apply id="S5.Thmtheorem4.p1.2.2.m2.2.3.2.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.2.2.m2.2.3.2.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem4.p1.2.2.m2.2.3.2.2.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2.2">𝜎</ci><cn id="S5.Thmtheorem4.p1.2.2.m2.2.3.2.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2.3">0</cn></apply><apply id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3"><times id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1"></times><ci id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2a.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.3.2"><ci id="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.1.1">𝐺</ci><ci id="S5.Thmtheorem4.p1.2.2.m2.2.2.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.2.2.m2.2c">\sigma_{0}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.2.2.m2.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math>, one can construct a solution <math alttext="\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.3.3.m3.2"><semantics id="S5.Thmtheorem4.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem4.p1.3.3.m3.2.3" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.cmml"><msubsup id="S5.Thmtheorem4.p1.3.3.m3.2.3.2" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2.cmml"><mi id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.2" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.3" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.3.cmml">0</mn><mo id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.3" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2.3.cmml">′</mo></msubsup><mo id="S5.Thmtheorem4.p1.3.3.m3.2.3.1" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem4.p1.3.3.m3.2.3.3" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.2" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.1" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.2" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.1.cmml"><mo id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem4.p1.3.3.m3.1.1" xref="S5.Thmtheorem4.p1.3.3.m3.1.1.cmml">G</mi><mo id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.2.2" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem4.p1.3.3.m3.2.2" xref="S5.Thmtheorem4.p1.3.3.m3.2.2.cmml">B</mi><mo id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.2.3" stretchy="false" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.3.3.m3.2b"><apply id="S5.Thmtheorem4.p1.3.3.m3.2.3.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3"><in id="S5.Thmtheorem4.p1.3.3.m3.2.3.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.1"></in><apply id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2">superscript</csymbol><apply id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.2">𝜎</ci><cn id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2.2.3">0</cn></apply><ci id="S5.Thmtheorem4.p1.3.3.m3.2.3.2.3.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.2.3">′</ci></apply><apply id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3"><times id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.1"></times><ci id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.2a.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.3.3.3.2"><ci id="S5.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.1">𝐺</ci><ci id="S5.Thmtheorem4.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.3.3.m3.2c">\sigma_{0}^{\prime}\in\mbox{SPS}(G,B)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.3.3.m3.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> that is Pareto-bounded, thus without cycles. Consider a set of witnesses <math alttext="\textsf{Wit}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.4.4.m4.1"><semantics id="S5.Thmtheorem4.p1.4.4.m4.1a"><msub id="S5.Thmtheorem4.p1.4.4.m4.1.1" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p1.4.4.m4.1.1.2" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.2a.cmml">Wit</mtext><msubsup id="S5.Thmtheorem4.p1.4.4.m4.1.1.3" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3.cmml"><mi id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.2" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.3" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml">0</mn><mo id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.3" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.4.4.m4.1b"><apply id="S5.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p1.4.4.m4.1.1.2a.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.2">Wit</mtext></ci><apply id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.1.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3">subscript</csymbol><apply id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.2.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.2">𝜎</ci><ci id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.3.cmml" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3.2.3">′</ci></apply><cn id="S5.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.4.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.4.4.m4.1c">\textsf{Wit}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.4.4.m4.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for this strategy, and the region decomposition of its witnesses. Along any witness <math alttext="\rho" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.5.5.m5.1"><semantics id="S5.Thmtheorem4.p1.5.5.m5.1a"><mi id="S5.Thmtheorem4.p1.5.5.m5.1.1" xref="S5.Thmtheorem4.p1.5.5.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.5.5.m5.1b"><ci id="S5.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem4.p1.5.5.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.5.5.m5.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.5.5.m5.1d">italic_ρ</annotation></semantics></math>, according to Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a>, it is thus impossible to eliminate cycles. Recall that Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a> only considers cycles implying histories in the same region, those cycles with a null weight before visiting Player <math alttext="0" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.6.6.m6.1"><semantics id="S5.Thmtheorem4.p1.6.6.m6.1a"><mn id="S5.Thmtheorem4.p1.6.6.m6.1.1" xref="S5.Thmtheorem4.p1.6.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.6.6.m6.1b"><cn id="S5.Thmtheorem4.p1.6.6.m6.1.1.cmml" type="integer" xref="S5.Thmtheorem4.p1.6.6.m6.1.1">0</cn></annotation-xml></semantics></math>’s target; moreover, the last region of <math alttext="\rho" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.7.7.m7.1"><semantics id="S5.Thmtheorem4.p1.7.7.m7.1a"><mi id="S5.Thmtheorem4.p1.7.7.m7.1.1" xref="S5.Thmtheorem4.p1.7.7.m7.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.7.7.m7.1b"><ci id="S5.Thmtheorem4.p1.7.7.m7.1.1.cmml" xref="S5.Thmtheorem4.p1.7.7.m7.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.7.7.m7.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.7.7.m7.1d">italic_ρ</annotation></semantics></math> is excluded. Let us study the memory used by <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.8.8.m8.1"><semantics id="S5.Thmtheorem4.p1.8.8.m8.1a"><msubsup id="S5.Thmtheorem4.p1.8.8.m8.1.1" xref="S5.Thmtheorem4.p1.8.8.m8.1.1.cmml"><mi id="S5.Thmtheorem4.p1.8.8.m8.1.1.2.2" xref="S5.Thmtheorem4.p1.8.8.m8.1.1.2.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p1.8.8.m8.1.1.3" xref="S5.Thmtheorem4.p1.8.8.m8.1.1.3.cmml">0</mn><mo id="S5.Thmtheorem4.p1.8.8.m8.1.1.2.3" xref="S5.Thmtheorem4.p1.8.8.m8.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.8.8.m8.1b"><apply id="S5.Thmtheorem4.p1.8.8.m8.1.1.cmml" xref="S5.Thmtheorem4.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.8.8.m8.1.1.1.cmml" xref="S5.Thmtheorem4.p1.8.8.m8.1.1">subscript</csymbol><apply id="S5.Thmtheorem4.p1.8.8.m8.1.1.2.cmml" xref="S5.Thmtheorem4.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.8.8.m8.1.1.2.1.cmml" xref="S5.Thmtheorem4.p1.8.8.m8.1.1">superscript</csymbol><ci id="S5.Thmtheorem4.p1.8.8.m8.1.1.2.2.cmml" xref="S5.Thmtheorem4.p1.8.8.m8.1.1.2.2">𝜎</ci><ci id="S5.Thmtheorem4.p1.8.8.m8.1.1.2.3.cmml" xref="S5.Thmtheorem4.p1.8.8.m8.1.1.2.3">′</ci></apply><cn id="S5.Thmtheorem4.p1.8.8.m8.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.8.8.m8.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.8.8.m8.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Along a witness <math alttext="\rho" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.9.9.m9.1"><semantics id="S5.Thmtheorem4.p1.9.9.m9.1a"><mi id="S5.Thmtheorem4.p1.9.9.m9.1.1" xref="S5.Thmtheorem4.p1.9.9.m9.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.9.9.m9.1b"><ci id="S5.Thmtheorem4.p1.9.9.m9.1.1.cmml" xref="S5.Thmtheorem4.p1.9.9.m9.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.9.9.m9.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.9.9.m9.1d">italic_ρ</annotation></semantics></math>:</span></p> <ul class="ltx_itemize" id="S5.I3"> <li class="ltx_item" id="S5.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I3.i1.p1"> <p class="ltx_p" id="S5.I3.i1.p1.7"><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.1">As </span><math alttext="\textsf{val}(\rho)\leq B" class="ltx_Math" display="inline" id="S5.I3.i1.p1.1.m1.1"><semantics id="S5.I3.i1.p1.1.m1.1a"><mrow id="S5.I3.i1.p1.1.m1.1.2" xref="S5.I3.i1.p1.1.m1.1.2.cmml"><mrow id="S5.I3.i1.p1.1.m1.1.2.2" xref="S5.I3.i1.p1.1.m1.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i1.p1.1.m1.1.2.2.2" xref="S5.I3.i1.p1.1.m1.1.2.2.2a.cmml">val</mtext><mo id="S5.I3.i1.p1.1.m1.1.2.2.1" xref="S5.I3.i1.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S5.I3.i1.p1.1.m1.1.2.2.3.2" xref="S5.I3.i1.p1.1.m1.1.2.2.cmml"><mo id="S5.I3.i1.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S5.I3.i1.p1.1.m1.1.2.2.cmml">(</mo><mi id="S5.I3.i1.p1.1.m1.1.1" xref="S5.I3.i1.p1.1.m1.1.1.cmml">ρ</mi><mo id="S5.I3.i1.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S5.I3.i1.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.I3.i1.p1.1.m1.1.2.1" xref="S5.I3.i1.p1.1.m1.1.2.1.cmml">≤</mo><mi id="S5.I3.i1.p1.1.m1.1.2.3" xref="S5.I3.i1.p1.1.m1.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.1.m1.1b"><apply id="S5.I3.i1.p1.1.m1.1.2.cmml" xref="S5.I3.i1.p1.1.m1.1.2"><leq id="S5.I3.i1.p1.1.m1.1.2.1.cmml" xref="S5.I3.i1.p1.1.m1.1.2.1"></leq><apply id="S5.I3.i1.p1.1.m1.1.2.2.cmml" xref="S5.I3.i1.p1.1.m1.1.2.2"><times id="S5.I3.i1.p1.1.m1.1.2.2.1.cmml" xref="S5.I3.i1.p1.1.m1.1.2.2.1"></times><ci id="S5.I3.i1.p1.1.m1.1.2.2.2a.cmml" xref="S5.I3.i1.p1.1.m1.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i1.p1.1.m1.1.2.2.2.cmml" xref="S5.I3.i1.p1.1.m1.1.2.2.2">val</mtext></ci><ci id="S5.I3.i1.p1.1.m1.1.1.cmml" xref="S5.I3.i1.p1.1.m1.1.1">𝜌</ci></apply><ci id="S5.I3.i1.p1.1.m1.1.2.3.cmml" xref="S5.I3.i1.p1.1.m1.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.1.m1.1c">\textsf{val}(\rho)\leq B</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.1.m1.1d">val ( italic_ρ ) ≤ italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.2"> and there is no cycle with a null weight before visiting Player </span><math alttext="0" class="ltx_Math" display="inline" id="S5.I3.i1.p1.2.m2.1"><semantics id="S5.I3.i1.p1.2.m2.1a"><mn id="S5.I3.i1.p1.2.m2.1.1" xref="S5.I3.i1.p1.2.m2.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.2.m2.1b"><cn id="S5.I3.i1.p1.2.m2.1.1.cmml" type="integer" xref="S5.I3.i1.p1.2.m2.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.3">’s target, it follows that the smallest prefix </span><math alttext="h" class="ltx_Math" display="inline" id="S5.I3.i1.p1.3.m3.1"><semantics id="S5.I3.i1.p1.3.m3.1a"><mi id="S5.I3.i1.p1.3.m3.1.1" xref="S5.I3.i1.p1.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.3.m3.1b"><ci id="S5.I3.i1.p1.3.m3.1.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.3.m3.1d">italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.4"> of </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I3.i1.p1.4.m4.1"><semantics id="S5.I3.i1.p1.4.m4.1a"><mi id="S5.I3.i1.p1.4.m4.1.1" xref="S5.I3.i1.p1.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.4.m4.1b"><ci id="S5.I3.i1.p1.4.m4.1.1.cmml" xref="S5.I3.i1.p1.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.4.m4.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.5"> such that </span><math alttext="\textsf{val}(h)=\textsf{val}(\rho)" class="ltx_Math" display="inline" id="S5.I3.i1.p1.5.m5.2"><semantics id="S5.I3.i1.p1.5.m5.2a"><mrow id="S5.I3.i1.p1.5.m5.2.3" xref="S5.I3.i1.p1.5.m5.2.3.cmml"><mrow id="S5.I3.i1.p1.5.m5.2.3.2" xref="S5.I3.i1.p1.5.m5.2.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i1.p1.5.m5.2.3.2.2" xref="S5.I3.i1.p1.5.m5.2.3.2.2a.cmml">val</mtext><mo id="S5.I3.i1.p1.5.m5.2.3.2.1" xref="S5.I3.i1.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S5.I3.i1.p1.5.m5.2.3.2.3.2" xref="S5.I3.i1.p1.5.m5.2.3.2.cmml"><mo id="S5.I3.i1.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="S5.I3.i1.p1.5.m5.2.3.2.cmml">(</mo><mi id="S5.I3.i1.p1.5.m5.1.1" xref="S5.I3.i1.p1.5.m5.1.1.cmml">h</mi><mo id="S5.I3.i1.p1.5.m5.2.3.2.3.2.2" stretchy="false" xref="S5.I3.i1.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.I3.i1.p1.5.m5.2.3.1" xref="S5.I3.i1.p1.5.m5.2.3.1.cmml">=</mo><mrow id="S5.I3.i1.p1.5.m5.2.3.3" xref="S5.I3.i1.p1.5.m5.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i1.p1.5.m5.2.3.3.2" xref="S5.I3.i1.p1.5.m5.2.3.3.2a.cmml">val</mtext><mo id="S5.I3.i1.p1.5.m5.2.3.3.1" xref="S5.I3.i1.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S5.I3.i1.p1.5.m5.2.3.3.3.2" xref="S5.I3.i1.p1.5.m5.2.3.3.cmml"><mo id="S5.I3.i1.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="S5.I3.i1.p1.5.m5.2.3.3.cmml">(</mo><mi id="S5.I3.i1.p1.5.m5.2.2" xref="S5.I3.i1.p1.5.m5.2.2.cmml">ρ</mi><mo id="S5.I3.i1.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="S5.I3.i1.p1.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.5.m5.2b"><apply id="S5.I3.i1.p1.5.m5.2.3.cmml" xref="S5.I3.i1.p1.5.m5.2.3"><eq id="S5.I3.i1.p1.5.m5.2.3.1.cmml" xref="S5.I3.i1.p1.5.m5.2.3.1"></eq><apply id="S5.I3.i1.p1.5.m5.2.3.2.cmml" xref="S5.I3.i1.p1.5.m5.2.3.2"><times id="S5.I3.i1.p1.5.m5.2.3.2.1.cmml" xref="S5.I3.i1.p1.5.m5.2.3.2.1"></times><ci id="S5.I3.i1.p1.5.m5.2.3.2.2a.cmml" xref="S5.I3.i1.p1.5.m5.2.3.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i1.p1.5.m5.2.3.2.2.cmml" xref="S5.I3.i1.p1.5.m5.2.3.2.2">val</mtext></ci><ci id="S5.I3.i1.p1.5.m5.1.1.cmml" xref="S5.I3.i1.p1.5.m5.1.1">ℎ</ci></apply><apply id="S5.I3.i1.p1.5.m5.2.3.3.cmml" xref="S5.I3.i1.p1.5.m5.2.3.3"><times id="S5.I3.i1.p1.5.m5.2.3.3.1.cmml" xref="S5.I3.i1.p1.5.m5.2.3.3.1"></times><ci id="S5.I3.i1.p1.5.m5.2.3.3.2a.cmml" xref="S5.I3.i1.p1.5.m5.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i1.p1.5.m5.2.3.3.2.cmml" xref="S5.I3.i1.p1.5.m5.2.3.3.2">val</mtext></ci><ci id="S5.I3.i1.p1.5.m5.2.2.cmml" xref="S5.I3.i1.p1.5.m5.2.2">𝜌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.5.m5.2c">\textsf{val}(h)=\textsf{val}(\rho)</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.5.m5.2d">val ( italic_h ) = val ( italic_ρ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.6"> has a length </span><math alttext="|h|\leq|V|\cdot B\cdot W" class="ltx_Math" display="inline" id="S5.I3.i1.p1.6.m6.2"><semantics id="S5.I3.i1.p1.6.m6.2a"><mrow id="S5.I3.i1.p1.6.m6.2.3" xref="S5.I3.i1.p1.6.m6.2.3.cmml"><mrow id="S5.I3.i1.p1.6.m6.2.3.2.2" xref="S5.I3.i1.p1.6.m6.2.3.2.1.cmml"><mo id="S5.I3.i1.p1.6.m6.2.3.2.2.1" stretchy="false" xref="S5.I3.i1.p1.6.m6.2.3.2.1.1.cmml">|</mo><mi id="S5.I3.i1.p1.6.m6.1.1" xref="S5.I3.i1.p1.6.m6.1.1.cmml">h</mi><mo id="S5.I3.i1.p1.6.m6.2.3.2.2.2" stretchy="false" xref="S5.I3.i1.p1.6.m6.2.3.2.1.1.cmml">|</mo></mrow><mo id="S5.I3.i1.p1.6.m6.2.3.1" xref="S5.I3.i1.p1.6.m6.2.3.1.cmml">≤</mo><mrow id="S5.I3.i1.p1.6.m6.2.3.3" xref="S5.I3.i1.p1.6.m6.2.3.3.cmml"><mrow id="S5.I3.i1.p1.6.m6.2.3.3.2.2" xref="S5.I3.i1.p1.6.m6.2.3.3.2.1.cmml"><mo id="S5.I3.i1.p1.6.m6.2.3.3.2.2.1" stretchy="false" xref="S5.I3.i1.p1.6.m6.2.3.3.2.1.1.cmml">|</mo><mi id="S5.I3.i1.p1.6.m6.2.2" xref="S5.I3.i1.p1.6.m6.2.2.cmml">V</mi><mo id="S5.I3.i1.p1.6.m6.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S5.I3.i1.p1.6.m6.2.3.3.2.1.1.cmml">|</mo></mrow><mo id="S5.I3.i1.p1.6.m6.2.3.3.1" rspace="0.222em" xref="S5.I3.i1.p1.6.m6.2.3.3.1.cmml">⋅</mo><mi id="S5.I3.i1.p1.6.m6.2.3.3.3" xref="S5.I3.i1.p1.6.m6.2.3.3.3.cmml">B</mi><mo id="S5.I3.i1.p1.6.m6.2.3.3.1a" lspace="0.222em" rspace="0.222em" xref="S5.I3.i1.p1.6.m6.2.3.3.1.cmml">⋅</mo><mi id="S5.I3.i1.p1.6.m6.2.3.3.4" xref="S5.I3.i1.p1.6.m6.2.3.3.4.cmml">W</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.6.m6.2b"><apply id="S5.I3.i1.p1.6.m6.2.3.cmml" xref="S5.I3.i1.p1.6.m6.2.3"><leq id="S5.I3.i1.p1.6.m6.2.3.1.cmml" xref="S5.I3.i1.p1.6.m6.2.3.1"></leq><apply id="S5.I3.i1.p1.6.m6.2.3.2.1.cmml" xref="S5.I3.i1.p1.6.m6.2.3.2.2"><abs id="S5.I3.i1.p1.6.m6.2.3.2.1.1.cmml" xref="S5.I3.i1.p1.6.m6.2.3.2.2.1"></abs><ci id="S5.I3.i1.p1.6.m6.1.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1">ℎ</ci></apply><apply id="S5.I3.i1.p1.6.m6.2.3.3.cmml" xref="S5.I3.i1.p1.6.m6.2.3.3"><ci id="S5.I3.i1.p1.6.m6.2.3.3.1.cmml" xref="S5.I3.i1.p1.6.m6.2.3.3.1">⋅</ci><apply id="S5.I3.i1.p1.6.m6.2.3.3.2.1.cmml" xref="S5.I3.i1.p1.6.m6.2.3.3.2.2"><abs id="S5.I3.i1.p1.6.m6.2.3.3.2.1.1.cmml" xref="S5.I3.i1.p1.6.m6.2.3.3.2.2.1"></abs><ci id="S5.I3.i1.p1.6.m6.2.2.cmml" xref="S5.I3.i1.p1.6.m6.2.2">𝑉</ci></apply><ci id="S5.I3.i1.p1.6.m6.2.3.3.3.cmml" xref="S5.I3.i1.p1.6.m6.2.3.3.3">𝐵</ci><ci id="S5.I3.i1.p1.6.m6.2.3.3.4.cmml" xref="S5.I3.i1.p1.6.m6.2.3.3.4">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.6.m6.2c">|h|\leq|V|\cdot B\cdot W</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.6.m6.2d">| italic_h | ≤ | italic_V | ⋅ italic_B ⋅ italic_W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.7"> (the presence of </span><math alttext="W" class="ltx_Math" display="inline" id="S5.I3.i1.p1.7.m7.1"><semantics id="S5.I3.i1.p1.7.m7.1a"><mi id="S5.I3.i1.p1.7.m7.1.1" xref="S5.I3.i1.p1.7.m7.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.7.m7.1b"><ci id="S5.I3.i1.p1.7.m7.1.1.cmml" xref="S5.I3.i1.p1.7.m7.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.7.m7.1c">W</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.7.m7.1d">italic_W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i1.p1.7.8"> comes from the fact that the arena has been made binary).</span></p> </div> </li> <li class="ltx_item" id="S5.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I3.i2.p1"> <p class="ltx_p" id="S5.I3.i2.p1.2"><span class="ltx_text ltx_font_italic" id="S5.I3.i2.p1.2.1">Once Player </span><math alttext="0" class="ltx_Math" display="inline" id="S5.I3.i2.p1.1.m1.1"><semantics id="S5.I3.i2.p1.1.m1.1a"><mn id="S5.I3.i2.p1.1.m1.1.1" xref="S5.I3.i2.p1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.1.m1.1b"><cn id="S5.I3.i2.p1.1.m1.1.1.cmml" type="integer" xref="S5.I3.i2.p1.1.m1.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i2.p1.2.2">’s target is visited and before the last region, </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.I3.i2.p1.2.m2.1"><semantics id="S5.I3.i2.p1.2.m2.1a"><msubsup id="S5.I3.i2.p1.2.m2.1.1" xref="S5.I3.i2.p1.2.m2.1.1.cmml"><mi id="S5.I3.i2.p1.2.m2.1.1.2.2" xref="S5.I3.i2.p1.2.m2.1.1.2.2.cmml">σ</mi><mn id="S5.I3.i2.p1.2.m2.1.1.3" xref="S5.I3.i2.p1.2.m2.1.1.3.cmml">0</mn><mo id="S5.I3.i2.p1.2.m2.1.1.2.3" xref="S5.I3.i2.p1.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.2.m2.1b"><apply id="S5.I3.i2.p1.2.m2.1.1.cmml" xref="S5.I3.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I3.i2.p1.2.m2.1.1.1.cmml" xref="S5.I3.i2.p1.2.m2.1.1">subscript</csymbol><apply id="S5.I3.i2.p1.2.m2.1.1.2.cmml" xref="S5.I3.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I3.i2.p1.2.m2.1.1.2.1.cmml" xref="S5.I3.i2.p1.2.m2.1.1">superscript</csymbol><ci id="S5.I3.i2.p1.2.m2.1.1.2.2.cmml" xref="S5.I3.i2.p1.2.m2.1.1.2.2">𝜎</ci><ci id="S5.I3.i2.p1.2.m2.1.1.2.3.cmml" xref="S5.I3.i2.p1.2.m2.1.1.2.3">′</ci></apply><cn id="S5.I3.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.I3.i2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.2.m2.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i2.p1.2.3"> is “locally” memoryless inside each region, as there is no cycle.</span></p> </div> </li> <li class="ltx_item" id="S5.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I3.i3.p1"> <p class="ltx_p" id="S5.I3.i3.p1.8"><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.1">In the last region, as soon as a vertex is repeated, we replace </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.I3.i3.p1.1.m1.1"><semantics id="S5.I3.i3.p1.1.m1.1a"><msubsup id="S5.I3.i3.p1.1.m1.1.1" xref="S5.I3.i3.p1.1.m1.1.1.cmml"><mi id="S5.I3.i3.p1.1.m1.1.1.2.2" xref="S5.I3.i3.p1.1.m1.1.1.2.2.cmml">σ</mi><mn id="S5.I3.i3.p1.1.m1.1.1.3" xref="S5.I3.i3.p1.1.m1.1.1.3.cmml">0</mn><mo id="S5.I3.i3.p1.1.m1.1.1.2.3" xref="S5.I3.i3.p1.1.m1.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.1.m1.1b"><apply id="S5.I3.i3.p1.1.m1.1.1.cmml" xref="S5.I3.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I3.i3.p1.1.m1.1.1.1.cmml" xref="S5.I3.i3.p1.1.m1.1.1">subscript</csymbol><apply id="S5.I3.i3.p1.1.m1.1.1.2.cmml" xref="S5.I3.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I3.i3.p1.1.m1.1.1.2.1.cmml" xref="S5.I3.i3.p1.1.m1.1.1">superscript</csymbol><ci id="S5.I3.i3.p1.1.m1.1.1.2.2.cmml" xref="S5.I3.i3.p1.1.m1.1.1.2.2">𝜎</ci><ci id="S5.I3.i3.p1.1.m1.1.1.2.3.cmml" xref="S5.I3.i3.p1.1.m1.1.1.2.3">′</ci></apply><cn id="S5.I3.i3.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.I3.i3.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.1.m1.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.2"> by a memoryless strategy that repeats this cycle forever. We then get a lasso replacing </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I3.i3.p1.2.m2.1"><semantics id="S5.I3.i3.p1.2.m2.1a"><mi id="S5.I3.i3.p1.2.m2.1.1" xref="S5.I3.i3.p1.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.2.m2.1b"><ci id="S5.I3.i3.p1.2.m2.1.1.cmml" xref="S5.I3.i3.p1.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.2.m2.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.3"> that has the same value and cost. We keep the same notation </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I3.i3.p1.3.m3.1"><semantics id="S5.I3.i3.p1.3.m3.1a"><mi id="S5.I3.i3.p1.3.m3.1.1" xref="S5.I3.i3.p1.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.3.m3.1b"><ci id="S5.I3.i3.p1.3.m3.1.1.cmml" xref="S5.I3.i3.p1.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.3.m3.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.4"> for this lasso. We also keep the notation </span><math alttext="\textsf{Wit}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S5.I3.i3.p1.4.m4.1"><semantics id="S5.I3.i3.p1.4.m4.1a"><msub id="S5.I3.i3.p1.4.m4.1.1" xref="S5.I3.i3.p1.4.m4.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.4.m4.1.1.2" xref="S5.I3.i3.p1.4.m4.1.1.2a.cmml">Wit</mtext><msubsup id="S5.I3.i3.p1.4.m4.1.1.3" xref="S5.I3.i3.p1.4.m4.1.1.3.cmml"><mi id="S5.I3.i3.p1.4.m4.1.1.3.2.2" xref="S5.I3.i3.p1.4.m4.1.1.3.2.2.cmml">σ</mi><mn id="S5.I3.i3.p1.4.m4.1.1.3.3" xref="S5.I3.i3.p1.4.m4.1.1.3.3.cmml">0</mn><mo id="S5.I3.i3.p1.4.m4.1.1.3.2.3" xref="S5.I3.i3.p1.4.m4.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.4.m4.1b"><apply id="S5.I3.i3.p1.4.m4.1.1.cmml" xref="S5.I3.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I3.i3.p1.4.m4.1.1.1.cmml" xref="S5.I3.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S5.I3.i3.p1.4.m4.1.1.2a.cmml" xref="S5.I3.i3.p1.4.m4.1.1.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.4.m4.1.1.2.cmml" xref="S5.I3.i3.p1.4.m4.1.1.2">Wit</mtext></ci><apply id="S5.I3.i3.p1.4.m4.1.1.3.cmml" xref="S5.I3.i3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.I3.i3.p1.4.m4.1.1.3.1.cmml" xref="S5.I3.i3.p1.4.m4.1.1.3">subscript</csymbol><apply id="S5.I3.i3.p1.4.m4.1.1.3.2.cmml" xref="S5.I3.i3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.I3.i3.p1.4.m4.1.1.3.2.1.cmml" xref="S5.I3.i3.p1.4.m4.1.1.3">superscript</csymbol><ci id="S5.I3.i3.p1.4.m4.1.1.3.2.2.cmml" xref="S5.I3.i3.p1.4.m4.1.1.3.2.2">𝜎</ci><ci id="S5.I3.i3.p1.4.m4.1.1.3.2.3.cmml" xref="S5.I3.i3.p1.4.m4.1.1.3.2.3">′</ci></apply><cn id="S5.I3.i3.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S5.I3.i3.p1.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.4.m4.1c">\textsf{Wit}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.4.m4.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.5"> for the set of lasso witnesses. Notice that for deviations </span><math alttext="hv" class="ltx_Math" display="inline" id="S5.I3.i3.p1.5.m5.1"><semantics id="S5.I3.i3.p1.5.m5.1a"><mrow id="S5.I3.i3.p1.5.m5.1.1" xref="S5.I3.i3.p1.5.m5.1.1.cmml"><mi id="S5.I3.i3.p1.5.m5.1.1.2" xref="S5.I3.i3.p1.5.m5.1.1.2.cmml">h</mi><mo id="S5.I3.i3.p1.5.m5.1.1.1" xref="S5.I3.i3.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S5.I3.i3.p1.5.m5.1.1.3" xref="S5.I3.i3.p1.5.m5.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.5.m5.1b"><apply id="S5.I3.i3.p1.5.m5.1.1.cmml" xref="S5.I3.i3.p1.5.m5.1.1"><times id="S5.I3.i3.p1.5.m5.1.1.1.cmml" xref="S5.I3.i3.p1.5.m5.1.1.1"></times><ci id="S5.I3.i3.p1.5.m5.1.1.2.cmml" xref="S5.I3.i3.p1.5.m5.1.1.2">ℎ</ci><ci id="S5.I3.i3.p1.5.m5.1.1.3.cmml" xref="S5.I3.i3.p1.5.m5.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.5.m5.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.5.m5.1d">italic_h italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.6"> such that </span><math alttext="h" class="ltx_Math" display="inline" id="S5.I3.i3.p1.6.m6.1"><semantics id="S5.I3.i3.p1.6.m6.1a"><mi id="S5.I3.i3.p1.6.m6.1.1" xref="S5.I3.i3.p1.6.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.6.m6.1b"><ci id="S5.I3.i3.p1.6.m6.1.1.cmml" xref="S5.I3.i3.p1.6.m6.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.6.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.6.m6.1d">italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.7"> belongs to the last region of a witness, the strategy </span><math alttext="\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}" class="ltx_Math" display="inline" id="S5.I3.i3.p1.7.m7.2"><semantics id="S5.I3.i3.p1.7.m7.2a"><msubsup id="S5.I3.i3.p1.7.m7.2.3" xref="S5.I3.i3.p1.7.m7.2.3.cmml"><mi id="S5.I3.i3.p1.7.m7.2.3.2.2" xref="S5.I3.i3.p1.7.m7.2.3.2.2.cmml">τ</mi><mrow id="S5.I3.i3.p1.7.m7.2.2.2.2" xref="S5.I3.i3.p1.7.m7.2.2.2.3.cmml"><mi id="S5.I3.i3.p1.7.m7.1.1.1.1" xref="S5.I3.i3.p1.7.m7.1.1.1.1.cmml">v</mi><mo id="S5.I3.i3.p1.7.m7.2.2.2.2.2" xref="S5.I3.i3.p1.7.m7.2.2.2.3.cmml">,</mo><mrow id="S5.I3.i3.p1.7.m7.2.2.2.2.1" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.7.m7.2.2.2.2.1.3" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.3a.cmml">rec</mtext><mo id="S5.I3.i3.p1.7.m7.2.2.2.2.1.2" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.cmml"><mo id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.cmml"><mi id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.2" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.2.cmml">h</mi><mo id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.1" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.3" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.7.m7.2.3.2.3" xref="S5.I3.i3.p1.7.m7.2.3.2.3a.cmml">Pun</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.7.m7.2b"><apply id="S5.I3.i3.p1.7.m7.2.3.cmml" xref="S5.I3.i3.p1.7.m7.2.3"><csymbol cd="ambiguous" id="S5.I3.i3.p1.7.m7.2.3.1.cmml" xref="S5.I3.i3.p1.7.m7.2.3">subscript</csymbol><apply id="S5.I3.i3.p1.7.m7.2.3.2.cmml" xref="S5.I3.i3.p1.7.m7.2.3"><csymbol cd="ambiguous" id="S5.I3.i3.p1.7.m7.2.3.2.1.cmml" xref="S5.I3.i3.p1.7.m7.2.3">superscript</csymbol><ci id="S5.I3.i3.p1.7.m7.2.3.2.2.cmml" xref="S5.I3.i3.p1.7.m7.2.3.2.2">𝜏</ci><ci id="S5.I3.i3.p1.7.m7.2.3.2.3a.cmml" xref="S5.I3.i3.p1.7.m7.2.3.2.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.7.m7.2.3.2.3.cmml" mathsize="70%" xref="S5.I3.i3.p1.7.m7.2.3.2.3">Pun</mtext></ci></apply><list id="S5.I3.i3.p1.7.m7.2.2.2.3.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2"><ci id="S5.I3.i3.p1.7.m7.1.1.1.1.cmml" xref="S5.I3.i3.p1.7.m7.1.1.1.1">𝑣</ci><apply id="S5.I3.i3.p1.7.m7.2.2.2.2.1.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1"><times id="S5.I3.i3.p1.7.m7.2.2.2.2.1.2.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.2"></times><ci id="S5.I3.i3.p1.7.m7.2.2.2.2.1.3a.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.7.m7.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1"><times id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.1.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.1"></times><ci id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.2.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.3.cmml" xref="S5.I3.i3.p1.7.m7.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.7.m7.2c">\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.7.m7.2d">italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.8"> is memoryless (as </span><math alttext="\textsf{val}(h)<\infty" class="ltx_Math" display="inline" id="S5.I3.i3.p1.8.m8.1"><semantics id="S5.I3.i3.p1.8.m8.1a"><mrow id="S5.I3.i3.p1.8.m8.1.2" xref="S5.I3.i3.p1.8.m8.1.2.cmml"><mrow id="S5.I3.i3.p1.8.m8.1.2.2" xref="S5.I3.i3.p1.8.m8.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.8.m8.1.2.2.2" xref="S5.I3.i3.p1.8.m8.1.2.2.2a.cmml">val</mtext><mo id="S5.I3.i3.p1.8.m8.1.2.2.1" xref="S5.I3.i3.p1.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S5.I3.i3.p1.8.m8.1.2.2.3.2" xref="S5.I3.i3.p1.8.m8.1.2.2.cmml"><mo id="S5.I3.i3.p1.8.m8.1.2.2.3.2.1" stretchy="false" xref="S5.I3.i3.p1.8.m8.1.2.2.cmml">(</mo><mi id="S5.I3.i3.p1.8.m8.1.1" xref="S5.I3.i3.p1.8.m8.1.1.cmml">h</mi><mo id="S5.I3.i3.p1.8.m8.1.2.2.3.2.2" stretchy="false" xref="S5.I3.i3.p1.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.I3.i3.p1.8.m8.1.2.1" xref="S5.I3.i3.p1.8.m8.1.2.1.cmml"><</mo><mi id="S5.I3.i3.p1.8.m8.1.2.3" mathvariant="normal" xref="S5.I3.i3.p1.8.m8.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.8.m8.1b"><apply id="S5.I3.i3.p1.8.m8.1.2.cmml" xref="S5.I3.i3.p1.8.m8.1.2"><lt id="S5.I3.i3.p1.8.m8.1.2.1.cmml" xref="S5.I3.i3.p1.8.m8.1.2.1"></lt><apply id="S5.I3.i3.p1.8.m8.1.2.2.cmml" xref="S5.I3.i3.p1.8.m8.1.2.2"><times id="S5.I3.i3.p1.8.m8.1.2.2.1.cmml" xref="S5.I3.i3.p1.8.m8.1.2.2.1"></times><ci id="S5.I3.i3.p1.8.m8.1.2.2.2a.cmml" xref="S5.I3.i3.p1.8.m8.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I3.i3.p1.8.m8.1.2.2.2.cmml" xref="S5.I3.i3.p1.8.m8.1.2.2.2">val</mtext></ci><ci id="S5.I3.i3.p1.8.m8.1.1.cmml" xref="S5.I3.i3.p1.8.m8.1.1">ℎ</ci></apply><infinity id="S5.I3.i3.p1.8.m8.1.2.3.cmml" xref="S5.I3.i3.p1.8.m8.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.8.m8.1c">\textsf{val}(h)<\infty</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.8.m8.1d">val ( italic_h ) < ∞</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.9">, see Lemma </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem2" title="Lemma 5.2 ‣ 5.2 Punishing Strategies ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.2</span></a><span class="ltx_text ltx_font_italic" id="S5.I3.i3.p1.8.10">). Hence the modification of the witnesses has no impact on the deviating strategies.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S5.Thmtheorem4.p2"> <p class="ltx_p" id="S5.Thmtheorem4.p2.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p2.2.2">Let us study the memory necessary for <math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p2.1.1.m1.1"><semantics id="S5.Thmtheorem4.p2.1.1.m1.1a"><msubsup id="S5.Thmtheorem4.p2.1.1.m1.1.1" xref="S5.Thmtheorem4.p2.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem4.p2.1.1.m1.1.1.2.2" xref="S5.Thmtheorem4.p2.1.1.m1.1.1.2.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p2.1.1.m1.1.1.3" xref="S5.Thmtheorem4.p2.1.1.m1.1.1.3.cmml">0</mn><mo id="S5.Thmtheorem4.p2.1.1.m1.1.1.2.3" xref="S5.Thmtheorem4.p2.1.1.m1.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p2.1.1.m1.1b"><apply id="S5.Thmtheorem4.p2.1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p2.1.1.m1.1.1"><csymbol 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alttext="\textsf{Wit}_{\sigma^{\prime}_{0}}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p2.2.2.m2.1"><semantics id="S5.Thmtheorem4.p2.2.2.m2.1a"><msub id="S5.Thmtheorem4.p2.2.2.m2.1.1" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p2.2.2.m2.1.1.2" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.2a.cmml">Wit</mtext><msubsup id="S5.Thmtheorem4.p2.2.2.m2.1.1.3" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3.cmml"><mi id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.2" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.3" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3.3.cmml">0</mn><mo id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.3" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.3.cmml">′</mo></msubsup></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p2.2.2.m2.1b"><apply id="S5.Thmtheorem4.p2.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p2.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p2.2.2.m2.1.1.2a.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p2.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.2">Wit</mtext></ci><apply id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.1.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3">subscript</csymbol><apply id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.1.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.2.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.2">𝜎</ci><ci id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.3.cmml" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3.2.3">′</ci></apply><cn id="S5.Thmtheorem4.p2.2.2.m2.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p2.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p2.2.2.m2.1c">\textsf{Wit}_{\sigma^{\prime}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p2.2.2.m2.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>:</span></p> <ul class="ltx_itemize" id="S5.I4"> <li class="ltx_item" id="S5.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I4.i1.p1"> <p class="ltx_p" id="S5.I4.i1.p1.8"><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.1">The number of regions traversed by a witness </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I4.i1.p1.1.m1.1"><semantics id="S5.I4.i1.p1.1.m1.1a"><mi id="S5.I4.i1.p1.1.m1.1.1" xref="S5.I4.i1.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.1.m1.1b"><ci id="S5.I4.i1.p1.1.m1.1.1.cmml" xref="S5.I4.i1.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.1.m1.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.2"> is bounded by </span><math alttext="(t+2)\cdot|\textsf{Wit}_{\sigma^{\prime}_{0}}|" class="ltx_Math" display="inline" id="S5.I4.i1.p1.2.m2.2"><semantics id="S5.I4.i1.p1.2.m2.2a"><mrow id="S5.I4.i1.p1.2.m2.2.2" xref="S5.I4.i1.p1.2.m2.2.2.cmml"><mrow id="S5.I4.i1.p1.2.m2.1.1.1.1" xref="S5.I4.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S5.I4.i1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S5.I4.i1.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.I4.i1.p1.2.m2.1.1.1.1.1" xref="S5.I4.i1.p1.2.m2.1.1.1.1.1.cmml"><mi id="S5.I4.i1.p1.2.m2.1.1.1.1.1.2" xref="S5.I4.i1.p1.2.m2.1.1.1.1.1.2.cmml">t</mi><mo 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xref="S5.I4.i1.p1.2.m2.2.2.2.1.1.3.3.cmml">0</mn><mo id="S5.I4.i1.p1.2.m2.2.2.2.1.1.3.2.3" xref="S5.I4.i1.p1.2.m2.2.2.2.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S5.I4.i1.p1.2.m2.2.2.2.1.3" stretchy="false" xref="S5.I4.i1.p1.2.m2.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.2.m2.2b"><apply id="S5.I4.i1.p1.2.m2.2.2.cmml" xref="S5.I4.i1.p1.2.m2.2.2"><ci id="S5.I4.i1.p1.2.m2.2.2.3.cmml" xref="S5.I4.i1.p1.2.m2.2.2.3">⋅</ci><apply id="S5.I4.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S5.I4.i1.p1.2.m2.1.1.1.1"><plus id="S5.I4.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.I4.i1.p1.2.m2.1.1.1.1.1.1"></plus><ci id="S5.I4.i1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S5.I4.i1.p1.2.m2.1.1.1.1.1.2">𝑡</ci><cn id="S5.I4.i1.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S5.I4.i1.p1.2.m2.1.1.1.1.1.3">2</cn></apply><apply id="S5.I4.i1.p1.2.m2.2.2.2.2.cmml" xref="S5.I4.i1.p1.2.m2.2.2.2.1"><abs id="S5.I4.i1.p1.2.m2.2.2.2.2.1.cmml" xref="S5.I4.i1.p1.2.m2.2.2.2.1.2"></abs><apply 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xref="S5.I4.i1.p1.2.m2.2.2.2.1.1.3.2.3">′</ci></apply><cn id="S5.I4.i1.p1.2.m2.2.2.2.1.1.3.3.cmml" type="integer" xref="S5.I4.i1.p1.2.m2.2.2.2.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.2.m2.2c">(t+2)\cdot|\textsf{Wit}_{\sigma^{\prime}_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.2.m2.2d">( italic_t + 2 ) ⋅ | Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.3">. Indeed, the number of visited targets increases from </span><math alttext="0" class="ltx_Math" display="inline" id="S5.I4.i1.p1.3.m3.1"><semantics id="S5.I4.i1.p1.3.m3.1a"><mn id="S5.I4.i1.p1.3.m3.1.1" xref="S5.I4.i1.p1.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.3.m3.1b"><cn id="S5.I4.i1.p1.3.m3.1.1.cmml" type="integer" xref="S5.I4.i1.p1.3.m3.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.4"> to </span><math alttext="t+1" class="ltx_Math" display="inline" id="S5.I4.i1.p1.4.m4.1"><semantics id="S5.I4.i1.p1.4.m4.1a"><mrow id="S5.I4.i1.p1.4.m4.1.1" xref="S5.I4.i1.p1.4.m4.1.1.cmml"><mi id="S5.I4.i1.p1.4.m4.1.1.2" xref="S5.I4.i1.p1.4.m4.1.1.2.cmml">t</mi><mo id="S5.I4.i1.p1.4.m4.1.1.1" xref="S5.I4.i1.p1.4.m4.1.1.1.cmml">+</mo><mn id="S5.I4.i1.p1.4.m4.1.1.3" xref="S5.I4.i1.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.4.m4.1b"><apply id="S5.I4.i1.p1.4.m4.1.1.cmml" xref="S5.I4.i1.p1.4.m4.1.1"><plus id="S5.I4.i1.p1.4.m4.1.1.1.cmml" xref="S5.I4.i1.p1.4.m4.1.1.1"></plus><ci id="S5.I4.i1.p1.4.m4.1.1.2.cmml" xref="S5.I4.i1.p1.4.m4.1.1.2">𝑡</ci><cn id="S5.I4.i1.p1.4.m4.1.1.3.cmml" type="integer" xref="S5.I4.i1.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.4.m4.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.4.m4.1d">italic_t + 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.5"> and the set </span><math alttext="\textsf{Wit}_{\sigma^{\prime}_{0}}(h)" class="ltx_Math" display="inline" id="S5.I4.i1.p1.5.m5.1"><semantics id="S5.I4.i1.p1.5.m5.1a"><mrow id="S5.I4.i1.p1.5.m5.1.2" xref="S5.I4.i1.p1.5.m5.1.2.cmml"><msub id="S5.I4.i1.p1.5.m5.1.2.2" xref="S5.I4.i1.p1.5.m5.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I4.i1.p1.5.m5.1.2.2.2" xref="S5.I4.i1.p1.5.m5.1.2.2.2a.cmml">Wit</mtext><msubsup id="S5.I4.i1.p1.5.m5.1.2.2.3" xref="S5.I4.i1.p1.5.m5.1.2.2.3.cmml"><mi id="S5.I4.i1.p1.5.m5.1.2.2.3.2.2" xref="S5.I4.i1.p1.5.m5.1.2.2.3.2.2.cmml">σ</mi><mn id="S5.I4.i1.p1.5.m5.1.2.2.3.3" xref="S5.I4.i1.p1.5.m5.1.2.2.3.3.cmml">0</mn><mo id="S5.I4.i1.p1.5.m5.1.2.2.3.2.3" xref="S5.I4.i1.p1.5.m5.1.2.2.3.2.3.cmml">′</mo></msubsup></msub><mo id="S5.I4.i1.p1.5.m5.1.2.1" xref="S5.I4.i1.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S5.I4.i1.p1.5.m5.1.2.3.2" xref="S5.I4.i1.p1.5.m5.1.2.cmml"><mo id="S5.I4.i1.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S5.I4.i1.p1.5.m5.1.2.cmml">(</mo><mi id="S5.I4.i1.p1.5.m5.1.1" xref="S5.I4.i1.p1.5.m5.1.1.cmml">h</mi><mo id="S5.I4.i1.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S5.I4.i1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.5.m5.1b"><apply id="S5.I4.i1.p1.5.m5.1.2.cmml" xref="S5.I4.i1.p1.5.m5.1.2"><times id="S5.I4.i1.p1.5.m5.1.2.1.cmml" xref="S5.I4.i1.p1.5.m5.1.2.1"></times><apply id="S5.I4.i1.p1.5.m5.1.2.2.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S5.I4.i1.p1.5.m5.1.2.2.1.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2">subscript</csymbol><ci id="S5.I4.i1.p1.5.m5.1.2.2.2a.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I4.i1.p1.5.m5.1.2.2.2.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.2">Wit</mtext></ci><apply id="S5.I4.i1.p1.5.m5.1.2.2.3.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.3"><csymbol cd="ambiguous" id="S5.I4.i1.p1.5.m5.1.2.2.3.1.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.3">subscript</csymbol><apply id="S5.I4.i1.p1.5.m5.1.2.2.3.2.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.3"><csymbol cd="ambiguous" id="S5.I4.i1.p1.5.m5.1.2.2.3.2.1.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.3">superscript</csymbol><ci id="S5.I4.i1.p1.5.m5.1.2.2.3.2.2.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.3.2.2">𝜎</ci><ci id="S5.I4.i1.p1.5.m5.1.2.2.3.2.3.cmml" xref="S5.I4.i1.p1.5.m5.1.2.2.3.2.3">′</ci></apply><cn id="S5.I4.i1.p1.5.m5.1.2.2.3.3.cmml" type="integer" xref="S5.I4.i1.p1.5.m5.1.2.2.3.3">0</cn></apply></apply><ci id="S5.I4.i1.p1.5.m5.1.1.cmml" xref="S5.I4.i1.p1.5.m5.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.5.m5.1c">\textsf{Wit}_{\sigma^{\prime}_{0}}(h)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.5.m5.1d">Wit start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_h )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.6"> decreases until being equal to </span><math alttext="\{\rho\}" class="ltx_Math" display="inline" id="S5.I4.i1.p1.6.m6.1"><semantics id="S5.I4.i1.p1.6.m6.1a"><mrow id="S5.I4.i1.p1.6.m6.1.2.2" xref="S5.I4.i1.p1.6.m6.1.2.1.cmml"><mo id="S5.I4.i1.p1.6.m6.1.2.2.1" stretchy="false" xref="S5.I4.i1.p1.6.m6.1.2.1.cmml">{</mo><mi id="S5.I4.i1.p1.6.m6.1.1" xref="S5.I4.i1.p1.6.m6.1.1.cmml">ρ</mi><mo id="S5.I4.i1.p1.6.m6.1.2.2.2" stretchy="false" xref="S5.I4.i1.p1.6.m6.1.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.6.m6.1b"><set id="S5.I4.i1.p1.6.m6.1.2.1.cmml" xref="S5.I4.i1.p1.6.m6.1.2.2"><ci id="S5.I4.i1.p1.6.m6.1.1.cmml" xref="S5.I4.i1.p1.6.m6.1.1">𝜌</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.6.m6.1c">\{\rho\}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.6.m6.1d">{ italic_ρ }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.7">. As there are </span><math alttext="|C_{\sigma^{\prime}_{0}}|" class="ltx_Math" display="inline" id="S5.I4.i1.p1.7.m7.1"><semantics id="S5.I4.i1.p1.7.m7.1a"><mrow id="S5.I4.i1.p1.7.m7.1.1.1" xref="S5.I4.i1.p1.7.m7.1.1.2.cmml"><mo id="S5.I4.i1.p1.7.m7.1.1.1.2" stretchy="false" xref="S5.I4.i1.p1.7.m7.1.1.2.1.cmml">|</mo><msub id="S5.I4.i1.p1.7.m7.1.1.1.1" xref="S5.I4.i1.p1.7.m7.1.1.1.1.cmml"><mi id="S5.I4.i1.p1.7.m7.1.1.1.1.2" xref="S5.I4.i1.p1.7.m7.1.1.1.1.2.cmml">C</mi><msubsup id="S5.I4.i1.p1.7.m7.1.1.1.1.3" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3.cmml"><mi id="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.2" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.2.cmml">σ</mi><mn id="S5.I4.i1.p1.7.m7.1.1.1.1.3.3" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3.3.cmml">0</mn><mo id="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.3" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S5.I4.i1.p1.7.m7.1.1.1.3" stretchy="false" xref="S5.I4.i1.p1.7.m7.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.7.m7.1b"><apply id="S5.I4.i1.p1.7.m7.1.1.2.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1"><abs id="S5.I4.i1.p1.7.m7.1.1.2.1.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.2"></abs><apply id="S5.I4.i1.p1.7.m7.1.1.1.1.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S5.I4.i1.p1.7.m7.1.1.1.1.1.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="S5.I4.i1.p1.7.m7.1.1.1.1.2.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1.2">𝐶</ci><apply id="S5.I4.i1.p1.7.m7.1.1.1.1.3.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.I4.i1.p1.7.m7.1.1.1.1.3.1.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3">subscript</csymbol><apply id="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.1.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3">superscript</csymbol><ci id="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.2.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.2">𝜎</ci><ci id="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.3.cmml" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3.2.3">′</ci></apply><cn id="S5.I4.i1.p1.7.m7.1.1.1.1.3.3.cmml" type="integer" xref="S5.I4.i1.p1.7.m7.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.7.m7.1c">|C_{\sigma^{\prime}_{0}}|</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.7.m7.1d">| italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.8"> witnesses, the total number of traversed regions is bounded by </span><math alttext="(t+2)\cdot|C_{\sigma^{\prime}_{0}}|^{2}" class="ltx_Math" display="inline" id="S5.I4.i1.p1.8.m8.2"><semantics id="S5.I4.i1.p1.8.m8.2a"><mrow id="S5.I4.i1.p1.8.m8.2.2" xref="S5.I4.i1.p1.8.m8.2.2.cmml"><mrow id="S5.I4.i1.p1.8.m8.1.1.1.1" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.cmml"><mo id="S5.I4.i1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S5.I4.i1.p1.8.m8.1.1.1.1.1" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S5.I4.i1.p1.8.m8.1.1.1.1.1.2" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.2.cmml">t</mi><mo id="S5.I4.i1.p1.8.m8.1.1.1.1.1.1" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.1.cmml">+</mo><mn id="S5.I4.i1.p1.8.m8.1.1.1.1.1.3" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S5.I4.i1.p1.8.m8.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.I4.i1.p1.8.m8.2.2.3" rspace="0.222em" xref="S5.I4.i1.p1.8.m8.2.2.3.cmml">⋅</mo><msup id="S5.I4.i1.p1.8.m8.2.2.2" xref="S5.I4.i1.p1.8.m8.2.2.2.cmml"><mrow id="S5.I4.i1.p1.8.m8.2.2.2.1.1" xref="S5.I4.i1.p1.8.m8.2.2.2.1.2.cmml"><mo id="S5.I4.i1.p1.8.m8.2.2.2.1.1.2" stretchy="false" xref="S5.I4.i1.p1.8.m8.2.2.2.1.2.1.cmml">|</mo><msub id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.cmml"><mi id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.2" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.2.cmml">C</mi><msubsup id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.cmml"><mi id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.2" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.2.cmml">σ</mi><mn id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.3" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.3.cmml">0</mn><mo id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.3" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S5.I4.i1.p1.8.m8.2.2.2.1.1.3" stretchy="false" xref="S5.I4.i1.p1.8.m8.2.2.2.1.2.1.cmml">|</mo></mrow><mn id="S5.I4.i1.p1.8.m8.2.2.2.3" xref="S5.I4.i1.p1.8.m8.2.2.2.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.8.m8.2b"><apply id="S5.I4.i1.p1.8.m8.2.2.cmml" xref="S5.I4.i1.p1.8.m8.2.2"><ci id="S5.I4.i1.p1.8.m8.2.2.3.cmml" xref="S5.I4.i1.p1.8.m8.2.2.3">⋅</ci><apply id="S5.I4.i1.p1.8.m8.1.1.1.1.1.cmml" xref="S5.I4.i1.p1.8.m8.1.1.1.1"><plus id="S5.I4.i1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.1"></plus><ci id="S5.I4.i1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.2">𝑡</ci><cn id="S5.I4.i1.p1.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="S5.I4.i1.p1.8.m8.1.1.1.1.1.3">2</cn></apply><apply id="S5.I4.i1.p1.8.m8.2.2.2.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2"><csymbol cd="ambiguous" id="S5.I4.i1.p1.8.m8.2.2.2.2.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2">superscript</csymbol><apply id="S5.I4.i1.p1.8.m8.2.2.2.1.2.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1"><abs id="S5.I4.i1.p1.8.m8.2.2.2.1.2.1.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.2"></abs><apply id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.1.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1">subscript</csymbol><ci id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.2.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.2">𝐶</ci><apply id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.1.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3">subscript</csymbol><apply id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.1.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3">superscript</csymbol><ci id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.2.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.2">𝜎</ci><ci id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.3.cmml" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.2.3">′</ci></apply><cn id="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S5.I4.i1.p1.8.m8.2.2.2.1.1.1.3.3">0</cn></apply></apply></apply><cn id="S5.I4.i1.p1.8.m8.2.2.2.3.cmml" type="integer" xref="S5.I4.i1.p1.8.m8.2.2.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.8.m8.2c">(t+2)\cdot|C_{\sigma^{\prime}_{0}}|^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.8.m8.2d">( italic_t + 2 ) ⋅ | italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.9"> which is exponential by Corollary </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem11" title="Corollary 4.11 ‣ 4.4 Proof of Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.11</span></a><span class="ltx_text ltx_font_italic" id="S5.I4.i1.p1.8.10">.</span></p> </div> </li> <li class="ltx_item" id="S5.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I4.i2.p1"> <p class="ltx_p" id="S5.I4.i2.p1.8"><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.1">On the first regions traversed by a witness </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I4.i2.p1.1.m1.1"><semantics id="S5.I4.i2.p1.1.m1.1a"><mi id="S5.I4.i2.p1.1.m1.1.1" xref="S5.I4.i2.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.1.m1.1b"><ci id="S5.I4.i2.p1.1.m1.1.1.cmml" xref="S5.I4.i2.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.1.m1.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.2">, </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.2.m2.1"><semantics id="S5.I4.i2.p1.2.m2.1a"><msubsup id="S5.I4.i2.p1.2.m2.1.1" xref="S5.I4.i2.p1.2.m2.1.1.cmml"><mi id="S5.I4.i2.p1.2.m2.1.1.2.2" xref="S5.I4.i2.p1.2.m2.1.1.2.2.cmml">σ</mi><mn id="S5.I4.i2.p1.2.m2.1.1.3" xref="S5.I4.i2.p1.2.m2.1.1.3.cmml">0</mn><mo id="S5.I4.i2.p1.2.m2.1.1.2.3" xref="S5.I4.i2.p1.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.2.m2.1b"><apply id="S5.I4.i2.p1.2.m2.1.1.cmml" xref="S5.I4.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.2.m2.1.1.1.cmml" xref="S5.I4.i2.p1.2.m2.1.1">subscript</csymbol><apply id="S5.I4.i2.p1.2.m2.1.1.2.cmml" xref="S5.I4.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.2.m2.1.1.2.1.cmml" xref="S5.I4.i2.p1.2.m2.1.1">superscript</csymbol><ci id="S5.I4.i2.p1.2.m2.1.1.2.2.cmml" xref="S5.I4.i2.p1.2.m2.1.1.2.2">𝜎</ci><ci id="S5.I4.i2.p1.2.m2.1.1.2.3.cmml" xref="S5.I4.i2.p1.2.m2.1.1.2.3">′</ci></apply><cn id="S5.I4.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.I4.i2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.2.m2.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.3"> has to memorize the current history </span><math alttext="h\rho" class="ltx_Math" display="inline" id="S5.I4.i2.p1.3.m3.1"><semantics id="S5.I4.i2.p1.3.m3.1a"><mrow id="S5.I4.i2.p1.3.m3.1.1" xref="S5.I4.i2.p1.3.m3.1.1.cmml"><mi id="S5.I4.i2.p1.3.m3.1.1.2" xref="S5.I4.i2.p1.3.m3.1.1.2.cmml">h</mi><mo id="S5.I4.i2.p1.3.m3.1.1.1" xref="S5.I4.i2.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S5.I4.i2.p1.3.m3.1.1.3" xref="S5.I4.i2.p1.3.m3.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.3.m3.1b"><apply id="S5.I4.i2.p1.3.m3.1.1.cmml" xref="S5.I4.i2.p1.3.m3.1.1"><times id="S5.I4.i2.p1.3.m3.1.1.1.cmml" xref="S5.I4.i2.p1.3.m3.1.1.1"></times><ci id="S5.I4.i2.p1.3.m3.1.1.2.cmml" xref="S5.I4.i2.p1.3.m3.1.1.2">ℎ</ci><ci id="S5.I4.i2.p1.3.m3.1.1.3.cmml" xref="S5.I4.i2.p1.3.m3.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.3.m3.1c">h\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.3.m3.1d">italic_h italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.4"> until </span><math alttext="|h|=|V|\cdot B\cdot W" class="ltx_Math" display="inline" id="S5.I4.i2.p1.4.m4.2"><semantics id="S5.I4.i2.p1.4.m4.2a"><mrow id="S5.I4.i2.p1.4.m4.2.3" xref="S5.I4.i2.p1.4.m4.2.3.cmml"><mrow id="S5.I4.i2.p1.4.m4.2.3.2.2" xref="S5.I4.i2.p1.4.m4.2.3.2.1.cmml"><mo id="S5.I4.i2.p1.4.m4.2.3.2.2.1" stretchy="false" xref="S5.I4.i2.p1.4.m4.2.3.2.1.1.cmml">|</mo><mi id="S5.I4.i2.p1.4.m4.1.1" xref="S5.I4.i2.p1.4.m4.1.1.cmml">h</mi><mo id="S5.I4.i2.p1.4.m4.2.3.2.2.2" stretchy="false" xref="S5.I4.i2.p1.4.m4.2.3.2.1.1.cmml">|</mo></mrow><mo id="S5.I4.i2.p1.4.m4.2.3.1" xref="S5.I4.i2.p1.4.m4.2.3.1.cmml">=</mo><mrow id="S5.I4.i2.p1.4.m4.2.3.3" xref="S5.I4.i2.p1.4.m4.2.3.3.cmml"><mrow id="S5.I4.i2.p1.4.m4.2.3.3.2.2" xref="S5.I4.i2.p1.4.m4.2.3.3.2.1.cmml"><mo id="S5.I4.i2.p1.4.m4.2.3.3.2.2.1" stretchy="false" xref="S5.I4.i2.p1.4.m4.2.3.3.2.1.1.cmml">|</mo><mi id="S5.I4.i2.p1.4.m4.2.2" xref="S5.I4.i2.p1.4.m4.2.2.cmml">V</mi><mo id="S5.I4.i2.p1.4.m4.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S5.I4.i2.p1.4.m4.2.3.3.2.1.1.cmml">|</mo></mrow><mo id="S5.I4.i2.p1.4.m4.2.3.3.1" rspace="0.222em" xref="S5.I4.i2.p1.4.m4.2.3.3.1.cmml">⋅</mo><mi id="S5.I4.i2.p1.4.m4.2.3.3.3" xref="S5.I4.i2.p1.4.m4.2.3.3.3.cmml">B</mi><mo id="S5.I4.i2.p1.4.m4.2.3.3.1a" lspace="0.222em" rspace="0.222em" xref="S5.I4.i2.p1.4.m4.2.3.3.1.cmml">⋅</mo><mi id="S5.I4.i2.p1.4.m4.2.3.3.4" xref="S5.I4.i2.p1.4.m4.2.3.3.4.cmml">W</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.4.m4.2b"><apply id="S5.I4.i2.p1.4.m4.2.3.cmml" xref="S5.I4.i2.p1.4.m4.2.3"><eq id="S5.I4.i2.p1.4.m4.2.3.1.cmml" xref="S5.I4.i2.p1.4.m4.2.3.1"></eq><apply id="S5.I4.i2.p1.4.m4.2.3.2.1.cmml" xref="S5.I4.i2.p1.4.m4.2.3.2.2"><abs id="S5.I4.i2.p1.4.m4.2.3.2.1.1.cmml" xref="S5.I4.i2.p1.4.m4.2.3.2.2.1"></abs><ci id="S5.I4.i2.p1.4.m4.1.1.cmml" xref="S5.I4.i2.p1.4.m4.1.1">ℎ</ci></apply><apply id="S5.I4.i2.p1.4.m4.2.3.3.cmml" xref="S5.I4.i2.p1.4.m4.2.3.3"><ci id="S5.I4.i2.p1.4.m4.2.3.3.1.cmml" xref="S5.I4.i2.p1.4.m4.2.3.3.1">⋅</ci><apply id="S5.I4.i2.p1.4.m4.2.3.3.2.1.cmml" xref="S5.I4.i2.p1.4.m4.2.3.3.2.2"><abs id="S5.I4.i2.p1.4.m4.2.3.3.2.1.1.cmml" xref="S5.I4.i2.p1.4.m4.2.3.3.2.2.1"></abs><ci id="S5.I4.i2.p1.4.m4.2.2.cmml" xref="S5.I4.i2.p1.4.m4.2.2">𝑉</ci></apply><ci id="S5.I4.i2.p1.4.m4.2.3.3.3.cmml" xref="S5.I4.i2.p1.4.m4.2.3.3.3">𝐵</ci><ci id="S5.I4.i2.p1.4.m4.2.3.3.4.cmml" xref="S5.I4.i2.p1.4.m4.2.3.3.4">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.4.m4.2c">|h|=|V|\cdot B\cdot W</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.4.m4.2d">| italic_h | = | italic_V | ⋅ italic_B ⋅ italic_W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.5">, which needs an exponential memory. Then, on each other region traversed by </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I4.i2.p1.5.m5.1"><semantics id="S5.I4.i2.p1.5.m5.1a"><mi id="S5.I4.i2.p1.5.m5.1.1" xref="S5.I4.i2.p1.5.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.5.m5.1b"><ci id="S5.I4.i2.p1.5.m5.1.1.cmml" xref="S5.I4.i2.p1.5.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.5.m5.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.5.m5.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.6">, </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.6.m6.1"><semantics id="S5.I4.i2.p1.6.m6.1a"><msubsup id="S5.I4.i2.p1.6.m6.1.1" xref="S5.I4.i2.p1.6.m6.1.1.cmml"><mi id="S5.I4.i2.p1.6.m6.1.1.2.2" xref="S5.I4.i2.p1.6.m6.1.1.2.2.cmml">σ</mi><mn id="S5.I4.i2.p1.6.m6.1.1.3" xref="S5.I4.i2.p1.6.m6.1.1.3.cmml">0</mn><mo id="S5.I4.i2.p1.6.m6.1.1.2.3" xref="S5.I4.i2.p1.6.m6.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.6.m6.1b"><apply id="S5.I4.i2.p1.6.m6.1.1.cmml" xref="S5.I4.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.6.m6.1.1.1.cmml" xref="S5.I4.i2.p1.6.m6.1.1">subscript</csymbol><apply id="S5.I4.i2.p1.6.m6.1.1.2.cmml" xref="S5.I4.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.6.m6.1.1.2.1.cmml" xref="S5.I4.i2.p1.6.m6.1.1">superscript</csymbol><ci id="S5.I4.i2.p1.6.m6.1.1.2.2.cmml" xref="S5.I4.i2.p1.6.m6.1.1.2.2">𝜎</ci><ci id="S5.I4.i2.p1.6.m6.1.1.2.3.cmml" xref="S5.I4.i2.p1.6.m6.1.1.2.3">′</ci></apply><cn id="S5.I4.i2.p1.6.m6.1.1.3.cmml" type="integer" xref="S5.I4.i2.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.6.m6.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.6.m6.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.7"> is locally memoryless. Therefore, the memory necessary for </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.7.m7.1"><semantics id="S5.I4.i2.p1.7.m7.1a"><msubsup id="S5.I4.i2.p1.7.m7.1.1" xref="S5.I4.i2.p1.7.m7.1.1.cmml"><mi id="S5.I4.i2.p1.7.m7.1.1.2.2" xref="S5.I4.i2.p1.7.m7.1.1.2.2.cmml">σ</mi><mn id="S5.I4.i2.p1.7.m7.1.1.3" xref="S5.I4.i2.p1.7.m7.1.1.3.cmml">0</mn><mo id="S5.I4.i2.p1.7.m7.1.1.2.3" xref="S5.I4.i2.p1.7.m7.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.7.m7.1b"><apply id="S5.I4.i2.p1.7.m7.1.1.cmml" xref="S5.I4.i2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.7.m7.1.1.1.cmml" xref="S5.I4.i2.p1.7.m7.1.1">subscript</csymbol><apply id="S5.I4.i2.p1.7.m7.1.1.2.cmml" xref="S5.I4.i2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.7.m7.1.1.2.1.cmml" xref="S5.I4.i2.p1.7.m7.1.1">superscript</csymbol><ci id="S5.I4.i2.p1.7.m7.1.1.2.2.cmml" xref="S5.I4.i2.p1.7.m7.1.1.2.2">𝜎</ci><ci id="S5.I4.i2.p1.7.m7.1.1.2.3.cmml" xref="S5.I4.i2.p1.7.m7.1.1.2.3">′</ci></apply><cn id="S5.I4.i2.p1.7.m7.1.1.3.cmml" type="integer" xref="S5.I4.i2.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.7.m7.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.7.m7.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.8"> to produce the lasso witnesses is exponential, as there is an exponential number of regions on which </span><math alttext="\sigma^{\prime}_{0}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.8.m8.1"><semantics id="S5.I4.i2.p1.8.m8.1a"><msubsup id="S5.I4.i2.p1.8.m8.1.1" xref="S5.I4.i2.p1.8.m8.1.1.cmml"><mi id="S5.I4.i2.p1.8.m8.1.1.2.2" xref="S5.I4.i2.p1.8.m8.1.1.2.2.cmml">σ</mi><mn id="S5.I4.i2.p1.8.m8.1.1.3" xref="S5.I4.i2.p1.8.m8.1.1.3.cmml">0</mn><mo id="S5.I4.i2.p1.8.m8.1.1.2.3" xref="S5.I4.i2.p1.8.m8.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.8.m8.1b"><apply id="S5.I4.i2.p1.8.m8.1.1.cmml" xref="S5.I4.i2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.8.m8.1.1.1.cmml" xref="S5.I4.i2.p1.8.m8.1.1">subscript</csymbol><apply id="S5.I4.i2.p1.8.m8.1.1.2.cmml" xref="S5.I4.i2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.8.m8.1.1.2.1.cmml" xref="S5.I4.i2.p1.8.m8.1.1">superscript</csymbol><ci id="S5.I4.i2.p1.8.m8.1.1.2.2.cmml" xref="S5.I4.i2.p1.8.m8.1.1.2.2">𝜎</ci><ci id="S5.I4.i2.p1.8.m8.1.1.2.3.cmml" xref="S5.I4.i2.p1.8.m8.1.1.2.3">′</ci></apply><cn id="S5.I4.i2.p1.8.m8.1.1.3.cmml" type="integer" xref="S5.I4.i2.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.8.m8.1c">\sigma^{\prime}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.8.m8.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I4.i2.p1.8.9"> needs an exponential memory.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S5.Thmtheorem4.p3"> <p class="ltx_p" id="S5.Thmtheorem4.p3.12"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p3.12.12">It remains to explain how to play with a finite-memory strategy in case of deviations from the witnesses. With each history <math alttext="g" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.1.1.m1.1"><semantics id="S5.Thmtheorem4.p3.1.1.m1.1a"><mi id="S5.Thmtheorem4.p3.1.1.m1.1.1" xref="S5.Thmtheorem4.p3.1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.1.1.m1.1b"><ci id="S5.Thmtheorem4.p3.1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p3.1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.1.1.m1.1d">italic_g</annotation></semantics></math>, we associate its record <math alttext="\textsf{rec}({g})" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.2.2.m2.1"><semantics id="S5.Thmtheorem4.p3.2.2.m2.1a"><mrow id="S5.Thmtheorem4.p3.2.2.m2.1.2" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.2.2.m2.1.2.2" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.2a.cmml">rec</mtext><mo id="S5.Thmtheorem4.p3.2.2.m2.1.2.1" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p3.2.2.m2.1.2.3.2" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.cmml"><mo id="S5.Thmtheorem4.p3.2.2.m2.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.cmml">(</mo><mi id="S5.Thmtheorem4.p3.2.2.m2.1.1" xref="S5.Thmtheorem4.p3.2.2.m2.1.1.cmml">g</mi><mo id="S5.Thmtheorem4.p3.2.2.m2.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.2.2.m2.1b"><apply id="S5.Thmtheorem4.p3.2.2.m2.1.2.cmml" xref="S5.Thmtheorem4.p3.2.2.m2.1.2"><times id="S5.Thmtheorem4.p3.2.2.m2.1.2.1.cmml" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.1"></times><ci id="S5.Thmtheorem4.p3.2.2.m2.1.2.2a.cmml" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.2.2.m2.1.2.2.cmml" xref="S5.Thmtheorem4.p3.2.2.m2.1.2.2">rec</mtext></ci><ci id="S5.Thmtheorem4.p3.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p3.2.2.m2.1.1">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.2.2.m2.1c">\textsf{rec}({g})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.2.2.m2.1d">rec ( italic_g )</annotation></semantics></math> whose second component indicates whether <math alttext="\textsf{val}(g)<\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.3.3.m3.1"><semantics id="S5.Thmtheorem4.p3.3.3.m3.1a"><mrow id="S5.Thmtheorem4.p3.3.3.m3.1.2" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.cmml"><mrow id="S5.Thmtheorem4.p3.3.3.m3.1.2.2" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.2" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.1" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.3.2" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.cmml"><mo id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem4.p3.3.3.m3.1.1" xref="S5.Thmtheorem4.p3.3.3.m3.1.1.cmml">g</mi><mo id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem4.p3.3.3.m3.1.2.1" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.1.cmml"><</mo><mi id="S5.Thmtheorem4.p3.3.3.m3.1.2.3" mathvariant="normal" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.3.3.m3.1b"><apply id="S5.Thmtheorem4.p3.3.3.m3.1.2.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.2"><lt id="S5.Thmtheorem4.p3.3.3.m3.1.2.1.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.1"></lt><apply id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2"><times id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.1.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.1"></times><ci id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.2a.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.3.3.m3.1.2.2.2.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem4.p3.3.3.m3.1.1.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.1">𝑔</ci></apply><infinity id="S5.Thmtheorem4.p3.3.3.m3.1.2.3.cmml" xref="S5.Thmtheorem4.p3.3.3.m3.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.3.3.m3.1c">\textsf{val}(g)<\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.3.3.m3.1d">val ( italic_g ) < ∞</annotation></semantics></math> or <math alttext="\textsf{val}(g)=\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.4.4.m4.1"><semantics id="S5.Thmtheorem4.p3.4.4.m4.1a"><mrow id="S5.Thmtheorem4.p3.4.4.m4.1.2" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.cmml"><mrow id="S5.Thmtheorem4.p3.4.4.m4.1.2.2" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.2" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.1" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.3.2" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.cmml"><mo id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem4.p3.4.4.m4.1.1" xref="S5.Thmtheorem4.p3.4.4.m4.1.1.cmml">g</mi><mo id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem4.p3.4.4.m4.1.2.1" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.1.cmml">=</mo><mi id="S5.Thmtheorem4.p3.4.4.m4.1.2.3" mathvariant="normal" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.4.4.m4.1b"><apply id="S5.Thmtheorem4.p3.4.4.m4.1.2.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.2"><eq id="S5.Thmtheorem4.p3.4.4.m4.1.2.1.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.1"></eq><apply id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2"><times id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.1.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.1"></times><ci id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.2a.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.4.4.m4.1.2.2.2.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem4.p3.4.4.m4.1.1.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.1">𝑔</ci></apply><infinity id="S5.Thmtheorem4.p3.4.4.m4.1.2.3.cmml" xref="S5.Thmtheorem4.p3.4.4.m4.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.4.4.m4.1c">\textsf{val}(g)=\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.4.4.m4.1d">val ( italic_g ) = ∞</annotation></semantics></math>. Let <math alttext="hv" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.5.5.m5.1"><semantics id="S5.Thmtheorem4.p3.5.5.m5.1a"><mrow id="S5.Thmtheorem4.p3.5.5.m5.1.1" xref="S5.Thmtheorem4.p3.5.5.m5.1.1.cmml"><mi id="S5.Thmtheorem4.p3.5.5.m5.1.1.2" xref="S5.Thmtheorem4.p3.5.5.m5.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem4.p3.5.5.m5.1.1.1" xref="S5.Thmtheorem4.p3.5.5.m5.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p3.5.5.m5.1.1.3" xref="S5.Thmtheorem4.p3.5.5.m5.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.5.5.m5.1b"><apply id="S5.Thmtheorem4.p3.5.5.m5.1.1.cmml" xref="S5.Thmtheorem4.p3.5.5.m5.1.1"><times id="S5.Thmtheorem4.p3.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem4.p3.5.5.m5.1.1.1"></times><ci id="S5.Thmtheorem4.p3.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem4.p3.5.5.m5.1.1.2">ℎ</ci><ci id="S5.Thmtheorem4.p3.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem4.p3.5.5.m5.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.5.5.m5.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.5.5.m5.1d">italic_h italic_v</annotation></semantics></math> be a deviation. We apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem2" title="Lemma 5.2 ‣ 5.2 Punishing Strategies ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.2</span></a> to replace <math alttext="\sigma^{\prime}_{0|hv}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.6.6.m6.1"><semantics id="S5.Thmtheorem4.p3.6.6.m6.1a"><msubsup id="S5.Thmtheorem4.p3.6.6.m6.1.1" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.cmml"><mi id="S5.Thmtheorem4.p3.6.6.m6.1.1.2.2" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.2.2.cmml">σ</mi><mrow id="S5.Thmtheorem4.p3.6.6.m6.1.1.3" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.cmml"><mn id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.2" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.2.cmml">0</mn><mo fence="false" id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.1" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.1.cmml">|</mo><mrow id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.cmml"><mi id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.2" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.2.cmml">h</mi><mo id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.1" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.3" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.3.cmml">v</mi></mrow></mrow><mo id="S5.Thmtheorem4.p3.6.6.m6.1.1.2.3" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.6.6.m6.1b"><apply id="S5.Thmtheorem4.p3.6.6.m6.1.1.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p3.6.6.m6.1.1.1.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1">subscript</csymbol><apply id="S5.Thmtheorem4.p3.6.6.m6.1.1.2.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p3.6.6.m6.1.1.2.1.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1">superscript</csymbol><ci id="S5.Thmtheorem4.p3.6.6.m6.1.1.2.2.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.2.2">𝜎</ci><ci id="S5.Thmtheorem4.p3.6.6.m6.1.1.2.3.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.2.3">′</ci></apply><apply id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3"><csymbol cd="latexml" id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.1.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.1">conditional</csymbol><cn id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.2">0</cn><apply id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3"><times id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.1.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.1"></times><ci id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.2.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.2">ℎ</ci><ci id="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.3.cmml" xref="S5.Thmtheorem4.p3.6.6.m6.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.6.6.m6.1c">\sigma^{\prime}_{0|hv}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.6.6.m6.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 | italic_h italic_v end_POSTSUBSCRIPT</annotation></semantics></math> by the punishing strategy <math alttext="\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.7.7.m7.2"><semantics id="S5.Thmtheorem4.p3.7.7.m7.2a"><msubsup id="S5.Thmtheorem4.p3.7.7.m7.2.3" xref="S5.Thmtheorem4.p3.7.7.m7.2.3.cmml"><mi id="S5.Thmtheorem4.p3.7.7.m7.2.3.2.2" xref="S5.Thmtheorem4.p3.7.7.m7.2.3.2.2.cmml">τ</mi><mrow id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.3.cmml"><mi id="S5.Thmtheorem4.p3.7.7.m7.1.1.1.1" xref="S5.Thmtheorem4.p3.7.7.m7.1.1.1.1.cmml">v</mi><mo id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.2" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.3.cmml">,</mo><mrow id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.3" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.3a.cmml">rec</mtext><mo id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.2" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.cmml"><mo id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.2" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.1" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.3" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.7.7.m7.2.3.2.3" xref="S5.Thmtheorem4.p3.7.7.m7.2.3.2.3a.cmml">Pun</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.7.7.m7.2b"><apply id="S5.Thmtheorem4.p3.7.7.m7.2.3.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p3.7.7.m7.2.3.1.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.3">subscript</csymbol><apply id="S5.Thmtheorem4.p3.7.7.m7.2.3.2.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p3.7.7.m7.2.3.2.1.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.3">superscript</csymbol><ci id="S5.Thmtheorem4.p3.7.7.m7.2.3.2.2.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.3.2.2">𝜏</ci><ci id="S5.Thmtheorem4.p3.7.7.m7.2.3.2.3a.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.3.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.7.7.m7.2.3.2.3.cmml" mathsize="70%" xref="S5.Thmtheorem4.p3.7.7.m7.2.3.2.3">Pun</mtext></ci></apply><list id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.3.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2"><ci id="S5.Thmtheorem4.p3.7.7.m7.1.1.1.1.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.1.1.1.1">𝑣</ci><apply id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1"><times id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.2.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.2"></times><ci id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.3a.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.3.cmml" mathsize="70%" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.3">rec</mtext></ci><apply id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1"><times id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.1"></times><ci id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem4.p3.7.7.m7.2.2.2.2.1.1.1.1.3">𝑣</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.7.7.m7.2c">\tau^{\textsf{Pun}}_{v,\textsf{rec}({hv})}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.7.7.m7.2d">italic_τ start_POSTSUPERSCRIPT Pun end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v , rec ( italic_h italic_v ) end_POSTSUBSCRIPT</annotation></semantics></math>. According to <math alttext="\textsf{val}(h)" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.8.8.m8.1"><semantics id="S5.Thmtheorem4.p3.8.8.m8.1a"><mrow id="S5.Thmtheorem4.p3.8.8.m8.1.2" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.8.8.m8.1.2.2" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem4.p3.8.8.m8.1.2.1" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p3.8.8.m8.1.2.3.2" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.cmml"><mo id="S5.Thmtheorem4.p3.8.8.m8.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.cmml">(</mo><mi id="S5.Thmtheorem4.p3.8.8.m8.1.1" xref="S5.Thmtheorem4.p3.8.8.m8.1.1.cmml">h</mi><mo id="S5.Thmtheorem4.p3.8.8.m8.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.8.8.m8.1b"><apply id="S5.Thmtheorem4.p3.8.8.m8.1.2.cmml" xref="S5.Thmtheorem4.p3.8.8.m8.1.2"><times id="S5.Thmtheorem4.p3.8.8.m8.1.2.1.cmml" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.1"></times><ci id="S5.Thmtheorem4.p3.8.8.m8.1.2.2a.cmml" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.8.8.m8.1.2.2.cmml" xref="S5.Thmtheorem4.p3.8.8.m8.1.2.2">val</mtext></ci><ci id="S5.Thmtheorem4.p3.8.8.m8.1.1.cmml" xref="S5.Thmtheorem4.p3.8.8.m8.1.1">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.8.8.m8.1c">\textsf{val}(h)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.8.8.m8.1d">val ( italic_h )</annotation></semantics></math>, this punishing strategy is either finite-memory with an exponential size or it is memoryless. Notice that given a deviation <math alttext="hv" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.9.9.m9.1"><semantics id="S5.Thmtheorem4.p3.9.9.m9.1a"><mrow id="S5.Thmtheorem4.p3.9.9.m9.1.1" xref="S5.Thmtheorem4.p3.9.9.m9.1.1.cmml"><mi id="S5.Thmtheorem4.p3.9.9.m9.1.1.2" xref="S5.Thmtheorem4.p3.9.9.m9.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem4.p3.9.9.m9.1.1.1" xref="S5.Thmtheorem4.p3.9.9.m9.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p3.9.9.m9.1.1.3" xref="S5.Thmtheorem4.p3.9.9.m9.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.9.9.m9.1b"><apply id="S5.Thmtheorem4.p3.9.9.m9.1.1.cmml" xref="S5.Thmtheorem4.p3.9.9.m9.1.1"><times id="S5.Thmtheorem4.p3.9.9.m9.1.1.1.cmml" xref="S5.Thmtheorem4.p3.9.9.m9.1.1.1"></times><ci id="S5.Thmtheorem4.p3.9.9.m9.1.1.2.cmml" xref="S5.Thmtheorem4.p3.9.9.m9.1.1.2">ℎ</ci><ci id="S5.Thmtheorem4.p3.9.9.m9.1.1.3.cmml" xref="S5.Thmtheorem4.p3.9.9.m9.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.9.9.m9.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.9.9.m9.1d">italic_h italic_v</annotation></semantics></math>, a punishing strategy only depends on the last vertex <math alttext="v" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.10.10.m10.1"><semantics id="S5.Thmtheorem4.p3.10.10.m10.1a"><mi id="S5.Thmtheorem4.p3.10.10.m10.1.1" xref="S5.Thmtheorem4.p3.10.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.10.10.m10.1b"><ci id="S5.Thmtheorem4.p3.10.10.m10.1.1.cmml" xref="S5.Thmtheorem4.p3.10.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.10.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.10.10.m10.1d">italic_v</annotation></semantics></math> of <math alttext="hv" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.11.11.m11.1"><semantics id="S5.Thmtheorem4.p3.11.11.m11.1a"><mrow id="S5.Thmtheorem4.p3.11.11.m11.1.1" xref="S5.Thmtheorem4.p3.11.11.m11.1.1.cmml"><mi id="S5.Thmtheorem4.p3.11.11.m11.1.1.2" xref="S5.Thmtheorem4.p3.11.11.m11.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem4.p3.11.11.m11.1.1.1" xref="S5.Thmtheorem4.p3.11.11.m11.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p3.11.11.m11.1.1.3" xref="S5.Thmtheorem4.p3.11.11.m11.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.11.11.m11.1b"><apply id="S5.Thmtheorem4.p3.11.11.m11.1.1.cmml" xref="S5.Thmtheorem4.p3.11.11.m11.1.1"><times id="S5.Thmtheorem4.p3.11.11.m11.1.1.1.cmml" xref="S5.Thmtheorem4.p3.11.11.m11.1.1.1"></times><ci id="S5.Thmtheorem4.p3.11.11.m11.1.1.2.cmml" xref="S5.Thmtheorem4.p3.11.11.m11.1.1.2">ℎ</ci><ci id="S5.Thmtheorem4.p3.11.11.m11.1.1.3.cmml" xref="S5.Thmtheorem4.p3.11.11.m11.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.11.11.m11.1c">hv</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.11.11.m11.1d">italic_h italic_v</annotation></semantics></math> and its record <math alttext="\textsf{rec}({hv})" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p3.12.12.m12.1"><semantics id="S5.Thmtheorem4.p3.12.12.m12.1a"><mrow id="S5.Thmtheorem4.p3.12.12.m12.1.1" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.12.12.m12.1.1.3" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.3a.cmml">rec</mtext><mo id="S5.Thmtheorem4.p3.12.12.m12.1.1.2" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.2" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.2.cmml">h</mi><mo id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.1" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.3" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p3.12.12.m12.1b"><apply id="S5.Thmtheorem4.p3.12.12.m12.1.1.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1"><times id="S5.Thmtheorem4.p3.12.12.m12.1.1.2.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.2"></times><ci id="S5.Thmtheorem4.p3.12.12.m12.1.1.3a.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p3.12.12.m12.1.1.3.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.3">rec</mtext></ci><apply id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1"><times id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.1"></times><ci id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.2">ℎ</ci><ci id="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem4.p3.12.12.m12.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p3.12.12.m12.1c">\textsf{rec}({hv})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p3.12.12.m12.1d">rec ( italic_h italic_v )</annotation></semantics></math>. Therefore, we have an exponential number of punishing strategies by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S4.Thmtheorem1" title="Theorem 4.1 ‣ 4 Bounding Pareto-Optimal Payoffs ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">4.1</span></a>.</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem4.p4"> <p class="ltx_p" id="S5.Thmtheorem4.p4.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p4.2.2">All in all, when <math alttext="C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p4.1.1.m1.4"><semantics id="S5.Thmtheorem4.p4.1.1.m1.4a"><mrow id="S5.Thmtheorem4.p4.1.1.m1.4.4" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.cmml"><msub id="S5.Thmtheorem4.p4.1.1.m1.4.4.3" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.cmml"><mi id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.2" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.2.cmml">C</mi><msub id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.cmml"><mi id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.2" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.3" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S5.Thmtheorem4.p4.1.1.m1.4.4.2" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.2.cmml">≠</mo><mrow id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.2.cmml"><mo id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.2.cmml">{</mo><mrow id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.2" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.1.cmml"><mo id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.2.1" stretchy="false" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.1.cmml">(</mo><mi id="S5.Thmtheorem4.p4.1.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem4.p4.1.1.m1.1.1.cmml">∞</mi><mo id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.2.2" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem4.p4.1.1.m1.2.2" mathvariant="normal" xref="S5.Thmtheorem4.p4.1.1.m1.2.2.cmml">…</mi><mo id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.2.3" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem4.p4.1.1.m1.3.3" mathvariant="normal" xref="S5.Thmtheorem4.p4.1.1.m1.3.3.cmml">∞</mi><mo id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.2.4" stretchy="false" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.3" stretchy="false" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p4.1.1.m1.4b"><apply id="S5.Thmtheorem4.p4.1.1.m1.4.4.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4"><neq id="S5.Thmtheorem4.p4.1.1.m1.4.4.2.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.2"></neq><apply id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.1.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3">subscript</csymbol><ci id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.2.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.2">𝐶</ci><apply id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.1.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3">subscript</csymbol><ci id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.2.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.2">𝜎</ci><cn id="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.3.3.3">0</cn></apply></apply><set id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.2.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1"><vector id="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.1.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.4.4.1.1.1.2"><infinity id="S5.Thmtheorem4.p4.1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.1.1"></infinity><ci id="S5.Thmtheorem4.p4.1.1.m1.2.2.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.2.2">…</ci><infinity id="S5.Thmtheorem4.p4.1.1.m1.3.3.cmml" xref="S5.Thmtheorem4.p4.1.1.m1.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p4.1.1.m1.4c">C_{\sigma_{0}}\neq\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p4.1.1.m1.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≠ { ( ∞ , … , ∞ ) }</annotation></semantics></math>, we get from <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p4.2.2.m2.1"><semantics id="S5.Thmtheorem4.p4.2.2.m2.1a"><msub id="S5.Thmtheorem4.p4.2.2.m2.1.1" xref="S5.Thmtheorem4.p4.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem4.p4.2.2.m2.1.1.2" xref="S5.Thmtheorem4.p4.2.2.m2.1.1.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p4.2.2.m2.1.1.3" xref="S5.Thmtheorem4.p4.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p4.2.2.m2.1b"><apply id="S5.Thmtheorem4.p4.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p4.2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p4.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem4.p4.2.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p4.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem4.p4.2.2.m2.1.1.2">𝜎</ci><cn id="S5.Thmtheorem4.p4.2.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p4.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p4.2.2.m2.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p4.2.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> a solution to the SPS problem that uses a memory of exponential size.</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem4.p5"> <br class="ltx_break"/> <p class="ltx_p" id="S5.Thmtheorem4.p5.21"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p5.21.21">(2) Suppose that <math alttext="C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.1.1.m1.4"><semantics id="S5.Thmtheorem4.p5.1.1.m1.4a"><mrow id="S5.Thmtheorem4.p5.1.1.m1.4.4" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.cmml"><msub id="S5.Thmtheorem4.p5.1.1.m1.4.4.3" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.cmml"><mi id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.2" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.2.cmml">C</mi><msub id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.cmml"><mi id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.2" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.3" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.3.cmml">0</mn></msub></msub><mo id="S5.Thmtheorem4.p5.1.1.m1.4.4.2" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.2.cmml">=</mo><mrow id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.2.cmml"><mo id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.2.cmml">{</mo><mrow id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.2" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.1.cmml"><mo id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.2.1" stretchy="false" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.1.cmml">(</mo><mi id="S5.Thmtheorem4.p5.1.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem4.p5.1.1.m1.1.1.cmml">∞</mi><mo id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.2.2" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem4.p5.1.1.m1.2.2" mathvariant="normal" xref="S5.Thmtheorem4.p5.1.1.m1.2.2.cmml">…</mi><mo id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.2.3" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.1.cmml">,</mo><mi id="S5.Thmtheorem4.p5.1.1.m1.3.3" mathvariant="normal" xref="S5.Thmtheorem4.p5.1.1.m1.3.3.cmml">∞</mi><mo id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.2.4" stretchy="false" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.3" stretchy="false" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.1.1.m1.4b"><apply id="S5.Thmtheorem4.p5.1.1.m1.4.4.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4"><eq id="S5.Thmtheorem4.p5.1.1.m1.4.4.2.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.2"></eq><apply id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.1.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3">subscript</csymbol><ci id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.2.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.2">𝐶</ci><apply id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.1.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3">subscript</csymbol><ci id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.2.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.2">𝜎</ci><cn id="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.3.3.3">0</cn></apply></apply><set id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.2.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1"><vector id="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.1.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.4.4.1.1.1.2"><infinity id="S5.Thmtheorem4.p5.1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.1.1"></infinity><ci id="S5.Thmtheorem4.p5.1.1.m1.2.2.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.2.2">…</ci><infinity id="S5.Thmtheorem4.p5.1.1.m1.3.3.cmml" xref="S5.Thmtheorem4.p5.1.1.m1.3.3"></infinity></vector></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.1.1.m1.4c">C_{\sigma_{0}}=\{(\infty,\ldots,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.1.1.m1.4d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { ( ∞ , … , ∞ ) }</annotation></semantics></math>. This means that each play <math alttext="\rho\in\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.2.2.m2.1"><semantics id="S5.Thmtheorem4.p5.2.2.m2.1a"><mrow id="S5.Thmtheorem4.p5.2.2.m2.1.1" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem4.p5.2.2.m2.1.1.2" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.2.cmml">ρ</mi><mo id="S5.Thmtheorem4.p5.2.2.m2.1.1.1" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem4.p5.2.2.m2.1.1.3" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.2" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.2a.cmml">Play</mtext><msub id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.cmml"><mi id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.2" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.3" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.2.2.m2.1b"><apply id="S5.Thmtheorem4.p5.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1"><in id="S5.Thmtheorem4.p5.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.1"></in><ci id="S5.Thmtheorem4.p5.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.2">𝜌</ci><apply id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.1.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.2a.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.2.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.2">Play</mtext></ci><apply id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.1.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.2.cmml" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.2">𝜎</ci><cn id="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.2.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.2.2.m2.1c">\rho\in\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.2.2.m2.1d">italic_ρ ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> visits no target of Player <math alttext="1" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.3.3.m3.1"><semantics id="S5.Thmtheorem4.p5.3.3.m3.1a"><mn id="S5.Thmtheorem4.p5.3.3.m3.1.1" xref="S5.Thmtheorem4.p5.3.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.3.3.m3.1b"><cn id="S5.Thmtheorem4.p5.3.3.m3.1.1.cmml" type="integer" xref="S5.Thmtheorem4.p5.3.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.3.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.3.3.m3.1d">1</annotation></semantics></math> and that <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.4.4.m4.1"><semantics id="S5.Thmtheorem4.p5.4.4.m4.1a"><msub id="S5.Thmtheorem4.p5.4.4.m4.1.1" xref="S5.Thmtheorem4.p5.4.4.m4.1.1.cmml"><mi id="S5.Thmtheorem4.p5.4.4.m4.1.1.2" xref="S5.Thmtheorem4.p5.4.4.m4.1.1.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p5.4.4.m4.1.1.3" xref="S5.Thmtheorem4.p5.4.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.4.4.m4.1b"><apply id="S5.Thmtheorem4.p5.4.4.m4.1.1.cmml" xref="S5.Thmtheorem4.p5.4.4.m4.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem4.p5.4.4.m4.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem4.p5.4.4.m4.1.1.2">𝜎</ci><cn id="S5.Thmtheorem4.p5.4.4.m4.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.4.4.m4.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.4.4.m4.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> can impose to each such <math alttext="\rho" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.5.5.m5.1"><semantics id="S5.Thmtheorem4.p5.5.5.m5.1a"><mi id="S5.Thmtheorem4.p5.5.5.m5.1.1" xref="S5.Thmtheorem4.p5.5.5.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.5.5.m5.1b"><ci id="S5.Thmtheorem4.p5.5.5.m5.1.1.cmml" xref="S5.Thmtheorem4.p5.5.5.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.5.5.m5.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.5.5.m5.1d">italic_ρ</annotation></semantics></math> to have a value <math alttext="\textsf{val}(\rho)\leq B" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.6.6.m6.1"><semantics id="S5.Thmtheorem4.p5.6.6.m6.1a"><mrow id="S5.Thmtheorem4.p5.6.6.m6.1.2" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.cmml"><mrow id="S5.Thmtheorem4.p5.6.6.m6.1.2.2" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.2" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.2a.cmml">val</mtext><mo id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.1" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.3.2" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.cmml"><mo id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem4.p5.6.6.m6.1.1" xref="S5.Thmtheorem4.p5.6.6.m6.1.1.cmml">ρ</mi><mo id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem4.p5.6.6.m6.1.2.1" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.1.cmml">≤</mo><mi id="S5.Thmtheorem4.p5.6.6.m6.1.2.3" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.6.6.m6.1b"><apply id="S5.Thmtheorem4.p5.6.6.m6.1.2.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.2"><leq id="S5.Thmtheorem4.p5.6.6.m6.1.2.1.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.1"></leq><apply id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2"><times id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.1.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.1"></times><ci id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.2a.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p5.6.6.m6.1.2.2.2.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.2.2">val</mtext></ci><ci id="S5.Thmtheorem4.p5.6.6.m6.1.1.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.1">𝜌</ci></apply><ci id="S5.Thmtheorem4.p5.6.6.m6.1.2.3.cmml" xref="S5.Thmtheorem4.p5.6.6.m6.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.6.6.m6.1c">\textsf{val}(\rho)\leq B</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.6.6.m6.1d">val ( italic_ρ ) ≤ italic_B</annotation></semantics></math>. As done in the proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem2" title="Lemma 5.2 ‣ 5.2 Punishing Strategies ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.2</span></a>, we can replace <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.7.7.m7.1"><semantics id="S5.Thmtheorem4.p5.7.7.m7.1a"><msub id="S5.Thmtheorem4.p5.7.7.m7.1.1" xref="S5.Thmtheorem4.p5.7.7.m7.1.1.cmml"><mi id="S5.Thmtheorem4.p5.7.7.m7.1.1.2" xref="S5.Thmtheorem4.p5.7.7.m7.1.1.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p5.7.7.m7.1.1.3" xref="S5.Thmtheorem4.p5.7.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.7.7.m7.1b"><apply id="S5.Thmtheorem4.p5.7.7.m7.1.1.cmml" xref="S5.Thmtheorem4.p5.7.7.m7.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.7.7.m7.1.1.1.cmml" xref="S5.Thmtheorem4.p5.7.7.m7.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.7.7.m7.1.1.2.cmml" xref="S5.Thmtheorem4.p5.7.7.m7.1.1.2">𝜎</ci><cn id="S5.Thmtheorem4.p5.7.7.m7.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.7.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.7.7.m7.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.7.7.m7.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> by a strategy corresponding to a winning strategy in a zero-sum game <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.8.8.m8.1"><semantics id="S5.Thmtheorem4.p5.8.8.m8.1a"><mi id="S5.Thmtheorem4.p5.8.8.m8.1.1" xref="S5.Thmtheorem4.p5.8.8.m8.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.8.8.m8.1b"><ci id="S5.Thmtheorem4.p5.8.8.m8.1.1.cmml" xref="S5.Thmtheorem4.p5.8.8.m8.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.8.8.m8.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.8.8.m8.1d">italic_H</annotation></semantics></math> with an exponential arena and a reachability objective. The arena of <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.9.9.m9.1"><semantics id="S5.Thmtheorem4.p5.9.9.m9.1a"><mi id="S5.Thmtheorem4.p5.9.9.m9.1.1" xref="S5.Thmtheorem4.p5.9.9.m9.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.9.9.m9.1b"><ci id="S5.Thmtheorem4.p5.9.9.m9.1.1.cmml" xref="S5.Thmtheorem4.p5.9.9.m9.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.9.9.m9.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.9.9.m9.1d">italic_H</annotation></semantics></math> has vertices of the form <math alttext="(v,(m_{1},m_{2}))" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.10.10.m10.2"><semantics id="S5.Thmtheorem4.p5.10.10.m10.2a"><mrow id="S5.Thmtheorem4.p5.10.10.m10.2.2.1" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.2.cmml"><mo id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.2" stretchy="false" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.2.cmml">(</mo><mi id="S5.Thmtheorem4.p5.10.10.m10.1.1" xref="S5.Thmtheorem4.p5.10.10.m10.1.1.cmml">v</mi><mo id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.3" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.2.cmml">,</mo><mrow id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.3.cmml"><mo id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.3" stretchy="false" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.3.cmml">(</mo><msub id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.2" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.3" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.4" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.3.cmml">,</mo><msub id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.cmml"><mi id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.2" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.3" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.3.cmml">2</mn></msub><mo id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.5" stretchy="false" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.3.cmml">)</mo></mrow><mo id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.4" stretchy="false" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.10.10.m10.2b"><interval closure="open" id="S5.Thmtheorem4.p5.10.10.m10.2.2.2.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1"><ci id="S5.Thmtheorem4.p5.10.10.m10.1.1.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.1.1">𝑣</ci><interval closure="open" id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.3.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2"><apply id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.1.1.3">1</cn></apply><apply id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.1.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.2.cmml" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.10.10.m10.2.2.1.1.2.2.3">2</cn></apply></interval></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.10.10.m10.2c">(v,(m_{1},m_{2}))</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.10.10.m10.2d">( italic_v , ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) )</annotation></semantics></math> with <math alttext="v\in V" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.11.11.m11.1"><semantics id="S5.Thmtheorem4.p5.11.11.m11.1a"><mrow id="S5.Thmtheorem4.p5.11.11.m11.1.1" xref="S5.Thmtheorem4.p5.11.11.m11.1.1.cmml"><mi id="S5.Thmtheorem4.p5.11.11.m11.1.1.2" xref="S5.Thmtheorem4.p5.11.11.m11.1.1.2.cmml">v</mi><mo id="S5.Thmtheorem4.p5.11.11.m11.1.1.1" xref="S5.Thmtheorem4.p5.11.11.m11.1.1.1.cmml">∈</mo><mi id="S5.Thmtheorem4.p5.11.11.m11.1.1.3" xref="S5.Thmtheorem4.p5.11.11.m11.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.11.11.m11.1b"><apply id="S5.Thmtheorem4.p5.11.11.m11.1.1.cmml" xref="S5.Thmtheorem4.p5.11.11.m11.1.1"><in id="S5.Thmtheorem4.p5.11.11.m11.1.1.1.cmml" xref="S5.Thmtheorem4.p5.11.11.m11.1.1.1"></in><ci id="S5.Thmtheorem4.p5.11.11.m11.1.1.2.cmml" xref="S5.Thmtheorem4.p5.11.11.m11.1.1.2">𝑣</ci><ci id="S5.Thmtheorem4.p5.11.11.m11.1.1.3.cmml" xref="S5.Thmtheorem4.p5.11.11.m11.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.11.11.m11.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.11.11.m11.1d">italic_v ∈ italic_V</annotation></semantics></math>, <math alttext="m_{1},m_{2}\in\{0,\ldots,B\}\cup\{\infty\}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.12.12.m12.6"><semantics id="S5.Thmtheorem4.p5.12.12.m12.6a"><mrow id="S5.Thmtheorem4.p5.12.12.m12.6.6" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.cmml"><mrow id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.3.cmml"><msub id="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1" xref="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.cmml"><mi id="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.2" xref="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.3" xref="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.3" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.3.cmml">,</mo><msub id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.cmml"><mi id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.2" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.3" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.3" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.3.cmml">∈</mo><mrow id="S5.Thmtheorem4.p5.12.12.m12.6.6.4" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.cmml"><mrow id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.2" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.1.cmml"><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.2.1" stretchy="false" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.1.cmml">{</mo><mn id="S5.Thmtheorem4.p5.12.12.m12.1.1" xref="S5.Thmtheorem4.p5.12.12.m12.1.1.cmml">0</mn><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.2.2" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.1.cmml">,</mo><mi id="S5.Thmtheorem4.p5.12.12.m12.2.2" mathvariant="normal" xref="S5.Thmtheorem4.p5.12.12.m12.2.2.cmml">…</mi><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.2.3" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.1.cmml">,</mo><mi id="S5.Thmtheorem4.p5.12.12.m12.3.3" xref="S5.Thmtheorem4.p5.12.12.m12.3.3.cmml">B</mi><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.2.4" stretchy="false" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.1.cmml">}</mo></mrow><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.1" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.1.cmml">∪</mo><mrow id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.2" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.1.cmml"><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.1.cmml">{</mo><mi id="S5.Thmtheorem4.p5.12.12.m12.4.4" mathvariant="normal" xref="S5.Thmtheorem4.p5.12.12.m12.4.4.cmml">∞</mi><mo id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.2.2" stretchy="false" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.12.12.m12.6b"><apply id="S5.Thmtheorem4.p5.12.12.m12.6.6.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6"><in id="S5.Thmtheorem4.p5.12.12.m12.6.6.3.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.3"></in><list id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.3.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2"><apply id="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.1.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.2.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.12.12.m12.5.5.1.1.1.3">1</cn></apply><apply id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.1.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.2.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.2.2.2.3">2</cn></apply></list><apply id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4"><union id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.1.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.1"></union><set id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.1.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.2.2"><cn id="S5.Thmtheorem4.p5.12.12.m12.1.1.cmml" type="integer" xref="S5.Thmtheorem4.p5.12.12.m12.1.1">0</cn><ci id="S5.Thmtheorem4.p5.12.12.m12.2.2.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.2.2">…</ci><ci id="S5.Thmtheorem4.p5.12.12.m12.3.3.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.3.3">𝐵</ci></set><set id="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.1.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.6.6.4.3.2"><infinity id="S5.Thmtheorem4.p5.12.12.m12.4.4.cmml" xref="S5.Thmtheorem4.p5.12.12.m12.4.4"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.12.12.m12.6c">m_{1},m_{2}\in\{0,\ldots,B\}\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.12.12.m12.6d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ { 0 , … , italic_B } ∪ { ∞ }</annotation></semantics></math>, such that, whenever <math alttext="v" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.13.13.m13.1"><semantics id="S5.Thmtheorem4.p5.13.13.m13.1a"><mi id="S5.Thmtheorem4.p5.13.13.m13.1.1" xref="S5.Thmtheorem4.p5.13.13.m13.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.13.13.m13.1b"><ci id="S5.Thmtheorem4.p5.13.13.m13.1.1.cmml" xref="S5.Thmtheorem4.p5.13.13.m13.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.13.13.m13.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.13.13.m13.1d">italic_v</annotation></semantics></math> belongs to Player <math alttext="0" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.14.14.m14.1"><semantics id="S5.Thmtheorem4.p5.14.14.m14.1a"><mn id="S5.Thmtheorem4.p5.14.14.m14.1.1" xref="S5.Thmtheorem4.p5.14.14.m14.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.14.14.m14.1b"><cn id="S5.Thmtheorem4.p5.14.14.m14.1.1.cmml" type="integer" xref="S5.Thmtheorem4.p5.14.14.m14.1.1">0</cn></annotation-xml></semantics></math>’s target, the weight component <math alttext="m_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.15.15.m15.1"><semantics id="S5.Thmtheorem4.p5.15.15.m15.1a"><msub id="S5.Thmtheorem4.p5.15.15.m15.1.1" xref="S5.Thmtheorem4.p5.15.15.m15.1.1.cmml"><mi id="S5.Thmtheorem4.p5.15.15.m15.1.1.2" xref="S5.Thmtheorem4.p5.15.15.m15.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.15.15.m15.1.1.3" xref="S5.Thmtheorem4.p5.15.15.m15.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.15.15.m15.1b"><apply id="S5.Thmtheorem4.p5.15.15.m15.1.1.cmml" xref="S5.Thmtheorem4.p5.15.15.m15.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.15.15.m15.1.1.1.cmml" xref="S5.Thmtheorem4.p5.15.15.m15.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.15.15.m15.1.1.2.cmml" xref="S5.Thmtheorem4.p5.15.15.m15.1.1.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.15.15.m15.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.15.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.15.15.m15.1c">m_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.15.15.m15.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> allows to update the <span class="ltx_text ltx_font_sansserif" id="S5.Thmtheorem4.p5.21.21.1">val</span> component <math alttext="m_{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.16.16.m16.1"><semantics id="S5.Thmtheorem4.p5.16.16.m16.1a"><msub id="S5.Thmtheorem4.p5.16.16.m16.1.1" xref="S5.Thmtheorem4.p5.16.16.m16.1.1.cmml"><mi id="S5.Thmtheorem4.p5.16.16.m16.1.1.2" xref="S5.Thmtheorem4.p5.16.16.m16.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.16.16.m16.1.1.3" xref="S5.Thmtheorem4.p5.16.16.m16.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.16.16.m16.1b"><apply id="S5.Thmtheorem4.p5.16.16.m16.1.1.cmml" xref="S5.Thmtheorem4.p5.16.16.m16.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.16.16.m16.1.1.1.cmml" xref="S5.Thmtheorem4.p5.16.16.m16.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.16.16.m16.1.1.2.cmml" xref="S5.Thmtheorem4.p5.16.16.m16.1.1.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.16.16.m16.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.16.16.m16.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.16.16.m16.1c">m_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.16.16.m16.1d">italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. The initial vertex of <math alttext="H" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.17.17.m17.1"><semantics id="S5.Thmtheorem4.p5.17.17.m17.1a"><mi id="S5.Thmtheorem4.p5.17.17.m17.1.1" xref="S5.Thmtheorem4.p5.17.17.m17.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.17.17.m17.1b"><ci id="S5.Thmtheorem4.p5.17.17.m17.1.1.cmml" xref="S5.Thmtheorem4.p5.17.17.m17.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.17.17.m17.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.17.17.m17.1d">italic_H</annotation></semantics></math> is equal to <math alttext="(v_{0},0,\textsf{val}(v_{0}))" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.18.18.m18.3"><semantics id="S5.Thmtheorem4.p5.18.18.m18.3a"><mrow id="S5.Thmtheorem4.p5.18.18.m18.3.3.2" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.3.cmml"><mo id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.3" stretchy="false" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.3.cmml">(</mo><msub id="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1" xref="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.cmml"><mi id="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.2" xref="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.2.cmml">v</mi><mn id="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.3" xref="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.3.cmml">0</mn></msub><mo id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.4" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.3.cmml">,</mo><mn id="S5.Thmtheorem4.p5.18.18.m18.1.1" xref="S5.Thmtheorem4.p5.18.18.m18.1.1.cmml">0</mn><mo id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.5" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.3.cmml">,</mo><mrow id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.3" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.3a.cmml">val</mtext><mo id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.2" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.2.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.cmml"><mo id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.cmml">(</mo><msub id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.2" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.2.cmml">v</mi><mn id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.3" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.3" stretchy="false" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.6" stretchy="false" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.18.18.m18.3b"><vector id="S5.Thmtheorem4.p5.18.18.m18.3.3.3.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2"><apply id="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.1.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.2.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.2">𝑣</ci><cn id="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.18.18.m18.2.2.1.1.3">0</cn></apply><cn id="S5.Thmtheorem4.p5.18.18.m18.1.1.cmml" type="integer" xref="S5.Thmtheorem4.p5.18.18.m18.1.1">0</cn><apply id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2"><times id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.2.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.2"></times><ci id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.3a.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.3.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.3">val</mtext></ci><apply id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.2">𝑣</ci><cn id="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.18.18.m18.3.3.2.2.1.1.1.3">0</cn></apply></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.18.18.m18.3c">(v_{0},0,\textsf{val}(v_{0}))</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.18.18.m18.3d">( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 0 , val ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) )</annotation></semantics></math>. This arena has an exponential size and its reachability objective is <math alttext="\{(v,(m_{1},m_{2}))\mid m_{2}\leq B\}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.19.19.m19.3"><semantics id="S5.Thmtheorem4.p5.19.19.m19.3a"><mrow id="S5.Thmtheorem4.p5.19.19.m19.3.3.2" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.3.cmml"><mo id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.3" stretchy="false" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.3.1.cmml">{</mo><mrow id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.2.cmml"><mo id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.2.cmml">(</mo><mi id="S5.Thmtheorem4.p5.19.19.m19.1.1" xref="S5.Thmtheorem4.p5.19.19.m19.1.1.cmml">v</mi><mo id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.3" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.2.cmml">,</mo><mrow id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.3.cmml"><mo id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.3" stretchy="false" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.3.cmml">(</mo><msub id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.2" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.3" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.4" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.3.cmml">,</mo><msub id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.cmml"><mi id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.2" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.3" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.3.cmml">2</mn></msub><mo id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.5" stretchy="false" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.3.cmml">)</mo></mrow><mo id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.4" stretchy="false" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.2.cmml">)</mo></mrow><mo fence="true" id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.4" lspace="0em" rspace="0em" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.3.1.cmml">∣</mo><mrow id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.cmml"><msub id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.cmml"><mi id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.2" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.2.cmml">m</mi><mn id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.3" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.1" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.1.cmml">≤</mo><mi id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.3" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.3.cmml">B</mi></mrow><mo id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.5" stretchy="false" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.19.19.m19.3b"><apply id="S5.Thmtheorem4.p5.19.19.m19.3.3.3.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2"><csymbol cd="latexml" id="S5.Thmtheorem4.p5.19.19.m19.3.3.3.1.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.3">conditional-set</csymbol><interval closure="open" id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.2.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1"><ci id="S5.Thmtheorem4.p5.19.19.m19.1.1.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.1.1">𝑣</ci><interval closure="open" id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2"><apply id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.1.1.3">1</cn></apply><apply id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.1.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.2.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.19.19.m19.2.2.1.1.1.1.2.2.3">2</cn></apply></interval></interval><apply id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2"><leq id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.1.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.1"></leq><apply id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.1.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.2.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.2">𝑚</ci><cn id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.2.3">2</cn></apply><ci id="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.3.cmml" xref="S5.Thmtheorem4.p5.19.19.m19.3.3.2.2.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.19.19.m19.3c">\{(v,(m_{1},m_{2}))\mid m_{2}\leq B\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.19.19.m19.3d">{ ( italic_v , ( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ∣ italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ italic_B }</annotation></semantics></math>. It is known that when there is a winning strategy in a zero-sum game with a reachability objective, then there is one that is memoryless <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib25" title="">25</a>]</cite>. Hence, coming back to <math alttext="G" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.20.20.m20.1"><semantics id="S5.Thmtheorem4.p5.20.20.m20.1a"><mi id="S5.Thmtheorem4.p5.20.20.m20.1.1" xref="S5.Thmtheorem4.p5.20.20.m20.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.20.20.m20.1b"><ci id="S5.Thmtheorem4.p5.20.20.m20.1.1.cmml" xref="S5.Thmtheorem4.p5.20.20.m20.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.20.20.m20.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.20.20.m20.1d">italic_G</annotation></semantics></math>, <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p5.21.21.m21.1"><semantics id="S5.Thmtheorem4.p5.21.21.m21.1a"><msub id="S5.Thmtheorem4.p5.21.21.m21.1.1" xref="S5.Thmtheorem4.p5.21.21.m21.1.1.cmml"><mi id="S5.Thmtheorem4.p5.21.21.m21.1.1.2" xref="S5.Thmtheorem4.p5.21.21.m21.1.1.2.cmml">σ</mi><mn id="S5.Thmtheorem4.p5.21.21.m21.1.1.3" xref="S5.Thmtheorem4.p5.21.21.m21.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p5.21.21.m21.1b"><apply id="S5.Thmtheorem4.p5.21.21.m21.1.1.cmml" xref="S5.Thmtheorem4.p5.21.21.m21.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p5.21.21.m21.1.1.1.cmml" xref="S5.Thmtheorem4.p5.21.21.m21.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p5.21.21.m21.1.1.2.cmml" xref="S5.Thmtheorem4.p5.21.21.m21.1.1.2">𝜎</ci><cn id="S5.Thmtheorem4.p5.21.21.m21.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p5.21.21.m21.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p5.21.21.m21.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p5.21.21.m21.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> can be replaced by a finite-memory strategy with exponential memory.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S5.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.4 </span><span class="ltx_text ltx_font_sansserif" id="S5.SS4.1.1">NEXPTIME</span>-Completeness</h3> <div class="ltx_para" id="S5.SS4.p1"> <p class="ltx_p" id="S5.SS4.p1.1">We finally prove that the SPS problem is <span class="ltx_text ltx_font_sansserif" id="S5.SS4.p1.1.1">NEXPTIME</span>-complete. For the <span class="ltx_text ltx_font_sansserif" id="S5.SS4.p1.1.2">NEXPTIME</span>-membership, the idea is to guess a strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.SS4.p1.1.m1.1"><semantics id="S5.SS4.p1.1.m1.1a"><msub id="S5.SS4.p1.1.m1.1.1" xref="S5.SS4.p1.1.m1.1.1.cmml"><mi id="S5.SS4.p1.1.m1.1.1.2" xref="S5.SS4.p1.1.m1.1.1.2.cmml">σ</mi><mn id="S5.SS4.p1.1.m1.1.1.3" xref="S5.SS4.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.1.m1.1b"><apply id="S5.SS4.p1.1.m1.1.1.cmml" xref="S5.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p1.1.m1.1.1.1.cmml" xref="S5.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS4.p1.1.m1.1.1.2.cmml" xref="S5.SS4.p1.1.m1.1.1.2">𝜎</ci><cn id="S5.SS4.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.SS4.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> given by a Mealy machine of exponential size (by Proposition <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem1" title="Proposition 5.1 ‣ 5.1 Finite-Memory Solutions ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.1</span></a>) and then to verify in exponential time that it is a solution. The <span class="ltx_text ltx_font_sansserif" id="S5.SS4.p1.1.3">NEXPTIME</span>-hardness follows from the <span class="ltx_text ltx_font_sansserif" id="S5.SS4.p1.1.4">NEXPTIME</span>-completeness of the Boolean variant of the SPS problem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S5.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.1.1.1">Proof 5.5</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.2.2"> </span>(Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem1" title="Theorem 2.1 ‣ Stackelberg-Pareto synthesis problem. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.1</span></a>)</h6> <div class="ltx_para" id="S5.Thmtheorem5.p1"> <p class="ltx_p" id="S5.Thmtheorem5.p1.7"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem5.p1.7.7">Let us first study the <span class="ltx_text ltx_font_sansserif" id="S5.Thmtheorem5.p1.7.7.1">NEXPTIME</span>-membership. Let <math alttext="G" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.1.1.m1.1"><semantics id="S5.Thmtheorem5.p1.1.1.m1.1a"><mi id="S5.Thmtheorem5.p1.1.1.m1.1.1" xref="S5.Thmtheorem5.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.1.1.m1.1b"><ci id="S5.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem5.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be an SP game and <math alttext="B\in{\rm Nature}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.2.2.m2.1"><semantics id="S5.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem5.p1.2.2.m2.1.1" xref="S5.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">B</mi><mo id="S5.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S5.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem5.p1.2.2.m2.1.1.3.cmml">Nature</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.2.2.m2.1b"><apply id="S5.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.1.1"><in id="S5.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.1.1.1"></in><ci id="S5.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.1.1.2">𝐵</ci><ci id="S5.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.1.1.3">Nature</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.2.2.m2.1c">B\in{\rm Nature}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.2.2.m2.1d">italic_B ∈ roman_Nature</annotation></semantics></math>. If there exists a solution, Proposition <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem1" title="Proposition 5.1 ‣ 5.1 Finite-Memory Solutions ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.1</span></a> states the existence of a solution <math alttext="\sigma_{0}\in\mbox{SPS}({G},{B})" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.3.3.m3.2"><semantics id="S5.Thmtheorem5.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem5.p1.3.3.m3.2.3" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.cmml"><msub id="S5.Thmtheorem5.p1.3.3.m3.2.3.2" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.2.cmml"><mi id="S5.Thmtheorem5.p1.3.3.m3.2.3.2.2" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.2.2.cmml">σ</mi><mn id="S5.Thmtheorem5.p1.3.3.m3.2.3.2.3" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.2.3.cmml">0</mn></msub><mo id="S5.Thmtheorem5.p1.3.3.m3.2.3.1" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem5.p1.3.3.m3.2.3.3" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.2" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.2a.cmml">SPS</mtext><mo id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.1" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.2" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.1.cmml"><mo id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem5.p1.3.3.m3.1.1" xref="S5.Thmtheorem5.p1.3.3.m3.1.1.cmml">G</mi><mo id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.2.2" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem5.p1.3.3.m3.2.2" xref="S5.Thmtheorem5.p1.3.3.m3.2.2.cmml">B</mi><mo id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.2.3" stretchy="false" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.3.3.m3.2b"><apply id="S5.Thmtheorem5.p1.3.3.m3.2.3.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3"><in id="S5.Thmtheorem5.p1.3.3.m3.2.3.1.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.1"></in><apply id="S5.Thmtheorem5.p1.3.3.m3.2.3.2.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.3.3.m3.2.3.2.1.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem5.p1.3.3.m3.2.3.2.2.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.2.2">𝜎</ci><cn id="S5.Thmtheorem5.p1.3.3.m3.2.3.2.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.2.3">0</cn></apply><apply id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3"><times id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.1.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.1"></times><ci id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.2a.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.2.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.2">SPS</mtext></ci><interval closure="open" id="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.1.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.3.3.3.2"><ci id="S5.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.1.1">𝐺</ci><ci id="S5.Thmtheorem5.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem5.p1.3.3.m3.2.2">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.3.3.m3.2c">\sigma_{0}\in\mbox{SPS}({G},{B})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.3.3.m3.2d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ SPS ( italic_G , italic_B )</annotation></semantics></math> that uses a finite memory bounded exponentially. We can guess such a strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.4.4.m4.1"><semantics id="S5.Thmtheorem5.p1.4.4.m4.1a"><msub id="S5.Thmtheorem5.p1.4.4.m4.1.1" xref="S5.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mi id="S5.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S5.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">σ</mi><mn id="S5.Thmtheorem5.p1.4.4.m4.1.1.3" xref="S5.Thmtheorem5.p1.4.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.4.4.m4.1b"><apply id="S5.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem5.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem5.p1.4.4.m4.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem5.p1.4.4.m4.1.1.2">𝜎</ci><cn id="S5.Thmtheorem5.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.4.4.m4.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.4.4.m4.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> as a Mealy machine <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.5.5.m5.1"><semantics id="S5.Thmtheorem5.p1.5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem5.p1.5.5.m5.1.1" xref="S5.Thmtheorem5.p1.5.5.m5.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.5.5.m5.1b"><ci id="S5.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem5.p1.5.5.m5.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.5.5.m5.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.5.5.m5.1d">caligraphic_M</annotation></semantics></math> with a set <math alttext="M" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.6.6.m6.1"><semantics id="S5.Thmtheorem5.p1.6.6.m6.1a"><mi id="S5.Thmtheorem5.p1.6.6.m6.1.1" xref="S5.Thmtheorem5.p1.6.6.m6.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.6.6.m6.1b"><ci id="S5.Thmtheorem5.p1.6.6.m6.1.1.cmml" xref="S5.Thmtheorem5.p1.6.6.m6.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.6.6.m6.1c">M</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.6.6.m6.1d">italic_M</annotation></semantics></math> of memory states at most exponential in the size of the instance. This can be done in exponential time. Let us explain how to verify in exponential time that the guessed strategy <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.7.7.m7.1"><semantics id="S5.Thmtheorem5.p1.7.7.m7.1a"><msub id="S5.Thmtheorem5.p1.7.7.m7.1.1" xref="S5.Thmtheorem5.p1.7.7.m7.1.1.cmml"><mi id="S5.Thmtheorem5.p1.7.7.m7.1.1.2" xref="S5.Thmtheorem5.p1.7.7.m7.1.1.2.cmml">σ</mi><mn id="S5.Thmtheorem5.p1.7.7.m7.1.1.3" xref="S5.Thmtheorem5.p1.7.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.7.7.m7.1b"><apply id="S5.Thmtheorem5.p1.7.7.m7.1.1.cmml" xref="S5.Thmtheorem5.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.7.7.m7.1.1.1.cmml" xref="S5.Thmtheorem5.p1.7.7.m7.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p1.7.7.m7.1.1.2.cmml" xref="S5.Thmtheorem5.p1.7.7.m7.1.1.2">𝜎</ci><cn id="S5.Thmtheorem5.p1.7.7.m7.1.1.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.7.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.7.7.m7.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.7.7.m7.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution to the SPS problem.</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem5.p2"> <p class="ltx_p" id="S5.Thmtheorem5.p2.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem5.p2.3.3">We make the following <em class="ltx_emph ltx_font_upright" id="S5.Thmtheorem5.p2.3.3.1">important observation</em>: it is enough to consider Pareto-optimal costs <math alttext="c" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p2.1.1.m1.1"><semantics id="S5.Thmtheorem5.p2.1.1.m1.1a"><mi id="S5.Thmtheorem5.p2.1.1.m1.1.1" xref="S5.Thmtheorem5.p2.1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p2.1.1.m1.1b"><ci id="S5.Thmtheorem5.p2.1.1.m1.1.1.cmml" xref="S5.Thmtheorem5.p2.1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p2.1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p2.1.1.m1.1d">italic_c</annotation></semantics></math> whose components <math alttext="c_{i}<\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p2.2.2.m2.1"><semantics id="S5.Thmtheorem5.p2.2.2.m2.1a"><mrow id="S5.Thmtheorem5.p2.2.2.m2.1.1" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.cmml"><msub id="S5.Thmtheorem5.p2.2.2.m2.1.1.2" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.2.cmml"><mi id="S5.Thmtheorem5.p2.2.2.m2.1.1.2.2" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.2.2.cmml">c</mi><mi id="S5.Thmtheorem5.p2.2.2.m2.1.1.2.3" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.Thmtheorem5.p2.2.2.m2.1.1.1" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.1.cmml"><</mo><mi id="S5.Thmtheorem5.p2.2.2.m2.1.1.3" mathvariant="normal" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p2.2.2.m2.1b"><apply id="S5.Thmtheorem5.p2.2.2.m2.1.1.cmml" xref="S5.Thmtheorem5.p2.2.2.m2.1.1"><lt id="S5.Thmtheorem5.p2.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.1"></lt><apply id="S5.Thmtheorem5.p2.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p2.2.2.m2.1.1.2.1.cmml" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.2">subscript</csymbol><ci id="S5.Thmtheorem5.p2.2.2.m2.1.1.2.2.cmml" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.2.2">𝑐</ci><ci id="S5.Thmtheorem5.p2.2.2.m2.1.1.2.3.cmml" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.2.3">𝑖</ci></apply><infinity id="S5.Thmtheorem5.p2.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem5.p2.2.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p2.2.2.m2.1c">c_{i}<\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p2.2.2.m2.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < ∞</annotation></semantics></math> belong to <math alttext="\{0,\ldots,|V|\cdot|M|\cdot t\cdot W\}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p2.3.3.m3.5"><semantics id="S5.Thmtheorem5.p2.3.3.m3.5a"><mrow id="S5.Thmtheorem5.p2.3.3.m3.5.5.1" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.2.cmml"><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.2" stretchy="false" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.2.cmml">{</mo><mn id="S5.Thmtheorem5.p2.3.3.m3.1.1" xref="S5.Thmtheorem5.p2.3.3.m3.1.1.cmml">0</mn><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.3" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.2.cmml">,</mo><mi id="S5.Thmtheorem5.p2.3.3.m3.2.2" mathvariant="normal" xref="S5.Thmtheorem5.p2.3.3.m3.2.2.cmml">…</mi><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.4" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.2.cmml">,</mo><mrow id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.cmml"><mrow id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.2" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.1.cmml"><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.2.1" stretchy="false" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.1.1.cmml">|</mo><mi id="S5.Thmtheorem5.p2.3.3.m3.3.3" xref="S5.Thmtheorem5.p2.3.3.m3.3.3.cmml">V</mi><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.2.2" rspace="0.055em" stretchy="false" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.1.1.cmml">|</mo></mrow><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1" rspace="0.222em" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1.cmml">⋅</mo><mrow id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.2" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.1.cmml"><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.2.1" stretchy="false" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.1.1.cmml">|</mo><mi id="S5.Thmtheorem5.p2.3.3.m3.4.4" xref="S5.Thmtheorem5.p2.3.3.m3.4.4.cmml">M</mi><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.2.2" rspace="0.055em" stretchy="false" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.1.1.cmml">|</mo></mrow><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1a" rspace="0.222em" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1.cmml">⋅</mo><mi id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.4" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.4.cmml">t</mi><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1b" lspace="0.222em" rspace="0.222em" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1.cmml">⋅</mo><mi id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.5" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.5.cmml">W</mi></mrow><mo id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.5" stretchy="false" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p2.3.3.m3.5b"><set id="S5.Thmtheorem5.p2.3.3.m3.5.5.2.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1"><cn id="S5.Thmtheorem5.p2.3.3.m3.1.1.cmml" type="integer" xref="S5.Thmtheorem5.p2.3.3.m3.1.1">0</cn><ci id="S5.Thmtheorem5.p2.3.3.m3.2.2.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.2.2">…</ci><apply id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1"><ci id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.1">⋅</ci><apply id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.1.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.2"><abs id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.1.1.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.2.2.1"></abs><ci id="S5.Thmtheorem5.p2.3.3.m3.3.3.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.3.3">𝑉</ci></apply><apply id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.1.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.2"><abs id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.1.1.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.3.2.1"></abs><ci id="S5.Thmtheorem5.p2.3.3.m3.4.4.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.4.4">𝑀</ci></apply><ci id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.4.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.4">𝑡</ci><ci id="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.5.cmml" xref="S5.Thmtheorem5.p2.3.3.m3.5.5.1.1.5">𝑊</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p2.3.3.m3.5c">\{0,\ldots,|V|\cdot|M|\cdot t\cdot W\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p2.3.3.m3.5d">{ 0 , … , | italic_V | ⋅ | italic_M | ⋅ italic_t ⋅ italic_W }</annotation></semantics></math>. Let us explain why:</span></p> <ul class="ltx_itemize" id="S5.I5"> <li class="ltx_item" id="S5.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I5.i1.p1"> <p class="ltx_p" id="S5.I5.i1.p1.7"><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.1">Consider the cartesian product </span><math alttext="G\times\mathcal{M}" class="ltx_Math" display="inline" id="S5.I5.i1.p1.1.m1.1"><semantics id="S5.I5.i1.p1.1.m1.1a"><mrow id="S5.I5.i1.p1.1.m1.1.1" xref="S5.I5.i1.p1.1.m1.1.1.cmml"><mi id="S5.I5.i1.p1.1.m1.1.1.2" xref="S5.I5.i1.p1.1.m1.1.1.2.cmml">G</mi><mo id="S5.I5.i1.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.I5.i1.p1.1.m1.1.1.1.cmml">×</mo><mi class="ltx_font_mathcaligraphic" id="S5.I5.i1.p1.1.m1.1.1.3" xref="S5.I5.i1.p1.1.m1.1.1.3.cmml">ℳ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.1.m1.1b"><apply id="S5.I5.i1.p1.1.m1.1.1.cmml" xref="S5.I5.i1.p1.1.m1.1.1"><times id="S5.I5.i1.p1.1.m1.1.1.1.cmml" xref="S5.I5.i1.p1.1.m1.1.1.1"></times><ci id="S5.I5.i1.p1.1.m1.1.1.2.cmml" xref="S5.I5.i1.p1.1.m1.1.1.2">𝐺</ci><ci id="S5.I5.i1.p1.1.m1.1.1.3.cmml" xref="S5.I5.i1.p1.1.m1.1.1.3">ℳ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.1.m1.1c">G\times\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.1.m1.1d">italic_G × caligraphic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.2"> whose infinite paths are exactly the plays consistent with </span><math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.I5.i1.p1.2.m2.1"><semantics id="S5.I5.i1.p1.2.m2.1a"><msub id="S5.I5.i1.p1.2.m2.1.1" xref="S5.I5.i1.p1.2.m2.1.1.cmml"><mi id="S5.I5.i1.p1.2.m2.1.1.2" xref="S5.I5.i1.p1.2.m2.1.1.2.cmml">σ</mi><mn id="S5.I5.i1.p1.2.m2.1.1.3" xref="S5.I5.i1.p1.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.2.m2.1b"><apply id="S5.I5.i1.p1.2.m2.1.1.cmml" xref="S5.I5.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I5.i1.p1.2.m2.1.1.1.cmml" xref="S5.I5.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I5.i1.p1.2.m2.1.1.2.cmml" xref="S5.I5.i1.p1.2.m2.1.1.2">𝜎</ci><cn id="S5.I5.i1.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.I5.i1.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.2.m2.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.3">. This product has an arena of size </span><math alttext="|V|\cdot|M|" class="ltx_Math" display="inline" id="S5.I5.i1.p1.3.m3.2"><semantics id="S5.I5.i1.p1.3.m3.2a"><mrow id="S5.I5.i1.p1.3.m3.2.3" xref="S5.I5.i1.p1.3.m3.2.3.cmml"><mrow id="S5.I5.i1.p1.3.m3.2.3.2.2" xref="S5.I5.i1.p1.3.m3.2.3.2.1.cmml"><mo id="S5.I5.i1.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S5.I5.i1.p1.3.m3.2.3.2.1.1.cmml">|</mo><mi id="S5.I5.i1.p1.3.m3.1.1" xref="S5.I5.i1.p1.3.m3.1.1.cmml">V</mi><mo id="S5.I5.i1.p1.3.m3.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S5.I5.i1.p1.3.m3.2.3.2.1.1.cmml">|</mo></mrow><mo id="S5.I5.i1.p1.3.m3.2.3.1" rspace="0.222em" xref="S5.I5.i1.p1.3.m3.2.3.1.cmml">⋅</mo><mrow id="S5.I5.i1.p1.3.m3.2.3.3.2" xref="S5.I5.i1.p1.3.m3.2.3.3.1.cmml"><mo id="S5.I5.i1.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S5.I5.i1.p1.3.m3.2.3.3.1.1.cmml">|</mo><mi id="S5.I5.i1.p1.3.m3.2.2" xref="S5.I5.i1.p1.3.m3.2.2.cmml">M</mi><mo id="S5.I5.i1.p1.3.m3.2.3.3.2.2" stretchy="false" xref="S5.I5.i1.p1.3.m3.2.3.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.3.m3.2b"><apply id="S5.I5.i1.p1.3.m3.2.3.cmml" xref="S5.I5.i1.p1.3.m3.2.3"><ci id="S5.I5.i1.p1.3.m3.2.3.1.cmml" xref="S5.I5.i1.p1.3.m3.2.3.1">⋅</ci><apply id="S5.I5.i1.p1.3.m3.2.3.2.1.cmml" xref="S5.I5.i1.p1.3.m3.2.3.2.2"><abs id="S5.I5.i1.p1.3.m3.2.3.2.1.1.cmml" xref="S5.I5.i1.p1.3.m3.2.3.2.2.1"></abs><ci id="S5.I5.i1.p1.3.m3.1.1.cmml" xref="S5.I5.i1.p1.3.m3.1.1">𝑉</ci></apply><apply id="S5.I5.i1.p1.3.m3.2.3.3.1.cmml" xref="S5.I5.i1.p1.3.m3.2.3.3.2"><abs id="S5.I5.i1.p1.3.m3.2.3.3.1.1.cmml" xref="S5.I5.i1.p1.3.m3.2.3.3.2.1"></abs><ci id="S5.I5.i1.p1.3.m3.2.2.cmml" xref="S5.I5.i1.p1.3.m3.2.2">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.3.m3.2c">|V|\cdot|M|</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.3.m3.2d">| italic_V | ⋅ | italic_M |</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.4"> where Player </span><math alttext="1" class="ltx_Math" display="inline" id="S5.I5.i1.p1.4.m4.1"><semantics id="S5.I5.i1.p1.4.m4.1a"><mn id="S5.I5.i1.p1.4.m4.1.1" xref="S5.I5.i1.p1.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.4.m4.1b"><cn id="S5.I5.i1.p1.4.m4.1.1.cmml" type="integer" xref="S5.I5.i1.p1.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.4.m4.1d">1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.5"> is the only player to play. The Pareto-optimal costs are among the costs of plays </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I5.i1.p1.5.m5.1"><semantics id="S5.I5.i1.p1.5.m5.1a"><mi id="S5.I5.i1.p1.5.m5.1.1" xref="S5.I5.i1.p1.5.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.5.m5.1b"><ci id="S5.I5.i1.p1.5.m5.1.1.cmml" xref="S5.I5.i1.p1.5.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.5.m5.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.5.m5.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.6"> in </span><math alttext="G\times\mathcal{M}" class="ltx_Math" display="inline" id="S5.I5.i1.p1.6.m6.1"><semantics id="S5.I5.i1.p1.6.m6.1a"><mrow id="S5.I5.i1.p1.6.m6.1.1" xref="S5.I5.i1.p1.6.m6.1.1.cmml"><mi id="S5.I5.i1.p1.6.m6.1.1.2" xref="S5.I5.i1.p1.6.m6.1.1.2.cmml">G</mi><mo id="S5.I5.i1.p1.6.m6.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.I5.i1.p1.6.m6.1.1.1.cmml">×</mo><mi class="ltx_font_mathcaligraphic" id="S5.I5.i1.p1.6.m6.1.1.3" xref="S5.I5.i1.p1.6.m6.1.1.3.cmml">ℳ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.6.m6.1b"><apply id="S5.I5.i1.p1.6.m6.1.1.cmml" xref="S5.I5.i1.p1.6.m6.1.1"><times id="S5.I5.i1.p1.6.m6.1.1.1.cmml" xref="S5.I5.i1.p1.6.m6.1.1.1"></times><ci id="S5.I5.i1.p1.6.m6.1.1.2.cmml" xref="S5.I5.i1.p1.6.m6.1.1.2">𝐺</ci><ci id="S5.I5.i1.p1.6.m6.1.1.3.cmml" xref="S5.I5.i1.p1.6.m6.1.1.3">ℳ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.6.m6.1c">G\times\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.6.m6.1d">italic_G × caligraphic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.7"> that have no cycle with positive weight between two consecutive visits of Player </span><math alttext="1" class="ltx_Math" display="inline" id="S5.I5.i1.p1.7.m7.1"><semantics id="S5.I5.i1.p1.7.m7.1a"><mn id="S5.I5.i1.p1.7.m7.1.1" xref="S5.I5.i1.p1.7.m7.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.7.m7.1b"><cn id="S5.I5.i1.p1.7.m7.1.1.cmml" type="integer" xref="S5.I5.i1.p1.7.m7.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.7.m7.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.7.m7.1d">1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.8">’s targets</span><span class="ltx_note ltx_role_footnote" id="footnote13"><sup class="ltx_note_mark">13</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">13</sup><span class="ltx_tag ltx_tag_note">13</span>or between the initial vertex and the first visit to some Player <math alttext="1" class="ltx_Math" display="inline" id="footnote13.m1.1"><semantics id="footnote13.m1.1b"><mn id="footnote13.m1.1.1" xref="footnote13.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="footnote13.m1.1c"><cn id="footnote13.m1.1.1.cmml" type="integer" xref="footnote13.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="footnote13.m1.1d">1</annotation><annotation encoding="application/x-llamapun" id="footnote13.m1.1e">1</annotation></semantics></math>’s target.</span></span></span><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.7.9">.</span></p> </div> </li> <li class="ltx_item" id="S5.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I5.i2.p1"> <p class="ltx_p" id="S5.I5.i2.p1.6"><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.6.1">Consider now a Pareto-optimal play </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I5.i2.p1.1.m1.1"><semantics id="S5.I5.i2.p1.1.m1.1a"><mi id="S5.I5.i2.p1.1.m1.1.1" xref="S5.I5.i2.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.1.m1.1b"><ci id="S5.I5.i2.p1.1.m1.1.1.cmml" xref="S5.I5.i2.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.1.m1.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.6.2"> with a cycle of null weight between two consecutive visits of Player </span><math alttext="1" class="ltx_Math" display="inline" id="S5.I5.i2.p1.2.m2.1"><semantics id="S5.I5.i2.p1.2.m2.1a"><mn id="S5.I5.i2.p1.2.m2.1.1" xref="S5.I5.i2.p1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.2.m2.1b"><cn id="S5.I5.i2.p1.2.m2.1.1.cmml" type="integer" xref="S5.I5.i2.p1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.2.m2.1d">1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.6.3">’s targets. Then there exists another play </span><math alttext="\rho^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i2.p1.3.m3.1"><semantics id="S5.I5.i2.p1.3.m3.1a"><msup id="S5.I5.i2.p1.3.m3.1.1" xref="S5.I5.i2.p1.3.m3.1.1.cmml"><mi id="S5.I5.i2.p1.3.m3.1.1.2" xref="S5.I5.i2.p1.3.m3.1.1.2.cmml">ρ</mi><mo id="S5.I5.i2.p1.3.m3.1.1.3" xref="S5.I5.i2.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.3.m3.1b"><apply id="S5.I5.i2.p1.3.m3.1.1.cmml" xref="S5.I5.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.I5.i2.p1.3.m3.1.1.1.cmml" xref="S5.I5.i2.p1.3.m3.1.1">superscript</csymbol><ci id="S5.I5.i2.p1.3.m3.1.1.2.cmml" xref="S5.I5.i2.p1.3.m3.1.1.2">𝜌</ci><ci id="S5.I5.i2.p1.3.m3.1.1.3.cmml" xref="S5.I5.i2.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.3.m3.1c">\rho^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.3.m3.1d">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.6.4">, with the same cost as </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I5.i2.p1.4.m4.1"><semantics id="S5.I5.i2.p1.4.m4.1a"><mi id="S5.I5.i2.p1.4.m4.1.1" xref="S5.I5.i2.p1.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.4.m4.1b"><ci id="S5.I5.i2.p1.4.m4.1.1.cmml" xref="S5.I5.i2.p1.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.4.m4.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.6.5">, that is obtained by removing this cycle. Moreover, if </span><math alttext="\textsf{val}(\rho)=\infty" class="ltx_Math" display="inline" id="S5.I5.i2.p1.5.m5.1"><semantics id="S5.I5.i2.p1.5.m5.1a"><mrow id="S5.I5.i2.p1.5.m5.1.2" xref="S5.I5.i2.p1.5.m5.1.2.cmml"><mrow id="S5.I5.i2.p1.5.m5.1.2.2" xref="S5.I5.i2.p1.5.m5.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i2.p1.5.m5.1.2.2.2" xref="S5.I5.i2.p1.5.m5.1.2.2.2a.cmml">val</mtext><mo id="S5.I5.i2.p1.5.m5.1.2.2.1" xref="S5.I5.i2.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S5.I5.i2.p1.5.m5.1.2.2.3.2" xref="S5.I5.i2.p1.5.m5.1.2.2.cmml"><mo id="S5.I5.i2.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S5.I5.i2.p1.5.m5.1.2.2.cmml">(</mo><mi id="S5.I5.i2.p1.5.m5.1.1" xref="S5.I5.i2.p1.5.m5.1.1.cmml">ρ</mi><mo id="S5.I5.i2.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S5.I5.i2.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.I5.i2.p1.5.m5.1.2.1" xref="S5.I5.i2.p1.5.m5.1.2.1.cmml">=</mo><mi id="S5.I5.i2.p1.5.m5.1.2.3" mathvariant="normal" xref="S5.I5.i2.p1.5.m5.1.2.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.5.m5.1b"><apply id="S5.I5.i2.p1.5.m5.1.2.cmml" xref="S5.I5.i2.p1.5.m5.1.2"><eq id="S5.I5.i2.p1.5.m5.1.2.1.cmml" xref="S5.I5.i2.p1.5.m5.1.2.1"></eq><apply id="S5.I5.i2.p1.5.m5.1.2.2.cmml" xref="S5.I5.i2.p1.5.m5.1.2.2"><times id="S5.I5.i2.p1.5.m5.1.2.2.1.cmml" xref="S5.I5.i2.p1.5.m5.1.2.2.1"></times><ci id="S5.I5.i2.p1.5.m5.1.2.2.2a.cmml" xref="S5.I5.i2.p1.5.m5.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i2.p1.5.m5.1.2.2.2.cmml" xref="S5.I5.i2.p1.5.m5.1.2.2.2">val</mtext></ci><ci id="S5.I5.i2.p1.5.m5.1.1.cmml" xref="S5.I5.i2.p1.5.m5.1.1">𝜌</ci></apply><infinity id="S5.I5.i2.p1.5.m5.1.2.3.cmml" xref="S5.I5.i2.p1.5.m5.1.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.5.m5.1c">\textsf{val}(\rho)=\infty</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.5.m5.1d">val ( italic_ρ ) = ∞</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.6.6">, then </span><math alttext="\textsf{val}(\rho^{\prime})=\infty" class="ltx_Math" display="inline" id="S5.I5.i2.p1.6.m6.1"><semantics id="S5.I5.i2.p1.6.m6.1a"><mrow id="S5.I5.i2.p1.6.m6.1.1" xref="S5.I5.i2.p1.6.m6.1.1.cmml"><mrow id="S5.I5.i2.p1.6.m6.1.1.1" xref="S5.I5.i2.p1.6.m6.1.1.1.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i2.p1.6.m6.1.1.1.3" xref="S5.I5.i2.p1.6.m6.1.1.1.3a.cmml">val</mtext><mo id="S5.I5.i2.p1.6.m6.1.1.1.2" xref="S5.I5.i2.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S5.I5.i2.p1.6.m6.1.1.1.1.1" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S5.I5.i2.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I5.i2.p1.6.m6.1.1.1.1.1.1" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.2" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.3" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I5.i2.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I5.i2.p1.6.m6.1.1.2" xref="S5.I5.i2.p1.6.m6.1.1.2.cmml">=</mo><mi id="S5.I5.i2.p1.6.m6.1.1.3" mathvariant="normal" xref="S5.I5.i2.p1.6.m6.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.6.m6.1b"><apply id="S5.I5.i2.p1.6.m6.1.1.cmml" xref="S5.I5.i2.p1.6.m6.1.1"><eq id="S5.I5.i2.p1.6.m6.1.1.2.cmml" xref="S5.I5.i2.p1.6.m6.1.1.2"></eq><apply id="S5.I5.i2.p1.6.m6.1.1.1.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1"><times id="S5.I5.i2.p1.6.m6.1.1.1.2.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1.2"></times><ci id="S5.I5.i2.p1.6.m6.1.1.1.3a.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1.3"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i2.p1.6.m6.1.1.1.3.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1.3">val</mtext></ci><apply id="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.2">𝜌</ci><ci id="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.I5.i2.p1.6.m6.1.1.1.1.1.1.3">′</ci></apply></apply><infinity id="S5.I5.i2.p1.6.m6.1.1.3.cmml" xref="S5.I5.i2.p1.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.6.m6.1c">\textsf{val}(\rho^{\prime})=\infty</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.6.m6.1d">val ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∞</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.6.7">.</span></p> </div> </li> <li class="ltx_item" id="S5.I5.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I5.i3.p1"> <p class="ltx_p" id="S5.I5.i3.p1.7"><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.1">Therefore, it is enough to consider plays </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I5.i3.p1.1.m1.1"><semantics id="S5.I5.i3.p1.1.m1.1a"><mi id="S5.I5.i3.p1.1.m1.1.1" xref="S5.I5.i3.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.1.m1.1b"><ci id="S5.I5.i3.p1.1.m1.1.1.cmml" xref="S5.I5.i3.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.1.m1.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.2"> such that for </span><math alttext="h\rho" class="ltx_Math" display="inline" id="S5.I5.i3.p1.2.m2.1"><semantics id="S5.I5.i3.p1.2.m2.1a"><mrow id="S5.I5.i3.p1.2.m2.1.1" xref="S5.I5.i3.p1.2.m2.1.1.cmml"><mi id="S5.I5.i3.p1.2.m2.1.1.2" xref="S5.I5.i3.p1.2.m2.1.1.2.cmml">h</mi><mo id="S5.I5.i3.p1.2.m2.1.1.1" xref="S5.I5.i3.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S5.I5.i3.p1.2.m2.1.1.3" xref="S5.I5.i3.p1.2.m2.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.2.m2.1b"><apply id="S5.I5.i3.p1.2.m2.1.1.cmml" xref="S5.I5.i3.p1.2.m2.1.1"><times id="S5.I5.i3.p1.2.m2.1.1.1.cmml" xref="S5.I5.i3.p1.2.m2.1.1.1"></times><ci id="S5.I5.i3.p1.2.m2.1.1.2.cmml" xref="S5.I5.i3.p1.2.m2.1.1.2">ℎ</ci><ci id="S5.I5.i3.p1.2.m2.1.1.3.cmml" xref="S5.I5.i3.p1.2.m2.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.2.m2.1c">h\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.2.m2.1d">italic_h italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.3"> with length </span><math alttext="|h|=|V|\cdot|M|\cdot t" class="ltx_Math" display="inline" id="S5.I5.i3.p1.3.m3.3"><semantics id="S5.I5.i3.p1.3.m3.3a"><mrow id="S5.I5.i3.p1.3.m3.3.4" xref="S5.I5.i3.p1.3.m3.3.4.cmml"><mrow id="S5.I5.i3.p1.3.m3.3.4.2.2" xref="S5.I5.i3.p1.3.m3.3.4.2.1.cmml"><mo id="S5.I5.i3.p1.3.m3.3.4.2.2.1" stretchy="false" xref="S5.I5.i3.p1.3.m3.3.4.2.1.1.cmml">|</mo><mi id="S5.I5.i3.p1.3.m3.1.1" xref="S5.I5.i3.p1.3.m3.1.1.cmml">h</mi><mo id="S5.I5.i3.p1.3.m3.3.4.2.2.2" stretchy="false" xref="S5.I5.i3.p1.3.m3.3.4.2.1.1.cmml">|</mo></mrow><mo id="S5.I5.i3.p1.3.m3.3.4.1" xref="S5.I5.i3.p1.3.m3.3.4.1.cmml">=</mo><mrow id="S5.I5.i3.p1.3.m3.3.4.3" xref="S5.I5.i3.p1.3.m3.3.4.3.cmml"><mrow id="S5.I5.i3.p1.3.m3.3.4.3.2.2" xref="S5.I5.i3.p1.3.m3.3.4.3.2.1.cmml"><mo id="S5.I5.i3.p1.3.m3.3.4.3.2.2.1" stretchy="false" xref="S5.I5.i3.p1.3.m3.3.4.3.2.1.1.cmml">|</mo><mi id="S5.I5.i3.p1.3.m3.2.2" xref="S5.I5.i3.p1.3.m3.2.2.cmml">V</mi><mo id="S5.I5.i3.p1.3.m3.3.4.3.2.2.2" rspace="0.055em" stretchy="false" xref="S5.I5.i3.p1.3.m3.3.4.3.2.1.1.cmml">|</mo></mrow><mo id="S5.I5.i3.p1.3.m3.3.4.3.1" rspace="0.222em" xref="S5.I5.i3.p1.3.m3.3.4.3.1.cmml">⋅</mo><mrow id="S5.I5.i3.p1.3.m3.3.4.3.3.2" xref="S5.I5.i3.p1.3.m3.3.4.3.3.1.cmml"><mo id="S5.I5.i3.p1.3.m3.3.4.3.3.2.1" stretchy="false" xref="S5.I5.i3.p1.3.m3.3.4.3.3.1.1.cmml">|</mo><mi id="S5.I5.i3.p1.3.m3.3.3" xref="S5.I5.i3.p1.3.m3.3.3.cmml">M</mi><mo id="S5.I5.i3.p1.3.m3.3.4.3.3.2.2" rspace="0.055em" stretchy="false" xref="S5.I5.i3.p1.3.m3.3.4.3.3.1.1.cmml">|</mo></mrow><mo id="S5.I5.i3.p1.3.m3.3.4.3.1a" rspace="0.222em" xref="S5.I5.i3.p1.3.m3.3.4.3.1.cmml">⋅</mo><mi id="S5.I5.i3.p1.3.m3.3.4.3.4" xref="S5.I5.i3.p1.3.m3.3.4.3.4.cmml">t</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.3.m3.3b"><apply id="S5.I5.i3.p1.3.m3.3.4.cmml" xref="S5.I5.i3.p1.3.m3.3.4"><eq id="S5.I5.i3.p1.3.m3.3.4.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.1"></eq><apply id="S5.I5.i3.p1.3.m3.3.4.2.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.2.2"><abs id="S5.I5.i3.p1.3.m3.3.4.2.1.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.2.2.1"></abs><ci id="S5.I5.i3.p1.3.m3.1.1.cmml" xref="S5.I5.i3.p1.3.m3.1.1">ℎ</ci></apply><apply id="S5.I5.i3.p1.3.m3.3.4.3.cmml" xref="S5.I5.i3.p1.3.m3.3.4.3"><ci id="S5.I5.i3.p1.3.m3.3.4.3.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.3.1">⋅</ci><apply id="S5.I5.i3.p1.3.m3.3.4.3.2.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.3.2.2"><abs id="S5.I5.i3.p1.3.m3.3.4.3.2.1.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.3.2.2.1"></abs><ci id="S5.I5.i3.p1.3.m3.2.2.cmml" xref="S5.I5.i3.p1.3.m3.2.2">𝑉</ci></apply><apply id="S5.I5.i3.p1.3.m3.3.4.3.3.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.3.3.2"><abs id="S5.I5.i3.p1.3.m3.3.4.3.3.1.1.cmml" xref="S5.I5.i3.p1.3.m3.3.4.3.3.2.1"></abs><ci id="S5.I5.i3.p1.3.m3.3.3.cmml" xref="S5.I5.i3.p1.3.m3.3.3">𝑀</ci></apply><ci id="S5.I5.i3.p1.3.m3.3.4.3.4.cmml" xref="S5.I5.i3.p1.3.m3.3.4.3.4">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.3.m3.3c">|h|=|V|\cdot|M|\cdot t</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.3.m3.3d">| italic_h | = | italic_V | ⋅ | italic_M | ⋅ italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.4">, we have </span><math alttext="\textsf{cost}({\rho})=\textsf{cost}({h})" class="ltx_Math" display="inline" id="S5.I5.i3.p1.4.m4.2"><semantics id="S5.I5.i3.p1.4.m4.2a"><mrow id="S5.I5.i3.p1.4.m4.2.3" xref="S5.I5.i3.p1.4.m4.2.3.cmml"><mrow id="S5.I5.i3.p1.4.m4.2.3.2" xref="S5.I5.i3.p1.4.m4.2.3.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i3.p1.4.m4.2.3.2.2" xref="S5.I5.i3.p1.4.m4.2.3.2.2a.cmml">cost</mtext><mo id="S5.I5.i3.p1.4.m4.2.3.2.1" xref="S5.I5.i3.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S5.I5.i3.p1.4.m4.2.3.2.3.2" xref="S5.I5.i3.p1.4.m4.2.3.2.cmml"><mo id="S5.I5.i3.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="S5.I5.i3.p1.4.m4.2.3.2.cmml">(</mo><mi id="S5.I5.i3.p1.4.m4.1.1" xref="S5.I5.i3.p1.4.m4.1.1.cmml">ρ</mi><mo id="S5.I5.i3.p1.4.m4.2.3.2.3.2.2" stretchy="false" xref="S5.I5.i3.p1.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.p1.4.m4.2.3.1" xref="S5.I5.i3.p1.4.m4.2.3.1.cmml">=</mo><mrow id="S5.I5.i3.p1.4.m4.2.3.3" xref="S5.I5.i3.p1.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i3.p1.4.m4.2.3.3.2" xref="S5.I5.i3.p1.4.m4.2.3.3.2a.cmml">cost</mtext><mo id="S5.I5.i3.p1.4.m4.2.3.3.1" xref="S5.I5.i3.p1.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S5.I5.i3.p1.4.m4.2.3.3.3.2" xref="S5.I5.i3.p1.4.m4.2.3.3.cmml"><mo id="S5.I5.i3.p1.4.m4.2.3.3.3.2.1" stretchy="false" xref="S5.I5.i3.p1.4.m4.2.3.3.cmml">(</mo><mi id="S5.I5.i3.p1.4.m4.2.2" xref="S5.I5.i3.p1.4.m4.2.2.cmml">h</mi><mo id="S5.I5.i3.p1.4.m4.2.3.3.3.2.2" stretchy="false" xref="S5.I5.i3.p1.4.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.4.m4.2b"><apply id="S5.I5.i3.p1.4.m4.2.3.cmml" xref="S5.I5.i3.p1.4.m4.2.3"><eq id="S5.I5.i3.p1.4.m4.2.3.1.cmml" xref="S5.I5.i3.p1.4.m4.2.3.1"></eq><apply id="S5.I5.i3.p1.4.m4.2.3.2.cmml" xref="S5.I5.i3.p1.4.m4.2.3.2"><times id="S5.I5.i3.p1.4.m4.2.3.2.1.cmml" xref="S5.I5.i3.p1.4.m4.2.3.2.1"></times><ci id="S5.I5.i3.p1.4.m4.2.3.2.2a.cmml" xref="S5.I5.i3.p1.4.m4.2.3.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i3.p1.4.m4.2.3.2.2.cmml" xref="S5.I5.i3.p1.4.m4.2.3.2.2">cost</mtext></ci><ci id="S5.I5.i3.p1.4.m4.1.1.cmml" xref="S5.I5.i3.p1.4.m4.1.1">𝜌</ci></apply><apply id="S5.I5.i3.p1.4.m4.2.3.3.cmml" xref="S5.I5.i3.p1.4.m4.2.3.3"><times id="S5.I5.i3.p1.4.m4.2.3.3.1.cmml" xref="S5.I5.i3.p1.4.m4.2.3.3.1"></times><ci id="S5.I5.i3.p1.4.m4.2.3.3.2a.cmml" xref="S5.I5.i3.p1.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i3.p1.4.m4.2.3.3.2.cmml" xref="S5.I5.i3.p1.4.m4.2.3.3.2">cost</mtext></ci><ci id="S5.I5.i3.p1.4.m4.2.2.cmml" xref="S5.I5.i3.p1.4.m4.2.2">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.4.m4.2c">\textsf{cost}({\rho})=\textsf{cost}({h})</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.4.m4.2d">cost ( italic_ρ ) = cost ( italic_h )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.5"> (the worst case happens when </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I5.i3.p1.5.m5.1"><semantics id="S5.I5.i3.p1.5.m5.1a"><mi id="S5.I5.i3.p1.5.m5.1.1" xref="S5.I5.i3.p1.5.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.5.m5.1b"><ci id="S5.I5.i3.p1.5.m5.1.1.cmml" xref="S5.I5.i3.p1.5.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.5.m5.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.5.m5.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.6"> visit all Player </span><math alttext="1" class="ltx_Math" display="inline" id="S5.I5.i3.p1.6.m6.1"><semantics id="S5.I5.i3.p1.6.m6.1a"><mn id="S5.I5.i3.p1.6.m6.1.1" xref="S5.I5.i3.p1.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.6.m6.1b"><cn id="S5.I5.i3.p1.6.m6.1.1.cmml" type="integer" xref="S5.I5.i3.p1.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.6.m6.1d">1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.7">’s targets each of them separated by a longest path without cycle). It follows that </span><math alttext="\textsf{cost}({\rho})\leq|V|\cdot|M|\cdot t\cdot W" class="ltx_Math" display="inline" id="S5.I5.i3.p1.7.m7.3"><semantics id="S5.I5.i3.p1.7.m7.3a"><mrow id="S5.I5.i3.p1.7.m7.3.4" xref="S5.I5.i3.p1.7.m7.3.4.cmml"><mrow id="S5.I5.i3.p1.7.m7.3.4.2" xref="S5.I5.i3.p1.7.m7.3.4.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i3.p1.7.m7.3.4.2.2" xref="S5.I5.i3.p1.7.m7.3.4.2.2a.cmml">cost</mtext><mo id="S5.I5.i3.p1.7.m7.3.4.2.1" xref="S5.I5.i3.p1.7.m7.3.4.2.1.cmml">⁢</mo><mrow id="S5.I5.i3.p1.7.m7.3.4.2.3.2" xref="S5.I5.i3.p1.7.m7.3.4.2.cmml"><mo id="S5.I5.i3.p1.7.m7.3.4.2.3.2.1" stretchy="false" xref="S5.I5.i3.p1.7.m7.3.4.2.cmml">(</mo><mi id="S5.I5.i3.p1.7.m7.1.1" xref="S5.I5.i3.p1.7.m7.1.1.cmml">ρ</mi><mo id="S5.I5.i3.p1.7.m7.3.4.2.3.2.2" stretchy="false" xref="S5.I5.i3.p1.7.m7.3.4.2.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.p1.7.m7.3.4.1" xref="S5.I5.i3.p1.7.m7.3.4.1.cmml">≤</mo><mrow id="S5.I5.i3.p1.7.m7.3.4.3" xref="S5.I5.i3.p1.7.m7.3.4.3.cmml"><mrow id="S5.I5.i3.p1.7.m7.3.4.3.2.2" xref="S5.I5.i3.p1.7.m7.3.4.3.2.1.cmml"><mo id="S5.I5.i3.p1.7.m7.3.4.3.2.2.1" stretchy="false" xref="S5.I5.i3.p1.7.m7.3.4.3.2.1.1.cmml">|</mo><mi id="S5.I5.i3.p1.7.m7.2.2" xref="S5.I5.i3.p1.7.m7.2.2.cmml">V</mi><mo id="S5.I5.i3.p1.7.m7.3.4.3.2.2.2" rspace="0.055em" stretchy="false" xref="S5.I5.i3.p1.7.m7.3.4.3.2.1.1.cmml">|</mo></mrow><mo id="S5.I5.i3.p1.7.m7.3.4.3.1" rspace="0.222em" xref="S5.I5.i3.p1.7.m7.3.4.3.1.cmml">⋅</mo><mrow id="S5.I5.i3.p1.7.m7.3.4.3.3.2" xref="S5.I5.i3.p1.7.m7.3.4.3.3.1.cmml"><mo id="S5.I5.i3.p1.7.m7.3.4.3.3.2.1" stretchy="false" xref="S5.I5.i3.p1.7.m7.3.4.3.3.1.1.cmml">|</mo><mi id="S5.I5.i3.p1.7.m7.3.3" xref="S5.I5.i3.p1.7.m7.3.3.cmml">M</mi><mo id="S5.I5.i3.p1.7.m7.3.4.3.3.2.2" rspace="0.055em" stretchy="false" xref="S5.I5.i3.p1.7.m7.3.4.3.3.1.1.cmml">|</mo></mrow><mo id="S5.I5.i3.p1.7.m7.3.4.3.1a" rspace="0.222em" xref="S5.I5.i3.p1.7.m7.3.4.3.1.cmml">⋅</mo><mi id="S5.I5.i3.p1.7.m7.3.4.3.4" xref="S5.I5.i3.p1.7.m7.3.4.3.4.cmml">t</mi><mo id="S5.I5.i3.p1.7.m7.3.4.3.1b" lspace="0.222em" rspace="0.222em" xref="S5.I5.i3.p1.7.m7.3.4.3.1.cmml">⋅</mo><mi id="S5.I5.i3.p1.7.m7.3.4.3.5" xref="S5.I5.i3.p1.7.m7.3.4.3.5.cmml">W</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.7.m7.3b"><apply id="S5.I5.i3.p1.7.m7.3.4.cmml" xref="S5.I5.i3.p1.7.m7.3.4"><leq id="S5.I5.i3.p1.7.m7.3.4.1.cmml" xref="S5.I5.i3.p1.7.m7.3.4.1"></leq><apply id="S5.I5.i3.p1.7.m7.3.4.2.cmml" xref="S5.I5.i3.p1.7.m7.3.4.2"><times id="S5.I5.i3.p1.7.m7.3.4.2.1.cmml" xref="S5.I5.i3.p1.7.m7.3.4.2.1"></times><ci id="S5.I5.i3.p1.7.m7.3.4.2.2a.cmml" xref="S5.I5.i3.p1.7.m7.3.4.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I5.i3.p1.7.m7.3.4.2.2.cmml" xref="S5.I5.i3.p1.7.m7.3.4.2.2">cost</mtext></ci><ci id="S5.I5.i3.p1.7.m7.1.1.cmml" xref="S5.I5.i3.p1.7.m7.1.1">𝜌</ci></apply><apply id="S5.I5.i3.p1.7.m7.3.4.3.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3"><ci id="S5.I5.i3.p1.7.m7.3.4.3.1.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3.1">⋅</ci><apply id="S5.I5.i3.p1.7.m7.3.4.3.2.1.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3.2.2"><abs id="S5.I5.i3.p1.7.m7.3.4.3.2.1.1.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3.2.2.1"></abs><ci id="S5.I5.i3.p1.7.m7.2.2.cmml" xref="S5.I5.i3.p1.7.m7.2.2">𝑉</ci></apply><apply id="S5.I5.i3.p1.7.m7.3.4.3.3.1.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3.3.2"><abs id="S5.I5.i3.p1.7.m7.3.4.3.3.1.1.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3.3.2.1"></abs><ci id="S5.I5.i3.p1.7.m7.3.3.cmml" xref="S5.I5.i3.p1.7.m7.3.3">𝑀</ci></apply><ci id="S5.I5.i3.p1.7.m7.3.4.3.4.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3.4">𝑡</ci><ci id="S5.I5.i3.p1.7.m7.3.4.3.5.cmml" xref="S5.I5.i3.p1.7.m7.3.4.3.5">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.7.m7.3c">\textsf{cost}({\rho})\leq|V|\cdot|M|\cdot t\cdot W</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.7.m7.3d">cost ( italic_ρ ) ≤ | italic_V | ⋅ | italic_M | ⋅ italic_t ⋅ italic_W</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i3.p1.7.8">.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S5.Thmtheorem5.p3"> <p class="ltx_p" id="S5.Thmtheorem5.p3.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem5.p3.2.2">From the previous observation, let us explain how to compute the set <math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p3.1.1.m1.1"><semantics id="S5.Thmtheorem5.p3.1.1.m1.1a"><msub id="S5.Thmtheorem5.p3.1.1.m1.1.1" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem5.p3.1.1.m1.1.1.2" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.2.cmml">C</mi><msub id="S5.Thmtheorem5.p3.1.1.m1.1.1.3" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.3.cmml"><mi id="S5.Thmtheorem5.p3.1.1.m1.1.1.3.2" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.3.2.cmml">σ</mi><mn id="S5.Thmtheorem5.p3.1.1.m1.1.1.3.3" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p3.1.1.m1.1b"><apply id="S5.Thmtheorem5.p3.1.1.m1.1.1.cmml" xref="S5.Thmtheorem5.p3.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p3.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem5.p3.1.1.m1.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p3.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.2">𝐶</ci><apply id="S5.Thmtheorem5.p3.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p3.1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem5.p3.1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.3.2">𝜎</ci><cn id="S5.Thmtheorem5.p3.1.1.m1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem5.p3.1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p3.1.1.m1.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p3.1.1.m1.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> of Pareto-optimal costs, and then how to check that <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p3.2.2.m2.1"><semantics id="S5.Thmtheorem5.p3.2.2.m2.1a"><msub id="S5.Thmtheorem5.p3.2.2.m2.1.1" xref="S5.Thmtheorem5.p3.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem5.p3.2.2.m2.1.1.2" xref="S5.Thmtheorem5.p3.2.2.m2.1.1.2.cmml">σ</mi><mn id="S5.Thmtheorem5.p3.2.2.m2.1.1.3" xref="S5.Thmtheorem5.p3.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p3.2.2.m2.1b"><apply id="S5.Thmtheorem5.p3.2.2.m2.1.1.cmml" xref="S5.Thmtheorem5.p3.2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p3.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem5.p3.2.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p3.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem5.p3.2.2.m2.1.1.2">𝜎</ci><cn id="S5.Thmtheorem5.p3.2.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem5.p3.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p3.2.2.m2.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p3.2.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a solution.</span></p> <ul class="ltx_itemize" id="S5.I6"> <li class="ltx_item" id="S5.I6.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I6.i1.p1"> <p class="ltx_p" id="S5.I6.i1.p1.16"><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.1">First, we further extend the vertices of </span><math alttext="G\times\mathcal{M}" class="ltx_Math" display="inline" id="S5.I6.i1.p1.1.m1.1"><semantics id="S5.I6.i1.p1.1.m1.1a"><mrow id="S5.I6.i1.p1.1.m1.1.1" xref="S5.I6.i1.p1.1.m1.1.1.cmml"><mi id="S5.I6.i1.p1.1.m1.1.1.2" xref="S5.I6.i1.p1.1.m1.1.1.2.cmml">G</mi><mo id="S5.I6.i1.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.I6.i1.p1.1.m1.1.1.1.cmml">×</mo><mi class="ltx_font_mathcaligraphic" id="S5.I6.i1.p1.1.m1.1.1.3" xref="S5.I6.i1.p1.1.m1.1.1.3.cmml">ℳ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.1.m1.1b"><apply id="S5.I6.i1.p1.1.m1.1.1.cmml" xref="S5.I6.i1.p1.1.m1.1.1"><times id="S5.I6.i1.p1.1.m1.1.1.1.cmml" xref="S5.I6.i1.p1.1.m1.1.1.1"></times><ci id="S5.I6.i1.p1.1.m1.1.1.2.cmml" xref="S5.I6.i1.p1.1.m1.1.1.2">𝐺</ci><ci id="S5.I6.i1.p1.1.m1.1.1.3.cmml" xref="S5.I6.i1.p1.1.m1.1.1.3">ℳ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.1.m1.1c">G\times\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.1.m1.1d">italic_G × caligraphic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.2"> to keep track of the weight, value and cost of the current history, truncated to </span><math alttext="\infty" class="ltx_Math" display="inline" id="S5.I6.i1.p1.2.m2.1"><semantics id="S5.I6.i1.p1.2.m2.1a"><mi id="S5.I6.i1.p1.2.m2.1.1" mathvariant="normal" xref="S5.I6.i1.p1.2.m2.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.2.m2.1b"><infinity id="S5.I6.i1.p1.2.m2.1.1.cmml" xref="S5.I6.i1.p1.2.m2.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.2.m2.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.2.m2.1d">∞</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.3"> when they are greater than </span><math alttext="\alpha=\max\{B,|V|\cdot|M|\cdot t\cdot W\}" class="ltx_Math" display="inline" id="S5.I6.i1.p1.3.m3.5"><semantics id="S5.I6.i1.p1.3.m3.5a"><mrow id="S5.I6.i1.p1.3.m3.5.5" xref="S5.I6.i1.p1.3.m3.5.5.cmml"><mi id="S5.I6.i1.p1.3.m3.5.5.3" xref="S5.I6.i1.p1.3.m3.5.5.3.cmml">α</mi><mo id="S5.I6.i1.p1.3.m3.5.5.2" xref="S5.I6.i1.p1.3.m3.5.5.2.cmml">=</mo><mrow id="S5.I6.i1.p1.3.m3.5.5.1.1" 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xref="S5.I6.i1.p1.3.m3.1.1">𝑉</ci></apply><apply id="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.3.1.cmml" xref="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.3.2"><abs id="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.3.1.1.cmml" xref="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.3.2.1"></abs><ci id="S5.I6.i1.p1.3.m3.2.2.cmml" xref="S5.I6.i1.p1.3.m3.2.2">𝑀</ci></apply><ci id="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.4.cmml" xref="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.4">𝑡</ci><ci id="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.5.cmml" xref="S5.I6.i1.p1.3.m3.5.5.1.1.1.1.5">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.3.m3.5c">\alpha=\max\{B,|V|\cdot|M|\cdot t\cdot W\}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.3.m3.5d">italic_α = roman_max { italic_B , | italic_V | ⋅ | italic_M | ⋅ italic_t ⋅ italic_W }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.4">). That is, we consider an arena </span><math alttext="H" class="ltx_Math" display="inline" id="S5.I6.i1.p1.4.m4.1"><semantics id="S5.I6.i1.p1.4.m4.1a"><mi id="S5.I6.i1.p1.4.m4.1.1" xref="S5.I6.i1.p1.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.4.m4.1b"><ci id="S5.I6.i1.p1.4.m4.1.1.cmml" xref="S5.I6.i1.p1.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.4.m4.1d">italic_H</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.5"> whose vertices are of the form </span><math alttext="(v,s,(m_{1},m_{2},m_{3}))" class="ltx_Math" display="inline" id="S5.I6.i1.p1.5.m5.3"><semantics id="S5.I6.i1.p1.5.m5.3a"><mrow id="S5.I6.i1.p1.5.m5.3.3.1" xref="S5.I6.i1.p1.5.m5.3.3.2.cmml"><mo id="S5.I6.i1.p1.5.m5.3.3.1.2" stretchy="false" xref="S5.I6.i1.p1.5.m5.3.3.2.cmml">(</mo><mi id="S5.I6.i1.p1.5.m5.1.1" 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id="S5.I6.i1.p1.9.m9.4.4.cmml" xref="S5.I6.i1.p1.9.m9.4.4"></infinity></set></apply><ci id="S5.I6.i1.p1.9.m9.5.5.1.3.cmml" xref="S5.I6.i1.p1.9.m9.5.5.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.9.m9.5c">m_{3}\in(\{0,\ldots,\alpha\}\cup\{\infty\})^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.9.m9.5d">italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ ( { 0 , … , italic_α } ∪ { ∞ } ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.10">. As in the proof of Lemma </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem2" title="Lemma 5.2 ‣ 5.2 Punishing Strategies ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.2</span></a><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.11">, whenever </span><math alttext="v" class="ltx_Math" display="inline" id="S5.I6.i1.p1.10.m10.1"><semantics id="S5.I6.i1.p1.10.m10.1a"><mi id="S5.I6.i1.p1.10.m10.1.1" xref="S5.I6.i1.p1.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.10.m10.1b"><ci id="S5.I6.i1.p1.10.m10.1.1.cmml" xref="S5.I6.i1.p1.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.10.m10.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.12"> belongs to some target, the weight component </span><math alttext="m_{1}" class="ltx_Math" display="inline" id="S5.I6.i1.p1.11.m11.1"><semantics id="S5.I6.i1.p1.11.m11.1a"><msub id="S5.I6.i1.p1.11.m11.1.1" xref="S5.I6.i1.p1.11.m11.1.1.cmml"><mi id="S5.I6.i1.p1.11.m11.1.1.2" xref="S5.I6.i1.p1.11.m11.1.1.2.cmml">m</mi><mn id="S5.I6.i1.p1.11.m11.1.1.3" xref="S5.I6.i1.p1.11.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.11.m11.1b"><apply id="S5.I6.i1.p1.11.m11.1.1.cmml" xref="S5.I6.i1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S5.I6.i1.p1.11.m11.1.1.1.cmml" xref="S5.I6.i1.p1.11.m11.1.1">subscript</csymbol><ci id="S5.I6.i1.p1.11.m11.1.1.2.cmml" xref="S5.I6.i1.p1.11.m11.1.1.2">𝑚</ci><cn id="S5.I6.i1.p1.11.m11.1.1.3.cmml" type="integer" xref="S5.I6.i1.p1.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.11.m11.1c">m_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.11.m11.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.13"> allows to update the </span><span class="ltx_text ltx_font_sansserif ltx_font_italic" id="S5.I6.i1.p1.16.14">val</span><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.15"> component </span><math alttext="m_{2}" class="ltx_Math" display="inline" id="S5.I6.i1.p1.12.m12.1"><semantics id="S5.I6.i1.p1.12.m12.1a"><msub id="S5.I6.i1.p1.12.m12.1.1" xref="S5.I6.i1.p1.12.m12.1.1.cmml"><mi id="S5.I6.i1.p1.12.m12.1.1.2" xref="S5.I6.i1.p1.12.m12.1.1.2.cmml">m</mi><mn id="S5.I6.i1.p1.12.m12.1.1.3" xref="S5.I6.i1.p1.12.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.12.m12.1b"><apply id="S5.I6.i1.p1.12.m12.1.1.cmml" xref="S5.I6.i1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S5.I6.i1.p1.12.m12.1.1.1.cmml" xref="S5.I6.i1.p1.12.m12.1.1">subscript</csymbol><ci id="S5.I6.i1.p1.12.m12.1.1.2.cmml" xref="S5.I6.i1.p1.12.m12.1.1.2">𝑚</ci><cn id="S5.I6.i1.p1.12.m12.1.1.3.cmml" type="integer" xref="S5.I6.i1.p1.12.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.12.m12.1c">m_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.12.m12.1d">italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.16"> and the </span><span class="ltx_text ltx_font_sansserif ltx_font_italic" id="S5.I6.i1.p1.16.17">cost</span><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.18"> component </span><math alttext="m_{3}" class="ltx_Math" display="inline" id="S5.I6.i1.p1.13.m13.1"><semantics id="S5.I6.i1.p1.13.m13.1a"><msub id="S5.I6.i1.p1.13.m13.1.1" xref="S5.I6.i1.p1.13.m13.1.1.cmml"><mi id="S5.I6.i1.p1.13.m13.1.1.2" xref="S5.I6.i1.p1.13.m13.1.1.2.cmml">m</mi><mn id="S5.I6.i1.p1.13.m13.1.1.3" xref="S5.I6.i1.p1.13.m13.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.13.m13.1b"><apply id="S5.I6.i1.p1.13.m13.1.1.cmml" xref="S5.I6.i1.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S5.I6.i1.p1.13.m13.1.1.1.cmml" xref="S5.I6.i1.p1.13.m13.1.1">subscript</csymbol><ci id="S5.I6.i1.p1.13.m13.1.1.2.cmml" xref="S5.I6.i1.p1.13.m13.1.1.2">𝑚</ci><cn id="S5.I6.i1.p1.13.m13.1.1.3.cmml" type="integer" xref="S5.I6.i1.p1.13.m13.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.13.m13.1c">m_{3}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.13.m13.1d">italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.19">. The initial vertex is </span><math alttext="(v_{0},s_{0},(0,\textsf{val}(v_{0}),\textsf{cost}({v_{0}})))" class="ltx_Math" display="inline" id="S5.I6.i1.p1.14.m14.4"><semantics id="S5.I6.i1.p1.14.m14.4a"><mrow id="S5.I6.i1.p1.14.m14.4.4.3" xref="S5.I6.i1.p1.14.m14.4.4.4.cmml"><mo id="S5.I6.i1.p1.14.m14.4.4.3.4" stretchy="false" xref="S5.I6.i1.p1.14.m14.4.4.4.cmml">(</mo><msub id="S5.I6.i1.p1.14.m14.2.2.1.1" xref="S5.I6.i1.p1.14.m14.2.2.1.1.cmml"><mi id="S5.I6.i1.p1.14.m14.2.2.1.1.2" xref="S5.I6.i1.p1.14.m14.2.2.1.1.2.cmml">v</mi><mn id="S5.I6.i1.p1.14.m14.2.2.1.1.3" xref="S5.I6.i1.p1.14.m14.2.2.1.1.3.cmml">0</mn></msub><mo id="S5.I6.i1.p1.14.m14.4.4.3.5" xref="S5.I6.i1.p1.14.m14.4.4.4.cmml">,</mo><msub id="S5.I6.i1.p1.14.m14.3.3.2.2" xref="S5.I6.i1.p1.14.m14.3.3.2.2.cmml"><mi id="S5.I6.i1.p1.14.m14.3.3.2.2.2" xref="S5.I6.i1.p1.14.m14.3.3.2.2.2.cmml">s</mi><mn id="S5.I6.i1.p1.14.m14.3.3.2.2.3" xref="S5.I6.i1.p1.14.m14.3.3.2.2.3.cmml">0</mn></msub><mo id="S5.I6.i1.p1.14.m14.4.4.3.6" 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class="ltx_mathvariant_sans-serif-italic" id="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.3.cmml" xref="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.3">cost</mtext></ci><apply id="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1.1.cmml" xref="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1.1.1.cmml" xref="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1">subscript</csymbol><ci id="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1.1.2.cmml" xref="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1.1.2">𝑣</ci><cn id="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1.1.3.cmml" type="integer" xref="S5.I6.i1.p1.14.m14.4.4.3.3.2.2.1.1.1.3">0</cn></apply></apply></vector></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.14.m14.4c">(v_{0},s_{0},(0,\textsf{val}(v_{0}),\textsf{cost}({v_{0}})))</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.14.m14.4d">( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , ( 0 , val ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , cost ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.20"> where </span><math alttext="s_{0}" class="ltx_Math" display="inline" id="S5.I6.i1.p1.15.m15.1"><semantics id="S5.I6.i1.p1.15.m15.1a"><msub id="S5.I6.i1.p1.15.m15.1.1" xref="S5.I6.i1.p1.15.m15.1.1.cmml"><mi id="S5.I6.i1.p1.15.m15.1.1.2" xref="S5.I6.i1.p1.15.m15.1.1.2.cmml">s</mi><mn id="S5.I6.i1.p1.15.m15.1.1.3" xref="S5.I6.i1.p1.15.m15.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.15.m15.1b"><apply id="S5.I6.i1.p1.15.m15.1.1.cmml" xref="S5.I6.i1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S5.I6.i1.p1.15.m15.1.1.1.cmml" xref="S5.I6.i1.p1.15.m15.1.1">subscript</csymbol><ci id="S5.I6.i1.p1.15.m15.1.1.2.cmml" xref="S5.I6.i1.p1.15.m15.1.1.2">𝑠</ci><cn id="S5.I6.i1.p1.15.m15.1.1.3.cmml" type="integer" xref="S5.I6.i1.p1.15.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.15.m15.1c">s_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.15.m15.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.21"> is the initial memory state of </span><math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S5.I6.i1.p1.16.m16.1"><semantics id="S5.I6.i1.p1.16.m16.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I6.i1.p1.16.m16.1.1" xref="S5.I6.i1.p1.16.m16.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i1.p1.16.m16.1b"><ci id="S5.I6.i1.p1.16.m16.1.1.cmml" xref="S5.I6.i1.p1.16.m16.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i1.p1.16.m16.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i1.p1.16.m16.1d">caligraphic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i1.p1.16.22">.</span></p> </div> </li> <li class="ltx_item" id="S5.I6.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I6.i2.p1"> <p class="ltx_p" id="S5.I6.i2.p1.13"><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.1">Then, to compute </span><math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.I6.i2.p1.1.m1.1"><semantics id="S5.I6.i2.p1.1.m1.1a"><msub id="S5.I6.i2.p1.1.m1.1.1" xref="S5.I6.i2.p1.1.m1.1.1.cmml"><mi id="S5.I6.i2.p1.1.m1.1.1.2" xref="S5.I6.i2.p1.1.m1.1.1.2.cmml">C</mi><msub id="S5.I6.i2.p1.1.m1.1.1.3" xref="S5.I6.i2.p1.1.m1.1.1.3.cmml"><mi id="S5.I6.i2.p1.1.m1.1.1.3.2" xref="S5.I6.i2.p1.1.m1.1.1.3.2.cmml">σ</mi><mn id="S5.I6.i2.p1.1.m1.1.1.3.3" xref="S5.I6.i2.p1.1.m1.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.1.m1.1b"><apply id="S5.I6.i2.p1.1.m1.1.1.cmml" xref="S5.I6.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I6.i2.p1.1.m1.1.1.1.cmml" xref="S5.I6.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I6.i2.p1.1.m1.1.1.2.cmml" xref="S5.I6.i2.p1.1.m1.1.1.2">𝐶</ci><apply id="S5.I6.i2.p1.1.m1.1.1.3.cmml" xref="S5.I6.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.I6.i2.p1.1.m1.1.1.3.1.cmml" xref="S5.I6.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.I6.i2.p1.1.m1.1.1.3.2.cmml" xref="S5.I6.i2.p1.1.m1.1.1.3.2">𝜎</ci><cn id="S5.I6.i2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S5.I6.i2.p1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.1.m1.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.1.m1.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.2">, we test for the existence of a play </span><math alttext="\rho\in\textsf{Play}_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.I6.i2.p1.2.m2.1"><semantics id="S5.I6.i2.p1.2.m2.1a"><mrow id="S5.I6.i2.p1.2.m2.1.1" xref="S5.I6.i2.p1.2.m2.1.1.cmml"><mi id="S5.I6.i2.p1.2.m2.1.1.2" xref="S5.I6.i2.p1.2.m2.1.1.2.cmml">ρ</mi><mo id="S5.I6.i2.p1.2.m2.1.1.1" xref="S5.I6.i2.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S5.I6.i2.p1.2.m2.1.1.3" xref="S5.I6.i2.p1.2.m2.1.1.3.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I6.i2.p1.2.m2.1.1.3.2" xref="S5.I6.i2.p1.2.m2.1.1.3.2a.cmml">Play</mtext><msub id="S5.I6.i2.p1.2.m2.1.1.3.3" xref="S5.I6.i2.p1.2.m2.1.1.3.3.cmml"><mi id="S5.I6.i2.p1.2.m2.1.1.3.3.2" xref="S5.I6.i2.p1.2.m2.1.1.3.3.2.cmml">σ</mi><mn id="S5.I6.i2.p1.2.m2.1.1.3.3.3" xref="S5.I6.i2.p1.2.m2.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.2.m2.1b"><apply id="S5.I6.i2.p1.2.m2.1.1.cmml" xref="S5.I6.i2.p1.2.m2.1.1"><in id="S5.I6.i2.p1.2.m2.1.1.1.cmml" xref="S5.I6.i2.p1.2.m2.1.1.1"></in><ci id="S5.I6.i2.p1.2.m2.1.1.2.cmml" xref="S5.I6.i2.p1.2.m2.1.1.2">𝜌</ci><apply id="S5.I6.i2.p1.2.m2.1.1.3.cmml" xref="S5.I6.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.I6.i2.p1.2.m2.1.1.3.1.cmml" xref="S5.I6.i2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.I6.i2.p1.2.m2.1.1.3.2a.cmml" xref="S5.I6.i2.p1.2.m2.1.1.3.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I6.i2.p1.2.m2.1.1.3.2.cmml" xref="S5.I6.i2.p1.2.m2.1.1.3.2">Play</mtext></ci><apply id="S5.I6.i2.p1.2.m2.1.1.3.3.cmml" xref="S5.I6.i2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.I6.i2.p1.2.m2.1.1.3.3.1.cmml" xref="S5.I6.i2.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.I6.i2.p1.2.m2.1.1.3.3.2.cmml" xref="S5.I6.i2.p1.2.m2.1.1.3.3.2">𝜎</ci><cn id="S5.I6.i2.p1.2.m2.1.1.3.3.3.cmml" type="integer" xref="S5.I6.i2.p1.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.2.m2.1c">\rho\in\textsf{Play}_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.2.m2.1d">italic_ρ ∈ Play start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.3"> with a given cost </span><math alttext="c=\textsf{cost}({\rho})\in(\{0,\ldots,\alpha\}\cup\{\infty\})^{t}" class="ltx_Math" display="inline" id="S5.I6.i2.p1.3.m3.6"><semantics id="S5.I6.i2.p1.3.m3.6a"><mrow id="S5.I6.i2.p1.3.m3.6.6" xref="S5.I6.i2.p1.3.m3.6.6.cmml"><mi id="S5.I6.i2.p1.3.m3.6.6.3" xref="S5.I6.i2.p1.3.m3.6.6.3.cmml">c</mi><mo id="S5.I6.i2.p1.3.m3.6.6.4" xref="S5.I6.i2.p1.3.m3.6.6.4.cmml">=</mo><mrow id="S5.I6.i2.p1.3.m3.6.6.5" xref="S5.I6.i2.p1.3.m3.6.6.5.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I6.i2.p1.3.m3.6.6.5.2" xref="S5.I6.i2.p1.3.m3.6.6.5.2a.cmml">cost</mtext><mo id="S5.I6.i2.p1.3.m3.6.6.5.1" 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xref="S5.I6.i2.p1.3.m3.3.3">…</ci><ci id="S5.I6.i2.p1.3.m3.4.4.cmml" xref="S5.I6.i2.p1.3.m3.4.4">𝛼</ci></set><set id="S5.I6.i2.p1.3.m3.6.6.1.1.1.1.3.1.cmml" xref="S5.I6.i2.p1.3.m3.6.6.1.1.1.1.3.2"><infinity id="S5.I6.i2.p1.3.m3.5.5.cmml" xref="S5.I6.i2.p1.3.m3.5.5"></infinity></set></apply><ci id="S5.I6.i2.p1.3.m3.6.6.1.3.cmml" xref="S5.I6.i2.p1.3.m3.6.6.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.3.m3.6c">c=\textsf{cost}({\rho})\in(\{0,\ldots,\alpha\}\cup\{\infty\})^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.3.m3.6d">italic_c = cost ( italic_ρ ) ∈ ( { 0 , … , italic_α } ∪ { ∞ } ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.4">, beginning with the smallest possible cost </span><math alttext="c=(0,\ldots,0)" class="ltx_Math" display="inline" id="S5.I6.i2.p1.4.m4.3"><semantics 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id="S5.I6.i2.p1.4.m4.3.4.cmml" xref="S5.I6.i2.p1.4.m4.3.4"><eq id="S5.I6.i2.p1.4.m4.3.4.1.cmml" xref="S5.I6.i2.p1.4.m4.3.4.1"></eq><ci id="S5.I6.i2.p1.4.m4.3.4.2.cmml" xref="S5.I6.i2.p1.4.m4.3.4.2">𝑐</ci><vector id="S5.I6.i2.p1.4.m4.3.4.3.1.cmml" xref="S5.I6.i2.p1.4.m4.3.4.3.2"><cn id="S5.I6.i2.p1.4.m4.1.1.cmml" type="integer" xref="S5.I6.i2.p1.4.m4.1.1">0</cn><ci id="S5.I6.i2.p1.4.m4.2.2.cmml" xref="S5.I6.i2.p1.4.m4.2.2">…</ci><cn id="S5.I6.i2.p1.4.m4.3.3.cmml" type="integer" xref="S5.I6.i2.p1.4.m4.3.3">0</cn></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.4.m4.3c">c=(0,\ldots,0)</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.4.m4.3d">italic_c = ( 0 , … , 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.5">, and finishing with the largest possible one </span><math alttext="c=(\infty,\ldots,\infty)" class="ltx_Math" display="inline" id="S5.I6.i2.p1.5.m5.3"><semantics id="S5.I6.i2.p1.5.m5.3a"><mrow id="S5.I6.i2.p1.5.m5.3.4" xref="S5.I6.i2.p1.5.m5.3.4.cmml"><mi id="S5.I6.i2.p1.5.m5.3.4.2" xref="S5.I6.i2.p1.5.m5.3.4.2.cmml">c</mi><mo id="S5.I6.i2.p1.5.m5.3.4.1" xref="S5.I6.i2.p1.5.m5.3.4.1.cmml">=</mo><mrow id="S5.I6.i2.p1.5.m5.3.4.3.2" xref="S5.I6.i2.p1.5.m5.3.4.3.1.cmml"><mo id="S5.I6.i2.p1.5.m5.3.4.3.2.1" stretchy="false" xref="S5.I6.i2.p1.5.m5.3.4.3.1.cmml">(</mo><mi id="S5.I6.i2.p1.5.m5.1.1" mathvariant="normal" xref="S5.I6.i2.p1.5.m5.1.1.cmml">∞</mi><mo id="S5.I6.i2.p1.5.m5.3.4.3.2.2" xref="S5.I6.i2.p1.5.m5.3.4.3.1.cmml">,</mo><mi id="S5.I6.i2.p1.5.m5.2.2" mathvariant="normal" xref="S5.I6.i2.p1.5.m5.2.2.cmml">…</mi><mo id="S5.I6.i2.p1.5.m5.3.4.3.2.3" xref="S5.I6.i2.p1.5.m5.3.4.3.1.cmml">,</mo><mi id="S5.I6.i2.p1.5.m5.3.3" mathvariant="normal" xref="S5.I6.i2.p1.5.m5.3.3.cmml">∞</mi><mo id="S5.I6.i2.p1.5.m5.3.4.3.2.4" stretchy="false" xref="S5.I6.i2.p1.5.m5.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.5.m5.3b"><apply id="S5.I6.i2.p1.5.m5.3.4.cmml" xref="S5.I6.i2.p1.5.m5.3.4"><eq id="S5.I6.i2.p1.5.m5.3.4.1.cmml" xref="S5.I6.i2.p1.5.m5.3.4.1"></eq><ci id="S5.I6.i2.p1.5.m5.3.4.2.cmml" xref="S5.I6.i2.p1.5.m5.3.4.2">𝑐</ci><vector id="S5.I6.i2.p1.5.m5.3.4.3.1.cmml" xref="S5.I6.i2.p1.5.m5.3.4.3.2"><infinity id="S5.I6.i2.p1.5.m5.1.1.cmml" xref="S5.I6.i2.p1.5.m5.1.1"></infinity><ci id="S5.I6.i2.p1.5.m5.2.2.cmml" xref="S5.I6.i2.p1.5.m5.2.2">…</ci><infinity id="S5.I6.i2.p1.5.m5.3.3.cmml" xref="S5.I6.i2.p1.5.m5.3.3"></infinity></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.5.m5.3c">c=(\infty,\ldots,\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.5.m5.3d">italic_c = ( ∞ , … , ∞ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.6">. Deciding the existence of such a play </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I6.i2.p1.6.m6.1"><semantics id="S5.I6.i2.p1.6.m6.1a"><mi id="S5.I6.i2.p1.6.m6.1.1" xref="S5.I6.i2.p1.6.m6.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.6.m6.1b"><ci id="S5.I6.i2.p1.6.m6.1.1.cmml" xref="S5.I6.i2.p1.6.m6.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.6.m6.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.6.m6.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.7"> for cost </span><math alttext="c" class="ltx_Math" display="inline" id="S5.I6.i2.p1.7.m7.1"><semantics id="S5.I6.i2.p1.7.m7.1a"><mi id="S5.I6.i2.p1.7.m7.1.1" xref="S5.I6.i2.p1.7.m7.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.7.m7.1b"><ci id="S5.I6.i2.p1.7.m7.1.1.cmml" xref="S5.I6.i2.p1.7.m7.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.7.m7.1c">c</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.7.m7.1d">italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.8"> corresponds to deciding the existence of a play in the extended arena </span><math alttext="H" class="ltx_Math" display="inline" id="S5.I6.i2.p1.8.m8.1"><semantics id="S5.I6.i2.p1.8.m8.1a"><mi id="S5.I6.i2.p1.8.m8.1.1" xref="S5.I6.i2.p1.8.m8.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.8.m8.1b"><ci id="S5.I6.i2.p1.8.m8.1.1.cmml" xref="S5.I6.i2.p1.8.m8.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.8.m8.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.8.m8.1d">italic_H</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.9"> that visits some vertex </span><math alttext="(v,s,(m_{1},m_{2},m_{3}))" class="ltx_Math" 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</span><math alttext="m_{3}=c" class="ltx_Math" display="inline" id="S5.I6.i2.p1.10.m10.1"><semantics id="S5.I6.i2.p1.10.m10.1a"><mrow id="S5.I6.i2.p1.10.m10.1.1" xref="S5.I6.i2.p1.10.m10.1.1.cmml"><msub id="S5.I6.i2.p1.10.m10.1.1.2" xref="S5.I6.i2.p1.10.m10.1.1.2.cmml"><mi id="S5.I6.i2.p1.10.m10.1.1.2.2" xref="S5.I6.i2.p1.10.m10.1.1.2.2.cmml">m</mi><mn id="S5.I6.i2.p1.10.m10.1.1.2.3" xref="S5.I6.i2.p1.10.m10.1.1.2.3.cmml">3</mn></msub><mo id="S5.I6.i2.p1.10.m10.1.1.1" xref="S5.I6.i2.p1.10.m10.1.1.1.cmml">=</mo><mi id="S5.I6.i2.p1.10.m10.1.1.3" xref="S5.I6.i2.p1.10.m10.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.10.m10.1b"><apply id="S5.I6.i2.p1.10.m10.1.1.cmml" xref="S5.I6.i2.p1.10.m10.1.1"><eq id="S5.I6.i2.p1.10.m10.1.1.1.cmml" xref="S5.I6.i2.p1.10.m10.1.1.1"></eq><apply id="S5.I6.i2.p1.10.m10.1.1.2.cmml" xref="S5.I6.i2.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.I6.i2.p1.10.m10.1.1.2.1.cmml" xref="S5.I6.i2.p1.10.m10.1.1.2">subscript</csymbol><ci id="S5.I6.i2.p1.10.m10.1.1.2.2.cmml" xref="S5.I6.i2.p1.10.m10.1.1.2.2">𝑚</ci><cn id="S5.I6.i2.p1.10.m10.1.1.2.3.cmml" type="integer" xref="S5.I6.i2.p1.10.m10.1.1.2.3">3</cn></apply><ci id="S5.I6.i2.p1.10.m10.1.1.3.cmml" xref="S5.I6.i2.p1.10.m10.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.10.m10.1c">m_{3}=c</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.10.m10.1d">italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.11">. This corresponds to a reachability objective that can be checked in polynomial time in the size of </span><math alttext="H" class="ltx_Math" display="inline" id="S5.I6.i2.p1.11.m11.1"><semantics id="S5.I6.i2.p1.11.m11.1a"><mi id="S5.I6.i2.p1.11.m11.1.1" xref="S5.I6.i2.p1.11.m11.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.11.m11.1b"><ci id="S5.I6.i2.p1.11.m11.1.1.cmml" xref="S5.I6.i2.p1.11.m11.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.11.m11.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.11.m11.1d">italic_H</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.12">, thus in exponential time in the size of the given instance. Therefore, as there is at most an exponential number of costs </span><math alttext="c" class="ltx_Math" display="inline" id="S5.I6.i2.p1.12.m12.1"><semantics id="S5.I6.i2.p1.12.m12.1a"><mi id="S5.I6.i2.p1.12.m12.1.1" xref="S5.I6.i2.p1.12.m12.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.12.m12.1b"><ci id="S5.I6.i2.p1.12.m12.1.1.cmml" xref="S5.I6.i2.p1.12.m12.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.12.m12.1c">c</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.12.m12.1d">italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.13"> to consider, the set </span><math alttext="C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.I6.i2.p1.13.m13.1"><semantics id="S5.I6.i2.p1.13.m13.1a"><msub id="S5.I6.i2.p1.13.m13.1.1" xref="S5.I6.i2.p1.13.m13.1.1.cmml"><mi id="S5.I6.i2.p1.13.m13.1.1.2" xref="S5.I6.i2.p1.13.m13.1.1.2.cmml">C</mi><msub id="S5.I6.i2.p1.13.m13.1.1.3" xref="S5.I6.i2.p1.13.m13.1.1.3.cmml"><mi id="S5.I6.i2.p1.13.m13.1.1.3.2" xref="S5.I6.i2.p1.13.m13.1.1.3.2.cmml">σ</mi><mn id="S5.I6.i2.p1.13.m13.1.1.3.3" xref="S5.I6.i2.p1.13.m13.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i2.p1.13.m13.1b"><apply id="S5.I6.i2.p1.13.m13.1.1.cmml" xref="S5.I6.i2.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S5.I6.i2.p1.13.m13.1.1.1.cmml" xref="S5.I6.i2.p1.13.m13.1.1">subscript</csymbol><ci id="S5.I6.i2.p1.13.m13.1.1.2.cmml" xref="S5.I6.i2.p1.13.m13.1.1.2">𝐶</ci><apply id="S5.I6.i2.p1.13.m13.1.1.3.cmml" xref="S5.I6.i2.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S5.I6.i2.p1.13.m13.1.1.3.1.cmml" xref="S5.I6.i2.p1.13.m13.1.1.3">subscript</csymbol><ci id="S5.I6.i2.p1.13.m13.1.1.3.2.cmml" xref="S5.I6.i2.p1.13.m13.1.1.3.2">𝜎</ci><cn id="S5.I6.i2.p1.13.m13.1.1.3.3.cmml" type="integer" xref="S5.I6.i2.p1.13.m13.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i2.p1.13.m13.1c">C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i2.p1.13.m13.1d">italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i2.p1.13.14"> can be computed in exponential time.</span></p> </div> </li> <li class="ltx_item" id="S5.I6.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I6.i3.p1"> <p class="ltx_p" id="S5.I6.i3.p1.10"><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.1">Finally, we check whether </span><math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.I6.i3.p1.1.m1.1"><semantics id="S5.I6.i3.p1.1.m1.1a"><msub id="S5.I6.i3.p1.1.m1.1.1" xref="S5.I6.i3.p1.1.m1.1.1.cmml"><mi id="S5.I6.i3.p1.1.m1.1.1.2" xref="S5.I6.i3.p1.1.m1.1.1.2.cmml">σ</mi><mn id="S5.I6.i3.p1.1.m1.1.1.3" xref="S5.I6.i3.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.1.m1.1b"><apply id="S5.I6.i3.p1.1.m1.1.1.cmml" xref="S5.I6.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I6.i3.p1.1.m1.1.1.1.cmml" xref="S5.I6.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I6.i3.p1.1.m1.1.1.2.cmml" xref="S5.I6.i3.p1.1.m1.1.1.2">𝜎</ci><cn id="S5.I6.i3.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.I6.i3.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.2"> is </span><em class="ltx_emph" id="S5.I6.i3.p1.10.3">not a solution</em><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.4">, i.e., there exists a play </span><math alttext="\rho" class="ltx_Math" display="inline" id="S5.I6.i3.p1.2.m2.1"><semantics id="S5.I6.i3.p1.2.m2.1a"><mi id="S5.I6.i3.p1.2.m2.1.1" xref="S5.I6.i3.p1.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.2.m2.1b"><ci id="S5.I6.i3.p1.2.m2.1.1.cmml" xref="S5.I6.i3.p1.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.2.m2.1d">italic_ρ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.5"> in </span><math alttext="H" class="ltx_Math" display="inline" id="S5.I6.i3.p1.3.m3.1"><semantics id="S5.I6.i3.p1.3.m3.1a"><mi id="S5.I6.i3.p1.3.m3.1.1" xref="S5.I6.i3.p1.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.3.m3.1b"><ci id="S5.I6.i3.p1.3.m3.1.1.cmml" xref="S5.I6.i3.p1.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.3.m3.1d">italic_H</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.6"> with a cost </span><math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.I6.i3.p1.4.m4.1"><semantics id="S5.I6.i3.p1.4.m4.1a"><mrow id="S5.I6.i3.p1.4.m4.1.1" xref="S5.I6.i3.p1.4.m4.1.1.cmml"><mi id="S5.I6.i3.p1.4.m4.1.1.2" xref="S5.I6.i3.p1.4.m4.1.1.2.cmml">c</mi><mo id="S5.I6.i3.p1.4.m4.1.1.1" xref="S5.I6.i3.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S5.I6.i3.p1.4.m4.1.1.3" xref="S5.I6.i3.p1.4.m4.1.1.3.cmml"><mi id="S5.I6.i3.p1.4.m4.1.1.3.2" xref="S5.I6.i3.p1.4.m4.1.1.3.2.cmml">C</mi><msub id="S5.I6.i3.p1.4.m4.1.1.3.3" xref="S5.I6.i3.p1.4.m4.1.1.3.3.cmml"><mi id="S5.I6.i3.p1.4.m4.1.1.3.3.2" xref="S5.I6.i3.p1.4.m4.1.1.3.3.2.cmml">σ</mi><mn id="S5.I6.i3.p1.4.m4.1.1.3.3.3" xref="S5.I6.i3.p1.4.m4.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.4.m4.1b"><apply id="S5.I6.i3.p1.4.m4.1.1.cmml" xref="S5.I6.i3.p1.4.m4.1.1"><in id="S5.I6.i3.p1.4.m4.1.1.1.cmml" xref="S5.I6.i3.p1.4.m4.1.1.1"></in><ci id="S5.I6.i3.p1.4.m4.1.1.2.cmml" xref="S5.I6.i3.p1.4.m4.1.1.2">𝑐</ci><apply id="S5.I6.i3.p1.4.m4.1.1.3.cmml" xref="S5.I6.i3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.I6.i3.p1.4.m4.1.1.3.1.cmml" xref="S5.I6.i3.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.I6.i3.p1.4.m4.1.1.3.2.cmml" xref="S5.I6.i3.p1.4.m4.1.1.3.2">𝐶</ci><apply id="S5.I6.i3.p1.4.m4.1.1.3.3.cmml" xref="S5.I6.i3.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S5.I6.i3.p1.4.m4.1.1.3.3.1.cmml" xref="S5.I6.i3.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S5.I6.i3.p1.4.m4.1.1.3.3.2.cmml" xref="S5.I6.i3.p1.4.m4.1.1.3.3.2">𝜎</ci><cn id="S5.I6.i3.p1.4.m4.1.1.3.3.3.cmml" type="integer" xref="S5.I6.i3.p1.4.m4.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.4.m4.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.4.m4.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.7"> such that </span><math alttext="\textsf{val}(\rho)>B" class="ltx_Math" display="inline" id="S5.I6.i3.p1.5.m5.1"><semantics id="S5.I6.i3.p1.5.m5.1a"><mrow id="S5.I6.i3.p1.5.m5.1.2" xref="S5.I6.i3.p1.5.m5.1.2.cmml"><mrow id="S5.I6.i3.p1.5.m5.1.2.2" xref="S5.I6.i3.p1.5.m5.1.2.2.cmml"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I6.i3.p1.5.m5.1.2.2.2" xref="S5.I6.i3.p1.5.m5.1.2.2.2a.cmml">val</mtext><mo id="S5.I6.i3.p1.5.m5.1.2.2.1" xref="S5.I6.i3.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S5.I6.i3.p1.5.m5.1.2.2.3.2" xref="S5.I6.i3.p1.5.m5.1.2.2.cmml"><mo id="S5.I6.i3.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S5.I6.i3.p1.5.m5.1.2.2.cmml">(</mo><mi id="S5.I6.i3.p1.5.m5.1.1" xref="S5.I6.i3.p1.5.m5.1.1.cmml">ρ</mi><mo id="S5.I6.i3.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S5.I6.i3.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.I6.i3.p1.5.m5.1.2.1" xref="S5.I6.i3.p1.5.m5.1.2.1.cmml">></mo><mi id="S5.I6.i3.p1.5.m5.1.2.3" xref="S5.I6.i3.p1.5.m5.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.5.m5.1b"><apply id="S5.I6.i3.p1.5.m5.1.2.cmml" xref="S5.I6.i3.p1.5.m5.1.2"><gt id="S5.I6.i3.p1.5.m5.1.2.1.cmml" xref="S5.I6.i3.p1.5.m5.1.2.1"></gt><apply id="S5.I6.i3.p1.5.m5.1.2.2.cmml" xref="S5.I6.i3.p1.5.m5.1.2.2"><times id="S5.I6.i3.p1.5.m5.1.2.2.1.cmml" xref="S5.I6.i3.p1.5.m5.1.2.2.1"></times><ci id="S5.I6.i3.p1.5.m5.1.2.2.2a.cmml" xref="S5.I6.i3.p1.5.m5.1.2.2.2"><mtext class="ltx_mathvariant_sans-serif-italic" id="S5.I6.i3.p1.5.m5.1.2.2.2.cmml" xref="S5.I6.i3.p1.5.m5.1.2.2.2">val</mtext></ci><ci id="S5.I6.i3.p1.5.m5.1.1.cmml" xref="S5.I6.i3.p1.5.m5.1.1">𝜌</ci></apply><ci id="S5.I6.i3.p1.5.m5.1.2.3.cmml" xref="S5.I6.i3.p1.5.m5.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.5.m5.1c">\textsf{val}(\rho)>B</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.5.m5.1d">val ( italic_ρ ) > italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.8">. We remove from </span><math alttext="H" class="ltx_Math" display="inline" id="S5.I6.i3.p1.6.m6.1"><semantics id="S5.I6.i3.p1.6.m6.1a"><mi id="S5.I6.i3.p1.6.m6.1.1" xref="S5.I6.i3.p1.6.m6.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.6.m6.1b"><ci id="S5.I6.i3.p1.6.m6.1.1.cmml" xref="S5.I6.i3.p1.6.m6.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.6.m6.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.6.m6.1d">italic_H</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.9"> all vertices </span><math alttext="(v,s,(m_{1},m_{2},m_{3}))" class="ltx_Math" display="inline" id="S5.I6.i3.p1.7.m7.3"><semantics id="S5.I6.i3.p1.7.m7.3a"><mrow id="S5.I6.i3.p1.7.m7.3.3.1" xref="S5.I6.i3.p1.7.m7.3.3.2.cmml"><mo id="S5.I6.i3.p1.7.m7.3.3.1.2" stretchy="false" xref="S5.I6.i3.p1.7.m7.3.3.2.cmml">(</mo><mi id="S5.I6.i3.p1.7.m7.1.1" xref="S5.I6.i3.p1.7.m7.1.1.cmml">v</mi><mo 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xref="S5.I6.i3.p1.8.m8.1.1.2.cmml"><mi id="S5.I6.i3.p1.8.m8.1.1.2.2" xref="S5.I6.i3.p1.8.m8.1.1.2.2.cmml">m</mi><mn id="S5.I6.i3.p1.8.m8.1.1.2.3" xref="S5.I6.i3.p1.8.m8.1.1.2.3.cmml">2</mn></msub><mo id="S5.I6.i3.p1.8.m8.1.1.1" xref="S5.I6.i3.p1.8.m8.1.1.1.cmml">≤</mo><mi id="S5.I6.i3.p1.8.m8.1.1.3" xref="S5.I6.i3.p1.8.m8.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.8.m8.1b"><apply id="S5.I6.i3.p1.8.m8.1.1.cmml" xref="S5.I6.i3.p1.8.m8.1.1"><leq id="S5.I6.i3.p1.8.m8.1.1.1.cmml" xref="S5.I6.i3.p1.8.m8.1.1.1"></leq><apply id="S5.I6.i3.p1.8.m8.1.1.2.cmml" xref="S5.I6.i3.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S5.I6.i3.p1.8.m8.1.1.2.1.cmml" xref="S5.I6.i3.p1.8.m8.1.1.2">subscript</csymbol><ci id="S5.I6.i3.p1.8.m8.1.1.2.2.cmml" xref="S5.I6.i3.p1.8.m8.1.1.2.2">𝑚</ci><cn id="S5.I6.i3.p1.8.m8.1.1.2.3.cmml" type="integer" xref="S5.I6.i3.p1.8.m8.1.1.2.3">2</cn></apply><ci id="S5.I6.i3.p1.8.m8.1.1.3.cmml" xref="S5.I6.i3.p1.8.m8.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.8.m8.1c">m_{2}\leq B</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.8.m8.1d">italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.11">, and we then check the existence of a play with a cost </span><math alttext="c\in C_{\sigma_{0}}" class="ltx_Math" display="inline" id="S5.I6.i3.p1.9.m9.1"><semantics id="S5.I6.i3.p1.9.m9.1a"><mrow id="S5.I6.i3.p1.9.m9.1.1" xref="S5.I6.i3.p1.9.m9.1.1.cmml"><mi id="S5.I6.i3.p1.9.m9.1.1.2" xref="S5.I6.i3.p1.9.m9.1.1.2.cmml">c</mi><mo id="S5.I6.i3.p1.9.m9.1.1.1" xref="S5.I6.i3.p1.9.m9.1.1.1.cmml">∈</mo><msub id="S5.I6.i3.p1.9.m9.1.1.3" xref="S5.I6.i3.p1.9.m9.1.1.3.cmml"><mi id="S5.I6.i3.p1.9.m9.1.1.3.2" xref="S5.I6.i3.p1.9.m9.1.1.3.2.cmml">C</mi><msub id="S5.I6.i3.p1.9.m9.1.1.3.3" xref="S5.I6.i3.p1.9.m9.1.1.3.3.cmml"><mi id="S5.I6.i3.p1.9.m9.1.1.3.3.2" xref="S5.I6.i3.p1.9.m9.1.1.3.3.2.cmml">σ</mi><mn id="S5.I6.i3.p1.9.m9.1.1.3.3.3" xref="S5.I6.i3.p1.9.m9.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.9.m9.1b"><apply id="S5.I6.i3.p1.9.m9.1.1.cmml" xref="S5.I6.i3.p1.9.m9.1.1"><in id="S5.I6.i3.p1.9.m9.1.1.1.cmml" xref="S5.I6.i3.p1.9.m9.1.1.1"></in><ci id="S5.I6.i3.p1.9.m9.1.1.2.cmml" xref="S5.I6.i3.p1.9.m9.1.1.2">𝑐</ci><apply id="S5.I6.i3.p1.9.m9.1.1.3.cmml" xref="S5.I6.i3.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S5.I6.i3.p1.9.m9.1.1.3.1.cmml" xref="S5.I6.i3.p1.9.m9.1.1.3">subscript</csymbol><ci id="S5.I6.i3.p1.9.m9.1.1.3.2.cmml" xref="S5.I6.i3.p1.9.m9.1.1.3.2">𝐶</ci><apply id="S5.I6.i3.p1.9.m9.1.1.3.3.cmml" xref="S5.I6.i3.p1.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S5.I6.i3.p1.9.m9.1.1.3.3.1.cmml" xref="S5.I6.i3.p1.9.m9.1.1.3.3">subscript</csymbol><ci id="S5.I6.i3.p1.9.m9.1.1.3.3.2.cmml" xref="S5.I6.i3.p1.9.m9.1.1.3.3.2">𝜎</ci><cn id="S5.I6.i3.p1.9.m9.1.1.3.3.3.cmml" type="integer" xref="S5.I6.i3.p1.9.m9.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.9.m9.1c">c\in C_{\sigma_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.9.m9.1d">italic_c ∈ italic_C start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.12"> as done in the previous item. As this can be done in exponential time, checking that </span><math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S5.I6.i3.p1.10.m10.1"><semantics id="S5.I6.i3.p1.10.m10.1a"><msub id="S5.I6.i3.p1.10.m10.1.1" xref="S5.I6.i3.p1.10.m10.1.1.cmml"><mi id="S5.I6.i3.p1.10.m10.1.1.2" xref="S5.I6.i3.p1.10.m10.1.1.2.cmml">σ</mi><mn id="S5.I6.i3.p1.10.m10.1.1.3" xref="S5.I6.i3.p1.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I6.i3.p1.10.m10.1b"><apply id="S5.I6.i3.p1.10.m10.1.1.cmml" xref="S5.I6.i3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S5.I6.i3.p1.10.m10.1.1.1.cmml" xref="S5.I6.i3.p1.10.m10.1.1">subscript</csymbol><ci id="S5.I6.i3.p1.10.m10.1.1.2.cmml" xref="S5.I6.i3.p1.10.m10.1.1.2">𝜎</ci><cn id="S5.I6.i3.p1.10.m10.1.1.3.cmml" type="integer" xref="S5.I6.i3.p1.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I6.i3.p1.10.m10.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I6.i3.p1.10.m10.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I6.i3.p1.10.13"> is a solution can thus be done in exponential time.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S5.Thmtheorem5.p4"> <p class="ltx_p" id="S5.Thmtheorem5.p4.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem5.p4.5.5">Let us finally study the <span class="ltx_text ltx_font_sansserif" id="S5.Thmtheorem5.p4.5.5.1">NEXPTIME</span>-hardness of the SPS problem. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite>, the Boolean variant of the SPS problem is proved to be <span class="ltx_text ltx_font_sansserif" id="S5.Thmtheorem5.p4.5.5.2">NEXPTIME</span>-complete. It can be reduced to its quantitative variant by labeling each edge with a weight equal to <math alttext="0" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p4.1.1.m1.1"><semantics id="S5.Thmtheorem5.p4.1.1.m1.1a"><mn id="S5.Thmtheorem5.p4.1.1.m1.1.1" xref="S5.Thmtheorem5.p4.1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p4.1.1.m1.1b"><cn id="S5.Thmtheorem5.p4.1.1.m1.1.1.cmml" type="integer" xref="S5.Thmtheorem5.p4.1.1.m1.1.1">0</cn></annotation-xml></semantics></math> and by considering a bound <math alttext="B" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p4.2.2.m2.1"><semantics id="S5.Thmtheorem5.p4.2.2.m2.1a"><mi id="S5.Thmtheorem5.p4.2.2.m2.1.1" xref="S5.Thmtheorem5.p4.2.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p4.2.2.m2.1b"><ci id="S5.Thmtheorem5.p4.2.2.m2.1.1.cmml" xref="S5.Thmtheorem5.p4.2.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p4.2.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p4.2.2.m2.1d">italic_B</annotation></semantics></math> equal to <math alttext="0" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p4.3.3.m3.1"><semantics id="S5.Thmtheorem5.p4.3.3.m3.1a"><mn id="S5.Thmtheorem5.p4.3.3.m3.1.1" xref="S5.Thmtheorem5.p4.3.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p4.3.3.m3.1b"><cn id="S5.Thmtheorem5.p4.3.3.m3.1.1.cmml" type="integer" xref="S5.Thmtheorem5.p4.3.3.m3.1.1">0</cn></annotation-xml></semantics></math>. Hence the <span class="ltx_text ltx_font_sansserif" id="S5.Thmtheorem5.p4.5.5.3">val</span> and <span class="ltx_text ltx_font_sansserif" id="S5.Thmtheorem5.p4.5.5.4">cost</span> components are either equal to <math alttext="0" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p4.4.4.m4.1"><semantics id="S5.Thmtheorem5.p4.4.4.m4.1a"><mn id="S5.Thmtheorem5.p4.4.4.m4.1.1" xref="S5.Thmtheorem5.p4.4.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p4.4.4.m4.1b"><cn id="S5.Thmtheorem5.p4.4.4.m4.1.1.cmml" type="integer" xref="S5.Thmtheorem5.p4.4.4.m4.1.1">0</cn></annotation-xml></semantics></math> or <math alttext="\infty" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p4.5.5.m5.1"><semantics id="S5.Thmtheorem5.p4.5.5.m5.1a"><mi id="S5.Thmtheorem5.p4.5.5.m5.1.1" mathvariant="normal" xref="S5.Thmtheorem5.p4.5.5.m5.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p4.5.5.m5.1b"><infinity id="S5.Thmtheorem5.p4.5.5.m5.1.1.cmml" xref="S5.Thmtheorem5.p4.5.5.m5.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p4.5.5.m5.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p4.5.5.m5.1d">∞</annotation></semantics></math>. It follows that the (quantitative) SPS problem is <span class="ltx_text ltx_font_sansserif" id="S5.Thmtheorem5.p4.5.5.5">NEXPTIME</span>-hard.</span></p> </div> </div> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Conclusion and Future Work</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib16" title="">16</a>]</cite>, the SPS problem is proved to be <span class="ltx_text ltx_font_sansserif" id="S6.p1.1.1">NEXPTIME</span>-complete for Boolean reachability. In this paper, we proved that the same result holds for quantitative reachability (with non-negative weights). The difficult part was to show that when there exists a solution to the SPS problem, there is one whose Pareto-optimal costs are exponentially bounded.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.17">In this paper, we suppose that there is one weight function <math alttext="w:E\rightarrow\mathbb{N}" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mrow id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml"><mi id="S6.p2.1.m1.1.1.2" xref="S6.p2.1.m1.1.1.2.cmml">w</mi><mo id="S6.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S6.p2.1.m1.1.1.3" xref="S6.p2.1.m1.1.1.3.cmml"><mi id="S6.p2.1.m1.1.1.3.2" xref="S6.p2.1.m1.1.1.3.2.cmml">E</mi><mo id="S6.p2.1.m1.1.1.3.1" stretchy="false" xref="S6.p2.1.m1.1.1.3.1.cmml">→</mo><mi id="S6.p2.1.m1.1.1.3.3" xref="S6.p2.1.m1.1.1.3.3.cmml">ℕ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><apply id="S6.p2.1.m1.1.1.cmml" 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We could have considered the more general case where there are <math alttext="t+1" class="ltx_Math" display="inline" id="S6.p2.5.m5.1"><semantics id="S6.p2.5.m5.1a"><mrow id="S6.p2.5.m5.1.1" xref="S6.p2.5.m5.1.1.cmml"><mi id="S6.p2.5.m5.1.1.2" xref="S6.p2.5.m5.1.1.2.cmml">t</mi><mo id="S6.p2.5.m5.1.1.1" xref="S6.p2.5.m5.1.1.1.cmml">+</mo><mn id="S6.p2.5.m5.1.1.3" xref="S6.p2.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.5.m5.1b"><apply id="S6.p2.5.m5.1.1.cmml" xref="S6.p2.5.m5.1.1"><plus id="S6.p2.5.m5.1.1.1.cmml" xref="S6.p2.5.m5.1.1.1"></plus><ci id="S6.p2.5.m5.1.1.2.cmml" xref="S6.p2.5.m5.1.1.2">𝑡</ci><cn id="S6.p2.5.m5.1.1.3.cmml" type="integer" xref="S6.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.5.m5.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S6.p2.5.m5.1d">italic_t + 1</annotation></semantics></math> weight functions <math alttext="w_{i}" class="ltx_Math" display="inline" id="S6.p2.6.m6.1"><semantics id="S6.p2.6.m6.1a"><msub id="S6.p2.6.m6.1.1" xref="S6.p2.6.m6.1.1.cmml"><mi id="S6.p2.6.m6.1.1.2" xref="S6.p2.6.m6.1.1.2.cmml">w</mi><mi id="S6.p2.6.m6.1.1.3" xref="S6.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p2.6.m6.1b"><apply id="S6.p2.6.m6.1.1.cmml" xref="S6.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.p2.6.m6.1.1.1.cmml" xref="S6.p2.6.m6.1.1">subscript</csymbol><ci id="S6.p2.6.m6.1.1.2.cmml" xref="S6.p2.6.m6.1.1.2">𝑤</ci><ci id="S6.p2.6.m6.1.1.3.cmml" xref="S6.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.6.m6.1c">w_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.6.m6.1d">italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="i\in\{0,1,\ldots t\}" class="ltx_Math" display="inline" id="S6.p2.7.m7.3"><semantics id="S6.p2.7.m7.3a"><mrow id="S6.p2.7.m7.3.3" xref="S6.p2.7.m7.3.3.cmml"><mi id="S6.p2.7.m7.3.3.3" xref="S6.p2.7.m7.3.3.3.cmml">i</mi><mo id="S6.p2.7.m7.3.3.2" xref="S6.p2.7.m7.3.3.2.cmml">∈</mo><mrow id="S6.p2.7.m7.3.3.1.1" xref="S6.p2.7.m7.3.3.1.2.cmml"><mo id="S6.p2.7.m7.3.3.1.1.2" stretchy="false" xref="S6.p2.7.m7.3.3.1.2.cmml">{</mo><mn id="S6.p2.7.m7.1.1" xref="S6.p2.7.m7.1.1.cmml">0</mn><mo id="S6.p2.7.m7.3.3.1.1.3" xref="S6.p2.7.m7.3.3.1.2.cmml">,</mo><mn id="S6.p2.7.m7.2.2" xref="S6.p2.7.m7.2.2.cmml">1</mn><mo id="S6.p2.7.m7.3.3.1.1.4" xref="S6.p2.7.m7.3.3.1.2.cmml">,</mo><mrow id="S6.p2.7.m7.3.3.1.1.1" xref="S6.p2.7.m7.3.3.1.1.1.cmml"><mi id="S6.p2.7.m7.3.3.1.1.1.2" mathvariant="normal" xref="S6.p2.7.m7.3.3.1.1.1.2.cmml">…</mi><mo id="S6.p2.7.m7.3.3.1.1.1.1" xref="S6.p2.7.m7.3.3.1.1.1.1.cmml">⁢</mo><mi id="S6.p2.7.m7.3.3.1.1.1.3" xref="S6.p2.7.m7.3.3.1.1.1.3.cmml">t</mi></mrow><mo id="S6.p2.7.m7.3.3.1.1.5" stretchy="false" xref="S6.p2.7.m7.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.7.m7.3b"><apply id="S6.p2.7.m7.3.3.cmml" xref="S6.p2.7.m7.3.3"><in id="S6.p2.7.m7.3.3.2.cmml" xref="S6.p2.7.m7.3.3.2"></in><ci id="S6.p2.7.m7.3.3.3.cmml" xref="S6.p2.7.m7.3.3.3">𝑖</ci><set id="S6.p2.7.m7.3.3.1.2.cmml" xref="S6.p2.7.m7.3.3.1.1"><cn id="S6.p2.7.m7.1.1.cmml" type="integer" xref="S6.p2.7.m7.1.1">0</cn><cn id="S6.p2.7.m7.2.2.cmml" type="integer" xref="S6.p2.7.m7.2.2">1</cn><apply id="S6.p2.7.m7.3.3.1.1.1.cmml" xref="S6.p2.7.m7.3.3.1.1.1"><times id="S6.p2.7.m7.3.3.1.1.1.1.cmml" xref="S6.p2.7.m7.3.3.1.1.1.1"></times><ci id="S6.p2.7.m7.3.3.1.1.1.2.cmml" xref="S6.p2.7.m7.3.3.1.1.1.2">…</ci><ci id="S6.p2.7.m7.3.3.1.1.1.3.cmml" xref="S6.p2.7.m7.3.3.1.1.1.3">𝑡</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.7.m7.3c">i\in\{0,1,\ldots t\}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.7.m7.3d">italic_i ∈ { 0 , 1 , … italic_t }</annotation></semantics></math>, and each cost <math alttext="\textsf{cost}_{T_{i}}(\rho))" class="ltx_math_unparsed" display="inline" id="S6.p2.8.m8.1"><semantics id="S6.p2.8.m8.1a"><mrow id="S6.p2.8.m8.1b"><msub id="S6.p2.8.m8.1.2"><mtext class="ltx_mathvariant_sans-serif" id="S6.p2.8.m8.1.2.2">cost</mtext><msub id="S6.p2.8.m8.1.2.3"><mi id="S6.p2.8.m8.1.2.3.2">T</mi><mi id="S6.p2.8.m8.1.2.3.3">i</mi></msub></msub><mrow id="S6.p2.8.m8.1.3"><mo id="S6.p2.8.m8.1.3.1" stretchy="false">(</mo><mi id="S6.p2.8.m8.1.1">ρ</mi><mo id="S6.p2.8.m8.1.3.2" stretchy="false">)</mo></mrow><mo id="S6.p2.8.m8.1.4" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S6.p2.8.m8.1c">\textsf{cost}_{T_{i}}(\rho))</annotation><annotation encoding="application/x-llamapun" id="S6.p2.8.m8.1d">cost start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) )</annotation></semantics></math> is computed thanks to <math alttext="w_{i}" class="ltx_Math" display="inline" id="S6.p2.9.m9.1"><semantics id="S6.p2.9.m9.1a"><msub id="S6.p2.9.m9.1.1" xref="S6.p2.9.m9.1.1.cmml"><mi id="S6.p2.9.m9.1.1.2" xref="S6.p2.9.m9.1.1.2.cmml">w</mi><mi id="S6.p2.9.m9.1.1.3" xref="S6.p2.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p2.9.m9.1b"><apply id="S6.p2.9.m9.1.1.cmml" xref="S6.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S6.p2.9.m9.1.1.1.cmml" xref="S6.p2.9.m9.1.1">subscript</csymbol><ci id="S6.p2.9.m9.1.1.2.cmml" xref="S6.p2.9.m9.1.1.2">𝑤</ci><ci id="S6.p2.9.m9.1.1.3.cmml" xref="S6.p2.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.9.m9.1c">w_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.9.m9.1d">italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Unfortunately, the techniques developed in this paper cannot be adapted to this case. Indeed, Lemma <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S3.Thmtheorem3" title="Lemma 3.3 ‣ 3.3 Eliminating a Cycle in a Witness ‣ 3 Improving a Solution ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">3.3</span></a> allows to eliminate cycles, with the restriction that the cycle has a null weight as long as Player <math alttext="0" class="ltx_Math" display="inline" id="S6.p2.10.m10.1"><semantics id="S6.p2.10.m10.1a"><mn id="S6.p2.10.m10.1.1" xref="S6.p2.10.m10.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.p2.10.m10.1b"><cn id="S6.p2.10.m10.1.1.cmml" type="integer" xref="S6.p2.10.m10.1.1">0</cn></annotation-xml></semantics></math>’s target <math alttext="T_{0}" class="ltx_Math" display="inline" id="S6.p2.11.m11.1"><semantics id="S6.p2.11.m11.1a"><msub id="S6.p2.11.m11.1.1" xref="S6.p2.11.m11.1.1.cmml"><mi id="S6.p2.11.m11.1.1.2" xref="S6.p2.11.m11.1.1.2.cmml">T</mi><mn id="S6.p2.11.m11.1.1.3" xref="S6.p2.11.m11.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.11.m11.1b"><apply id="S6.p2.11.m11.1.1.cmml" xref="S6.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S6.p2.11.m11.1.1.1.cmml" xref="S6.p2.11.m11.1.1">subscript</csymbol><ci id="S6.p2.11.m11.1.1.2.cmml" xref="S6.p2.11.m11.1.1.2">𝑇</ci><cn id="S6.p2.11.m11.1.1.3.cmml" type="integer" xref="S6.p2.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.11.m11.1c">T_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.11.m11.1d">italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is not visited yet. It follows that in the proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S5.Thmtheorem1" title="Proposition 5.1 ‣ 5.1 Finite-Memory Solutions ‣ 5 Complexity of the SPS Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">5.1</span></a> where we transform each witness into a lasso by eliminating cycles, we can guarantee that this lasso visits <math alttext="T_{0}" class="ltx_Math" display="inline" id="S6.p2.12.m12.1"><semantics id="S6.p2.12.m12.1a"><msub id="S6.p2.12.m12.1.1" xref="S6.p2.12.m12.1.1.cmml"><mi id="S6.p2.12.m12.1.1.2" xref="S6.p2.12.m12.1.1.2.cmml">T</mi><mn id="S6.p2.12.m12.1.1.3" xref="S6.p2.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.12.m12.1b"><apply id="S6.p2.12.m12.1.1.cmml" xref="S6.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S6.p2.12.m12.1.1.1.cmml" xref="S6.p2.12.m12.1.1">subscript</csymbol><ci id="S6.p2.12.m12.1.1.2.cmml" xref="S6.p2.12.m12.1.1.2">𝑇</ci><cn id="S6.p2.12.m12.1.1.3.cmml" type="integer" xref="S6.p2.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.12.m12.1c">T_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.12.m12.1d">italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> along its prefix <math alttext="h" class="ltx_Math" display="inline" id="S6.p2.13.m13.1"><semantics id="S6.p2.13.m13.1a"><mi id="S6.p2.13.m13.1.1" xref="S6.p2.13.m13.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S6.p2.13.m13.1b"><ci id="S6.p2.13.m13.1.1.cmml" xref="S6.p2.13.m13.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.13.m13.1c">h</annotation><annotation encoding="application/x-llamapun" id="S6.p2.13.m13.1d">italic_h</annotation></semantics></math> with length <math alttext="|V|\cdot B\cdot W" class="ltx_Math" display="inline" id="S6.p2.14.m14.1"><semantics id="S6.p2.14.m14.1a"><mrow id="S6.p2.14.m14.1.2" xref="S6.p2.14.m14.1.2.cmml"><mrow id="S6.p2.14.m14.1.2.2.2" xref="S6.p2.14.m14.1.2.2.1.cmml"><mo id="S6.p2.14.m14.1.2.2.2.1" stretchy="false" xref="S6.p2.14.m14.1.2.2.1.1.cmml">|</mo><mi id="S6.p2.14.m14.1.1" xref="S6.p2.14.m14.1.1.cmml">V</mi><mo id="S6.p2.14.m14.1.2.2.2.2" rspace="0.055em" stretchy="false" xref="S6.p2.14.m14.1.2.2.1.1.cmml">|</mo></mrow><mo id="S6.p2.14.m14.1.2.1" rspace="0.222em" xref="S6.p2.14.m14.1.2.1.cmml">⋅</mo><mi id="S6.p2.14.m14.1.2.3" xref="S6.p2.14.m14.1.2.3.cmml">B</mi><mo id="S6.p2.14.m14.1.2.1a" lspace="0.222em" rspace="0.222em" xref="S6.p2.14.m14.1.2.1.cmml">⋅</mo><mi id="S6.p2.14.m14.1.2.4" xref="S6.p2.14.m14.1.2.4.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.14.m14.1b"><apply id="S6.p2.14.m14.1.2.cmml" xref="S6.p2.14.m14.1.2"><ci id="S6.p2.14.m14.1.2.1.cmml" xref="S6.p2.14.m14.1.2.1">⋅</ci><apply id="S6.p2.14.m14.1.2.2.1.cmml" xref="S6.p2.14.m14.1.2.2.2"><abs id="S6.p2.14.m14.1.2.2.1.1.cmml" xref="S6.p2.14.m14.1.2.2.2.1"></abs><ci id="S6.p2.14.m14.1.1.cmml" xref="S6.p2.14.m14.1.1">𝑉</ci></apply><ci id="S6.p2.14.m14.1.2.3.cmml" xref="S6.p2.14.m14.1.2.3">𝐵</ci><ci id="S6.p2.14.m14.1.2.4.cmml" xref="S6.p2.14.m14.1.2.4">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.14.m14.1c">|V|\cdot B\cdot W</annotation><annotation encoding="application/x-llamapun" id="S6.p2.14.m14.1d">| italic_V | ⋅ italic_B ⋅ italic_W</annotation></semantics></math>. In the case of games with <math alttext="t+1" class="ltx_Math" display="inline" id="S6.p2.15.m15.1"><semantics id="S6.p2.15.m15.1a"><mrow id="S6.p2.15.m15.1.1" xref="S6.p2.15.m15.1.1.cmml"><mi id="S6.p2.15.m15.1.1.2" xref="S6.p2.15.m15.1.1.2.cmml">t</mi><mo id="S6.p2.15.m15.1.1.1" xref="S6.p2.15.m15.1.1.1.cmml">+</mo><mn id="S6.p2.15.m15.1.1.3" xref="S6.p2.15.m15.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.15.m15.1b"><apply id="S6.p2.15.m15.1.1.cmml" xref="S6.p2.15.m15.1.1"><plus id="S6.p2.15.m15.1.1.1.cmml" xref="S6.p2.15.m15.1.1.1"></plus><ci id="S6.p2.15.m15.1.1.2.cmml" xref="S6.p2.15.m15.1.1.2">𝑡</ci><cn id="S6.p2.15.m15.1.1.3.cmml" type="integer" xref="S6.p2.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.15.m15.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S6.p2.15.m15.1d">italic_t + 1</annotation></semantics></math> (dissociated) weight functions <math alttext="w_{i}" class="ltx_Math" display="inline" id="S6.p2.16.m16.1"><semantics id="S6.p2.16.m16.1a"><msub id="S6.p2.16.m16.1.1" xref="S6.p2.16.m16.1.1.cmml"><mi id="S6.p2.16.m16.1.1.2" xref="S6.p2.16.m16.1.1.2.cmml">w</mi><mi id="S6.p2.16.m16.1.1.3" xref="S6.p2.16.m16.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p2.16.m16.1b"><apply id="S6.p2.16.m16.1.1.cmml" xref="S6.p2.16.m16.1.1"><csymbol cd="ambiguous" id="S6.p2.16.m16.1.1.1.cmml" xref="S6.p2.16.m16.1.1">subscript</csymbol><ci id="S6.p2.16.m16.1.1.2.cmml" xref="S6.p2.16.m16.1.1.2">𝑤</ci><ci id="S6.p2.16.m16.1.1.3.cmml" xref="S6.p2.16.m16.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.16.m16.1c">w_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.16.m16.1d">italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we do not see how to bound the length of such a prefix <math alttext="h" class="ltx_Math" display="inline" id="S6.p2.17.m17.1"><semantics id="S6.p2.17.m17.1a"><mi id="S6.p2.17.m17.1.1" xref="S6.p2.17.m17.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S6.p2.17.m17.1b"><ci id="S6.p2.17.m17.1.1.cmml" xref="S6.p2.17.m17.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.17.m17.1c">h</annotation><annotation encoding="application/x-llamapun" id="S6.p2.17.m17.1d">italic_h</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">This seems like a challenging problem to solve, in view of some very recent results obtained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib14" title="">14</a>]</cite>. The <em class="ltx_emph ltx_font_italic" id="S6.p3.1.1">rational synthesis</em> problem is investigated in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib14" title="">14</a>]</cite> such that the environment is composed of several players whose rational responses are to play an NE (instead of one player with several targets following a play with a Pareto-optimal cost). Both concepts of rationally coincide in the following case: on one hand, an environment composed of one player playing an NE, on the other hand an environment composed of one player with one target following a play with a Pareto-optimal cost. The reference <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib14" title="">14</a>]</cite> uses several weight functions, one for each player, and the obtained results are: (1) when the environment is composed of one player, the rational synthesis problem is <span class="ltx_text ltx_font_sansserif" id="S6.p3.1.2">PSPACE</span>-hard and in <span class="ltx_text ltx_font_sansserif" id="S6.p3.1.3">EXPTIME</span>, (2) this problem is in general <span class="ltx_text ltx_font_sansserif" id="S6.p3.1.4">EXPTIME</span>-hard and it is unknown whether it is decidable. For the SPS problem with several weight functions, we thus have the following results: (1) when Player <math alttext="1" class="ltx_Math" display="inline" id="S6.p3.1.m1.1"><semantics id="S6.p3.1.m1.1a"><mn id="S6.p3.1.m1.1.1" xref="S6.p3.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.p3.1.m1.1b"><cn id="S6.p3.1.m1.1.1.cmml" type="integer" xref="S6.p3.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S6.p3.1.m1.1d">1</annotation></semantics></math> has one target, the SPS problem is <span class="ltx_text ltx_font_sansserif" id="S6.p3.1.5">PSPACE</span>-hard and in <span class="ltx_text ltx_font_sansserif" id="S6.p3.1.6">EXPTIME</span>, (2) this problem is in general <span class="ltx_text ltx_font_sansserif" id="S6.p3.1.7">NEXPTIME</span>-hard (as a consequence of our Theorem <a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#S2.Thmtheorem1" title="Theorem 2.1 ‣ Stackelberg-Pareto synthesis problem. ‣ 2.2 Stackelberg-Pareto Synthesis Problem ‣ 2 Preliminaries and Studied Problem ‣ Stackelberg-Pareto Synthesis with Quantitative Reachability Objectives"><span class="ltx_text ltx_ref_tag">2.1</span></a>.</p> </div> <div class="ltx_para" id="S6.p4"> <p class="ltx_p" id="S6.p4.1">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2308.09443v2#bib.bib14" title="">14</a>]</cite>, weight functions with positive and negative weights have been investigated: it is proved that in both cases of rationality (NE/Pareto-optimality), the synthesis problem becomes undecidable as soon as negative weights are allowed. The proof requires to use several weight functions. It would be interesting to study whether this problem remains undecidable in case of one weight function <math alttext="w" class="ltx_Math" display="inline" id="S6.p4.1.m1.1"><semantics id="S6.p4.1.m1.1a"><mi id="S6.p4.1.m1.1.1" xref="S6.p4.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S6.p4.1.m1.1b"><ci id="S6.p4.1.m1.1.1.cmml" xref="S6.p4.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S6.p4.1.m1.1d">italic_w</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.p5"> <p class="ltx_p" id="S6.p5.1">Considering multiple objectives for Player <math alttext="0" class="ltx_Math" display="inline" id="S6.p5.1.m1.1"><semantics id="S6.p5.1.m1.1a"><mn id="S6.p5.1.m1.1.1" xref="S6.p5.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.p5.1.m1.1b"><cn id="S6.p5.1.m1.1.1.cmml" type="integer" xref="S6.p5.1.m1.1.1">0</cn></annotation-xml></semantics></math> (instead of one) is also an interesting problem to study. The order on the tuples of values becomes partial. In this case, we could impose a threshold on each value component.</p> </div> <div class="ltx_para" id="S6.p6"> <p class="ltx_p" id="S6.p6.1">It is well-known that quantitative objectives make it possible to model richer properties than with Boolean objectives. This paper studied quantitative reachability. 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