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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Learning a Game by Paying the Agents</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/2503.01976v1/"/></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/2503.01976v1#S1" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S1.SS1" title="In 1 Introduction ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Overview of our results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S1.SS2" title="In 1 Introduction ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Related research</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Notation and Preliminaries</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.SS1" title="In 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Normal form games</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.SS2" title="In 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>No-regret learning</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.SS3" title="In 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Realized and in-expectation regret</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Our Setting</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/2503.01976v1#S3.SS1" title="In 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Behavioral models</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/2503.01976v1#S3.SS1.SSS0.Px1" title="In 3.1 Behavioral models ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Rationalizable model.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS1.SSS0.Px2" title="In 3.1 Behavioral models ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- No-regret model.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS2" title="In 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Remarks about the model</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/2503.01976v1#S3.SS2.SSS0.Px1" title="In 3.2 Remarks about the model ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Boundedness of payment functions</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS2.SSS0.Px2" title="In 3.2 Remarks about the model ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Nondeterminism of models.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS2.SSS0.Px3" title="In 3.2 Remarks about the model ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Agents’ no-regret algorithms.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS2.SSS0.Px4" title="In 3.2 Remarks about the model ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Anytime regret bound.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS2.SSS0.Px5" title="In 3.2 Remarks about the model ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Deterministic regret bounds</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS3" title="In 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Game equivalence and formal goal statement</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/2503.01976v1#S3.SS3.SSS0.Px1" title="In 3.3 Game equivalence and formal goal statement ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">Goal.</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Learning the Utility Function in the Rationalizable Model</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/2503.01976v1#S4.SS1" title="In 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>The single-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.SS2" title="In 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>The multi-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.SS3" title="In 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Lower bound</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Learning the Utility in the No-Regret Model</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/2503.01976v1#S5.SS1" title="In 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>The single-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.SS2" title="In 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>The multi-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.SS3" title="In 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Lower bound</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S6" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Minimizing Payment</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/2503.01976v1#S6.SS0.SSS0.Px1" title="In 6 Minimizing Payment ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">Single rational agent</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Steering No-Regret Learners by Learning Utilities</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/2503.01976v1#S7.SS1" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.1 </span>Correlated signals</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.SS2" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.2 </span>What outcome should we steer to?</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.SS3" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.3 </span>Properties of correlated equilibria with payments</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.SS4" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.4 </span>CEPs and optimal steering</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S8" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8 </span>Conclusions and Future Research</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/2503.01976v1#S8.SS0.SSS0.Px1" title="In 8 Conclusions and Future Research ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Closing the polynomial gaps</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S8.SS0.SSS0.Px2" title="In 8 Conclusions and Future Research ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Extending beyond normal-form games</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S8.SS0.SSS0.Px3" title="In 8 Conclusions and Future Research ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title">- Round complexity of steering without utilities</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A1" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>An anytime Azuma-Hoeffding bound</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Details omitted from <span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">7</span></span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2.SS1" title="In Appendix B Details omitted from Section 7 ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.1 </span>Completed proof of <span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">7.4</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2.SS2" title="In Appendix B Details omitted from Section 7 ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.2 </span>Revelation principle for CEPs</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A3" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Other omitted proofs</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A3.SS1" title="In Appendix C Other omitted proofs ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.1 </span><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">2.1</span></span></a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Learning a Game by Paying the Agents</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Brian Hu Zhang <sup class="ltx_sup" id="id9.2.id1"><span class="ltx_text ltx_font_italic" id="id9.2.id1.1">1</span></sup> </span><span class="ltx_author_notes">Equal contribution, author order randomized</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Tao Lin<sup class="ltx_sup" id="id10.2.id1"><span class="ltx_text ltx_font_italic" id="id10.2.id1.1">∗2</span></sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Yiling Chen<sup class="ltx_sup" id="id11.2.id1"><span class="ltx_text ltx_font_italic" id="id11.2.id1.1">2</span></sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Tuomas Sandholm<sup class="ltx_sup" id="id12.2.id1"><span class="ltx_text ltx_font_italic" id="id12.2.id1.1">1,3</span></sup> </span></span> </div> <div class="ltx_dates">(<sup class="ltx_sup" id="id13.id1"><span class="ltx_text" id="id13.id1.1" style="font-size:90%;">1</span></sup><span class="ltx_text" id="id7.7.2" style="font-size:90%;">Carnegie Mellon University <br class="ltx_break"/><sup class="ltx_sup" id="id7.7.2.1">2</sup>Harvard University <br class="ltx_break"/><sup class="ltx_sup" id="id7.7.2.2">3</sup>Additional affiliations: Strategy Robot, Inc., Strategic Machine, Inc., Optimized Markets, Inc <br class="ltx_break"/> </span>March 3, 2025)</div> <div class="ltx_abstract"> <p class="ltx_p" id="id8.1">We study the problem of learning the utility functions of agents in a normal-form game by observing the agents play the game repeatedly. Differing from most prior literature, we introduce a <span class="ltx_text ltx_font_italic" id="id8.1.1">principal</span> with the power to observe the agents playing the game, send the agents signals, and send the agents payments as a function of their actions. Under reasonable behavioral models for the agents such as iterated dominated action removal or a no-regret assumption, we show that the principal can, using a number of rounds polynomial in the size of the game, learn the utility functions of all agents to any desirable precision <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="id8.1.m1.1"><semantics id="id8.1.m1.1a"><mrow id="id8.1.m1.1.1" xref="id8.1.m1.1.1.cmml"><mi id="id8.1.m1.1.1.2" xref="id8.1.m1.1.1.2.cmml">ε</mi><mo id="id8.1.m1.1.1.1" xref="id8.1.m1.1.1.1.cmml">></mo><mn id="id8.1.m1.1.1.3" xref="id8.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="id8.1.m1.1b"><apply id="id8.1.m1.1.1.cmml" xref="id8.1.m1.1.1"><gt id="id8.1.m1.1.1.1.cmml" xref="id8.1.m1.1.1.1"></gt><ci id="id8.1.m1.1.1.2.cmml" xref="id8.1.m1.1.1.2">𝜀</ci><cn id="id8.1.m1.1.1.3.cmml" type="integer" xref="id8.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="id8.1.m1.1d">italic_ε > 0</annotation></semantics></math>. We also show lower bounds in both models, which nearly match the upper bounds in the former model and also strictly separate the two models: the principal can learn strictly faster in the iterated dominance model. Finally, we discuss implications for the problem of <span class="ltx_text ltx_font_italic" id="id8.1.2">steering</span> agents to a desired equilibrium: in particular, we introduce, using our utility-learning algorithm as a subroutine, the first algorithm for steering learning agents without prior knowledge of their utilities.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S1" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S1.SS1" title="In 1 Introduction ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Overview of our results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S1.SS2" title="In 1 Introduction ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Related research</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Notation and Preliminaries</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.SS1" title="In 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Normal form games</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.SS2" title="In 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>No-regret learning</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.SS3" title="In 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Realized and in-expectation regret</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Our Setting</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/2503.01976v1#S3.SS1" title="In 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Behavioral models</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS2" title="In 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Remarks about the model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS3" title="In 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Game equivalence and formal goal statement</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Learning the Utility Function in the Rationalizable Model</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/2503.01976v1#S4.SS1" title="In 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>The single-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.SS2" title="In 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>The multi-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.SS3" title="In 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Lower bound</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Learning the Utility in the No-Regret Model</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/2503.01976v1#S5.SS1" title="In 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>The single-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.SS2" title="In 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>The multi-agent case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.SS3" title="In 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Lower bound</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S6" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Minimizing Payment</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Steering No-Regret Learners by Learning Utilities</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/2503.01976v1#S7.SS1" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.1 </span>Correlated signals</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.SS2" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.2 </span>What outcome should we steer to?</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.SS3" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.3 </span>Properties of correlated equilibria with payments</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.SS4" title="In 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.4 </span>CEPs and optimal steering</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S8" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8 </span>Conclusions and Future Research</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A1" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>An anytime Azuma-Hoeffding bound</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Details omitted from <span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">7</span></span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2.SS1" title="In Appendix B Details omitted from Section 7 ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.1 </span>Completed proof of <span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">7.4</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2.SS2" title="In Appendix B Details omitted from Section 7 ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.2 </span>Revelation principle for CEPs</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A3" title="In Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Other omitted proofs</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A3.SS1" title="In Appendix C Other omitted proofs ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.1 </span><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">2.1</span></span></a></li> </ol> </li> </ol></nav> <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">Most literature on game theory considers the problem of understanding the behavior of agents in a game given the preferences of the agents. That is, <span class="ltx_text ltx_font_italic" id="S1.p1.1.1">knowing</span> what the agents want, how will they act? In this paper, we consider the inverse of this problem: if all we can observe is how agents act, can we infer what they want, that is, their utility functions? The problem of infering agents’ utility functions from their behavior is known as “learning from revealed preferences” (e.g., <cite class="ltx_cite ltx_citemacro_citet">Beigman and Vohra (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib4" title="">2006</a>); Zadimoghaddam and Roth (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib23" title="">2012</a>)</cite>) or “inverse game theory” (e.g., <cite class="ltx_cite ltx_citemacro_citet">Kuleshov and Schrijvers (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib15" title="">2015</a>)</cite>) in the literature. Despite a long history, these two lines of work have some limitations. First, they often assume that the observed behavior of the agents is <em class="ltx_emph ltx_font_italic" id="S1.p1.1.2">(Nash) equilibrium behavior</em>. This is arguably unrealistic, especially because Nash equilibria are <span class="ltx_ERROR undefined" id="S1.p1.1.3">\PPAD</span>-hard to compute <cite class="ltx_cite ltx_citemacro_cite">Daskalakis et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib8" title="">2006</a>); Chen et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib7" title="">2009</a>)</cite>. Second, they mostly focus on <em class="ltx_emph ltx_font_italic" id="S1.p1.1.4">static</em> problems, aiming to learn agents’ utility functions from a fixed set of behavior data. Often, this creates trivial impossibility results, stemming from the fact that the behavioral data available is simply not enough to determine the utility functions.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">In this paper, we consider a <span class="ltx_text ltx_font_italic" id="S1.p2.1.1">dynamic, non-equilibrium</span> model in which agents play the game repeatedly over many rounds, and their behavior may not correspond to any equilibrium. A principal can observe the actions of the agents in each round, and seeks to learn the utilities from only those observations. The principal is empowered to give <span class="ltx_text ltx_font_italic" id="S1.p2.1.2">payments</span> to the agents as a function of their play, as well as <span class="ltx_text ltx_font_italic" id="S1.p2.1.3">signals</span>, in the spirit of correlated equilibria <cite class="ltx_cite ltx_citemacro_cite">Aumann (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib1" title="">1974</a>)</cite>. The signaling scheme and payment schemes can change from round to round, depending on the players’ past actions. We do not assume that the agents will play equilibria. Instead, we consider two permissive and popular behavior models for the agents: 1) they can play any iteratively undominated (rationalizable) actions or 2) they use no-regret learning algorithms to pick actions. Under this setup, we ask:</p> <blockquote class="ltx_quote" id="S1.p2.2"> <p class="ltx_p" id="S1.p2.2.1"><span class="ltx_text ltx_font_italic" id="S1.p2.2.1.1">Can a principal completely learn the utility functions of the agents?</span></p> </blockquote> <p class="ltx_p" id="S1.p2.3">We will give a positive answer to this question, under both behavioral models. Then, we will apply our algorithm to the problem of <span class="ltx_text ltx_font_italic" id="S1.p2.3.1">steering</span> no-regret learning agents toward desirable outcomes, introduced by <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite>. Building on their results, we will show that it is possible to steer agents to optimal outcomes even without prior knowledge of their utilities, and give an exact characterization of what these optimal outcomes are.</p> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Overview of our results</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.3">Here, we summarize our results. There is a fixed normal-form game <math alttext="\Gamma" class="ltx_Math" display="inline" id="S1.SS1.p1.1.m1.1"><semantics id="S1.SS1.p1.1.m1.1a"><mi id="S1.SS1.p1.1.m1.1.1" mathvariant="normal" xref="S1.SS1.p1.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.1.m1.1b"><ci id="S1.SS1.p1.1.m1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.1d">roman_Γ</annotation></semantics></math>, which the agents play repeatedly over rounds <math alttext="t=1,\dots,T" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.3"><semantics id="S1.SS1.p1.2.m2.3a"><mrow id="S1.SS1.p1.2.m2.3.4" xref="S1.SS1.p1.2.m2.3.4.cmml"><mi id="S1.SS1.p1.2.m2.3.4.2" xref="S1.SS1.p1.2.m2.3.4.2.cmml">t</mi><mo id="S1.SS1.p1.2.m2.3.4.1" xref="S1.SS1.p1.2.m2.3.4.1.cmml">=</mo><mrow id="S1.SS1.p1.2.m2.3.4.3.2" xref="S1.SS1.p1.2.m2.3.4.3.1.cmml"><mn id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml">1</mn><mo id="S1.SS1.p1.2.m2.3.4.3.2.1" xref="S1.SS1.p1.2.m2.3.4.3.1.cmml">,</mo><mi id="S1.SS1.p1.2.m2.2.2" mathvariant="normal" xref="S1.SS1.p1.2.m2.2.2.cmml">…</mi><mo id="S1.SS1.p1.2.m2.3.4.3.2.2" xref="S1.SS1.p1.2.m2.3.4.3.1.cmml">,</mo><mi id="S1.SS1.p1.2.m2.3.3" xref="S1.SS1.p1.2.m2.3.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.3b"><apply id="S1.SS1.p1.2.m2.3.4.cmml" xref="S1.SS1.p1.2.m2.3.4"><eq id="S1.SS1.p1.2.m2.3.4.1.cmml" xref="S1.SS1.p1.2.m2.3.4.1"></eq><ci id="S1.SS1.p1.2.m2.3.4.2.cmml" xref="S1.SS1.p1.2.m2.3.4.2">𝑡</ci><list id="S1.SS1.p1.2.m2.3.4.3.1.cmml" xref="S1.SS1.p1.2.m2.3.4.3.2"><cn id="S1.SS1.p1.2.m2.1.1.cmml" type="integer" xref="S1.SS1.p1.2.m2.1.1">1</cn><ci id="S1.SS1.p1.2.m2.2.2.cmml" xref="S1.SS1.p1.2.m2.2.2">…</ci><ci id="S1.SS1.p1.2.m2.3.3.cmml" xref="S1.SS1.p1.2.m2.3.3">𝑇</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.3c">t=1,\dots,T</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.3d">italic_t = 1 , … , italic_T</annotation></semantics></math>. A principal attempts to learn the utility functions of all the agents in the game to a given target precision <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S1.SS1.p1.3.m3.1"><semantics id="S1.SS1.p1.3.m3.1a"><mrow id="S1.SS1.p1.3.m3.1.1" xref="S1.SS1.p1.3.m3.1.1.cmml"><mi id="S1.SS1.p1.3.m3.1.1.2" xref="S1.SS1.p1.3.m3.1.1.2.cmml">ε</mi><mo id="S1.SS1.p1.3.m3.1.1.1" xref="S1.SS1.p1.3.m3.1.1.1.cmml">></mo><mn id="S1.SS1.p1.3.m3.1.1.3" xref="S1.SS1.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.3.m3.1b"><apply id="S1.SS1.p1.3.m3.1.1.cmml" xref="S1.SS1.p1.3.m3.1.1"><gt id="S1.SS1.p1.3.m3.1.1.1.cmml" xref="S1.SS1.p1.3.m3.1.1.1"></gt><ci id="S1.SS1.p1.3.m3.1.1.2.cmml" xref="S1.SS1.p1.3.m3.1.1.2">𝜀</ci><cn id="S1.SS1.p1.3.m3.1.1.3.cmml" type="integer" xref="S1.SS1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.3.m3.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.3.m3.1d">italic_ε > 0</annotation></semantics></math>. The principal initially knows nothing about the utilities, and can only infer information about them by watching agents play the game. In order to influence the agents, the principal can give non-negative <span class="ltx_text ltx_font_italic" id="S1.SS1.p1.3.1">payments</span> to the agents, that get added to the players’ utilities.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.1">Learning the utility completely is impossible, because agents’ incentives only depend on <span class="ltx_text ltx_font_italic" id="S1.SS1.p2.1.1">relative</span> utilities between their actions, not <span class="ltx_text ltx_font_italic" id="S1.SS1.p2.1.2">absolute</span> utilities. Thus, we only demand that the principal output utility functions that yield a <span class="ltx_text ltx_font_italic" id="S1.SS1.p2.1.3">strategically-equivalent</span> game, that is, one in which all agents’ relative utilities are identical to those in the true game. Equivalently, we identify each agent’s utility function up to a term that does not depend on that agent’s action.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.4">Under both rationalizable and no-regret learning behavioral models, we give both upper- and lower-bounds on the number of rounds the principal must take in order to learn the game. In what follows, <math alttext="n" class="ltx_Math" display="inline" id="S1.SS1.p3.1.m1.1"><semantics id="S1.SS1.p3.1.m1.1a"><mi id="S1.SS1.p3.1.m1.1.1" xref="S1.SS1.p3.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.1.m1.1b"><ci id="S1.SS1.p3.1.m1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.1.m1.1d">italic_n</annotation></semantics></math> is the number of agents, <math alttext="m_{i}" class="ltx_Math" display="inline" id="S1.SS1.p3.2.m2.1"><semantics id="S1.SS1.p3.2.m2.1a"><msub id="S1.SS1.p3.2.m2.1.1" xref="S1.SS1.p3.2.m2.1.1.cmml"><mi id="S1.SS1.p3.2.m2.1.1.2" xref="S1.SS1.p3.2.m2.1.1.2.cmml">m</mi><mi id="S1.SS1.p3.2.m2.1.1.3" xref="S1.SS1.p3.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.2.m2.1b"><apply id="S1.SS1.p3.2.m2.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.2.m2.1.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1">subscript</csymbol><ci id="S1.SS1.p3.2.m2.1.1.2.cmml" xref="S1.SS1.p3.2.m2.1.1.2">𝑚</ci><ci id="S1.SS1.p3.2.m2.1.1.3.cmml" xref="S1.SS1.p3.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.2.m2.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.2.m2.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the number of actions of each agent <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p3.3.m3.1"><semantics id="S1.SS1.p3.3.m3.1a"><mi id="S1.SS1.p3.3.m3.1.1" xref="S1.SS1.p3.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.3.m3.1b"><ci id="S1.SS1.p3.3.m3.1.1.cmml" xref="S1.SS1.p3.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.3.m3.1d">italic_i</annotation></semantics></math>, and <math alttext="M=\prod_{i}m_{i}" class="ltx_Math" display="inline" id="S1.SS1.p3.4.m4.1"><semantics id="S1.SS1.p3.4.m4.1a"><mrow id="S1.SS1.p3.4.m4.1.1" xref="S1.SS1.p3.4.m4.1.1.cmml"><mi id="S1.SS1.p3.4.m4.1.1.2" xref="S1.SS1.p3.4.m4.1.1.2.cmml">M</mi><mo id="S1.SS1.p3.4.m4.1.1.1" rspace="0.111em" xref="S1.SS1.p3.4.m4.1.1.1.cmml">=</mo><mrow id="S1.SS1.p3.4.m4.1.1.3" xref="S1.SS1.p3.4.m4.1.1.3.cmml"><msub id="S1.SS1.p3.4.m4.1.1.3.1" xref="S1.SS1.p3.4.m4.1.1.3.1.cmml"><mo id="S1.SS1.p3.4.m4.1.1.3.1.2" xref="S1.SS1.p3.4.m4.1.1.3.1.2.cmml">∏</mo><mi id="S1.SS1.p3.4.m4.1.1.3.1.3" xref="S1.SS1.p3.4.m4.1.1.3.1.3.cmml">i</mi></msub><msub id="S1.SS1.p3.4.m4.1.1.3.2" xref="S1.SS1.p3.4.m4.1.1.3.2.cmml"><mi id="S1.SS1.p3.4.m4.1.1.3.2.2" xref="S1.SS1.p3.4.m4.1.1.3.2.2.cmml">m</mi><mi id="S1.SS1.p3.4.m4.1.1.3.2.3" xref="S1.SS1.p3.4.m4.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.4.m4.1b"><apply id="S1.SS1.p3.4.m4.1.1.cmml" xref="S1.SS1.p3.4.m4.1.1"><eq id="S1.SS1.p3.4.m4.1.1.1.cmml" xref="S1.SS1.p3.4.m4.1.1.1"></eq><ci id="S1.SS1.p3.4.m4.1.1.2.cmml" xref="S1.SS1.p3.4.m4.1.1.2">𝑀</ci><apply id="S1.SS1.p3.4.m4.1.1.3.cmml" xref="S1.SS1.p3.4.m4.1.1.3"><apply id="S1.SS1.p3.4.m4.1.1.3.1.cmml" xref="S1.SS1.p3.4.m4.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.p3.4.m4.1.1.3.1.1.cmml" xref="S1.SS1.p3.4.m4.1.1.3.1">subscript</csymbol><csymbol cd="latexml" id="S1.SS1.p3.4.m4.1.1.3.1.2.cmml" xref="S1.SS1.p3.4.m4.1.1.3.1.2">product</csymbol><ci id="S1.SS1.p3.4.m4.1.1.3.1.3.cmml" xref="S1.SS1.p3.4.m4.1.1.3.1.3">𝑖</ci></apply><apply id="S1.SS1.p3.4.m4.1.1.3.2.cmml" xref="S1.SS1.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS1.p3.4.m4.1.1.3.2.1.cmml" xref="S1.SS1.p3.4.m4.1.1.3.2">subscript</csymbol><ci id="S1.SS1.p3.4.m4.1.1.3.2.2.cmml" xref="S1.SS1.p3.4.m4.1.1.3.2.2">𝑚</ci><ci id="S1.SS1.p3.4.m4.1.1.3.2.3.cmml" xref="S1.SS1.p3.4.m4.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.4.m4.1c">M=\prod_{i}m_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.4.m4.1d">italic_M = ∏ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the total number of action profiles.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem1.1.1.1">Theorem 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2"> </span>(Informal summary of results in the rationalizable model)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.3"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem1.p1.3.3">In the rationalizable model, there exists an algorithm for the principal that can learn a game to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.1.m1.1"><semantics id="S1.Thmtheorem1.p1.1.1.m1.1a"><mi id="S1.Thmtheorem1.p1.1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.1.m1.1b"><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.1.m1.1d">italic_ε</annotation></semantics></math> in <math alttext="{\mathcal{O}}(nM\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.2.m2.3"><semantics id="S1.Thmtheorem1.p1.2.2.m2.3a"><mrow id="S1.Thmtheorem1.p1.2.2.m2.3.3" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.2.2.m2.3.3.3" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.3.cmml">𝒪</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.3.3.2" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.2.cmml">⁢</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.cmml"><mo id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.2" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.cmml"><mi id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.2" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.2.cmml">n</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.3" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.3.cmml">M</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.1a" lspace="0.167em" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.2.2.4" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.3.cmml"><mi id="S1.Thmtheorem1.p1.2.2.m2.2.2.2.2" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.3.1.cmml">log</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.2.2.4a" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.3.1.cmml">⁡</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.2.2.4.1" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.3.cmml"><mo id="S1.Thmtheorem1.p1.2.2.m2.2.2.4.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.3.1.cmml">(</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.cmml"><mn id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2.cmml">1</mn><mo id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml">/</mo><mi id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.Thmtheorem1.p1.2.2.m2.2.2.4.1.2" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.3" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.2.m2.3b"><apply id="S1.Thmtheorem1.p1.2.2.m2.3.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3"><times id="S1.Thmtheorem1.p1.2.2.m2.3.3.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.2"></times><ci id="S1.Thmtheorem1.p1.2.2.m2.3.3.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.3">𝒪</ci><apply id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1"><times id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.1"></times><ci id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.2">𝑛</ci><ci id="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.1.1.1.3">𝑀</ci><apply id="S1.Thmtheorem1.p1.2.2.m2.2.2.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.4"><log id="S1.Thmtheorem1.p1.2.2.m2.2.2.3.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.2.2"></log><apply id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1"><divide id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1"></divide><cn id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2">1</cn><ci id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.2.m2.3c">{\mathcal{O}}(nM\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.2.m2.3d">caligraphic_O ( italic_n italic_M roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> rounds. This is tight up to <math alttext="\log(M)" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.3.3.m3.2"><semantics id="S1.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S1.Thmtheorem1.p1.3.3.m3.2.2.4" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.3.cmml"><mi id="S1.Thmtheorem1.p1.3.3.m3.2.2.2.2" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.3.1.cmml">log</mi><mo id="S1.Thmtheorem1.p1.3.3.m3.2.2.4a" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.3.1.cmml">⁡</mo><mrow id="S1.Thmtheorem1.p1.3.3.m3.2.2.4.1" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.3.cmml"><mo id="S1.Thmtheorem1.p1.3.3.m3.2.2.4.1.1" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.3.1.cmml">(</mo><mi id="S1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="S1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml">M</mi><mo id="S1.Thmtheorem1.p1.3.3.m3.2.2.4.1.2" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.3.3.m3.2b"><apply id="S1.Thmtheorem1.p1.3.3.m3.2.2.3.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.4"><log id="S1.Thmtheorem1.p1.3.3.m3.2.2.3.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.2.2"></log><ci id="S1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.3.3.m3.2c">\log(M)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.3.3.m3.2d">roman_log ( start_ARG italic_M end_ARG )</annotation></semantics></math> factors.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.8">Intuitively, the algorithm works by running a binary search for each agent <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p4.1.m1.1"><semantics id="S1.SS1.p4.1.m1.1a"><mi id="S1.SS1.p4.1.m1.1.1" xref="S1.SS1.p4.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.1.m1.1b"><ci id="S1.SS1.p4.1.m1.1.1.cmml" xref="S1.SS1.p4.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.1.m1.1d">italic_i</annotation></semantics></math> and each action profile <math alttext="a=(a_{1},\dots,a_{n})" class="ltx_Math" display="inline" id="S1.SS1.p4.2.m2.3"><semantics id="S1.SS1.p4.2.m2.3a"><mrow id="S1.SS1.p4.2.m2.3.3" xref="S1.SS1.p4.2.m2.3.3.cmml"><mi id="S1.SS1.p4.2.m2.3.3.4" xref="S1.SS1.p4.2.m2.3.3.4.cmml">a</mi><mo id="S1.SS1.p4.2.m2.3.3.3" xref="S1.SS1.p4.2.m2.3.3.3.cmml">=</mo><mrow id="S1.SS1.p4.2.m2.3.3.2.2" xref="S1.SS1.p4.2.m2.3.3.2.3.cmml"><mo id="S1.SS1.p4.2.m2.3.3.2.2.3" stretchy="false" xref="S1.SS1.p4.2.m2.3.3.2.3.cmml">(</mo><msub id="S1.SS1.p4.2.m2.2.2.1.1.1" xref="S1.SS1.p4.2.m2.2.2.1.1.1.cmml"><mi id="S1.SS1.p4.2.m2.2.2.1.1.1.2" xref="S1.SS1.p4.2.m2.2.2.1.1.1.2.cmml">a</mi><mn id="S1.SS1.p4.2.m2.2.2.1.1.1.3" xref="S1.SS1.p4.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS1.p4.2.m2.3.3.2.2.4" xref="S1.SS1.p4.2.m2.3.3.2.3.cmml">,</mo><mi 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xref="S1.SS1.p4.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S1.SS1.p4.2.m2.2.2.1.1.1.2.cmml" xref="S1.SS1.p4.2.m2.2.2.1.1.1.2">𝑎</ci><cn id="S1.SS1.p4.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S1.SS1.p4.2.m2.2.2.1.1.1.3">1</cn></apply><ci id="S1.SS1.p4.2.m2.1.1.cmml" xref="S1.SS1.p4.2.m2.1.1">…</ci><apply id="S1.SS1.p4.2.m2.3.3.2.2.2.cmml" xref="S1.SS1.p4.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.SS1.p4.2.m2.3.3.2.2.2.1.cmml" xref="S1.SS1.p4.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S1.SS1.p4.2.m2.3.3.2.2.2.2.cmml" xref="S1.SS1.p4.2.m2.3.3.2.2.2.2">𝑎</ci><ci id="S1.SS1.p4.2.m2.3.3.2.2.2.3.cmml" xref="S1.SS1.p4.2.m2.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.2.m2.3c">a=(a_{1},\dots,a_{n})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.2.m2.3d">italic_a = ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>, to identify the smallest possible payment that agent <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p4.3.m3.1"><semantics id="S1.SS1.p4.3.m3.1a"><mi id="S1.SS1.p4.3.m3.1.1" xref="S1.SS1.p4.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.3.m3.1b"><ci id="S1.SS1.p4.3.m3.1.1.cmml" xref="S1.SS1.p4.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.3.m3.1d">italic_i</annotation></semantics></math> needs to be motivated to play <math alttext="a_{i}" class="ltx_Math" display="inline" id="S1.SS1.p4.4.m4.1"><semantics id="S1.SS1.p4.4.m4.1a"><msub id="S1.SS1.p4.4.m4.1.1" xref="S1.SS1.p4.4.m4.1.1.cmml"><mi id="S1.SS1.p4.4.m4.1.1.2" xref="S1.SS1.p4.4.m4.1.1.2.cmml">a</mi><mi id="S1.SS1.p4.4.m4.1.1.3" xref="S1.SS1.p4.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.4.m4.1b"><apply id="S1.SS1.p4.4.m4.1.1.cmml" xref="S1.SS1.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS1.p4.4.m4.1.1.1.cmml" xref="S1.SS1.p4.4.m4.1.1">subscript</csymbol><ci id="S1.SS1.p4.4.m4.1.1.2.cmml" xref="S1.SS1.p4.4.m4.1.1.2">𝑎</ci><ci id="S1.SS1.p4.4.m4.1.1.3.cmml" xref="S1.SS1.p4.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.4.m4.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. The lower bound is information-theoretic: learning a game to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS1.p4.5.m5.1"><semantics id="S1.SS1.p4.5.m5.1a"><mi id="S1.SS1.p4.5.m5.1.1" xref="S1.SS1.p4.5.m5.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.5.m5.1b"><ci id="S1.SS1.p4.5.m5.1.1.cmml" xref="S1.SS1.p4.5.m5.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.5.m5.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.5.m5.1d">italic_ε</annotation></semantics></math> requires learning <math alttext="nM\log_{2}(1/\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.p4.6.m6.2"><semantics id="S1.SS1.p4.6.m6.2a"><mrow id="S1.SS1.p4.6.m6.2.2" xref="S1.SS1.p4.6.m6.2.2.cmml"><mi id="S1.SS1.p4.6.m6.2.2.4" xref="S1.SS1.p4.6.m6.2.2.4.cmml">n</mi><mo id="S1.SS1.p4.6.m6.2.2.3" xref="S1.SS1.p4.6.m6.2.2.3.cmml">⁢</mo><mi id="S1.SS1.p4.6.m6.2.2.5" xref="S1.SS1.p4.6.m6.2.2.5.cmml">M</mi><mo id="S1.SS1.p4.6.m6.2.2.3a" lspace="0.167em" xref="S1.SS1.p4.6.m6.2.2.3.cmml">⁢</mo><mrow id="S1.SS1.p4.6.m6.2.2.2.2" xref="S1.SS1.p4.6.m6.2.2.2.3.cmml"><msub id="S1.SS1.p4.6.m6.1.1.1.1.1" xref="S1.SS1.p4.6.m6.1.1.1.1.1.cmml"><mi id="S1.SS1.p4.6.m6.1.1.1.1.1.2" xref="S1.SS1.p4.6.m6.1.1.1.1.1.2.cmml">log</mi><mn id="S1.SS1.p4.6.m6.1.1.1.1.1.3" xref="S1.SS1.p4.6.m6.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S1.SS1.p4.6.m6.2.2.2.2a" xref="S1.SS1.p4.6.m6.2.2.2.3.cmml">⁡</mo><mrow id="S1.SS1.p4.6.m6.2.2.2.2.2" xref="S1.SS1.p4.6.m6.2.2.2.3.cmml"><mo id="S1.SS1.p4.6.m6.2.2.2.2.2.2" stretchy="false" xref="S1.SS1.p4.6.m6.2.2.2.3.cmml">(</mo><mrow id="S1.SS1.p4.6.m6.2.2.2.2.2.1" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1.cmml"><mn id="S1.SS1.p4.6.m6.2.2.2.2.2.1.2" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1.2.cmml">1</mn><mo id="S1.SS1.p4.6.m6.2.2.2.2.2.1.1" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1.1.cmml">/</mo><mi id="S1.SS1.p4.6.m6.2.2.2.2.2.1.3" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.p4.6.m6.2.2.2.2.2.3" stretchy="false" xref="S1.SS1.p4.6.m6.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.6.m6.2b"><apply id="S1.SS1.p4.6.m6.2.2.cmml" xref="S1.SS1.p4.6.m6.2.2"><times id="S1.SS1.p4.6.m6.2.2.3.cmml" xref="S1.SS1.p4.6.m6.2.2.3"></times><ci id="S1.SS1.p4.6.m6.2.2.4.cmml" xref="S1.SS1.p4.6.m6.2.2.4">𝑛</ci><ci id="S1.SS1.p4.6.m6.2.2.5.cmml" xref="S1.SS1.p4.6.m6.2.2.5">𝑀</ci><apply id="S1.SS1.p4.6.m6.2.2.2.3.cmml" xref="S1.SS1.p4.6.m6.2.2.2.2"><apply id="S1.SS1.p4.6.m6.1.1.1.1.1.cmml" xref="S1.SS1.p4.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p4.6.m6.1.1.1.1.1.1.cmml" xref="S1.SS1.p4.6.m6.1.1.1.1.1">subscript</csymbol><log id="S1.SS1.p4.6.m6.1.1.1.1.1.2.cmml" xref="S1.SS1.p4.6.m6.1.1.1.1.1.2"></log><cn id="S1.SS1.p4.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS1.p4.6.m6.1.1.1.1.1.3">2</cn></apply><apply id="S1.SS1.p4.6.m6.2.2.2.2.2.1.cmml" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1"><divide id="S1.SS1.p4.6.m6.2.2.2.2.2.1.1.cmml" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1.1"></divide><cn id="S1.SS1.p4.6.m6.2.2.2.2.2.1.2.cmml" type="integer" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1.2">1</cn><ci id="S1.SS1.p4.6.m6.2.2.2.2.2.1.3.cmml" xref="S1.SS1.p4.6.m6.2.2.2.2.2.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.6.m6.2c">nM\log_{2}(1/\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.6.m6.2d">italic_n italic_M roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 / italic_ε )</annotation></semantics></math> bits of information, but on each round, the principal only observes a single action profile out of a possible <math alttext="M" class="ltx_Math" display="inline" id="S1.SS1.p4.7.m7.1"><semantics id="S1.SS1.p4.7.m7.1a"><mi id="S1.SS1.p4.7.m7.1.1" xref="S1.SS1.p4.7.m7.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.7.m7.1b"><ci id="S1.SS1.p4.7.m7.1.1.cmml" xref="S1.SS1.p4.7.m7.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.7.m7.1c">M</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.7.m7.1d">italic_M</annotation></semantics></math> action profiles, which conveys only <math alttext="\log(M)" class="ltx_Math" display="inline" id="S1.SS1.p4.8.m8.2"><semantics id="S1.SS1.p4.8.m8.2a"><mrow id="S1.SS1.p4.8.m8.2.2.4" xref="S1.SS1.p4.8.m8.2.2.3.cmml"><mi id="S1.SS1.p4.8.m8.2.2.2.2" xref="S1.SS1.p4.8.m8.2.2.3.1.cmml">log</mi><mo id="S1.SS1.p4.8.m8.2.2.4a" xref="S1.SS1.p4.8.m8.2.2.3.1.cmml">⁡</mo><mrow id="S1.SS1.p4.8.m8.2.2.4.1" xref="S1.SS1.p4.8.m8.2.2.3.cmml"><mo id="S1.SS1.p4.8.m8.2.2.4.1.1" xref="S1.SS1.p4.8.m8.2.2.3.1.cmml">(</mo><mi id="S1.SS1.p4.8.m8.1.1.1.1.1" xref="S1.SS1.p4.8.m8.1.1.1.1.1.cmml">M</mi><mo id="S1.SS1.p4.8.m8.2.2.4.1.2" xref="S1.SS1.p4.8.m8.2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.8.m8.2b"><apply id="S1.SS1.p4.8.m8.2.2.3.cmml" xref="S1.SS1.p4.8.m8.2.2.4"><log id="S1.SS1.p4.8.m8.2.2.3.1.cmml" xref="S1.SS1.p4.8.m8.2.2.2.2"></log><ci id="S1.SS1.p4.8.m8.1.1.1.1.1.cmml" xref="S1.SS1.p4.8.m8.1.1.1.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.8.m8.2c">\log(M)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.8.m8.2d">roman_log ( start_ARG italic_M end_ARG )</annotation></semantics></math> bits of information.</p> </div> <div class="ltx_para" id="S1.SS1.p5"> <p class="ltx_p" id="S1.SS1.p5.1">In the no-regret model, we derive an analogous result.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem2.1.1.1">Theorem 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2"> </span>(Informal summary of results in the no-regret model)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.3.3">In the no-regret model, there exists an algorithm for the principal that can learn a game to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.1.m1.1"><semantics id="S1.Thmtheorem2.p1.1.1.m1.1a"><mi id="S1.Thmtheorem2.p1.1.1.m1.1.1" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.1.1.m1.1b"><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.1.1.m1.1d">italic_ε</annotation></semantics></math> in <math alttext="M^{{\mathcal{O}}(1)}/\varepsilon^{2}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.2.2.m2.1"><semantics id="S1.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem2.p1.2.2.m2.1.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.cmml"><msup id="S1.Thmtheorem2.p1.2.2.m2.1.2.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.2.cmml"><mi id="S1.Thmtheorem2.p1.2.2.m2.1.2.2.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.2.2.cmml">M</mi><mrow id="S1.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.3" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.3.cmml">𝒪</mi><mo id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.4.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml"><mo id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.4.2.1" stretchy="false" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">(</mo><mn id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.1.cmml">1</mn><mo id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.4.2.2" stretchy="false" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S1.Thmtheorem2.p1.2.2.m2.1.2.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.1.cmml">/</mo><msup id="S1.Thmtheorem2.p1.2.2.m2.1.2.3" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.3.cmml"><mi id="S1.Thmtheorem2.p1.2.2.m2.1.2.3.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.3.2.cmml">ε</mi><mn id="S1.Thmtheorem2.p1.2.2.m2.1.2.3.3" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.2.2.m2.1b"><apply id="S1.Thmtheorem2.p1.2.2.m2.1.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2"><divide id="S1.Thmtheorem2.p1.2.2.m2.1.2.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.1"></divide><apply id="S1.Thmtheorem2.p1.2.2.m2.1.2.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.2.2.m2.1.2.2.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.2">superscript</csymbol><ci id="S1.Thmtheorem2.p1.2.2.m2.1.2.2.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.2.2">𝑀</ci><apply id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1"><times id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.2"></times><ci id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.3">𝒪</ci><cn id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.1.cmml" type="integer" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.1.1">1</cn></apply></apply><apply id="S1.Thmtheorem2.p1.2.2.m2.1.2.3.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.2.2.m2.1.2.3.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.3">superscript</csymbol><ci id="S1.Thmtheorem2.p1.2.2.m2.1.2.3.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.3.2">𝜀</ci><cn id="S1.Thmtheorem2.p1.2.2.m2.1.2.3.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.2.2.m2.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.2.2.m2.1c">M^{{\mathcal{O}}(1)}/\varepsilon^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.2.2.m2.1d">italic_M start_POSTSUPERSCRIPT caligraphic_O ( 1 ) end_POSTSUPERSCRIPT / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> rounds. This is tight up to the exponent hidden by the <math alttext="{\mathcal{O}}(1)" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.3.3.m3.1"><semantics id="S1.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.3.3.m3.1.2.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.2.cmml">𝒪</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.2.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.2.3.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml"><mo id="S1.Thmtheorem2.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml">(</mo><mn id="S1.Thmtheorem2.p1.3.3.m3.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml">1</mn><mo id="S1.Thmtheorem2.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.3.3.m3.1b"><apply id="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2"><times id="S1.Thmtheorem2.p1.3.3.m3.1.2.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.1"></times><ci id="S1.Thmtheorem2.p1.3.3.m3.1.2.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.2">𝒪</ci><cn id="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml" type="integer" xref="S1.Thmtheorem2.p1.3.3.m3.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.3.3.m3.1c">{\mathcal{O}}(1)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.3.3.m3.1d">caligraphic_O ( 1 )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p6"> <p class="ltx_p" id="S1.SS1.p6.1">Critically, the no-regret algorithms, at least in the multi-agent case, require <span class="ltx_text ltx_font_italic" id="S1.SS1.p6.1.1">signaling</span>: the principal must have the power not only to pay the agents but also to send them signals from a finite signal set <math alttext="S_{i}" class="ltx_Math" display="inline" id="S1.SS1.p6.1.m1.1"><semantics id="S1.SS1.p6.1.m1.1a"><msub id="S1.SS1.p6.1.m1.1.1" xref="S1.SS1.p6.1.m1.1.1.cmml"><mi id="S1.SS1.p6.1.m1.1.1.2" xref="S1.SS1.p6.1.m1.1.1.2.cmml">S</mi><mi id="S1.SS1.p6.1.m1.1.1.3" xref="S1.SS1.p6.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.1.m1.1b"><apply id="S1.SS1.p6.1.m1.1.1.cmml" xref="S1.SS1.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p6.1.m1.1.1.1.cmml" xref="S1.SS1.p6.1.m1.1.1">subscript</csymbol><ci id="S1.SS1.p6.1.m1.1.1.2.cmml" xref="S1.SS1.p6.1.m1.1.1.2">𝑆</ci><ci id="S1.SS1.p6.1.m1.1.1.3.cmml" xref="S1.SS1.p6.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.1.m1.1c">S_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.1.m1.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in each round. However, our main motivation, which is to <span class="ltx_text ltx_font_italic" id="S1.SS1.p6.1.2">steer</span> agents toward desirable outcomes without prior knowledge of utilities, already requires signaling.<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>This was shown already by <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite>.</span></span></span> As such, we consider the use of signaling to learn utilities in the no-regret model to be reasonable.</p> </div> <div class="ltx_para" id="S1.SS1.p7"> <p class="ltx_p" id="S1.SS1.p7.1">We also consider the problem of minimizing the <span class="ltx_text ltx_font_italic" id="S1.SS1.p7.1.1">total payment</span> instead of the number of rounds. In the single-agent rationalizable setting, we show the following result, which implies that minimizing payments is not the same as minimizing rounds:</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S1.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="S1.Thmtheorem3.1.1.1">Proposition 1.3</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.2.2"> </span>(Informal version of <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S6.Thmtheorem1" title="Proposition 6.1. ‣ Single rational agent ‣ 6 Minimizing Payment ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">6.1</span></a>)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem3.p1"> <p class="ltx_p" id="S1.Thmtheorem3.p1.3"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem3.p1.3.3">In a single-agent game with <math alttext="m" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.1.1.m1.1"><semantics id="S1.Thmtheorem3.p1.1.1.m1.1a"><mi id="S1.Thmtheorem3.p1.1.1.m1.1.1" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.1.1.m1.1b"><ci id="S1.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.1.1.m1.1d">italic_m</annotation></semantics></math> actions, there exists an algorithm in the rationalizable model that learns a game to any precision <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.2.2.m2.1"><semantics id="S1.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem3.p1.2.2.m2.1.1" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S1.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S1.Thmtheorem3.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S1.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.2.2.m2.1b"><apply id="S1.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1"><gt id="S1.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.1"></gt><ci id="S1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math> using total payment bounded by only a game-dependent constant (and not <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.3.3.m3.1"><semantics id="S1.Thmtheorem3.p1.3.3.m3.1a"><mi id="S1.Thmtheorem3.p1.3.3.m3.1.1" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.3.3.m3.1b"><ci id="S1.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.3.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.3.3.m3.1d">italic_ε</annotation></semantics></math>).</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p8"> <p class="ltx_p" id="S1.SS1.p8.1">The above algorithm takes <math alttext="O(m/\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.p8.1.m1.1"><semantics id="S1.SS1.p8.1.m1.1a"><mrow id="S1.SS1.p8.1.m1.1.1" xref="S1.SS1.p8.1.m1.1.1.cmml"><mi id="S1.SS1.p8.1.m1.1.1.3" xref="S1.SS1.p8.1.m1.1.1.3.cmml">O</mi><mo id="S1.SS1.p8.1.m1.1.1.2" xref="S1.SS1.p8.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p8.1.m1.1.1.1.1" xref="S1.SS1.p8.1.m1.1.1.1.1.1.cmml"><mo id="S1.SS1.p8.1.m1.1.1.1.1.2" stretchy="false" xref="S1.SS1.p8.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.p8.1.m1.1.1.1.1.1" xref="S1.SS1.p8.1.m1.1.1.1.1.1.cmml"><mi id="S1.SS1.p8.1.m1.1.1.1.1.1.2" xref="S1.SS1.p8.1.m1.1.1.1.1.1.2.cmml">m</mi><mo id="S1.SS1.p8.1.m1.1.1.1.1.1.1" xref="S1.SS1.p8.1.m1.1.1.1.1.1.1.cmml">/</mo><mi id="S1.SS1.p8.1.m1.1.1.1.1.1.3" xref="S1.SS1.p8.1.m1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.p8.1.m1.1.1.1.1.3" stretchy="false" xref="S1.SS1.p8.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.1.m1.1b"><apply id="S1.SS1.p8.1.m1.1.1.cmml" xref="S1.SS1.p8.1.m1.1.1"><times id="S1.SS1.p8.1.m1.1.1.2.cmml" xref="S1.SS1.p8.1.m1.1.1.2"></times><ci id="S1.SS1.p8.1.m1.1.1.3.cmml" xref="S1.SS1.p8.1.m1.1.1.3">𝑂</ci><apply id="S1.SS1.p8.1.m1.1.1.1.1.1.cmml" xref="S1.SS1.p8.1.m1.1.1.1.1"><divide id="S1.SS1.p8.1.m1.1.1.1.1.1.1.cmml" xref="S1.SS1.p8.1.m1.1.1.1.1.1.1"></divide><ci id="S1.SS1.p8.1.m1.1.1.1.1.1.2.cmml" xref="S1.SS1.p8.1.m1.1.1.1.1.1.2">𝑚</ci><ci id="S1.SS1.p8.1.m1.1.1.1.1.1.3.cmml" xref="S1.SS1.p8.1.m1.1.1.1.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.1.m1.1c">O(m/\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.1.m1.1d">italic_O ( italic_m / italic_ε )</annotation></semantics></math> rounds, thereby making it worse in terms of number of rounds than the algorithm in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S1.Thmtheorem1" title="Theorem 1.1 (Informal summary of results in the rationalizable model). ‣ 1.1 Overview of our results ‣ 1 Introduction ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.1</span></a>. However, on most rounds, no payment will be given, so the total payment can be bounded. Intuitively, the algorithm works by slowly increasing the payment on each action until the agent switches actions.</p> </div> <div class="ltx_para" id="S1.SS1.p9"> <p class="ltx_p" id="S1.SS1.p9.1">However, with multiple agents, we show that minimizing payment is essentially equivalent to minimizing rounds. In particular, we have the following informal result.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S1.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="S1.Thmtheorem4.1.1.1">Proposition 1.4</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.2.2"> </span>(Informal version of <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S6.Thmtheorem2" title="Proposition 6.2. ‣ Single rational agent ‣ 6 Minimizing Payment ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">6.2</span></a>)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem4.p1"> <p class="ltx_p" id="S1.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem4.p1.2.2">In both the no-regret and the realizable models, the amount of <span class="ltx_text ltx_font_normal" id="S1.Thmtheorem4.p1.2.2.1">payment</span> required to learn an <math alttext="n" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.1.1.m1.1"><semantics id="S1.Thmtheorem4.p1.1.1.m1.1a"><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.1.1.m1.1b"><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.1.1.m1.1d">italic_n</annotation></semantics></math>-agent game is lower-bounded by the number of <span class="ltx_text ltx_font_normal" id="S1.Thmtheorem4.p1.2.2.2">rounds</span> required to learn an <math alttext="(n-1)" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.2.2.m2.1"><semantics id="S1.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.cmml"><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2" stretchy="false" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.cmml"><mi id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.2" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.2.cmml">n</mi><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.3" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3" stretchy="false" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.2.2.m2.1b"><apply id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1"><minus id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1"></minus><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.2">𝑛</ci><cn id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.3.cmml" type="integer" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.2.2.m2.1c">(n-1)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.2.2.m2.1d">( italic_n - 1 )</annotation></semantics></math>-agent game.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p10"> <p class="ltx_p" id="S1.SS1.p10.8">The proof, informally, works by making one of the agents, say agent <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p10.1.m1.1"><semantics id="S1.SS1.p10.1.m1.1a"><mi id="S1.SS1.p10.1.m1.1.1" xref="S1.SS1.p10.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.1.m1.1b"><ci id="S1.SS1.p10.1.m1.1.1.cmml" xref="S1.SS1.p10.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.1.m1.1d">italic_i</annotation></semantics></math>, have a dominated action <math alttext="\tilde{a}_{i}" class="ltx_Math" display="inline" id="S1.SS1.p10.2.m2.1"><semantics id="S1.SS1.p10.2.m2.1a"><msub id="S1.SS1.p10.2.m2.1.1" xref="S1.SS1.p10.2.m2.1.1.cmml"><mover accent="true" id="S1.SS1.p10.2.m2.1.1.2" xref="S1.SS1.p10.2.m2.1.1.2.cmml"><mi id="S1.SS1.p10.2.m2.1.1.2.2" xref="S1.SS1.p10.2.m2.1.1.2.2.cmml">a</mi><mo id="S1.SS1.p10.2.m2.1.1.2.1" xref="S1.SS1.p10.2.m2.1.1.2.1.cmml">~</mo></mover><mi id="S1.SS1.p10.2.m2.1.1.3" xref="S1.SS1.p10.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.2.m2.1b"><apply id="S1.SS1.p10.2.m2.1.1.cmml" xref="S1.SS1.p10.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p10.2.m2.1.1.1.cmml" xref="S1.SS1.p10.2.m2.1.1">subscript</csymbol><apply id="S1.SS1.p10.2.m2.1.1.2.cmml" xref="S1.SS1.p10.2.m2.1.1.2"><ci id="S1.SS1.p10.2.m2.1.1.2.1.cmml" xref="S1.SS1.p10.2.m2.1.1.2.1">~</ci><ci id="S1.SS1.p10.2.m2.1.1.2.2.cmml" xref="S1.SS1.p10.2.m2.1.1.2.2">𝑎</ci></apply><ci id="S1.SS1.p10.2.m2.1.1.3.cmml" xref="S1.SS1.p10.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.2.m2.1c">\tilde{a}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.2.m2.1d">over~ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. In order to learn the utilities in the <math alttext="(n-1)" class="ltx_Math" display="inline" id="S1.SS1.p10.3.m3.1"><semantics id="S1.SS1.p10.3.m3.1a"><mrow id="S1.SS1.p10.3.m3.1.1.1" xref="S1.SS1.p10.3.m3.1.1.1.1.cmml"><mo id="S1.SS1.p10.3.m3.1.1.1.2" stretchy="false" xref="S1.SS1.p10.3.m3.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.p10.3.m3.1.1.1.1" xref="S1.SS1.p10.3.m3.1.1.1.1.cmml"><mi id="S1.SS1.p10.3.m3.1.1.1.1.2" xref="S1.SS1.p10.3.m3.1.1.1.1.2.cmml">n</mi><mo id="S1.SS1.p10.3.m3.1.1.1.1.1" xref="S1.SS1.p10.3.m3.1.1.1.1.1.cmml">−</mo><mn id="S1.SS1.p10.3.m3.1.1.1.1.3" xref="S1.SS1.p10.3.m3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.SS1.p10.3.m3.1.1.1.3" stretchy="false" xref="S1.SS1.p10.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.3.m3.1b"><apply id="S1.SS1.p10.3.m3.1.1.1.1.cmml" xref="S1.SS1.p10.3.m3.1.1.1"><minus id="S1.SS1.p10.3.m3.1.1.1.1.1.cmml" xref="S1.SS1.p10.3.m3.1.1.1.1.1"></minus><ci id="S1.SS1.p10.3.m3.1.1.1.1.2.cmml" xref="S1.SS1.p10.3.m3.1.1.1.1.2">𝑛</ci><cn id="S1.SS1.p10.3.m3.1.1.1.1.3.cmml" type="integer" xref="S1.SS1.p10.3.m3.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.3.m3.1c">(n-1)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.3.m3.1d">( italic_n - 1 )</annotation></semantics></math>-player subgame induced by agent <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p10.4.m4.1"><semantics id="S1.SS1.p10.4.m4.1a"><mi id="S1.SS1.p10.4.m4.1.1" xref="S1.SS1.p10.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.4.m4.1b"><ci id="S1.SS1.p10.4.m4.1.1.cmml" xref="S1.SS1.p10.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.4.m4.1d">italic_i</annotation></semantics></math> playing <math alttext="\tilde{a}_{i}" class="ltx_Math" display="inline" id="S1.SS1.p10.5.m5.1"><semantics id="S1.SS1.p10.5.m5.1a"><msub id="S1.SS1.p10.5.m5.1.1" xref="S1.SS1.p10.5.m5.1.1.cmml"><mover accent="true" id="S1.SS1.p10.5.m5.1.1.2" xref="S1.SS1.p10.5.m5.1.1.2.cmml"><mi id="S1.SS1.p10.5.m5.1.1.2.2" xref="S1.SS1.p10.5.m5.1.1.2.2.cmml">a</mi><mo id="S1.SS1.p10.5.m5.1.1.2.1" xref="S1.SS1.p10.5.m5.1.1.2.1.cmml">~</mo></mover><mi id="S1.SS1.p10.5.m5.1.1.3" xref="S1.SS1.p10.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.5.m5.1b"><apply id="S1.SS1.p10.5.m5.1.1.cmml" xref="S1.SS1.p10.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS1.p10.5.m5.1.1.1.cmml" xref="S1.SS1.p10.5.m5.1.1">subscript</csymbol><apply id="S1.SS1.p10.5.m5.1.1.2.cmml" xref="S1.SS1.p10.5.m5.1.1.2"><ci id="S1.SS1.p10.5.m5.1.1.2.1.cmml" xref="S1.SS1.p10.5.m5.1.1.2.1">~</ci><ci id="S1.SS1.p10.5.m5.1.1.2.2.cmml" xref="S1.SS1.p10.5.m5.1.1.2.2">𝑎</ci></apply><ci id="S1.SS1.p10.5.m5.1.1.3.cmml" xref="S1.SS1.p10.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.5.m5.1c">\tilde{a}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.5.m5.1d">over~ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the principal must pay <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p10.6.m6.1"><semantics id="S1.SS1.p10.6.m6.1a"><mi id="S1.SS1.p10.6.m6.1.1" xref="S1.SS1.p10.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.6.m6.1b"><ci id="S1.SS1.p10.6.m6.1.1.cmml" xref="S1.SS1.p10.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.6.m6.1d">italic_i</annotation></semantics></math> to play <math alttext="\tilde{a}_{i}" class="ltx_Math" display="inline" id="S1.SS1.p10.7.m7.1"><semantics id="S1.SS1.p10.7.m7.1a"><msub id="S1.SS1.p10.7.m7.1.1" xref="S1.SS1.p10.7.m7.1.1.cmml"><mover accent="true" id="S1.SS1.p10.7.m7.1.1.2" xref="S1.SS1.p10.7.m7.1.1.2.cmml"><mi id="S1.SS1.p10.7.m7.1.1.2.2" xref="S1.SS1.p10.7.m7.1.1.2.2.cmml">a</mi><mo id="S1.SS1.p10.7.m7.1.1.2.1" xref="S1.SS1.p10.7.m7.1.1.2.1.cmml">~</mo></mover><mi id="S1.SS1.p10.7.m7.1.1.3" xref="S1.SS1.p10.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.7.m7.1b"><apply id="S1.SS1.p10.7.m7.1.1.cmml" xref="S1.SS1.p10.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SS1.p10.7.m7.1.1.1.cmml" xref="S1.SS1.p10.7.m7.1.1">subscript</csymbol><apply id="S1.SS1.p10.7.m7.1.1.2.cmml" xref="S1.SS1.p10.7.m7.1.1.2"><ci id="S1.SS1.p10.7.m7.1.1.2.1.cmml" xref="S1.SS1.p10.7.m7.1.1.2.1">~</ci><ci id="S1.SS1.p10.7.m7.1.1.2.2.cmml" xref="S1.SS1.p10.7.m7.1.1.2.2">𝑎</ci></apply><ci id="S1.SS1.p10.7.m7.1.1.3.cmml" xref="S1.SS1.p10.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.7.m7.1c">\tilde{a}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.7.m7.1d">over~ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for as many rounds as it takes to learn the <math alttext="(n-1)" class="ltx_Math" display="inline" id="S1.SS1.p10.8.m8.1"><semantics id="S1.SS1.p10.8.m8.1a"><mrow id="S1.SS1.p10.8.m8.1.1.1" xref="S1.SS1.p10.8.m8.1.1.1.1.cmml"><mo id="S1.SS1.p10.8.m8.1.1.1.2" stretchy="false" xref="S1.SS1.p10.8.m8.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.p10.8.m8.1.1.1.1" xref="S1.SS1.p10.8.m8.1.1.1.1.cmml"><mi id="S1.SS1.p10.8.m8.1.1.1.1.2" xref="S1.SS1.p10.8.m8.1.1.1.1.2.cmml">n</mi><mo id="S1.SS1.p10.8.m8.1.1.1.1.1" xref="S1.SS1.p10.8.m8.1.1.1.1.1.cmml">−</mo><mn id="S1.SS1.p10.8.m8.1.1.1.1.3" xref="S1.SS1.p10.8.m8.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.SS1.p10.8.m8.1.1.1.3" stretchy="false" xref="S1.SS1.p10.8.m8.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.8.m8.1b"><apply id="S1.SS1.p10.8.m8.1.1.1.1.cmml" xref="S1.SS1.p10.8.m8.1.1.1"><minus id="S1.SS1.p10.8.m8.1.1.1.1.1.cmml" xref="S1.SS1.p10.8.m8.1.1.1.1.1"></minus><ci id="S1.SS1.p10.8.m8.1.1.1.1.2.cmml" xref="S1.SS1.p10.8.m8.1.1.1.1.2">𝑛</ci><cn id="S1.SS1.p10.8.m8.1.1.1.1.3.cmml" type="integer" xref="S1.SS1.p10.8.m8.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.8.m8.1c">(n-1)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.8.m8.1d">( italic_n - 1 )</annotation></semantics></math>-player subgame.</p> </div> <div class="ltx_para" id="S1.SS1.p11"> <p class="ltx_p" id="S1.SS1.p11.2">Finally, we turn to a motivating application, which, as discussed above, is the problem of <span class="ltx_text ltx_font_italic" id="S1.SS1.p11.2.1">steering</span> no-regret learners to desirable outcomes, introduced by <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite>. Departing from their paper, our results do not require the principal to have prior knowledge of the agents’ utility functions. We define a solution concept called <span class="ltx_text ltx_font_italic" id="S1.SS1.p11.2.2">correlated equilibrium with payments</span> (CEP), in which the principal has a utility function, and wishes to optimize its utility minus the amount of payment that it must give. Departing from <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite> again, it is possible for the total amount of payment to be nonzero in equilibrium (<span class="ltx_text ltx_font_italic" id="S1.SS1.p11.2.3">i.e.</span>, linearly increasing in <math alttext="T" class="ltx_Math" display="inline" id="S1.SS1.p11.1.m1.1"><semantics id="S1.SS1.p11.1.m1.1a"><mi id="S1.SS1.p11.1.m1.1.1" xref="S1.SS1.p11.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p11.1.m1.1b"><ci id="S1.SS1.p11.1.m1.1.1.cmml" xref="S1.SS1.p11.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p11.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p11.1.m1.1d">italic_T</annotation></semantics></math>), so long as the corresponding increase in principal utility is large enough to justify the payments. We then show that the principal-optimal CEP exactly characterizes the value (averaged across timesteps) that the principal can achieve in the limit <math alttext="T\to\infty" class="ltx_Math" display="inline" id="S1.SS1.p11.2.m2.1"><semantics id="S1.SS1.p11.2.m2.1a"><mrow id="S1.SS1.p11.2.m2.1.1" xref="S1.SS1.p11.2.m2.1.1.cmml"><mi id="S1.SS1.p11.2.m2.1.1.2" xref="S1.SS1.p11.2.m2.1.1.2.cmml">T</mi><mo id="S1.SS1.p11.2.m2.1.1.1" stretchy="false" xref="S1.SS1.p11.2.m2.1.1.1.cmml">→</mo><mi id="S1.SS1.p11.2.m2.1.1.3" mathvariant="normal" xref="S1.SS1.p11.2.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p11.2.m2.1b"><apply id="S1.SS1.p11.2.m2.1.1.cmml" xref="S1.SS1.p11.2.m2.1.1"><ci id="S1.SS1.p11.2.m2.1.1.1.cmml" xref="S1.SS1.p11.2.m2.1.1.1">→</ci><ci id="S1.SS1.p11.2.m2.1.1.2.cmml" xref="S1.SS1.p11.2.m2.1.1.2">𝑇</ci><infinity id="S1.SS1.p11.2.m2.1.1.3.cmml" xref="S1.SS1.p11.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p11.2.m2.1c">T\to\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p11.2.m2.1d">italic_T → ∞</annotation></semantics></math>:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem5.1.1.1">Theorem 1.5</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem5.2.2"> </span>(Informal summary of steering results)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem5.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem5.p1"> <p class="ltx_p" id="S1.Thmtheorem5.p1.1"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem5.p1.1.1">Let <math alttext="F^{*}" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.1.1.m1.1"><semantics id="S1.Thmtheorem5.p1.1.1.m1.1a"><msup id="S1.Thmtheorem5.p1.1.1.m1.1.1" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem5.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.2.cmml">F</mi><mo id="S1.Thmtheorem5.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.1.1.m1.1b"><apply id="S1.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem5.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1">superscript</csymbol><ci id="S1.Thmtheorem5.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.2">𝐹</ci><times id="S1.Thmtheorem5.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.1.1.m1.1c">F^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.1.1.m1.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be the objective value for the principal in the principal-optimal CEP. Then, in the no-regret model,</span></p> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.2"><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.2.1">no principal – even if the principal knows the game </span><math alttext="\Gamma" class="ltx_Math" display="inline" id="S1.I1.i1.p1.1.m1.1"><semantics id="S1.I1.i1.p1.1.m1.1a"><mi id="S1.I1.i1.p1.1.m1.1.1" mathvariant="normal" xref="S1.I1.i1.p1.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.1.m1.1b"><ci id="S1.I1.i1.p1.1.m1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.1.m1.1d">roman_Γ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.2.2"> exactly on the first round – can achieve time-averaged value better than </span><math alttext="F^{*}+\poly(M)\cdot T^{-1/2}" class="ltx_Math" display="inline" id="S1.I1.i1.p1.2.m2.1"><semantics id="S1.I1.i1.p1.2.m2.1a"><mrow id="S1.I1.i1.p1.2.m2.1.2" xref="S1.I1.i1.p1.2.m2.1.2.cmml"><msup id="S1.I1.i1.p1.2.m2.1.2.2" xref="S1.I1.i1.p1.2.m2.1.2.2.cmml"><mi id="S1.I1.i1.p1.2.m2.1.2.2.2" xref="S1.I1.i1.p1.2.m2.1.2.2.2.cmml">F</mi><mo id="S1.I1.i1.p1.2.m2.1.2.2.3" xref="S1.I1.i1.p1.2.m2.1.2.2.3.cmml">∗</mo></msup><mo id="S1.I1.i1.p1.2.m2.1.2.1" xref="S1.I1.i1.p1.2.m2.1.2.1.cmml">+</mo><mrow id="S1.I1.i1.p1.2.m2.1.2.3" xref="S1.I1.i1.p1.2.m2.1.2.3.cmml"><mrow id="S1.I1.i1.p1.2.m2.1.2.3.2" xref="S1.I1.i1.p1.2.m2.1.2.3.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S1.I1.i1.p1.2.m2.1.2.3.2.2" xref="S1.I1.i1.p1.2.m2.1.2.3.2.2b.cmml"><mtext id="S1.I1.i1.p1.2.m2.1.2.3.2.2a" xref="S1.I1.i1.p1.2.m2.1.2.3.2.2b.cmml">\poly</mtext></merror><mo id="S1.I1.i1.p1.2.m2.1.2.3.2.1" xref="S1.I1.i1.p1.2.m2.1.2.3.2.1.cmml">⁢</mo><mrow id="S1.I1.i1.p1.2.m2.1.2.3.2.3.2" xref="S1.I1.i1.p1.2.m2.1.2.3.2.cmml"><mo id="S1.I1.i1.p1.2.m2.1.2.3.2.3.2.1" stretchy="false" xref="S1.I1.i1.p1.2.m2.1.2.3.2.cmml">(</mo><mi id="S1.I1.i1.p1.2.m2.1.1" xref="S1.I1.i1.p1.2.m2.1.1.cmml">M</mi><mo id="S1.I1.i1.p1.2.m2.1.2.3.2.3.2.2" rspace="0.055em" stretchy="false" xref="S1.I1.i1.p1.2.m2.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S1.I1.i1.p1.2.m2.1.2.3.1" rspace="0.222em" xref="S1.I1.i1.p1.2.m2.1.2.3.1.cmml">⋅</mo><msup id="S1.I1.i1.p1.2.m2.1.2.3.3" xref="S1.I1.i1.p1.2.m2.1.2.3.3.cmml"><mi id="S1.I1.i1.p1.2.m2.1.2.3.3.2" xref="S1.I1.i1.p1.2.m2.1.2.3.3.2.cmml">T</mi><mrow id="S1.I1.i1.p1.2.m2.1.2.3.3.3" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.cmml"><mo id="S1.I1.i1.p1.2.m2.1.2.3.3.3a" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.cmml">−</mo><mrow id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.cmml"><mn id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.2" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.2.cmml">1</mn><mo id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.1" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.1.cmml">/</mo><mn id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.3" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.3.cmml">2</mn></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.2.m2.1b"><apply id="S1.I1.i1.p1.2.m2.1.2.cmml" xref="S1.I1.i1.p1.2.m2.1.2"><plus id="S1.I1.i1.p1.2.m2.1.2.1.cmml" xref="S1.I1.i1.p1.2.m2.1.2.1"></plus><apply id="S1.I1.i1.p1.2.m2.1.2.2.cmml" xref="S1.I1.i1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S1.I1.i1.p1.2.m2.1.2.2.1.cmml" xref="S1.I1.i1.p1.2.m2.1.2.2">superscript</csymbol><ci id="S1.I1.i1.p1.2.m2.1.2.2.2.cmml" xref="S1.I1.i1.p1.2.m2.1.2.2.2">𝐹</ci><times id="S1.I1.i1.p1.2.m2.1.2.2.3.cmml" xref="S1.I1.i1.p1.2.m2.1.2.2.3"></times></apply><apply id="S1.I1.i1.p1.2.m2.1.2.3.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3"><ci id="S1.I1.i1.p1.2.m2.1.2.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.1">⋅</ci><apply id="S1.I1.i1.p1.2.m2.1.2.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.2"><times id="S1.I1.i1.p1.2.m2.1.2.3.2.1.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.2.1"></times><ci id="S1.I1.i1.p1.2.m2.1.2.3.2.2b.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.2.2"><merror class="ltx_ERROR undefined undefined" id="S1.I1.i1.p1.2.m2.1.2.3.2.2.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.2.2"><mtext id="S1.I1.i1.p1.2.m2.1.2.3.2.2a.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.2.2">\poly</mtext></merror></ci><ci id="S1.I1.i1.p1.2.m2.1.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1">𝑀</ci></apply><apply id="S1.I1.i1.p1.2.m2.1.2.3.3.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.3"><csymbol cd="ambiguous" id="S1.I1.i1.p1.2.m2.1.2.3.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.3">superscript</csymbol><ci id="S1.I1.i1.p1.2.m2.1.2.3.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.3.2">𝑇</ci><apply id="S1.I1.i1.p1.2.m2.1.2.3.3.3.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3"><minus id="S1.I1.i1.p1.2.m2.1.2.3.3.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3"></minus><apply id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2"><divide id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.1.cmml" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.1"></divide><cn id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.2.cmml" type="integer" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.2">1</cn><cn id="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.3.cmml" type="integer" xref="S1.I1.i1.p1.2.m2.1.2.3.3.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.2.m2.1c">F^{*}+\poly(M)\cdot T^{-1/2}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.2.m2.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT + ( italic_M ) ⋅ italic_T start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.2.3">, and</span></p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1"><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.1.1">there exists a principal that can achieve time-averaged value at least </span><math alttext="F^{*}-\poly(M)\cdot T^{-1/4}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.1.m1.1"><semantics id="S1.I1.i2.p1.1.m1.1a"><mrow id="S1.I1.i2.p1.1.m1.1.2" xref="S1.I1.i2.p1.1.m1.1.2.cmml"><msup id="S1.I1.i2.p1.1.m1.1.2.2" xref="S1.I1.i2.p1.1.m1.1.2.2.cmml"><mi id="S1.I1.i2.p1.1.m1.1.2.2.2" xref="S1.I1.i2.p1.1.m1.1.2.2.2.cmml">F</mi><mo id="S1.I1.i2.p1.1.m1.1.2.2.3" xref="S1.I1.i2.p1.1.m1.1.2.2.3.cmml">∗</mo></msup><mo id="S1.I1.i2.p1.1.m1.1.2.1" xref="S1.I1.i2.p1.1.m1.1.2.1.cmml">−</mo><mrow id="S1.I1.i2.p1.1.m1.1.2.3" xref="S1.I1.i2.p1.1.m1.1.2.3.cmml"><mrow id="S1.I1.i2.p1.1.m1.1.2.3.2" xref="S1.I1.i2.p1.1.m1.1.2.3.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S1.I1.i2.p1.1.m1.1.2.3.2.2" xref="S1.I1.i2.p1.1.m1.1.2.3.2.2b.cmml"><mtext id="S1.I1.i2.p1.1.m1.1.2.3.2.2a" 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</ul> </div> </div> <div class="ltx_para" id="S1.SS1.p12"> <p class="ltx_p" id="S1.SS1.p12.1">All our algorithms are implementable by the principal with <math alttext="\poly(M)" class="ltx_Math" display="inline" id="S1.SS1.p12.1.m1.1"><semantics id="S1.SS1.p12.1.m1.1a"><mrow id="S1.SS1.p12.1.m1.1.2" xref="S1.SS1.p12.1.m1.1.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S1.SS1.p12.1.m1.1.2.2" xref="S1.SS1.p12.1.m1.1.2.2b.cmml"><mtext id="S1.SS1.p12.1.m1.1.2.2a" xref="S1.SS1.p12.1.m1.1.2.2b.cmml">\poly</mtext></merror><mo id="S1.SS1.p12.1.m1.1.2.1" xref="S1.SS1.p12.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p12.1.m1.1.2.3.2" xref="S1.SS1.p12.1.m1.1.2.cmml"><mo id="S1.SS1.p12.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS1.p12.1.m1.1.2.cmml">(</mo><mi id="S1.SS1.p12.1.m1.1.1" xref="S1.SS1.p12.1.m1.1.1.cmml">M</mi><mo id="S1.SS1.p12.1.m1.1.2.3.2.2" stretchy="false" xref="S1.SS1.p12.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" 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To our knowledge, all our results are the first results for their respective problems: our utility-learning results are the first in their models, and is the first result for steering agents to desirable outcomes without prior knowledge of agents’ utilities, and the first exact characterization of the optimal value achievable by the principal in the steering setting.</p> </div> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Related research</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.1">As discussed above, the closest paper to ours is that of <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite>, who introduced the problem of steering no-regret learners to desirable outcomes. However, critically, their paper assumes that the principal 1) knows the utility functions of all agents, and 2) knows a desirable outcome to which it wishes to steer the players. Here, we make neither of these assumptions; indeed, the main results of the paper show how to circumvent these requirements. However, unlike their paper, which deals with extensive-form games, our results are only for normal-form games. We leave it as an interesting future direction to extend our results to the extensive-form setting.</p> </div> <div class="ltx_para" id="S1.SS2.p2"> <p class="ltx_p" id="S1.SS2.p2.3"><cite class="ltx_cite ltx_citemacro_citet">Monderer and Tennenholtz (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib19" title="">2003</a>)</cite> showed how a principal can motivate agents to play any correlated equilibrium using signals and appropriate payments. Our concept of correlated equilibrium with payments is closely related to their concept of <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.p2.1.m1.1"><semantics id="S1.SS2.p2.1.m1.1a"><mi id="S1.SS2.p2.1.m1.1.1" xref="S1.SS2.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.1.m1.1b"><ci id="S1.SS2.p2.1.m1.1.1.cmml" xref="S1.SS2.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.1.m1.1d">italic_k</annotation></semantics></math>-implementable correlated equilibrium: <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.p2.2.m2.1"><semantics id="S1.SS2.p2.2.m2.1a"><mi id="S1.SS2.p2.2.m2.1.1" xref="S1.SS2.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.2.m2.1b"><ci id="S1.SS2.p2.2.m2.1.1.cmml" xref="S1.SS2.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.2.m2.1d">italic_k</annotation></semantics></math> is the amount of payment in equilibrium required to sustain the correlated equilibrium. As with the line of work on inverse game theory, our work differs from <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.p2.3.m3.1"><semantics id="S1.SS2.p2.3.m3.1a"><mi id="S1.SS2.p2.3.m3.1.1" xref="S1.SS2.p2.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.3.m3.1b"><ci id="S1.SS2.p2.3.m3.1.1.cmml" xref="S1.SS2.p2.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.3.m3.1d">italic_k</annotation></semantics></math>-implementation in that we study a dynamic model instead of a static one, and do not assume that the principal has prior knowledge of the utilities.</p> </div> <div class="ltx_para" id="S1.SS2.p3"> <p class="ltx_p" id="S1.SS2.p3.1">Our work, especially the results on steering agents toward desired outcomes, carries resemblance to <span class="ltx_text ltx_font_italic" id="S1.SS2.p3.1.1">Bayesian persuasion</span> (<span class="ltx_text ltx_font_italic" id="S1.SS2.p3.1.2">e.g.</span>, <cite class="ltx_cite ltx_citemacro_citet">Kamenica and Gentzkow (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib14" title="">2011</a>)</cite>). Perhaps most closely related, <cite class="ltx_cite ltx_citemacro_citet">Feng et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib10" title="">2022</a>); Bacchiocchi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib3" title="">2024</a>)</cite> studied the problem of persuading an agent without prior knowledge of the agent’s utility. A similar line of work considers the problem of principal learning in <span class="ltx_text ltx_font_italic" id="S1.SS2.p3.1.3">Stackelberg games</span>, where the principal cannot pay the agent(s) but can attempt to influence the agents’ actions by changing its own strategy (<span class="ltx_text ltx_font_italic" id="S1.SS2.p3.1.4">e.g.</span>, <cite class="ltx_cite ltx_citemacro_cite">Letchford et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib16" title="">2009</a>); Haghtalab et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib12" title="">2022</a>)</cite>), and the agents are best-responding, myopically or not. These settings are related but distinct from ours: both settings involve a principal, but our principal only gives payments and does not take actions, whereas their setting has the opposite. Moreover, our agents do not necessarily best respond; they may instead be no-regret learning algorithms.</p> </div> <div class="ltx_para" id="S1.SS2.p4"> <p class="ltx_p" id="S1.SS2.p4.1">Our problem of learning utility functions from no-regret agents and steering them is related to the line of work on playing against no-regret agents <cite class="ltx_cite ltx_citemacro_cite">Deng et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib9" title="">2019</a>); Mansour et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib18" title="">2022</a>); Lin and Chen (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib17" title="">2025</a>)</cite>. That literature assumes that the principal knows not only the utilities but also the specific algorithm being used by the agent (or at least some properties of the algorithm). For example, <cite class="ltx_cite ltx_citemacro_citet">Deng et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib9" title="">2019</a>)</cite> show that, for a popular class of no-regret algorithms known as <span class="ltx_text ltx_font_italic" id="S1.SS2.p4.1.1">mean-based</span> algorithms, a principal can gain <span class="ltx_text ltx_font_italic" id="S1.SS2.p4.1.2">more</span> than the Stackelberg value in a Stackelberg game. Our algorithms and setting, on the other hand, consider <span class="ltx_text ltx_font_italic" id="S1.SS2.p4.1.3">worst-case</span> no-regret agent behaviors. Finally, all of the papers cited in the previous two paragraphs consider only principal-agent problems with a single agent, whereas we consider arbitrary multi-agent (normal-form) games.</p> </div> <div class="ltx_para" id="S1.SS2.p5"> <p class="ltx_p" id="S1.SS2.p5.1">As we will discuss more in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5" title="5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">5</span></a>, a key challenge in learning from no-regret agents by paying them is that the agents’ no-regret algorithms can be non-forgetful. The payment given to the agents in the past affect the agents’ behavior in the future. Non-forgetfulness is a known obstacle to the fast convergence of multi-agent dynamics <cite class="ltx_cite ltx_citemacro_cite">Wu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib22" title="">2022</a>); Cai et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib6" title="">2024</a>); Scheid et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib21" title="">2024</a>)</cite>. We overcome this obstacle by designing a zero-sum-game-based learning algorithm for the principal and using signals to influence agents, without requiring the agents’ algorithm to be forgetful.</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>Notation and Preliminaries</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.18">Throughout this paper, <math alttext="\tilde{\mathcal{O}}" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mover accent="true" id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p1.1.m1.1.1.2" xref="S2.p1.1.m1.1.1.2.cmml">𝒪</mi><mo id="S2.p1.1.m1.1.1.1" xref="S2.p1.1.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><apply id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1"><ci id="S2.p1.1.m1.1.1.1.cmml" xref="S2.p1.1.m1.1.1.1">~</ci><ci id="S2.p1.1.m1.1.1.2.cmml" xref="S2.p1.1.m1.1.1.2">𝒪</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">\tilde{\mathcal{O}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">over~ start_ARG caligraphic_O end_ARG</annotation></semantics></math> and <math alttext="\tilde{\Omega}" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mover accent="true" id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml"><mi id="S2.p1.2.m2.1.1.2" mathvariant="normal" xref="S2.p1.2.m2.1.1.2.cmml">Ω</mi><mo id="S2.p1.2.m2.1.1.1" xref="S2.p1.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><apply id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1"><ci id="S2.p1.2.m2.1.1.1.cmml" xref="S2.p1.2.m2.1.1.1">~</ci><ci id="S2.p1.2.m2.1.1.2.cmml" xref="S2.p1.2.m2.1.1.2">Ω</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">\tilde{\Omega}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">over~ start_ARG roman_Ω end_ARG</annotation></semantics></math> hide factors logarithmic in their argument. The symbol <math alttext="[n]" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><mrow id="S2.p1.3.m3.1.2.2" xref="S2.p1.3.m3.1.2.1.cmml"><mo id="S2.p1.3.m3.1.2.2.1" stretchy="false" xref="S2.p1.3.m3.1.2.1.1.cmml">[</mo><mi id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml">n</mi><mo id="S2.p1.3.m3.1.2.2.2" stretchy="false" xref="S2.p1.3.m3.1.2.1.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><apply id="S2.p1.3.m3.1.2.1.cmml" xref="S2.p1.3.m3.1.2.2"><csymbol cd="latexml" id="S2.p1.3.m3.1.2.1.1.cmml" xref="S2.p1.3.m3.1.2.2.1">delimited-[]</csymbol><ci id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">[n]</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">[ italic_n ]</annotation></semantics></math> denotes the set of positive integers <math alttext="\{1,\dots,n\}" class="ltx_Math" display="inline" id="S2.p1.4.m4.3"><semantics id="S2.p1.4.m4.3a"><mrow id="S2.p1.4.m4.3.4.2" xref="S2.p1.4.m4.3.4.1.cmml"><mo id="S2.p1.4.m4.3.4.2.1" stretchy="false" xref="S2.p1.4.m4.3.4.1.cmml">{</mo><mn id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml">1</mn><mo id="S2.p1.4.m4.3.4.2.2" xref="S2.p1.4.m4.3.4.1.cmml">,</mo><mi id="S2.p1.4.m4.2.2" mathvariant="normal" xref="S2.p1.4.m4.2.2.cmml">…</mi><mo id="S2.p1.4.m4.3.4.2.3" xref="S2.p1.4.m4.3.4.1.cmml">,</mo><mi id="S2.p1.4.m4.3.3" xref="S2.p1.4.m4.3.3.cmml">n</mi><mo id="S2.p1.4.m4.3.4.2.4" stretchy="false" xref="S2.p1.4.m4.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.3b"><set id="S2.p1.4.m4.3.4.1.cmml" xref="S2.p1.4.m4.3.4.2"><cn id="S2.p1.4.m4.1.1.cmml" type="integer" xref="S2.p1.4.m4.1.1">1</cn><ci id="S2.p1.4.m4.2.2.cmml" xref="S2.p1.4.m4.2.2">…</ci><ci id="S2.p1.4.m4.3.3.cmml" xref="S2.p1.4.m4.3.3">𝑛</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.3c">\{1,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.3d">{ 1 , … , italic_n }</annotation></semantics></math>. The notation <math alttext="f\lesssim g" class="ltx_Math" display="inline" id="S2.p1.5.m5.1"><semantics id="S2.p1.5.m5.1a"><mrow id="S2.p1.5.m5.1.1" xref="S2.p1.5.m5.1.1.cmml"><mi id="S2.p1.5.m5.1.1.2" xref="S2.p1.5.m5.1.1.2.cmml">f</mi><mo id="S2.p1.5.m5.1.1.1" xref="S2.p1.5.m5.1.1.1.cmml">≲</mo><mi id="S2.p1.5.m5.1.1.3" xref="S2.p1.5.m5.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m5.1b"><apply id="S2.p1.5.m5.1.1.cmml" xref="S2.p1.5.m5.1.1"><csymbol cd="latexml" id="S2.p1.5.m5.1.1.1.cmml" xref="S2.p1.5.m5.1.1.1">less-than-or-similar-to</csymbol><ci id="S2.p1.5.m5.1.1.2.cmml" xref="S2.p1.5.m5.1.1.2">𝑓</ci><ci id="S2.p1.5.m5.1.1.3.cmml" xref="S2.p1.5.m5.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m5.1c">f\lesssim g</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m5.1d">italic_f ≲ italic_g</annotation></semantics></math> means <math alttext="f\leq{\mathcal{O}}(g)" class="ltx_Math" display="inline" id="S2.p1.6.m6.1"><semantics id="S2.p1.6.m6.1a"><mrow id="S2.p1.6.m6.1.2" xref="S2.p1.6.m6.1.2.cmml"><mi id="S2.p1.6.m6.1.2.2" xref="S2.p1.6.m6.1.2.2.cmml">f</mi><mo id="S2.p1.6.m6.1.2.1" xref="S2.p1.6.m6.1.2.1.cmml">≤</mo><mrow id="S2.p1.6.m6.1.2.3" xref="S2.p1.6.m6.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p1.6.m6.1.2.3.2" xref="S2.p1.6.m6.1.2.3.2.cmml">𝒪</mi><mo id="S2.p1.6.m6.1.2.3.1" xref="S2.p1.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S2.p1.6.m6.1.2.3.3.2" xref="S2.p1.6.m6.1.2.3.cmml"><mo id="S2.p1.6.m6.1.2.3.3.2.1" stretchy="false" xref="S2.p1.6.m6.1.2.3.cmml">(</mo><mi id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml">g</mi><mo id="S2.p1.6.m6.1.2.3.3.2.2" stretchy="false" xref="S2.p1.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.1b"><apply id="S2.p1.6.m6.1.2.cmml" xref="S2.p1.6.m6.1.2"><leq id="S2.p1.6.m6.1.2.1.cmml" xref="S2.p1.6.m6.1.2.1"></leq><ci id="S2.p1.6.m6.1.2.2.cmml" xref="S2.p1.6.m6.1.2.2">𝑓</ci><apply id="S2.p1.6.m6.1.2.3.cmml" xref="S2.p1.6.m6.1.2.3"><times id="S2.p1.6.m6.1.2.3.1.cmml" xref="S2.p1.6.m6.1.2.3.1"></times><ci id="S2.p1.6.m6.1.2.3.2.cmml" xref="S2.p1.6.m6.1.2.3.2">𝒪</ci><ci id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.1c">f\leq{\mathcal{O}}(g)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.1d">italic_f ≤ caligraphic_O ( italic_g )</annotation></semantics></math>, and <math alttext="f\gtrsim g" class="ltx_Math" display="inline" id="S2.p1.7.m7.1"><semantics id="S2.p1.7.m7.1a"><mrow id="S2.p1.7.m7.1.1" xref="S2.p1.7.m7.1.1.cmml"><mi id="S2.p1.7.m7.1.1.2" xref="S2.p1.7.m7.1.1.2.cmml">f</mi><mo id="S2.p1.7.m7.1.1.1" xref="S2.p1.7.m7.1.1.1.cmml">≳</mo><mi id="S2.p1.7.m7.1.1.3" xref="S2.p1.7.m7.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.m7.1b"><apply id="S2.p1.7.m7.1.1.cmml" xref="S2.p1.7.m7.1.1"><csymbol cd="latexml" id="S2.p1.7.m7.1.1.1.cmml" xref="S2.p1.7.m7.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S2.p1.7.m7.1.1.2.cmml" xref="S2.p1.7.m7.1.1.2">𝑓</ci><ci id="S2.p1.7.m7.1.1.3.cmml" xref="S2.p1.7.m7.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m7.1c">f\gtrsim g</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m7.1d">italic_f ≳ italic_g</annotation></semantics></math> means <math alttext="f\geq\Omega(g)" class="ltx_Math" display="inline" id="S2.p1.8.m8.1"><semantics id="S2.p1.8.m8.1a"><mrow id="S2.p1.8.m8.1.2" xref="S2.p1.8.m8.1.2.cmml"><mi id="S2.p1.8.m8.1.2.2" xref="S2.p1.8.m8.1.2.2.cmml">f</mi><mo id="S2.p1.8.m8.1.2.1" xref="S2.p1.8.m8.1.2.1.cmml">≥</mo><mrow id="S2.p1.8.m8.1.2.3" xref="S2.p1.8.m8.1.2.3.cmml"><mi id="S2.p1.8.m8.1.2.3.2" mathvariant="normal" xref="S2.p1.8.m8.1.2.3.2.cmml">Ω</mi><mo id="S2.p1.8.m8.1.2.3.1" xref="S2.p1.8.m8.1.2.3.1.cmml">⁢</mo><mrow id="S2.p1.8.m8.1.2.3.3.2" xref="S2.p1.8.m8.1.2.3.cmml"><mo id="S2.p1.8.m8.1.2.3.3.2.1" stretchy="false" xref="S2.p1.8.m8.1.2.3.cmml">(</mo><mi id="S2.p1.8.m8.1.1" xref="S2.p1.8.m8.1.1.cmml">g</mi><mo id="S2.p1.8.m8.1.2.3.3.2.2" stretchy="false" xref="S2.p1.8.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.8.m8.1b"><apply id="S2.p1.8.m8.1.2.cmml" xref="S2.p1.8.m8.1.2"><geq id="S2.p1.8.m8.1.2.1.cmml" xref="S2.p1.8.m8.1.2.1"></geq><ci id="S2.p1.8.m8.1.2.2.cmml" xref="S2.p1.8.m8.1.2.2">𝑓</ci><apply id="S2.p1.8.m8.1.2.3.cmml" xref="S2.p1.8.m8.1.2.3"><times id="S2.p1.8.m8.1.2.3.1.cmml" xref="S2.p1.8.m8.1.2.3.1"></times><ci id="S2.p1.8.m8.1.2.3.2.cmml" xref="S2.p1.8.m8.1.2.3.2">Ω</ci><ci id="S2.p1.8.m8.1.1.cmml" xref="S2.p1.8.m8.1.1">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m8.1c">f\geq\Omega(g)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m8.1d">italic_f ≥ roman_Ω ( italic_g )</annotation></semantics></math>. For a vector <math alttext="{\bm{v}}\in\mathbb{R}^{m}" class="ltx_Math" display="inline" id="S2.p1.9.m9.1"><semantics id="S2.p1.9.m9.1a"><mrow id="S2.p1.9.m9.1.1" xref="S2.p1.9.m9.1.1.cmml"><mi id="S2.p1.9.m9.1.1.2" xref="S2.p1.9.m9.1.1.2.cmml">𝒗</mi><mo id="S2.p1.9.m9.1.1.1" xref="S2.p1.9.m9.1.1.1.cmml">∈</mo><msup id="S2.p1.9.m9.1.1.3" xref="S2.p1.9.m9.1.1.3.cmml"><mi id="S2.p1.9.m9.1.1.3.2" xref="S2.p1.9.m9.1.1.3.2.cmml">ℝ</mi><mi id="S2.p1.9.m9.1.1.3.3" xref="S2.p1.9.m9.1.1.3.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m9.1b"><apply id="S2.p1.9.m9.1.1.cmml" xref="S2.p1.9.m9.1.1"><in id="S2.p1.9.m9.1.1.1.cmml" xref="S2.p1.9.m9.1.1.1"></in><ci id="S2.p1.9.m9.1.1.2.cmml" xref="S2.p1.9.m9.1.1.2">𝒗</ci><apply id="S2.p1.9.m9.1.1.3.cmml" xref="S2.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.p1.9.m9.1.1.3.1.cmml" xref="S2.p1.9.m9.1.1.3">superscript</csymbol><ci id="S2.p1.9.m9.1.1.3.2.cmml" xref="S2.p1.9.m9.1.1.3.2">ℝ</ci><ci id="S2.p1.9.m9.1.1.3.3.cmml" xref="S2.p1.9.m9.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m9.1c">{\bm{v}}\in\mathbb{R}^{m}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m9.1d">bold_italic_v ∈ blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="{\bm{v}}[i]" class="ltx_Math" display="inline" id="S2.p1.10.m10.1"><semantics id="S2.p1.10.m10.1a"><mrow id="S2.p1.10.m10.1.2" xref="S2.p1.10.m10.1.2.cmml"><mi id="S2.p1.10.m10.1.2.2" xref="S2.p1.10.m10.1.2.2.cmml">𝒗</mi><mo id="S2.p1.10.m10.1.2.1" xref="S2.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.p1.10.m10.1.2.3.2" xref="S2.p1.10.m10.1.2.3.1.cmml"><mo id="S2.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S2.p1.10.m10.1.2.3.1.1.cmml">[</mo><mi id="S2.p1.10.m10.1.1" xref="S2.p1.10.m10.1.1.cmml">i</mi><mo id="S2.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S2.p1.10.m10.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.10.m10.1b"><apply id="S2.p1.10.m10.1.2.cmml" xref="S2.p1.10.m10.1.2"><times id="S2.p1.10.m10.1.2.1.cmml" xref="S2.p1.10.m10.1.2.1"></times><ci id="S2.p1.10.m10.1.2.2.cmml" xref="S2.p1.10.m10.1.2.2">𝒗</ci><apply id="S2.p1.10.m10.1.2.3.1.cmml" xref="S2.p1.10.m10.1.2.3.2"><csymbol cd="latexml" id="S2.p1.10.m10.1.2.3.1.1.cmml" xref="S2.p1.10.m10.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.p1.10.m10.1.1.cmml" xref="S2.p1.10.m10.1.1">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m10.1c">{\bm{v}}[i]</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m10.1d">bold_italic_v [ italic_i ]</annotation></semantics></math> denotes its <math alttext="i" class="ltx_Math" display="inline" id="S2.p1.11.m11.1"><semantics id="S2.p1.11.m11.1a"><mi id="S2.p1.11.m11.1.1" xref="S2.p1.11.m11.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.p1.11.m11.1b"><ci id="S2.p1.11.m11.1.1.cmml" xref="S2.p1.11.m11.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m11.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m11.1d">italic_i</annotation></semantics></math>-th component. <math alttext="\quantity{\cdot}" class="ltx_Math" display="inline" id="S2.p1.12.m12.1"><semantics id="S2.p1.12.m12.1a"><mrow id="S2.p1.12.m12.1.1.3"><mo id="S2.p1.12.m12.1.1.3.1">{</mo><mo id="S2.p1.12.m12.1.1.1.1.1" lspace="0em" rspace="0em" xref="S2.p1.12.m12.1.1.1.1.1.cmml">⋅</mo><mo id="S2.p1.12.m12.1.1.3.2">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m12.1b"><ci id="S2.p1.12.m12.1.1.1.1.1.cmml" xref="S2.p1.12.m12.1.1.1.1.1">⋅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m12.1c">\quantity{\cdot}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m12.1d">{ start_ARG ⋅ end_ARG }</annotation></semantics></math> denotes an indicator, <span class="ltx_text ltx_font_italic" id="S2.p1.18.1">i.e.</span>, for a statement <math alttext="p" class="ltx_Math" display="inline" id="S2.p1.13.m13.1"><semantics id="S2.p1.13.m13.1a"><mi id="S2.p1.13.m13.1.1" xref="S2.p1.13.m13.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.p1.13.m13.1b"><ci id="S2.p1.13.m13.1.1.cmml" xref="S2.p1.13.m13.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m13.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m13.1d">italic_p</annotation></semantics></math>, <math alttext="\quantity{p}=1" class="ltx_Math" display="inline" id="S2.p1.14.m14.1"><semantics id="S2.p1.14.m14.1a"><mrow id="S2.p1.14.m14.1.2" xref="S2.p1.14.m14.1.2.cmml"><mrow id="S2.p1.14.m14.1.1.3" xref="S2.p1.14.m14.1.2.cmml"><mo id="S2.p1.14.m14.1.1.3.1" xref="S2.p1.14.m14.1.2.cmml">{</mo><mi id="S2.p1.14.m14.1.1.1.1.1" xref="S2.p1.14.m14.1.1.1.1.1.cmml">p</mi><mo id="S2.p1.14.m14.1.1.3.2" xref="S2.p1.14.m14.1.2.cmml">}</mo></mrow><mo id="S2.p1.14.m14.1.2.1" xref="S2.p1.14.m14.1.2.1.cmml">=</mo><mn id="S2.p1.14.m14.1.2.2" xref="S2.p1.14.m14.1.2.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.14.m14.1b"><apply id="S2.p1.14.m14.1.2.cmml" xref="S2.p1.14.m14.1.2"><eq id="S2.p1.14.m14.1.2.1.cmml" xref="S2.p1.14.m14.1.2.1"></eq><ci id="S2.p1.14.m14.1.1.1.1.1.cmml" xref="S2.p1.14.m14.1.1.1.1.1">𝑝</ci><cn id="S2.p1.14.m14.1.2.2.cmml" type="integer" xref="S2.p1.14.m14.1.2.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m14.1c">\quantity{p}=1</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m14.1d">{ start_ARG italic_p end_ARG } = 1</annotation></semantics></math> if <math alttext="p" class="ltx_Math" display="inline" id="S2.p1.15.m15.1"><semantics id="S2.p1.15.m15.1a"><mi id="S2.p1.15.m15.1.1" xref="S2.p1.15.m15.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.p1.15.m15.1b"><ci id="S2.p1.15.m15.1.1.cmml" xref="S2.p1.15.m15.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m15.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m15.1d">italic_p</annotation></semantics></math> is true and <math alttext="0" class="ltx_Math" display="inline" id="S2.p1.16.m16.1"><semantics id="S2.p1.16.m16.1a"><mn id="S2.p1.16.m16.1.1" xref="S2.p1.16.m16.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.p1.16.m16.1b"><cn id="S2.p1.16.m16.1.1.cmml" type="integer" xref="S2.p1.16.m16.1.1">0</cn></annotation-xml></semantics></math> if <math alttext="p" class="ltx_Math" display="inline" id="S2.p1.17.m17.1"><semantics id="S2.p1.17.m17.1a"><mi id="S2.p1.17.m17.1.1" xref="S2.p1.17.m17.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.p1.17.m17.1b"><ci id="S2.p1.17.m17.1.1.cmml" xref="S2.p1.17.m17.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.17.m17.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.p1.17.m17.1d">italic_p</annotation></semantics></math> is false. <math alttext="\mathbb{R}_{+}" class="ltx_Math" display="inline" id="S2.p1.18.m18.1"><semantics id="S2.p1.18.m18.1a"><msub id="S2.p1.18.m18.1.1" xref="S2.p1.18.m18.1.1.cmml"><mi id="S2.p1.18.m18.1.1.2" xref="S2.p1.18.m18.1.1.2.cmml">ℝ</mi><mo id="S2.p1.18.m18.1.1.3" xref="S2.p1.18.m18.1.1.3.cmml">+</mo></msub><annotation-xml encoding="MathML-Content" id="S2.p1.18.m18.1b"><apply id="S2.p1.18.m18.1.1.cmml" xref="S2.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S2.p1.18.m18.1.1.1.cmml" xref="S2.p1.18.m18.1.1">subscript</csymbol><ci id="S2.p1.18.m18.1.1.2.cmml" xref="S2.p1.18.m18.1.1.2">ℝ</ci><plus id="S2.p1.18.m18.1.1.3.cmml" xref="S2.p1.18.m18.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.18.m18.1c">\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.18.m18.1d">blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> is the set of nonnegative real numbers.</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>Normal form games</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.26">A normal-form game <math alttext="\Gamma=(n,A,U)" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.3"><semantics id="S2.SS1.p1.1.m1.3a"><mrow id="S2.SS1.p1.1.m1.3.4" xref="S2.SS1.p1.1.m1.3.4.cmml"><mi id="S2.SS1.p1.1.m1.3.4.2" mathvariant="normal" xref="S2.SS1.p1.1.m1.3.4.2.cmml">Γ</mi><mo id="S2.SS1.p1.1.m1.3.4.1" xref="S2.SS1.p1.1.m1.3.4.1.cmml">=</mo><mrow id="S2.SS1.p1.1.m1.3.4.3.2" xref="S2.SS1.p1.1.m1.3.4.3.1.cmml"><mo id="S2.SS1.p1.1.m1.3.4.3.2.1" stretchy="false" xref="S2.SS1.p1.1.m1.3.4.3.1.cmml">(</mo><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">n</mi><mo id="S2.SS1.p1.1.m1.3.4.3.2.2" xref="S2.SS1.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S2.SS1.p1.1.m1.2.2" xref="S2.SS1.p1.1.m1.2.2.cmml">A</mi><mo id="S2.SS1.p1.1.m1.3.4.3.2.3" xref="S2.SS1.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S2.SS1.p1.1.m1.3.3" xref="S2.SS1.p1.1.m1.3.3.cmml">U</mi><mo id="S2.SS1.p1.1.m1.3.4.3.2.4" stretchy="false" xref="S2.SS1.p1.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.3b"><apply id="S2.SS1.p1.1.m1.3.4.cmml" xref="S2.SS1.p1.1.m1.3.4"><eq id="S2.SS1.p1.1.m1.3.4.1.cmml" xref="S2.SS1.p1.1.m1.3.4.1"></eq><ci id="S2.SS1.p1.1.m1.3.4.2.cmml" xref="S2.SS1.p1.1.m1.3.4.2">Γ</ci><vector id="S2.SS1.p1.1.m1.3.4.3.1.cmml" xref="S2.SS1.p1.1.m1.3.4.3.2"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑛</ci><ci id="S2.SS1.p1.1.m1.2.2.cmml" xref="S2.SS1.p1.1.m1.2.2">𝐴</ci><ci id="S2.SS1.p1.1.m1.3.3.cmml" xref="S2.SS1.p1.1.m1.3.3">𝑈</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.3c">\Gamma=(n,A,U)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.3d">roman_Γ = ( italic_n , italic_A , italic_U )</annotation></semantics></math> consists of a set of <math alttext="n" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><ci id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_n</annotation></semantics></math> <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.26.1">agents</span>, or <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.26.2">players</span>, which we will identify with the set of integers <math alttext="[n]" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><mrow id="S2.SS1.p1.3.m3.1.2.2" xref="S2.SS1.p1.3.m3.1.2.1.cmml"><mo id="S2.SS1.p1.3.m3.1.2.2.1" stretchy="false" xref="S2.SS1.p1.3.m3.1.2.1.1.cmml">[</mo><mi id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">n</mi><mo id="S2.SS1.p1.3.m3.1.2.2.2" stretchy="false" xref="S2.SS1.p1.3.m3.1.2.1.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.2.1.cmml" xref="S2.SS1.p1.3.m3.1.2.2"><csymbol cd="latexml" id="S2.SS1.p1.3.m3.1.2.1.1.cmml" xref="S2.SS1.p1.3.m3.1.2.2.1">delimited-[]</csymbol><ci id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">[n]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">[ italic_n ]</annotation></semantics></math>. Each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mi id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><ci id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_i</annotation></semantics></math> has an action set <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><msub id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml"><mi id="S2.SS1.p1.5.m5.1.1.2" xref="S2.SS1.p1.5.m5.1.1.2.cmml">A</mi><mi id="S2.SS1.p1.5.m5.1.1.3" xref="S2.SS1.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m5.1.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p1.5.m5.1.1.2.cmml" xref="S2.SS1.p1.5.m5.1.1.2">𝐴</ci><ci id="S2.SS1.p1.5.m5.1.1.3.cmml" xref="S2.SS1.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> of size <math alttext="m_{i}\geq 2" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><mrow id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml"><msub id="S2.SS1.p1.6.m6.1.1.2" xref="S2.SS1.p1.6.m6.1.1.2.cmml"><mi id="S2.SS1.p1.6.m6.1.1.2.2" xref="S2.SS1.p1.6.m6.1.1.2.2.cmml">m</mi><mi id="S2.SS1.p1.6.m6.1.1.2.3" xref="S2.SS1.p1.6.m6.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p1.6.m6.1.1.1" xref="S2.SS1.p1.6.m6.1.1.1.cmml">≥</mo><mn id="S2.SS1.p1.6.m6.1.1.3" xref="S2.SS1.p1.6.m6.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><apply id="S2.SS1.p1.6.m6.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1"><geq id="S2.SS1.p1.6.m6.1.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1.1"></geq><apply id="S2.SS1.p1.6.m6.1.1.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.2.1.cmml" xref="S2.SS1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.2.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2.2">𝑚</ci><ci id="S2.SS1.p1.6.m6.1.1.2.3.cmml" xref="S2.SS1.p1.6.m6.1.1.2.3">𝑖</ci></apply><cn id="S2.SS1.p1.6.m6.1.1.3.cmml" type="integer" xref="S2.SS1.p1.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.1c">m_{i}\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 2</annotation></semantics></math>. We will let <math alttext="m:=\max_{i}m_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><mrow id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.2" xref="S2.SS1.p1.7.m7.1.1.2.cmml">m</mi><mo id="S2.SS1.p1.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p1.7.m7.1.1.1.cmml">:=</mo><mrow id="S2.SS1.p1.7.m7.1.1.3" xref="S2.SS1.p1.7.m7.1.1.3.cmml"><msub id="S2.SS1.p1.7.m7.1.1.3.1" xref="S2.SS1.p1.7.m7.1.1.3.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.3.1.2" xref="S2.SS1.p1.7.m7.1.1.3.1.2.cmml">max</mi><mi id="S2.SS1.p1.7.m7.1.1.3.1.3" xref="S2.SS1.p1.7.m7.1.1.3.1.3.cmml">i</mi></msub><mo id="S2.SS1.p1.7.m7.1.1.3a" lspace="0.167em" xref="S2.SS1.p1.7.m7.1.1.3.cmml">⁡</mo><msub id="S2.SS1.p1.7.m7.1.1.3.2" xref="S2.SS1.p1.7.m7.1.1.3.2.cmml"><mi id="S2.SS1.p1.7.m7.1.1.3.2.2" xref="S2.SS1.p1.7.m7.1.1.3.2.2.cmml">m</mi><mi id="S2.SS1.p1.7.m7.1.1.3.2.3" xref="S2.SS1.p1.7.m7.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><apply id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1"><csymbol cd="latexml" id="S2.SS1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1.1">assign</csymbol><ci id="S2.SS1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.2">𝑚</ci><apply id="S2.SS1.p1.7.m7.1.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3"><apply id="S2.SS1.p1.7.m7.1.1.3.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.3.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1">subscript</csymbol><max id="S2.SS1.p1.7.m7.1.1.3.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1.2"></max><ci id="S2.SS1.p1.7.m7.1.1.3.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1.3">𝑖</ci></apply><apply id="S2.SS1.p1.7.m7.1.1.3.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.3.2.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.3.2.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2.2">𝑚</ci><ci id="S2.SS1.p1.7.m7.1.1.3.2.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">m:=\max_{i}m_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">italic_m := roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="M=\prod_{i}m_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><mrow id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml"><mi id="S2.SS1.p1.8.m8.1.1.2" xref="S2.SS1.p1.8.m8.1.1.2.cmml">M</mi><mo id="S2.SS1.p1.8.m8.1.1.1" rspace="0.111em" xref="S2.SS1.p1.8.m8.1.1.1.cmml">=</mo><mrow id="S2.SS1.p1.8.m8.1.1.3" xref="S2.SS1.p1.8.m8.1.1.3.cmml"><msub id="S2.SS1.p1.8.m8.1.1.3.1" xref="S2.SS1.p1.8.m8.1.1.3.1.cmml"><mo id="S2.SS1.p1.8.m8.1.1.3.1.2" xref="S2.SS1.p1.8.m8.1.1.3.1.2.cmml">∏</mo><mi id="S2.SS1.p1.8.m8.1.1.3.1.3" xref="S2.SS1.p1.8.m8.1.1.3.1.3.cmml">i</mi></msub><msub id="S2.SS1.p1.8.m8.1.1.3.2" xref="S2.SS1.p1.8.m8.1.1.3.2.cmml"><mi id="S2.SS1.p1.8.m8.1.1.3.2.2" xref="S2.SS1.p1.8.m8.1.1.3.2.2.cmml">m</mi><mi id="S2.SS1.p1.8.m8.1.1.3.2.3" xref="S2.SS1.p1.8.m8.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><apply id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1"><eq id="S2.SS1.p1.8.m8.1.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1.1"></eq><ci id="S2.SS1.p1.8.m8.1.1.2.cmml" xref="S2.SS1.p1.8.m8.1.1.2">𝑀</ci><apply id="S2.SS1.p1.8.m8.1.1.3.cmml" xref="S2.SS1.p1.8.m8.1.1.3"><apply id="S2.SS1.p1.8.m8.1.1.3.1.cmml" xref="S2.SS1.p1.8.m8.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS1.p1.8.m8.1.1.3.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1.3.1">subscript</csymbol><csymbol cd="latexml" id="S2.SS1.p1.8.m8.1.1.3.1.2.cmml" xref="S2.SS1.p1.8.m8.1.1.3.1.2">product</csymbol><ci id="S2.SS1.p1.8.m8.1.1.3.1.3.cmml" xref="S2.SS1.p1.8.m8.1.1.3.1.3">𝑖</ci></apply><apply id="S2.SS1.p1.8.m8.1.1.3.2.cmml" xref="S2.SS1.p1.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.8.m8.1.1.3.2.1.cmml" xref="S2.SS1.p1.8.m8.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p1.8.m8.1.1.3.2.2.cmml" xref="S2.SS1.p1.8.m8.1.1.3.2.2">𝑚</ci><ci id="S2.SS1.p1.8.m8.1.1.3.2.3.cmml" xref="S2.SS1.p1.8.m8.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">M=\prod_{i}m_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">italic_M = ∏ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. A tuple <math alttext="a\in A:=A_{1}\times\dots\times A_{n}" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m9.1"><semantics id="S2.SS1.p1.9.m9.1a"><mrow id="S2.SS1.p1.9.m9.1.1" xref="S2.SS1.p1.9.m9.1.1.cmml"><mi id="S2.SS1.p1.9.m9.1.1.2" xref="S2.SS1.p1.9.m9.1.1.2.cmml">a</mi><mo id="S2.SS1.p1.9.m9.1.1.3" xref="S2.SS1.p1.9.m9.1.1.3.cmml">∈</mo><mi id="S2.SS1.p1.9.m9.1.1.4" xref="S2.SS1.p1.9.m9.1.1.4.cmml">A</mi><mo id="S2.SS1.p1.9.m9.1.1.5" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p1.9.m9.1.1.5.cmml">:=</mo><mrow id="S2.SS1.p1.9.m9.1.1.6" xref="S2.SS1.p1.9.m9.1.1.6.cmml"><msub id="S2.SS1.p1.9.m9.1.1.6.2" xref="S2.SS1.p1.9.m9.1.1.6.2.cmml"><mi id="S2.SS1.p1.9.m9.1.1.6.2.2" xref="S2.SS1.p1.9.m9.1.1.6.2.2.cmml">A</mi><mn id="S2.SS1.p1.9.m9.1.1.6.2.3" xref="S2.SS1.p1.9.m9.1.1.6.2.3.cmml">1</mn></msub><mo id="S2.SS1.p1.9.m9.1.1.6.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.9.m9.1.1.6.1.cmml">×</mo><mi id="S2.SS1.p1.9.m9.1.1.6.3" mathvariant="normal" xref="S2.SS1.p1.9.m9.1.1.6.3.cmml">⋯</mi><mo id="S2.SS1.p1.9.m9.1.1.6.1a" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.9.m9.1.1.6.1.cmml">×</mo><msub id="S2.SS1.p1.9.m9.1.1.6.4" xref="S2.SS1.p1.9.m9.1.1.6.4.cmml"><mi id="S2.SS1.p1.9.m9.1.1.6.4.2" xref="S2.SS1.p1.9.m9.1.1.6.4.2.cmml">A</mi><mi id="S2.SS1.p1.9.m9.1.1.6.4.3" xref="S2.SS1.p1.9.m9.1.1.6.4.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m9.1b"><apply id="S2.SS1.p1.9.m9.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1"><and id="S2.SS1.p1.9.m9.1.1a.cmml" xref="S2.SS1.p1.9.m9.1.1"></and><apply id="S2.SS1.p1.9.m9.1.1b.cmml" xref="S2.SS1.p1.9.m9.1.1"><in id="S2.SS1.p1.9.m9.1.1.3.cmml" xref="S2.SS1.p1.9.m9.1.1.3"></in><ci id="S2.SS1.p1.9.m9.1.1.2.cmml" xref="S2.SS1.p1.9.m9.1.1.2">𝑎</ci><ci id="S2.SS1.p1.9.m9.1.1.4.cmml" xref="S2.SS1.p1.9.m9.1.1.4">𝐴</ci></apply><apply id="S2.SS1.p1.9.m9.1.1c.cmml" xref="S2.SS1.p1.9.m9.1.1"><csymbol cd="latexml" id="S2.SS1.p1.9.m9.1.1.5.cmml" xref="S2.SS1.p1.9.m9.1.1.5">assign</csymbol><share href="https://arxiv.org/html/2503.01976v1#S2.SS1.p1.9.m9.1.1.4.cmml" id="S2.SS1.p1.9.m9.1.1d.cmml" xref="S2.SS1.p1.9.m9.1.1"></share><apply id="S2.SS1.p1.9.m9.1.1.6.cmml" xref="S2.SS1.p1.9.m9.1.1.6"><times id="S2.SS1.p1.9.m9.1.1.6.1.cmml" xref="S2.SS1.p1.9.m9.1.1.6.1"></times><apply id="S2.SS1.p1.9.m9.1.1.6.2.cmml" xref="S2.SS1.p1.9.m9.1.1.6.2"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.1.1.6.2.1.cmml" xref="S2.SS1.p1.9.m9.1.1.6.2">subscript</csymbol><ci id="S2.SS1.p1.9.m9.1.1.6.2.2.cmml" xref="S2.SS1.p1.9.m9.1.1.6.2.2">𝐴</ci><cn id="S2.SS1.p1.9.m9.1.1.6.2.3.cmml" type="integer" xref="S2.SS1.p1.9.m9.1.1.6.2.3">1</cn></apply><ci id="S2.SS1.p1.9.m9.1.1.6.3.cmml" xref="S2.SS1.p1.9.m9.1.1.6.3">⋯</ci><apply id="S2.SS1.p1.9.m9.1.1.6.4.cmml" xref="S2.SS1.p1.9.m9.1.1.6.4"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.1.1.6.4.1.cmml" xref="S2.SS1.p1.9.m9.1.1.6.4">subscript</csymbol><ci id="S2.SS1.p1.9.m9.1.1.6.4.2.cmml" xref="S2.SS1.p1.9.m9.1.1.6.4.2">𝐴</ci><ci id="S2.SS1.p1.9.m9.1.1.6.4.3.cmml" xref="S2.SS1.p1.9.m9.1.1.6.4.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m9.1c">a\in A:=A_{1}\times\dots\times A_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m9.1d">italic_a ∈ italic_A := italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × ⋯ × italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is an <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.26.3">action profile</span>. Each agent has a <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.26.4">utility function</span> <math alttext="U_{i}:A\to[0,1]" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m10.2"><semantics id="S2.SS1.p1.10.m10.2a"><mrow id="S2.SS1.p1.10.m10.2.3" xref="S2.SS1.p1.10.m10.2.3.cmml"><msub id="S2.SS1.p1.10.m10.2.3.2" xref="S2.SS1.p1.10.m10.2.3.2.cmml"><mi id="S2.SS1.p1.10.m10.2.3.2.2" xref="S2.SS1.p1.10.m10.2.3.2.2.cmml">U</mi><mi id="S2.SS1.p1.10.m10.2.3.2.3" xref="S2.SS1.p1.10.m10.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS1.p1.10.m10.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p1.10.m10.2.3.1.cmml">:</mo><mrow id="S2.SS1.p1.10.m10.2.3.3" xref="S2.SS1.p1.10.m10.2.3.3.cmml"><mi id="S2.SS1.p1.10.m10.2.3.3.2" xref="S2.SS1.p1.10.m10.2.3.3.2.cmml">A</mi><mo id="S2.SS1.p1.10.m10.2.3.3.1" stretchy="false" xref="S2.SS1.p1.10.m10.2.3.3.1.cmml">→</mo><mrow id="S2.SS1.p1.10.m10.2.3.3.3.2" xref="S2.SS1.p1.10.m10.2.3.3.3.1.cmml"><mo id="S2.SS1.p1.10.m10.2.3.3.3.2.1" stretchy="false" xref="S2.SS1.p1.10.m10.2.3.3.3.1.cmml">[</mo><mn id="S2.SS1.p1.10.m10.1.1" xref="S2.SS1.p1.10.m10.1.1.cmml">0</mn><mo id="S2.SS1.p1.10.m10.2.3.3.3.2.2" xref="S2.SS1.p1.10.m10.2.3.3.3.1.cmml">,</mo><mn id="S2.SS1.p1.10.m10.2.2" xref="S2.SS1.p1.10.m10.2.2.cmml">1</mn><mo id="S2.SS1.p1.10.m10.2.3.3.3.2.3" stretchy="false" xref="S2.SS1.p1.10.m10.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.10.m10.2b"><apply id="S2.SS1.p1.10.m10.2.3.cmml" xref="S2.SS1.p1.10.m10.2.3"><ci id="S2.SS1.p1.10.m10.2.3.1.cmml" xref="S2.SS1.p1.10.m10.2.3.1">:</ci><apply id="S2.SS1.p1.10.m10.2.3.2.cmml" xref="S2.SS1.p1.10.m10.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.10.m10.2.3.2.1.cmml" xref="S2.SS1.p1.10.m10.2.3.2">subscript</csymbol><ci id="S2.SS1.p1.10.m10.2.3.2.2.cmml" xref="S2.SS1.p1.10.m10.2.3.2.2">𝑈</ci><ci id="S2.SS1.p1.10.m10.2.3.2.3.cmml" xref="S2.SS1.p1.10.m10.2.3.2.3">𝑖</ci></apply><apply id="S2.SS1.p1.10.m10.2.3.3.cmml" xref="S2.SS1.p1.10.m10.2.3.3"><ci id="S2.SS1.p1.10.m10.2.3.3.1.cmml" xref="S2.SS1.p1.10.m10.2.3.3.1">→</ci><ci id="S2.SS1.p1.10.m10.2.3.3.2.cmml" xref="S2.SS1.p1.10.m10.2.3.3.2">𝐴</ci><interval closure="closed" id="S2.SS1.p1.10.m10.2.3.3.3.1.cmml" xref="S2.SS1.p1.10.m10.2.3.3.3.2"><cn id="S2.SS1.p1.10.m10.1.1.cmml" type="integer" xref="S2.SS1.p1.10.m10.1.1">0</cn><cn id="S2.SS1.p1.10.m10.2.2.cmml" type="integer" xref="S2.SS1.p1.10.m10.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m10.2c">U_{i}:A\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m10.2d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A → [ 0 , 1 ]</annotation></semantics></math>, denoting the utility for agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.11.m11.1"><semantics id="S2.SS1.p1.11.m11.1a"><mi id="S2.SS1.p1.11.m11.1.1" xref="S2.SS1.p1.11.m11.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.11.m11.1b"><ci id="S2.SS1.p1.11.m11.1.1.cmml" xref="S2.SS1.p1.11.m11.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.11.m11.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.11.m11.1d">italic_i</annotation></semantics></math> when the agents play action profile <math alttext="a\in A" class="ltx_Math" display="inline" id="S2.SS1.p1.12.m12.1"><semantics id="S2.SS1.p1.12.m12.1a"><mrow id="S2.SS1.p1.12.m12.1.1" xref="S2.SS1.p1.12.m12.1.1.cmml"><mi id="S2.SS1.p1.12.m12.1.1.2" xref="S2.SS1.p1.12.m12.1.1.2.cmml">a</mi><mo id="S2.SS1.p1.12.m12.1.1.1" xref="S2.SS1.p1.12.m12.1.1.1.cmml">∈</mo><mi id="S2.SS1.p1.12.m12.1.1.3" xref="S2.SS1.p1.12.m12.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.12.m12.1b"><apply id="S2.SS1.p1.12.m12.1.1.cmml" xref="S2.SS1.p1.12.m12.1.1"><in id="S2.SS1.p1.12.m12.1.1.1.cmml" xref="S2.SS1.p1.12.m12.1.1.1"></in><ci id="S2.SS1.p1.12.m12.1.1.2.cmml" xref="S2.SS1.p1.12.m12.1.1.2">𝑎</ci><ci id="S2.SS1.p1.12.m12.1.1.3.cmml" xref="S2.SS1.p1.12.m12.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.12.m12.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.12.m12.1d">italic_a ∈ italic_A</annotation></semantics></math>. A <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.26.5">mixed strategy</span> of agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.13.m13.1"><semantics id="S2.SS1.p1.13.m13.1a"><mi id="S2.SS1.p1.13.m13.1.1" xref="S2.SS1.p1.13.m13.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.13.m13.1b"><ci id="S2.SS1.p1.13.m13.1.1.cmml" xref="S2.SS1.p1.13.m13.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.13.m13.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.13.m13.1d">italic_i</annotation></semantics></math> is a distribution <math alttext="x_{i}\in\Delta(A_{i})" class="ltx_Math" display="inline" id="S2.SS1.p1.14.m14.1"><semantics id="S2.SS1.p1.14.m14.1a"><mrow id="S2.SS1.p1.14.m14.1.1" xref="S2.SS1.p1.14.m14.1.1.cmml"><msub id="S2.SS1.p1.14.m14.1.1.3" xref="S2.SS1.p1.14.m14.1.1.3.cmml"><mi id="S2.SS1.p1.14.m14.1.1.3.2" xref="S2.SS1.p1.14.m14.1.1.3.2.cmml">x</mi><mi id="S2.SS1.p1.14.m14.1.1.3.3" xref="S2.SS1.p1.14.m14.1.1.3.3.cmml">i</mi></msub><mo id="S2.SS1.p1.14.m14.1.1.2" xref="S2.SS1.p1.14.m14.1.1.2.cmml">∈</mo><mrow id="S2.SS1.p1.14.m14.1.1.1" xref="S2.SS1.p1.14.m14.1.1.1.cmml"><mi id="S2.SS1.p1.14.m14.1.1.1.3" mathvariant="normal" xref="S2.SS1.p1.14.m14.1.1.1.3.cmml">Δ</mi><mo id="S2.SS1.p1.14.m14.1.1.1.2" xref="S2.SS1.p1.14.m14.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.14.m14.1.1.1.1.1" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p1.14.m14.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p1.14.m14.1.1.1.1.1.1" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.14.m14.1.1.1.1.1.1.2" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.SS1.p1.14.m14.1.1.1.1.1.1.3" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS1.p1.14.m14.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.14.m14.1b"><apply id="S2.SS1.p1.14.m14.1.1.cmml" xref="S2.SS1.p1.14.m14.1.1"><in id="S2.SS1.p1.14.m14.1.1.2.cmml" xref="S2.SS1.p1.14.m14.1.1.2"></in><apply id="S2.SS1.p1.14.m14.1.1.3.cmml" xref="S2.SS1.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.14.m14.1.1.3.1.cmml" xref="S2.SS1.p1.14.m14.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.14.m14.1.1.3.2.cmml" xref="S2.SS1.p1.14.m14.1.1.3.2">𝑥</ci><ci id="S2.SS1.p1.14.m14.1.1.3.3.cmml" xref="S2.SS1.p1.14.m14.1.1.3.3">𝑖</ci></apply><apply id="S2.SS1.p1.14.m14.1.1.1.cmml" xref="S2.SS1.p1.14.m14.1.1.1"><times id="S2.SS1.p1.14.m14.1.1.1.2.cmml" xref="S2.SS1.p1.14.m14.1.1.1.2"></times><ci id="S2.SS1.p1.14.m14.1.1.1.3.cmml" xref="S2.SS1.p1.14.m14.1.1.1.3">Δ</ci><apply id="S2.SS1.p1.14.m14.1.1.1.1.1.1.cmml" xref="S2.SS1.p1.14.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.14.m14.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p1.14.m14.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.14.m14.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p1.14.m14.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p1.14.m14.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.14.m14.1c">x_{i}\in\Delta(A_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.14.m14.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ roman_Δ ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. We will overload the utility function <math alttext="U_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.15.m15.1"><semantics id="S2.SS1.p1.15.m15.1a"><msub id="S2.SS1.p1.15.m15.1.1" xref="S2.SS1.p1.15.m15.1.1.cmml"><mi id="S2.SS1.p1.15.m15.1.1.2" xref="S2.SS1.p1.15.m15.1.1.2.cmml">U</mi><mi id="S2.SS1.p1.15.m15.1.1.3" xref="S2.SS1.p1.15.m15.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.15.m15.1b"><apply id="S2.SS1.p1.15.m15.1.1.cmml" xref="S2.SS1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.15.m15.1.1.1.cmml" xref="S2.SS1.p1.15.m15.1.1">subscript</csymbol><ci id="S2.SS1.p1.15.m15.1.1.2.cmml" xref="S2.SS1.p1.15.m15.1.1.2">𝑈</ci><ci id="S2.SS1.p1.15.m15.1.1.3.cmml" xref="S2.SS1.p1.15.m15.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.15.m15.1c">U_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.15.m15.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to accept mixed strategies, so that <math alttext="U_{i}(x_{1},\dots,x_{n})" class="ltx_Math" display="inline" id="S2.SS1.p1.16.m16.3"><semantics id="S2.SS1.p1.16.m16.3a"><mrow id="S2.SS1.p1.16.m16.3.3" xref="S2.SS1.p1.16.m16.3.3.cmml"><msub id="S2.SS1.p1.16.m16.3.3.4" xref="S2.SS1.p1.16.m16.3.3.4.cmml"><mi id="S2.SS1.p1.16.m16.3.3.4.2" xref="S2.SS1.p1.16.m16.3.3.4.2.cmml">U</mi><mi id="S2.SS1.p1.16.m16.3.3.4.3" xref="S2.SS1.p1.16.m16.3.3.4.3.cmml">i</mi></msub><mo id="S2.SS1.p1.16.m16.3.3.3" xref="S2.SS1.p1.16.m16.3.3.3.cmml">⁢</mo><mrow id="S2.SS1.p1.16.m16.3.3.2.2" xref="S2.SS1.p1.16.m16.3.3.2.3.cmml"><mo id="S2.SS1.p1.16.m16.3.3.2.2.3" stretchy="false" xref="S2.SS1.p1.16.m16.3.3.2.3.cmml">(</mo><msub id="S2.SS1.p1.16.m16.2.2.1.1.1" xref="S2.SS1.p1.16.m16.2.2.1.1.1.cmml"><mi id="S2.SS1.p1.16.m16.2.2.1.1.1.2" xref="S2.SS1.p1.16.m16.2.2.1.1.1.2.cmml">x</mi><mn id="S2.SS1.p1.16.m16.2.2.1.1.1.3" xref="S2.SS1.p1.16.m16.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p1.16.m16.3.3.2.2.4" xref="S2.SS1.p1.16.m16.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p1.16.m16.1.1" mathvariant="normal" xref="S2.SS1.p1.16.m16.1.1.cmml">…</mi><mo id="S2.SS1.p1.16.m16.3.3.2.2.5" xref="S2.SS1.p1.16.m16.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p1.16.m16.3.3.2.2.2" xref="S2.SS1.p1.16.m16.3.3.2.2.2.cmml"><mi id="S2.SS1.p1.16.m16.3.3.2.2.2.2" xref="S2.SS1.p1.16.m16.3.3.2.2.2.2.cmml">x</mi><mi id="S2.SS1.p1.16.m16.3.3.2.2.2.3" xref="S2.SS1.p1.16.m16.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S2.SS1.p1.16.m16.3.3.2.2.6" stretchy="false" xref="S2.SS1.p1.16.m16.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.16.m16.3b"><apply id="S2.SS1.p1.16.m16.3.3.cmml" xref="S2.SS1.p1.16.m16.3.3"><times id="S2.SS1.p1.16.m16.3.3.3.cmml" xref="S2.SS1.p1.16.m16.3.3.3"></times><apply id="S2.SS1.p1.16.m16.3.3.4.cmml" xref="S2.SS1.p1.16.m16.3.3.4"><csymbol cd="ambiguous" id="S2.SS1.p1.16.m16.3.3.4.1.cmml" xref="S2.SS1.p1.16.m16.3.3.4">subscript</csymbol><ci id="S2.SS1.p1.16.m16.3.3.4.2.cmml" xref="S2.SS1.p1.16.m16.3.3.4.2">𝑈</ci><ci id="S2.SS1.p1.16.m16.3.3.4.3.cmml" xref="S2.SS1.p1.16.m16.3.3.4.3">𝑖</ci></apply><vector id="S2.SS1.p1.16.m16.3.3.2.3.cmml" xref="S2.SS1.p1.16.m16.3.3.2.2"><apply id="S2.SS1.p1.16.m16.2.2.1.1.1.cmml" xref="S2.SS1.p1.16.m16.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.16.m16.2.2.1.1.1.1.cmml" xref="S2.SS1.p1.16.m16.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.16.m16.2.2.1.1.1.2.cmml" xref="S2.SS1.p1.16.m16.2.2.1.1.1.2">𝑥</ci><cn id="S2.SS1.p1.16.m16.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.16.m16.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS1.p1.16.m16.1.1.cmml" xref="S2.SS1.p1.16.m16.1.1">…</ci><apply id="S2.SS1.p1.16.m16.3.3.2.2.2.cmml" xref="S2.SS1.p1.16.m16.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.16.m16.3.3.2.2.2.1.cmml" xref="S2.SS1.p1.16.m16.3.3.2.2.2">subscript</csymbol><ci id="S2.SS1.p1.16.m16.3.3.2.2.2.2.cmml" xref="S2.SS1.p1.16.m16.3.3.2.2.2.2">𝑥</ci><ci id="S2.SS1.p1.16.m16.3.3.2.2.2.3.cmml" xref="S2.SS1.p1.16.m16.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.16.m16.3c">U_{i}(x_{1},\dots,x_{n})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.16.m16.3d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> is the expected utility for agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.17.m17.1"><semantics id="S2.SS1.p1.17.m17.1a"><mi id="S2.SS1.p1.17.m17.1.1" xref="S2.SS1.p1.17.m17.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.17.m17.1b"><ci id="S2.SS1.p1.17.m17.1.1.cmml" xref="S2.SS1.p1.17.m17.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.17.m17.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.17.m17.1d">italic_i</annotation></semantics></math> when every agent <math alttext="j\in[n]" class="ltx_Math" display="inline" id="S2.SS1.p1.18.m18.1"><semantics id="S2.SS1.p1.18.m18.1a"><mrow id="S2.SS1.p1.18.m18.1.2" xref="S2.SS1.p1.18.m18.1.2.cmml"><mi id="S2.SS1.p1.18.m18.1.2.2" xref="S2.SS1.p1.18.m18.1.2.2.cmml">j</mi><mo id="S2.SS1.p1.18.m18.1.2.1" xref="S2.SS1.p1.18.m18.1.2.1.cmml">∈</mo><mrow id="S2.SS1.p1.18.m18.1.2.3.2" xref="S2.SS1.p1.18.m18.1.2.3.1.cmml"><mo id="S2.SS1.p1.18.m18.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.18.m18.1.2.3.1.1.cmml">[</mo><mi id="S2.SS1.p1.18.m18.1.1" xref="S2.SS1.p1.18.m18.1.1.cmml">n</mi><mo id="S2.SS1.p1.18.m18.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.18.m18.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.18.m18.1b"><apply id="S2.SS1.p1.18.m18.1.2.cmml" xref="S2.SS1.p1.18.m18.1.2"><in id="S2.SS1.p1.18.m18.1.2.1.cmml" xref="S2.SS1.p1.18.m18.1.2.1"></in><ci id="S2.SS1.p1.18.m18.1.2.2.cmml" xref="S2.SS1.p1.18.m18.1.2.2">𝑗</ci><apply id="S2.SS1.p1.18.m18.1.2.3.1.cmml" xref="S2.SS1.p1.18.m18.1.2.3.2"><csymbol cd="latexml" id="S2.SS1.p1.18.m18.1.2.3.1.1.cmml" xref="S2.SS1.p1.18.m18.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS1.p1.18.m18.1.1.cmml" xref="S2.SS1.p1.18.m18.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.18.m18.1c">j\in[n]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.18.m18.1d">italic_j ∈ [ italic_n ]</annotation></semantics></math> independently samples <math alttext="a_{j}\sim x_{j}" class="ltx_Math" display="inline" id="S2.SS1.p1.19.m19.1"><semantics id="S2.SS1.p1.19.m19.1a"><mrow id="S2.SS1.p1.19.m19.1.1" xref="S2.SS1.p1.19.m19.1.1.cmml"><msub id="S2.SS1.p1.19.m19.1.1.2" xref="S2.SS1.p1.19.m19.1.1.2.cmml"><mi id="S2.SS1.p1.19.m19.1.1.2.2" xref="S2.SS1.p1.19.m19.1.1.2.2.cmml">a</mi><mi id="S2.SS1.p1.19.m19.1.1.2.3" xref="S2.SS1.p1.19.m19.1.1.2.3.cmml">j</mi></msub><mo id="S2.SS1.p1.19.m19.1.1.1" xref="S2.SS1.p1.19.m19.1.1.1.cmml">∼</mo><msub id="S2.SS1.p1.19.m19.1.1.3" xref="S2.SS1.p1.19.m19.1.1.3.cmml"><mi id="S2.SS1.p1.19.m19.1.1.3.2" xref="S2.SS1.p1.19.m19.1.1.3.2.cmml">x</mi><mi id="S2.SS1.p1.19.m19.1.1.3.3" xref="S2.SS1.p1.19.m19.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.19.m19.1b"><apply id="S2.SS1.p1.19.m19.1.1.cmml" xref="S2.SS1.p1.19.m19.1.1"><csymbol cd="latexml" id="S2.SS1.p1.19.m19.1.1.1.cmml" xref="S2.SS1.p1.19.m19.1.1.1">similar-to</csymbol><apply id="S2.SS1.p1.19.m19.1.1.2.cmml" xref="S2.SS1.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.19.m19.1.1.2.1.cmml" xref="S2.SS1.p1.19.m19.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.19.m19.1.1.2.2.cmml" xref="S2.SS1.p1.19.m19.1.1.2.2">𝑎</ci><ci id="S2.SS1.p1.19.m19.1.1.2.3.cmml" xref="S2.SS1.p1.19.m19.1.1.2.3">𝑗</ci></apply><apply id="S2.SS1.p1.19.m19.1.1.3.cmml" xref="S2.SS1.p1.19.m19.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.19.m19.1.1.3.1.cmml" xref="S2.SS1.p1.19.m19.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.19.m19.1.1.3.2.cmml" xref="S2.SS1.p1.19.m19.1.1.3.2">𝑥</ci><ci id="S2.SS1.p1.19.m19.1.1.3.3.cmml" xref="S2.SS1.p1.19.m19.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.19.m19.1c">a_{j}\sim x_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.19.m19.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∼ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. As is standard in game theory, we will use <math alttext="-i" class="ltx_Math" display="inline" id="S2.SS1.p1.20.m20.1"><semantics id="S2.SS1.p1.20.m20.1a"><mrow id="S2.SS1.p1.20.m20.1.1" xref="S2.SS1.p1.20.m20.1.1.cmml"><mo id="S2.SS1.p1.20.m20.1.1a" xref="S2.SS1.p1.20.m20.1.1.cmml">−</mo><mi id="S2.SS1.p1.20.m20.1.1.2" xref="S2.SS1.p1.20.m20.1.1.2.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.20.m20.1b"><apply id="S2.SS1.p1.20.m20.1.1.cmml" xref="S2.SS1.p1.20.m20.1.1"><minus id="S2.SS1.p1.20.m20.1.1.1.cmml" xref="S2.SS1.p1.20.m20.1.1"></minus><ci id="S2.SS1.p1.20.m20.1.1.2.cmml" xref="S2.SS1.p1.20.m20.1.1.2">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.20.m20.1c">-i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.20.m20.1d">- italic_i</annotation></semantics></math> to refer to the tuple of all agents except <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.21.m21.1"><semantics id="S2.SS1.p1.21.m21.1a"><mi id="S2.SS1.p1.21.m21.1.1" xref="S2.SS1.p1.21.m21.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.21.m21.1b"><ci id="S2.SS1.p1.21.m21.1.1.cmml" xref="S2.SS1.p1.21.m21.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.21.m21.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.21.m21.1d">italic_i</annotation></semantics></math>, so that, for instance, <math alttext="U_{i}(a_{i}^{\prime},a_{-i})" class="ltx_Math" display="inline" id="S2.SS1.p1.22.m22.2"><semantics id="S2.SS1.p1.22.m22.2a"><mrow id="S2.SS1.p1.22.m22.2.2" xref="S2.SS1.p1.22.m22.2.2.cmml"><msub id="S2.SS1.p1.22.m22.2.2.4" xref="S2.SS1.p1.22.m22.2.2.4.cmml"><mi id="S2.SS1.p1.22.m22.2.2.4.2" xref="S2.SS1.p1.22.m22.2.2.4.2.cmml">U</mi><mi id="S2.SS1.p1.22.m22.2.2.4.3" xref="S2.SS1.p1.22.m22.2.2.4.3.cmml">i</mi></msub><mo id="S2.SS1.p1.22.m22.2.2.3" xref="S2.SS1.p1.22.m22.2.2.3.cmml">⁢</mo><mrow id="S2.SS1.p1.22.m22.2.2.2.2" xref="S2.SS1.p1.22.m22.2.2.2.3.cmml"><mo id="S2.SS1.p1.22.m22.2.2.2.2.3" stretchy="false" xref="S2.SS1.p1.22.m22.2.2.2.3.cmml">(</mo><msubsup id="S2.SS1.p1.22.m22.1.1.1.1.1" xref="S2.SS1.p1.22.m22.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.22.m22.1.1.1.1.1.2.2" xref="S2.SS1.p1.22.m22.1.1.1.1.1.2.2.cmml">a</mi><mi id="S2.SS1.p1.22.m22.1.1.1.1.1.2.3" xref="S2.SS1.p1.22.m22.1.1.1.1.1.2.3.cmml">i</mi><mo id="S2.SS1.p1.22.m22.1.1.1.1.1.3" xref="S2.SS1.p1.22.m22.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="S2.SS1.p1.22.m22.2.2.2.2.4" xref="S2.SS1.p1.22.m22.2.2.2.3.cmml">,</mo><msub id="S2.SS1.p1.22.m22.2.2.2.2.2" xref="S2.SS1.p1.22.m22.2.2.2.2.2.cmml"><mi id="S2.SS1.p1.22.m22.2.2.2.2.2.2" xref="S2.SS1.p1.22.m22.2.2.2.2.2.2.cmml">a</mi><mrow id="S2.SS1.p1.22.m22.2.2.2.2.2.3" xref="S2.SS1.p1.22.m22.2.2.2.2.2.3.cmml"><mo id="S2.SS1.p1.22.m22.2.2.2.2.2.3a" xref="S2.SS1.p1.22.m22.2.2.2.2.2.3.cmml">−</mo><mi id="S2.SS1.p1.22.m22.2.2.2.2.2.3.2" xref="S2.SS1.p1.22.m22.2.2.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="S2.SS1.p1.22.m22.2.2.2.2.5" stretchy="false" xref="S2.SS1.p1.22.m22.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.22.m22.2b"><apply id="S2.SS1.p1.22.m22.2.2.cmml" xref="S2.SS1.p1.22.m22.2.2"><times id="S2.SS1.p1.22.m22.2.2.3.cmml" xref="S2.SS1.p1.22.m22.2.2.3"></times><apply id="S2.SS1.p1.22.m22.2.2.4.cmml" xref="S2.SS1.p1.22.m22.2.2.4"><csymbol cd="ambiguous" id="S2.SS1.p1.22.m22.2.2.4.1.cmml" xref="S2.SS1.p1.22.m22.2.2.4">subscript</csymbol><ci id="S2.SS1.p1.22.m22.2.2.4.2.cmml" xref="S2.SS1.p1.22.m22.2.2.4.2">𝑈</ci><ci id="S2.SS1.p1.22.m22.2.2.4.3.cmml" xref="S2.SS1.p1.22.m22.2.2.4.3">𝑖</ci></apply><interval closure="open" id="S2.SS1.p1.22.m22.2.2.2.3.cmml" xref="S2.SS1.p1.22.m22.2.2.2.2"><apply id="S2.SS1.p1.22.m22.1.1.1.1.1.cmml" xref="S2.SS1.p1.22.m22.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.22.m22.1.1.1.1.1.1.cmml" xref="S2.SS1.p1.22.m22.1.1.1.1.1">superscript</csymbol><apply id="S2.SS1.p1.22.m22.1.1.1.1.1.2.cmml" xref="S2.SS1.p1.22.m22.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.22.m22.1.1.1.1.1.2.1.cmml" xref="S2.SS1.p1.22.m22.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.22.m22.1.1.1.1.1.2.2.cmml" xref="S2.SS1.p1.22.m22.1.1.1.1.1.2.2">𝑎</ci><ci id="S2.SS1.p1.22.m22.1.1.1.1.1.2.3.cmml" xref="S2.SS1.p1.22.m22.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p1.22.m22.1.1.1.1.1.3.cmml" xref="S2.SS1.p1.22.m22.1.1.1.1.1.3">′</ci></apply><apply id="S2.SS1.p1.22.m22.2.2.2.2.2.cmml" xref="S2.SS1.p1.22.m22.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.22.m22.2.2.2.2.2.1.cmml" xref="S2.SS1.p1.22.m22.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p1.22.m22.2.2.2.2.2.2.cmml" xref="S2.SS1.p1.22.m22.2.2.2.2.2.2">𝑎</ci><apply id="S2.SS1.p1.22.m22.2.2.2.2.2.3.cmml" xref="S2.SS1.p1.22.m22.2.2.2.2.2.3"><minus id="S2.SS1.p1.22.m22.2.2.2.2.2.3.1.cmml" xref="S2.SS1.p1.22.m22.2.2.2.2.2.3"></minus><ci id="S2.SS1.p1.22.m22.2.2.2.2.2.3.2.cmml" xref="S2.SS1.p1.22.m22.2.2.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.22.m22.2c">U_{i}(a_{i}^{\prime},a_{-i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.22.m22.2d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is the utility of agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.23.m23.1"><semantics id="S2.SS1.p1.23.m23.1a"><mi id="S2.SS1.p1.23.m23.1.1" xref="S2.SS1.p1.23.m23.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.23.m23.1b"><ci id="S2.SS1.p1.23.m23.1.1.cmml" xref="S2.SS1.p1.23.m23.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.23.m23.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.23.m23.1d">italic_i</annotation></semantics></math> when agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.24.m24.1"><semantics id="S2.SS1.p1.24.m24.1a"><mi id="S2.SS1.p1.24.m24.1.1" xref="S2.SS1.p1.24.m24.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.24.m24.1b"><ci id="S2.SS1.p1.24.m24.1.1.cmml" xref="S2.SS1.p1.24.m24.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.24.m24.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.24.m24.1d">italic_i</annotation></semantics></math> plays action <math alttext="a_{i}^{\prime}\in A_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.25.m25.1"><semantics id="S2.SS1.p1.25.m25.1a"><mrow id="S2.SS1.p1.25.m25.1.1" xref="S2.SS1.p1.25.m25.1.1.cmml"><msubsup id="S2.SS1.p1.25.m25.1.1.2" xref="S2.SS1.p1.25.m25.1.1.2.cmml"><mi id="S2.SS1.p1.25.m25.1.1.2.2.2" xref="S2.SS1.p1.25.m25.1.1.2.2.2.cmml">a</mi><mi id="S2.SS1.p1.25.m25.1.1.2.2.3" xref="S2.SS1.p1.25.m25.1.1.2.2.3.cmml">i</mi><mo id="S2.SS1.p1.25.m25.1.1.2.3" xref="S2.SS1.p1.25.m25.1.1.2.3.cmml">′</mo></msubsup><mo id="S2.SS1.p1.25.m25.1.1.1" xref="S2.SS1.p1.25.m25.1.1.1.cmml">∈</mo><msub id="S2.SS1.p1.25.m25.1.1.3" xref="S2.SS1.p1.25.m25.1.1.3.cmml"><mi id="S2.SS1.p1.25.m25.1.1.3.2" xref="S2.SS1.p1.25.m25.1.1.3.2.cmml">A</mi><mi id="S2.SS1.p1.25.m25.1.1.3.3" xref="S2.SS1.p1.25.m25.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.25.m25.1b"><apply id="S2.SS1.p1.25.m25.1.1.cmml" xref="S2.SS1.p1.25.m25.1.1"><in id="S2.SS1.p1.25.m25.1.1.1.cmml" xref="S2.SS1.p1.25.m25.1.1.1"></in><apply id="S2.SS1.p1.25.m25.1.1.2.cmml" xref="S2.SS1.p1.25.m25.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.25.m25.1.1.2.1.cmml" xref="S2.SS1.p1.25.m25.1.1.2">superscript</csymbol><apply id="S2.SS1.p1.25.m25.1.1.2.2.cmml" xref="S2.SS1.p1.25.m25.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.25.m25.1.1.2.2.1.cmml" xref="S2.SS1.p1.25.m25.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.25.m25.1.1.2.2.2.cmml" xref="S2.SS1.p1.25.m25.1.1.2.2.2">𝑎</ci><ci id="S2.SS1.p1.25.m25.1.1.2.2.3.cmml" xref="S2.SS1.p1.25.m25.1.1.2.2.3">𝑖</ci></apply><ci id="S2.SS1.p1.25.m25.1.1.2.3.cmml" xref="S2.SS1.p1.25.m25.1.1.2.3">′</ci></apply><apply id="S2.SS1.p1.25.m25.1.1.3.cmml" xref="S2.SS1.p1.25.m25.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.25.m25.1.1.3.1.cmml" xref="S2.SS1.p1.25.m25.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.25.m25.1.1.3.2.cmml" xref="S2.SS1.p1.25.m25.1.1.3.2">𝐴</ci><ci id="S2.SS1.p1.25.m25.1.1.3.3.cmml" xref="S2.SS1.p1.25.m25.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.25.m25.1c">a_{i}^{\prime}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.25.m25.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and every other agent plays according to profile <math alttext="a_{-i}\in\bigtimes_{j\neq i}A_{j}" class="ltx_math_unparsed" display="inline" id="S2.SS1.p1.26.m26.1"><semantics id="S2.SS1.p1.26.m26.1a"><mrow id="S2.SS1.p1.26.m26.1b"><msub id="S2.SS1.p1.26.m26.1.1"><mi id="S2.SS1.p1.26.m26.1.1.2">a</mi><mrow id="S2.SS1.p1.26.m26.1.1.3"><mo id="S2.SS1.p1.26.m26.1.1.3a">−</mo><mi id="S2.SS1.p1.26.m26.1.1.3.2">i</mi></mrow></msub><mo id="S2.SS1.p1.26.m26.1.2" rspace="0em">∈</mo><msub id="S2.SS1.p1.26.m26.1.3"><mo id="S2.SS1.p1.26.m26.1.3.2" lspace="0em" mathsize="160%" rspace="0.222em">×</mo><mrow id="S2.SS1.p1.26.m26.1.3.3"><mi id="S2.SS1.p1.26.m26.1.3.3.2">j</mi><mo id="S2.SS1.p1.26.m26.1.3.3.1">≠</mo><mi id="S2.SS1.p1.26.m26.1.3.3.3">i</mi></mrow></msub><msub id="S2.SS1.p1.26.m26.1.4"><mi id="S2.SS1.p1.26.m26.1.4.2">A</mi><mi id="S2.SS1.p1.26.m26.1.4.3">j</mi></msub></mrow><annotation encoding="application/x-tex" id="S2.SS1.p1.26.m26.1c">a_{-i}\in\bigtimes_{j\neq i}A_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.26.m26.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ∈ × start_POSTSUBSCRIPT italic_j ≠ italic_i end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </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>No-regret learning</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.8">In <span class="ltx_text ltx_font_italic" id="S2.SS2.p1.8.1">no-regret learning</span>, a learner has a convex compact strategy set <math alttext="{\mathcal{X}}\subset\mathbb{R}^{m}" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mrow id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.1.m1.1.1.2" xref="S2.SS2.p1.1.m1.1.1.2.cmml">𝒳</mi><mo id="S2.SS2.p1.1.m1.1.1.1" xref="S2.SS2.p1.1.m1.1.1.1.cmml">⊂</mo><msup id="S2.SS2.p1.1.m1.1.1.3" xref="S2.SS2.p1.1.m1.1.1.3.cmml"><mi id="S2.SS2.p1.1.m1.1.1.3.2" xref="S2.SS2.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS2.p1.1.m1.1.1.3.3" xref="S2.SS2.p1.1.m1.1.1.3.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><apply id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1"><subset id="S2.SS2.p1.1.m1.1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1.1"></subset><ci id="S2.SS2.p1.1.m1.1.1.2.cmml" xref="S2.SS2.p1.1.m1.1.1.2">𝒳</ci><apply id="S2.SS2.p1.1.m1.1.1.3.cmml" xref="S2.SS2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS2.p1.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS2.p1.1.m1.1.1.3.2.cmml" xref="S2.SS2.p1.1.m1.1.1.3.2">ℝ</ci><ci id="S2.SS2.p1.1.m1.1.1.3.3.cmml" xref="S2.SS2.p1.1.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">{\mathcal{X}}\subset\mathbb{R}^{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">caligraphic_X ⊂ blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math>, and interacts with a possibly adversarial environment. On each timestep <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><mi id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><ci id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">italic_t</annotation></semantics></math>, the learner selects a strategy <math alttext="{\bm{x}}^{t}\in{\mathcal{X}}" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.1"><semantics id="S2.SS2.p1.3.m3.1a"><mrow id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml"><msup id="S2.SS2.p1.3.m3.1.1.2" xref="S2.SS2.p1.3.m3.1.1.2.cmml"><mi id="S2.SS2.p1.3.m3.1.1.2.2" xref="S2.SS2.p1.3.m3.1.1.2.2.cmml">𝒙</mi><mi id="S2.SS2.p1.3.m3.1.1.2.3" xref="S2.SS2.p1.3.m3.1.1.2.3.cmml">t</mi></msup><mo id="S2.SS2.p1.3.m3.1.1.1" xref="S2.SS2.p1.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.3.m3.1.1.3" xref="S2.SS2.p1.3.m3.1.1.3.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.1b"><apply id="S2.SS2.p1.3.m3.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1"><in id="S2.SS2.p1.3.m3.1.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1.1"></in><apply id="S2.SS2.p1.3.m3.1.1.2.cmml" xref="S2.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.3.m3.1.1.2.1.cmml" xref="S2.SS2.p1.3.m3.1.1.2">superscript</csymbol><ci id="S2.SS2.p1.3.m3.1.1.2.2.cmml" xref="S2.SS2.p1.3.m3.1.1.2.2">𝒙</ci><ci id="S2.SS2.p1.3.m3.1.1.2.3.cmml" xref="S2.SS2.p1.3.m3.1.1.2.3">𝑡</ci></apply><ci id="S2.SS2.p1.3.m3.1.1.3.cmml" xref="S2.SS2.p1.3.m3.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.1c">{\bm{x}}^{t}\in{\mathcal{X}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.1d">bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∈ caligraphic_X</annotation></semantics></math>. Simultaneously, the environment, possibly adversarially, selects a linear utility vector <math alttext="{\bm{u}}^{t}\in\mathbb{R}^{m}" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m4.1"><semantics id="S2.SS2.p1.4.m4.1a"><mrow id="S2.SS2.p1.4.m4.1.1" xref="S2.SS2.p1.4.m4.1.1.cmml"><msup id="S2.SS2.p1.4.m4.1.1.2" xref="S2.SS2.p1.4.m4.1.1.2.cmml"><mi id="S2.SS2.p1.4.m4.1.1.2.2" xref="S2.SS2.p1.4.m4.1.1.2.2.cmml">𝒖</mi><mi id="S2.SS2.p1.4.m4.1.1.2.3" xref="S2.SS2.p1.4.m4.1.1.2.3.cmml">t</mi></msup><mo id="S2.SS2.p1.4.m4.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S2.SS2.p1.4.m4.1.1.3" xref="S2.SS2.p1.4.m4.1.1.3.cmml"><mi id="S2.SS2.p1.4.m4.1.1.3.2" xref="S2.SS2.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS2.p1.4.m4.1.1.3.3" xref="S2.SS2.p1.4.m4.1.1.3.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m4.1b"><apply id="S2.SS2.p1.4.m4.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1"><in id="S2.SS2.p1.4.m4.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1"></in><apply id="S2.SS2.p1.4.m4.1.1.2.cmml" xref="S2.SS2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.2.1.cmml" xref="S2.SS2.p1.4.m4.1.1.2">superscript</csymbol><ci id="S2.SS2.p1.4.m4.1.1.2.2.cmml" xref="S2.SS2.p1.4.m4.1.1.2.2">𝒖</ci><ci id="S2.SS2.p1.4.m4.1.1.2.3.cmml" xref="S2.SS2.p1.4.m4.1.1.2.3">𝑡</ci></apply><apply id="S2.SS2.p1.4.m4.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.3.1.cmml" xref="S2.SS2.p1.4.m4.1.1.3">superscript</csymbol><ci id="S2.SS2.p1.4.m4.1.1.3.2.cmml" xref="S2.SS2.p1.4.m4.1.1.3.2">ℝ</ci><ci id="S2.SS2.p1.4.m4.1.1.3.3.cmml" xref="S2.SS2.p1.4.m4.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m4.1c">{\bm{u}}^{t}\in\mathbb{R}^{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m4.1d">bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math>, which we will assume to be bounded: <math alttext="|\expectationvalue*{{\bm{u}}^{t},{\bm{x}}}|\lesssim 1" class="ltx_Math" display="inline" id="S2.SS2.p1.5.m5.1"><semantics id="S2.SS2.p1.5.m5.1a"><mrow id="S2.SS2.p1.5.m5.1.2" xref="S2.SS2.p1.5.m5.1.2.cmml"><mrow id="S2.SS2.p1.5.m5.1.2.2.2" xref="S2.SS2.p1.5.m5.1.2.2.1.cmml"><mo id="S2.SS2.p1.5.m5.1.2.2.2.1" stretchy="false" xref="S2.SS2.p1.5.m5.1.2.2.1.1.cmml">|</mo><mrow id="S2.SS2.p1.5.m5.1.1.3" xref="S2.SS2.p1.5.m5.1.1.2.cmml"><mo id="S2.SS2.p1.5.m5.1.1.3.1" stretchy="false" xref="S2.SS2.p1.5.m5.1.1.2.1.cmml">⟨</mo><mrow id="S2.SS2.p1.5.m5.1.1.1.1.1.2" xref="S2.SS2.p1.5.m5.1.1.1.1.1.3.cmml"><msup id="S2.SS2.p1.5.m5.1.1.1.1.1.2.1" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.cmml"><mi id="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.2" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.2.cmml">𝒖</mi><mi id="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.3" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.3.cmml">t</mi></msup><mo id="S2.SS2.p1.5.m5.1.1.1.1.1.2.2" xref="S2.SS2.p1.5.m5.1.1.1.1.1.3.cmml">,</mo><mi id="S2.SS2.p1.5.m5.1.1.1.1.1.1" xref="S2.SS2.p1.5.m5.1.1.1.1.1.1.cmml">𝒙</mi></mrow><mo id="S2.SS2.p1.5.m5.1.1.3.2" stretchy="false" xref="S2.SS2.p1.5.m5.1.1.2.1.cmml">⟩</mo></mrow><mo id="S2.SS2.p1.5.m5.1.2.2.2.2" stretchy="false" xref="S2.SS2.p1.5.m5.1.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS2.p1.5.m5.1.2.1" xref="S2.SS2.p1.5.m5.1.2.1.cmml">≲</mo><mn id="S2.SS2.p1.5.m5.1.2.3" xref="S2.SS2.p1.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.5.m5.1b"><apply id="S2.SS2.p1.5.m5.1.2.cmml" xref="S2.SS2.p1.5.m5.1.2"><csymbol cd="latexml" id="S2.SS2.p1.5.m5.1.2.1.cmml" xref="S2.SS2.p1.5.m5.1.2.1">less-than-or-similar-to</csymbol><apply id="S2.SS2.p1.5.m5.1.2.2.1.cmml" xref="S2.SS2.p1.5.m5.1.2.2.2"><abs id="S2.SS2.p1.5.m5.1.2.2.1.1.cmml" xref="S2.SS2.p1.5.m5.1.2.2.2.1"></abs><apply id="S2.SS2.p1.5.m5.1.1.2.cmml" xref="S2.SS2.p1.5.m5.1.1.3"><csymbol cd="latexml" id="S2.SS2.p1.5.m5.1.1.2.1.cmml" xref="S2.SS2.p1.5.m5.1.1.3.1">expectation-value</csymbol><list id="S2.SS2.p1.5.m5.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2"><apply id="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2.1">superscript</csymbol><ci id="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.2.cmml" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.2">𝒖</ci><ci id="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.3.cmml" xref="S2.SS2.p1.5.m5.1.1.1.1.1.2.1.3">𝑡</ci></apply><ci id="S2.SS2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1.1.1.1.1">𝒙</ci></list></apply></apply><cn id="S2.SS2.p1.5.m5.1.2.3.cmml" type="integer" xref="S2.SS2.p1.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.5.m5.1c">|\expectationvalue*{{\bm{u}}^{t},{\bm{x}}}|\lesssim 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.5.m5.1d">| ⟨ start_ARG bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , bold_italic_x end_ARG ⟩ | ≲ 1</annotation></semantics></math> for every time <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p1.6.m6.1"><semantics id="S2.SS2.p1.6.m6.1a"><mi id="S2.SS2.p1.6.m6.1.1" xref="S2.SS2.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.6.m6.1b"><ci id="S2.SS2.p1.6.m6.1.1.cmml" xref="S2.SS2.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.6.m6.1d">italic_t</annotation></semantics></math> and strategy <math alttext="{\bm{x}}" class="ltx_Math" display="inline" id="S2.SS2.p1.7.m7.1"><semantics id="S2.SS2.p1.7.m7.1a"><mi id="S2.SS2.p1.7.m7.1.1" xref="S2.SS2.p1.7.m7.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.7.m7.1b"><ci id="S2.SS2.p1.7.m7.1.1.cmml" xref="S2.SS2.p1.7.m7.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.7.m7.1c">{\bm{x}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.7.m7.1d">bold_italic_x</annotation></semantics></math>. The learner’s <span class="ltx_text ltx_font_italic" id="S2.SS2.p1.8.2">regret</span> after <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p1.8.m8.1"><semantics id="S2.SS2.p1.8.m8.1a"><mi id="S2.SS2.p1.8.m8.1.1" xref="S2.SS2.p1.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.8.m8.1b"><ci id="S2.SS2.p1.8.m8.1.1.cmml" xref="S2.SS2.p1.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.8.m8.1d">italic_t</annotation></semantics></math> timesteps is defined as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.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 R(T):=\max_{{\bm{x}}\in{\mathcal{X}}}\sum_{t=1}^{T}% \expectationvalue{{\bm{u}}^{t},{\bm{x}}-{\bm{x}}^{t}}." class="ltx_Math" display="inline" id="S2.E1.m1.3"><semantics id="S2.E1.m1.3a"><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mi id="S2.E1.m1.3.3.1.1.2.2" xref="S2.E1.m1.3.3.1.1.2.2.cmml">R</mi><mo id="S2.E1.m1.3.3.1.1.2.1" xref="S2.E1.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.2.3.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mo id="S2.E1.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S2.E1.m1.3.3.1.1.2.cmml">(</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">T</mi><mo id="S2.E1.m1.3.3.1.1.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.E1.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.3.3.1.1.1" rspace="0.278em" xref="S2.E1.m1.3.3.1.1.1.cmml">:=</mo><mrow id="S2.E1.m1.3.3.1.1.3" xref="S2.E1.m1.3.3.1.1.3.cmml"><munder id="S2.E1.m1.3.3.1.1.3.2" xref="S2.E1.m1.3.3.1.1.3.2.cmml"><mi id="S2.E1.m1.3.3.1.1.3.2.2" xref="S2.E1.m1.3.3.1.1.3.2.2.cmml">max</mi><mrow id="S2.E1.m1.3.3.1.1.3.2.3" xref="S2.E1.m1.3.3.1.1.3.2.3.cmml"><mi id="S2.E1.m1.3.3.1.1.3.2.3.2" xref="S2.E1.m1.3.3.1.1.3.2.3.2.cmml">𝒙</mi><mo id="S2.E1.m1.3.3.1.1.3.2.3.1" xref="S2.E1.m1.3.3.1.1.3.2.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.3.3.1.1.3.2.3.3" xref="S2.E1.m1.3.3.1.1.3.2.3.3.cmml">𝒳</mi></mrow></munder><mo id="S2.E1.m1.3.3.1.1.3.1" lspace="0.167em" xref="S2.E1.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.3.3" xref="S2.E1.m1.3.3.1.1.3.3.cmml"><mstyle displaystyle="true" id="S2.E1.m1.3.3.1.1.3.3.1" xref="S2.E1.m1.3.3.1.1.3.3.1.cmml"><munderover id="S2.E1.m1.3.3.1.1.3.3.1a" xref="S2.E1.m1.3.3.1.1.3.3.1.cmml"><mo id="S2.E1.m1.3.3.1.1.3.3.1.2.2" movablelimits="false" xref="S2.E1.m1.3.3.1.1.3.3.1.2.2.cmml">∑</mo><mrow id="S2.E1.m1.3.3.1.1.3.3.1.2.3" xref="S2.E1.m1.3.3.1.1.3.3.1.2.3.cmml"><mi id="S2.E1.m1.3.3.1.1.3.3.1.2.3.2" 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\expectationvalue{{\bm{u}}^{t},{\bm{x}}-{\bm{x}}^{t}}.</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.3d">italic_R ( italic_T ) := roman_max start_POSTSUBSCRIPT bold_italic_x ∈ caligraphic_X end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ⟨ start_ARG bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , bold_italic_x - bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_ARG ⟩ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.11">We say that the learner is running a <span class="ltx_text ltx_font_italic" id="S2.SS2.p1.11.1">no-regret algorithm</span> if it guarantees <math alttext="R(T)/T\to 0" class="ltx_Math" 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xref="S2.SS2.p1.10.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.10.m2.1c">T\to\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.10.m2.1d">italic_T → ∞</annotation></semantics></math> for all possible sequences <math alttext="\bm{u}^{1},\ldots,\bm{u}^{T}" class="ltx_Math" display="inline" id="S2.SS2.p1.11.m3.3"><semantics id="S2.SS2.p1.11.m3.3a"><mrow id="S2.SS2.p1.11.m3.3.3.2" xref="S2.SS2.p1.11.m3.3.3.3.cmml"><msup id="S2.SS2.p1.11.m3.2.2.1.1" xref="S2.SS2.p1.11.m3.2.2.1.1.cmml"><mi id="S2.SS2.p1.11.m3.2.2.1.1.2" xref="S2.SS2.p1.11.m3.2.2.1.1.2.cmml">𝒖</mi><mn id="S2.SS2.p1.11.m3.2.2.1.1.3" xref="S2.SS2.p1.11.m3.2.2.1.1.3.cmml">1</mn></msup><mo id="S2.SS2.p1.11.m3.3.3.2.3" xref="S2.SS2.p1.11.m3.3.3.3.cmml">,</mo><mi id="S2.SS2.p1.11.m3.1.1" mathvariant="normal" xref="S2.SS2.p1.11.m3.1.1.cmml">…</mi><mo id="S2.SS2.p1.11.m3.3.3.2.4" xref="S2.SS2.p1.11.m3.3.3.3.cmml">,</mo><msup id="S2.SS2.p1.11.m3.3.3.2.2" xref="S2.SS2.p1.11.m3.3.3.2.2.cmml"><mi id="S2.SS2.p1.11.m3.3.3.2.2.2" xref="S2.SS2.p1.11.m3.3.3.2.2.2.cmml">𝒖</mi><mi id="S2.SS2.p1.11.m3.3.3.2.2.3" xref="S2.SS2.p1.11.m3.3.3.2.2.3.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.11.m3.3b"><list id="S2.SS2.p1.11.m3.3.3.3.cmml" xref="S2.SS2.p1.11.m3.3.3.2"><apply id="S2.SS2.p1.11.m3.2.2.1.1.cmml" xref="S2.SS2.p1.11.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.11.m3.2.2.1.1.1.cmml" xref="S2.SS2.p1.11.m3.2.2.1.1">superscript</csymbol><ci id="S2.SS2.p1.11.m3.2.2.1.1.2.cmml" xref="S2.SS2.p1.11.m3.2.2.1.1.2">𝒖</ci><cn id="S2.SS2.p1.11.m3.2.2.1.1.3.cmml" type="integer" xref="S2.SS2.p1.11.m3.2.2.1.1.3">1</cn></apply><ci id="S2.SS2.p1.11.m3.1.1.cmml" xref="S2.SS2.p1.11.m3.1.1">…</ci><apply id="S2.SS2.p1.11.m3.3.3.2.2.cmml" xref="S2.SS2.p1.11.m3.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.11.m3.3.3.2.2.1.cmml" xref="S2.SS2.p1.11.m3.3.3.2.2">superscript</csymbol><ci 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xref="S2.SS2.p2.1.m1.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">{\mathcal{X}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">caligraphic_X</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S2.SS2.p2.16.1">projected gradient descent/ascent</span> <cite class="ltx_cite ltx_citemacro_cite">Zinkevich (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib25" title="">2003</a>)</cite>. Projected gradient ascent selects <math alttext="{\bm{x}}^{1}\in{\mathcal{X}}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><mrow id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml"><msup id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml"><mi id="S2.SS2.p2.2.m2.1.1.2.2" xref="S2.SS2.p2.2.m2.1.1.2.2.cmml">𝒙</mi><mn id="S2.SS2.p2.2.m2.1.1.2.3" xref="S2.SS2.p2.2.m2.1.1.2.3.cmml">1</mn></msup><mo id="S2.SS2.p2.2.m2.1.1.1" xref="S2.SS2.p2.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.1b"><apply id="S2.SS2.p2.2.m2.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1"><in id="S2.SS2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1"></in><apply id="S2.SS2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.2.m2.1.1.2.1.cmml" xref="S2.SS2.p2.2.m2.1.1.2">superscript</csymbol><ci id="S2.SS2.p2.2.m2.1.1.2.2.cmml" xref="S2.SS2.p2.2.m2.1.1.2.2">𝒙</ci><cn id="S2.SS2.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S2.SS2.p2.2.m2.1.1.2.3">1</cn></apply><ci id="S2.SS2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.p2.2.m2.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">{\bm{x}}^{1}\in{\mathcal{X}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">bold_italic_x start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ caligraphic_X</annotation></semantics></math> arbitrarily, and for every timestep <math alttext="t>1" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.1"><semantics id="S2.SS2.p2.3.m3.1a"><mrow id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml"><mi id="S2.SS2.p2.3.m3.1.1.2" xref="S2.SS2.p2.3.m3.1.1.2.cmml">t</mi><mo id="S2.SS2.p2.3.m3.1.1.1" xref="S2.SS2.p2.3.m3.1.1.1.cmml">></mo><mn id="S2.SS2.p2.3.m3.1.1.3" xref="S2.SS2.p2.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml 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id="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.2.cmml" xref="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.2">𝒖</ci><apply id="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3.cmml" xref="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3"><minus id="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3.1.cmml" xref="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3.1"></minus><ci id="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3.2.cmml" xref="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3.2">𝑡</ci><cn id="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS2.p2.4.m4.1.1.1.1.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">{\bm{x}}^{t}=\Pi_{\mathcal{X}}[{\bm{x}}^{t-1}+\eta{\bm{u}}^{t-1}]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = roman_Π start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT [ bold_italic_x start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT + italic_η bold_italic_u start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT ]</annotation></semantics></math> where <math alttext="\Pi_{\mathcal{X}}" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m5.1"><semantics id="S2.SS2.p2.5.m5.1a"><msub id="S2.SS2.p2.5.m5.1.1" xref="S2.SS2.p2.5.m5.1.1.cmml"><mi id="S2.SS2.p2.5.m5.1.1.2" mathvariant="normal" xref="S2.SS2.p2.5.m5.1.1.2.cmml">Π</mi><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.5.m5.1.1.3" xref="S2.SS2.p2.5.m5.1.1.3.cmml">𝒳</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m5.1b"><apply id="S2.SS2.p2.5.m5.1.1.cmml" xref="S2.SS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m5.1.1.1.cmml" xref="S2.SS2.p2.5.m5.1.1">subscript</csymbol><ci id="S2.SS2.p2.5.m5.1.1.2.cmml" xref="S2.SS2.p2.5.m5.1.1.2">Π</ci><ci id="S2.SS2.p2.5.m5.1.1.3.cmml" xref="S2.SS2.p2.5.m5.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m5.1c">\Pi_{\mathcal{X}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m5.1d">roman_Π start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT</annotation></semantics></math> is the projection operator into <math alttext="{\mathcal{X}}" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m6.1"><semantics id="S2.SS2.p2.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.6.m6.1.1" xref="S2.SS2.p2.6.m6.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m6.1b"><ci id="S2.SS2.p2.6.m6.1.1.cmml" xref="S2.SS2.p2.6.m6.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m6.1c">{\mathcal{X}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m6.1d">caligraphic_X</annotation></semantics></math>. The step size <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m7.1"><semantics id="S2.SS2.p2.7.m7.1a"><mi id="S2.SS2.p2.7.m7.1.1" xref="S2.SS2.p2.7.m7.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m7.1b"><ci id="S2.SS2.p2.7.m7.1.1.cmml" xref="S2.SS2.p2.7.m7.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m7.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m7.1d">italic_η</annotation></semantics></math> is chosen to be <math alttext="\eta=B/(G\sqrt{T})" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m8.1"><semantics id="S2.SS2.p2.8.m8.1a"><mrow id="S2.SS2.p2.8.m8.1.1" xref="S2.SS2.p2.8.m8.1.1.cmml"><mi id="S2.SS2.p2.8.m8.1.1.3" xref="S2.SS2.p2.8.m8.1.1.3.cmml">η</mi><mo id="S2.SS2.p2.8.m8.1.1.2" xref="S2.SS2.p2.8.m8.1.1.2.cmml">=</mo><mrow id="S2.SS2.p2.8.m8.1.1.1" xref="S2.SS2.p2.8.m8.1.1.1.cmml"><mi id="S2.SS2.p2.8.m8.1.1.1.3" xref="S2.SS2.p2.8.m8.1.1.1.3.cmml">B</mi><mo id="S2.SS2.p2.8.m8.1.1.1.2" xref="S2.SS2.p2.8.m8.1.1.1.2.cmml">/</mo><mrow id="S2.SS2.p2.8.m8.1.1.1.1.1" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p2.8.m8.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p2.8.m8.1.1.1.1.1.1" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.8.m8.1.1.1.1.1.1.2" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.2.cmml">G</mi><mo id="S2.SS2.p2.8.m8.1.1.1.1.1.1.1" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.1.cmml">⁢</mo><msqrt id="S2.SS2.p2.8.m8.1.1.1.1.1.1.3" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p2.8.m8.1.1.1.1.1.1.3.2" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.3.2.cmml">T</mi></msqrt></mrow><mo id="S2.SS2.p2.8.m8.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m8.1b"><apply id="S2.SS2.p2.8.m8.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1"><eq id="S2.SS2.p2.8.m8.1.1.2.cmml" xref="S2.SS2.p2.8.m8.1.1.2"></eq><ci id="S2.SS2.p2.8.m8.1.1.3.cmml" xref="S2.SS2.p2.8.m8.1.1.3">𝜂</ci><apply id="S2.SS2.p2.8.m8.1.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1.1"><divide id="S2.SS2.p2.8.m8.1.1.1.2.cmml" xref="S2.SS2.p2.8.m8.1.1.1.2"></divide><ci id="S2.SS2.p2.8.m8.1.1.1.3.cmml" xref="S2.SS2.p2.8.m8.1.1.1.3">𝐵</ci><apply id="S2.SS2.p2.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1.1.1.1"><times id="S2.SS2.p2.8.m8.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.1"></times><ci id="S2.SS2.p2.8.m8.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.2">𝐺</ci><apply id="S2.SS2.p2.8.m8.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.3"><root id="S2.SS2.p2.8.m8.1.1.1.1.1.1.3a.cmml" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.3"></root><ci id="S2.SS2.p2.8.m8.1.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p2.8.m8.1.1.1.1.1.1.3.2">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m8.1c">\eta=B/(G\sqrt{T})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m8.1d">italic_η = italic_B / ( italic_G square-root start_ARG italic_T end_ARG )</annotation></semantics></math>, where <math alttext="B" class="ltx_Math" display="inline" id="S2.SS2.p2.9.m9.1"><semantics id="S2.SS2.p2.9.m9.1a"><mi id="S2.SS2.p2.9.m9.1.1" xref="S2.SS2.p2.9.m9.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.9.m9.1b"><ci id="S2.SS2.p2.9.m9.1.1.cmml" xref="S2.SS2.p2.9.m9.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.9.m9.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.9.m9.1d">italic_B</annotation></semantics></math> is the <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.10.m10.1"><semantics id="S2.SS2.p2.10.m10.1a"><msub id="S2.SS2.p2.10.m10.1.1" xref="S2.SS2.p2.10.m10.1.1.cmml"><mi id="S2.SS2.p2.10.m10.1.1.2" mathvariant="normal" xref="S2.SS2.p2.10.m10.1.1.2.cmml">ℓ</mi><mn id="S2.SS2.p2.10.m10.1.1.3" xref="S2.SS2.p2.10.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.10.m10.1b"><apply id="S2.SS2.p2.10.m10.1.1.cmml" xref="S2.SS2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.10.m10.1.1.1.cmml" xref="S2.SS2.p2.10.m10.1.1">subscript</csymbol><ci id="S2.SS2.p2.10.m10.1.1.2.cmml" xref="S2.SS2.p2.10.m10.1.1.2">ℓ</ci><cn id="S2.SS2.p2.10.m10.1.1.3.cmml" type="integer" xref="S2.SS2.p2.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.10.m10.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.10.m10.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-diameter of <math alttext="{\mathcal{X}}" class="ltx_Math" display="inline" id="S2.SS2.p2.11.m11.1"><semantics id="S2.SS2.p2.11.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.11.m11.1.1" xref="S2.SS2.p2.11.m11.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.11.m11.1b"><ci id="S2.SS2.p2.11.m11.1.1.cmml" xref="S2.SS2.p2.11.m11.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.11.m11.1c">{\mathcal{X}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.11.m11.1d">caligraphic_X</annotation></semantics></math>, and <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.p2.12.m12.1"><semantics id="S2.SS2.p2.12.m12.1a"><mi id="S2.SS2.p2.12.m12.1.1" xref="S2.SS2.p2.12.m12.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.12.m12.1b"><ci id="S2.SS2.p2.12.m12.1.1.cmml" xref="S2.SS2.p2.12.m12.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.12.m12.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.12.m12.1d">italic_G</annotation></semantics></math> bounds the <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.13.m13.1"><semantics id="S2.SS2.p2.13.m13.1a"><msub id="S2.SS2.p2.13.m13.1.1" xref="S2.SS2.p2.13.m13.1.1.cmml"><mi id="S2.SS2.p2.13.m13.1.1.2" mathvariant="normal" xref="S2.SS2.p2.13.m13.1.1.2.cmml">ℓ</mi><mn id="S2.SS2.p2.13.m13.1.1.3" xref="S2.SS2.p2.13.m13.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.13.m13.1b"><apply id="S2.SS2.p2.13.m13.1.1.cmml" xref="S2.SS2.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.13.m13.1.1.1.cmml" xref="S2.SS2.p2.13.m13.1.1">subscript</csymbol><ci id="S2.SS2.p2.13.m13.1.1.2.cmml" xref="S2.SS2.p2.13.m13.1.1.2">ℓ</ci><cn id="S2.SS2.p2.13.m13.1.1.3.cmml" type="integer" xref="S2.SS2.p2.13.m13.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.13.m13.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.13.m13.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-norm of <math alttext="{\bm{u}}^{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.14.m14.1"><semantics id="S2.SS2.p2.14.m14.1a"><msup id="S2.SS2.p2.14.m14.1.1" xref="S2.SS2.p2.14.m14.1.1.cmml"><mi id="S2.SS2.p2.14.m14.1.1.2" xref="S2.SS2.p2.14.m14.1.1.2.cmml">𝒖</mi><mi id="S2.SS2.p2.14.m14.1.1.3" xref="S2.SS2.p2.14.m14.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.14.m14.1b"><apply id="S2.SS2.p2.14.m14.1.1.cmml" xref="S2.SS2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.14.m14.1.1.1.cmml" xref="S2.SS2.p2.14.m14.1.1">superscript</csymbol><ci id="S2.SS2.p2.14.m14.1.1.2.cmml" xref="S2.SS2.p2.14.m14.1.1.2">𝒖</ci><ci id="S2.SS2.p2.14.m14.1.1.3.cmml" xref="S2.SS2.p2.14.m14.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.14.m14.1c">{\bm{u}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.14.m14.1d">bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> for all <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p2.15.m15.1"><semantics id="S2.SS2.p2.15.m15.1a"><mi id="S2.SS2.p2.15.m15.1.1" xref="S2.SS2.p2.15.m15.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.15.m15.1b"><ci id="S2.SS2.p2.15.m15.1.1.cmml" xref="S2.SS2.p2.15.m15.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.15.m15.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.15.m15.1d">italic_t</annotation></semantics></math>. This achieves a regret guarantee of <math alttext="R(T)\lesssim BG\sqrt{T}" class="ltx_Math" display="inline" id="S2.SS2.p2.16.m16.1"><semantics id="S2.SS2.p2.16.m16.1a"><mrow id="S2.SS2.p2.16.m16.1.2" xref="S2.SS2.p2.16.m16.1.2.cmml"><mrow id="S2.SS2.p2.16.m16.1.2.2" xref="S2.SS2.p2.16.m16.1.2.2.cmml"><mi id="S2.SS2.p2.16.m16.1.2.2.2" xref="S2.SS2.p2.16.m16.1.2.2.2.cmml">R</mi><mo id="S2.SS2.p2.16.m16.1.2.2.1" xref="S2.SS2.p2.16.m16.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.16.m16.1.2.2.3.2" xref="S2.SS2.p2.16.m16.1.2.2.cmml"><mo id="S2.SS2.p2.16.m16.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p2.16.m16.1.2.2.cmml">(</mo><mi id="S2.SS2.p2.16.m16.1.1" xref="S2.SS2.p2.16.m16.1.1.cmml">T</mi><mo id="S2.SS2.p2.16.m16.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p2.16.m16.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.16.m16.1.2.1" xref="S2.SS2.p2.16.m16.1.2.1.cmml">≲</mo><mrow id="S2.SS2.p2.16.m16.1.2.3" xref="S2.SS2.p2.16.m16.1.2.3.cmml"><mi id="S2.SS2.p2.16.m16.1.2.3.2" xref="S2.SS2.p2.16.m16.1.2.3.2.cmml">B</mi><mo id="S2.SS2.p2.16.m16.1.2.3.1" xref="S2.SS2.p2.16.m16.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.16.m16.1.2.3.3" xref="S2.SS2.p2.16.m16.1.2.3.3.cmml">G</mi><mo id="S2.SS2.p2.16.m16.1.2.3.1a" xref="S2.SS2.p2.16.m16.1.2.3.1.cmml">⁢</mo><msqrt id="S2.SS2.p2.16.m16.1.2.3.4" xref="S2.SS2.p2.16.m16.1.2.3.4.cmml"><mi id="S2.SS2.p2.16.m16.1.2.3.4.2" xref="S2.SS2.p2.16.m16.1.2.3.4.2.cmml">T</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.16.m16.1b"><apply id="S2.SS2.p2.16.m16.1.2.cmml" xref="S2.SS2.p2.16.m16.1.2"><csymbol cd="latexml" id="S2.SS2.p2.16.m16.1.2.1.cmml" xref="S2.SS2.p2.16.m16.1.2.1">less-than-or-similar-to</csymbol><apply id="S2.SS2.p2.16.m16.1.2.2.cmml" xref="S2.SS2.p2.16.m16.1.2.2"><times id="S2.SS2.p2.16.m16.1.2.2.1.cmml" xref="S2.SS2.p2.16.m16.1.2.2.1"></times><ci id="S2.SS2.p2.16.m16.1.2.2.2.cmml" xref="S2.SS2.p2.16.m16.1.2.2.2">𝑅</ci><ci id="S2.SS2.p2.16.m16.1.1.cmml" xref="S2.SS2.p2.16.m16.1.1">𝑇</ci></apply><apply id="S2.SS2.p2.16.m16.1.2.3.cmml" xref="S2.SS2.p2.16.m16.1.2.3"><times id="S2.SS2.p2.16.m16.1.2.3.1.cmml" xref="S2.SS2.p2.16.m16.1.2.3.1"></times><ci id="S2.SS2.p2.16.m16.1.2.3.2.cmml" xref="S2.SS2.p2.16.m16.1.2.3.2">𝐵</ci><ci id="S2.SS2.p2.16.m16.1.2.3.3.cmml" xref="S2.SS2.p2.16.m16.1.2.3.3">𝐺</ci><apply id="S2.SS2.p2.16.m16.1.2.3.4.cmml" xref="S2.SS2.p2.16.m16.1.2.3.4"><root id="S2.SS2.p2.16.m16.1.2.3.4a.cmml" xref="S2.SS2.p2.16.m16.1.2.3.4"></root><ci id="S2.SS2.p2.16.m16.1.2.3.4.2.cmml" xref="S2.SS2.p2.16.m16.1.2.3.4.2">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.16.m16.1c">R(T)\lesssim BG\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.16.m16.1d">italic_R ( italic_T ) ≲ italic_B italic_G square-root start_ARG italic_T end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.6">To apply no-regret learning algorithms in games, each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mi id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><ci id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">italic_i</annotation></semantics></math> needs to run no-regret algorithms over their strategy sets <math alttext="{\mathcal{X}}=\Delta(m_{i})" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><mrow id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.2.m2.1.1.3" xref="S2.SS2.p3.2.m2.1.1.3.cmml">𝒳</mi><mo id="S2.SS2.p3.2.m2.1.1.2" xref="S2.SS2.p3.2.m2.1.1.2.cmml">=</mo><mrow 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id="S2.SS2.p3.2.m2.1.1.3.cmml" xref="S2.SS2.p3.2.m2.1.1.3">𝒳</ci><apply id="S2.SS2.p3.2.m2.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1"><times id="S2.SS2.p3.2.m2.1.1.1.2.cmml" xref="S2.SS2.p3.2.m2.1.1.1.2"></times><ci id="S2.SS2.p3.2.m2.1.1.1.3.cmml" xref="S2.SS2.p3.2.m2.1.1.1.3">Δ</ci><apply id="S2.SS2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p3.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.2">𝑚</ci><ci id="S2.SS2.p3.2.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">{\mathcal{X}}=\Delta(m_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">caligraphic_X = roman_Δ ( italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. For this special case, the most common algorithm for learning in games is the <span class="ltx_text ltx_font_italic" id="S2.SS2.p3.6.1">multiplicative weights update</span> (MWU) algorithm (<span class="ltx_text ltx_font_italic" id="S2.SS2.p3.6.2">e.g.</span>, <cite class="ltx_cite ltx_citemacro_citet">Freund and Schapire (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib11" title="">1999</a>)</cite>), which sets <math alttext="{\bm{x}}^{t}\propto\exp(\eta\sum_{\tau<t}{\bm{u}}^{\tau})" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.2"><semantics id="S2.SS2.p3.3.m3.2a"><mrow id="S2.SS2.p3.3.m3.2.3" xref="S2.SS2.p3.3.m3.2.3.cmml"><msup id="S2.SS2.p3.3.m3.2.3.2" xref="S2.SS2.p3.3.m3.2.3.2.cmml"><mi id="S2.SS2.p3.3.m3.2.3.2.2" xref="S2.SS2.p3.3.m3.2.3.2.2.cmml">𝒙</mi><mi id="S2.SS2.p3.3.m3.2.3.2.3" xref="S2.SS2.p3.3.m3.2.3.2.3.cmml">t</mi></msup><mo id="S2.SS2.p3.3.m3.2.3.1" xref="S2.SS2.p3.3.m3.2.3.1.cmml">∝</mo><mrow id="S2.SS2.p3.3.m3.2.2.4" 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roman_exp ( start_ARG italic_η ∑ start_POSTSUBSCRIPT italic_τ < italic_t end_POSTSUBSCRIPT bold_italic_u start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math>, where <math alttext="\eta=\sqrt{\log(m_{i})/T}" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m4.2"><semantics id="S2.SS2.p3.4.m4.2a"><mrow id="S2.SS2.p3.4.m4.2.3" xref="S2.SS2.p3.4.m4.2.3.cmml"><mi id="S2.SS2.p3.4.m4.2.3.2" xref="S2.SS2.p3.4.m4.2.3.2.cmml">η</mi><mo id="S2.SS2.p3.4.m4.2.3.1" xref="S2.SS2.p3.4.m4.2.3.1.cmml">=</mo><msqrt id="S2.SS2.p3.4.m4.2.2" xref="S2.SS2.p3.4.m4.2.2.cmml"><mrow id="S2.SS2.p3.4.m4.2.2.2" xref="S2.SS2.p3.4.m4.2.2.2.cmml"><mrow id="S2.SS2.p3.4.m4.2.2.2.2.4" xref="S2.SS2.p3.4.m4.2.2.2.2.3.cmml"><mi id="S2.SS2.p3.4.m4.2.2.2.2.2.2" xref="S2.SS2.p3.4.m4.2.2.2.2.3.1.cmml">log</mi><mo id="S2.SS2.p3.4.m4.2.2.2.2.4a" xref="S2.SS2.p3.4.m4.2.2.2.2.3.1.cmml">⁡</mo><mrow id="S2.SS2.p3.4.m4.2.2.2.2.4.1" xref="S2.SS2.p3.4.m4.2.2.2.2.3.cmml"><mo 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id="S2.SS2.p3.4.m4.2d">italic_η = square-root start_ARG roman_log ( start_ARG italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) / italic_T end_ARG</annotation></semantics></math> is an appropriately-chosen step size and <math alttext="\exp(\cdot)" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m5.2"><semantics id="S2.SS2.p3.5.m5.2a"><mrow id="S2.SS2.p3.5.m5.2.2.4" xref="S2.SS2.p3.5.m5.2.2.3.cmml"><mi id="S2.SS2.p3.5.m5.2.2.2.2" xref="S2.SS2.p3.5.m5.2.2.3.1.cmml">exp</mi><mo id="S2.SS2.p3.5.m5.2.2.4a" xref="S2.SS2.p3.5.m5.2.2.3.1.cmml">⁡</mo><mrow id="S2.SS2.p3.5.m5.2.2.4.1" xref="S2.SS2.p3.5.m5.2.2.3.cmml"><mo id="S2.SS2.p3.5.m5.2.2.4.1.1" xref="S2.SS2.p3.5.m5.2.2.3.1.cmml">(</mo><mo id="S2.SS2.p3.5.m5.1.1.1.1.1" lspace="0em" rspace="0em" xref="S2.SS2.p3.5.m5.1.1.1.1.1.cmml">⋅</mo><mo id="S2.SS2.p3.5.m5.2.2.4.1.2" xref="S2.SS2.p3.5.m5.2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m5.2b"><apply id="S2.SS2.p3.5.m5.2.2.3.cmml" xref="S2.SS2.p3.5.m5.2.2.4"><exp id="S2.SS2.p3.5.m5.2.2.3.1.cmml" xref="S2.SS2.p3.5.m5.2.2.2.2"></exp><ci id="S2.SS2.p3.5.m5.1.1.1.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1.1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m5.2c">\exp(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m5.2d">roman_exp ( start_ARG ⋅ end_ARG )</annotation></semantics></math> is the element-wise exponential function. Multiplicative weights achieves regret bound <math alttext="R(T)\lesssim\sqrt{T\log m_{i}}" class="ltx_Math" display="inline" id="S2.SS2.p3.6.m6.1"><semantics id="S2.SS2.p3.6.m6.1a"><mrow id="S2.SS2.p3.6.m6.1.2" xref="S2.SS2.p3.6.m6.1.2.cmml"><mrow id="S2.SS2.p3.6.m6.1.2.2" xref="S2.SS2.p3.6.m6.1.2.2.cmml"><mi id="S2.SS2.p3.6.m6.1.2.2.2" xref="S2.SS2.p3.6.m6.1.2.2.2.cmml">R</mi><mo id="S2.SS2.p3.6.m6.1.2.2.1" xref="S2.SS2.p3.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p3.6.m6.1.2.2.3.2" xref="S2.SS2.p3.6.m6.1.2.2.cmml"><mo id="S2.SS2.p3.6.m6.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p3.6.m6.1.2.2.cmml">(</mo><mi id="S2.SS2.p3.6.m6.1.1" xref="S2.SS2.p3.6.m6.1.1.cmml">T</mi><mo id="S2.SS2.p3.6.m6.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p3.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p3.6.m6.1.2.1" xref="S2.SS2.p3.6.m6.1.2.1.cmml">≲</mo><msqrt id="S2.SS2.p3.6.m6.1.2.3" xref="S2.SS2.p3.6.m6.1.2.3.cmml"><mrow id="S2.SS2.p3.6.m6.1.2.3.2" xref="S2.SS2.p3.6.m6.1.2.3.2.cmml"><mi id="S2.SS2.p3.6.m6.1.2.3.2.2" xref="S2.SS2.p3.6.m6.1.2.3.2.2.cmml">T</mi><mo id="S2.SS2.p3.6.m6.1.2.3.2.1" lspace="0.167em" xref="S2.SS2.p3.6.m6.1.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS2.p3.6.m6.1.2.3.2.3" xref="S2.SS2.p3.6.m6.1.2.3.2.3.cmml"><mi id="S2.SS2.p3.6.m6.1.2.3.2.3.1" xref="S2.SS2.p3.6.m6.1.2.3.2.3.1.cmml">log</mi><mo id="S2.SS2.p3.6.m6.1.2.3.2.3a" lspace="0.167em" xref="S2.SS2.p3.6.m6.1.2.3.2.3.cmml">⁡</mo><msub id="S2.SS2.p3.6.m6.1.2.3.2.3.2" xref="S2.SS2.p3.6.m6.1.2.3.2.3.2.cmml"><mi id="S2.SS2.p3.6.m6.1.2.3.2.3.2.2" xref="S2.SS2.p3.6.m6.1.2.3.2.3.2.2.cmml">m</mi><mi id="S2.SS2.p3.6.m6.1.2.3.2.3.2.3" xref="S2.SS2.p3.6.m6.1.2.3.2.3.2.3.cmml">i</mi></msub></mrow></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.6.m6.1b"><apply id="S2.SS2.p3.6.m6.1.2.cmml" xref="S2.SS2.p3.6.m6.1.2"><csymbol cd="latexml" id="S2.SS2.p3.6.m6.1.2.1.cmml" xref="S2.SS2.p3.6.m6.1.2.1">less-than-or-similar-to</csymbol><apply id="S2.SS2.p3.6.m6.1.2.2.cmml" xref="S2.SS2.p3.6.m6.1.2.2"><times id="S2.SS2.p3.6.m6.1.2.2.1.cmml" xref="S2.SS2.p3.6.m6.1.2.2.1"></times><ci id="S2.SS2.p3.6.m6.1.2.2.2.cmml" xref="S2.SS2.p3.6.m6.1.2.2.2">𝑅</ci><ci id="S2.SS2.p3.6.m6.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1">𝑇</ci></apply><apply id="S2.SS2.p3.6.m6.1.2.3.cmml" xref="S2.SS2.p3.6.m6.1.2.3"><root id="S2.SS2.p3.6.m6.1.2.3a.cmml" xref="S2.SS2.p3.6.m6.1.2.3"></root><apply id="S2.SS2.p3.6.m6.1.2.3.2.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2"><times id="S2.SS2.p3.6.m6.1.2.3.2.1.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.1"></times><ci id="S2.SS2.p3.6.m6.1.2.3.2.2.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.2">𝑇</ci><apply id="S2.SS2.p3.6.m6.1.2.3.2.3.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.3"><log id="S2.SS2.p3.6.m6.1.2.3.2.3.1.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.3.1"></log><apply id="S2.SS2.p3.6.m6.1.2.3.2.3.2.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p3.6.m6.1.2.3.2.3.2.1.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.3.2">subscript</csymbol><ci id="S2.SS2.p3.6.m6.1.2.3.2.3.2.2.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.3.2.2">𝑚</ci><ci id="S2.SS2.p3.6.m6.1.2.3.2.3.2.3.cmml" xref="S2.SS2.p3.6.m6.1.2.3.2.3.2.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.6.m6.1c">R(T)\lesssim\sqrt{T\log m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.6.m6.1d">italic_R ( italic_T ) ≲ square-root start_ARG italic_T roman_log italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>.</p> </div> </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>Realized and in-expectation regret</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.7">Perhaps unusually in the literature of learning in games, we will require that no-regret learning agents actually choose an <span class="ltx_text ltx_font_italic" id="S2.SS3.p1.7.1">action</span>, rather than a <span class="ltx_text ltx_font_italic" id="S2.SS3.p1.7.2">mixed strategy</span>, at every timestep. To do this, the agent can simply sample an action <math alttext="a^{t}\sim{\bm{x}}^{t}" class="ltx_Math" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mrow id="S2.SS3.p1.1.m1.1.1" xref="S2.SS3.p1.1.m1.1.1.cmml"><msup id="S2.SS3.p1.1.m1.1.1.2" xref="S2.SS3.p1.1.m1.1.1.2.cmml"><mi id="S2.SS3.p1.1.m1.1.1.2.2" xref="S2.SS3.p1.1.m1.1.1.2.2.cmml">a</mi><mi id="S2.SS3.p1.1.m1.1.1.2.3" xref="S2.SS3.p1.1.m1.1.1.2.3.cmml">t</mi></msup><mo id="S2.SS3.p1.1.m1.1.1.1" xref="S2.SS3.p1.1.m1.1.1.1.cmml">∼</mo><msup id="S2.SS3.p1.1.m1.1.1.3" xref="S2.SS3.p1.1.m1.1.1.3.cmml"><mi id="S2.SS3.p1.1.m1.1.1.3.2" xref="S2.SS3.p1.1.m1.1.1.3.2.cmml">𝒙</mi><mi id="S2.SS3.p1.1.m1.1.1.3.3" xref="S2.SS3.p1.1.m1.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.1.m1.1b"><apply id="S2.SS3.p1.1.m1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1"><csymbol cd="latexml" id="S2.SS3.p1.1.m1.1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1.1">similar-to</csymbol><apply id="S2.SS3.p1.1.m1.1.1.2.cmml" xref="S2.SS3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p1.1.m1.1.1.2.1.cmml" xref="S2.SS3.p1.1.m1.1.1.2">superscript</csymbol><ci id="S2.SS3.p1.1.m1.1.1.2.2.cmml" xref="S2.SS3.p1.1.m1.1.1.2.2">𝑎</ci><ci id="S2.SS3.p1.1.m1.1.1.2.3.cmml" xref="S2.SS3.p1.1.m1.1.1.2.3">𝑡</ci></apply><apply id="S2.SS3.p1.1.m1.1.1.3.cmml" xref="S2.SS3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p1.1.m1.1.1.3.1.cmml" xref="S2.SS3.p1.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS3.p1.1.m1.1.1.3.2.cmml" xref="S2.SS3.p1.1.m1.1.1.3.2">𝒙</ci><ci id="S2.SS3.p1.1.m1.1.1.3.3.cmml" xref="S2.SS3.p1.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">a^{t}\sim{\bm{x}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∼ bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> from a strategy <math alttext="{\bm{x}}^{t}" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m2.1"><semantics id="S2.SS3.p1.2.m2.1a"><msup id="S2.SS3.p1.2.m2.1.1" xref="S2.SS3.p1.2.m2.1.1.cmml"><mi id="S2.SS3.p1.2.m2.1.1.2" xref="S2.SS3.p1.2.m2.1.1.2.cmml">𝒙</mi><mi id="S2.SS3.p1.2.m2.1.1.3" xref="S2.SS3.p1.2.m2.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.2.m2.1b"><apply id="S2.SS3.p1.2.m2.1.1.cmml" xref="S2.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.2.m2.1.1.1.cmml" xref="S2.SS3.p1.2.m2.1.1">superscript</csymbol><ci id="S2.SS3.p1.2.m2.1.1.2.cmml" xref="S2.SS3.p1.2.m2.1.1.2">𝒙</ci><ci id="S2.SS3.p1.2.m2.1.1.3.cmml" xref="S2.SS3.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.2.m2.1c">{\bm{x}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m2.1d">bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> chosen by a no-regret algorithm such as multiplicative weights. The adversary’s utility vector <math alttext="{\bm{u}}^{t}" class="ltx_Math" display="inline" id="S2.SS3.p1.3.m3.1"><semantics id="S2.SS3.p1.3.m3.1a"><msup id="S2.SS3.p1.3.m3.1.1" xref="S2.SS3.p1.3.m3.1.1.cmml"><mi id="S2.SS3.p1.3.m3.1.1.2" xref="S2.SS3.p1.3.m3.1.1.2.cmml">𝒖</mi><mi id="S2.SS3.p1.3.m3.1.1.3" xref="S2.SS3.p1.3.m3.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.3.m3.1b"><apply id="S2.SS3.p1.3.m3.1.1.cmml" xref="S2.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.3.m3.1.1.1.cmml" xref="S2.SS3.p1.3.m3.1.1">superscript</csymbol><ci id="S2.SS3.p1.3.m3.1.1.2.cmml" xref="S2.SS3.p1.3.m3.1.1.2">𝒖</ci><ci id="S2.SS3.p1.3.m3.1.1.3.cmml" xref="S2.SS3.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m3.1c">{\bm{u}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m3.1d">bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> may depend on <math alttext="{\bm{x}}^{t}" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m4.1"><semantics id="S2.SS3.p1.4.m4.1a"><msup id="S2.SS3.p1.4.m4.1.1" xref="S2.SS3.p1.4.m4.1.1.cmml"><mi id="S2.SS3.p1.4.m4.1.1.2" xref="S2.SS3.p1.4.m4.1.1.2.cmml">𝒙</mi><mi id="S2.SS3.p1.4.m4.1.1.3" xref="S2.SS3.p1.4.m4.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m4.1b"><apply id="S2.SS3.p1.4.m4.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.4.m4.1.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1">superscript</csymbol><ci id="S2.SS3.p1.4.m4.1.1.2.cmml" xref="S2.SS3.p1.4.m4.1.1.2">𝒙</ci><ci id="S2.SS3.p1.4.m4.1.1.3.cmml" xref="S2.SS3.p1.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m4.1c">{\bm{x}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m4.1d">bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, but <span class="ltx_text ltx_font_italic" id="S2.SS3.p1.7.3">not</span> on the sample <math alttext="a^{t}" class="ltx_Math" display="inline" id="S2.SS3.p1.5.m5.1"><semantics id="S2.SS3.p1.5.m5.1a"><msup id="S2.SS3.p1.5.m5.1.1" xref="S2.SS3.p1.5.m5.1.1.cmml"><mi id="S2.SS3.p1.5.m5.1.1.2" xref="S2.SS3.p1.5.m5.1.1.2.cmml">a</mi><mi id="S2.SS3.p1.5.m5.1.1.3" xref="S2.SS3.p1.5.m5.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.5.m5.1b"><apply id="S2.SS3.p1.5.m5.1.1.cmml" xref="S2.SS3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.5.m5.1.1.1.cmml" xref="S2.SS3.p1.5.m5.1.1">superscript</csymbol><ci id="S2.SS3.p1.5.m5.1.1.2.cmml" xref="S2.SS3.p1.5.m5.1.1.2">𝑎</ci><ci id="S2.SS3.p1.5.m5.1.1.3.cmml" xref="S2.SS3.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.5.m5.1c">a^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.5.m5.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. Since the agent is now randomizing its action, the regret bound becomes probabilistic, and we need to distinguish between the <span class="ltx_text ltx_font_italic" id="S2.SS3.p1.7.4">realized</span> regret, defined using the sampled action <math alttext="a^{t}" class="ltx_Math" display="inline" id="S2.SS3.p1.6.m6.1"><semantics id="S2.SS3.p1.6.m6.1a"><msup id="S2.SS3.p1.6.m6.1.1" xref="S2.SS3.p1.6.m6.1.1.cmml"><mi id="S2.SS3.p1.6.m6.1.1.2" xref="S2.SS3.p1.6.m6.1.1.2.cmml">a</mi><mi id="S2.SS3.p1.6.m6.1.1.3" xref="S2.SS3.p1.6.m6.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.6.m6.1b"><apply id="S2.SS3.p1.6.m6.1.1.cmml" xref="S2.SS3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.6.m6.1.1.1.cmml" xref="S2.SS3.p1.6.m6.1.1">superscript</csymbol><ci id="S2.SS3.p1.6.m6.1.1.2.cmml" xref="S2.SS3.p1.6.m6.1.1.2">𝑎</ci><ci id="S2.SS3.p1.6.m6.1.1.3.cmml" xref="S2.SS3.p1.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.6.m6.1c">a^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.6.m6.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, and the <span class="ltx_text ltx_font_italic" id="S2.SS3.p1.7.5">in-expectation</span> regret, defined by the mixed strategy <math alttext="{\bm{x}}^{t}" class="ltx_Math" display="inline" id="S2.SS3.p1.7.m7.1"><semantics id="S2.SS3.p1.7.m7.1a"><msup id="S2.SS3.p1.7.m7.1.1" xref="S2.SS3.p1.7.m7.1.1.cmml"><mi id="S2.SS3.p1.7.m7.1.1.2" xref="S2.SS3.p1.7.m7.1.1.2.cmml">𝒙</mi><mi id="S2.SS3.p1.7.m7.1.1.3" xref="S2.SS3.p1.7.m7.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.7.m7.1b"><apply id="S2.SS3.p1.7.m7.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.7.m7.1.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1">superscript</csymbol><ci id="S2.SS3.p1.7.m7.1.1.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2">𝒙</ci><ci id="S2.SS3.p1.7.m7.1.1.3.cmml" xref="S2.SS3.p1.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.7.m7.1c">{\bm{x}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.7.m7.1d">bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" 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">Proposition 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.10"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p1.10.10">Let <math alttext="{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.1.m1.1"><semantics id="S2.Thmtheorem1.p1.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.1.m1.1b"><ci id="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.1.m1.1c">{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.1.m1.1d">caligraphic_R</annotation></semantics></math> be a no-regret algorithm on the <math alttext="m" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.2.m2.1a"><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.2.m2.1b"><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.2.m2.1d">italic_m</annotation></semantics></math>-simplex <math alttext="\Delta(m)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.3.m3.1"><semantics id="S2.Thmtheorem1.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem1.p1.3.3.m3.1.2" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.cmml"><mi id="S2.Thmtheorem1.p1.3.3.m3.1.2.2" mathvariant="normal" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.2.cmml">Δ</mi><mo id="S2.Thmtheorem1.p1.3.3.m3.1.2.1" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.3.3.m3.1.2.3.2" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.cmml"><mo id="S2.Thmtheorem1.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.cmml">(</mo><mi id="S2.Thmtheorem1.p1.3.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml">m</mi><mo id="S2.Thmtheorem1.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.3.m3.1b"><apply id="S2.Thmtheorem1.p1.3.3.m3.1.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.2"><times id="S2.Thmtheorem1.p1.3.3.m3.1.2.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.1"></times><ci id="S2.Thmtheorem1.p1.3.3.m3.1.2.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.2.2">Δ</ci><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.3.m3.1c">\Delta(m)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.3.m3.1d">roman_Δ ( italic_m )</annotation></semantics></math> with regret <math alttext="R(T)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.4.m4.1"><semantics id="S2.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem1.p1.4.4.m4.1.2" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mi id="S2.Thmtheorem1.p1.4.4.m4.1.2.2" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.2.cmml">R</mi><mo id="S2.Thmtheorem1.p1.4.4.m4.1.2.1" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.4.4.m4.1.2.3.2" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mo id="S2.Thmtheorem1.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.cmml">(</mo><mi id="S2.Thmtheorem1.p1.4.4.m4.1.1" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.cmml">T</mi><mo id="S2.Thmtheorem1.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.4.m4.1b"><apply id="S2.Thmtheorem1.p1.4.4.m4.1.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.2"><times id="S2.Thmtheorem1.p1.4.4.m4.1.2.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.1"></times><ci id="S2.Thmtheorem1.p1.4.4.m4.1.2.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.2.2">𝑅</ci><ci id="S2.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.4.m4.1c">R(T)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.4.m4.1d">italic_R ( italic_T )</annotation></semantics></math> after <math alttext="T" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.5.m5.1"><semantics id="S2.Thmtheorem1.p1.5.5.m5.1a"><mi id="S2.Thmtheorem1.p1.5.5.m5.1.1" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.5.5.m5.1b"><ci id="S2.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.5.m5.1d">italic_T</annotation></semantics></math> timesteps. Let <math alttext="\hat{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.6.6.m6.1"><semantics id="S2.Thmtheorem1.p1.6.6.m6.1a"><mover accent="true" id="S2.Thmtheorem1.p1.6.6.m6.1.1" xref="S2.Thmtheorem1.p1.6.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.6.6.m6.1.1.2" xref="S2.Thmtheorem1.p1.6.6.m6.1.1.2.cmml">ℛ</mi><mo id="S2.Thmtheorem1.p1.6.6.m6.1.1.1" xref="S2.Thmtheorem1.p1.6.6.m6.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.6.6.m6.1b"><apply id="S2.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.1"><ci id="S2.Thmtheorem1.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.1.1">^</ci><ci id="S2.Thmtheorem1.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.1.2">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.6.6.m6.1c">\hat{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.6.6.m6.1d">over^ start_ARG caligraphic_R end_ARG</annotation></semantics></math> be the algorithm that samples <math alttext="a^{t}\sim{\bm{x}}^{t}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.7.7.m7.1"><semantics id="S2.Thmtheorem1.p1.7.7.m7.1a"><mrow id="S2.Thmtheorem1.p1.7.7.m7.1.1" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.cmml"><msup id="S2.Thmtheorem1.p1.7.7.m7.1.1.2" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.2.cmml"><mi id="S2.Thmtheorem1.p1.7.7.m7.1.1.2.2" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.2.2.cmml">a</mi><mi id="S2.Thmtheorem1.p1.7.7.m7.1.1.2.3" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.2.3.cmml">t</mi></msup><mo id="S2.Thmtheorem1.p1.7.7.m7.1.1.1" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.1.cmml">∼</mo><msup id="S2.Thmtheorem1.p1.7.7.m7.1.1.3" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.7.7.m7.1.1.3.2" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.3.2.cmml">𝐱</mi><mi id="S2.Thmtheorem1.p1.7.7.m7.1.1.3.3" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.7.7.m7.1b"><apply id="S2.Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1"><csymbol cd="latexml" id="S2.Thmtheorem1.p1.7.7.m7.1.1.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.1">similar-to</csymbol><apply id="S2.Thmtheorem1.p1.7.7.m7.1.1.2.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.7.7.m7.1.1.2.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem1.p1.7.7.m7.1.1.2.2.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.2.2">𝑎</ci><ci id="S2.Thmtheorem1.p1.7.7.m7.1.1.2.3.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.2.3">𝑡</ci></apply><apply id="S2.Thmtheorem1.p1.7.7.m7.1.1.3.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.7.7.m7.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem1.p1.7.7.m7.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.3.2">𝐱</ci><ci id="S2.Thmtheorem1.p1.7.7.m7.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.7.7.m7.1c">a^{t}\sim{\bm{x}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.7.7.m7.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∼ bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> from strategies <math alttext="{\bm{x}}^{t}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.8.8.m8.1"><semantics id="S2.Thmtheorem1.p1.8.8.m8.1a"><msup id="S2.Thmtheorem1.p1.8.8.m8.1.1" xref="S2.Thmtheorem1.p1.8.8.m8.1.1.cmml"><mi id="S2.Thmtheorem1.p1.8.8.m8.1.1.2" xref="S2.Thmtheorem1.p1.8.8.m8.1.1.2.cmml">𝐱</mi><mi id="S2.Thmtheorem1.p1.8.8.m8.1.1.3" xref="S2.Thmtheorem1.p1.8.8.m8.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.8.8.m8.1b"><apply id="S2.Thmtheorem1.p1.8.8.m8.1.1.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.8.8.m8.1.1.1.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.1.1">superscript</csymbol><ci id="S2.Thmtheorem1.p1.8.8.m8.1.1.2.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.1.1.2">𝐱</ci><ci id="S2.Thmtheorem1.p1.8.8.m8.1.1.3.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.8.8.m8.1c">{\bm{x}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.8.8.m8.1d">bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> chosen by <math alttext="{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.9.9.m9.1"><semantics id="S2.Thmtheorem1.p1.9.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.9.9.m9.1.1" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.9.9.m9.1b"><ci id="S2.Thmtheorem1.p1.9.9.m9.1.1.cmml" xref="S2.Thmtheorem1.p1.9.9.m9.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.9.9.m9.1c">{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.9.9.m9.1d">caligraphic_R</annotation></semantics></math>. Then, for all possible adversaries, <math alttext="\hat{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.10.10.m10.1"><semantics id="S2.Thmtheorem1.p1.10.10.m10.1a"><mover accent="true" id="S2.Thmtheorem1.p1.10.10.m10.1.1" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.10.10.m10.1.1.2" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.2.cmml">ℛ</mi><mo id="S2.Thmtheorem1.p1.10.10.m10.1.1.1" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.10.10.m10.1b"><apply id="S2.Thmtheorem1.p1.10.10.m10.1.1.cmml" xref="S2.Thmtheorem1.p1.10.10.m10.1.1"><ci id="S2.Thmtheorem1.p1.10.10.m10.1.1.1.cmml" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.1">^</ci><ci id="S2.Thmtheorem1.p1.10.10.m10.1.1.2.cmml" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.2">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.10.10.m10.1c">\hat{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.10.10.m10.1d">over^ start_ARG caligraphic_R end_ARG</annotation></semantics></math> achieves the regret guarantee</span></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="\hat{R}(T)~{}:=\max_{a\in A}\sum_{t=1}^{T}\quantity({\bm{u}}^{t}[a]-{\bm{u}}^{% t}[a^{t}])~{}\leq~{}R(T)+{\mathcal{O}}\quantity(\sqrt{T\log\frac{1}{\delta}})" class="ltx_Math" display="block" id="S2.Ex1.m1.4"><semantics id="S2.Ex1.m1.4a"><mrow id="S2.Ex1.m1.4.5" xref="S2.Ex1.m1.4.5.cmml"><mrow id="S2.Ex1.m1.4.5.2" xref="S2.Ex1.m1.4.5.2.cmml"><mover accent="true" id="S2.Ex1.m1.4.5.2.2" xref="S2.Ex1.m1.4.5.2.2.cmml"><mi id="S2.Ex1.m1.4.5.2.2.2" xref="S2.Ex1.m1.4.5.2.2.2.cmml">R</mi><mo 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id="S2.Ex1.m1.2.2.1.1.1a.cmml" xref="S2.Ex1.m1.2.2.3"></root><apply id="S2.Ex1.m1.2.2.1.1.1.2.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2"><times id="S2.Ex1.m1.2.2.1.1.1.2.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.1"></times><ci id="S2.Ex1.m1.2.2.1.1.1.2.2.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.2">𝑇</ci><apply id="S2.Ex1.m1.2.2.1.1.1.2.3.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.3"><log id="S2.Ex1.m1.2.2.1.1.1.2.3.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.3.1"></log><apply id="S2.Ex1.m1.2.2.1.1.1.2.3.2.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.3.2"><divide id="S2.Ex1.m1.2.2.1.1.1.2.3.2.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.3.2"></divide><cn id="S2.Ex1.m1.2.2.1.1.1.2.3.2.2.cmml" type="integer" xref="S2.Ex1.m1.2.2.1.1.1.2.3.2.2">1</cn><ci id="S2.Ex1.m1.2.2.1.1.1.2.3.2.3.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.3.2.3">𝛿</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.4c">\hat{R}(T)~{}:=\max_{a\in A}\sum_{t=1}^{T}\quantity({\bm{u}}^{t}[a]-{\bm{u}}^{% t}[a^{t}])~{}\leq~{}R(T)+{\mathcal{O}}\quantity(\sqrt{T\log\frac{1}{\delta}})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.4d">over^ start_ARG italic_R end_ARG ( italic_T ) := roman_max start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( start_ARG bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_a ] - bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ] end_ARG ) ≤ italic_R ( italic_T ) + caligraphic_O ( start_ARG square-root start_ARG italic_T roman_log divide start_ARG 1 end_ARG start_ARG italic_δ end_ARG end_ARG end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem1.p1.11"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p1.11.1">with probability at least <math alttext="1-\delta" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.11.1.m1.1"><semantics id="S2.Thmtheorem1.p1.11.1.m1.1a"><mrow id="S2.Thmtheorem1.p1.11.1.m1.1.1" xref="S2.Thmtheorem1.p1.11.1.m1.1.1.cmml"><mn id="S2.Thmtheorem1.p1.11.1.m1.1.1.2" xref="S2.Thmtheorem1.p1.11.1.m1.1.1.2.cmml">1</mn><mo id="S2.Thmtheorem1.p1.11.1.m1.1.1.1" xref="S2.Thmtheorem1.p1.11.1.m1.1.1.1.cmml">−</mo><mi id="S2.Thmtheorem1.p1.11.1.m1.1.1.3" xref="S2.Thmtheorem1.p1.11.1.m1.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.11.1.m1.1b"><apply id="S2.Thmtheorem1.p1.11.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.11.1.m1.1.1"><minus id="S2.Thmtheorem1.p1.11.1.m1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.11.1.m1.1.1.1"></minus><cn id="S2.Thmtheorem1.p1.11.1.m1.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.11.1.m1.1.1.2">1</cn><ci id="S2.Thmtheorem1.p1.11.1.m1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.11.1.m1.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.11.1.m1.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.11.1.m1.1d">1 - italic_δ</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.3">(Omitted proofs, unless otherwise stated, can be found in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A3" title="Appendix C Other omitted proofs ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">C</span></a>.) Thus, in particular, if <math alttext="{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">caligraphic_R</annotation></semantics></math> is MWU, we have <math alttext="\hat{R}(T)\lesssim\sqrt{T\log(m/\delta)}" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.3"><semantics id="S2.SS3.p2.2.m2.3a"><mrow id="S2.SS3.p2.2.m2.3.4" xref="S2.SS3.p2.2.m2.3.4.cmml"><mrow id="S2.SS3.p2.2.m2.3.4.2" xref="S2.SS3.p2.2.m2.3.4.2.cmml"><mover accent="true" id="S2.SS3.p2.2.m2.3.4.2.2" xref="S2.SS3.p2.2.m2.3.4.2.2.cmml"><mi id="S2.SS3.p2.2.m2.3.4.2.2.2" xref="S2.SS3.p2.2.m2.3.4.2.2.2.cmml">R</mi><mo id="S2.SS3.p2.2.m2.3.4.2.2.1" xref="S2.SS3.p2.2.m2.3.4.2.2.1.cmml">^</mo></mover><mo id="S2.SS3.p2.2.m2.3.4.2.1" xref="S2.SS3.p2.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.2.m2.3.4.2.3.2" xref="S2.SS3.p2.2.m2.3.4.2.cmml"><mo id="S2.SS3.p2.2.m2.3.4.2.3.2.1" stretchy="false" xref="S2.SS3.p2.2.m2.3.4.2.cmml">(</mo><mi id="S2.SS3.p2.2.m2.3.3" xref="S2.SS3.p2.2.m2.3.3.cmml">T</mi><mo id="S2.SS3.p2.2.m2.3.4.2.3.2.2" stretchy="false" xref="S2.SS3.p2.2.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p2.2.m2.3.4.1" xref="S2.SS3.p2.2.m2.3.4.1.cmml">≲</mo><msqrt id="S2.SS3.p2.2.m2.2.2" xref="S2.SS3.p2.2.m2.2.2.cmml"><mrow id="S2.SS3.p2.2.m2.2.2.2" xref="S2.SS3.p2.2.m2.2.2.2.cmml"><mi id="S2.SS3.p2.2.m2.2.2.2.4" xref="S2.SS3.p2.2.m2.2.2.2.4.cmml">T</mi><mo id="S2.SS3.p2.2.m2.2.2.2.3" lspace="0.167em" xref="S2.SS3.p2.2.m2.2.2.2.3.cmml">⁢</mo><mrow id="S2.SS3.p2.2.m2.2.2.2.2.4" xref="S2.SS3.p2.2.m2.2.2.2.2.3.cmml"><mi id="S2.SS3.p2.2.m2.2.2.2.2.2.2" xref="S2.SS3.p2.2.m2.2.2.2.2.3.1.cmml">log</mi><mo id="S2.SS3.p2.2.m2.2.2.2.2.4a" xref="S2.SS3.p2.2.m2.2.2.2.2.3.1.cmml">⁡</mo><mrow id="S2.SS3.p2.2.m2.2.2.2.2.4.1" xref="S2.SS3.p2.2.m2.2.2.2.2.3.cmml"><mo id="S2.SS3.p2.2.m2.2.2.2.2.4.1.1" xref="S2.SS3.p2.2.m2.2.2.2.2.3.1.cmml">(</mo><mrow id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.cmml"><mi id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.2" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.2.cmml">m</mi><mo id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.1" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.3" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.3.cmml">δ</mi></mrow><mo id="S2.SS3.p2.2.m2.2.2.2.2.4.1.2" xref="S2.SS3.p2.2.m2.2.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.3b"><apply id="S2.SS3.p2.2.m2.3.4.cmml" xref="S2.SS3.p2.2.m2.3.4"><csymbol cd="latexml" id="S2.SS3.p2.2.m2.3.4.1.cmml" 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xref="S2.SS3.p2.2.m2.2.2.2.2.2.2"></log><apply id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1"><divide id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.1"></divide><ci id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.2.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.2">𝑚</ci><ci id="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1.1.1.1.3">𝛿</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.3c">\hat{R}(T)\lesssim\sqrt{T\log(m/\delta)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.3d">over^ start_ARG italic_R end_ARG ( italic_T ) ≲ square-root start_ARG italic_T roman_log ( start_ARG italic_m / italic_δ end_ARG ) end_ARG</annotation></semantics></math> with probability at least <math alttext="1-\delta" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><mrow id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml"><mn id="S2.SS3.p2.3.m3.1.1.2" xref="S2.SS3.p2.3.m3.1.1.2.cmml">1</mn><mo id="S2.SS3.p2.3.m3.1.1.1" xref="S2.SS3.p2.3.m3.1.1.1.cmml">−</mo><mi id="S2.SS3.p2.3.m3.1.1.3" xref="S2.SS3.p2.3.m3.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><apply id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1"><minus id="S2.SS3.p2.3.m3.1.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1.1"></minus><cn id="S2.SS3.p2.3.m3.1.1.2.cmml" type="integer" xref="S2.SS3.p2.3.m3.1.1.2">1</cn><ci id="S2.SS3.p2.3.m3.1.1.3.cmml" xref="S2.SS3.p2.3.m3.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">1 - italic_δ</annotation></semantics></math>.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Our Setting</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.3">We consider a setting in which a principal oversees the <math alttext="n" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_n</annotation></semantics></math>-agent game repeatedly being played over <math alttext="T" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mi id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><ci id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_T</annotation></semantics></math> rounds <math alttext="t=1,\dots T" class="ltx_Math" display="inline" id="S3.p1.3.m3.2"><semantics id="S3.p1.3.m3.2a"><mrow id="S3.p1.3.m3.2.2" xref="S3.p1.3.m3.2.2.cmml"><mi id="S3.p1.3.m3.2.2.3" xref="S3.p1.3.m3.2.2.3.cmml">t</mi><mo id="S3.p1.3.m3.2.2.2" xref="S3.p1.3.m3.2.2.2.cmml">=</mo><mrow id="S3.p1.3.m3.2.2.1.1" xref="S3.p1.3.m3.2.2.1.2.cmml"><mn id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">1</mn><mo id="S3.p1.3.m3.2.2.1.1.2" xref="S3.p1.3.m3.2.2.1.2.cmml">,</mo><mrow id="S3.p1.3.m3.2.2.1.1.1" xref="S3.p1.3.m3.2.2.1.1.1.cmml"><mi id="S3.p1.3.m3.2.2.1.1.1.2" mathvariant="normal" xref="S3.p1.3.m3.2.2.1.1.1.2.cmml">…</mi><mo id="S3.p1.3.m3.2.2.1.1.1.1" xref="S3.p1.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mi id="S3.p1.3.m3.2.2.1.1.1.3" xref="S3.p1.3.m3.2.2.1.1.1.3.cmml">T</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.2b"><apply id="S3.p1.3.m3.2.2.cmml" xref="S3.p1.3.m3.2.2"><eq id="S3.p1.3.m3.2.2.2.cmml" xref="S3.p1.3.m3.2.2.2"></eq><ci id="S3.p1.3.m3.2.2.3.cmml" xref="S3.p1.3.m3.2.2.3">𝑡</ci><list id="S3.p1.3.m3.2.2.1.2.cmml" xref="S3.p1.3.m3.2.2.1.1"><cn id="S3.p1.3.m3.1.1.cmml" type="integer" xref="S3.p1.3.m3.1.1">1</cn><apply id="S3.p1.3.m3.2.2.1.1.1.cmml" xref="S3.p1.3.m3.2.2.1.1.1"><times id="S3.p1.3.m3.2.2.1.1.1.1.cmml" xref="S3.p1.3.m3.2.2.1.1.1.1"></times><ci id="S3.p1.3.m3.2.2.1.1.1.2.cmml" xref="S3.p1.3.m3.2.2.1.1.1.2">…</ci><ci id="S3.p1.3.m3.2.2.1.1.1.3.cmml" xref="S3.p1.3.m3.2.2.1.1.1.3">𝑇</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.2c">t=1,\dots T</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.2d">italic_t = 1 , … italic_T</annotation></semantics></math>. In each round, the following events happen, in order:</p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.6">The principal selects <span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.6.1">payment function</span> <math alttext="P_{i}^{t}:A_{i}\to\mathbb{R}_{+}" 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"><msubsup id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2.2.2" xref="S3.I1.i1.p1.1.m1.1.1.2.2.2.cmml">P</mi><mi id="S3.I1.i1.p1.1.m1.1.1.2.2.3" xref="S3.I1.i1.p1.1.m1.1.1.2.2.3.cmml">i</mi><mi id="S3.I1.i1.p1.1.m1.1.1.2.3" xref="S3.I1.i1.p1.1.m1.1.1.2.3.cmml">t</mi></msubsup><mo id="S3.I1.i1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.I1.i1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml"><msub id="S3.I1.i1.p1.1.m1.1.1.3.2" xref="S3.I1.i1.p1.1.m1.1.1.3.2.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.3.2.2" xref="S3.I1.i1.p1.1.m1.1.1.3.2.2.cmml">A</mi><mi id="S3.I1.i1.p1.1.m1.1.1.3.2.3" xref="S3.I1.i1.p1.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S3.I1.i1.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.I1.i1.p1.1.m1.1.1.3.1.cmml">→</mo><msub id="S3.I1.i1.p1.1.m1.1.1.3.3" xref="S3.I1.i1.p1.1.m1.1.1.3.3.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.3.3.2" xref="S3.I1.i1.p1.1.m1.1.1.3.3.2.cmml">ℝ</mi><mo id="S3.I1.i1.p1.1.m1.1.1.3.3.3" xref="S3.I1.i1.p1.1.m1.1.1.3.3.3.cmml">+</mo></msub></mrow></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"><ci id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1">:</ci><apply id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">superscript</csymbol><apply id="S3.I1.i1.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.2.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.2.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2.2.2">𝑃</ci><ci id="S3.I1.i1.p1.1.m1.1.1.2.2.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2.2.3">𝑖</ci></apply><ci id="S3.I1.i1.p1.1.m1.1.1.2.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2.3">𝑡</ci></apply><apply id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3"><ci id="S3.I1.i1.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.1">→</ci><apply id="S3.I1.i1.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.3.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.3.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2.2">𝐴</ci><ci id="S3.I1.i1.p1.1.m1.1.1.3.2.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S3.I1.i1.p1.1.m1.1.1.3.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.3.3.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.3.3.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3.2">ℝ</ci><plus id="S3.I1.i1.p1.1.m1.1.1.3.3.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">P_{i}^{t}:A_{i}\to\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT : italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> for each agent <math alttext="i" 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">i</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">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">italic_i</annotation></semantics></math>. The payment <math alttext="P_{i}^{t}(a_{i})" class="ltx_Math" display="inline" id="S3.I1.i1.p1.3.m3.1"><semantics id="S3.I1.i1.p1.3.m3.1a"><mrow id="S3.I1.i1.p1.3.m3.1.1" xref="S3.I1.i1.p1.3.m3.1.1.cmml"><msubsup id="S3.I1.i1.p1.3.m3.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.3.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.3.2.2" xref="S3.I1.i1.p1.3.m3.1.1.3.2.2.cmml">P</mi><mi id="S3.I1.i1.p1.3.m3.1.1.3.2.3" xref="S3.I1.i1.p1.3.m3.1.1.3.2.3.cmml">i</mi><mi id="S3.I1.i1.p1.3.m3.1.1.3.3" xref="S3.I1.i1.p1.3.m3.1.1.3.3.cmml">t</mi></msubsup><mo id="S3.I1.i1.p1.3.m3.1.1.2" xref="S3.I1.i1.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.I1.i1.p1.3.m3.1.1.1.1" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.I1.i1.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.cmml">(</mo><msub id="S3.I1.i1.p1.3.m3.1.1.1.1.1" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.1.1.1.2" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.2.cmml">a</mi><mi id="S3.I1.i1.p1.3.m3.1.1.1.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.I1.i1.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><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"><times id="S3.I1.i1.p1.3.m3.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2"></times><apply id="S3.I1.i1.p1.3.m3.1.1.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.3.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3">superscript</csymbol><apply id="S3.I1.i1.p1.3.m3.1.1.3.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.3.2.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.I1.i1.p1.3.m3.1.1.3.2.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.2.2">𝑃</ci><ci id="S3.I1.i1.p1.3.m3.1.1.3.2.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.2.3">𝑖</ci></apply><ci id="S3.I1.i1.p1.3.m3.1.1.3.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.3">𝑡</ci></apply><apply id="S3.I1.i1.p1.3.m3.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.2">𝑎</ci><ci id="S3.I1.i1.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">P_{i}^{t}(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is added to agent <math alttext="i" class="ltx_Math" display="inline" id="S3.I1.i1.p1.4.m4.1"><semantics id="S3.I1.i1.p1.4.m4.1a"><mi id="S3.I1.i1.p1.4.m4.1.1" xref="S3.I1.i1.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.4.m4.1b"><ci id="S3.I1.i1.p1.4.m4.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.4.m4.1d">italic_i</annotation></semantics></math>’s reward, creating a new game <math alttext="\Gamma^{t}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.5.m5.1"><semantics id="S3.I1.i1.p1.5.m5.1a"><msup id="S3.I1.i1.p1.5.m5.1.1" xref="S3.I1.i1.p1.5.m5.1.1.cmml"><mi id="S3.I1.i1.p1.5.m5.1.1.2" mathvariant="normal" xref="S3.I1.i1.p1.5.m5.1.1.2.cmml">Γ</mi><mi id="S3.I1.i1.p1.5.m5.1.1.3" xref="S3.I1.i1.p1.5.m5.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.5.m5.1b"><apply id="S3.I1.i1.p1.5.m5.1.1.cmml" xref="S3.I1.i1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.5.m5.1.1.1.cmml" xref="S3.I1.i1.p1.5.m5.1.1">superscript</csymbol><ci id="S3.I1.i1.p1.5.m5.1.1.2.cmml" xref="S3.I1.i1.p1.5.m5.1.1.2">Γ</ci><ci id="S3.I1.i1.p1.5.m5.1.1.3.cmml" xref="S3.I1.i1.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.5.m5.1c">\Gamma^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.5.m5.1d">roman_Γ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> with utility functions given by <math alttext="U^{t}_{i}(a):=U_{i}(a)+P^{t}_{i}(a_{i})" class="ltx_Math" display="inline" id="S3.I1.i1.p1.6.m6.3"><semantics id="S3.I1.i1.p1.6.m6.3a"><mrow id="S3.I1.i1.p1.6.m6.3.3" xref="S3.I1.i1.p1.6.m6.3.3.cmml"><mrow id="S3.I1.i1.p1.6.m6.3.3.3" xref="S3.I1.i1.p1.6.m6.3.3.3.cmml"><msubsup id="S3.I1.i1.p1.6.m6.3.3.3.2" xref="S3.I1.i1.p1.6.m6.3.3.3.2.cmml"><mi id="S3.I1.i1.p1.6.m6.3.3.3.2.2.2" xref="S3.I1.i1.p1.6.m6.3.3.3.2.2.2.cmml">U</mi><mi id="S3.I1.i1.p1.6.m6.3.3.3.2.3" xref="S3.I1.i1.p1.6.m6.3.3.3.2.3.cmml">i</mi><mi id="S3.I1.i1.p1.6.m6.3.3.3.2.2.3" xref="S3.I1.i1.p1.6.m6.3.3.3.2.2.3.cmml">t</mi></msubsup><mo id="S3.I1.i1.p1.6.m6.3.3.3.1" xref="S3.I1.i1.p1.6.m6.3.3.3.1.cmml">⁢</mo><mrow id="S3.I1.i1.p1.6.m6.3.3.3.3.2" xref="S3.I1.i1.p1.6.m6.3.3.3.cmml"><mo id="S3.I1.i1.p1.6.m6.3.3.3.3.2.1" stretchy="false" xref="S3.I1.i1.p1.6.m6.3.3.3.cmml">(</mo><mi id="S3.I1.i1.p1.6.m6.1.1" xref="S3.I1.i1.p1.6.m6.1.1.cmml">a</mi><mo id="S3.I1.i1.p1.6.m6.3.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S3.I1.i1.p1.6.m6.3.3.3.cmml">)</mo></mrow></mrow><mo id="S3.I1.i1.p1.6.m6.3.3.2" rspace="0.278em" xref="S3.I1.i1.p1.6.m6.3.3.2.cmml">:=</mo><mrow id="S3.I1.i1.p1.6.m6.3.3.1" xref="S3.I1.i1.p1.6.m6.3.3.1.cmml"><mrow id="S3.I1.i1.p1.6.m6.3.3.1.3" xref="S3.I1.i1.p1.6.m6.3.3.1.3.cmml"><msub id="S3.I1.i1.p1.6.m6.3.3.1.3.2" xref="S3.I1.i1.p1.6.m6.3.3.1.3.2.cmml"><mi id="S3.I1.i1.p1.6.m6.3.3.1.3.2.2" xref="S3.I1.i1.p1.6.m6.3.3.1.3.2.2.cmml">U</mi><mi id="S3.I1.i1.p1.6.m6.3.3.1.3.2.3" xref="S3.I1.i1.p1.6.m6.3.3.1.3.2.3.cmml">i</mi></msub><mo id="S3.I1.i1.p1.6.m6.3.3.1.3.1" xref="S3.I1.i1.p1.6.m6.3.3.1.3.1.cmml">⁢</mo><mrow id="S3.I1.i1.p1.6.m6.3.3.1.3.3.2" xref="S3.I1.i1.p1.6.m6.3.3.1.3.cmml"><mo id="S3.I1.i1.p1.6.m6.3.3.1.3.3.2.1" stretchy="false" xref="S3.I1.i1.p1.6.m6.3.3.1.3.cmml">(</mo><mi id="S3.I1.i1.p1.6.m6.2.2" xref="S3.I1.i1.p1.6.m6.2.2.cmml">a</mi><mo id="S3.I1.i1.p1.6.m6.3.3.1.3.3.2.2" stretchy="false" xref="S3.I1.i1.p1.6.m6.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S3.I1.i1.p1.6.m6.3.3.1.2" xref="S3.I1.i1.p1.6.m6.3.3.1.2.cmml">+</mo><mrow id="S3.I1.i1.p1.6.m6.3.3.1.1" xref="S3.I1.i1.p1.6.m6.3.3.1.1.cmml"><msubsup id="S3.I1.i1.p1.6.m6.3.3.1.1.3" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3.cmml"><mi id="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.2" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.2.cmml">P</mi><mi id="S3.I1.i1.p1.6.m6.3.3.1.1.3.3" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3.3.cmml">i</mi><mi id="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.3" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.3.cmml">t</mi></msubsup><mo id="S3.I1.i1.p1.6.m6.3.3.1.1.2" xref="S3.I1.i1.p1.6.m6.3.3.1.1.2.cmml">⁢</mo><mrow id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.cmml"><mo id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.2" stretchy="false" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.cmml">(</mo><msub id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.cmml"><mi id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.2" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.2.cmml">a</mi><mi id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.3" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.3" stretchy="false" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.6.m6.3b"><apply id="S3.I1.i1.p1.6.m6.3.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3"><csymbol cd="latexml" id="S3.I1.i1.p1.6.m6.3.3.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.2">assign</csymbol><apply id="S3.I1.i1.p1.6.m6.3.3.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3"><times id="S3.I1.i1.p1.6.m6.3.3.3.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.1"></times><apply id="S3.I1.i1.p1.6.m6.3.3.3.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.3.3.3.2.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.2">subscript</csymbol><apply id="S3.I1.i1.p1.6.m6.3.3.3.2.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.3.3.3.2.2.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.2">superscript</csymbol><ci id="S3.I1.i1.p1.6.m6.3.3.3.2.2.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.2.2.2">𝑈</ci><ci id="S3.I1.i1.p1.6.m6.3.3.3.2.2.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.2.2.3">𝑡</ci></apply><ci id="S3.I1.i1.p1.6.m6.3.3.3.2.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.3.2.3">𝑖</ci></apply><ci id="S3.I1.i1.p1.6.m6.1.1.cmml" xref="S3.I1.i1.p1.6.m6.1.1">𝑎</ci></apply><apply id="S3.I1.i1.p1.6.m6.3.3.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1"><plus id="S3.I1.i1.p1.6.m6.3.3.1.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.2"></plus><apply id="S3.I1.i1.p1.6.m6.3.3.1.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.3"><times id="S3.I1.i1.p1.6.m6.3.3.1.3.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.3.1"></times><apply id="S3.I1.i1.p1.6.m6.3.3.1.3.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.3.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.3.3.1.3.2.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.3.2">subscript</csymbol><ci id="S3.I1.i1.p1.6.m6.3.3.1.3.2.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.3.2.2">𝑈</ci><ci id="S3.I1.i1.p1.6.m6.3.3.1.3.2.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.3.2.3">𝑖</ci></apply><ci id="S3.I1.i1.p1.6.m6.2.2.cmml" xref="S3.I1.i1.p1.6.m6.2.2">𝑎</ci></apply><apply id="S3.I1.i1.p1.6.m6.3.3.1.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1"><times id="S3.I1.i1.p1.6.m6.3.3.1.1.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.2"></times><apply id="S3.I1.i1.p1.6.m6.3.3.1.1.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.3.3.1.1.3.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3">subscript</csymbol><apply id="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3">superscript</csymbol><ci id="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.2">𝑃</ci><ci id="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3.2.3">𝑡</ci></apply><ci id="S3.I1.i1.p1.6.m6.3.3.1.1.3.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.3.3">𝑖</ci></apply><apply id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.2">𝑎</ci><ci id="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.3.cmml" xref="S3.I1.i1.p1.6.m6.3.3.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.6.m6.3c">U^{t}_{i}(a):=U_{i}(a)+P^{t}_{i}(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.6.m6.3d">italic_U start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a ) := italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a ) + italic_P start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.2">Each agent <math alttext="i" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><mi id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><ci id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_i</annotation></semantics></math> simultaneously selects an action <math alttext="a^{t}_{i}\in A_{i}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><mrow id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml"><msubsup id="S3.I1.i2.p1.2.m2.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.2.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.2.2.2" xref="S3.I1.i2.p1.2.m2.1.1.2.2.2.cmml">a</mi><mi id="S3.I1.i2.p1.2.m2.1.1.2.3" xref="S3.I1.i2.p1.2.m2.1.1.2.3.cmml">i</mi><mi id="S3.I1.i2.p1.2.m2.1.1.2.2.3" xref="S3.I1.i2.p1.2.m2.1.1.2.2.3.cmml">t</mi></msubsup><mo id="S3.I1.i2.p1.2.m2.1.1.1" xref="S3.I1.i2.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S3.I1.i2.p1.2.m2.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.3.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.3.2" xref="S3.I1.i2.p1.2.m2.1.1.3.2.cmml">A</mi><mi id="S3.I1.i2.p1.2.m2.1.1.3.3" xref="S3.I1.i2.p1.2.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><apply id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1"><in id="S3.I1.i2.p1.2.m2.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1"></in><apply id="S3.I1.i2.p1.2.m2.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.2.m2.1.1.2.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2">subscript</csymbol><apply id="S3.I1.i2.p1.2.m2.1.1.2.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.2.m2.1.1.2.2.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2">superscript</csymbol><ci id="S3.I1.i2.p1.2.m2.1.1.2.2.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2.2.2">𝑎</ci><ci id="S3.I1.i2.p1.2.m2.1.1.2.2.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2.2.3">𝑡</ci></apply><ci id="S3.I1.i2.p1.2.m2.1.1.2.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S3.I1.i2.p1.2.m2.1.1.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.2.m2.1.1.3.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3.2">𝐴</ci><ci id="S3.I1.i2.p1.2.m2.1.1.3.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">a^{t}_{i}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.3">The principal observes the joint strategy <math alttext="a^{t}_{i}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.1"><semantics id="S3.I1.i3.p1.1.m1.1a"><msubsup id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml"><mi id="S3.I1.i3.p1.1.m1.1.1.2.2" xref="S3.I1.i3.p1.1.m1.1.1.2.2.cmml">a</mi><mi id="S3.I1.i3.p1.1.m1.1.1.3" xref="S3.I1.i3.p1.1.m1.1.1.3.cmml">i</mi><mi id="S3.I1.i3.p1.1.m1.1.1.2.3" xref="S3.I1.i3.p1.1.m1.1.1.2.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.1b"><apply id="S3.I1.i3.p1.1.m1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">subscript</csymbol><apply id="S3.I1.i3.p1.1.m1.1.1.2.cmml" xref="S3.I1.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">superscript</csymbol><ci id="S3.I1.i3.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i3.p1.1.m1.1.1.2.2">𝑎</ci><ci id="S3.I1.i3.p1.1.m1.1.1.2.3.cmml" xref="S3.I1.i3.p1.1.m1.1.1.2.3">𝑡</ci></apply><ci id="S3.I1.i3.p1.1.m1.1.1.3.cmml" xref="S3.I1.i3.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.1c">a^{t}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.m1.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Each agent <math alttext="i" class="ltx_Math" display="inline" id="S3.I1.i3.p1.2.m2.1"><semantics id="S3.I1.i3.p1.2.m2.1a"><mi id="S3.I1.i3.p1.2.m2.1.1" xref="S3.I1.i3.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.2.m2.1b"><ci id="S3.I1.i3.p1.2.m2.1.1.cmml" xref="S3.I1.i3.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.2.m2.1d">italic_i</annotation></semantics></math> gets reward <math alttext="U_{i}^{t}(a^{t})" class="ltx_Math" display="inline" id="S3.I1.i3.p1.3.m3.1"><semantics id="S3.I1.i3.p1.3.m3.1a"><mrow id="S3.I1.i3.p1.3.m3.1.1" xref="S3.I1.i3.p1.3.m3.1.1.cmml"><msubsup id="S3.I1.i3.p1.3.m3.1.1.3" xref="S3.I1.i3.p1.3.m3.1.1.3.cmml"><mi id="S3.I1.i3.p1.3.m3.1.1.3.2.2" xref="S3.I1.i3.p1.3.m3.1.1.3.2.2.cmml">U</mi><mi id="S3.I1.i3.p1.3.m3.1.1.3.2.3" xref="S3.I1.i3.p1.3.m3.1.1.3.2.3.cmml">i</mi><mi id="S3.I1.i3.p1.3.m3.1.1.3.3" xref="S3.I1.i3.p1.3.m3.1.1.3.3.cmml">t</mi></msubsup><mo id="S3.I1.i3.p1.3.m3.1.1.2" xref="S3.I1.i3.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.I1.i3.p1.3.m3.1.1.1.1" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.I1.i3.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S3.I1.i3.p1.3.m3.1.1.1.1.1" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.cmml"><mi id="S3.I1.i3.p1.3.m3.1.1.1.1.1.2" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.2.cmml">a</mi><mi id="S3.I1.i3.p1.3.m3.1.1.1.1.1.3" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.3.cmml">t</mi></msup><mo id="S3.I1.i3.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.3.m3.1b"><apply id="S3.I1.i3.p1.3.m3.1.1.cmml" xref="S3.I1.i3.p1.3.m3.1.1"><times id="S3.I1.i3.p1.3.m3.1.1.2.cmml" xref="S3.I1.i3.p1.3.m3.1.1.2"></times><apply id="S3.I1.i3.p1.3.m3.1.1.3.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i3.p1.3.m3.1.1.3.1.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3">superscript</csymbol><apply id="S3.I1.i3.p1.3.m3.1.1.3.2.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i3.p1.3.m3.1.1.3.2.1.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.I1.i3.p1.3.m3.1.1.3.2.2.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3.2.2">𝑈</ci><ci id="S3.I1.i3.p1.3.m3.1.1.3.2.3.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3.2.3">𝑖</ci></apply><ci id="S3.I1.i3.p1.3.m3.1.1.3.3.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3.3">𝑡</ci></apply><apply id="S3.I1.i3.p1.3.m3.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.3.m3.1.1.1.1">superscript</csymbol><ci id="S3.I1.i3.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.2">𝑎</ci><ci id="S3.I1.i3.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.I1.i3.p1.3.m3.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.3.m3.1c">U_{i}^{t}(a^{t})</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.3.m3.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.2">We assume that the principal initially does not know anything about the agents’ utility functions <math alttext="U_{i}" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><msub id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">U</mi><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">𝑈</ci><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">U_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> except for boundedness, but does know the action sets <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><msub id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml"><mi id="S3.p2.2.m2.1.1.2" xref="S3.p2.2.m2.1.1.2.cmml">A</mi><mi id="S3.p2.2.m2.1.1.3" xref="S3.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p2.2.m2.1.1.1.cmml" xref="S3.p2.2.m2.1.1">subscript</csymbol><ci id="S3.p2.2.m2.1.1.2.cmml" xref="S3.p2.2.m2.1.1.2">𝐴</ci><ci id="S3.p2.2.m2.1.1.3.cmml" xref="S3.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. The principal’s goal is to (approximately) learn the utility functions.</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>Behavioral models</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">Since the only method that the principal has for learning about the utilities is by observing how the agents play the game, we must stipulate a <span class="ltx_text ltx_font_italic" id="S3.SS1.p1.1.1">behavioral model</span> for the agents, that is, we must impose some conditions on how they choose their actions <math alttext="a_{i}^{t}" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><msubsup id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2.2" xref="S3.SS1.p1.1.m1.1.1.2.2.cmml">a</mi><mi id="S3.SS1.p1.1.m1.1.1.2.3" xref="S3.SS1.p1.1.m1.1.1.2.3.cmml">i</mi><mi id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">superscript</csymbol><apply id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.1.m1.1.1.2.1.cmml" xref="S3.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.p1.1.m1.1.1.2.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2.2">𝑎</ci><ci id="S3.SS1.p1.1.m1.1.1.2.3.cmml" xref="S3.SS1.p1.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">a_{i}^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. In this paper, we will introduce two possible behavioral models.</p> </div> <section class="ltx_paragraph" id="S3.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">- Rationalizable model.</h4> <div class="ltx_para" id="S3.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p1.7">An action <math alttext="a_{i}\in A_{i}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.1.m1.1"><semantics id="S3.SS1.SSS0.Px1.p1.1.m1.1a"><mrow id="S3.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.cmml"><msub id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml"><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.2" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.2.cmml">a</mi><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.3" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.1" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.2" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.2.cmml">A</mi><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.3" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.1.m1.1b"><apply id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1"><in id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.1"></in><apply id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.2">𝑎</ci><ci id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.2">𝐴</ci><ci id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.1.m1.1c">a_{i}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for agent <math alttext="i" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.2.m2.1"><semantics id="S3.SS1.SSS0.Px1.p1.2.m2.1a"><mi id="S3.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S3.SS1.SSS0.Px1.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.2.m2.1b"><ci id="S3.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.2.m2.1d">italic_i</annotation></semantics></math> in a game <math alttext="\Gamma^{t}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.3.m3.1"><semantics id="S3.SS1.SSS0.Px1.p1.3.m3.1a"><msup id="S3.SS1.SSS0.Px1.p1.3.m3.1.1" xref="S3.SS1.SSS0.Px1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.3.m3.1.1.2" mathvariant="normal" xref="S3.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml">Γ</mi><mi id="S3.SS1.SSS0.Px1.p1.3.m3.1.1.3" xref="S3.SS1.SSS0.Px1.p1.3.m3.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.3.m3.1b"><apply id="S3.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m3.1.1">superscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m3.1.1.2">Γ</ci><ci id="S3.SS1.SSS0.Px1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.3.m3.1c">\Gamma^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.3.m3.1d">roman_Γ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px1.p1.7.1">(strictly) dominated</span> if there is a (possibly mixed) strategy <math alttext="x_{i}\in\Delta(A_{i})" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.4.m4.1"><semantics id="S3.SS1.SSS0.Px1.p1.4.m4.1a"><mrow id="S3.SS1.SSS0.Px1.p1.4.m4.1.1" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.cmml"><msub id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.3" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.3.2" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.3.2.cmml">x</mi><mi id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.3.3" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.3.3.cmml">i</mi></msub><mo id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.2" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.2.cmml">∈</mo><mrow id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.3" mathvariant="normal" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.3.cmml">Δ</mi><mo id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.2" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.2" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.2.cmml">A</mi><mi id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.3" xref="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3" stretchy="false" 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id="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.1.cmml" xref="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.2">𝑎</ci><apply id="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.3"><minus id="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.3"></minus><ci id="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.3.2.cmml" xref="S3.SS1.SSS0.Px1.p1.5.m5.4.4.4.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.5.m5.4c">U_{i}^{t}(x_{i},a_{-i})>U_{i}^{t}(a_{i},a_{-i})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.5.m5.4d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) > italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> for every action profile <math alttext="a_{-i}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.6.m6.1"><semantics id="S3.SS1.SSS0.Px1.p1.6.m6.1a"><msub id="S3.SS1.SSS0.Px1.p1.6.m6.1.1" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.2" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml">a</mi><mrow id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml"><mo id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3a" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml">−</mo><mi id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.2" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.6.m6.1b"><apply id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.2">𝑎</ci><apply id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3"><minus id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3"></minus><ci id="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.2.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m6.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.6.m6.1c">a_{-i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.6.m6.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. The process of <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px1.p1.7.2">iterated elimination of dominated strategies</span> removes dominated actions from <math alttext="\Gamma^{t}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.7.m7.1"><semantics id="S3.SS1.SSS0.Px1.p1.7.m7.1a"><msup id="S3.SS1.SSS0.Px1.p1.7.m7.1.1" xref="S3.SS1.SSS0.Px1.p1.7.m7.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.7.m7.1.1.2" mathvariant="normal" xref="S3.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml">Γ</mi><mi id="S3.SS1.SSS0.Px1.p1.7.m7.1.1.3" xref="S3.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.7.m7.1b"><apply id="S3.SS1.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m7.1.1">superscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m7.1.1.2">Γ</ci><ci id="S3.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.7.m7.1c">\Gamma^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.7.m7.1d">roman_Γ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> until no such actions remain. An action is <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px1.p1.7.3">rationalizable</span> <cite class="ltx_cite ltx_citemacro_cite">Pearce (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib20" title="">1984</a>); Bernheim (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib5" title="">1984</a>)</cite> if it survives this process. In the <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px1.p1.7.4">rationalizable</span> model, we assume that all agents play rationalizable actions.</p> </div> </section> <section class="ltx_paragraph" id="S3.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">- No-regret model.</h4> <div class="ltx_para" id="S3.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S3.SS1.SSS0.Px2.p1.5">In the <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px2.p1.5.1">no-regret model</span>, we assume that each agent selects <math alttext="a_{i}^{t}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S3.SS1.SSS0.Px2.p1.1.m1.1a"><msubsup id="S3.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.2" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.2.cmml">a</mi><mi id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.3" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.3.cmml">i</mi><mi id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.3" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.1.m1.1b"><apply id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1">superscript</csymbol><apply id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.2">𝑎</ci><ci id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.1.m1.1c">a_{i}^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> in such a way that their regret is bounded. We will also assume in this model that the principal can send <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px2.p1.5.2">signals</span> to the agents that affect their actions, and that each agent’s <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px2.p1.5.3">realized</span> regret is bounded <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px2.p1.5.4">for every signal separately</span> and <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px2.p1.5.5">at every timestep</span> <math alttext="t\leq T" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S3.SS1.SSS0.Px2.p1.2.m2.1a"><mrow id="S3.SS1.SSS0.Px2.p1.2.m2.1.1" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.2" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml">t</mi><mo id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.cmml">≤</mo><mi id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.2.m2.1b"><apply id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1"><leq id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1"></leq><ci id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.2">𝑡</ci><ci id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.2.m2.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.2.m2.1d">italic_t ≤ italic_T</annotation></semantics></math>. More precisely, we assume that there is a (possibly game-dependent) constant <math alttext="C\leq\poly(M)" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="S3.SS1.SSS0.Px2.p1.3.m3.1a"><mrow id="S3.SS1.SSS0.Px2.p1.3.m3.1.2" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.cmml"><mi id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.2" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.2.cmml">C</mi><mo id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.1" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.1.cmml">≤</mo><mrow id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.cmml"><merror class="ltx_ERROR undefined undefined" id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2b.cmml"><mtext id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2a" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2b.cmml">\poly</mtext></merror><mo id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.1" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.3.2" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.cmml"><mo id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.cmml">(</mo><mi id="S3.SS1.SSS0.Px2.p1.3.m3.1.1" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.cmml">M</mi><mo id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.3.m3.1b"><apply id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2"><leq id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.1"></leq><ci id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.2">𝐶</ci><apply id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3"><times id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.1.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.1"></times><ci id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2b.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2"><merror class="ltx_ERROR undefined undefined" id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2"><mtext id="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2a.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.2.3.2">\poly</mtext></merror></ci><ci id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.3.m3.1c">C\leq\poly(M)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.3.m3.1d">italic_C ≤ ( italic_M )</annotation></semantics></math> such that, for every agent <math alttext="i" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p1.4.m4.1"><semantics id="S3.SS1.SSS0.Px2.p1.4.m4.1a"><mi id="S3.SS1.SSS0.Px2.p1.4.m4.1.1" xref="S3.SS1.SSS0.Px2.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.4.m4.1b"><ci id="S3.SS1.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.4.m4.1d">italic_i</annotation></semantics></math> and signal <math alttext="s_{i}\in S_{i}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p1.5.m5.1"><semantics id="S3.SS1.SSS0.Px2.p1.5.m5.1a"><mrow id="S3.SS1.SSS0.Px2.p1.5.m5.1.1" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.cmml"><msub id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.cmml"><mi id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.2" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.2.cmml">s</mi><mi id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.3" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.1" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.cmml"><mi id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.2" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.2.cmml">S</mi><mi id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.3" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.5.m5.1b"><apply id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1"><in id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.1"></in><apply id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.2">𝑠</ci><ci id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.1.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.2.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.2">𝑆</ci><ci id="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.5.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.5.m5.1c">s_{i}\in S_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.5.m5.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx2"> <tbody id="S3.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\hat{R}_{i}(t,s_{i}):=\max_{a_{i}\in A_{i}}\sum_{\begin{subarray}% {c}\tau\leq t\\ s_{i}^{\tau}=s_{i}\end{subarray}}\quantity[U_{i}^{\tau}(a_{i},a_{-i}^{\tau})-U% _{i}^{\tau}(a^{\tau})]\leq C\sqrt{T}." class="ltx_Math" display="inline" id="S3.E2.m1.4"><semantics id="S3.E2.m1.4a"><mrow id="S3.E2.m1.4.4.1" xref="S3.E2.m1.4.4.1.1.cmml"><mrow id="S3.E2.m1.4.4.1.1" xref="S3.E2.m1.4.4.1.1.cmml"><mrow id="S3.E2.m1.4.4.1.1.1" xref="S3.E2.m1.4.4.1.1.1.cmml"><msub id="S3.E2.m1.4.4.1.1.1.3" xref="S3.E2.m1.4.4.1.1.1.3.cmml"><mover accent="true" id="S3.E2.m1.4.4.1.1.1.3.2" xref="S3.E2.m1.4.4.1.1.1.3.2.cmml"><mi id="S3.E2.m1.4.4.1.1.1.3.2.2" xref="S3.E2.m1.4.4.1.1.1.3.2.2.cmml">R</mi><mo id="S3.E2.m1.4.4.1.1.1.3.2.1" xref="S3.E2.m1.4.4.1.1.1.3.2.1.cmml">^</mo></mover><mi id="S3.E2.m1.4.4.1.1.1.3.3" xref="S3.E2.m1.4.4.1.1.1.3.3.cmml">i</mi></msub><mo id="S3.E2.m1.4.4.1.1.1.2" xref="S3.E2.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.E2.m1.4.4.1.1.1.1.1" xref="S3.E2.m1.4.4.1.1.1.1.2.cmml"><mo id="S3.E2.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.E2.m1.4.4.1.1.1.1.2.cmml">(</mo><mi id="S3.E2.m1.3.3" xref="S3.E2.m1.3.3.cmml">t</mi><mo id="S3.E2.m1.4.4.1.1.1.1.1.3" xref="S3.E2.m1.4.4.1.1.1.1.2.cmml">,</mo><msub id="S3.E2.m1.4.4.1.1.1.1.1.1" xref="S3.E2.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S3.E2.m1.4.4.1.1.1.1.1.1.2" xref="S3.E2.m1.4.4.1.1.1.1.1.1.2.cmml">s</mi><mi id="S3.E2.m1.4.4.1.1.1.1.1.1.3" xref="S3.E2.m1.4.4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.E2.m1.4.4.1.1.1.1.1.4" rspace="0.278em" stretchy="false" xref="S3.E2.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E2.m1.4.4.1.1.3" rspace="0.278em" xref="S3.E2.m1.4.4.1.1.3.cmml">:=</mo><mrow id="S3.E2.m1.4.4.1.1.4" xref="S3.E2.m1.4.4.1.1.4.cmml"><munder id="S3.E2.m1.4.4.1.1.4.2" xref="S3.E2.m1.4.4.1.1.4.2.cmml"><mi id="S3.E2.m1.4.4.1.1.4.2.2" xref="S3.E2.m1.4.4.1.1.4.2.2.cmml">max</mi><mrow id="S3.E2.m1.4.4.1.1.4.2.3" xref="S3.E2.m1.4.4.1.1.4.2.3.cmml"><msub id="S3.E2.m1.4.4.1.1.4.2.3.2" xref="S3.E2.m1.4.4.1.1.4.2.3.2.cmml"><mi id="S3.E2.m1.4.4.1.1.4.2.3.2.2" xref="S3.E2.m1.4.4.1.1.4.2.3.2.2.cmml">a</mi><mi id="S3.E2.m1.4.4.1.1.4.2.3.2.3" xref="S3.E2.m1.4.4.1.1.4.2.3.2.3.cmml">i</mi></msub><mo id="S3.E2.m1.4.4.1.1.4.2.3.1" xref="S3.E2.m1.4.4.1.1.4.2.3.1.cmml">∈</mo><msub id="S3.E2.m1.4.4.1.1.4.2.3.3" xref="S3.E2.m1.4.4.1.1.4.2.3.3.cmml"><mi id="S3.E2.m1.4.4.1.1.4.2.3.3.2" xref="S3.E2.m1.4.4.1.1.4.2.3.3.2.cmml">A</mi><mi id="S3.E2.m1.4.4.1.1.4.2.3.3.3" xref="S3.E2.m1.4.4.1.1.4.2.3.3.3.cmml">i</mi></msub></mrow></munder><mo id="S3.E2.m1.4.4.1.1.4.1" lspace="0.167em" xref="S3.E2.m1.4.4.1.1.4.1.cmml">⁢</mo><mrow id="S3.E2.m1.4.4.1.1.4.3" xref="S3.E2.m1.4.4.1.1.4.3.cmml"><mstyle displaystyle="true" id="S3.E2.m1.4.4.1.1.4.3.1" xref="S3.E2.m1.4.4.1.1.4.3.1.cmml"><munder id="S3.E2.m1.4.4.1.1.4.3.1a" xref="S3.E2.m1.4.4.1.1.4.3.1.cmml"><mo id="S3.E2.m1.4.4.1.1.4.3.1.2" movablelimits="false" xref="S3.E2.m1.4.4.1.1.4.3.1.2.cmml">∑</mo><mtable id="S3.E2.m1.1.1.1.1.1.1" rowspacing="0pt" xref="S3.E2.m1.1.1.1a.2.cmml"><mtr id="S3.E2.m1.1.1.1.1.1.1a" xref="S3.E2.m1.1.1.1a.2.cmml"><mtd id="S3.E2.m1.1.1.1.1.1.1b" xref="S3.E2.m1.1.1.1a.2.cmml"><mrow id="S3.E2.m1.1.1.1.1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E2.m1.1.1.1.1.1.1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.1.1.1.1.1.2.cmml">τ</mi><mo id="S3.E2.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.1.1.1.1.cmml">≤</mo><mi id="S3.E2.m1.1.1.1.1.1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.1.1.1.1.1.3.cmml">t</mi></mrow></mtd></mtr><mtr id="S3.E2.m1.1.1.1.1.1.1c" xref="S3.E2.m1.1.1.1a.2.cmml"><mtd id="S3.E2.m1.1.1.1.1.1.1d" xref="S3.E2.m1.1.1.1a.2.cmml"><mrow id="S3.E2.m1.1.1.1.1.1.1.2.1.1" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.cmml"><msubsup id="S3.E2.m1.1.1.1.1.1.1.2.1.1.2" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.2.cmml"><mi id="S3.E2.m1.1.1.1.1.1.1.2.1.1.2.2.2" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.2.2.2.cmml">s</mi><mi id="S3.E2.m1.1.1.1.1.1.1.2.1.1.2.2.3" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.2.2.3.cmml">i</mi><mi id="S3.E2.m1.1.1.1.1.1.1.2.1.1.2.3" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.2.3.cmml">τ</mi></msubsup><mo id="S3.E2.m1.1.1.1.1.1.1.2.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.1.cmml">=</mo><msub id="S3.E2.m1.1.1.1.1.1.1.2.1.1.3" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.3.cmml"><mi id="S3.E2.m1.1.1.1.1.1.1.2.1.1.3.2" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.3.2.cmml">s</mi><mi id="S3.E2.m1.1.1.1.1.1.1.2.1.1.3.3" xref="S3.E2.m1.1.1.1.1.1.1.2.1.1.3.3.cmml">i</mi></msub></mrow></mtd></mtr></mtable></munder></mstyle><mrow id="S3.E2.m1.2.2a.3" xref="S3.E2.m1.2.2.1.1.1.cmml"><mo id="S3.E2.m1.2.2a.3.1" xref="S3.E2.m1.2.2.1.1.1.cmml">[</mo><mrow id="S3.E2.m1.2.2.1.1.1" xref="S3.E2.m1.2.2.1.1.1.cmml"><mrow id="S3.E2.m1.2.2.1.1.1.2" xref="S3.E2.m1.2.2.1.1.1.2.cmml"><msubsup id="S3.E2.m1.2.2.1.1.1.2.4" xref="S3.E2.m1.2.2.1.1.1.2.4.cmml"><mi id="S3.E2.m1.2.2.1.1.1.2.4.2.2" xref="S3.E2.m1.2.2.1.1.1.2.4.2.2.cmml">U</mi><mi id="S3.E2.m1.2.2.1.1.1.2.4.2.3" xref="S3.E2.m1.2.2.1.1.1.2.4.2.3.cmml">i</mi><mi id="S3.E2.m1.2.2.1.1.1.2.4.3" xref="S3.E2.m1.2.2.1.1.1.2.4.3.cmml">τ</mi></msubsup><mo id="S3.E2.m1.2.2.1.1.1.2.3" xref="S3.E2.m1.2.2.1.1.1.2.3.cmml">⁢</mo><mrow id="S3.E2.m1.2.2.1.1.1.2.2.2" xref="S3.E2.m1.2.2.1.1.1.2.2.3.cmml"><mo 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italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ) - italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ) end_ARG ] ≤ italic_C square-root start_ARG italic_T end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS0.Px2.p1.8">One way to achieve this guarantee is for each agent to 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<h4 class="ltx_title ltx_title_paragraph">- Boundedness of payment functions</h4> <div class="ltx_para" id="S3.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px1.p1.2">We defined the payment functions as maps <math alttext="P^{t}_{i}:A\to\mathbb{R}_{+}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S3.SS2.SSS0.Px1.p1.1.m1.1a"><mrow id="S3.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.cmml"><msubsup id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml"><mi id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.2.2" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.2.2.cmml">P</mi><mi id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.3" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.3.cmml">i</mi><mi id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.2.3" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.2.3.cmml">t</mi></msubsup><mo id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3" 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id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3.3.2.cmml" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3.3.2">ℝ</ci><plus id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3.3.3.cmml" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px1.p1.1.m1.1c">P^{t}_{i}:A\to\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px1.p1.1.m1.1d">italic_P start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>: However, all our upper bounds for learning utilities will have payment functions for each agent that are upper-bounded by an absolute constant, namely <math alttext="2" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px1.p1.2.m2.1"><semantics id="S3.SS2.SSS0.Px1.p1.2.m2.1a"><mn id="S3.SS2.SSS0.Px1.p1.2.m2.1.1" xref="S3.SS2.SSS0.Px1.p1.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px1.p1.2.m2.1b"><cn id="S3.SS2.SSS0.Px1.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS2.SSS0.Px1.p1.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px1.p1.2.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px1.p1.2.m2.1d">2</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S3.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">- Nondeterminism of models.</h4> <div class="ltx_para" id="S3.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px2.p1.1">Neither of our behavioral models completely determines the behavior of the agents: in the rationalizable model, agents may choose arbitrarily among their undominated actions; in the no-regret model, agents may play arbitrarily so long as they maintain the no-regret property.</p> </div> <div class="ltx_para" id="S3.SS2.SSS0.Px2.p2"> <p class="ltx_p" id="S3.SS2.SSS0.Px2.p2.1">In particular, we do not require the agents to play a Nash or even correlated equilibrium in the game <math alttext="\Gamma^{t}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p2.1.m1.1"><semantics id="S3.SS2.SSS0.Px2.p2.1.m1.1a"><msup id="S3.SS2.SSS0.Px2.p2.1.m1.1.1" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.cmml">Γ</mi><mi id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p2.1.m1.1b"><apply id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1">superscript</csymbol><ci id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2">Γ</ci><ci id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p2.1.m1.1c">\Gamma^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p2.1.m1.1d">roman_Γ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. It is, in fact unnecessary for us to define a Nash equilibrium model. Since Nash equilibria involve only rationalizable actions and do not result in any regret for the agents, a Nash equilibrium model would be stronger than both our models, and so the upper bounds that we present would also apply to Nash equilibria. Conversely, our lower bounds on the rationalizable setting apply information-theoretically to <span class="ltx_text ltx_font_italic" id="S3.SS2.SSS0.Px2.p2.1.1">any</span> possible behavior model, and thus apply to the Nash model as well. Thus, for our purposes, a Nash equilibrium model would be essentially equivalent to the rationalizable model.</p> </div> </section> <section class="ltx_paragraph" id="S3.SS2.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">- Agents’ no-regret algorithms.</h4> <div class="ltx_para" id="S3.SS2.SSS0.Px3.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px3.p1.1">We do not distinguish between full-feedback and bandit-feedback no-regret learning algorithms in the no-regret model. Indeed, we are not even assuming that the agents run independent learning algorithms <span class="ltx_text ltx_font_italic" id="S3.SS2.SSS0.Px3.p1.1.1">per se</span>; they could even choose their actions using some centralized algorithm. We impose only the condition that each agent’s regret is small.</p> </div> </section> <section class="ltx_paragraph" id="S3.SS2.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">- Anytime regret bound.</h4> <div class="ltx_para" id="S3.SS2.SSS0.Px4.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px4.p1.5">Our condition on no-regret learning is that, for every signal <math alttext="s_{i}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.p1.1.m1.1"><semantics id="S3.SS2.SSS0.Px4.p1.1.m1.1a"><msub id="S3.SS2.SSS0.Px4.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px4.p1.1.m1.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.p1.1.m1.1.1.2" xref="S3.SS2.SSS0.Px4.p1.1.m1.1.1.2.cmml">s</mi><mi id="S3.SS2.SSS0.Px4.p1.1.m1.1.1.3" xref="S3.SS2.SSS0.Px4.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.p1.1.m1.1b"><apply id="S3.SS2.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.p1.1.m1.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS0.Px4.p1.1.m1.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.p1.1.m1.1.1.2">𝑠</ci><ci id="S3.SS2.SSS0.Px4.p1.1.m1.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.p1.1.m1.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.p1.1.m1.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (here omitted as a superscript for notational clarity), the regret <math alttext="R_{i}(t)" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.p1.2.m2.1"><semantics id="S3.SS2.SSS0.Px4.p1.2.m2.1a"><mrow id="S3.SS2.SSS0.Px4.p1.2.m2.1.2" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.cmml"><msub id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.cmml"><mi id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.2" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.2.cmml">R</mi><mi id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.3" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.1" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.3.2" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.cmml"><mo id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.cmml">(</mo><mi id="S3.SS2.SSS0.Px4.p1.2.m2.1.1" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.1.cmml">t</mi><mo id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.p1.2.m2.1b"><apply id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.cmml" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2"><times id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.1.cmml" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.1"></times><apply id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.cmml" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.1.cmml" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2">subscript</csymbol><ci id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.2.cmml" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.2">𝑅</ci><ci id="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.3.cmml" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.2.2.3">𝑖</ci></apply><ci id="S3.SS2.SSS0.Px4.p1.2.m2.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.p1.2.m2.1c">R_{i}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.p1.2.m2.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is bounded by <math alttext="C\sqrt{T}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.p1.3.m3.1"><semantics id="S3.SS2.SSS0.Px4.p1.3.m3.1a"><mrow id="S3.SS2.SSS0.Px4.p1.3.m3.1.1" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.2" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.2.cmml">C</mi><mo id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.1" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.1.cmml">⁢</mo><msqrt id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3.cmml"><mi id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3.2" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3.2.cmml">T</mi></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.p1.3.m3.1b"><apply id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1"><times id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.1"></times><ci id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.2">𝐶</ci><apply id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3"><root id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3a.cmml" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3"></root><ci id="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3.2.cmml" xref="S3.SS2.SSS0.Px4.p1.3.m3.1.1.3.2">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.p1.3.m3.1c">C\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.p1.3.m3.1d">italic_C square-root start_ARG italic_T end_ARG</annotation></semantics></math> for <span class="ltx_text ltx_font_italic" id="S3.SS2.SSS0.Px4.p1.5.1">every</span> timestep <math alttext="t\leq T" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.p1.4.m4.1"><semantics id="S3.SS2.SSS0.Px4.p1.4.m4.1a"><mrow id="S3.SS2.SSS0.Px4.p1.4.m4.1.1" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.p1.4.m4.1.1.2" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1.2.cmml">t</mi><mo id="S3.SS2.SSS0.Px4.p1.4.m4.1.1.1" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1.1.cmml">≤</mo><mi id="S3.SS2.SSS0.Px4.p1.4.m4.1.1.3" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.p1.4.m4.1b"><apply id="S3.SS2.SSS0.Px4.p1.4.m4.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1"><leq id="S3.SS2.SSS0.Px4.p1.4.m4.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1.1"></leq><ci id="S3.SS2.SSS0.Px4.p1.4.m4.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1.2">𝑡</ci><ci id="S3.SS2.SSS0.Px4.p1.4.m4.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.p1.4.m4.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.p1.4.m4.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.p1.4.m4.1d">italic_t ≤ italic_T</annotation></semantics></math>, not just at <math alttext="t=T" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.p1.5.m5.1"><semantics id="S3.SS2.SSS0.Px4.p1.5.m5.1a"><mrow id="S3.SS2.SSS0.Px4.p1.5.m5.1.1" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.p1.5.m5.1.1.2" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1.2.cmml">t</mi><mo id="S3.SS2.SSS0.Px4.p1.5.m5.1.1.1" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1.1.cmml">=</mo><mi id="S3.SS2.SSS0.Px4.p1.5.m5.1.1.3" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.p1.5.m5.1b"><apply id="S3.SS2.SSS0.Px4.p1.5.m5.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1"><eq id="S3.SS2.SSS0.Px4.p1.5.m5.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1.1"></eq><ci id="S3.SS2.SSS0.Px4.p1.5.m5.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1.2">𝑡</ci><ci id="S3.SS2.SSS0.Px4.p1.5.m5.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.p1.5.m5.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.p1.5.m5.1c">t=T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.p1.5.m5.1d">italic_t = italic_T</annotation></semantics></math> as is conventionally required by adversarial no-regret algorithms. This is not a significantly stronger requirement:</p> </div> <div class="ltx_theorem ltx_theorem_proposition" 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">Proposition 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.5.5">If a (possibly randomized) adversarial no-regret algorithm satisfies <math alttext="R_{i}(T)\leq B" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem1.p1.1.1.m1.1.2" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.cmml"><mrow id="S3.Thmtheorem1.p1.1.1.m1.1.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.cmml"><msub id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.cmml"><mi id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.2.cmml">R</mi><mi id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.3" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.3.cmml">i</mi></msub><mo id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.3.2" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.cmml"><mo id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml">T</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p1.1.1.m1.1.2.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.1.cmml">≤</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.1.2.3" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.1b"><apply id="S3.Thmtheorem1.p1.1.1.m1.1.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2"><leq id="S3.Thmtheorem1.p1.1.1.m1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.1"></leq><apply id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2"><times id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.1"></times><apply id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.2">𝑅</ci><ci id="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.3.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.2.2.3">𝑖</ci></apply><ci id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1">𝑇</ci></apply><ci id="S3.Thmtheorem1.p1.1.1.m1.1.2.3.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.1c">R_{i}(T)\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_T ) ≤ italic_B</annotation></semantics></math> with probability <math alttext="1-\delta" 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"><mn id="S3.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">1</mn><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">δ</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"><minus id="S3.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.1"></minus><cn id="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" type="integer" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2">1</cn><ci id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.2.m2.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.2.m2.1d">1 - italic_δ</annotation></semantics></math> against any adversary, then with probability <math alttext="1-\delta" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.3.m3.1"><semantics id="S3.Thmtheorem1.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem1.p1.3.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mn id="S3.Thmtheorem1.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2.cmml">1</mn><mo id="S3.Thmtheorem1.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.1.cmml">−</mo><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.3.m3.1b"><apply id="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1"><minus id="S3.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.1"></minus><cn id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" type="integer" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2">1</cn><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.3.m3.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.3.m3.1d">1 - italic_δ</annotation></semantics></math> it also satisfies <math alttext="R_{i}(t)\leq B" 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.m4.1.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mrow id="S3.Thmtheorem1.p1.4.4.m4.1.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.cmml"><msub id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.2.cmml">R</mi><mi id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.3" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.3.cmml">i</mi></msub><mo id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.3.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.cmml"><mo id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml">t</mi><mo id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p1.4.4.m4.1.2.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.1.cmml">≤</mo><mi id="S3.Thmtheorem1.p1.4.4.m4.1.2.3" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.4.4.m4.1b"><apply id="S3.Thmtheorem1.p1.4.4.m4.1.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2"><leq id="S3.Thmtheorem1.p1.4.4.m4.1.2.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.1"></leq><apply id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2"><times id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.1"></times><apply id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.2">𝑅</ci><ci id="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.2.2.3">𝑖</ci></apply><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1">𝑡</ci></apply><ci id="S3.Thmtheorem1.p1.4.4.m4.1.2.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.4.m4.1c">R_{i}(t)\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.4.m4.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ≤ italic_B</annotation></semantics></math> simultaneously for all <math alttext="t\leq T" 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"><mi id="S3.Thmtheorem1.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2.cmml">t</mi><mo id="S3.Thmtheorem1.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.1.cmml">≤</mo><mi id="S3.Thmtheorem1.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.cmml">T</mi></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"><leq id="S3.Thmtheorem1.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.1"></leq><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2">𝑡</ci><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.5.5.m5.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.5.5.m5.1d">italic_t ≤ italic_T</annotation></semantics></math> against any adversary.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS2.SSS0.Px4.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS2.SSS0.Px4.1.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px4.1.p1.12">Suppose not, <span class="ltx_text ltx_font_italic" id="S3.SS2.SSS0.Px4.1.p1.12.1">i.e.</span>, suppose that there is some adversary <math alttext="{\mathcal{A}}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.1.m1.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.SSS0.Px4.1.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.1.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.1.m1.1b"><ci id="S3.SS2.SSS0.Px4.1.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.1.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.1.m1.1c">{\mathcal{A}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.1.m1.1d">caligraphic_A</annotation></semantics></math> such that, with probability <math alttext=">\delta" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.2.m2.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.2.m2.1a"><mrow id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.2.cmml"></mi><mo id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.1.cmml">></mo><mi id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.2.m2.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1"><gt id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.1"></gt><csymbol cd="latexml" id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.2">absent</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.2.m2.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.2.m2.1c">>\delta</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.2.m2.1d">> italic_δ</annotation></semantics></math>, there exists some <math alttext="t\leq T" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.3.m3.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.3.m3.1a"><mrow id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.2.cmml">t</mi><mo id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.1.cmml">≤</mo><mi id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.3.m3.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1"><leq id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.1"></leq><ci id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.2">𝑡</ci><ci id="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.3.m3.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.3.m3.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.3.m3.1d">italic_t ≤ italic_T</annotation></semantics></math> for which <math alttext="R_{i}(t)>B" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.4.m4.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.4.m4.1a"><mrow id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.cmml"><mrow id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.cmml"><msub id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.cmml"><mi id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.2" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.2.cmml">R</mi><mi id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.3" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.3.cmml">i</mi></msub><mo id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.1" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.3.2" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.cmml"><mo id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.cmml">(</mo><mi id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.1" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.1.cmml">t</mi><mo id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.1" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.1.cmml">></mo><mi id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.3" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.4.m4.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2"><gt id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.1"></gt><apply id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2"><times id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.1"></times><apply id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2">subscript</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.2">𝑅</ci><ci id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.2.2.3">𝑖</ci></apply><ci id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.1">𝑡</ci></apply><ci id="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.4.m4.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.4.m4.1c">R_{i}(t)>B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.4.m4.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) > italic_B</annotation></semantics></math>. Then consider the adversary <math alttext="{\mathcal{A}}^{\prime}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.5.m5.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.5.m5.1a"><msup id="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1" xref="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.2.cmml">𝒜</mi><mo id="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.5.m5.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1">superscript</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.2">𝒜</ci><ci id="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.5.m5.1c">{\mathcal{A}}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.5.m5.1d">caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> that acts as follows. At every time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.6.m6.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.6.m6.1a"><mi id="S3.SS2.SSS0.Px4.1.p1.6.m6.1.1" xref="S3.SS2.SSS0.Px4.1.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.6.m6.1b"><ci id="S3.SS2.SSS0.Px4.1.p1.6.m6.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.6.m6.1d">italic_t</annotation></semantics></math>, if <math alttext="R_{i}(t-1)\leq B" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.7.m7.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.7.m7.1a"><mrow id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.cmml"><mrow id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.cmml"><msub id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.cmml"><mi id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.2" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.2.cmml">R</mi><mi id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.3" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.3.cmml">i</mi></msub><mo id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.2.cmml">≤</mo><mi id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.7.m7.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1"><leq id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.2"></leq><apply id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1"><times id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.2"></times><apply id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3">subscript</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.2">𝑅</ci><ci id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.3.3">𝑖</ci></apply><apply id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1"><minus id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.1"></minus><ci id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.2">𝑡</ci><cn id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.7.m7.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.7.m7.1c">R_{i}(t-1)\leq B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.7.m7.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t - 1 ) ≤ italic_B</annotation></semantics></math>, it copies <math alttext="{\mathcal{A}}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.8.m8.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.SSS0.Px4.1.p1.8.m8.1.1" xref="S3.SS2.SSS0.Px4.1.p1.8.m8.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.8.m8.1b"><ci id="S3.SS2.SSS0.Px4.1.p1.8.m8.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.8.m8.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.8.m8.1c">{\mathcal{A}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.8.m8.1d">caligraphic_A</annotation></semantics></math>. Otherwise, it outputs <math alttext="{\bm{u}}^{t}=\bm{0}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.9.m9.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.9.m9.1a"><mrow id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.cmml"><msup id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.cmml"><mi id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.2" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.2.cmml">𝒖</mi><mi id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.3" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.3.cmml">t</mi></msup><mo id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.1.cmml">=</mo><mn id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.9.m9.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1"><eq id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.1"></eq><apply id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2">superscript</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.2">𝒖</ci><ci id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.2.3">𝑡</ci></apply><cn id="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.3.cmml" type="integer" xref="S3.SS2.SSS0.Px4.1.p1.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.9.m9.1c">{\bm{u}}^{t}=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.9.m9.1d">bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = bold_0</annotation></semantics></math> for all remaining timesteps. In the latter case, which by definition occurs with probability <math alttext=">\delta" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.10.m10.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.10.m10.1a"><mrow id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.cmml"><mi id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.2.cmml"></mi><mo id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.1" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.1.cmml">></mo><mi id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.10.m10.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1"><gt id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.1"></gt><csymbol cd="latexml" id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.2">absent</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.10.m10.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.10.m10.1c">>\delta</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.10.m10.1d">> italic_δ</annotation></semantics></math>, adversary <math alttext="{\mathcal{A}}^{\prime}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.11.m11.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.11.m11.1a"><msup id="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1" xref="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.2" xref="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.2.cmml">𝒜</mi><mo id="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.3" xref="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.11.m11.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1">superscript</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.2">𝒜</ci><ci id="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.11.m11.1c">{\mathcal{A}}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.11.m11.1d">caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> will also achieve <math alttext="R_{i}(T)>B" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px4.1.p1.12.m12.1"><semantics id="S3.SS2.SSS0.Px4.1.p1.12.m12.1a"><mrow id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.cmml"><mrow id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.cmml"><msub 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xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px4.1.p1.12.m12.1b"><apply id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2"><gt id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.1"></gt><apply id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2"><times id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.1"></times><apply id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2">subscript</csymbol><ci id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2.2.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2.2">𝑅</ci><ci id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.2.2.3">𝑖</ci></apply><ci id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.1.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.1">𝑇</ci></apply><ci id="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.3.cmml" xref="S3.SS2.SSS0.Px4.1.p1.12.m12.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px4.1.p1.12.m12.1c">R_{i}(T)>B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px4.1.p1.12.m12.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_T ) > italic_B</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_paragraph" id="S3.SS2.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">- Deterministic regret bounds</h4> <div class="ltx_para" id="S3.SS2.SSS0.Px5.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px5.p1.5">Since the bounds in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS1" title="3.1 Behavioral models ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.1</span></a> are phrased <span class="ltx_text ltx_font_italic" id="S3.SS2.SSS0.Px5.p1.5.1">deterministically</span>, in the case where the agents are running randomized algorithms, one should think of any guarantees that we show as conditional on the event <math alttext="R_{i}(t)\leq C\sqrt{T}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px5.p1.1.m1.1"><semantics id="S3.SS2.SSS0.Px5.p1.1.m1.1a"><mrow id="S3.SS2.SSS0.Px5.p1.1.m1.1.2" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.cmml"><mrow id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.cmml"><msub id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.cmml"><mi id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.2" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.2.cmml">R</mi><mi id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.3" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.3.cmml">i</mi></msub><mo id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.1" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.3.2" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.cmml"><mo id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.cmml">(</mo><mi id="S3.SS2.SSS0.Px5.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.1.cmml">t</mi><mo id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.1" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.1.cmml">≤</mo><mrow id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.cmml"><mi id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.2" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.2.cmml">C</mi><mo id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.1" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.1.cmml">⁢</mo><msqrt id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3.cmml"><mi id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3.2" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3.2.cmml">T</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px5.p1.1.m1.1b"><apply id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2"><leq id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.1.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.1"></leq><apply id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2"><times id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.1.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.1"></times><apply id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.1.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2">subscript</csymbol><ci id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.2.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.2">𝑅</ci><ci id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.3.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.2.2.3">𝑖</ci></apply><ci id="S3.SS2.SSS0.Px5.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.1">𝑡</ci></apply><apply id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3"><times id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.1.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.1"></times><ci id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.2.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.2">𝐶</ci><apply id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3"><root id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3a.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3"></root><ci id="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3.2.cmml" xref="S3.SS2.SSS0.Px5.p1.1.m1.1.2.3.3.2">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px5.p1.1.m1.1c">R_{i}(t)\leq C\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px5.p1.1.m1.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ≤ italic_C square-root start_ARG italic_T end_ARG</annotation></semantics></math> for all agents <math alttext="i" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px5.p1.2.m2.1"><semantics id="S3.SS2.SSS0.Px5.p1.2.m2.1a"><mi id="S3.SS2.SSS0.Px5.p1.2.m2.1.1" xref="S3.SS2.SSS0.Px5.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px5.p1.2.m2.1b"><ci id="S3.SS2.SSS0.Px5.p1.2.m2.1.1.cmml" xref="S3.SS2.SSS0.Px5.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px5.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px5.p1.2.m2.1d">italic_i</annotation></semantics></math> and times <math alttext="t\leq T" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px5.p1.3.m3.1"><semantics id="S3.SS2.SSS0.Px5.p1.3.m3.1a"><mrow id="S3.SS2.SSS0.Px5.p1.3.m3.1.1" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1.cmml"><mi id="S3.SS2.SSS0.Px5.p1.3.m3.1.1.2" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1.2.cmml">t</mi><mo id="S3.SS2.SSS0.Px5.p1.3.m3.1.1.1" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1.1.cmml">≤</mo><mi id="S3.SS2.SSS0.Px5.p1.3.m3.1.1.3" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px5.p1.3.m3.1b"><apply id="S3.SS2.SSS0.Px5.p1.3.m3.1.1.cmml" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1"><leq id="S3.SS2.SSS0.Px5.p1.3.m3.1.1.1.cmml" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1.1"></leq><ci id="S3.SS2.SSS0.Px5.p1.3.m3.1.1.2.cmml" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1.2">𝑡</ci><ci id="S3.SS2.SSS0.Px5.p1.3.m3.1.1.3.cmml" xref="S3.SS2.SSS0.Px5.p1.3.m3.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px5.p1.3.m3.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px5.p1.3.m3.1d">italic_t ≤ italic_T</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.Thmtheorem1" title="Proposition 2.1. ‣ 2.3 Realized and in-expectation regret ‣ 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Propositions</span> <span class="ltx_text ltx_ref_tag">2.1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.Thmtheorem1" title="Proposition 3.1. ‣ - Anytime regret bound. ‣ 3.2 Remarks about the model ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">3.1</span></a>, this happens with probability at least <math alttext="1-\delta" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px5.p1.4.m4.1"><semantics id="S3.SS2.SSS0.Px5.p1.4.m4.1a"><mrow id="S3.SS2.SSS0.Px5.p1.4.m4.1.1" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1.cmml"><mn id="S3.SS2.SSS0.Px5.p1.4.m4.1.1.2" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1.2.cmml">1</mn><mo id="S3.SS2.SSS0.Px5.p1.4.m4.1.1.1" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1.1.cmml">−</mo><mi id="S3.SS2.SSS0.Px5.p1.4.m4.1.1.3" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px5.p1.4.m4.1b"><apply id="S3.SS2.SSS0.Px5.p1.4.m4.1.1.cmml" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1"><minus id="S3.SS2.SSS0.Px5.p1.4.m4.1.1.1.cmml" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1.1"></minus><cn id="S3.SS2.SSS0.Px5.p1.4.m4.1.1.2.cmml" type="integer" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1.2">1</cn><ci id="S3.SS2.SSS0.Px5.p1.4.m4.1.1.3.cmml" xref="S3.SS2.SSS0.Px5.p1.4.m4.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px5.p1.4.m4.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px5.p1.4.m4.1d">1 - italic_δ</annotation></semantics></math> for <math alttext="C=\sqrt{\log(mn/\delta)}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px5.p1.5.m5.2"><semantics id="S3.SS2.SSS0.Px5.p1.5.m5.2a"><mrow 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xref="S3.SS2.SSS0.Px5.p1.5.m5.1.1.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S3.SS2.SSS0.Px5.p1.5.m5.1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.SSS0.Px5.p1.5.m5.1.1.1.1.1.1.1.3">𝛿</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px5.p1.5.m5.2c">C=\sqrt{\log(mn/\delta)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px5.p1.5.m5.2d">italic_C = square-root start_ARG roman_log ( start_ARG italic_m italic_n / italic_δ end_ARG ) end_ARG</annotation></semantics></math> if all agents use MWU.</p> </div> </section> </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>Game equivalence and formal goal statement</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.3">Our goal is to design algorithms for the principal to learn the agents’ utility functions <math alttext="U_{i}" class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><msub id="S3.SS3.p1.1.m1.1.1" xref="S3.SS3.p1.1.m1.1.1.cmml"><mi id="S3.SS3.p1.1.m1.1.1.2" xref="S3.SS3.p1.1.m1.1.1.2.cmml">U</mi><mi id="S3.SS3.p1.1.m1.1.1.3" xref="S3.SS3.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><apply id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.1.m1.1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p1.1.m1.1.1.2.cmml" xref="S3.SS3.p1.1.m1.1.1.2">𝑈</ci><ci id="S3.SS3.p1.1.m1.1.1.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.1.m1.1c">U_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.1.m1.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> by designing the payment functions <math alttext="P_{i}^{t}" class="ltx_Math" display="inline" id="S3.SS3.p1.2.m2.1"><semantics id="S3.SS3.p1.2.m2.1a"><msubsup id="S3.SS3.p1.2.m2.1.1" xref="S3.SS3.p1.2.m2.1.1.cmml"><mi id="S3.SS3.p1.2.m2.1.1.2.2" xref="S3.SS3.p1.2.m2.1.1.2.2.cmml">P</mi><mi id="S3.SS3.p1.2.m2.1.1.2.3" xref="S3.SS3.p1.2.m2.1.1.2.3.cmml">i</mi><mi id="S3.SS3.p1.2.m2.1.1.3" xref="S3.SS3.p1.2.m2.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.2.m2.1b"><apply id="S3.SS3.p1.2.m2.1.1.cmml" xref="S3.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.2.m2.1.1.1.cmml" xref="S3.SS3.p1.2.m2.1.1">superscript</csymbol><apply id="S3.SS3.p1.2.m2.1.1.2.cmml" xref="S3.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.2.m2.1.1.2.1.cmml" xref="S3.SS3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p1.2.m2.1.1.2.2.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2">𝑃</ci><ci id="S3.SS3.p1.2.m2.1.1.2.3.cmml" xref="S3.SS3.p1.2.m2.1.1.2.3">𝑖</ci></apply><ci id="S3.SS3.p1.2.m2.1.1.3.cmml" xref="S3.SS3.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.2.m2.1c">P_{i}^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> and sending signals <math alttext="s_{i}^{t}" class="ltx_Math" display="inline" id="S3.SS3.p1.3.m3.1"><semantics id="S3.SS3.p1.3.m3.1a"><msubsup id="S3.SS3.p1.3.m3.1.1" xref="S3.SS3.p1.3.m3.1.1.cmml"><mi id="S3.SS3.p1.3.m3.1.1.2.2" xref="S3.SS3.p1.3.m3.1.1.2.2.cmml">s</mi><mi id="S3.SS3.p1.3.m3.1.1.2.3" xref="S3.SS3.p1.3.m3.1.1.2.3.cmml">i</mi><mi id="S3.SS3.p1.3.m3.1.1.3" xref="S3.SS3.p1.3.m3.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.3.m3.1b"><apply id="S3.SS3.p1.3.m3.1.1.cmml" xref="S3.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.3.m3.1.1.1.cmml" xref="S3.SS3.p1.3.m3.1.1">superscript</csymbol><apply id="S3.SS3.p1.3.m3.1.1.2.cmml" xref="S3.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.3.m3.1.1.2.1.cmml" xref="S3.SS3.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.p1.3.m3.1.1.2.2.cmml" xref="S3.SS3.p1.3.m3.1.1.2.2">𝑠</ci><ci id="S3.SS3.p1.3.m3.1.1.2.3.cmml" xref="S3.SS3.p1.3.m3.1.1.2.3">𝑖</ci></apply><ci id="S3.SS3.p1.3.m3.1.1.3.cmml" xref="S3.SS3.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.3.m3.1c">s_{i}^{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.3.m3.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> for all agents, under different behavior models we introduced.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.9">This goal as currently stated is impossible. To see this, note that agents’ actions in all the behavioral models are only affected by their utility <span class="ltx_text ltx_font_italic" id="S3.SS3.p2.9.1">differences</span>, that is, the differences <math alttext="U_{i}(a_{i},a_{-i})-U_{i}(a_{i}^{\prime},a_{-i})" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.4"><semantics id="S3.SS3.p2.1.m1.4a"><mrow id="S3.SS3.p2.1.m1.4.4" xref="S3.SS3.p2.1.m1.4.4.cmml"><mrow id="S3.SS3.p2.1.m1.2.2.2" xref="S3.SS3.p2.1.m1.2.2.2.cmml"><msub id="S3.SS3.p2.1.m1.2.2.2.4" xref="S3.SS3.p2.1.m1.2.2.2.4.cmml"><mi id="S3.SS3.p2.1.m1.2.2.2.4.2" xref="S3.SS3.p2.1.m1.2.2.2.4.2.cmml">U</mi><mi id="S3.SS3.p2.1.m1.2.2.2.4.3" xref="S3.SS3.p2.1.m1.2.2.2.4.3.cmml">i</mi></msub><mo id="S3.SS3.p2.1.m1.2.2.2.3" xref="S3.SS3.p2.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS3.p2.1.m1.2.2.2.2.2" xref="S3.SS3.p2.1.m1.2.2.2.2.3.cmml"><mo id="S3.SS3.p2.1.m1.2.2.2.2.2.3" stretchy="false" xref="S3.SS3.p2.1.m1.2.2.2.2.3.cmml">(</mo><msub id="S3.SS3.p2.1.m1.1.1.1.1.1.1" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p2.1.m1.1.1.1.1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS3.p2.1.m1.1.1.1.1.1.1.3" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS3.p2.1.m1.2.2.2.2.2.4" xref="S3.SS3.p2.1.m1.2.2.2.2.3.cmml">,</mo><msub id="S3.SS3.p2.1.m1.2.2.2.2.2.2" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.cmml"><mi id="S3.SS3.p2.1.m1.2.2.2.2.2.2.2" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.2.cmml">a</mi><mrow id="S3.SS3.p2.1.m1.2.2.2.2.2.2.3" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.cmml"><mo id="S3.SS3.p2.1.m1.2.2.2.2.2.2.3a" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.cmml">−</mo><mi id="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.2" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="S3.SS3.p2.1.m1.2.2.2.2.2.5" stretchy="false" xref="S3.SS3.p2.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p2.1.m1.4.4.5" xref="S3.SS3.p2.1.m1.4.4.5.cmml">−</mo><mrow id="S3.SS3.p2.1.m1.4.4.4" xref="S3.SS3.p2.1.m1.4.4.4.cmml"><msub 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id="S3.SS3.p2.1.m1.4.4.4.2.2.2.2" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.2.cmml">a</mi><mrow id="S3.SS3.p2.1.m1.4.4.4.2.2.2.3" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.cmml"><mo id="S3.SS3.p2.1.m1.4.4.4.2.2.2.3a" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.cmml">−</mo><mi id="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.2" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="S3.SS3.p2.1.m1.4.4.4.2.2.5" stretchy="false" xref="S3.SS3.p2.1.m1.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.4b"><apply id="S3.SS3.p2.1.m1.4.4.cmml" xref="S3.SS3.p2.1.m1.4.4"><minus id="S3.SS3.p2.1.m1.4.4.5.cmml" xref="S3.SS3.p2.1.m1.4.4.5"></minus><apply id="S3.SS3.p2.1.m1.2.2.2.cmml" xref="S3.SS3.p2.1.m1.2.2.2"><times id="S3.SS3.p2.1.m1.2.2.2.3.cmml" xref="S3.SS3.p2.1.m1.2.2.2.3"></times><apply id="S3.SS3.p2.1.m1.2.2.2.4.cmml" xref="S3.SS3.p2.1.m1.2.2.2.4"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.2.2.2.4.1.cmml" xref="S3.SS3.p2.1.m1.2.2.2.4">subscript</csymbol><ci id="S3.SS3.p2.1.m1.2.2.2.4.2.cmml" xref="S3.SS3.p2.1.m1.2.2.2.4.2">𝑈</ci><ci id="S3.SS3.p2.1.m1.2.2.2.4.3.cmml" xref="S3.SS3.p2.1.m1.2.2.2.4.3">𝑖</ci></apply><interval closure="open" id="S3.SS3.p2.1.m1.2.2.2.2.3.cmml" xref="S3.SS3.p2.1.m1.2.2.2.2.2"><apply id="S3.SS3.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS3.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.SS3.p2.1.m1.2.2.2.2.2.2.cmml" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.2.2.2.2.2.2.1.cmml" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S3.SS3.p2.1.m1.2.2.2.2.2.2.2.cmml" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.2">𝑎</ci><apply id="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.cmml" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.3"><minus id="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.1.cmml" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.3"></minus><ci id="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.2.cmml" xref="S3.SS3.p2.1.m1.2.2.2.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply><apply id="S3.SS3.p2.1.m1.4.4.4.cmml" xref="S3.SS3.p2.1.m1.4.4.4"><times id="S3.SS3.p2.1.m1.4.4.4.3.cmml" xref="S3.SS3.p2.1.m1.4.4.4.3"></times><apply id="S3.SS3.p2.1.m1.4.4.4.4.cmml" xref="S3.SS3.p2.1.m1.4.4.4.4"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.4.4.4.4.1.cmml" xref="S3.SS3.p2.1.m1.4.4.4.4">subscript</csymbol><ci id="S3.SS3.p2.1.m1.4.4.4.4.2.cmml" xref="S3.SS3.p2.1.m1.4.4.4.4.2">𝑈</ci><ci id="S3.SS3.p2.1.m1.4.4.4.4.3.cmml" xref="S3.SS3.p2.1.m1.4.4.4.4.3">𝑖</ci></apply><interval closure="open" id="S3.SS3.p2.1.m1.4.4.4.2.3.cmml" xref="S3.SS3.p2.1.m1.4.4.4.2.2"><apply id="S3.SS3.p2.1.m1.3.3.3.1.1.1.cmml" xref="S3.SS3.p2.1.m1.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.3.3.3.1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.3.3.3.1.1.1">superscript</csymbol><apply id="S3.SS3.p2.1.m1.3.3.3.1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.3.3.3.1.1.1.2.1.cmml" xref="S3.SS3.p2.1.m1.3.3.3.1.1.1">subscript</csymbol><ci id="S3.SS3.p2.1.m1.3.3.3.1.1.1.2.2.cmml" xref="S3.SS3.p2.1.m1.3.3.3.1.1.1.2.2">𝑎</ci><ci id="S3.SS3.p2.1.m1.3.3.3.1.1.1.2.3.cmml" xref="S3.SS3.p2.1.m1.3.3.3.1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS3.p2.1.m1.3.3.3.1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.3.3.3.1.1.1.3">′</ci></apply><apply id="S3.SS3.p2.1.m1.4.4.4.2.2.2.cmml" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.4.4.4.2.2.2.1.cmml" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2">subscript</csymbol><ci id="S3.SS3.p2.1.m1.4.4.4.2.2.2.2.cmml" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.2">𝑎</ci><apply id="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.cmml" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.3"><minus id="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.1.cmml" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.3"></minus><ci id="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.2.cmml" xref="S3.SS3.p2.1.m1.4.4.4.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.4c">U_{i}(a_{i},a_{-i})-U_{i}(a_{i}^{\prime},a_{-i})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.4d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) - italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. In other words, if we create another game <math alttext="\Gamma^{\prime}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><msup id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml"><mi id="S3.SS3.p2.2.m2.1.1.2" mathvariant="normal" xref="S3.SS3.p2.2.m2.1.1.2.cmml">Γ</mi><mo id="S3.SS3.p2.2.m2.1.1.3" xref="S3.SS3.p2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><apply id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.2.m2.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1">superscript</csymbol><ci id="S3.SS3.p2.2.m2.1.1.2.cmml" xref="S3.SS3.p2.2.m2.1.1.2">Γ</ci><ci id="S3.SS3.p2.2.m2.1.1.3.cmml" xref="S3.SS3.p2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.1c">\Gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.1d">roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="U_{i}^{\prime}(a_{i},a_{-i})=U_{i}(a_{i},a_{-i})+W_{i}(a_{-i})" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m3.5"><semantics id="S3.SS3.p2.3.m3.5a"><mrow id="S3.SS3.p2.3.m3.5.5" xref="S3.SS3.p2.3.m3.5.5.cmml"><mrow id="S3.SS3.p2.3.m3.2.2.2" xref="S3.SS3.p2.3.m3.2.2.2.cmml"><msubsup id="S3.SS3.p2.3.m3.2.2.2.4" xref="S3.SS3.p2.3.m3.2.2.2.4.cmml"><mi id="S3.SS3.p2.3.m3.2.2.2.4.2.2" xref="S3.SS3.p2.3.m3.2.2.2.4.2.2.cmml">U</mi><mi id="S3.SS3.p2.3.m3.2.2.2.4.2.3" xref="S3.SS3.p2.3.m3.2.2.2.4.2.3.cmml">i</mi><mo id="S3.SS3.p2.3.m3.2.2.2.4.3" xref="S3.SS3.p2.3.m3.2.2.2.4.3.cmml">′</mo></msubsup><mo id="S3.SS3.p2.3.m3.2.2.2.3" xref="S3.SS3.p2.3.m3.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS3.p2.3.m3.2.2.2.2.2" xref="S3.SS3.p2.3.m3.2.2.2.2.3.cmml"><mo id="S3.SS3.p2.3.m3.2.2.2.2.2.3" stretchy="false" xref="S3.SS3.p2.3.m3.2.2.2.2.3.cmml">(</mo><msub id="S3.SS3.p2.3.m3.1.1.1.1.1.1" xref="S3.SS3.p2.3.m3.1.1.1.1.1.1.cmml"><mi 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start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) = italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) + italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> for all <math alttext="a\in A" 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"><mi id="S3.SS3.p2.4.m4.1.1.2" xref="S3.SS3.p2.4.m4.1.1.2.cmml">a</mi><mo id="S3.SS3.p2.4.m4.1.1.1" xref="S3.SS3.p2.4.m4.1.1.1.cmml">∈</mo><mi id="S3.SS3.p2.4.m4.1.1.3" xref="S3.SS3.p2.4.m4.1.1.3.cmml">A</mi></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"><in id="S3.SS3.p2.4.m4.1.1.1.cmml" xref="S3.SS3.p2.4.m4.1.1.1"></in><ci id="S3.SS3.p2.4.m4.1.1.2.cmml" xref="S3.SS3.p2.4.m4.1.1.2">𝑎</ci><ci id="S3.SS3.p2.4.m4.1.1.3.cmml" xref="S3.SS3.p2.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.4.m4.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.4.m4.1d">italic_a ∈ italic_A</annotation></semantics></math>, where <math alttext="W_{i}:A_{-i}\to\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS3.p2.5.m5.1"><semantics id="S3.SS3.p2.5.m5.1a"><mrow id="S3.SS3.p2.5.m5.1.1" xref="S3.SS3.p2.5.m5.1.1.cmml"><msub id="S3.SS3.p2.5.m5.1.1.2" xref="S3.SS3.p2.5.m5.1.1.2.cmml"><mi id="S3.SS3.p2.5.m5.1.1.2.2" xref="S3.SS3.p2.5.m5.1.1.2.2.cmml">W</mi><mi id="S3.SS3.p2.5.m5.1.1.2.3" xref="S3.SS3.p2.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS3.p2.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.p2.5.m5.1.1.1.cmml">:</mo><mrow id="S3.SS3.p2.5.m5.1.1.3" xref="S3.SS3.p2.5.m5.1.1.3.cmml"><msub id="S3.SS3.p2.5.m5.1.1.3.2" xref="S3.SS3.p2.5.m5.1.1.3.2.cmml"><mi id="S3.SS3.p2.5.m5.1.1.3.2.2" xref="S3.SS3.p2.5.m5.1.1.3.2.2.cmml">A</mi><mrow id="S3.SS3.p2.5.m5.1.1.3.2.3" xref="S3.SS3.p2.5.m5.1.1.3.2.3.cmml"><mo id="S3.SS3.p2.5.m5.1.1.3.2.3a" xref="S3.SS3.p2.5.m5.1.1.3.2.3.cmml">−</mo><mi id="S3.SS3.p2.5.m5.1.1.3.2.3.2" xref="S3.SS3.p2.5.m5.1.1.3.2.3.2.cmml">i</mi></mrow></msub><mo id="S3.SS3.p2.5.m5.1.1.3.1" stretchy="false" 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">ℝ</mi></mrow></mrow><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"><ci id="S3.SS3.p2.5.m5.1.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1.1">:</ci><apply id="S3.SS3.p2.5.m5.1.1.2.cmml" xref="S3.SS3.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.1.1.2.1.cmml" xref="S3.SS3.p2.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS3.p2.5.m5.1.1.2.2.cmml" xref="S3.SS3.p2.5.m5.1.1.2.2">𝑊</ci><ci id="S3.SS3.p2.5.m5.1.1.2.3.cmml" xref="S3.SS3.p2.5.m5.1.1.2.3">𝑖</ci></apply><apply id="S3.SS3.p2.5.m5.1.1.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3"><ci id="S3.SS3.p2.5.m5.1.1.3.1.cmml" xref="S3.SS3.p2.5.m5.1.1.3.1">→</ci><apply id="S3.SS3.p2.5.m5.1.1.3.2.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.1.1.3.2.1.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2">subscript</csymbol><ci id="S3.SS3.p2.5.m5.1.1.3.2.2.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2.2">𝐴</ci><apply id="S3.SS3.p2.5.m5.1.1.3.2.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2.3"><minus id="S3.SS3.p2.5.m5.1.1.3.2.3.1.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2.3"></minus><ci id="S3.SS3.p2.5.m5.1.1.3.2.3.2.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2.3.2">𝑖</ci></apply></apply><ci id="S3.SS3.p2.5.m5.1.1.3.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.5.m5.1c">W_{i}:A_{-i}\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.5.m5.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT → blackboard_R</annotation></semantics></math> is an arbitrary function not depending on <math alttext="i" class="ltx_Math" display="inline" id="S3.SS3.p2.6.m6.1"><semantics id="S3.SS3.p2.6.m6.1a"><mi id="S3.SS3.p2.6.m6.1.1" xref="S3.SS3.p2.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.6.m6.1b"><ci id="S3.SS3.p2.6.m6.1.1.cmml" xref="S3.SS3.p2.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.6.m6.1d">italic_i</annotation></semantics></math>’s action, there is no way to distinguish <math alttext="\Gamma" class="ltx_Math" display="inline" id="S3.SS3.p2.7.m7.1"><semantics id="S3.SS3.p2.7.m7.1a"><mi id="S3.SS3.p2.7.m7.1.1" mathvariant="normal" xref="S3.SS3.p2.7.m7.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.7.m7.1b"><ci id="S3.SS3.p2.7.m7.1.1.cmml" xref="S3.SS3.p2.7.m7.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.7.m7.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.7.m7.1d">roman_Γ</annotation></semantics></math> from <math alttext="\Gamma^{\prime}" class="ltx_Math" display="inline" id="S3.SS3.p2.8.m8.1"><semantics id="S3.SS3.p2.8.m8.1a"><msup id="S3.SS3.p2.8.m8.1.1" xref="S3.SS3.p2.8.m8.1.1.cmml"><mi id="S3.SS3.p2.8.m8.1.1.2" mathvariant="normal" xref="S3.SS3.p2.8.m8.1.1.2.cmml">Γ</mi><mo id="S3.SS3.p2.8.m8.1.1.3" xref="S3.SS3.p2.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.8.m8.1b"><apply id="S3.SS3.p2.8.m8.1.1.cmml" xref="S3.SS3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.8.m8.1.1.1.cmml" xref="S3.SS3.p2.8.m8.1.1">superscript</csymbol><ci id="S3.SS3.p2.8.m8.1.1.2.cmml" xref="S3.SS3.p2.8.m8.1.1.2">Γ</ci><ci id="S3.SS3.p2.8.m8.1.1.3.cmml" xref="S3.SS3.p2.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.8.m8.1c">\Gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.8.m8.1d">roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> using only behavioral data. Thus, we can only determine utility functions <span class="ltx_text ltx_font_italic" id="S3.SS3.p2.9.2">up to</span> additive <math alttext="W_{i}" class="ltx_Math" display="inline" id="S3.SS3.p2.9.m9.1"><semantics id="S3.SS3.p2.9.m9.1a"><msub id="S3.SS3.p2.9.m9.1.1" xref="S3.SS3.p2.9.m9.1.1.cmml"><mi id="S3.SS3.p2.9.m9.1.1.2" xref="S3.SS3.p2.9.m9.1.1.2.cmml">W</mi><mi id="S3.SS3.p2.9.m9.1.1.3" xref="S3.SS3.p2.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.9.m9.1b"><apply id="S3.SS3.p2.9.m9.1.1.cmml" xref="S3.SS3.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.9.m9.1.1.1.cmml" xref="S3.SS3.p2.9.m9.1.1">subscript</csymbol><ci id="S3.SS3.p2.9.m9.1.1.2.cmml" xref="S3.SS3.p2.9.m9.1.1.2">𝑊</ci><ci id="S3.SS3.p2.9.m9.1.1.3.cmml" xref="S3.SS3.p2.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.9.m9.1c">W_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.9.m9.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> terms. We can thus formally state our goal as follows.</p> </div> <section class="ltx_paragraph" id="S3.SS3.SSS0.Px1"> <h4 class="ltx_title ltx_font_bold ltx_title_paragraph">Goal.</h4> <div class="ltx_para" id="S3.SS3.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS3.SSS0.Px1.p1.6">Given a game <math alttext="\Gamma" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.1.m1.1"><semantics id="S3.SS3.SSS0.Px1.p1.1.m1.1a"><mi id="S3.SS3.SSS0.Px1.p1.1.m1.1.1" mathvariant="normal" xref="S3.SS3.SSS0.Px1.p1.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.1.m1.1b"><ci id="S3.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.1.m1.1d">roman_Γ</annotation></semantics></math> and precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.2.m2.1"><semantics id="S3.SS3.SSS0.Px1.p1.2.m2.1a"><mi id="S3.SS3.SSS0.Px1.p1.2.m2.1.1" xref="S3.SS3.SSS0.Px1.p1.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.2.m2.1b"><ci id="S3.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.2.m2.1d">italic_ε</annotation></semantics></math>, we say that the principal <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.3.m3.1"><semantics id="S3.SS3.SSS0.Px1.p1.3.m3.1a"><mi id="S3.SS3.SSS0.Px1.p1.3.m3.1.1" xref="S3.SS3.SSS0.Px1.p1.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.3.m3.1b"><ci id="S3.SS3.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.3.m3.1d">italic_ε</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.SS3.SSS0.Px1.p1.6.1">-learns</span> the game <math alttext="\Gamma" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.4.m4.1"><semantics id="S3.SS3.SSS0.Px1.p1.4.m4.1a"><mi id="S3.SS3.SSS0.Px1.p1.4.m4.1.1" mathvariant="normal" xref="S3.SS3.SSS0.Px1.p1.4.m4.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.4.m4.1b"><ci id="S3.SS3.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.4.m4.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.4.m4.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.4.m4.1d">roman_Γ</annotation></semantics></math> if it outputs utility functions <math alttext="\tilde{U}_{i}:A\to\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.5.m5.1"><semantics id="S3.SS3.SSS0.Px1.p1.5.m5.1a"><mrow id="S3.SS3.SSS0.Px1.p1.5.m5.1.1" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.cmml"><msub id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.cmml"><mover accent="true" id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.cmml"><mi id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.2" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.2.cmml">U</mi><mo id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.1" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.1.cmml">~</mo></mover><mi id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.3" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.1.cmml">:</mo><mrow id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.cmml"><mi id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.2" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.2.cmml">A</mi><mo id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.1" stretchy="false" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.1.cmml">→</mo><mi id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.3" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.5.m5.1b"><apply id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1"><ci id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.1">:</ci><apply id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.1.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2">subscript</csymbol><apply id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2"><ci id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.1.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.1">~</ci><ci id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.2.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.2.2">𝑈</ci></apply><ci id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.3.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.2.3">𝑖</ci></apply><apply id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3"><ci id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.1.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.1">→</ci><ci id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.2.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.2">𝐴</ci><ci id="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.3.cmml" xref="S3.SS3.SSS0.Px1.p1.5.m5.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.5.m5.1c">\tilde{U}_{i}:A\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.5.m5.1d">over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A → blackboard_R</annotation></semantics></math> such that there exist functions <math alttext="W_{i}:A_{-i}\to\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.6.m6.1"><semantics id="S3.SS3.SSS0.Px1.p1.6.m6.1a"><mrow id="S3.SS3.SSS0.Px1.p1.6.m6.1.1" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.cmml"><msub id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.cmml"><mi id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.2" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.2.cmml">W</mi><mi id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.3" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.1.cmml">:</mo><mrow id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.cmml"><msub id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.cmml"><mi id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.2" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.2.cmml">A</mi><mrow id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.cmml"><mo id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3a" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.cmml">−</mo><mi id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.2" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.2.cmml">i</mi></mrow></msub><mo id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.1" stretchy="false" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.1.cmml">→</mo><mi id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.3" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.6.m6.1b"><apply id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1"><ci id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.1">:</ci><apply id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.1.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.2.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.2">𝑊</ci><ci id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.3.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.2.3">𝑖</ci></apply><apply id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3"><ci id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.1.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.1">→</ci><apply id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.1.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2">subscript</csymbol><ci id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.2.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.2">𝐴</ci><apply id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3"><minus id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.1.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3"></minus><ci id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.2.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.2.3.2">𝑖</ci></apply></apply><ci id="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.3.cmml" xref="S3.SS3.SSS0.Px1.p1.6.m6.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.6.m6.1c">W_{i}:A_{-i}\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.6.m6.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT → blackboard_R</annotation></semantics></math> satisfying</p> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <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="\absolutevalue{U_{i}(a)+W_{i}(a_{-i})-\tilde{U}_{i}(a)}\leq\varepsilon" class="ltx_Math" display="block" id="S3.E3.m1.1"><semantics id="S3.E3.m1.1a"><mrow id="S3.E3.m1.1.2" xref="S3.E3.m1.1.2.cmml"><mrow id="S3.E3.m1.1.1.3" xref="S3.E3.m1.1.1.2.cmml"><mo id="S3.E3.m1.1.1.3.1" xref="S3.E3.m1.1.1.2.1.cmml">|</mo><mrow id="S3.E3.m1.1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.cmml"><mrow id="S3.E3.m1.1.1.1.1.1.3" xref="S3.E3.m1.1.1.1.1.1.3.cmml"><mrow id="S3.E3.m1.1.1.1.1.1.3.3" xref="S3.E3.m1.1.1.1.1.1.3.3.cmml"><msub id="S3.E3.m1.1.1.1.1.1.3.3.2" xref="S3.E3.m1.1.1.1.1.1.3.3.2.cmml"><mi id="S3.E3.m1.1.1.1.1.1.3.3.2.2" xref="S3.E3.m1.1.1.1.1.1.3.3.2.2.cmml">U</mi><mi id="S3.E3.m1.1.1.1.1.1.3.3.2.3" xref="S3.E3.m1.1.1.1.1.1.3.3.2.3.cmml">i</mi></msub><mo id="S3.E3.m1.1.1.1.1.1.3.3.1" xref="S3.E3.m1.1.1.1.1.1.3.3.1.cmml">⁢</mo><mrow id="S3.E3.m1.1.1.1.1.1.3.3.3.2" xref="S3.E3.m1.1.1.1.1.1.3.3.cmml"><mo id="S3.E3.m1.1.1.1.1.1.3.3.3.2.1" stretchy="false" xref="S3.E3.m1.1.1.1.1.1.3.3.cmml">(</mo><mi id="S3.E3.m1.1.1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.1.cmml">a</mi><mo id="S3.E3.m1.1.1.1.1.1.3.3.3.2.2" stretchy="false" xref="S3.E3.m1.1.1.1.1.1.3.3.cmml">)</mo></mrow></mrow><mo id="S3.E3.m1.1.1.1.1.1.3.2" xref="S3.E3.m1.1.1.1.1.1.3.2.cmml">+</mo><mrow id="S3.E3.m1.1.1.1.1.1.3.1" xref="S3.E3.m1.1.1.1.1.1.3.1.cmml"><msub id="S3.E3.m1.1.1.1.1.1.3.1.3" xref="S3.E3.m1.1.1.1.1.1.3.1.3.cmml"><mi id="S3.E3.m1.1.1.1.1.1.3.1.3.2" xref="S3.E3.m1.1.1.1.1.1.3.1.3.2.cmml">W</mi><mi id="S3.E3.m1.1.1.1.1.1.3.1.3.3" xref="S3.E3.m1.1.1.1.1.1.3.1.3.3.cmml">i</mi></msub><mo id="S3.E3.m1.1.1.1.1.1.3.1.2" xref="S3.E3.m1.1.1.1.1.1.3.1.2.cmml">⁢</mo><mrow id="S3.E3.m1.1.1.1.1.1.3.1.1.1" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.cmml"><mo id="S3.E3.m1.1.1.1.1.1.3.1.1.1.2" stretchy="false" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.cmml">(</mo><msub id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.cmml"><mi id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.2" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.2.cmml">a</mi><mrow id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.3" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.3.cmml"><mo id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.3a" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.3.cmml">−</mo><mi id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.3.2" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.3.2.cmml">i</mi></mrow></msub><mo id="S3.E3.m1.1.1.1.1.1.3.1.1.1.3" stretchy="false" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E3.m1.1.1.1.1.1.4" xref="S3.E3.m1.1.1.1.1.1.4.cmml">−</mo><mrow id="S3.E3.m1.1.1.1.1.1.5" xref="S3.E3.m1.1.1.1.1.1.5.cmml"><msub id="S3.E3.m1.1.1.1.1.1.5.2" xref="S3.E3.m1.1.1.1.1.1.5.2.cmml"><mover accent="true" id="S3.E3.m1.1.1.1.1.1.5.2.2" xref="S3.E3.m1.1.1.1.1.1.5.2.2.cmml"><mi id="S3.E3.m1.1.1.1.1.1.5.2.2.2" xref="S3.E3.m1.1.1.1.1.1.5.2.2.2.cmml">U</mi><mo id="S3.E3.m1.1.1.1.1.1.5.2.2.1" xref="S3.E3.m1.1.1.1.1.1.5.2.2.1.cmml">~</mo></mover><mi id="S3.E3.m1.1.1.1.1.1.5.2.3" xref="S3.E3.m1.1.1.1.1.1.5.2.3.cmml">i</mi></msub><mo id="S3.E3.m1.1.1.1.1.1.5.1" xref="S3.E3.m1.1.1.1.1.1.5.1.cmml">⁢</mo><mrow id="S3.E3.m1.1.1.1.1.1.5.3.2" xref="S3.E3.m1.1.1.1.1.1.5.cmml"><mo id="S3.E3.m1.1.1.1.1.1.5.3.2.1" stretchy="false" xref="S3.E3.m1.1.1.1.1.1.5.cmml">(</mo><mi id="S3.E3.m1.1.1.1.1.1.2" xref="S3.E3.m1.1.1.1.1.1.2.cmml">a</mi><mo id="S3.E3.m1.1.1.1.1.1.5.3.2.2" stretchy="false" xref="S3.E3.m1.1.1.1.1.1.5.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E3.m1.1.1.3.2" xref="S3.E3.m1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.E3.m1.1.2.1" xref="S3.E3.m1.1.2.1.cmml">≤</mo><mi id="S3.E3.m1.1.2.2" xref="S3.E3.m1.1.2.2.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.E3.m1.1b"><apply id="S3.E3.m1.1.2.cmml" xref="S3.E3.m1.1.2"><leq id="S3.E3.m1.1.2.1.cmml" xref="S3.E3.m1.1.2.1"></leq><apply id="S3.E3.m1.1.1.2.cmml" xref="S3.E3.m1.1.1.3"><abs id="S3.E3.m1.1.1.2.1.cmml" xref="S3.E3.m1.1.1.3.1"></abs><apply id="S3.E3.m1.1.1.1.1.1.cmml" xref="S3.E3.m1.1.1.1.1.1"><minus id="S3.E3.m1.1.1.1.1.1.4.cmml" xref="S3.E3.m1.1.1.1.1.1.4"></minus><apply id="S3.E3.m1.1.1.1.1.1.3.cmml" xref="S3.E3.m1.1.1.1.1.1.3"><plus id="S3.E3.m1.1.1.1.1.1.3.2.cmml" xref="S3.E3.m1.1.1.1.1.1.3.2"></plus><apply id="S3.E3.m1.1.1.1.1.1.3.3.cmml" xref="S3.E3.m1.1.1.1.1.1.3.3"><times id="S3.E3.m1.1.1.1.1.1.3.3.1.cmml" xref="S3.E3.m1.1.1.1.1.1.3.3.1"></times><apply id="S3.E3.m1.1.1.1.1.1.3.3.2.cmml" xref="S3.E3.m1.1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.1.3.3.2.1.cmml" xref="S3.E3.m1.1.1.1.1.1.3.3.2">subscript</csymbol><ci id="S3.E3.m1.1.1.1.1.1.3.3.2.2.cmml" xref="S3.E3.m1.1.1.1.1.1.3.3.2.2">𝑈</ci><ci id="S3.E3.m1.1.1.1.1.1.3.3.2.3.cmml" xref="S3.E3.m1.1.1.1.1.1.3.3.2.3">𝑖</ci></apply><ci id="S3.E3.m1.1.1.1.1.1.1.cmml" xref="S3.E3.m1.1.1.1.1.1.1">𝑎</ci></apply><apply id="S3.E3.m1.1.1.1.1.1.3.1.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1"><times id="S3.E3.m1.1.1.1.1.1.3.1.2.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1.2"></times><apply id="S3.E3.m1.1.1.1.1.1.3.1.3.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1.3"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.1.3.1.3.1.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1.3">subscript</csymbol><ci id="S3.E3.m1.1.1.1.1.1.3.1.3.2.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1.3.2">𝑊</ci><ci id="S3.E3.m1.1.1.1.1.1.3.1.3.3.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1.3.3">𝑖</ci></apply><apply id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.1.cmml" xref="S3.E3.m1.1.1.1.1.1.3.1.1.1">subscript</csymbol><ci id="S3.E3.m1.1.1.1.1.1.3.1.1.1.1.2.cmml" 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</tr></tbody> </table> <p class="ltx_p" id="S3.SS3.SSS0.Px1.p1.10">for all agents <math alttext="i" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.7.m1.1"><semantics id="S3.SS3.SSS0.Px1.p1.7.m1.1a"><mi id="S3.SS3.SSS0.Px1.p1.7.m1.1.1" xref="S3.SS3.SSS0.Px1.p1.7.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.7.m1.1b"><ci id="S3.SS3.SSS0.Px1.p1.7.m1.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.7.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.7.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.7.m1.1d">italic_i</annotation></semantics></math> and action profiles <math alttext="a\in A" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.8.m2.1"><semantics id="S3.SS3.SSS0.Px1.p1.8.m2.1a"><mrow id="S3.SS3.SSS0.Px1.p1.8.m2.1.1" xref="S3.SS3.SSS0.Px1.p1.8.m2.1.1.cmml"><mi id="S3.SS3.SSS0.Px1.p1.8.m2.1.1.2" xref="S3.SS3.SSS0.Px1.p1.8.m2.1.1.2.cmml">a</mi><mo 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The goal of the principal is to <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.9.m3.1"><semantics id="S3.SS3.SSS0.Px1.p1.9.m3.1a"><mi id="S3.SS3.SSS0.Px1.p1.9.m3.1.1" xref="S3.SS3.SSS0.Px1.p1.9.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.9.m3.1b"><ci id="S3.SS3.SSS0.Px1.p1.9.m3.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.9.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.9.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.9.m3.1d">italic_ε</annotation></semantics></math>-learn <math alttext="\Gamma" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p1.10.m4.1"><semantics id="S3.SS3.SSS0.Px1.p1.10.m4.1a"><mi id="S3.SS3.SSS0.Px1.p1.10.m4.1.1" mathvariant="normal" xref="S3.SS3.SSS0.Px1.p1.10.m4.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS0.Px1.p1.10.m4.1b"><ci id="S3.SS3.SSS0.Px1.p1.10.m4.1.1.cmml" xref="S3.SS3.SSS0.Px1.p1.10.m4.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS0.Px1.p1.10.m4.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS0.Px1.p1.10.m4.1d">roman_Γ</annotation></semantics></math> in as few rounds as possible.</p> </div> <div class="ltx_para" id="S3.SS3.SSS0.Px1.p2"> <p class="ltx_p" id="S3.SS3.SSS0.Px1.p2.1">For most of the paper, we will be focused on the problem of minimizing the number of <span class="ltx_text ltx_font_italic" id="S3.SS3.SSS0.Px1.p2.1.1">rounds</span> it takes to learn the utility functions. In <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S6" title="6 Minimizing Payment ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a>, we will instead focus on the problem of minimizing the amount of <span class="ltx_text ltx_font_italic" id="S3.SS3.SSS0.Px1.p2.1.2">total payment</span>, <math alttext="\sum_{t\in[T],i\in[n]}P_{i}^{t}(a_{i}^{t})" class="ltx_Math" display="inline" id="S3.SS3.SSS0.Px1.p2.1.m1.5"><semantics id="S3.SS3.SSS0.Px1.p2.1.m1.5a"><mrow id="S3.SS3.SSS0.Px1.p2.1.m1.5.5" xref="S3.SS3.SSS0.Px1.p2.1.m1.5.5.cmml"><msub id="S3.SS3.SSS0.Px1.p2.1.m1.5.5.2" xref="S3.SS3.SSS0.Px1.p2.1.m1.5.5.2.cmml"><mo id="S3.SS3.SSS0.Px1.p2.1.m1.5.5.2.2" xref="S3.SS3.SSS0.Px1.p2.1.m1.5.5.2.2.cmml">∑</mo><mrow id="S3.SS3.SSS0.Px1.p2.1.m1.4.4.4.4" xref="S3.SS3.SSS0.Px1.p2.1.m1.4.4.4.5.cmml"><mrow id="S3.SS3.SSS0.Px1.p2.1.m1.3.3.3.3.1" xref="S3.SS3.SSS0.Px1.p2.1.m1.3.3.3.3.1.cmml"><mi 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </section> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Learning the Utility Function in the Rationalizable Model</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">We start by studying the rationalizable model. We will first study the simpler, but illustrative, single-agent setting, before moving on to the multi-agent case.</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>The single-agent case</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.2">We will show that it is possible to learn the utility function <math alttext="U:A\to[0,1]" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.2"><semantics id="S4.SS1.p1.1.m1.2a"><mrow id="S4.SS1.p1.1.m1.2.3" xref="S4.SS1.p1.1.m1.2.3.cmml"><mi id="S4.SS1.p1.1.m1.2.3.2" xref="S4.SS1.p1.1.m1.2.3.2.cmml">U</mi><mo id="S4.SS1.p1.1.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p1.1.m1.2.3.1.cmml">:</mo><mrow id="S4.SS1.p1.1.m1.2.3.3" xref="S4.SS1.p1.1.m1.2.3.3.cmml"><mi id="S4.SS1.p1.1.m1.2.3.3.2" xref="S4.SS1.p1.1.m1.2.3.3.2.cmml">A</mi><mo id="S4.SS1.p1.1.m1.2.3.3.1" stretchy="false" xref="S4.SS1.p1.1.m1.2.3.3.1.cmml">→</mo><mrow id="S4.SS1.p1.1.m1.2.3.3.3.2" xref="S4.SS1.p1.1.m1.2.3.3.3.1.cmml"><mo id="S4.SS1.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.p1.1.m1.2.3.3.3.1.cmml">[</mo><mn id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">0</mn><mo id="S4.SS1.p1.1.m1.2.3.3.3.2.2" xref="S4.SS1.p1.1.m1.2.3.3.3.1.cmml">,</mo><mn id="S4.SS1.p1.1.m1.2.2" xref="S4.SS1.p1.1.m1.2.2.cmml">1</mn><mo id="S4.SS1.p1.1.m1.2.3.3.3.2.3" stretchy="false" xref="S4.SS1.p1.1.m1.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.2b"><apply id="S4.SS1.p1.1.m1.2.3.cmml" xref="S4.SS1.p1.1.m1.2.3"><ci id="S4.SS1.p1.1.m1.2.3.1.cmml" xref="S4.SS1.p1.1.m1.2.3.1">:</ci><ci id="S4.SS1.p1.1.m1.2.3.2.cmml" xref="S4.SS1.p1.1.m1.2.3.2">𝑈</ci><apply id="S4.SS1.p1.1.m1.2.3.3.cmml" xref="S4.SS1.p1.1.m1.2.3.3"><ci id="S4.SS1.p1.1.m1.2.3.3.1.cmml" xref="S4.SS1.p1.1.m1.2.3.3.1">→</ci><ci id="S4.SS1.p1.1.m1.2.3.3.2.cmml" xref="S4.SS1.p1.1.m1.2.3.3.2">𝐴</ci><interval closure="closed" id="S4.SS1.p1.1.m1.2.3.3.3.1.cmml" xref="S4.SS1.p1.1.m1.2.3.3.3.2"><cn id="S4.SS1.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS1.p1.1.m1.1.1">0</cn><cn id="S4.SS1.p1.1.m1.2.2.cmml" type="integer" xref="S4.SS1.p1.1.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.2c">U:A\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.2d">italic_U : italic_A → [ 0 , 1 ]</annotation></semantics></math> in both the rationalizable and no-regret models, then show nearly-matching lower bounds. Note that, in the case of a single agent, the rationalizable model simply amounts to the agent always picking an action maximizing <math alttext="U^{t}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><msup id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml"><mi id="S4.SS1.p1.2.m2.1.1.2" xref="S4.SS1.p1.2.m2.1.1.2.cmml">U</mi><mi id="S4.SS1.p1.2.m2.1.1.3" xref="S4.SS1.p1.2.m2.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><apply id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">superscript</csymbol><ci id="S4.SS1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.2">𝑈</ci><ci id="S4.SS1.p1.2.m2.1.1.3.cmml" xref="S4.SS1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">U^{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">italic_U start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. We will therefore refer to it as the <span class="ltx_text ltx_font_italic" id="S4.SS1.p1.2.1">optimal action model</span>.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.10">Our first main algorithm shows the possibility of learning the utility of a single agent in the optimal action model. The algorithm operates as follows. For each action <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_a</annotation></semantics></math>, our goal is to find the smallest value <math alttext="P^{*}(a)\in[0,1]" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.3"><semantics id="S4.SS1.p2.2.m2.3a"><mrow id="S4.SS1.p2.2.m2.3.4" xref="S4.SS1.p2.2.m2.3.4.cmml"><mrow id="S4.SS1.p2.2.m2.3.4.2" xref="S4.SS1.p2.2.m2.3.4.2.cmml"><msup id="S4.SS1.p2.2.m2.3.4.2.2" xref="S4.SS1.p2.2.m2.3.4.2.2.cmml"><mi id="S4.SS1.p2.2.m2.3.4.2.2.2" xref="S4.SS1.p2.2.m2.3.4.2.2.2.cmml">P</mi><mo id="S4.SS1.p2.2.m2.3.4.2.2.3" xref="S4.SS1.p2.2.m2.3.4.2.2.3.cmml">∗</mo></msup><mo id="S4.SS1.p2.2.m2.3.4.2.1" xref="S4.SS1.p2.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.2.m2.3.4.2.3.2" xref="S4.SS1.p2.2.m2.3.4.2.cmml"><mo id="S4.SS1.p2.2.m2.3.4.2.3.2.1" stretchy="false" xref="S4.SS1.p2.2.m2.3.4.2.cmml">(</mo><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">a</mi><mo id="S4.SS1.p2.2.m2.3.4.2.3.2.2" stretchy="false" xref="S4.SS1.p2.2.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.2.m2.3.4.1" xref="S4.SS1.p2.2.m2.3.4.1.cmml">∈</mo><mrow id="S4.SS1.p2.2.m2.3.4.3.2" xref="S4.SS1.p2.2.m2.3.4.3.1.cmml"><mo id="S4.SS1.p2.2.m2.3.4.3.2.1" stretchy="false" xref="S4.SS1.p2.2.m2.3.4.3.1.cmml">[</mo><mn id="S4.SS1.p2.2.m2.2.2" xref="S4.SS1.p2.2.m2.2.2.cmml">0</mn><mo id="S4.SS1.p2.2.m2.3.4.3.2.2" xref="S4.SS1.p2.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.SS1.p2.2.m2.3.3" xref="S4.SS1.p2.2.m2.3.3.cmml">1</mn><mo id="S4.SS1.p2.2.m2.3.4.3.2.3" stretchy="false" xref="S4.SS1.p2.2.m2.3.4.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.3b"><apply id="S4.SS1.p2.2.m2.3.4.cmml" xref="S4.SS1.p2.2.m2.3.4"><in id="S4.SS1.p2.2.m2.3.4.1.cmml" xref="S4.SS1.p2.2.m2.3.4.1"></in><apply id="S4.SS1.p2.2.m2.3.4.2.cmml" xref="S4.SS1.p2.2.m2.3.4.2"><times id="S4.SS1.p2.2.m2.3.4.2.1.cmml" xref="S4.SS1.p2.2.m2.3.4.2.1"></times><apply id="S4.SS1.p2.2.m2.3.4.2.2.cmml" xref="S4.SS1.p2.2.m2.3.4.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.3.4.2.2.1.cmml" xref="S4.SS1.p2.2.m2.3.4.2.2">superscript</csymbol><ci id="S4.SS1.p2.2.m2.3.4.2.2.2.cmml" xref="S4.SS1.p2.2.m2.3.4.2.2.2">𝑃</ci><times id="S4.SS1.p2.2.m2.3.4.2.2.3.cmml" xref="S4.SS1.p2.2.m2.3.4.2.2.3"></times></apply><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝑎</ci></apply><interval closure="closed" id="S4.SS1.p2.2.m2.3.4.3.1.cmml" xref="S4.SS1.p2.2.m2.3.4.3.2"><cn id="S4.SS1.p2.2.m2.2.2.cmml" type="integer" xref="S4.SS1.p2.2.m2.2.2">0</cn><cn id="S4.SS1.p2.2.m2.3.3.cmml" type="integer" xref="S4.SS1.p2.2.m2.3.3">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.3c">P^{*}(a)\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.3d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a ) ∈ [ 0 , 1 ]</annotation></semantics></math> such that the agent plays action <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mi id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">italic_a</annotation></semantics></math> when given payment <math alttext="P^{*}(a)" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mrow id="S4.SS1.p2.4.m4.1.2" xref="S4.SS1.p2.4.m4.1.2.cmml"><msup id="S4.SS1.p2.4.m4.1.2.2" xref="S4.SS1.p2.4.m4.1.2.2.cmml"><mi id="S4.SS1.p2.4.m4.1.2.2.2" xref="S4.SS1.p2.4.m4.1.2.2.2.cmml">P</mi><mo id="S4.SS1.p2.4.m4.1.2.2.3" xref="S4.SS1.p2.4.m4.1.2.2.3.cmml">∗</mo></msup><mo id="S4.SS1.p2.4.m4.1.2.1" xref="S4.SS1.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.4.m4.1.2.3.2" xref="S4.SS1.p2.4.m4.1.2.cmml"><mo id="S4.SS1.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S4.SS1.p2.4.m4.1.2.cmml">(</mo><mi id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">a</mi><mo id="S4.SS1.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S4.SS1.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.2.cmml" xref="S4.SS1.p2.4.m4.1.2"><times id="S4.SS1.p2.4.m4.1.2.1.cmml" xref="S4.SS1.p2.4.m4.1.2.1"></times><apply id="S4.SS1.p2.4.m4.1.2.2.cmml" xref="S4.SS1.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.2.2.1.cmml" xref="S4.SS1.p2.4.m4.1.2.2">superscript</csymbol><ci id="S4.SS1.p2.4.m4.1.2.2.2.cmml" xref="S4.SS1.p2.4.m4.1.2.2.2">𝑃</ci><times id="S4.SS1.p2.4.m4.1.2.2.3.cmml" xref="S4.SS1.p2.4.m4.1.2.2.3"></times></apply><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">P^{*}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> for action <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_a</annotation></semantics></math> and <math alttext="0" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mn id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><cn id="S4.SS1.p2.6.m6.1.1.cmml" type="integer" xref="S4.SS1.p2.6.m6.1.1">0</cn></annotation-xml></semantics></math> for other actions. By construction of the optimal action model, this value is exactly <math alttext="P^{*}(a)=\max_{a^{\prime}}\{U(a^{\prime})-U(a)\}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.4"><semantics id="S4.SS1.p2.7.m7.4a"><mrow id="S4.SS1.p2.7.m7.4.4" xref="S4.SS1.p2.7.m7.4.4.cmml"><mrow id="S4.SS1.p2.7.m7.4.4.4" xref="S4.SS1.p2.7.m7.4.4.4.cmml"><msup id="S4.SS1.p2.7.m7.4.4.4.2" xref="S4.SS1.p2.7.m7.4.4.4.2.cmml"><mi id="S4.SS1.p2.7.m7.4.4.4.2.2" xref="S4.SS1.p2.7.m7.4.4.4.2.2.cmml">P</mi><mo id="S4.SS1.p2.7.m7.4.4.4.2.3" xref="S4.SS1.p2.7.m7.4.4.4.2.3.cmml">∗</mo></msup><mo id="S4.SS1.p2.7.m7.4.4.4.1" xref="S4.SS1.p2.7.m7.4.4.4.1.cmml">⁢</mo><mrow id="S4.SS1.p2.7.m7.4.4.4.3.2" xref="S4.SS1.p2.7.m7.4.4.4.cmml"><mo id="S4.SS1.p2.7.m7.4.4.4.3.2.1" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.4.cmml">(</mo><mi id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml">a</mi><mo id="S4.SS1.p2.7.m7.4.4.4.3.2.2" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.4.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.7.m7.4.4.3" xref="S4.SS1.p2.7.m7.4.4.3.cmml">=</mo><mrow id="S4.SS1.p2.7.m7.4.4.2.2" xref="S4.SS1.p2.7.m7.4.4.2.3.cmml"><msub id="S4.SS1.p2.7.m7.3.3.1.1.1" xref="S4.SS1.p2.7.m7.3.3.1.1.1.cmml"><mi id="S4.SS1.p2.7.m7.3.3.1.1.1.2" xref="S4.SS1.p2.7.m7.3.3.1.1.1.2.cmml">max</mi><msup id="S4.SS1.p2.7.m7.3.3.1.1.1.3" xref="S4.SS1.p2.7.m7.3.3.1.1.1.3.cmml"><mi id="S4.SS1.p2.7.m7.3.3.1.1.1.3.2" xref="S4.SS1.p2.7.m7.3.3.1.1.1.3.2.cmml">a</mi><mo id="S4.SS1.p2.7.m7.3.3.1.1.1.3.3" xref="S4.SS1.p2.7.m7.3.3.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S4.SS1.p2.7.m7.4.4.2.2a" xref="S4.SS1.p2.7.m7.4.4.2.3.cmml">⁡</mo><mrow id="S4.SS1.p2.7.m7.4.4.2.2.2" xref="S4.SS1.p2.7.m7.4.4.2.3.cmml"><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.2" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.2.3.cmml">{</mo><mrow id="S4.SS1.p2.7.m7.4.4.2.2.2.1" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.cmml"><mrow id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.cmml"><mi id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.3" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.3.cmml">U</mi><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.2" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.cmml"><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.2" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.cmml"><mi id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.2" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.2.cmml">a</mi><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.3" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.3" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.2" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.2.cmml">−</mo><mrow id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.cmml"><mi id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.2" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.2.cmml">U</mi><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.1" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.3.2" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.cmml"><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.3.2.1" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.cmml">(</mo><mi id="S4.SS1.p2.7.m7.2.2" xref="S4.SS1.p2.7.m7.2.2.cmml">a</mi><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.3.2.2" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS1.p2.7.m7.4.4.2.2.2.3" stretchy="false" xref="S4.SS1.p2.7.m7.4.4.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.4b"><apply id="S4.SS1.p2.7.m7.4.4.cmml" xref="S4.SS1.p2.7.m7.4.4"><eq id="S4.SS1.p2.7.m7.4.4.3.cmml" xref="S4.SS1.p2.7.m7.4.4.3"></eq><apply id="S4.SS1.p2.7.m7.4.4.4.cmml" xref="S4.SS1.p2.7.m7.4.4.4"><times id="S4.SS1.p2.7.m7.4.4.4.1.cmml" xref="S4.SS1.p2.7.m7.4.4.4.1"></times><apply id="S4.SS1.p2.7.m7.4.4.4.2.cmml" xref="S4.SS1.p2.7.m7.4.4.4.2"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m7.4.4.4.2.1.cmml" xref="S4.SS1.p2.7.m7.4.4.4.2">superscript</csymbol><ci id="S4.SS1.p2.7.m7.4.4.4.2.2.cmml" xref="S4.SS1.p2.7.m7.4.4.4.2.2">𝑃</ci><times id="S4.SS1.p2.7.m7.4.4.4.2.3.cmml" xref="S4.SS1.p2.7.m7.4.4.4.2.3"></times></apply><ci id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1">𝑎</ci></apply><apply id="S4.SS1.p2.7.m7.4.4.2.3.cmml" xref="S4.SS1.p2.7.m7.4.4.2.2"><apply id="S4.SS1.p2.7.m7.3.3.1.1.1.cmml" xref="S4.SS1.p2.7.m7.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m7.3.3.1.1.1.1.cmml" xref="S4.SS1.p2.7.m7.3.3.1.1.1">subscript</csymbol><max id="S4.SS1.p2.7.m7.3.3.1.1.1.2.cmml" xref="S4.SS1.p2.7.m7.3.3.1.1.1.2"></max><apply id="S4.SS1.p2.7.m7.3.3.1.1.1.3.cmml" xref="S4.SS1.p2.7.m7.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m7.3.3.1.1.1.3.1.cmml" xref="S4.SS1.p2.7.m7.3.3.1.1.1.3">superscript</csymbol><ci id="S4.SS1.p2.7.m7.3.3.1.1.1.3.2.cmml" 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xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.cmml" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3"><times id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.1.cmml" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.1"></times><ci id="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.2.cmml" xref="S4.SS1.p2.7.m7.4.4.2.2.2.1.3.2">𝑈</ci><ci id="S4.SS1.p2.7.m7.2.2.cmml" xref="S4.SS1.p2.7.m7.2.2">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.4c">P^{*}(a)=\max_{a^{\prime}}\{U(a^{\prime})-U(a)\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.4d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a ) = roman_max start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT { italic_U ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) - italic_U ( italic_a ) }</annotation></semantics></math>. Moreover, the value <math alttext="P^{*}(a)" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m8.1"><semantics id="S4.SS1.p2.8.m8.1a"><mrow id="S4.SS1.p2.8.m8.1.2" xref="S4.SS1.p2.8.m8.1.2.cmml"><msup id="S4.SS1.p2.8.m8.1.2.2" xref="S4.SS1.p2.8.m8.1.2.2.cmml"><mi id="S4.SS1.p2.8.m8.1.2.2.2" xref="S4.SS1.p2.8.m8.1.2.2.2.cmml">P</mi><mo id="S4.SS1.p2.8.m8.1.2.2.3" xref="S4.SS1.p2.8.m8.1.2.2.3.cmml">∗</mo></msup><mo id="S4.SS1.p2.8.m8.1.2.1" xref="S4.SS1.p2.8.m8.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.8.m8.1.2.3.2" xref="S4.SS1.p2.8.m8.1.2.cmml"><mo id="S4.SS1.p2.8.m8.1.2.3.2.1" stretchy="false" xref="S4.SS1.p2.8.m8.1.2.cmml">(</mo><mi id="S4.SS1.p2.8.m8.1.1" xref="S4.SS1.p2.8.m8.1.1.cmml">a</mi><mo id="S4.SS1.p2.8.m8.1.2.3.2.2" stretchy="false" xref="S4.SS1.p2.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m8.1b"><apply id="S4.SS1.p2.8.m8.1.2.cmml" xref="S4.SS1.p2.8.m8.1.2"><times id="S4.SS1.p2.8.m8.1.2.1.cmml" xref="S4.SS1.p2.8.m8.1.2.1"></times><apply id="S4.SS1.p2.8.m8.1.2.2.cmml" xref="S4.SS1.p2.8.m8.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.8.m8.1.2.2.1.cmml" xref="S4.SS1.p2.8.m8.1.2.2">superscript</csymbol><ci id="S4.SS1.p2.8.m8.1.2.2.2.cmml" xref="S4.SS1.p2.8.m8.1.2.2.2">𝑃</ci><times id="S4.SS1.p2.8.m8.1.2.2.3.cmml" xref="S4.SS1.p2.8.m8.1.2.2.3"></times></apply><ci id="S4.SS1.p2.8.m8.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m8.1c">P^{*}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m8.1d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> can be approximated to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.p2.9.m9.1"><semantics id="S4.SS1.p2.9.m9.1a"><mi id="S4.SS1.p2.9.m9.1.1" xref="S4.SS1.p2.9.m9.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.9.m9.1b"><ci id="S4.SS1.p2.9.m9.1.1.cmml" xref="S4.SS1.p2.9.m9.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.9.m9.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.9.m9.1d">italic_ε</annotation></semantics></math> using <math alttext="{\mathcal{O}}(\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S4.SS1.p2.10.m10.2"><semantics id="S4.SS1.p2.10.m10.2a"><mrow id="S4.SS1.p2.10.m10.2.3" xref="S4.SS1.p2.10.m10.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.10.m10.2.3.2" xref="S4.SS1.p2.10.m10.2.3.2.cmml">𝒪</mi><mo id="S4.SS1.p2.10.m10.2.3.1" xref="S4.SS1.p2.10.m10.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.10.m10.2.3.3.2" xref="S4.SS1.p2.10.m10.2.3.cmml"><mo id="S4.SS1.p2.10.m10.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.10.m10.2.3.cmml">(</mo><mrow id="S4.SS1.p2.10.m10.2.2.4" xref="S4.SS1.p2.10.m10.2.2.3.cmml"><mi id="S4.SS1.p2.10.m10.2.2.2.2" xref="S4.SS1.p2.10.m10.2.2.3.1.cmml">log</mi><mo id="S4.SS1.p2.10.m10.2.2.4a" xref="S4.SS1.p2.10.m10.2.2.3.1.cmml">⁡</mo><mrow id="S4.SS1.p2.10.m10.2.2.4.1" xref="S4.SS1.p2.10.m10.2.2.3.cmml"><mo id="S4.SS1.p2.10.m10.2.2.4.1.1" xref="S4.SS1.p2.10.m10.2.2.3.1.cmml">(</mo><mrow id="S4.SS1.p2.10.m10.1.1.1.1.1" xref="S4.SS1.p2.10.m10.1.1.1.1.1.cmml"><mn id="S4.SS1.p2.10.m10.1.1.1.1.1.2" xref="S4.SS1.p2.10.m10.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS1.p2.10.m10.1.1.1.1.1.1" xref="S4.SS1.p2.10.m10.1.1.1.1.1.1.cmml">/</mo><mi id="S4.SS1.p2.10.m10.1.1.1.1.1.3" xref="S4.SS1.p2.10.m10.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.SS1.p2.10.m10.2.2.4.1.2" xref="S4.SS1.p2.10.m10.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.10.m10.2.3.3.2.2" stretchy="false" xref="S4.SS1.p2.10.m10.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.10.m10.2b"><apply id="S4.SS1.p2.10.m10.2.3.cmml" xref="S4.SS1.p2.10.m10.2.3"><times id="S4.SS1.p2.10.m10.2.3.1.cmml" xref="S4.SS1.p2.10.m10.2.3.1"></times><ci id="S4.SS1.p2.10.m10.2.3.2.cmml" xref="S4.SS1.p2.10.m10.2.3.2">𝒪</ci><apply id="S4.SS1.p2.10.m10.2.2.3.cmml" xref="S4.SS1.p2.10.m10.2.2.4"><log id="S4.SS1.p2.10.m10.2.2.3.1.cmml" xref="S4.SS1.p2.10.m10.2.2.2.2"></log><apply id="S4.SS1.p2.10.m10.1.1.1.1.1.cmml" xref="S4.SS1.p2.10.m10.1.1.1.1.1"><divide id="S4.SS1.p2.10.m10.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.10.m10.1.1.1.1.1.1"></divide><cn id="S4.SS1.p2.10.m10.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.p2.10.m10.1.1.1.1.1.2">1</cn><ci id="S4.SS1.p2.10.m10.1.1.1.1.1.3.cmml" xref="S4.SS1.p2.10.m10.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.10.m10.2c">{\mathcal{O}}(\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.10.m10.2d">caligraphic_O ( roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> iterations of binary search. This idea is formalized in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg1" title="In 4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <div class="ltx_listing ltx_listing" id="alg1.2"> <div class="ltx_listingline" id="alg0.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg0.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text ltx_font_bold" id="alg0.l1.2">for</span> each action <math alttext="a\in A" class="ltx_Math" display="inline" id="alg0.l1.m1.1"><semantics id="alg0.l1.m1.1a"><mrow id="alg0.l1.m1.1.1" xref="alg0.l1.m1.1.1.cmml"><mi id="alg0.l1.m1.1.1.2" xref="alg0.l1.m1.1.1.2.cmml">a</mi><mo id="alg0.l1.m1.1.1.1" xref="alg0.l1.m1.1.1.1.cmml">∈</mo><mi id="alg0.l1.m1.1.1.3" xref="alg0.l1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="alg0.l1.m1.1b"><apply id="alg0.l1.m1.1.1.cmml" xref="alg0.l1.m1.1.1"><in id="alg0.l1.m1.1.1.1.cmml" xref="alg0.l1.m1.1.1.1"></in><ci id="alg0.l1.m1.1.1.2.cmml" xref="alg0.l1.m1.1.1.2">𝑎</ci><ci id="alg0.l1.m1.1.1.3.cmml" xref="alg0.l1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l1.m1.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="alg0.l1.m1.1d">italic_a ∈ italic_A</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg0.l1.3">do</span> </div> <div class="ltx_listingline" id="alg0.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg0.l2.1.1.1" style="font-size:80%;">2:</span></span> using <math alttext="{\mathcal{O}}(\log(1/\varepsilon))" class="ltx_Math" display="inline" id="alg0.l2.m1.2"><semantics 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xref="alg0.l2.m1.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="alg0.l2.m1.2.3.3.2.2" stretchy="false" xref="alg0.l2.m1.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg0.l2.m1.2b"><apply id="alg0.l2.m1.2.3.cmml" xref="alg0.l2.m1.2.3"><times id="alg0.l2.m1.2.3.1.cmml" xref="alg0.l2.m1.2.3.1"></times><ci id="alg0.l2.m1.2.3.2.cmml" xref="alg0.l2.m1.2.3.2">𝒪</ci><apply id="alg0.l2.m1.2.2.3.cmml" xref="alg0.l2.m1.2.2.4"><log id="alg0.l2.m1.2.2.3.1.cmml" xref="alg0.l2.m1.2.2.2.2"></log><apply id="alg0.l2.m1.1.1.1.1.1.cmml" xref="alg0.l2.m1.1.1.1.1.1"><divide id="alg0.l2.m1.1.1.1.1.1.1.cmml" xref="alg0.l2.m1.1.1.1.1.1.1"></divide><cn id="alg0.l2.m1.1.1.1.1.1.2.cmml" type="integer" xref="alg0.l2.m1.1.1.1.1.1.2">1</cn><ci id="alg0.l2.m1.1.1.1.1.1.3.cmml" xref="alg0.l2.m1.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l2.m1.2c">{\mathcal{O}}(\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="alg0.l2.m1.2d">caligraphic_O ( roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> iterations of binary search, </div> <div class="ltx_listingline" id="alg0.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg0.l3.1.1.1" style="font-size:80%;">3:</span></span> approximate to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="alg0.l3.m1.1"><semantics id="alg0.l3.m1.1a"><mi id="alg0.l3.m1.1.1" xref="alg0.l3.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="alg0.l3.m1.1b"><ci id="alg0.l3.m1.1.1.cmml" xref="alg0.l3.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l3.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="alg0.l3.m1.1d">italic_ε</annotation></semantics></math> the smallest value <math alttext="P^{*}(a)\in[0,1]" class="ltx_Math" display="inline" id="alg0.l3.m2.3"><semantics id="alg0.l3.m2.3a"><mrow id="alg0.l3.m2.3.4" xref="alg0.l3.m2.3.4.cmml"><mrow id="alg0.l3.m2.3.4.2" xref="alg0.l3.m2.3.4.2.cmml"><msup id="alg0.l3.m2.3.4.2.2" xref="alg0.l3.m2.3.4.2.2.cmml"><mi id="alg0.l3.m2.3.4.2.2.2" xref="alg0.l3.m2.3.4.2.2.2.cmml">P</mi><mo id="alg0.l3.m2.3.4.2.2.3" xref="alg0.l3.m2.3.4.2.2.3.cmml">∗</mo></msup><mo id="alg0.l3.m2.3.4.2.1" xref="alg0.l3.m2.3.4.2.1.cmml">⁢</mo><mrow id="alg0.l3.m2.3.4.2.3.2" xref="alg0.l3.m2.3.4.2.cmml"><mo id="alg0.l3.m2.3.4.2.3.2.1" stretchy="false" xref="alg0.l3.m2.3.4.2.cmml">(</mo><mi id="alg0.l3.m2.1.1" xref="alg0.l3.m2.1.1.cmml">a</mi><mo id="alg0.l3.m2.3.4.2.3.2.2" stretchy="false" xref="alg0.l3.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="alg0.l3.m2.3.4.1" xref="alg0.l3.m2.3.4.1.cmml">∈</mo><mrow id="alg0.l3.m2.3.4.3.2" xref="alg0.l3.m2.3.4.3.1.cmml"><mo id="alg0.l3.m2.3.4.3.2.1" stretchy="false" xref="alg0.l3.m2.3.4.3.1.cmml">[</mo><mn id="alg0.l3.m2.2.2" xref="alg0.l3.m2.2.2.cmml">0</mn><mo id="alg0.l3.m2.3.4.3.2.2" xref="alg0.l3.m2.3.4.3.1.cmml">,</mo><mn id="alg0.l3.m2.3.3" xref="alg0.l3.m2.3.3.cmml">1</mn><mo id="alg0.l3.m2.3.4.3.2.3" stretchy="false" xref="alg0.l3.m2.3.4.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg0.l3.m2.3b"><apply id="alg0.l3.m2.3.4.cmml" xref="alg0.l3.m2.3.4"><in id="alg0.l3.m2.3.4.1.cmml" xref="alg0.l3.m2.3.4.1"></in><apply id="alg0.l3.m2.3.4.2.cmml" xref="alg0.l3.m2.3.4.2"><times id="alg0.l3.m2.3.4.2.1.cmml" xref="alg0.l3.m2.3.4.2.1"></times><apply id="alg0.l3.m2.3.4.2.2.cmml" xref="alg0.l3.m2.3.4.2.2"><csymbol cd="ambiguous" id="alg0.l3.m2.3.4.2.2.1.cmml" xref="alg0.l3.m2.3.4.2.2">superscript</csymbol><ci id="alg0.l3.m2.3.4.2.2.2.cmml" xref="alg0.l3.m2.3.4.2.2.2">𝑃</ci><times id="alg0.l3.m2.3.4.2.2.3.cmml" xref="alg0.l3.m2.3.4.2.2.3"></times></apply><ci id="alg0.l3.m2.1.1.cmml" xref="alg0.l3.m2.1.1">𝑎</ci></apply><interval closure="closed" id="alg0.l3.m2.3.4.3.1.cmml" xref="alg0.l3.m2.3.4.3.2"><cn id="alg0.l3.m2.2.2.cmml" type="integer" xref="alg0.l3.m2.2.2">0</cn><cn id="alg0.l3.m2.3.3.cmml" type="integer" xref="alg0.l3.m2.3.3">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l3.m2.3c">P^{*}(a)\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="alg0.l3.m2.3d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a ) ∈ [ 0 , 1 ]</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg0.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg0.l4.1.1.1" style="font-size:80%;">4:</span></span> such that the agent plays <math alttext="a" class="ltx_Math" display="inline" id="alg0.l4.m1.1"><semantics id="alg0.l4.m1.1a"><mi id="alg0.l4.m1.1.1" xref="alg0.l4.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="alg0.l4.m1.1b"><ci id="alg0.l4.m1.1.1.cmml" xref="alg0.l4.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l4.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="alg0.l4.m1.1d">italic_a</annotation></semantics></math> when given payment <math alttext="P^{*}(a)" class="ltx_Math" display="inline" id="alg0.l4.m2.1"><semantics id="alg0.l4.m2.1a"><mrow id="alg0.l4.m2.1.2" xref="alg0.l4.m2.1.2.cmml"><msup id="alg0.l4.m2.1.2.2" xref="alg0.l4.m2.1.2.2.cmml"><mi id="alg0.l4.m2.1.2.2.2" xref="alg0.l4.m2.1.2.2.2.cmml">P</mi><mo id="alg0.l4.m2.1.2.2.3" xref="alg0.l4.m2.1.2.2.3.cmml">∗</mo></msup><mo id="alg0.l4.m2.1.2.1" xref="alg0.l4.m2.1.2.1.cmml">⁢</mo><mrow id="alg0.l4.m2.1.2.3.2" xref="alg0.l4.m2.1.2.cmml"><mo id="alg0.l4.m2.1.2.3.2.1" stretchy="false" xref="alg0.l4.m2.1.2.cmml">(</mo><mi id="alg0.l4.m2.1.1" xref="alg0.l4.m2.1.1.cmml">a</mi><mo id="alg0.l4.m2.1.2.3.2.2" stretchy="false" xref="alg0.l4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg0.l4.m2.1b"><apply id="alg0.l4.m2.1.2.cmml" xref="alg0.l4.m2.1.2"><times id="alg0.l4.m2.1.2.1.cmml" xref="alg0.l4.m2.1.2.1"></times><apply id="alg0.l4.m2.1.2.2.cmml" xref="alg0.l4.m2.1.2.2"><csymbol cd="ambiguous" id="alg0.l4.m2.1.2.2.1.cmml" xref="alg0.l4.m2.1.2.2">superscript</csymbol><ci id="alg0.l4.m2.1.2.2.2.cmml" xref="alg0.l4.m2.1.2.2.2">𝑃</ci><times id="alg0.l4.m2.1.2.2.3.cmml" xref="alg0.l4.m2.1.2.2.3"></times></apply><ci id="alg0.l4.m2.1.1.cmml" xref="alg0.l4.m2.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l4.m2.1c">P^{*}(a)</annotation><annotation encoding="application/x-llamapun" id="alg0.l4.m2.1d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> for action <math alttext="a" class="ltx_Math" display="inline" id="alg0.l4.m3.1"><semantics id="alg0.l4.m3.1a"><mi id="alg0.l4.m3.1.1" xref="alg0.l4.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="alg0.l4.m3.1b"><ci id="alg0.l4.m3.1.1.cmml" xref="alg0.l4.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l4.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="alg0.l4.m3.1d">italic_a</annotation></semantics></math> and <math alttext="0" class="ltx_Math" display="inline" id="alg0.l4.m4.1"><semantics id="alg0.l4.m4.1a"><mn id="alg0.l4.m4.1.1" xref="alg0.l4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="alg0.l4.m4.1b"><cn id="alg0.l4.m4.1.1.cmml" type="integer" xref="alg0.l4.m4.1.1">0</cn></annotation-xml></semantics></math> elsewhere </div> <div class="ltx_listingline" id="alg0.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg0.l5.1.1.1" style="font-size:80%;">5:</span></span><span class="ltx_text ltx_font_bold" id="alg0.l5.2">return</span> <math alttext="\tilde{U}:=-P^{*}" class="ltx_Math" display="inline" id="alg0.l5.m1.1"><semantics id="alg0.l5.m1.1a"><mrow id="alg0.l5.m1.1.1" xref="alg0.l5.m1.1.1.cmml"><mover accent="true" id="alg0.l5.m1.1.1.2" xref="alg0.l5.m1.1.1.2.cmml"><mi id="alg0.l5.m1.1.1.2.2" xref="alg0.l5.m1.1.1.2.2.cmml">U</mi><mo id="alg0.l5.m1.1.1.2.1" xref="alg0.l5.m1.1.1.2.1.cmml">~</mo></mover><mo id="alg0.l5.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="alg0.l5.m1.1.1.1.cmml">:=</mo><mrow id="alg0.l5.m1.1.1.3" xref="alg0.l5.m1.1.1.3.cmml"><mo id="alg0.l5.m1.1.1.3a" xref="alg0.l5.m1.1.1.3.cmml">−</mo><msup id="alg0.l5.m1.1.1.3.2" xref="alg0.l5.m1.1.1.3.2.cmml"><mi id="alg0.l5.m1.1.1.3.2.2" xref="alg0.l5.m1.1.1.3.2.2.cmml">P</mi><mo id="alg0.l5.m1.1.1.3.2.3" xref="alg0.l5.m1.1.1.3.2.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg0.l5.m1.1b"><apply id="alg0.l5.m1.1.1.cmml" xref="alg0.l5.m1.1.1"><csymbol cd="latexml" id="alg0.l5.m1.1.1.1.cmml" xref="alg0.l5.m1.1.1.1">assign</csymbol><apply id="alg0.l5.m1.1.1.2.cmml" xref="alg0.l5.m1.1.1.2"><ci id="alg0.l5.m1.1.1.2.1.cmml" xref="alg0.l5.m1.1.1.2.1">~</ci><ci id="alg0.l5.m1.1.1.2.2.cmml" xref="alg0.l5.m1.1.1.2.2">𝑈</ci></apply><apply id="alg0.l5.m1.1.1.3.cmml" xref="alg0.l5.m1.1.1.3"><minus id="alg0.l5.m1.1.1.3.1.cmml" xref="alg0.l5.m1.1.1.3"></minus><apply id="alg0.l5.m1.1.1.3.2.cmml" xref="alg0.l5.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg0.l5.m1.1.1.3.2.1.cmml" xref="alg0.l5.m1.1.1.3.2">superscript</csymbol><ci id="alg0.l5.m1.1.1.3.2.2.cmml" xref="alg0.l5.m1.1.1.3.2.2">𝑃</ci><times id="alg0.l5.m1.1.1.3.2.3.cmml" xref="alg0.l5.m1.1.1.3.2.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l5.m1.1c">\tilde{U}:=-P^{*}</annotation><annotation encoding="application/x-llamapun" id="alg0.l5.m1.1d">over~ start_ARG italic_U end_ARG := - italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.3.1.1">Algorithm 1</span> </span> Principal’s algorithm for learning a single-agent game in the optimal action model</figcaption> </figure> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.1">We thus have the following result:</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><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.3"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2503.01976v1#alg1" title="In 4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.3.3"> <math alttext="\varepsilon" class="ltx_Math" display="inline" 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xref="S4.Thmtheorem1.p1.2.2.m2.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.2.m2.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">roman_Γ</annotation></semantics></math> using <math alttext="{\mathcal{O}}(m\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.3.m3.3"><semantics id="S4.Thmtheorem1.p1.3.3.m3.3a"><mrow id="S4.Thmtheorem1.p1.3.3.m3.3.3" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.3.3.m3.3.3.3" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.3.cmml">𝒪</mi><mo id="S4.Thmtheorem1.p1.3.3.m3.3.3.2" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.2.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.cmml"><mo id="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.2" stretchy="false" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1" 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xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml">/</mo><mi id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.Thmtheorem1.p1.3.3.m3.2.2.4.1.2" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.3" stretchy="false" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.3.m3.3b"><apply id="S4.Thmtheorem1.p1.3.3.m3.3.3.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.3.3"><times id="S4.Thmtheorem1.p1.3.3.m3.3.3.2.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.2"></times><ci id="S4.Thmtheorem1.p1.3.3.m3.3.3.3.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.3">𝒪</ci><apply id="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1"><times id="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.1"></times><ci id="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.3.3.1.1.1.2">𝑚</ci><apply id="S4.Thmtheorem1.p1.3.3.m3.2.2.3.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.4"><log id="S4.Thmtheorem1.p1.3.3.m3.2.2.3.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2"></log><apply id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"><divide id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1"></divide><cn id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2">1</cn><ci id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.3.m3.3c">{\mathcal{O}}(m\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.3.m3.3d">caligraphic_O ( italic_m roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> rounds.</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>The multi-agent case</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.4">We now extend the results of the previous section to the multi-agent case. Intuitively, our multi-agent algorithms will work as follows. We will learn the agents’ utility functions one by one. For each agent <math alttext="i" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_i</annotation></semantics></math> and each strategy profile <math alttext="a_{-i}\in A_{-i}" 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"><msub id="S4.SS2.p1.2.m2.1.1.2" xref="S4.SS2.p1.2.m2.1.1.2.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2.2" xref="S4.SS2.p1.2.m2.1.1.2.2.cmml">a</mi><mrow id="S4.SS2.p1.2.m2.1.1.2.3" xref="S4.SS2.p1.2.m2.1.1.2.3.cmml"><mo id="S4.SS2.p1.2.m2.1.1.2.3a" xref="S4.SS2.p1.2.m2.1.1.2.3.cmml">−</mo><mi id="S4.SS2.p1.2.m2.1.1.2.3.2" xref="S4.SS2.p1.2.m2.1.1.2.3.2.cmml">i</mi></mrow></msub><mo id="S4.SS2.p1.2.m2.1.1.1" xref="S4.SS2.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml"><mi id="S4.SS2.p1.2.m2.1.1.3.2" xref="S4.SS2.p1.2.m2.1.1.3.2.cmml">A</mi><mrow id="S4.SS2.p1.2.m2.1.1.3.3" xref="S4.SS2.p1.2.m2.1.1.3.3.cmml"><mo id="S4.SS2.p1.2.m2.1.1.3.3a" xref="S4.SS2.p1.2.m2.1.1.3.3.cmml">−</mo><mi id="S4.SS2.p1.2.m2.1.1.3.3.2" xref="S4.SS2.p1.2.m2.1.1.3.3.2.cmml">i</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><in id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1.1"></in><apply id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.2.1.cmml" xref="S4.SS2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.2.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2.2">𝑎</ci><apply id="S4.SS2.p1.2.m2.1.1.2.3.cmml" xref="S4.SS2.p1.2.m2.1.1.2.3"><minus id="S4.SS2.p1.2.m2.1.1.2.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.2.3"></minus><ci id="S4.SS2.p1.2.m2.1.1.2.3.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2.3.2">𝑖</ci></apply></apply><apply id="S4.SS2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.p1.2.m2.1.1.3.2">𝐴</ci><apply id="S4.SS2.p1.2.m2.1.1.3.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3.3"><minus id="S4.SS2.p1.2.m2.1.1.3.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.3.3"></minus><ci id="S4.SS2.p1.2.m2.1.1.3.3.2.cmml" xref="S4.SS2.p1.2.m2.1.1.3.3.2">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">a_{-i}\in A_{-i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we will induce the other agents to play according to <math alttext="a_{-i}" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><msub id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml"><mi id="S4.SS2.p1.3.m3.1.1.2" xref="S4.SS2.p1.3.m3.1.1.2.cmml">a</mi><mrow id="S4.SS2.p1.3.m3.1.1.3" xref="S4.SS2.p1.3.m3.1.1.3.cmml"><mo id="S4.SS2.p1.3.m3.1.1.3a" xref="S4.SS2.p1.3.m3.1.1.3.cmml">−</mo><mi id="S4.SS2.p1.3.m3.1.1.3.2" xref="S4.SS2.p1.3.m3.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><apply id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2">𝑎</ci><apply id="S4.SS2.p1.3.m3.1.1.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3"><minus id="S4.SS2.p1.3.m3.1.1.3.1.cmml" xref="S4.SS2.p1.3.m3.1.1.3"></minus><ci id="S4.SS2.p1.3.m3.1.1.3.2.cmml" xref="S4.SS2.p1.3.m3.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">a_{-i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for some number of rounds, by giving them a large payment. During these rounds, we will use the single-agent learning algorithm from the previous section to learn the values <math alttext="U_{i}(\cdot,a_{-i})" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.2"><semantics id="S4.SS2.p1.4.m4.2a"><mrow id="S4.SS2.p1.4.m4.2.2" xref="S4.SS2.p1.4.m4.2.2.cmml"><msub id="S4.SS2.p1.4.m4.2.2.3" xref="S4.SS2.p1.4.m4.2.2.3.cmml"><mi id="S4.SS2.p1.4.m4.2.2.3.2" xref="S4.SS2.p1.4.m4.2.2.3.2.cmml">U</mi><mi id="S4.SS2.p1.4.m4.2.2.3.3" xref="S4.SS2.p1.4.m4.2.2.3.3.cmml">i</mi></msub><mo id="S4.SS2.p1.4.m4.2.2.2" xref="S4.SS2.p1.4.m4.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.p1.4.m4.2.2.1.1" xref="S4.SS2.p1.4.m4.2.2.1.2.cmml"><mo id="S4.SS2.p1.4.m4.2.2.1.1.2" stretchy="false" xref="S4.SS2.p1.4.m4.2.2.1.2.cmml">(</mo><mo id="S4.SS2.p1.4.m4.1.1" lspace="0em" rspace="0em" xref="S4.SS2.p1.4.m4.1.1.cmml">⋅</mo><mo id="S4.SS2.p1.4.m4.2.2.1.1.3" xref="S4.SS2.p1.4.m4.2.2.1.2.cmml">,</mo><msub id="S4.SS2.p1.4.m4.2.2.1.1.1" xref="S4.SS2.p1.4.m4.2.2.1.1.1.cmml"><mi id="S4.SS2.p1.4.m4.2.2.1.1.1.2" xref="S4.SS2.p1.4.m4.2.2.1.1.1.2.cmml">a</mi><mrow id="S4.SS2.p1.4.m4.2.2.1.1.1.3" xref="S4.SS2.p1.4.m4.2.2.1.1.1.3.cmml"><mo id="S4.SS2.p1.4.m4.2.2.1.1.1.3a" xref="S4.SS2.p1.4.m4.2.2.1.1.1.3.cmml">−</mo><mi id="S4.SS2.p1.4.m4.2.2.1.1.1.3.2" xref="S4.SS2.p1.4.m4.2.2.1.1.1.3.2.cmml">i</mi></mrow></msub><mo id="S4.SS2.p1.4.m4.2.2.1.1.4" stretchy="false" xref="S4.SS2.p1.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.2b"><apply id="S4.SS2.p1.4.m4.2.2.cmml" xref="S4.SS2.p1.4.m4.2.2"><times id="S4.SS2.p1.4.m4.2.2.2.cmml" xref="S4.SS2.p1.4.m4.2.2.2"></times><apply id="S4.SS2.p1.4.m4.2.2.3.cmml" xref="S4.SS2.p1.4.m4.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.2.2.3.1.cmml" xref="S4.SS2.p1.4.m4.2.2.3">subscript</csymbol><ci id="S4.SS2.p1.4.m4.2.2.3.2.cmml" xref="S4.SS2.p1.4.m4.2.2.3.2">𝑈</ci><ci id="S4.SS2.p1.4.m4.2.2.3.3.cmml" xref="S4.SS2.p1.4.m4.2.2.3.3">𝑖</ci></apply><interval closure="open" id="S4.SS2.p1.4.m4.2.2.1.2.cmml" xref="S4.SS2.p1.4.m4.2.2.1.1"><ci id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1">⋅</ci><apply id="S4.SS2.p1.4.m4.2.2.1.1.1.cmml" xref="S4.SS2.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.2.2.1.1.1.1.cmml" xref="S4.SS2.p1.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.p1.4.m4.2.2.1.1.1.2.cmml" xref="S4.SS2.p1.4.m4.2.2.1.1.1.2">𝑎</ci><apply id="S4.SS2.p1.4.m4.2.2.1.1.1.3.cmml" xref="S4.SS2.p1.4.m4.2.2.1.1.1.3"><minus id="S4.SS2.p1.4.m4.2.2.1.1.1.3.1.cmml" xref="S4.SS2.p1.4.m4.2.2.1.1.1.3"></minus><ci id="S4.SS2.p1.4.m4.2.2.1.1.1.3.2.cmml" xref="S4.SS2.p1.4.m4.2.2.1.1.1.3.2">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.2c">U_{i}(\cdot,a_{-i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.2d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">This process naturally takes time proportional to the number of action profiles in the game, which we will define as <math alttext="M:=\prod_{i=1}^{n}m_{i}" 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">M</mi><mo id="S4.SS2.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.111em" xref="S4.SS2.p2.1.m1.1.1.1.cmml">:=</mo><mrow id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml"><msubsup id="S4.SS2.p2.1.m1.1.1.3.1" xref="S4.SS2.p2.1.m1.1.1.3.1.cmml"><mo id="S4.SS2.p2.1.m1.1.1.3.1.2.2" xref="S4.SS2.p2.1.m1.1.1.3.1.2.2.cmml">∏</mo><mrow id="S4.SS2.p2.1.m1.1.1.3.1.2.3" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3.cmml"><mi id="S4.SS2.p2.1.m1.1.1.3.1.2.3.2" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3.2.cmml">i</mi><mo id="S4.SS2.p2.1.m1.1.1.3.1.2.3.1" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.SS2.p2.1.m1.1.1.3.1.2.3.3" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S4.SS2.p2.1.m1.1.1.3.1.3" xref="S4.SS2.p2.1.m1.1.1.3.1.3.cmml">n</mi></msubsup><msub id="S4.SS2.p2.1.m1.1.1.3.2" xref="S4.SS2.p2.1.m1.1.1.3.2.cmml"><mi id="S4.SS2.p2.1.m1.1.1.3.2.2" xref="S4.SS2.p2.1.m1.1.1.3.2.2.cmml">m</mi><mi id="S4.SS2.p2.1.m1.1.1.3.2.3" xref="S4.SS2.p2.1.m1.1.1.3.2.3.cmml">i</mi></msub></mrow></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"><csymbol cd="latexml" id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1">assign</csymbol><ci id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2">𝑀</ci><apply id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3"><apply id="S4.SS2.p2.1.m1.1.1.3.1.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.3.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1">superscript</csymbol><apply id="S4.SS2.p2.1.m1.1.1.3.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.3.1.2.1.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1">subscript</csymbol><csymbol cd="latexml" id="S4.SS2.p2.1.m1.1.1.3.1.2.2.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1.2.2">product</csymbol><apply id="S4.SS2.p2.1.m1.1.1.3.1.2.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3"><eq id="S4.SS2.p2.1.m1.1.1.3.1.2.3.1.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3.1"></eq><ci id="S4.SS2.p2.1.m1.1.1.3.1.2.3.2.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3.2">𝑖</ci><cn id="S4.SS2.p2.1.m1.1.1.3.1.2.3.3.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1.3.1.2.3.3">1</cn></apply></apply><ci id="S4.SS2.p2.1.m1.1.1.3.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3.1.3">𝑛</ci></apply><apply id="S4.SS2.p2.1.m1.1.1.3.2.cmml" xref="S4.SS2.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.3.2.1.cmml" xref="S4.SS2.p2.1.m1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.3.2.2.cmml" xref="S4.SS2.p2.1.m1.1.1.3.2.2">𝑚</ci><ci id="S4.SS2.p2.1.m1.1.1.3.2.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">M:=\prod_{i=1}^{n}m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_M := ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Although this construction seems naïve, we will show, as in the single-agent case, that it is essentially optimal. The above idea is formalized in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg2" title="In 4.2 The multi-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg2"> <div class="ltx_listing ltx_listing" id="alg2.2"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l1.2">for</span> each agent <math alttext="i=1,\dots,n" class="ltx_Math" display="inline" id="alg1.l1.m1.3"><semantics id="alg1.l1.m1.3a"><mrow id="alg1.l1.m1.3.4" xref="alg1.l1.m1.3.4.cmml"><mi id="alg1.l1.m1.3.4.2" xref="alg1.l1.m1.3.4.2.cmml">i</mi><mo id="alg1.l1.m1.3.4.1" xref="alg1.l1.m1.3.4.1.cmml">=</mo><mrow id="alg1.l1.m1.3.4.3.2" xref="alg1.l1.m1.3.4.3.1.cmml"><mn id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml">1</mn><mo id="alg1.l1.m1.3.4.3.2.1" xref="alg1.l1.m1.3.4.3.1.cmml">,</mo><mi id="alg1.l1.m1.2.2" mathvariant="normal" xref="alg1.l1.m1.2.2.cmml">…</mi><mo id="alg1.l1.m1.3.4.3.2.2" xref="alg1.l1.m1.3.4.3.1.cmml">,</mo><mi id="alg1.l1.m1.3.3" xref="alg1.l1.m1.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.3b"><apply id="alg1.l1.m1.3.4.cmml" xref="alg1.l1.m1.3.4"><eq id="alg1.l1.m1.3.4.1.cmml" xref="alg1.l1.m1.3.4.1"></eq><ci id="alg1.l1.m1.3.4.2.cmml" xref="alg1.l1.m1.3.4.2">𝑖</ci><list id="alg1.l1.m1.3.4.3.1.cmml" xref="alg1.l1.m1.3.4.3.2"><cn id="alg1.l1.m1.1.1.cmml" type="integer" xref="alg1.l1.m1.1.1">1</cn><ci id="alg1.l1.m1.2.2.cmml" xref="alg1.l1.m1.2.2">…</ci><ci id="alg1.l1.m1.3.3.cmml" xref="alg1.l1.m1.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.3c">i=1,\dots,n</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.3d">italic_i = 1 , … , italic_n</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l1.3">do</span> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l2.2">for</span> each action profile <math alttext="a_{-i}\in A_{-i}" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><mrow id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><msub id="alg1.l2.m1.1.1.2" xref="alg1.l2.m1.1.1.2.cmml"><mi id="alg1.l2.m1.1.1.2.2" xref="alg1.l2.m1.1.1.2.2.cmml">a</mi><mrow id="alg1.l2.m1.1.1.2.3" xref="alg1.l2.m1.1.1.2.3.cmml"><mo id="alg1.l2.m1.1.1.2.3a" xref="alg1.l2.m1.1.1.2.3.cmml">−</mo><mi id="alg1.l2.m1.1.1.2.3.2" xref="alg1.l2.m1.1.1.2.3.2.cmml">i</mi></mrow></msub><mo id="alg1.l2.m1.1.1.1" xref="alg1.l2.m1.1.1.1.cmml">∈</mo><msub id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml"><mi id="alg1.l2.m1.1.1.3.2" xref="alg1.l2.m1.1.1.3.2.cmml">A</mi><mrow id="alg1.l2.m1.1.1.3.3" xref="alg1.l2.m1.1.1.3.3.cmml"><mo id="alg1.l2.m1.1.1.3.3a" xref="alg1.l2.m1.1.1.3.3.cmml">−</mo><mi id="alg1.l2.m1.1.1.3.3.2" xref="alg1.l2.m1.1.1.3.3.2.cmml">i</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><in id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1.1"></in><apply id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.2.1.cmml" xref="alg1.l2.m1.1.1.2">subscript</csymbol><ci id="alg1.l2.m1.1.1.2.2.cmml" xref="alg1.l2.m1.1.1.2.2">𝑎</ci><apply id="alg1.l2.m1.1.1.2.3.cmml" xref="alg1.l2.m1.1.1.2.3"><minus id="alg1.l2.m1.1.1.2.3.1.cmml" xref="alg1.l2.m1.1.1.2.3"></minus><ci id="alg1.l2.m1.1.1.2.3.2.cmml" xref="alg1.l2.m1.1.1.2.3.2">𝑖</ci></apply></apply><apply id="alg1.l2.m1.1.1.3.cmml" xref="alg1.l2.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.3.1.cmml" xref="alg1.l2.m1.1.1.3">subscript</csymbol><ci id="alg1.l2.m1.1.1.3.2.cmml" xref="alg1.l2.m1.1.1.3.2">𝐴</ci><apply id="alg1.l2.m1.1.1.3.3.cmml" xref="alg1.l2.m1.1.1.3.3"><minus id="alg1.l2.m1.1.1.3.3.1.cmml" xref="alg1.l2.m1.1.1.3.3"></minus><ci id="alg1.l2.m1.1.1.3.3.2.cmml" xref="alg1.l2.m1.1.1.3.3.2">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">a_{-i}\in A_{-i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l2.3">do</span> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.1.1.1" style="font-size:80%;">3:</span></span> principal runs <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg1" title="In 4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> to learn an estimate <math alttext="\tilde{U}_{i}(\cdot,a_{-i})" class="ltx_Math" display="inline" id="alg1.l3.m1.2"><semantics id="alg1.l3.m1.2a"><mrow id="alg1.l3.m1.2.2" xref="alg1.l3.m1.2.2.cmml"><msub id="alg1.l3.m1.2.2.3" xref="alg1.l3.m1.2.2.3.cmml"><mover accent="true" id="alg1.l3.m1.2.2.3.2" xref="alg1.l3.m1.2.2.3.2.cmml"><mi id="alg1.l3.m1.2.2.3.2.2" xref="alg1.l3.m1.2.2.3.2.2.cmml">U</mi><mo id="alg1.l3.m1.2.2.3.2.1" xref="alg1.l3.m1.2.2.3.2.1.cmml">~</mo></mover><mi id="alg1.l3.m1.2.2.3.3" xref="alg1.l3.m1.2.2.3.3.cmml">i</mi></msub><mo id="alg1.l3.m1.2.2.2" xref="alg1.l3.m1.2.2.2.cmml">⁢</mo><mrow id="alg1.l3.m1.2.2.1.1" xref="alg1.l3.m1.2.2.1.2.cmml"><mo id="alg1.l3.m1.2.2.1.1.2" stretchy="false" xref="alg1.l3.m1.2.2.1.2.cmml">(</mo><mo id="alg1.l3.m1.1.1" lspace="0em" rspace="0em" xref="alg1.l3.m1.1.1.cmml">⋅</mo><mo id="alg1.l3.m1.2.2.1.1.3" xref="alg1.l3.m1.2.2.1.2.cmml">,</mo><msub id="alg1.l3.m1.2.2.1.1.1" xref="alg1.l3.m1.2.2.1.1.1.cmml"><mi id="alg1.l3.m1.2.2.1.1.1.2" xref="alg1.l3.m1.2.2.1.1.1.2.cmml">a</mi><mrow id="alg1.l3.m1.2.2.1.1.1.3" xref="alg1.l3.m1.2.2.1.1.1.3.cmml"><mo id="alg1.l3.m1.2.2.1.1.1.3a" xref="alg1.l3.m1.2.2.1.1.1.3.cmml">−</mo><mi id="alg1.l3.m1.2.2.1.1.1.3.2" xref="alg1.l3.m1.2.2.1.1.1.3.2.cmml">i</mi></mrow></msub><mo id="alg1.l3.m1.2.2.1.1.4" stretchy="false" xref="alg1.l3.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.2b"><apply id="alg1.l3.m1.2.2.cmml" xref="alg1.l3.m1.2.2"><times id="alg1.l3.m1.2.2.2.cmml" xref="alg1.l3.m1.2.2.2"></times><apply id="alg1.l3.m1.2.2.3.cmml" xref="alg1.l3.m1.2.2.3"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.3.1.cmml" xref="alg1.l3.m1.2.2.3">subscript</csymbol><apply id="alg1.l3.m1.2.2.3.2.cmml" xref="alg1.l3.m1.2.2.3.2"><ci id="alg1.l3.m1.2.2.3.2.1.cmml" xref="alg1.l3.m1.2.2.3.2.1">~</ci><ci id="alg1.l3.m1.2.2.3.2.2.cmml" xref="alg1.l3.m1.2.2.3.2.2">𝑈</ci></apply><ci id="alg1.l3.m1.2.2.3.3.cmml" xref="alg1.l3.m1.2.2.3.3">𝑖</ci></apply><interval closure="open" id="alg1.l3.m1.2.2.1.2.cmml" xref="alg1.l3.m1.2.2.1.1"><ci id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1">⋅</ci><apply id="alg1.l3.m1.2.2.1.1.1.cmml" xref="alg1.l3.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.1.1.1.1.cmml" xref="alg1.l3.m1.2.2.1.1.1">subscript</csymbol><ci id="alg1.l3.m1.2.2.1.1.1.2.cmml" xref="alg1.l3.m1.2.2.1.1.1.2">𝑎</ci><apply id="alg1.l3.m1.2.2.1.1.1.3.cmml" xref="alg1.l3.m1.2.2.1.1.1.3"><minus id="alg1.l3.m1.2.2.1.1.1.3.1.cmml" xref="alg1.l3.m1.2.2.1.1.1.3"></minus><ci id="alg1.l3.m1.2.2.1.1.1.3.2.cmml" xref="alg1.l3.m1.2.2.1.1.1.3.2">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.2c">\tilde{U}_{i}(\cdot,a_{-i})</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.2d">over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> of <math alttext="U_{i}(\cdot,a_{-i})" class="ltx_Math" display="inline" id="alg1.l3.m2.2"><semantics id="alg1.l3.m2.2a"><mrow id="alg1.l3.m2.2.2" xref="alg1.l3.m2.2.2.cmml"><msub id="alg1.l3.m2.2.2.3" xref="alg1.l3.m2.2.2.3.cmml"><mi id="alg1.l3.m2.2.2.3.2" xref="alg1.l3.m2.2.2.3.2.cmml">U</mi><mi id="alg1.l3.m2.2.2.3.3" xref="alg1.l3.m2.2.2.3.3.cmml">i</mi></msub><mo id="alg1.l3.m2.2.2.2" xref="alg1.l3.m2.2.2.2.cmml">⁢</mo><mrow id="alg1.l3.m2.2.2.1.1" xref="alg1.l3.m2.2.2.1.2.cmml"><mo id="alg1.l3.m2.2.2.1.1.2" stretchy="false" xref="alg1.l3.m2.2.2.1.2.cmml">(</mo><mo id="alg1.l3.m2.1.1" lspace="0em" rspace="0em" xref="alg1.l3.m2.1.1.cmml">⋅</mo><mo id="alg1.l3.m2.2.2.1.1.3" xref="alg1.l3.m2.2.2.1.2.cmml">,</mo><msub id="alg1.l3.m2.2.2.1.1.1" xref="alg1.l3.m2.2.2.1.1.1.cmml"><mi id="alg1.l3.m2.2.2.1.1.1.2" xref="alg1.l3.m2.2.2.1.1.1.2.cmml">a</mi><mrow id="alg1.l3.m2.2.2.1.1.1.3" xref="alg1.l3.m2.2.2.1.1.1.3.cmml"><mo id="alg1.l3.m2.2.2.1.1.1.3a" xref="alg1.l3.m2.2.2.1.1.1.3.cmml">−</mo><mi id="alg1.l3.m2.2.2.1.1.1.3.2" xref="alg1.l3.m2.2.2.1.1.1.3.2.cmml">i</mi></mrow></msub><mo id="alg1.l3.m2.2.2.1.1.4" stretchy="false" xref="alg1.l3.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m2.2b"><apply id="alg1.l3.m2.2.2.cmml" xref="alg1.l3.m2.2.2"><times id="alg1.l3.m2.2.2.2.cmml" xref="alg1.l3.m2.2.2.2"></times><apply id="alg1.l3.m2.2.2.3.cmml" xref="alg1.l3.m2.2.2.3"><csymbol cd="ambiguous" id="alg1.l3.m2.2.2.3.1.cmml" xref="alg1.l3.m2.2.2.3">subscript</csymbol><ci id="alg1.l3.m2.2.2.3.2.cmml" xref="alg1.l3.m2.2.2.3.2">𝑈</ci><ci id="alg1.l3.m2.2.2.3.3.cmml" xref="alg1.l3.m2.2.2.3.3">𝑖</ci></apply><interval closure="open" id="alg1.l3.m2.2.2.1.2.cmml" xref="alg1.l3.m2.2.2.1.1"><ci id="alg1.l3.m2.1.1.cmml" xref="alg1.l3.m2.1.1">⋅</ci><apply id="alg1.l3.m2.2.2.1.1.1.cmml" xref="alg1.l3.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m2.2.2.1.1.1.1.cmml" xref="alg1.l3.m2.2.2.1.1.1">subscript</csymbol><ci id="alg1.l3.m2.2.2.1.1.1.2.cmml" xref="alg1.l3.m2.2.2.1.1.1.2">𝑎</ci><apply id="alg1.l3.m2.2.2.1.1.1.3.cmml" xref="alg1.l3.m2.2.2.1.1.1.3"><minus id="alg1.l3.m2.2.2.1.1.1.3.1.cmml" xref="alg1.l3.m2.2.2.1.1.1.3"></minus><ci id="alg1.l3.m2.2.2.1.1.1.3.2.cmml" xref="alg1.l3.m2.2.2.1.1.1.3.2">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m2.2c">U_{i}(\cdot,a_{-i})</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m2.2d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="alg1.l3.m3.1"><semantics id="alg1.l3.m3.1a"><mi id="alg1.l3.m3.1.1" xref="alg1.l3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="alg1.l3.m3.1b"><ci id="alg1.l3.m3.1.1.cmml" xref="alg1.l3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m3.1d">italic_ε</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span> (during these rounds, principal sets <math alttext="P_{j}^{t}(a_{j}^{\prime})=2\cdot\{a^{\prime}_{j}=a_{j}\}" class="ltx_Math" display="inline" id="alg1.l4.m1.2"><semantics id="alg1.l4.m1.2a"><mrow id="alg1.l4.m1.2.2" xref="alg1.l4.m1.2.2.cmml"><mrow id="alg1.l4.m1.1.1.1" xref="alg1.l4.m1.1.1.1.cmml"><msubsup id="alg1.l4.m1.1.1.1.3" xref="alg1.l4.m1.1.1.1.3.cmml"><mi id="alg1.l4.m1.1.1.1.3.2.2" xref="alg1.l4.m1.1.1.1.3.2.2.cmml">P</mi><mi id="alg1.l4.m1.1.1.1.3.2.3" xref="alg1.l4.m1.1.1.1.3.2.3.cmml">j</mi><mi id="alg1.l4.m1.1.1.1.3.3" xref="alg1.l4.m1.1.1.1.3.3.cmml">t</mi></msubsup><mo id="alg1.l4.m1.1.1.1.2" xref="alg1.l4.m1.1.1.1.2.cmml">⁢</mo><mrow id="alg1.l4.m1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.1.cmml"><mo id="alg1.l4.m1.1.1.1.1.1.2" stretchy="false" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">(</mo><msubsup id="alg1.l4.m1.1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.1.cmml"><mi id="alg1.l4.m1.1.1.1.1.1.1.2.2" xref="alg1.l4.m1.1.1.1.1.1.1.2.2.cmml">a</mi><mi id="alg1.l4.m1.1.1.1.1.1.1.2.3" xref="alg1.l4.m1.1.1.1.1.1.1.2.3.cmml">j</mi><mo id="alg1.l4.m1.1.1.1.1.1.1.3" xref="alg1.l4.m1.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="alg1.l4.m1.1.1.1.1.1.3" stretchy="false" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l4.m1.2.2.3" xref="alg1.l4.m1.2.2.3.cmml">=</mo><mrow id="alg1.l4.m1.2.2.2" xref="alg1.l4.m1.2.2.2.cmml"><mn id="alg1.l4.m1.2.2.2.3" xref="alg1.l4.m1.2.2.2.3.cmml">2</mn><mo id="alg1.l4.m1.2.2.2.2" lspace="0.222em" rspace="0.222em" xref="alg1.l4.m1.2.2.2.2.cmml">⋅</mo><mrow id="alg1.l4.m1.2.2.2.1.1" xref="alg1.l4.m1.2.2.2.1.2.cmml"><mo id="alg1.l4.m1.2.2.2.1.1.2" stretchy="false" xref="alg1.l4.m1.2.2.2.1.2.cmml">{</mo><mrow id="alg1.l4.m1.2.2.2.1.1.1" xref="alg1.l4.m1.2.2.2.1.1.1.cmml"><msubsup id="alg1.l4.m1.2.2.2.1.1.1.2" xref="alg1.l4.m1.2.2.2.1.1.1.2.cmml"><mi id="alg1.l4.m1.2.2.2.1.1.1.2.2.2" xref="alg1.l4.m1.2.2.2.1.1.1.2.2.2.cmml">a</mi><mi id="alg1.l4.m1.2.2.2.1.1.1.2.3" xref="alg1.l4.m1.2.2.2.1.1.1.2.3.cmml">j</mi><mo id="alg1.l4.m1.2.2.2.1.1.1.2.2.3" xref="alg1.l4.m1.2.2.2.1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="alg1.l4.m1.2.2.2.1.1.1.1" xref="alg1.l4.m1.2.2.2.1.1.1.1.cmml">=</mo><msub id="alg1.l4.m1.2.2.2.1.1.1.3" xref="alg1.l4.m1.2.2.2.1.1.1.3.cmml"><mi id="alg1.l4.m1.2.2.2.1.1.1.3.2" xref="alg1.l4.m1.2.2.2.1.1.1.3.2.cmml">a</mi><mi id="alg1.l4.m1.2.2.2.1.1.1.3.3" xref="alg1.l4.m1.2.2.2.1.1.1.3.3.cmml">j</mi></msub></mrow><mo id="alg1.l4.m1.2.2.2.1.1.3" stretchy="false" xref="alg1.l4.m1.2.2.2.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.2b"><apply id="alg1.l4.m1.2.2.cmml" xref="alg1.l4.m1.2.2"><eq id="alg1.l4.m1.2.2.3.cmml" xref="alg1.l4.m1.2.2.3"></eq><apply id="alg1.l4.m1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1"><times id="alg1.l4.m1.1.1.1.2.cmml" xref="alg1.l4.m1.1.1.1.2"></times><apply id="alg1.l4.m1.1.1.1.3.cmml" xref="alg1.l4.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.1.3.1.cmml" xref="alg1.l4.m1.1.1.1.3">superscript</csymbol><apply id="alg1.l4.m1.1.1.1.3.2.cmml" xref="alg1.l4.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.1.3.2.1.cmml" xref="alg1.l4.m1.1.1.1.3">subscript</csymbol><ci id="alg1.l4.m1.1.1.1.3.2.2.cmml" xref="alg1.l4.m1.1.1.1.3.2.2">𝑃</ci><ci id="alg1.l4.m1.1.1.1.3.2.3.cmml" xref="alg1.l4.m1.1.1.1.3.2.3">𝑗</ci></apply><ci id="alg1.l4.m1.1.1.1.3.3.cmml" xref="alg1.l4.m1.1.1.1.3.3">𝑡</ci></apply><apply id="alg1.l4.m1.1.1.1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.1.1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1.1.1">superscript</csymbol><apply id="alg1.l4.m1.1.1.1.1.1.1.2.cmml" xref="alg1.l4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.1.1.1.1.2.1.cmml" xref="alg1.l4.m1.1.1.1.1.1">subscript</csymbol><ci id="alg1.l4.m1.1.1.1.1.1.1.2.2.cmml" xref="alg1.l4.m1.1.1.1.1.1.1.2.2">𝑎</ci><ci id="alg1.l4.m1.1.1.1.1.1.1.2.3.cmml" xref="alg1.l4.m1.1.1.1.1.1.1.2.3">𝑗</ci></apply><ci id="alg1.l4.m1.1.1.1.1.1.1.3.cmml" xref="alg1.l4.m1.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="alg1.l4.m1.2.2.2.cmml" xref="alg1.l4.m1.2.2.2"><ci id="alg1.l4.m1.2.2.2.2.cmml" xref="alg1.l4.m1.2.2.2.2">⋅</ci><cn id="alg1.l4.m1.2.2.2.3.cmml" type="integer" xref="alg1.l4.m1.2.2.2.3">2</cn><set id="alg1.l4.m1.2.2.2.1.2.cmml" xref="alg1.l4.m1.2.2.2.1.1"><apply id="alg1.l4.m1.2.2.2.1.1.1.cmml" xref="alg1.l4.m1.2.2.2.1.1.1"><eq id="alg1.l4.m1.2.2.2.1.1.1.1.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.1"></eq><apply id="alg1.l4.m1.2.2.2.1.1.1.2.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l4.m1.2.2.2.1.1.1.2.1.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.2">subscript</csymbol><apply id="alg1.l4.m1.2.2.2.1.1.1.2.2.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l4.m1.2.2.2.1.1.1.2.2.1.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.2">superscript</csymbol><ci id="alg1.l4.m1.2.2.2.1.1.1.2.2.2.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.2.2.2">𝑎</ci><ci id="alg1.l4.m1.2.2.2.1.1.1.2.2.3.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.2.2.3">′</ci></apply><ci id="alg1.l4.m1.2.2.2.1.1.1.2.3.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.2.3">𝑗</ci></apply><apply id="alg1.l4.m1.2.2.2.1.1.1.3.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l4.m1.2.2.2.1.1.1.3.1.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.3">subscript</csymbol><ci id="alg1.l4.m1.2.2.2.1.1.1.3.2.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.3.2">𝑎</ci><ci id="alg1.l4.m1.2.2.2.1.1.1.3.3.cmml" xref="alg1.l4.m1.2.2.2.1.1.1.3.3">𝑗</ci></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.2c">P_{j}^{t}(a_{j}^{\prime})=2\cdot\{a^{\prime}_{j}=a_{j}\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.2d">italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = 2 ⋅ { italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT }</annotation></semantics></math> for every <math alttext="j\neq i" class="ltx_Math" display="inline" id="alg1.l4.m2.1"><semantics id="alg1.l4.m2.1a"><mrow id="alg1.l4.m2.1.1" xref="alg1.l4.m2.1.1.cmml"><mi id="alg1.l4.m2.1.1.2" xref="alg1.l4.m2.1.1.2.cmml">j</mi><mo id="alg1.l4.m2.1.1.1" xref="alg1.l4.m2.1.1.1.cmml">≠</mo><mi id="alg1.l4.m2.1.1.3" xref="alg1.l4.m2.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m2.1b"><apply id="alg1.l4.m2.1.1.cmml" xref="alg1.l4.m2.1.1"><neq id="alg1.l4.m2.1.1.1.cmml" xref="alg1.l4.m2.1.1.1"></neq><ci id="alg1.l4.m2.1.1.2.cmml" xref="alg1.l4.m2.1.1.2">𝑗</ci><ci id="alg1.l4.m2.1.1.3.cmml" xref="alg1.l4.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m2.1c">j\neq i</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m2.1d">italic_j ≠ italic_i</annotation></semantics></math>) </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l5.5.1.1" style="font-size:80%;">5:</span></span> <math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l5.m1.1"><semantics id="alg1.l5.m1.1a"><mo id="alg1.l5.m1.1.1" mathcolor="#808080" xref="alg1.l5.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.1b"><ci id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.1d">▷</annotation></semantics></math><span class="ltx_text" id="alg1.l5.4" style="color:#808080;"> <math alttext="a_{j}" class="ltx_Math" display="inline" id="alg1.l5.1.m1.1"><semantics id="alg1.l5.1.m1.1a"><msub id="alg1.l5.1.m1.1.1" xref="alg1.l5.1.m1.1.1.cmml"><mi id="alg1.l5.1.m1.1.1.2" mathcolor="#808080" xref="alg1.l5.1.m1.1.1.2.cmml">a</mi><mi id="alg1.l5.1.m1.1.1.3" mathcolor="#808080" xref="alg1.l5.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l5.1.m1.1b"><apply id="alg1.l5.1.m1.1.1.cmml" xref="alg1.l5.1.m1.1.1"><csymbol cd="ambiguous" id="alg1.l5.1.m1.1.1.1.cmml" xref="alg1.l5.1.m1.1.1">subscript</csymbol><ci id="alg1.l5.1.m1.1.1.2.cmml" xref="alg1.l5.1.m1.1.1.2">𝑎</ci><ci id="alg1.l5.1.m1.1.1.3.cmml" xref="alg1.l5.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.1.m1.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is agent <math alttext="j" class="ltx_Math" display="inline" id="alg1.l5.2.m2.1"><semantics id="alg1.l5.2.m2.1a"><mi id="alg1.l5.2.m2.1.1" mathcolor="#808080" xref="alg1.l5.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="alg1.l5.2.m2.1b"><ci id="alg1.l5.2.m2.1.1.cmml" xref="alg1.l5.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.2.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.2.m2.1d">italic_j</annotation></semantics></math>’s action in <math alttext="a_{-i}" class="ltx_Math" display="inline" id="alg1.l5.3.m3.1"><semantics id="alg1.l5.3.m3.1a"><msub id="alg1.l5.3.m3.1.1" xref="alg1.l5.3.m3.1.1.cmml"><mi id="alg1.l5.3.m3.1.1.2" mathcolor="#808080" xref="alg1.l5.3.m3.1.1.2.cmml">a</mi><mrow id="alg1.l5.3.m3.1.1.3" xref="alg1.l5.3.m3.1.1.3.cmml"><mo id="alg1.l5.3.m3.1.1.3a" mathcolor="#808080" xref="alg1.l5.3.m3.1.1.3.cmml">−</mo><mi id="alg1.l5.3.m3.1.1.3.2" mathcolor="#808080" xref="alg1.l5.3.m3.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l5.3.m3.1b"><apply id="alg1.l5.3.m3.1.1.cmml" xref="alg1.l5.3.m3.1.1"><csymbol cd="ambiguous" id="alg1.l5.3.m3.1.1.1.cmml" xref="alg1.l5.3.m3.1.1">subscript</csymbol><ci id="alg1.l5.3.m3.1.1.2.cmml" xref="alg1.l5.3.m3.1.1.2">𝑎</ci><apply id="alg1.l5.3.m3.1.1.3.cmml" xref="alg1.l5.3.m3.1.1.3"><minus id="alg1.l5.3.m3.1.1.3.1.cmml" xref="alg1.l5.3.m3.1.1.3"></minus><ci id="alg1.l5.3.m3.1.1.3.2.cmml" xref="alg1.l5.3.m3.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.3.m3.1c">a_{-i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.3.m3.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\quantity{\cdot}" class="ltx_Math" display="inline" id="alg1.l5.4.m4.1"><semantics id="alg1.l5.4.m4.1a"><mrow id="alg1.l5.4.m4.1.1.3"><mo id="alg1.l5.4.m4.1.1.3.1" mathcolor="#808080">{</mo><mo id="alg1.l5.4.m4.1.1.1.1.1" lspace="0em" mathcolor="#808080" rspace="0em" xref="alg1.l5.4.m4.1.1.1.1.1.cmml">⋅</mo><mo id="alg1.l5.4.m4.1.1.3.2" mathcolor="#808080">}</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.4.m4.1b"><ci id="alg1.l5.4.m4.1.1.1.1.1.cmml" xref="alg1.l5.4.m4.1.1.1.1.1">⋅</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.4.m4.1c">\quantity{\cdot}</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.4.m4.1d">{ start_ARG ⋅ end_ARG }</annotation></semantics></math> is an indicator function.</span> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l6.2">return</span> <math alttext="\tilde{U}" class="ltx_Math" display="inline" id="alg1.l6.m1.1"><semantics id="alg1.l6.m1.1a"><mover accent="true" id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><mi id="alg1.l6.m1.1.1.2" xref="alg1.l6.m1.1.1.2.cmml">U</mi><mo id="alg1.l6.m1.1.1.1" xref="alg1.l6.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1"><ci id="alg1.l6.m1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1">~</ci><ci id="alg1.l6.m1.1.1.2.cmml" xref="alg1.l6.m1.1.1.2">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">\tilde{U}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.1d">over~ start_ARG italic_U end_ARG</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg2.3.1.1">Algorithm 2</span> </span> Principal’s algorithm for learning a multi-agent game in the rationalizable model</figcaption> </figure> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.1"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg2" title="In 4.2 The multi-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a> yields the following performance guarantee.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.2"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2503.01976v1#alg2" title="In 4.2 The multi-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.2.2"> <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.1.m1.1a"><mi id="S4.Thmtheorem2.p1.1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.1.m1.1b"><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.1d">italic_ε</annotation></semantics></math>-learns any game in <math alttext="{\mathcal{O}}(nM\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.3"><semantics id="S4.Thmtheorem2.p1.2.2.m2.3a"><mrow id="S4.Thmtheorem2.p1.2.2.m2.3.3" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem2.p1.2.2.m2.3.3.3" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.3.cmml">𝒪</mi><mo id="S4.Thmtheorem2.p1.2.2.m2.3.3.2" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.2" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.3" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.3.cmml">M</mi><mo id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.1a" lspace="0.167em" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.2.2.m2.2.2.4" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.3.cmml"><mi id="S4.Thmtheorem2.p1.2.2.m2.2.2.2.2" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.3.1.cmml">log</mi><mo id="S4.Thmtheorem2.p1.2.2.m2.2.2.4a" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.3.1.cmml">⁡</mo><mrow id="S4.Thmtheorem2.p1.2.2.m2.2.2.4.1" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.3.cmml"><mo id="S4.Thmtheorem2.p1.2.2.m2.2.2.4.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.3.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.cmml"><mn id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.2" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.2.cmml">1</mn><mo id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.cmml">/</mo><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.3" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.Thmtheorem2.p1.2.2.m2.2.2.4.1.2" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.2.m2.3b"><apply id="S4.Thmtheorem2.p1.2.2.m2.3.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.3.3"><times id="S4.Thmtheorem2.p1.2.2.m2.3.3.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.2"></times><ci id="S4.Thmtheorem2.p1.2.2.m2.3.3.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.3">𝒪</ci><apply id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1"><times id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.1"></times><ci id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.2">𝑛</ci><ci id="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.3.3.1.1.1.3">𝑀</ci><apply id="S4.Thmtheorem2.p1.2.2.m2.2.2.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.4"><log id="S4.Thmtheorem2.p1.2.2.m2.2.2.3.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.2.2.2.2"></log><apply id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1"><divide id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.1"></divide><cn id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.2">1</cn><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.3c">{\mathcal{O}}(nM\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.3d">caligraphic_O ( italic_n italic_M roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> rounds.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.1.p1"> <p class="ltx_p" id="S4.SS2.1.p1.7">Fix an agent <math alttext="i" class="ltx_Math" display="inline" id="S4.SS2.1.p1.1.m1.1"><semantics id="S4.SS2.1.p1.1.m1.1a"><mi id="S4.SS2.1.p1.1.m1.1.1" xref="S4.SS2.1.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.1.m1.1b"><ci id="S4.SS2.1.p1.1.m1.1.1.cmml" xref="S4.SS2.1.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.1.m1.1d">italic_i</annotation></semantics></math> and action profile <math alttext="a_{-i}" class="ltx_Math" display="inline" id="S4.SS2.1.p1.2.m2.1"><semantics id="S4.SS2.1.p1.2.m2.1a"><msub id="S4.SS2.1.p1.2.m2.1.1" xref="S4.SS2.1.p1.2.m2.1.1.cmml"><mi id="S4.SS2.1.p1.2.m2.1.1.2" xref="S4.SS2.1.p1.2.m2.1.1.2.cmml">a</mi><mrow id="S4.SS2.1.p1.2.m2.1.1.3" xref="S4.SS2.1.p1.2.m2.1.1.3.cmml"><mo id="S4.SS2.1.p1.2.m2.1.1.3a" xref="S4.SS2.1.p1.2.m2.1.1.3.cmml">−</mo><mi id="S4.SS2.1.p1.2.m2.1.1.3.2" xref="S4.SS2.1.p1.2.m2.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.2.m2.1b"><apply id="S4.SS2.1.p1.2.m2.1.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.1.p1.2.m2.1.1.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.1.p1.2.m2.1.1.2.cmml" xref="S4.SS2.1.p1.2.m2.1.1.2">𝑎</ci><apply id="S4.SS2.1.p1.2.m2.1.1.3.cmml" xref="S4.SS2.1.p1.2.m2.1.1.3"><minus id="S4.SS2.1.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1.3"></minus><ci id="S4.SS2.1.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.1.p1.2.m2.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.2.m2.1c">a_{-i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.2.m2.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <span class="ltx_text ltx_font_italic" id="S4.SS2.1.p1.7.1">i.e.</span>, an iteration of the inner loop of <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg2" title="In 4.2 The multi-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a>. The algorithm uses <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg1" title="In 4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> to learn <math alttext="U_{i}(\cdot,a_{-i})" class="ltx_Math" display="inline" id="S4.SS2.1.p1.3.m3.2"><semantics id="S4.SS2.1.p1.3.m3.2a"><mrow id="S4.SS2.1.p1.3.m3.2.2" xref="S4.SS2.1.p1.3.m3.2.2.cmml"><msub id="S4.SS2.1.p1.3.m3.2.2.3" xref="S4.SS2.1.p1.3.m3.2.2.3.cmml"><mi id="S4.SS2.1.p1.3.m3.2.2.3.2" xref="S4.SS2.1.p1.3.m3.2.2.3.2.cmml">U</mi><mi id="S4.SS2.1.p1.3.m3.2.2.3.3" xref="S4.SS2.1.p1.3.m3.2.2.3.3.cmml">i</mi></msub><mo id="S4.SS2.1.p1.3.m3.2.2.2" xref="S4.SS2.1.p1.3.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.1.p1.3.m3.2.2.1.1" xref="S4.SS2.1.p1.3.m3.2.2.1.2.cmml"><mo id="S4.SS2.1.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S4.SS2.1.p1.3.m3.2.2.1.2.cmml">(</mo><mo id="S4.SS2.1.p1.3.m3.1.1" lspace="0em" rspace="0em" xref="S4.SS2.1.p1.3.m3.1.1.cmml">⋅</mo><mo id="S4.SS2.1.p1.3.m3.2.2.1.1.3" xref="S4.SS2.1.p1.3.m3.2.2.1.2.cmml">,</mo><msub id="S4.SS2.1.p1.3.m3.2.2.1.1.1" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.cmml"><mi id="S4.SS2.1.p1.3.m3.2.2.1.1.1.2" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.2.cmml">a</mi><mrow id="S4.SS2.1.p1.3.m3.2.2.1.1.1.3" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.cmml"><mo id="S4.SS2.1.p1.3.m3.2.2.1.1.1.3a" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.cmml">−</mo><mi id="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.2" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.2.cmml">i</mi></mrow></msub><mo id="S4.SS2.1.p1.3.m3.2.2.1.1.4" stretchy="false" xref="S4.SS2.1.p1.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.3.m3.2b"><apply id="S4.SS2.1.p1.3.m3.2.2.cmml" xref="S4.SS2.1.p1.3.m3.2.2"><times id="S4.SS2.1.p1.3.m3.2.2.2.cmml" xref="S4.SS2.1.p1.3.m3.2.2.2"></times><apply id="S4.SS2.1.p1.3.m3.2.2.3.cmml" xref="S4.SS2.1.p1.3.m3.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.1.p1.3.m3.2.2.3.1.cmml" xref="S4.SS2.1.p1.3.m3.2.2.3">subscript</csymbol><ci id="S4.SS2.1.p1.3.m3.2.2.3.2.cmml" xref="S4.SS2.1.p1.3.m3.2.2.3.2">𝑈</ci><ci id="S4.SS2.1.p1.3.m3.2.2.3.3.cmml" xref="S4.SS2.1.p1.3.m3.2.2.3.3">𝑖</ci></apply><interval closure="open" id="S4.SS2.1.p1.3.m3.2.2.1.2.cmml" xref="S4.SS2.1.p1.3.m3.2.2.1.1"><ci id="S4.SS2.1.p1.3.m3.1.1.cmml" xref="S4.SS2.1.p1.3.m3.1.1">⋅</ci><apply id="S4.SS2.1.p1.3.m3.2.2.1.1.1.cmml" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.1.p1.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.1.p1.3.m3.2.2.1.1.1.2.cmml" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.2">𝑎</ci><apply id="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.cmml" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.3"><minus id="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.1.cmml" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.3"></minus><ci id="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.2.cmml" xref="S4.SS2.1.p1.3.m3.2.2.1.1.1.3.2">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.3.m3.2c">U_{i}(\cdot,a_{-i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.3.m3.2d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS2.1.p1.4.m4.1"><semantics id="S4.SS2.1.p1.4.m4.1a"><mi id="S4.SS2.1.p1.4.m4.1.1" xref="S4.SS2.1.p1.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.4.m4.1b"><ci id="S4.SS2.1.p1.4.m4.1.1.cmml" xref="S4.SS2.1.p1.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.4.m4.1d">italic_ε</annotation></semantics></math>, which takes <math alttext="{\mathcal{O}}(m_{i}\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S4.SS2.1.p1.5.m5.3"><semantics id="S4.SS2.1.p1.5.m5.3a"><mrow id="S4.SS2.1.p1.5.m5.3.3" xref="S4.SS2.1.p1.5.m5.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.1.p1.5.m5.3.3.3" xref="S4.SS2.1.p1.5.m5.3.3.3.cmml">𝒪</mi><mo id="S4.SS2.1.p1.5.m5.3.3.2" xref="S4.SS2.1.p1.5.m5.3.3.2.cmml">⁢</mo><mrow id="S4.SS2.1.p1.5.m5.3.3.1.1" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.cmml"><mo id="S4.SS2.1.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS2.1.p1.5.m5.3.3.1.1.1" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.cmml"><msub id="S4.SS2.1.p1.5.m5.3.3.1.1.1.2" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.cmml"><mi id="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.2" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.2.cmml">m</mi><mi id="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.3" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS2.1.p1.5.m5.3.3.1.1.1.1" lspace="0.167em" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.5.m5.2.2.4" xref="S4.SS2.1.p1.5.m5.2.2.3.cmml"><mi id="S4.SS2.1.p1.5.m5.2.2.2.2" xref="S4.SS2.1.p1.5.m5.2.2.3.1.cmml">log</mi><mo id="S4.SS2.1.p1.5.m5.2.2.4a" xref="S4.SS2.1.p1.5.m5.2.2.3.1.cmml">⁡</mo><mrow id="S4.SS2.1.p1.5.m5.2.2.4.1" xref="S4.SS2.1.p1.5.m5.2.2.3.cmml"><mo id="S4.SS2.1.p1.5.m5.2.2.4.1.1" xref="S4.SS2.1.p1.5.m5.2.2.3.1.cmml">(</mo><mrow id="S4.SS2.1.p1.5.m5.1.1.1.1.1" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1.cmml"><mn id="S4.SS2.1.p1.5.m5.1.1.1.1.1.2" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.1.p1.5.m5.1.1.1.1.1.1" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1.1.cmml">/</mo><mi id="S4.SS2.1.p1.5.m5.1.1.1.1.1.3" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.SS2.1.p1.5.m5.2.2.4.1.2" xref="S4.SS2.1.p1.5.m5.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.1.p1.5.m5.3.3.1.1.3" stretchy="false" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.5.m5.3b"><apply id="S4.SS2.1.p1.5.m5.3.3.cmml" xref="S4.SS2.1.p1.5.m5.3.3"><times id="S4.SS2.1.p1.5.m5.3.3.2.cmml" xref="S4.SS2.1.p1.5.m5.3.3.2"></times><ci id="S4.SS2.1.p1.5.m5.3.3.3.cmml" xref="S4.SS2.1.p1.5.m5.3.3.3">𝒪</ci><apply id="S4.SS2.1.p1.5.m5.3.3.1.1.1.cmml" xref="S4.SS2.1.p1.5.m5.3.3.1.1"><times id="S4.SS2.1.p1.5.m5.3.3.1.1.1.1.cmml" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.1"></times><apply id="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.cmml" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.1.cmml" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.2">subscript</csymbol><ci id="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.2.cmml" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.2">𝑚</ci><ci id="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.3.cmml" xref="S4.SS2.1.p1.5.m5.3.3.1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS2.1.p1.5.m5.2.2.3.cmml" xref="S4.SS2.1.p1.5.m5.2.2.4"><log id="S4.SS2.1.p1.5.m5.2.2.3.1.cmml" xref="S4.SS2.1.p1.5.m5.2.2.2.2"></log><apply id="S4.SS2.1.p1.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1"><divide id="S4.SS2.1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1.1"></divide><cn id="S4.SS2.1.p1.5.m5.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1.2">1</cn><ci id="S4.SS2.1.p1.5.m5.1.1.1.1.1.3.cmml" xref="S4.SS2.1.p1.5.m5.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.5.m5.3c">{\mathcal{O}}(m_{i}\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.5.m5.3d">caligraphic_O ( italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> rounds. Thus, each outer-loop iteration takes <math alttext="{\mathcal{O}}(M\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S4.SS2.1.p1.6.m6.3"><semantics id="S4.SS2.1.p1.6.m6.3a"><mrow id="S4.SS2.1.p1.6.m6.3.3" xref="S4.SS2.1.p1.6.m6.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.1.p1.6.m6.3.3.3" xref="S4.SS2.1.p1.6.m6.3.3.3.cmml">𝒪</mi><mo id="S4.SS2.1.p1.6.m6.3.3.2" xref="S4.SS2.1.p1.6.m6.3.3.2.cmml">⁢</mo><mrow id="S4.SS2.1.p1.6.m6.3.3.1.1" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.cmml"><mo id="S4.SS2.1.p1.6.m6.3.3.1.1.2" stretchy="false" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS2.1.p1.6.m6.3.3.1.1.1" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.cmml"><mi id="S4.SS2.1.p1.6.m6.3.3.1.1.1.2" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.2.cmml">M</mi><mo id="S4.SS2.1.p1.6.m6.3.3.1.1.1.1" lspace="0.167em" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.6.m6.2.2.4" xref="S4.SS2.1.p1.6.m6.2.2.3.cmml"><mi id="S4.SS2.1.p1.6.m6.2.2.2.2" xref="S4.SS2.1.p1.6.m6.2.2.3.1.cmml">log</mi><mo id="S4.SS2.1.p1.6.m6.2.2.4a" xref="S4.SS2.1.p1.6.m6.2.2.3.1.cmml">⁡</mo><mrow id="S4.SS2.1.p1.6.m6.2.2.4.1" xref="S4.SS2.1.p1.6.m6.2.2.3.cmml"><mo id="S4.SS2.1.p1.6.m6.2.2.4.1.1" xref="S4.SS2.1.p1.6.m6.2.2.3.1.cmml">(</mo><mrow id="S4.SS2.1.p1.6.m6.1.1.1.1.1" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1.cmml"><mn id="S4.SS2.1.p1.6.m6.1.1.1.1.1.2" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.1.p1.6.m6.1.1.1.1.1.1" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1.1.cmml">/</mo><mi id="S4.SS2.1.p1.6.m6.1.1.1.1.1.3" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.SS2.1.p1.6.m6.2.2.4.1.2" xref="S4.SS2.1.p1.6.m6.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.1.p1.6.m6.3.3.1.1.3" stretchy="false" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.6.m6.3b"><apply id="S4.SS2.1.p1.6.m6.3.3.cmml" xref="S4.SS2.1.p1.6.m6.3.3"><times id="S4.SS2.1.p1.6.m6.3.3.2.cmml" xref="S4.SS2.1.p1.6.m6.3.3.2"></times><ci id="S4.SS2.1.p1.6.m6.3.3.3.cmml" xref="S4.SS2.1.p1.6.m6.3.3.3">𝒪</ci><apply id="S4.SS2.1.p1.6.m6.3.3.1.1.1.cmml" xref="S4.SS2.1.p1.6.m6.3.3.1.1"><times id="S4.SS2.1.p1.6.m6.3.3.1.1.1.1.cmml" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.1"></times><ci id="S4.SS2.1.p1.6.m6.3.3.1.1.1.2.cmml" xref="S4.SS2.1.p1.6.m6.3.3.1.1.1.2">𝑀</ci><apply id="S4.SS2.1.p1.6.m6.2.2.3.cmml" xref="S4.SS2.1.p1.6.m6.2.2.4"><log id="S4.SS2.1.p1.6.m6.2.2.3.1.cmml" xref="S4.SS2.1.p1.6.m6.2.2.2.2"></log><apply id="S4.SS2.1.p1.6.m6.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1"><divide id="S4.SS2.1.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1.1"></divide><cn id="S4.SS2.1.p1.6.m6.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1.2">1</cn><ci id="S4.SS2.1.p1.6.m6.1.1.1.1.1.3.cmml" xref="S4.SS2.1.p1.6.m6.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.6.m6.3c">{\mathcal{O}}(M\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.6.m6.3d">caligraphic_O ( italic_M roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> rounds, so the overall algorithm takes <math alttext="{\mathcal{O}}(nM\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S4.SS2.1.p1.7.m7.3"><semantics id="S4.SS2.1.p1.7.m7.3a"><mrow id="S4.SS2.1.p1.7.m7.3.3" xref="S4.SS2.1.p1.7.m7.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.1.p1.7.m7.3.3.3" xref="S4.SS2.1.p1.7.m7.3.3.3.cmml">𝒪</mi><mo id="S4.SS2.1.p1.7.m7.3.3.2" xref="S4.SS2.1.p1.7.m7.3.3.2.cmml">⁢</mo><mrow id="S4.SS2.1.p1.7.m7.3.3.1.1" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.cmml"><mo id="S4.SS2.1.p1.7.m7.3.3.1.1.2" stretchy="false" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS2.1.p1.7.m7.3.3.1.1.1" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.cmml"><mi id="S4.SS2.1.p1.7.m7.3.3.1.1.1.2" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.2.cmml">n</mi><mo id="S4.SS2.1.p1.7.m7.3.3.1.1.1.1" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.1.p1.7.m7.3.3.1.1.1.3" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.3.cmml">M</mi><mo id="S4.SS2.1.p1.7.m7.3.3.1.1.1.1a" lspace="0.167em" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.7.m7.2.2.4" xref="S4.SS2.1.p1.7.m7.2.2.3.cmml"><mi id="S4.SS2.1.p1.7.m7.2.2.2.2" xref="S4.SS2.1.p1.7.m7.2.2.3.1.cmml">log</mi><mo id="S4.SS2.1.p1.7.m7.2.2.4a" xref="S4.SS2.1.p1.7.m7.2.2.3.1.cmml">⁡</mo><mrow id="S4.SS2.1.p1.7.m7.2.2.4.1" xref="S4.SS2.1.p1.7.m7.2.2.3.cmml"><mo id="S4.SS2.1.p1.7.m7.2.2.4.1.1" xref="S4.SS2.1.p1.7.m7.2.2.3.1.cmml">(</mo><mrow id="S4.SS2.1.p1.7.m7.1.1.1.1.1" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.cmml"><mn id="S4.SS2.1.p1.7.m7.1.1.1.1.1.2" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.cmml">/</mo><mi id="S4.SS2.1.p1.7.m7.1.1.1.1.1.3" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.SS2.1.p1.7.m7.2.2.4.1.2" xref="S4.SS2.1.p1.7.m7.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.1.p1.7.m7.3.3.1.1.3" stretchy="false" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.7.m7.3b"><apply id="S4.SS2.1.p1.7.m7.3.3.cmml" xref="S4.SS2.1.p1.7.m7.3.3"><times id="S4.SS2.1.p1.7.m7.3.3.2.cmml" xref="S4.SS2.1.p1.7.m7.3.3.2"></times><ci id="S4.SS2.1.p1.7.m7.3.3.3.cmml" xref="S4.SS2.1.p1.7.m7.3.3.3">𝒪</ci><apply id="S4.SS2.1.p1.7.m7.3.3.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.3.3.1.1"><times id="S4.SS2.1.p1.7.m7.3.3.1.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.1"></times><ci id="S4.SS2.1.p1.7.m7.3.3.1.1.1.2.cmml" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.2">𝑛</ci><ci id="S4.SS2.1.p1.7.m7.3.3.1.1.1.3.cmml" xref="S4.SS2.1.p1.7.m7.3.3.1.1.1.3">𝑀</ci><apply id="S4.SS2.1.p1.7.m7.2.2.3.cmml" xref="S4.SS2.1.p1.7.m7.2.2.4"><log id="S4.SS2.1.p1.7.m7.2.2.3.1.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2"></log><apply id="S4.SS2.1.p1.7.m7.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1"><divide id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1"></divide><cn id="S4.SS2.1.p1.7.m7.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.2">1</cn><ci id="S4.SS2.1.p1.7.m7.1.1.1.1.1.3.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.7.m7.3c">{\mathcal{O}}(nM\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.7.m7.3d">caligraphic_O ( italic_n italic_M roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> rounds. ∎</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>Lower bound</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">We now prove a lower bound that nearly matches <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorems</span> <span class="ltx_text ltx_ref_tag">4.1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.Thmtheorem2" title="Theorem 4.2. ‣ 4.2 The multi-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">4.2</span></a>, up to logarithmic factors.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.2.2">Regardless of the behavioral model, any algorithm that <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.1.m1.1a"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.1b"><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.1d">italic_ε</annotation></semantics></math>-learns a multi-agent game must take at least <math alttext="\tilde{\Omega}(M)\cdot n\log(1/\varepsilon)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.3"><semantics id="S4.Thmtheorem3.p1.2.2.m2.3a"><mrow id="S4.Thmtheorem3.p1.2.2.m2.3.4" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.cmml"><mrow id="S4.Thmtheorem3.p1.2.2.m2.3.4.2" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.cmml"><mrow id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.cmml"><mover accent="true" id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.cmml"><mi id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.2" mathvariant="normal" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.2.cmml">Ω</mi><mo id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.1" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.1.cmml">~</mo></mover><mo id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.1" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.3.2" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.cmml"><mo id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.cmml">(</mo><mi id="S4.Thmtheorem3.p1.2.2.m2.3.3" xref="S4.Thmtheorem3.p1.2.2.m2.3.3.cmml">M</mi><mo id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.1" rspace="0.222em" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.1.cmml">⋅</mo><mi id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.3" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.3.cmml">n</mi></mrow><mo id="S4.Thmtheorem3.p1.2.2.m2.3.4.1" lspace="0.167em" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.2.2.m2.2.2.4" xref="S4.Thmtheorem3.p1.2.2.m2.2.2.3.cmml"><mi id="S4.Thmtheorem3.p1.2.2.m2.2.2.2.2" xref="S4.Thmtheorem3.p1.2.2.m2.2.2.3.1.cmml">log</mi><mo id="S4.Thmtheorem3.p1.2.2.m2.2.2.4a" xref="S4.Thmtheorem3.p1.2.2.m2.2.2.3.1.cmml">⁡</mo><mrow id="S4.Thmtheorem3.p1.2.2.m2.2.2.4.1" xref="S4.Thmtheorem3.p1.2.2.m2.2.2.3.cmml"><mo id="S4.Thmtheorem3.p1.2.2.m2.2.2.4.1.1" 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xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.1">⋅</ci><apply id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2"><times id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.1"></times><apply id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2"><ci id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.1">~</ci><ci id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.2.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.2.2.2">Ω</ci></apply><ci id="S4.Thmtheorem3.p1.2.2.m2.3.3.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.3.3">𝑀</ci></apply><ci id="S4.Thmtheorem3.p1.2.2.m2.3.4.2.3.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.3.4.2.3">𝑛</ci></apply><apply id="S4.Thmtheorem3.p1.2.2.m2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.2.2.4"><log id="S4.Thmtheorem3.p1.2.2.m2.2.2.3.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.2.2.2.2"></log><apply id="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1"><divide id="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1"></divide><cn id="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1.2">1</cn><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.3c">\tilde{\Omega}(M)\cdot n\log(1/\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.3d">over~ start_ARG roman_Ω end_ARG ( italic_M ) ⋅ italic_n roman_log ( start_ARG 1 / italic_ε end_ARG )</annotation></semantics></math> rounds.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS3.1.p1"> <p class="ltx_p" id="S4.SS3.1.p1.7">Suppose that <math alttext="U_{i}(1,\cdot)=0" class="ltx_Math" display="inline" id="S4.SS3.1.p1.1.m1.2"><semantics id="S4.SS3.1.p1.1.m1.2a"><mrow id="S4.SS3.1.p1.1.m1.2.3" xref="S4.SS3.1.p1.1.m1.2.3.cmml"><mrow id="S4.SS3.1.p1.1.m1.2.3.2" xref="S4.SS3.1.p1.1.m1.2.3.2.cmml"><msub id="S4.SS3.1.p1.1.m1.2.3.2.2" xref="S4.SS3.1.p1.1.m1.2.3.2.2.cmml"><mi id="S4.SS3.1.p1.1.m1.2.3.2.2.2" xref="S4.SS3.1.p1.1.m1.2.3.2.2.2.cmml">U</mi><mi id="S4.SS3.1.p1.1.m1.2.3.2.2.3" xref="S4.SS3.1.p1.1.m1.2.3.2.2.3.cmml">i</mi></msub><mo id="S4.SS3.1.p1.1.m1.2.3.2.1" xref="S4.SS3.1.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS3.1.p1.1.m1.2.3.2.3.2" xref="S4.SS3.1.p1.1.m1.2.3.2.3.1.cmml"><mo id="S4.SS3.1.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.SS3.1.p1.1.m1.2.3.2.3.1.cmml">(</mo><mn id="S4.SS3.1.p1.1.m1.1.1" xref="S4.SS3.1.p1.1.m1.1.1.cmml">1</mn><mo id="S4.SS3.1.p1.1.m1.2.3.2.3.2.2" rspace="0em" xref="S4.SS3.1.p1.1.m1.2.3.2.3.1.cmml">,</mo><mo id="S4.SS3.1.p1.1.m1.2.2" lspace="0em" rspace="0em" xref="S4.SS3.1.p1.1.m1.2.2.cmml">⋅</mo><mo id="S4.SS3.1.p1.1.m1.2.3.2.3.2.3" stretchy="false" xref="S4.SS3.1.p1.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.1.p1.1.m1.2.3.1" xref="S4.SS3.1.p1.1.m1.2.3.1.cmml">=</mo><mn id="S4.SS3.1.p1.1.m1.2.3.3" xref="S4.SS3.1.p1.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.1.m1.2b"><apply id="S4.SS3.1.p1.1.m1.2.3.cmml" xref="S4.SS3.1.p1.1.m1.2.3"><eq id="S4.SS3.1.p1.1.m1.2.3.1.cmml" xref="S4.SS3.1.p1.1.m1.2.3.1"></eq><apply id="S4.SS3.1.p1.1.m1.2.3.2.cmml" xref="S4.SS3.1.p1.1.m1.2.3.2"><times id="S4.SS3.1.p1.1.m1.2.3.2.1.cmml" xref="S4.SS3.1.p1.1.m1.2.3.2.1"></times><apply id="S4.SS3.1.p1.1.m1.2.3.2.2.cmml" xref="S4.SS3.1.p1.1.m1.2.3.2.2"><csymbol cd="ambiguous" id="S4.SS3.1.p1.1.m1.2.3.2.2.1.cmml" xref="S4.SS3.1.p1.1.m1.2.3.2.2">subscript</csymbol><ci id="S4.SS3.1.p1.1.m1.2.3.2.2.2.cmml" xref="S4.SS3.1.p1.1.m1.2.3.2.2.2">𝑈</ci><ci id="S4.SS3.1.p1.1.m1.2.3.2.2.3.cmml" xref="S4.SS3.1.p1.1.m1.2.3.2.2.3">𝑖</ci></apply><interval closure="open" id="S4.SS3.1.p1.1.m1.2.3.2.3.1.cmml" xref="S4.SS3.1.p1.1.m1.2.3.2.3.2"><cn id="S4.SS3.1.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS3.1.p1.1.m1.1.1">1</cn><ci id="S4.SS3.1.p1.1.m1.2.2.cmml" xref="S4.SS3.1.p1.1.m1.2.2">⋅</ci></interval></apply><cn id="S4.SS3.1.p1.1.m1.2.3.3.cmml" type="integer" xref="S4.SS3.1.p1.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.1.m1.2c">U_{i}(1,\cdot)=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.1.m1.2d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 , ⋅ ) = 0</annotation></semantics></math> for all agents <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.1.p1.2.m2.1"><semantics id="S4.SS3.1.p1.2.m2.1a"><mi id="S4.SS3.1.p1.2.m2.1.1" xref="S4.SS3.1.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.2.m2.1b"><ci id="S4.SS3.1.p1.2.m2.1.1.cmml" xref="S4.SS3.1.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.2.m2.1d">italic_i</annotation></semantics></math>, and <math alttext="U_{i}(a_{i},a_{-i})\sim\{0,2\varepsilon,4\varepsilon,\dots,1\}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.3.m3.7"><semantics id="S4.SS3.1.p1.3.m3.7a"><mrow id="S4.SS3.1.p1.3.m3.7.7" xref="S4.SS3.1.p1.3.m3.7.7.cmml"><mrow id="S4.SS3.1.p1.3.m3.5.5.2" xref="S4.SS3.1.p1.3.m3.5.5.2.cmml"><msub id="S4.SS3.1.p1.3.m3.5.5.2.4" xref="S4.SS3.1.p1.3.m3.5.5.2.4.cmml"><mi id="S4.SS3.1.p1.3.m3.5.5.2.4.2" xref="S4.SS3.1.p1.3.m3.5.5.2.4.2.cmml">U</mi><mi id="S4.SS3.1.p1.3.m3.5.5.2.4.3" xref="S4.SS3.1.p1.3.m3.5.5.2.4.3.cmml">i</mi></msub><mo id="S4.SS3.1.p1.3.m3.5.5.2.3" xref="S4.SS3.1.p1.3.m3.5.5.2.3.cmml">⁢</mo><mrow id="S4.SS3.1.p1.3.m3.5.5.2.2.2" xref="S4.SS3.1.p1.3.m3.5.5.2.2.3.cmml"><mo id="S4.SS3.1.p1.3.m3.5.5.2.2.2.3" 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xref="S4.SS3.1.p1.3.m3.5.5.2.2.2.2">subscript</csymbol><ci id="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.2.cmml" xref="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.2">𝑎</ci><apply id="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.3.cmml" xref="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.3"><minus id="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.3.1.cmml" xref="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.3"></minus><ci id="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.3.2.cmml" xref="S4.SS3.1.p1.3.m3.5.5.2.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply><set id="S4.SS3.1.p1.3.m3.7.7.4.3.cmml" xref="S4.SS3.1.p1.3.m3.7.7.4.2"><cn id="S4.SS3.1.p1.3.m3.1.1.cmml" type="integer" xref="S4.SS3.1.p1.3.m3.1.1">0</cn><apply id="S4.SS3.1.p1.3.m3.6.6.3.1.1.cmml" xref="S4.SS3.1.p1.3.m3.6.6.3.1.1"><times id="S4.SS3.1.p1.3.m3.6.6.3.1.1.1.cmml" xref="S4.SS3.1.p1.3.m3.6.6.3.1.1.1"></times><cn id="S4.SS3.1.p1.3.m3.6.6.3.1.1.2.cmml" type="integer" xref="S4.SS3.1.p1.3.m3.6.6.3.1.1.2">2</cn><ci id="S4.SS3.1.p1.3.m3.6.6.3.1.1.3.cmml" xref="S4.SS3.1.p1.3.m3.6.6.3.1.1.3">𝜀</ci></apply><apply id="S4.SS3.1.p1.3.m3.7.7.4.2.2.cmml" xref="S4.SS3.1.p1.3.m3.7.7.4.2.2"><times id="S4.SS3.1.p1.3.m3.7.7.4.2.2.1.cmml" xref="S4.SS3.1.p1.3.m3.7.7.4.2.2.1"></times><cn id="S4.SS3.1.p1.3.m3.7.7.4.2.2.2.cmml" type="integer" xref="S4.SS3.1.p1.3.m3.7.7.4.2.2.2">4</cn><ci id="S4.SS3.1.p1.3.m3.7.7.4.2.2.3.cmml" xref="S4.SS3.1.p1.3.m3.7.7.4.2.2.3">𝜀</ci></apply><ci id="S4.SS3.1.p1.3.m3.2.2.cmml" xref="S4.SS3.1.p1.3.m3.2.2">…</ci><cn id="S4.SS3.1.p1.3.m3.3.3.cmml" type="integer" xref="S4.SS3.1.p1.3.m3.3.3">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.3.m3.7c">U_{i}(a_{i},a_{-i})\sim\{0,2\varepsilon,4\varepsilon,\dots,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.3.m3.7d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) ∼ { 0 , 2 italic_ε , 4 italic_ε , … , 1 }</annotation></semantics></math> i.i.d. for <math alttext="2\leq a\leq m_{i}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.4.m4.1"><semantics id="S4.SS3.1.p1.4.m4.1a"><mrow id="S4.SS3.1.p1.4.m4.1.1" xref="S4.SS3.1.p1.4.m4.1.1.cmml"><mn id="S4.SS3.1.p1.4.m4.1.1.2" xref="S4.SS3.1.p1.4.m4.1.1.2.cmml">2</mn><mo id="S4.SS3.1.p1.4.m4.1.1.3" xref="S4.SS3.1.p1.4.m4.1.1.3.cmml">≤</mo><mi id="S4.SS3.1.p1.4.m4.1.1.4" xref="S4.SS3.1.p1.4.m4.1.1.4.cmml">a</mi><mo id="S4.SS3.1.p1.4.m4.1.1.5" xref="S4.SS3.1.p1.4.m4.1.1.5.cmml">≤</mo><msub id="S4.SS3.1.p1.4.m4.1.1.6" xref="S4.SS3.1.p1.4.m4.1.1.6.cmml"><mi id="S4.SS3.1.p1.4.m4.1.1.6.2" xref="S4.SS3.1.p1.4.m4.1.1.6.2.cmml">m</mi><mi id="S4.SS3.1.p1.4.m4.1.1.6.3" xref="S4.SS3.1.p1.4.m4.1.1.6.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.4.m4.1b"><apply id="S4.SS3.1.p1.4.m4.1.1.cmml" xref="S4.SS3.1.p1.4.m4.1.1"><and id="S4.SS3.1.p1.4.m4.1.1a.cmml" xref="S4.SS3.1.p1.4.m4.1.1"></and><apply id="S4.SS3.1.p1.4.m4.1.1b.cmml" xref="S4.SS3.1.p1.4.m4.1.1"><leq id="S4.SS3.1.p1.4.m4.1.1.3.cmml" xref="S4.SS3.1.p1.4.m4.1.1.3"></leq><cn id="S4.SS3.1.p1.4.m4.1.1.2.cmml" type="integer" xref="S4.SS3.1.p1.4.m4.1.1.2">2</cn><ci id="S4.SS3.1.p1.4.m4.1.1.4.cmml" xref="S4.SS3.1.p1.4.m4.1.1.4">𝑎</ci></apply><apply id="S4.SS3.1.p1.4.m4.1.1c.cmml" xref="S4.SS3.1.p1.4.m4.1.1"><leq id="S4.SS3.1.p1.4.m4.1.1.5.cmml" xref="S4.SS3.1.p1.4.m4.1.1.5"></leq><share href="https://arxiv.org/html/2503.01976v1#S4.SS3.1.p1.4.m4.1.1.4.cmml" id="S4.SS3.1.p1.4.m4.1.1d.cmml" xref="S4.SS3.1.p1.4.m4.1.1"></share><apply id="S4.SS3.1.p1.4.m4.1.1.6.cmml" xref="S4.SS3.1.p1.4.m4.1.1.6"><csymbol cd="ambiguous" id="S4.SS3.1.p1.4.m4.1.1.6.1.cmml" xref="S4.SS3.1.p1.4.m4.1.1.6">subscript</csymbol><ci id="S4.SS3.1.p1.4.m4.1.1.6.2.cmml" xref="S4.SS3.1.p1.4.m4.1.1.6.2">𝑚</ci><ci id="S4.SS3.1.p1.4.m4.1.1.6.3.cmml" xref="S4.SS3.1.p1.4.m4.1.1.6.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.4.m4.1c">2\leq a\leq m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.4.m4.1d">2 ≤ italic_a ≤ italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="a_{-i}\in A_{-i}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.5.m5.1"><semantics id="S4.SS3.1.p1.5.m5.1a"><mrow id="S4.SS3.1.p1.5.m5.1.1" xref="S4.SS3.1.p1.5.m5.1.1.cmml"><msub id="S4.SS3.1.p1.5.m5.1.1.2" xref="S4.SS3.1.p1.5.m5.1.1.2.cmml"><mi id="S4.SS3.1.p1.5.m5.1.1.2.2" xref="S4.SS3.1.p1.5.m5.1.1.2.2.cmml">a</mi><mrow id="S4.SS3.1.p1.5.m5.1.1.2.3" xref="S4.SS3.1.p1.5.m5.1.1.2.3.cmml"><mo id="S4.SS3.1.p1.5.m5.1.1.2.3a" xref="S4.SS3.1.p1.5.m5.1.1.2.3.cmml">−</mo><mi id="S4.SS3.1.p1.5.m5.1.1.2.3.2" xref="S4.SS3.1.p1.5.m5.1.1.2.3.2.cmml">i</mi></mrow></msub><mo id="S4.SS3.1.p1.5.m5.1.1.1" xref="S4.SS3.1.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S4.SS3.1.p1.5.m5.1.1.3" xref="S4.SS3.1.p1.5.m5.1.1.3.cmml"><mi id="S4.SS3.1.p1.5.m5.1.1.3.2" xref="S4.SS3.1.p1.5.m5.1.1.3.2.cmml">A</mi><mrow id="S4.SS3.1.p1.5.m5.1.1.3.3" xref="S4.SS3.1.p1.5.m5.1.1.3.3.cmml"><mo id="S4.SS3.1.p1.5.m5.1.1.3.3a" xref="S4.SS3.1.p1.5.m5.1.1.3.3.cmml">−</mo><mi id="S4.SS3.1.p1.5.m5.1.1.3.3.2" xref="S4.SS3.1.p1.5.m5.1.1.3.3.2.cmml">i</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.5.m5.1b"><apply id="S4.SS3.1.p1.5.m5.1.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1"><in id="S4.SS3.1.p1.5.m5.1.1.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1.1"></in><apply id="S4.SS3.1.p1.5.m5.1.1.2.cmml" xref="S4.SS3.1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.1.p1.5.m5.1.1.2.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS3.1.p1.5.m5.1.1.2.2.cmml" xref="S4.SS3.1.p1.5.m5.1.1.2.2">𝑎</ci><apply id="S4.SS3.1.p1.5.m5.1.1.2.3.cmml" xref="S4.SS3.1.p1.5.m5.1.1.2.3"><minus id="S4.SS3.1.p1.5.m5.1.1.2.3.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1.2.3"></minus><ci id="S4.SS3.1.p1.5.m5.1.1.2.3.2.cmml" xref="S4.SS3.1.p1.5.m5.1.1.2.3.2">𝑖</ci></apply></apply><apply id="S4.SS3.1.p1.5.m5.1.1.3.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.1.p1.5.m5.1.1.3.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS3.1.p1.5.m5.1.1.3.2.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3.2">𝐴</ci><apply id="S4.SS3.1.p1.5.m5.1.1.3.3.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3.3"><minus id="S4.SS3.1.p1.5.m5.1.1.3.3.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3.3"></minus><ci id="S4.SS3.1.p1.5.m5.1.1.3.3.2.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3.3.2">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.5.m5.1c">a_{-i}\in A_{-i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.5.m5.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Thus the utility <math alttext="U" class="ltx_Math" display="inline" id="S4.SS3.1.p1.6.m6.1"><semantics id="S4.SS3.1.p1.6.m6.1a"><mi id="S4.SS3.1.p1.6.m6.1.1" xref="S4.SS3.1.p1.6.m6.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.6.m6.1b"><ci id="S4.SS3.1.p1.6.m6.1.1.cmml" xref="S4.SS3.1.p1.6.m6.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.6.m6.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.6.m6.1d">italic_U</annotation></semantics></math> is uniformly sampled from a set of <math alttext="\Omega(1/\varepsilon)^{K}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.7.m7.1"><semantics id="S4.SS3.1.p1.7.m7.1a"><mrow id="S4.SS3.1.p1.7.m7.1.1" xref="S4.SS3.1.p1.7.m7.1.1.cmml"><mi id="S4.SS3.1.p1.7.m7.1.1.3" mathvariant="normal" xref="S4.SS3.1.p1.7.m7.1.1.3.cmml">Ω</mi><mo id="S4.SS3.1.p1.7.m7.1.1.2" xref="S4.SS3.1.p1.7.m7.1.1.2.cmml">⁢</mo><msup id="S4.SS3.1.p1.7.m7.1.1.1" xref="S4.SS3.1.p1.7.m7.1.1.1.cmml"><mrow id="S4.SS3.1.p1.7.m7.1.1.1.1.1" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S4.SS3.1.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml"><mn id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.2" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.1" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.1.cmml">/</mo><mi id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.3" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.SS3.1.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S4.SS3.1.p1.7.m7.1.1.1.3" xref="S4.SS3.1.p1.7.m7.1.1.1.3.cmml">K</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.7.m7.1b"><apply id="S4.SS3.1.p1.7.m7.1.1.cmml" xref="S4.SS3.1.p1.7.m7.1.1"><times id="S4.SS3.1.p1.7.m7.1.1.2.cmml" xref="S4.SS3.1.p1.7.m7.1.1.2"></times><ci id="S4.SS3.1.p1.7.m7.1.1.3.cmml" xref="S4.SS3.1.p1.7.m7.1.1.3">Ω</ci><apply id="S4.SS3.1.p1.7.m7.1.1.1.cmml" xref="S4.SS3.1.p1.7.m7.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.1.p1.7.m7.1.1.1.2.cmml" xref="S4.SS3.1.p1.7.m7.1.1.1">superscript</csymbol><apply id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1"><divide id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.1"></divide><cn id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.2">1</cn><ci id="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.3.cmml" xref="S4.SS3.1.p1.7.m7.1.1.1.1.1.1.3">𝜀</ci></apply><ci id="S4.SS3.1.p1.7.m7.1.1.1.3.cmml" xref="S4.SS3.1.p1.7.m7.1.1.1.3">𝐾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.7.m7.1c">\Omega(1/\varepsilon)^{K}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.7.m7.1d">roman_Ω ( 1 / italic_ε ) start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT</annotation></semantics></math> possible utilities, where</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="K=\sum_{i=1}^{n}\quantity((m_{i}-1)\prod_{j\neq i}m_{j})\geq\frac{nM}{2}." class="ltx_Math" display="block" id="S4.Ex2.m1.2"><semantics id="S4.Ex2.m1.2a"><mrow id="S4.Ex2.m1.2.2.1" xref="S4.Ex2.m1.2.2.1.1.cmml"><mrow id="S4.Ex2.m1.2.2.1.1" xref="S4.Ex2.m1.2.2.1.1.cmml"><mi id="S4.Ex2.m1.2.2.1.1.2" xref="S4.Ex2.m1.2.2.1.1.2.cmml">K</mi><mo id="S4.Ex2.m1.2.2.1.1.3" rspace="0.111em" xref="S4.Ex2.m1.2.2.1.1.3.cmml">=</mo><mrow id="S4.Ex2.m1.2.2.1.1.4" xref="S4.Ex2.m1.2.2.1.1.4.cmml"><munderover id="S4.Ex2.m1.2.2.1.1.4.1" xref="S4.Ex2.m1.2.2.1.1.4.1.cmml"><mo id="S4.Ex2.m1.2.2.1.1.4.1.2.2" movablelimits="false" rspace="0em" xref="S4.Ex2.m1.2.2.1.1.4.1.2.2.cmml">∑</mo><mrow id="S4.Ex2.m1.2.2.1.1.4.1.2.3" xref="S4.Ex2.m1.2.2.1.1.4.1.2.3.cmml"><mi id="S4.Ex2.m1.2.2.1.1.4.1.2.3.2" xref="S4.Ex2.m1.2.2.1.1.4.1.2.3.2.cmml">i</mi><mo id="S4.Ex2.m1.2.2.1.1.4.1.2.3.1" xref="S4.Ex2.m1.2.2.1.1.4.1.2.3.1.cmml">=</mo><mn id="S4.Ex2.m1.2.2.1.1.4.1.2.3.3" xref="S4.Ex2.m1.2.2.1.1.4.1.2.3.3.cmml">1</mn></mrow><mi id="S4.Ex2.m1.2.2.1.1.4.1.3" xref="S4.Ex2.m1.2.2.1.1.4.1.3.cmml">n</mi></munderover><mrow id="S4.Ex2.m1.1.1.3" xref="S4.Ex2.m1.1.1.1.1.1.cmml"><mo id="S4.Ex2.m1.1.1.3.1" xref="S4.Ex2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex2.m1.1.1.1.1.1" xref="S4.Ex2.m1.1.1.1.1.1.cmml"><mrow id="S4.Ex2.m1.1.1.1.1.1.1.1" xref="S4.Ex2.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex2.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex2.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex2.m1.1.1.1.1.1.1.1.1" xref="S4.Ex2.m1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex2.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex2.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex2.m1.1.1.1.1.1.1.1.1.2.2" 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type="integer" xref="S4.Ex2.m1.2.2.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m1.2c">K=\sum_{i=1}^{n}\quantity((m_{i}-1)\prod_{j\neq i}m_{j})\geq\frac{nM}{2}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m1.2d">italic_K = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( start_ARG ( italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1 ) ∏ start_POSTSUBSCRIPT italic_j ≠ italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) ≥ divide start_ARG italic_n italic_M end_ARG start_ARG 2 end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS3.1.p1.13">As before, since each utility function differs by <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS3.1.p1.8.m1.1"><semantics id="S4.SS3.1.p1.8.m1.1a"><mi id="S4.SS3.1.p1.8.m1.1.1" xref="S4.SS3.1.p1.8.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.8.m1.1b"><ci id="S4.SS3.1.p1.8.m1.1.1.cmml" xref="S4.SS3.1.p1.8.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.8.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.8.m1.1d">italic_ε</annotation></semantics></math>, it follows that <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS3.1.p1.9.m2.1"><semantics id="S4.SS3.1.p1.9.m2.1a"><mi id="S4.SS3.1.p1.9.m2.1.1" xref="S4.SS3.1.p1.9.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.9.m2.1b"><ci id="S4.SS3.1.p1.9.m2.1.1.cmml" xref="S4.SS3.1.p1.9.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.9.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.9.m2.1d">italic_ε</annotation></semantics></math>-learning a game sampled from this family entails exactly outputting the utility <math alttext="U" class="ltx_Math" display="inline" id="S4.SS3.1.p1.10.m3.1"><semantics id="S4.SS3.1.p1.10.m3.1a"><mi id="S4.SS3.1.p1.10.m3.1.1" xref="S4.SS3.1.p1.10.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.10.m3.1b"><ci id="S4.SS3.1.p1.10.m3.1.1.cmml" xref="S4.SS3.1.p1.10.m3.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.10.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.10.m3.1d">italic_U</annotation></semantics></math>. On each round, the principal only observes a single action profile <math alttext="a\in A" class="ltx_Math" display="inline" id="S4.SS3.1.p1.11.m4.1"><semantics id="S4.SS3.1.p1.11.m4.1a"><mrow id="S4.SS3.1.p1.11.m4.1.1" xref="S4.SS3.1.p1.11.m4.1.1.cmml"><mi id="S4.SS3.1.p1.11.m4.1.1.2" xref="S4.SS3.1.p1.11.m4.1.1.2.cmml">a</mi><mo id="S4.SS3.1.p1.11.m4.1.1.1" xref="S4.SS3.1.p1.11.m4.1.1.1.cmml">∈</mo><mi id="S4.SS3.1.p1.11.m4.1.1.3" xref="S4.SS3.1.p1.11.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.11.m4.1b"><apply id="S4.SS3.1.p1.11.m4.1.1.cmml" xref="S4.SS3.1.p1.11.m4.1.1"><in id="S4.SS3.1.p1.11.m4.1.1.1.cmml" xref="S4.SS3.1.p1.11.m4.1.1.1"></in><ci id="S4.SS3.1.p1.11.m4.1.1.2.cmml" xref="S4.SS3.1.p1.11.m4.1.1.2">𝑎</ci><ci id="S4.SS3.1.p1.11.m4.1.1.3.cmml" xref="S4.SS3.1.p1.11.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.11.m4.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.11.m4.1d">italic_a ∈ italic_A</annotation></semantics></math>, which only conveys <math alttext="\log M" class="ltx_Math" display="inline" id="S4.SS3.1.p1.12.m5.1"><semantics id="S4.SS3.1.p1.12.m5.1a"><mrow id="S4.SS3.1.p1.12.m5.1.1" xref="S4.SS3.1.p1.12.m5.1.1.cmml"><mi id="S4.SS3.1.p1.12.m5.1.1.1" xref="S4.SS3.1.p1.12.m5.1.1.1.cmml">log</mi><mo id="S4.SS3.1.p1.12.m5.1.1a" lspace="0.167em" xref="S4.SS3.1.p1.12.m5.1.1.cmml">⁡</mo><mi id="S4.SS3.1.p1.12.m5.1.1.2" xref="S4.SS3.1.p1.12.m5.1.1.2.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.12.m5.1b"><apply id="S4.SS3.1.p1.12.m5.1.1.cmml" xref="S4.SS3.1.p1.12.m5.1.1"><log id="S4.SS3.1.p1.12.m5.1.1.1.cmml" xref="S4.SS3.1.p1.12.m5.1.1.1"></log><ci id="S4.SS3.1.p1.12.m5.1.1.2.cmml" xref="S4.SS3.1.p1.12.m5.1.1.2">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.12.m5.1c">\log M</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.12.m5.1d">roman_log italic_M</annotation></semantics></math> bits of information. Therefore, <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS3.1.p1.13.m6.1"><semantics id="S4.SS3.1.p1.13.m6.1a"><mi id="S4.SS3.1.p1.13.m6.1.1" xref="S4.SS3.1.p1.13.m6.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.13.m6.1b"><ci id="S4.SS3.1.p1.13.m6.1.1.cmml" xref="S4.SS3.1.p1.13.m6.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.13.m6.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.13.m6.1d">italic_ε</annotation></semantics></math>-learning the game takes</p> <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <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="\frac{K\log(\Omega(1/\varepsilon))}{\log M}\gtrsim\frac{nM\log(1/\varepsilon)}% {\log M}" class="ltx_Math" display="block" id="S4.E4.m1.4"><semantics id="S4.E4.m1.4a"><mrow 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xref="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E4.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mn id="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.E4.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E4.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E4.m1.3.3.2.2.4.1.2" xref="S4.E4.m1.3.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mrow id="S4.E4.m1.3.3.4" xref="S4.E4.m1.3.3.4.cmml"><mi id="S4.E4.m1.3.3.4.1" xref="S4.E4.m1.3.3.4.1.cmml">log</mi><mo id="S4.E4.m1.3.3.4a" lspace="0.167em" xref="S4.E4.m1.3.3.4.cmml">⁡</mo><mi id="S4.E4.m1.3.3.4.2" xref="S4.E4.m1.3.3.4.2.cmml">M</mi></mrow></mfrac><mo id="S4.E4.m1.4.5.1" xref="S4.E4.m1.4.5.1.cmml">≳</mo><mfrac id="S4.E4.m1.4.4" xref="S4.E4.m1.4.4.cmml"><mrow id="S4.E4.m1.4.4.2" xref="S4.E4.m1.4.4.2.cmml"><mi id="S4.E4.m1.4.4.2.4" xref="S4.E4.m1.4.4.2.4.cmml">n</mi><mo id="S4.E4.m1.4.4.2.3" xref="S4.E4.m1.4.4.2.3.cmml">⁢</mo><mi id="S4.E4.m1.4.4.2.5" xref="S4.E4.m1.4.4.2.5.cmml">M</mi><mo id="S4.E4.m1.4.4.2.3a" lspace="0.167em" xref="S4.E4.m1.4.4.2.3.cmml">⁢</mo><mrow id="S4.E4.m1.4.4.2.2.4" xref="S4.E4.m1.4.4.2.2.3.cmml"><mi id="S4.E4.m1.4.4.2.2.2.2" xref="S4.E4.m1.4.4.2.2.3.1.cmml">log</mi><mo id="S4.E4.m1.4.4.2.2.4a" xref="S4.E4.m1.4.4.2.2.3.1.cmml">⁡</mo><mrow id="S4.E4.m1.4.4.2.2.4.1" xref="S4.E4.m1.4.4.2.2.3.cmml"><mo id="S4.E4.m1.4.4.2.2.4.1.1" xref="S4.E4.m1.4.4.2.2.3.1.cmml">(</mo><mrow id="S4.E4.m1.2.2.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.1.cmml"><mn id="S4.E4.m1.2.2.1.1.1.1.1.2" xref="S4.E4.m1.2.2.1.1.1.1.1.2.cmml">1</mn><mo id="S4.E4.m1.2.2.1.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.1.1.cmml">/</mo><mi id="S4.E4.m1.2.2.1.1.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S4.E4.m1.4.4.2.2.4.1.2" xref="S4.E4.m1.4.4.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mrow id="S4.E4.m1.4.4.4" xref="S4.E4.m1.4.4.4.cmml"><mi id="S4.E4.m1.4.4.4.1" xref="S4.E4.m1.4.4.4.1.cmml">log</mi><mo id="S4.E4.m1.4.4.4a" lspace="0.167em" xref="S4.E4.m1.4.4.4.cmml">⁡</mo><mi id="S4.E4.m1.4.4.4.2" xref="S4.E4.m1.4.4.4.2.cmml">M</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.E4.m1.4b"><apply id="S4.E4.m1.4.5.cmml" xref="S4.E4.m1.4.5"><csymbol cd="latexml" id="S4.E4.m1.4.5.1.cmml" xref="S4.E4.m1.4.5.1">greater-than-or-equivalent-to</csymbol><apply id="S4.E4.m1.3.3.cmml" xref="S4.E4.m1.3.3"><divide id="S4.E4.m1.3.3.3.cmml" xref="S4.E4.m1.3.3"></divide><apply id="S4.E4.m1.3.3.2.cmml" xref="S4.E4.m1.3.3.2"><times id="S4.E4.m1.3.3.2.3.cmml" xref="S4.E4.m1.3.3.2.3"></times><ci id="S4.E4.m1.3.3.2.4.cmml" xref="S4.E4.m1.3.3.2.4">𝐾</ci><apply id="S4.E4.m1.3.3.2.2.3.cmml" xref="S4.E4.m1.3.3.2.2.4"><log id="S4.E4.m1.3.3.2.2.3.1.cmml" xref="S4.E4.m1.3.3.2.2.2.2"></log><apply 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id="S4.E4.m1.4.4.2.3.cmml" xref="S4.E4.m1.4.4.2.3"></times><ci id="S4.E4.m1.4.4.2.4.cmml" xref="S4.E4.m1.4.4.2.4">𝑛</ci><ci id="S4.E4.m1.4.4.2.5.cmml" xref="S4.E4.m1.4.4.2.5">𝑀</ci><apply id="S4.E4.m1.4.4.2.2.3.cmml" xref="S4.E4.m1.4.4.2.2.4"><log id="S4.E4.m1.4.4.2.2.3.1.cmml" xref="S4.E4.m1.4.4.2.2.2.2"></log><apply id="S4.E4.m1.2.2.1.1.1.1.1.cmml" xref="S4.E4.m1.2.2.1.1.1.1.1"><divide id="S4.E4.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E4.m1.2.2.1.1.1.1.1.1"></divide><cn id="S4.E4.m1.2.2.1.1.1.1.1.2.cmml" type="integer" xref="S4.E4.m1.2.2.1.1.1.1.1.2">1</cn><ci id="S4.E4.m1.2.2.1.1.1.1.1.3.cmml" xref="S4.E4.m1.2.2.1.1.1.1.1.3">𝜀</ci></apply></apply></apply><apply id="S4.E4.m1.4.4.4.cmml" xref="S4.E4.m1.4.4.4"><log id="S4.E4.m1.4.4.4.1.cmml" xref="S4.E4.m1.4.4.4.1"></log><ci id="S4.E4.m1.4.4.4.2.cmml" xref="S4.E4.m1.4.4.4.2">𝑀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.4c">\frac{K\log(\Omega(1/\varepsilon))}{\log M}\gtrsim\frac{nM\log(1/\varepsilon)}% {\log M}</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.4d">divide start_ARG italic_K roman_log ( start_ARG roman_Ω ( 1 / italic_ε ) end_ARG ) end_ARG start_ARG roman_log italic_M end_ARG ≳ divide start_ARG italic_n italic_M roman_log ( start_ARG 1 / italic_ε end_ARG ) end_ARG start_ARG roman_log italic_M end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS3.1.p1.14">rounds, as desired. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Learning the Utility in the No-Regret Model</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We now study the more difficult no-regret model. As before, we will start with the single-agent case as a warm-up, before proceeding to the multi-agent case.</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>The single-agent case</h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.7">In this subsection, it will be convenient to view the single agent’s utility function as a vector <math alttext="{\bm{u}}\in[0,1]^{m}" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.2"><semantics id="S5.SS1.p1.1.m1.2a"><mrow id="S5.SS1.p1.1.m1.2.3" xref="S5.SS1.p1.1.m1.2.3.cmml"><mi id="S5.SS1.p1.1.m1.2.3.2" xref="S5.SS1.p1.1.m1.2.3.2.cmml">𝒖</mi><mo id="S5.SS1.p1.1.m1.2.3.1" xref="S5.SS1.p1.1.m1.2.3.1.cmml">∈</mo><msup id="S5.SS1.p1.1.m1.2.3.3" xref="S5.SS1.p1.1.m1.2.3.3.cmml"><mrow id="S5.SS1.p1.1.m1.2.3.3.2.2" xref="S5.SS1.p1.1.m1.2.3.3.2.1.cmml"><mo id="S5.SS1.p1.1.m1.2.3.3.2.2.1" stretchy="false" 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id="S5.SS1.p1.1.m1.1.1.cmml" type="integer" xref="S5.SS1.p1.1.m1.1.1">0</cn><cn id="S5.SS1.p1.1.m1.2.2.cmml" type="integer" xref="S5.SS1.p1.1.m1.2.2">1</cn></interval><ci id="S5.SS1.p1.1.m1.2.3.3.3.cmml" xref="S5.SS1.p1.1.m1.2.3.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.2c">{\bm{u}}\in[0,1]^{m}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.2d">bold_italic_u ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math>, and similarly the payment <math alttext="P^{t}:[m]\to\mathbb{R}" class="ltx_Math" display="inline" id="S5.SS1.p1.2.m2.1"><semantics id="S5.SS1.p1.2.m2.1a"><mrow id="S5.SS1.p1.2.m2.1.2" xref="S5.SS1.p1.2.m2.1.2.cmml"><msup id="S5.SS1.p1.2.m2.1.2.2" xref="S5.SS1.p1.2.m2.1.2.2.cmml"><mi id="S5.SS1.p1.2.m2.1.2.2.2" xref="S5.SS1.p1.2.m2.1.2.2.2.cmml">P</mi><mi id="S5.SS1.p1.2.m2.1.2.2.3" xref="S5.SS1.p1.2.m2.1.2.2.3.cmml">t</mi></msup><mo id="S5.SS1.p1.2.m2.1.2.1" lspace="0.278em" rspace="0.278em" xref="S5.SS1.p1.2.m2.1.2.1.cmml">:</mo><mrow id="S5.SS1.p1.2.m2.1.2.3" xref="S5.SS1.p1.2.m2.1.2.3.cmml"><mrow id="S5.SS1.p1.2.m2.1.2.3.2.2" xref="S5.SS1.p1.2.m2.1.2.3.2.1.cmml"><mo id="S5.SS1.p1.2.m2.1.2.3.2.2.1" stretchy="false" xref="S5.SS1.p1.2.m2.1.2.3.2.1.1.cmml">[</mo><mi id="S5.SS1.p1.2.m2.1.1" xref="S5.SS1.p1.2.m2.1.1.cmml">m</mi><mo id="S5.SS1.p1.2.m2.1.2.3.2.2.2" stretchy="false" xref="S5.SS1.p1.2.m2.1.2.3.2.1.1.cmml">]</mo></mrow><mo id="S5.SS1.p1.2.m2.1.2.3.1" stretchy="false" xref="S5.SS1.p1.2.m2.1.2.3.1.cmml">→</mo><mi id="S5.SS1.p1.2.m2.1.2.3.3" xref="S5.SS1.p1.2.m2.1.2.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.2.m2.1b"><apply id="S5.SS1.p1.2.m2.1.2.cmml" xref="S5.SS1.p1.2.m2.1.2"><ci id="S5.SS1.p1.2.m2.1.2.1.cmml" xref="S5.SS1.p1.2.m2.1.2.1">:</ci><apply id="S5.SS1.p1.2.m2.1.2.2.cmml" xref="S5.SS1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S5.SS1.p1.2.m2.1.2.2.1.cmml" xref="S5.SS1.p1.2.m2.1.2.2">superscript</csymbol><ci id="S5.SS1.p1.2.m2.1.2.2.2.cmml" xref="S5.SS1.p1.2.m2.1.2.2.2">𝑃</ci><ci id="S5.SS1.p1.2.m2.1.2.2.3.cmml" xref="S5.SS1.p1.2.m2.1.2.2.3">𝑡</ci></apply><apply id="S5.SS1.p1.2.m2.1.2.3.cmml" xref="S5.SS1.p1.2.m2.1.2.3"><ci id="S5.SS1.p1.2.m2.1.2.3.1.cmml" xref="S5.SS1.p1.2.m2.1.2.3.1">→</ci><apply id="S5.SS1.p1.2.m2.1.2.3.2.1.cmml" xref="S5.SS1.p1.2.m2.1.2.3.2.2"><csymbol cd="latexml" id="S5.SS1.p1.2.m2.1.2.3.2.1.1.cmml" xref="S5.SS1.p1.2.m2.1.2.3.2.2.1">delimited-[]</csymbol><ci id="S5.SS1.p1.2.m2.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1">𝑚</ci></apply><ci id="S5.SS1.p1.2.m2.1.2.3.3.cmml" xref="S5.SS1.p1.2.m2.1.2.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.2.m2.1c">P^{t}:[m]\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.2.m2.1d">italic_P start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT : [ italic_m ] → blackboard_R</annotation></semantics></math> as vector <math alttext="{\bm{p}}^{t}\in\mathbb{R}^{m}" class="ltx_Math" display="inline" id="S5.SS1.p1.3.m3.1"><semantics id="S5.SS1.p1.3.m3.1a"><mrow id="S5.SS1.p1.3.m3.1.1" xref="S5.SS1.p1.3.m3.1.1.cmml"><msup id="S5.SS1.p1.3.m3.1.1.2" xref="S5.SS1.p1.3.m3.1.1.2.cmml"><mi id="S5.SS1.p1.3.m3.1.1.2.2" xref="S5.SS1.p1.3.m3.1.1.2.2.cmml">𝒑</mi><mi id="S5.SS1.p1.3.m3.1.1.2.3" xref="S5.SS1.p1.3.m3.1.1.2.3.cmml">t</mi></msup><mo id="S5.SS1.p1.3.m3.1.1.1" xref="S5.SS1.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S5.SS1.p1.3.m3.1.1.3" xref="S5.SS1.p1.3.m3.1.1.3.cmml"><mi id="S5.SS1.p1.3.m3.1.1.3.2" xref="S5.SS1.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S5.SS1.p1.3.m3.1.1.3.3" xref="S5.SS1.p1.3.m3.1.1.3.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.3.m3.1b"><apply id="S5.SS1.p1.3.m3.1.1.cmml" xref="S5.SS1.p1.3.m3.1.1"><in id="S5.SS1.p1.3.m3.1.1.1.cmml" xref="S5.SS1.p1.3.m3.1.1.1"></in><apply id="S5.SS1.p1.3.m3.1.1.2.cmml" xref="S5.SS1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.1.1.2.1.cmml" xref="S5.SS1.p1.3.m3.1.1.2">superscript</csymbol><ci id="S5.SS1.p1.3.m3.1.1.2.2.cmml" xref="S5.SS1.p1.3.m3.1.1.2.2">𝒑</ci><ci id="S5.SS1.p1.3.m3.1.1.2.3.cmml" xref="S5.SS1.p1.3.m3.1.1.2.3">𝑡</ci></apply><apply id="S5.SS1.p1.3.m3.1.1.3.cmml" xref="S5.SS1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.1.1.3.1.cmml" xref="S5.SS1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S5.SS1.p1.3.m3.1.1.3.2.cmml" xref="S5.SS1.p1.3.m3.1.1.3.2">ℝ</ci><ci id="S5.SS1.p1.3.m3.1.1.3.3.cmml" xref="S5.SS1.p1.3.m3.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.3.m3.1c">{\bm{p}}^{t}\in\mathbb{R}^{m}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.3.m3.1d">bold_italic_p start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> and total utility <math alttext="{\bm{u}}^{t}:={\bm{u}}+{\bm{p}}^{t}" class="ltx_Math" display="inline" id="S5.SS1.p1.4.m4.1"><semantics id="S5.SS1.p1.4.m4.1a"><mrow id="S5.SS1.p1.4.m4.1.1" xref="S5.SS1.p1.4.m4.1.1.cmml"><msup id="S5.SS1.p1.4.m4.1.1.2" xref="S5.SS1.p1.4.m4.1.1.2.cmml"><mi id="S5.SS1.p1.4.m4.1.1.2.2" xref="S5.SS1.p1.4.m4.1.1.2.2.cmml">𝒖</mi><mi id="S5.SS1.p1.4.m4.1.1.2.3" xref="S5.SS1.p1.4.m4.1.1.2.3.cmml">t</mi></msup><mo id="S5.SS1.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.SS1.p1.4.m4.1.1.1.cmml">:=</mo><mrow id="S5.SS1.p1.4.m4.1.1.3" xref="S5.SS1.p1.4.m4.1.1.3.cmml"><mi id="S5.SS1.p1.4.m4.1.1.3.2" xref="S5.SS1.p1.4.m4.1.1.3.2.cmml">𝒖</mi><mo id="S5.SS1.p1.4.m4.1.1.3.1" xref="S5.SS1.p1.4.m4.1.1.3.1.cmml">+</mo><msup id="S5.SS1.p1.4.m4.1.1.3.3" xref="S5.SS1.p1.4.m4.1.1.3.3.cmml"><mi id="S5.SS1.p1.4.m4.1.1.3.3.2" xref="S5.SS1.p1.4.m4.1.1.3.3.2.cmml">𝒑</mi><mi id="S5.SS1.p1.4.m4.1.1.3.3.3" xref="S5.SS1.p1.4.m4.1.1.3.3.3.cmml">t</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.4.m4.1b"><apply id="S5.SS1.p1.4.m4.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1"><csymbol cd="latexml" id="S5.SS1.p1.4.m4.1.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1">assign</csymbol><apply id="S5.SS1.p1.4.m4.1.1.2.cmml" xref="S5.SS1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p1.4.m4.1.1.2.1.cmml" xref="S5.SS1.p1.4.m4.1.1.2">superscript</csymbol><ci id="S5.SS1.p1.4.m4.1.1.2.2.cmml" xref="S5.SS1.p1.4.m4.1.1.2.2">𝒖</ci><ci id="S5.SS1.p1.4.m4.1.1.2.3.cmml" xref="S5.SS1.p1.4.m4.1.1.2.3">𝑡</ci></apply><apply id="S5.SS1.p1.4.m4.1.1.3.cmml" xref="S5.SS1.p1.4.m4.1.1.3"><plus id="S5.SS1.p1.4.m4.1.1.3.1.cmml" xref="S5.SS1.p1.4.m4.1.1.3.1"></plus><ci id="S5.SS1.p1.4.m4.1.1.3.2.cmml" xref="S5.SS1.p1.4.m4.1.1.3.2">𝒖</ci><apply id="S5.SS1.p1.4.m4.1.1.3.3.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS1.p1.4.m4.1.1.3.3.1.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3">superscript</csymbol><ci id="S5.SS1.p1.4.m4.1.1.3.3.2.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.2">𝒑</ci><ci id="S5.SS1.p1.4.m4.1.1.3.3.3.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.4.m4.1c">{\bm{u}}^{t}:={\bm{u}}+{\bm{p}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.4.m4.1d">bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT := bold_italic_u + bold_italic_p start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. To simplify notations, we subtract the average utility of all actions from the utility of each action: <math alttext="{\bm{u}}\leftarrow{\bm{u}}-\expectationvalue{\bm{1},{\bm{u}}}\bm{1}/m" class="ltx_Math" display="inline" id="S5.SS1.p1.5.m5.1"><semantics id="S5.SS1.p1.5.m5.1a"><mrow id="S5.SS1.p1.5.m5.1.2" xref="S5.SS1.p1.5.m5.1.2.cmml"><mi id="S5.SS1.p1.5.m5.1.2.2" xref="S5.SS1.p1.5.m5.1.2.2.cmml">𝒖</mi><mo id="S5.SS1.p1.5.m5.1.2.1" stretchy="false" xref="S5.SS1.p1.5.m5.1.2.1.cmml">←</mo><mrow id="S5.SS1.p1.5.m5.1.2.3" xref="S5.SS1.p1.5.m5.1.2.3.cmml"><mi id="S5.SS1.p1.5.m5.1.2.3.2" xref="S5.SS1.p1.5.m5.1.2.3.2.cmml">𝒖</mi><mo id="S5.SS1.p1.5.m5.1.2.3.1" xref="S5.SS1.p1.5.m5.1.2.3.1.cmml">−</mo><mrow id="S5.SS1.p1.5.m5.1.2.3.3" xref="S5.SS1.p1.5.m5.1.2.3.3.cmml"><mrow id="S5.SS1.p1.5.m5.1.2.3.3.2" xref="S5.SS1.p1.5.m5.1.2.3.3.2.cmml"><mrow id="S5.SS1.p1.5.m5.1.1.3" xref="S5.SS1.p1.5.m5.1.1.2.cmml"><mo id="S5.SS1.p1.5.m5.1.1.3.1" xref="S5.SS1.p1.5.m5.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.p1.5.m5.1.1.1.1.1.4" xref="S5.SS1.p1.5.m5.1.1.1.1.1.3.cmml"><mn id="S5.SS1.p1.5.m5.1.1.1.1.1.1" xref="S5.SS1.p1.5.m5.1.1.1.1.1.1.cmml">𝟏</mn><mo id="S5.SS1.p1.5.m5.1.1.1.1.1.4.1" xref="S5.SS1.p1.5.m5.1.1.1.1.1.3.cmml">,</mo><mi id="S5.SS1.p1.5.m5.1.1.1.1.1.2" xref="S5.SS1.p1.5.m5.1.1.1.1.1.2.cmml">𝒖</mi></mrow><mo id="S5.SS1.p1.5.m5.1.1.3.2" xref="S5.SS1.p1.5.m5.1.1.2.1.cmml">⟩</mo></mrow><mo id="S5.SS1.p1.5.m5.1.2.3.3.2.1" xref="S5.SS1.p1.5.m5.1.2.3.3.2.1.cmml">⁢</mo><mn id="S5.SS1.p1.5.m5.1.2.3.3.2.2" xref="S5.SS1.p1.5.m5.1.2.3.3.2.2.cmml">𝟏</mn></mrow><mo id="S5.SS1.p1.5.m5.1.2.3.3.1" xref="S5.SS1.p1.5.m5.1.2.3.3.1.cmml">/</mo><mi id="S5.SS1.p1.5.m5.1.2.3.3.3" xref="S5.SS1.p1.5.m5.1.2.3.3.3.cmml">m</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.5.m5.1b"><apply id="S5.SS1.p1.5.m5.1.2.cmml" xref="S5.SS1.p1.5.m5.1.2"><ci id="S5.SS1.p1.5.m5.1.2.1.cmml" xref="S5.SS1.p1.5.m5.1.2.1">←</ci><ci id="S5.SS1.p1.5.m5.1.2.2.cmml" xref="S5.SS1.p1.5.m5.1.2.2">𝒖</ci><apply id="S5.SS1.p1.5.m5.1.2.3.cmml" xref="S5.SS1.p1.5.m5.1.2.3"><minus id="S5.SS1.p1.5.m5.1.2.3.1.cmml" xref="S5.SS1.p1.5.m5.1.2.3.1"></minus><ci id="S5.SS1.p1.5.m5.1.2.3.2.cmml" xref="S5.SS1.p1.5.m5.1.2.3.2">𝒖</ci><apply id="S5.SS1.p1.5.m5.1.2.3.3.cmml" xref="S5.SS1.p1.5.m5.1.2.3.3"><divide id="S5.SS1.p1.5.m5.1.2.3.3.1.cmml" xref="S5.SS1.p1.5.m5.1.2.3.3.1"></divide><apply id="S5.SS1.p1.5.m5.1.2.3.3.2.cmml" xref="S5.SS1.p1.5.m5.1.2.3.3.2"><times id="S5.SS1.p1.5.m5.1.2.3.3.2.1.cmml" xref="S5.SS1.p1.5.m5.1.2.3.3.2.1"></times><apply id="S5.SS1.p1.5.m5.1.1.2.cmml" xref="S5.SS1.p1.5.m5.1.1.3"><csymbol cd="latexml" id="S5.SS1.p1.5.m5.1.1.2.1.cmml" xref="S5.SS1.p1.5.m5.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.p1.5.m5.1.1.1.1.1.3.cmml" xref="S5.SS1.p1.5.m5.1.1.1.1.1.4"><cn id="S5.SS1.p1.5.m5.1.1.1.1.1.1.cmml" type="integer" xref="S5.SS1.p1.5.m5.1.1.1.1.1.1">1</cn><ci id="S5.SS1.p1.5.m5.1.1.1.1.1.2.cmml" xref="S5.SS1.p1.5.m5.1.1.1.1.1.2">𝒖</ci></list></apply><cn id="S5.SS1.p1.5.m5.1.2.3.3.2.2.cmml" type="integer" xref="S5.SS1.p1.5.m5.1.2.3.3.2.2">1</cn></apply><ci id="S5.SS1.p1.5.m5.1.2.3.3.3.cmml" xref="S5.SS1.p1.5.m5.1.2.3.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.5.m5.1c">{\bm{u}}\leftarrow{\bm{u}}-\expectationvalue{\bm{1},{\bm{u}}}\bm{1}/m</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.5.m5.1d">bold_italic_u ← bold_italic_u - ⟨ start_ARG bold_1 , bold_italic_u end_ARG ⟩ bold_1 / italic_m</annotation></semantics></math>, so that <math alttext="{\bm{u}}\in[-1,1]^{m}" class="ltx_Math" display="inline" id="S5.SS1.p1.6.m6.2"><semantics id="S5.SS1.p1.6.m6.2a"><mrow id="S5.SS1.p1.6.m6.2.2" xref="S5.SS1.p1.6.m6.2.2.cmml"><mi id="S5.SS1.p1.6.m6.2.2.3" xref="S5.SS1.p1.6.m6.2.2.3.cmml">𝒖</mi><mo id="S5.SS1.p1.6.m6.2.2.2" xref="S5.SS1.p1.6.m6.2.2.2.cmml">∈</mo><msup id="S5.SS1.p1.6.m6.2.2.1" xref="S5.SS1.p1.6.m6.2.2.1.cmml"><mrow id="S5.SS1.p1.6.m6.2.2.1.1.1" xref="S5.SS1.p1.6.m6.2.2.1.1.2.cmml"><mo id="S5.SS1.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="S5.SS1.p1.6.m6.2.2.1.1.2.cmml">[</mo><mrow id="S5.SS1.p1.6.m6.2.2.1.1.1.1" xref="S5.SS1.p1.6.m6.2.2.1.1.1.1.cmml"><mo id="S5.SS1.p1.6.m6.2.2.1.1.1.1a" xref="S5.SS1.p1.6.m6.2.2.1.1.1.1.cmml">−</mo><mn id="S5.SS1.p1.6.m6.2.2.1.1.1.1.2" xref="S5.SS1.p1.6.m6.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S5.SS1.p1.6.m6.2.2.1.1.1.3" xref="S5.SS1.p1.6.m6.2.2.1.1.2.cmml">,</mo><mn id="S5.SS1.p1.6.m6.1.1" xref="S5.SS1.p1.6.m6.1.1.cmml">1</mn><mo id="S5.SS1.p1.6.m6.2.2.1.1.1.4" stretchy="false" xref="S5.SS1.p1.6.m6.2.2.1.1.2.cmml">]</mo></mrow><mi id="S5.SS1.p1.6.m6.2.2.1.3" xref="S5.SS1.p1.6.m6.2.2.1.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.6.m6.2b"><apply id="S5.SS1.p1.6.m6.2.2.cmml" xref="S5.SS1.p1.6.m6.2.2"><in id="S5.SS1.p1.6.m6.2.2.2.cmml" xref="S5.SS1.p1.6.m6.2.2.2"></in><ci id="S5.SS1.p1.6.m6.2.2.3.cmml" xref="S5.SS1.p1.6.m6.2.2.3">𝒖</ci><apply id="S5.SS1.p1.6.m6.2.2.1.cmml" xref="S5.SS1.p1.6.m6.2.2.1"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.2.2.1.2.cmml" xref="S5.SS1.p1.6.m6.2.2.1">superscript</csymbol><interval closure="closed" id="S5.SS1.p1.6.m6.2.2.1.1.2.cmml" xref="S5.SS1.p1.6.m6.2.2.1.1.1"><apply id="S5.SS1.p1.6.m6.2.2.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.2.2.1.1.1.1"><minus id="S5.SS1.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.2.2.1.1.1.1"></minus><cn id="S5.SS1.p1.6.m6.2.2.1.1.1.1.2.cmml" type="integer" xref="S5.SS1.p1.6.m6.2.2.1.1.1.1.2">1</cn></apply><cn id="S5.SS1.p1.6.m6.1.1.cmml" type="integer" xref="S5.SS1.p1.6.m6.1.1">1</cn></interval><ci id="S5.SS1.p1.6.m6.2.2.1.3.cmml" xref="S5.SS1.p1.6.m6.2.2.1.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.6.m6.2c">{\bm{u}}\in[-1,1]^{m}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.6.m6.2d">bold_italic_u ∈ [ - 1 , 1 ] start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\expectationvalue{\bm{1},{\bm{u}}}=0" class="ltx_Math" display="inline" id="S5.SS1.p1.7.m7.1"><semantics id="S5.SS1.p1.7.m7.1a"><mrow id="S5.SS1.p1.7.m7.1.2" xref="S5.SS1.p1.7.m7.1.2.cmml"><mrow id="S5.SS1.p1.7.m7.1.1.3" xref="S5.SS1.p1.7.m7.1.1.2.cmml"><mo id="S5.SS1.p1.7.m7.1.1.3.1" xref="S5.SS1.p1.7.m7.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.p1.7.m7.1.1.1.1.1.4" xref="S5.SS1.p1.7.m7.1.1.1.1.1.3.cmml"><mn id="S5.SS1.p1.7.m7.1.1.1.1.1.1" xref="S5.SS1.p1.7.m7.1.1.1.1.1.1.cmml">𝟏</mn><mo id="S5.SS1.p1.7.m7.1.1.1.1.1.4.1" xref="S5.SS1.p1.7.m7.1.1.1.1.1.3.cmml">,</mo><mi id="S5.SS1.p1.7.m7.1.1.1.1.1.2" xref="S5.SS1.p1.7.m7.1.1.1.1.1.2.cmml">𝒖</mi></mrow><mo id="S5.SS1.p1.7.m7.1.1.3.2" xref="S5.SS1.p1.7.m7.1.1.2.1.cmml">⟩</mo></mrow><mo id="S5.SS1.p1.7.m7.1.2.1" xref="S5.SS1.p1.7.m7.1.2.1.cmml">=</mo><mn id="S5.SS1.p1.7.m7.1.2.2" xref="S5.SS1.p1.7.m7.1.2.2.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.7.m7.1b"><apply id="S5.SS1.p1.7.m7.1.2.cmml" xref="S5.SS1.p1.7.m7.1.2"><eq id="S5.SS1.p1.7.m7.1.2.1.cmml" xref="S5.SS1.p1.7.m7.1.2.1"></eq><apply id="S5.SS1.p1.7.m7.1.1.2.cmml" xref="S5.SS1.p1.7.m7.1.1.3"><csymbol cd="latexml" id="S5.SS1.p1.7.m7.1.1.2.1.cmml" xref="S5.SS1.p1.7.m7.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.p1.7.m7.1.1.1.1.1.3.cmml" xref="S5.SS1.p1.7.m7.1.1.1.1.1.4"><cn id="S5.SS1.p1.7.m7.1.1.1.1.1.1.cmml" type="integer" xref="S5.SS1.p1.7.m7.1.1.1.1.1.1">1</cn><ci id="S5.SS1.p1.7.m7.1.1.1.1.1.2.cmml" xref="S5.SS1.p1.7.m7.1.1.1.1.1.2">𝒖</ci></list></apply><cn id="S5.SS1.p1.7.m7.1.2.2.cmml" type="integer" xref="S5.SS1.p1.7.m7.1.2.2">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.7.m7.1c">\expectationvalue{\bm{1},{\bm{u}}}=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.7.m7.1d">⟨ start_ARG bold_1 , bold_italic_u end_ARG ⟩ = 0</annotation></semantics></math>. By the discussion in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S3.SS3" title="3.3 Game equivalence and formal goal statement ‣ 3 Our Setting ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.3</span></a>, this does not change the principal’s learning problem.</p> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.2">The no-regret model presents challenges that do not appear in the rationalizable model. In particular, the binary search algorithm (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg1" title="In 4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a>) cannot be applied, because no-regret agents may not respond instantaneously to changes to the payment function. Historical payments affect the agent’s future behavior. Moreover, the agent may incur <span class="ltx_text ltx_font_italic" id="S5.SS1.p2.2.1">negative</span> regret over time, making it difficult to learn anything from the agent’s behavior on subsequent rounds. (For example, if an agent has regret <math alttext="-K" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.1"><semantics id="S5.SS1.p2.1.m1.1a"><mrow id="S5.SS1.p2.1.m1.1.1" xref="S5.SS1.p2.1.m1.1.1.cmml"><mo id="S5.SS1.p2.1.m1.1.1a" xref="S5.SS1.p2.1.m1.1.1.cmml">−</mo><mi id="S5.SS1.p2.1.m1.1.1.2" xref="S5.SS1.p2.1.m1.1.1.2.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.1b"><apply id="S5.SS1.p2.1.m1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1"><minus id="S5.SS1.p2.1.m1.1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1"></minus><ci id="S5.SS1.p2.1.m1.1.1.2.cmml" xref="S5.SS1.p2.1.m1.1.1.2">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.1c">-K</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.1d">- italic_K</annotation></semantics></math> for all actions, then one cannot say anything at all about how the agent will behave in the next <math alttext="K" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m2.1"><semantics id="S5.SS1.p2.2.m2.1a"><mi id="S5.SS1.p2.2.m2.1.1" xref="S5.SS1.p2.2.m2.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m2.1b"><ci id="S5.SS1.p2.2.m2.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m2.1c">K</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m2.1d">italic_K</annotation></semantics></math> rounds.)</p> </div> <div class="ltx_para" id="S5.SS1.p3"> <p class="ltx_p" id="S5.SS1.p3.1">For single-agent games, we will also not use signaling, <span class="ltx_text ltx_font_italic" id="S5.SS1.p3.1.1">i.e.</span>, it will be enough to set <math alttext="|S_{i}|=1" class="ltx_Math" display="inline" id="S5.SS1.p3.1.m1.1"><semantics id="S5.SS1.p3.1.m1.1a"><mrow id="S5.SS1.p3.1.m1.1.1" xref="S5.SS1.p3.1.m1.1.1.cmml"><mrow id="S5.SS1.p3.1.m1.1.1.1.1" xref="S5.SS1.p3.1.m1.1.1.1.2.cmml"><mo id="S5.SS1.p3.1.m1.1.1.1.1.2" stretchy="false" xref="S5.SS1.p3.1.m1.1.1.1.2.1.cmml">|</mo><msub id="S5.SS1.p3.1.m1.1.1.1.1.1" xref="S5.SS1.p3.1.m1.1.1.1.1.1.cmml"><mi id="S5.SS1.p3.1.m1.1.1.1.1.1.2" xref="S5.SS1.p3.1.m1.1.1.1.1.1.2.cmml">S</mi><mi id="S5.SS1.p3.1.m1.1.1.1.1.1.3" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS1.p3.1.m1.1.1.1.1.3" stretchy="false" xref="S5.SS1.p3.1.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS1.p3.1.m1.1.1.2" xref="S5.SS1.p3.1.m1.1.1.2.cmml">=</mo><mn id="S5.SS1.p3.1.m1.1.1.3" xref="S5.SS1.p3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.1.m1.1b"><apply id="S5.SS1.p3.1.m1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1"><eq id="S5.SS1.p3.1.m1.1.1.2.cmml" xref="S5.SS1.p3.1.m1.1.1.2"></eq><apply id="S5.SS1.p3.1.m1.1.1.1.2.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1"><abs id="S5.SS1.p3.1.m1.1.1.1.2.1.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.2"></abs><apply id="S5.SS1.p3.1.m1.1.1.1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p3.1.m1.1.1.1.1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS1.p3.1.m1.1.1.1.1.1.2.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1.2">𝑆</ci><ci id="S5.SS1.p3.1.m1.1.1.1.1.1.3.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S5.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S5.SS1.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.1.m1.1c">|S_{i}|=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.1.m1.1d">| italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS1.p4"> <p class="ltx_p" id="S5.SS1.p4.7">The key idea in our algorithm is to imagine the principal and agent as playing a zero-sum game where the principal selects the payment function <math alttext="{\bm{p}}" class="ltx_Math" display="inline" id="S5.SS1.p4.1.m1.1"><semantics id="S5.SS1.p4.1.m1.1a"><mi id="S5.SS1.p4.1.m1.1.1" xref="S5.SS1.p4.1.m1.1.1.cmml">𝒑</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.1.m1.1b"><ci id="S5.SS1.p4.1.m1.1.1.cmml" xref="S5.SS1.p4.1.m1.1.1">𝒑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.1.m1.1c">{\bm{p}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.1.m1.1d">bold_italic_p</annotation></semantics></math> from some set <math alttext="{\mathcal{P}}" class="ltx_Math" display="inline" id="S5.SS1.p4.2.m2.1"><semantics id="S5.SS1.p4.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.p4.2.m2.1.1" xref="S5.SS1.p4.2.m2.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.2.m2.1b"><ci id="S5.SS1.p4.2.m2.1.1.cmml" xref="S5.SS1.p4.2.m2.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.2.m2.1c">{\mathcal{P}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.2.m2.1d">caligraphic_P</annotation></semantics></math> to be specified later, the agent selects <math alttext="{\bm{x}}\in\Delta(m)" class="ltx_Math" display="inline" id="S5.SS1.p4.3.m3.1"><semantics id="S5.SS1.p4.3.m3.1a"><mrow id="S5.SS1.p4.3.m3.1.2" xref="S5.SS1.p4.3.m3.1.2.cmml"><mi id="S5.SS1.p4.3.m3.1.2.2" xref="S5.SS1.p4.3.m3.1.2.2.cmml">𝒙</mi><mo id="S5.SS1.p4.3.m3.1.2.1" xref="S5.SS1.p4.3.m3.1.2.1.cmml">∈</mo><mrow id="S5.SS1.p4.3.m3.1.2.3" xref="S5.SS1.p4.3.m3.1.2.3.cmml"><mi id="S5.SS1.p4.3.m3.1.2.3.2" mathvariant="normal" xref="S5.SS1.p4.3.m3.1.2.3.2.cmml">Δ</mi><mo id="S5.SS1.p4.3.m3.1.2.3.1" xref="S5.SS1.p4.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS1.p4.3.m3.1.2.3.3.2" xref="S5.SS1.p4.3.m3.1.2.3.cmml"><mo id="S5.SS1.p4.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.SS1.p4.3.m3.1.2.3.cmml">(</mo><mi id="S5.SS1.p4.3.m3.1.1" xref="S5.SS1.p4.3.m3.1.1.cmml">m</mi><mo id="S5.SS1.p4.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.SS1.p4.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.3.m3.1b"><apply id="S5.SS1.p4.3.m3.1.2.cmml" xref="S5.SS1.p4.3.m3.1.2"><in id="S5.SS1.p4.3.m3.1.2.1.cmml" xref="S5.SS1.p4.3.m3.1.2.1"></in><ci id="S5.SS1.p4.3.m3.1.2.2.cmml" xref="S5.SS1.p4.3.m3.1.2.2">𝒙</ci><apply id="S5.SS1.p4.3.m3.1.2.3.cmml" xref="S5.SS1.p4.3.m3.1.2.3"><times id="S5.SS1.p4.3.m3.1.2.3.1.cmml" xref="S5.SS1.p4.3.m3.1.2.3.1"></times><ci id="S5.SS1.p4.3.m3.1.2.3.2.cmml" xref="S5.SS1.p4.3.m3.1.2.3.2">Δ</ci><ci id="S5.SS1.p4.3.m3.1.1.cmml" xref="S5.SS1.p4.3.m3.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.3.m3.1c">{\bm{x}}\in\Delta(m)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.3.m3.1d">bold_italic_x ∈ roman_Δ ( italic_m )</annotation></semantics></math>, the agent’s utility is given by <math alttext="\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}" class="ltx_Math" display="inline" id="S5.SS1.p4.4.m4.1"><semantics id="S5.SS1.p4.4.m4.1a"><mrow id="S5.SS1.p4.4.m4.1.1.3" xref="S5.SS1.p4.4.m4.1.1.2.cmml"><mo id="S5.SS1.p4.4.m4.1.1.3.1" stretchy="false" xref="S5.SS1.p4.4.m4.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.p4.4.m4.1.1.1.1.1.2" xref="S5.SS1.p4.4.m4.1.1.1.1.1.3.cmml"><mrow id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.cmml"><mi id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.2" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.2.cmml">𝒖</mi><mo id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.1" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.1.cmml">+</mo><mi id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.3" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.3.cmml">𝒑</mi></mrow><mo id="S5.SS1.p4.4.m4.1.1.1.1.1.2.2" xref="S5.SS1.p4.4.m4.1.1.1.1.1.3.cmml">,</mo><mi id="S5.SS1.p4.4.m4.1.1.1.1.1.1" xref="S5.SS1.p4.4.m4.1.1.1.1.1.1.cmml">𝒙</mi></mrow><mo id="S5.SS1.p4.4.m4.1.1.3.2" stretchy="false" xref="S5.SS1.p4.4.m4.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.4.m4.1b"><apply id="S5.SS1.p4.4.m4.1.1.2.cmml" xref="S5.SS1.p4.4.m4.1.1.3"><csymbol cd="latexml" id="S5.SS1.p4.4.m4.1.1.2.1.cmml" xref="S5.SS1.p4.4.m4.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.p4.4.m4.1.1.1.1.1.3.cmml" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2"><apply id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.cmml" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1"><plus id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.1.cmml" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.1"></plus><ci id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.2.cmml" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.2">𝒖</ci><ci id="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.3.cmml" xref="S5.SS1.p4.4.m4.1.1.1.1.1.2.1.3">𝒑</ci></apply><ci id="S5.SS1.p4.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS1.p4.4.m4.1.1.1.1.1.1">𝒙</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.4.m4.1c">\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.4.m4.1d">⟨ start_ARG bold_italic_u + bold_italic_p , bold_italic_x end_ARG ⟩</annotation></semantics></math>, and the principal’s utility is <math alttext="-\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}" class="ltx_Math" display="inline" id="S5.SS1.p4.5.m5.1"><semantics id="S5.SS1.p4.5.m5.1a"><mrow id="S5.SS1.p4.5.m5.1.2" xref="S5.SS1.p4.5.m5.1.2.cmml"><mo id="S5.SS1.p4.5.m5.1.2a" xref="S5.SS1.p4.5.m5.1.2.cmml">−</mo><mrow id="S5.SS1.p4.5.m5.1.1.3" xref="S5.SS1.p4.5.m5.1.1.2.cmml"><mo id="S5.SS1.p4.5.m5.1.1.3.1" stretchy="false" xref="S5.SS1.p4.5.m5.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.p4.5.m5.1.1.1.1.1.2" xref="S5.SS1.p4.5.m5.1.1.1.1.1.3.cmml"><mrow id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.cmml"><mi id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.2" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.2.cmml">𝒖</mi><mo id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.1" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.1.cmml">+</mo><mi id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.3" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.3.cmml">𝒑</mi></mrow><mo id="S5.SS1.p4.5.m5.1.1.1.1.1.2.2" xref="S5.SS1.p4.5.m5.1.1.1.1.1.3.cmml">,</mo><mi id="S5.SS1.p4.5.m5.1.1.1.1.1.1" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.cmml">𝒙</mi></mrow><mo id="S5.SS1.p4.5.m5.1.1.3.2" stretchy="false" xref="S5.SS1.p4.5.m5.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.5.m5.1b"><apply id="S5.SS1.p4.5.m5.1.2.cmml" xref="S5.SS1.p4.5.m5.1.2"><minus id="S5.SS1.p4.5.m5.1.2.1.cmml" xref="S5.SS1.p4.5.m5.1.2"></minus><apply id="S5.SS1.p4.5.m5.1.1.2.cmml" xref="S5.SS1.p4.5.m5.1.1.3"><csymbol cd="latexml" id="S5.SS1.p4.5.m5.1.1.2.1.cmml" xref="S5.SS1.p4.5.m5.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.p4.5.m5.1.1.1.1.1.3.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2"><apply id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1"><plus id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.1"></plus><ci id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.2.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.2">𝒖</ci><ci id="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.3.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.2.1.3">𝒑</ci></apply><ci id="S5.SS1.p4.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1">𝒙</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.5.m5.1c">-\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.5.m5.1d">- ⟨ start_ARG bold_italic_u + bold_italic_p , bold_italic_x end_ARG ⟩</annotation></semantics></math>. Call this game <math alttext="\Gamma_{0}" class="ltx_Math" display="inline" id="S5.SS1.p4.6.m6.1"><semantics id="S5.SS1.p4.6.m6.1a"><msub id="S5.SS1.p4.6.m6.1.1" xref="S5.SS1.p4.6.m6.1.1.cmml"><mi id="S5.SS1.p4.6.m6.1.1.2" mathvariant="normal" xref="S5.SS1.p4.6.m6.1.1.2.cmml">Γ</mi><mn id="S5.SS1.p4.6.m6.1.1.3" xref="S5.SS1.p4.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.6.m6.1b"><apply id="S5.SS1.p4.6.m6.1.1.cmml" xref="S5.SS1.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.6.m6.1.1.1.cmml" xref="S5.SS1.p4.6.m6.1.1">subscript</csymbol><ci id="S5.SS1.p4.6.m6.1.1.2.cmml" xref="S5.SS1.p4.6.m6.1.1.2">Γ</ci><cn id="S5.SS1.p4.6.m6.1.1.3.cmml" type="integer" xref="S5.SS1.p4.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.6.m6.1c">\Gamma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.6.m6.1d">roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. In particular, if we set <math alttext="{\mathcal{P}}=\{{\bm{p}}\in[0,2]^{m}:\expectationvalue{\bm{1},{\bm{p}}}=m\}" class="ltx_Math" display="inline" id="S5.SS1.p4.7.m7.5"><semantics id="S5.SS1.p4.7.m7.5a"><mrow id="S5.SS1.p4.7.m7.5.5" xref="S5.SS1.p4.7.m7.5.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.p4.7.m7.5.5.4" xref="S5.SS1.p4.7.m7.5.5.4.cmml">𝒫</mi><mo id="S5.SS1.p4.7.m7.5.5.3" xref="S5.SS1.p4.7.m7.5.5.3.cmml">=</mo><mrow id="S5.SS1.p4.7.m7.5.5.2.2" xref="S5.SS1.p4.7.m7.5.5.2.3.cmml"><mo id="S5.SS1.p4.7.m7.5.5.2.2.3" stretchy="false" xref="S5.SS1.p4.7.m7.5.5.2.3.1.cmml">{</mo><mrow id="S5.SS1.p4.7.m7.4.4.1.1.1" xref="S5.SS1.p4.7.m7.4.4.1.1.1.cmml"><mi id="S5.SS1.p4.7.m7.4.4.1.1.1.2" xref="S5.SS1.p4.7.m7.4.4.1.1.1.2.cmml">𝒑</mi><mo id="S5.SS1.p4.7.m7.4.4.1.1.1.1" xref="S5.SS1.p4.7.m7.4.4.1.1.1.1.cmml">∈</mo><msup id="S5.SS1.p4.7.m7.4.4.1.1.1.3" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.cmml"><mrow id="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.2" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.1.cmml"><mo id="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.2.1" stretchy="false" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.1.cmml">[</mo><mn id="S5.SS1.p4.7.m7.2.2" xref="S5.SS1.p4.7.m7.2.2.cmml">0</mn><mo id="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.2.2" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.1.cmml">,</mo><mn id="S5.SS1.p4.7.m7.3.3" xref="S5.SS1.p4.7.m7.3.3.cmml">2</mn><mo id="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.2.3" rspace="0.278em" stretchy="false" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.1.cmml">]</mo></mrow><mi id="S5.SS1.p4.7.m7.4.4.1.1.1.3.3" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.3.cmml">m</mi></msup></mrow><mo id="S5.SS1.p4.7.m7.5.5.2.2.4" rspace="0.278em" xref="S5.SS1.p4.7.m7.5.5.2.3.1.cmml">:</mo><mrow id="S5.SS1.p4.7.m7.5.5.2.2.2" xref="S5.SS1.p4.7.m7.5.5.2.2.2.cmml"><mrow id="S5.SS1.p4.7.m7.1.1.3" xref="S5.SS1.p4.7.m7.1.1.2.cmml"><mo id="S5.SS1.p4.7.m7.1.1.3.1" xref="S5.SS1.p4.7.m7.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.p4.7.m7.1.1.1.1.1.4" xref="S5.SS1.p4.7.m7.1.1.1.1.1.3.cmml"><mn id="S5.SS1.p4.7.m7.1.1.1.1.1.1" xref="S5.SS1.p4.7.m7.1.1.1.1.1.1.cmml">𝟏</mn><mo id="S5.SS1.p4.7.m7.1.1.1.1.1.4.1" xref="S5.SS1.p4.7.m7.1.1.1.1.1.3.cmml">,</mo><mi id="S5.SS1.p4.7.m7.1.1.1.1.1.2" xref="S5.SS1.p4.7.m7.1.1.1.1.1.2.cmml">𝒑</mi></mrow><mo id="S5.SS1.p4.7.m7.1.1.3.2" xref="S5.SS1.p4.7.m7.1.1.2.1.cmml">⟩</mo></mrow><mo id="S5.SS1.p4.7.m7.5.5.2.2.2.1" xref="S5.SS1.p4.7.m7.5.5.2.2.2.1.cmml">=</mo><mi id="S5.SS1.p4.7.m7.5.5.2.2.2.2" xref="S5.SS1.p4.7.m7.5.5.2.2.2.2.cmml">m</mi></mrow><mo id="S5.SS1.p4.7.m7.5.5.2.2.5" stretchy="false" xref="S5.SS1.p4.7.m7.5.5.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.7.m7.5b"><apply id="S5.SS1.p4.7.m7.5.5.cmml" xref="S5.SS1.p4.7.m7.5.5"><eq id="S5.SS1.p4.7.m7.5.5.3.cmml" xref="S5.SS1.p4.7.m7.5.5.3"></eq><ci id="S5.SS1.p4.7.m7.5.5.4.cmml" xref="S5.SS1.p4.7.m7.5.5.4">𝒫</ci><apply id="S5.SS1.p4.7.m7.5.5.2.3.cmml" xref="S5.SS1.p4.7.m7.5.5.2.2"><csymbol cd="latexml" id="S5.SS1.p4.7.m7.5.5.2.3.1.cmml" xref="S5.SS1.p4.7.m7.5.5.2.2.3">conditional-set</csymbol><apply id="S5.SS1.p4.7.m7.4.4.1.1.1.cmml" xref="S5.SS1.p4.7.m7.4.4.1.1.1"><in id="S5.SS1.p4.7.m7.4.4.1.1.1.1.cmml" xref="S5.SS1.p4.7.m7.4.4.1.1.1.1"></in><ci id="S5.SS1.p4.7.m7.4.4.1.1.1.2.cmml" xref="S5.SS1.p4.7.m7.4.4.1.1.1.2">𝒑</ci><apply id="S5.SS1.p4.7.m7.4.4.1.1.1.3.cmml" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p4.7.m7.4.4.1.1.1.3.1.cmml" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3">superscript</csymbol><interval closure="closed" id="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.1.cmml" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.2.2"><cn id="S5.SS1.p4.7.m7.2.2.cmml" type="integer" xref="S5.SS1.p4.7.m7.2.2">0</cn><cn id="S5.SS1.p4.7.m7.3.3.cmml" type="integer" xref="S5.SS1.p4.7.m7.3.3">2</cn></interval><ci id="S5.SS1.p4.7.m7.4.4.1.1.1.3.3.cmml" xref="S5.SS1.p4.7.m7.4.4.1.1.1.3.3">𝑚</ci></apply></apply><apply id="S5.SS1.p4.7.m7.5.5.2.2.2.cmml" xref="S5.SS1.p4.7.m7.5.5.2.2.2"><eq id="S5.SS1.p4.7.m7.5.5.2.2.2.1.cmml" xref="S5.SS1.p4.7.m7.5.5.2.2.2.1"></eq><apply id="S5.SS1.p4.7.m7.1.1.2.cmml" xref="S5.SS1.p4.7.m7.1.1.3"><csymbol cd="latexml" id="S5.SS1.p4.7.m7.1.1.2.1.cmml" xref="S5.SS1.p4.7.m7.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.p4.7.m7.1.1.1.1.1.3.cmml" xref="S5.SS1.p4.7.m7.1.1.1.1.1.4"><cn id="S5.SS1.p4.7.m7.1.1.1.1.1.1.cmml" type="integer" xref="S5.SS1.p4.7.m7.1.1.1.1.1.1">1</cn><ci id="S5.SS1.p4.7.m7.1.1.1.1.1.2.cmml" xref="S5.SS1.p4.7.m7.1.1.1.1.1.2">𝒑</ci></list></apply><ci id="S5.SS1.p4.7.m7.5.5.2.2.2.2.cmml" xref="S5.SS1.p4.7.m7.5.5.2.2.2.2">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.7.m7.5c">{\mathcal{P}}=\{{\bm{p}}\in[0,2]^{m}:\expectationvalue{\bm{1},{\bm{p}}}=m\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.7.m7.5d">caligraphic_P = { bold_italic_p ∈ [ 0 , 2 ] start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT : ⟨ start_ARG bold_1 , bold_italic_p end_ARG ⟩ = italic_m }</annotation></semantics></math>, we have the following:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.4"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p1.4.4">In the zero-sum game <math alttext="\Gamma_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.1.m1.1"><semantics id="S5.Thmtheorem1.p1.1.1.m1.1a"><msub id="S5.Thmtheorem1.p1.1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.2" mathvariant="normal" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">Γ</mi><mn id="S5.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.1.m1.1b"><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.2">Γ</ci><cn id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.1.1.m1.1c">\Gamma_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.1.1.m1.1d">roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, every <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.2.2.m2.1"><semantics id="S5.Thmtheorem1.p1.2.2.m2.1a"><mi id="S5.Thmtheorem1.p1.2.2.m2.1.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.2.2.m2.1b"><ci id="S5.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.2.m2.1d">italic_ε</annotation></semantics></math>-Nash equilibrium strategy for the principal has the form <math alttext="{\bm{p}}=\bm{1}-\bm{u}+{\bm{z}}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.3.3.m3.1"><semantics id="S5.Thmtheorem1.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.2.cmml">𝐩</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.cmml">=</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.cmml"><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.cmml"><mn id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.2.cmml">𝟏</mn><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.1.cmml">−</mo><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.3.cmml">𝐮</mi></mrow><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.1.cmml">+</mo><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.3.cmml">𝐳</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.3.m3.1b"><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1"><eq id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1"></eq><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.2">𝐩</ci><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3"><plus id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.1"></plus><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2"><minus id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.1"></minus><cn id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.2">1</cn><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.2.3">𝐮</ci></apply><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.3.3">𝐳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.3.m3.1c">{\bm{p}}=\bm{1}-\bm{u}+{\bm{z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.3.m3.1d">bold_italic_p = bold_1 - bold_italic_u + bold_italic_z</annotation></semantics></math>, where <math alttext="\norm{{\bm{z}}}_{1}\leq 4m\varepsilon" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.4.4.m4.1"><semantics id="S5.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S5.Thmtheorem1.p1.4.4.m4.1.2" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.cmml"><msub id="S5.Thmtheorem1.p1.4.4.m4.1.2.2" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.2.cmml"><mrow id="S5.Thmtheorem1.p1.4.4.m4.1.1.3" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.2.cmml"><mo id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.2.1.cmml">‖</mo><mi id="S5.Thmtheorem1.p1.4.4.m4.1.1.1.1.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.cmml">𝐳</mi><mo id="S5.Thmtheorem1.p1.4.4.m4.1.1.3.2" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.2.1.cmml">‖</mo></mrow><mn id="S5.Thmtheorem1.p1.4.4.m4.1.2.2.2" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.2.2.cmml">1</mn></msub><mo id="S5.Thmtheorem1.p1.4.4.m4.1.2.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.1.cmml">≤</mo><mrow id="S5.Thmtheorem1.p1.4.4.m4.1.2.3" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.cmml"><mn id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.2" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.2.cmml">4</mn><mo id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.1" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.3" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.3.cmml">m</mi><mo id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.1a" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.4" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.4.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.4.4.m4.1b"><apply id="S5.Thmtheorem1.p1.4.4.m4.1.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2"><leq id="S5.Thmtheorem1.p1.4.4.m4.1.2.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.1"></leq><apply id="S5.Thmtheorem1.p1.4.4.m4.1.2.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.4.4.m4.1.2.2.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.2">subscript</csymbol><apply id="S5.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3"><csymbol cd="latexml" id="S5.Thmtheorem1.p1.4.4.m4.1.1.2.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.3.1">norm</csymbol><ci id="S5.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.1.1.1.1">𝐳</ci></apply><cn id="S5.Thmtheorem1.p1.4.4.m4.1.2.2.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.2.2">1</cn></apply><apply id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3"><times id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.1.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.1"></times><cn id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.2">4</cn><ci id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.3.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.3">𝑚</ci><ci id="S5.Thmtheorem1.p1.4.4.m4.1.2.3.4.cmml" xref="S5.Thmtheorem1.p1.4.4.m4.1.2.3.4">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.4.4.m4.1c">\norm{{\bm{z}}}_{1}\leq 4m\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.4.4.m4.1d">∥ start_ARG bold_italic_z end_ARG ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ 4 italic_m italic_ε</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS1.1.p1"> <p class="ltx_p" id="S5.SS1.1.p1.14">Setting <math alttext="{\bm{p}}=\bm{1}-\bm{u}" class="ltx_Math" display="inline" id="S5.SS1.1.p1.1.m1.1"><semantics id="S5.SS1.1.p1.1.m1.1a"><mrow id="S5.SS1.1.p1.1.m1.1.1" xref="S5.SS1.1.p1.1.m1.1.1.cmml"><mi id="S5.SS1.1.p1.1.m1.1.1.2" xref="S5.SS1.1.p1.1.m1.1.1.2.cmml">𝒑</mi><mo id="S5.SS1.1.p1.1.m1.1.1.1" xref="S5.SS1.1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S5.SS1.1.p1.1.m1.1.1.3" xref="S5.SS1.1.p1.1.m1.1.1.3.cmml"><mn id="S5.SS1.1.p1.1.m1.1.1.3.2" xref="S5.SS1.1.p1.1.m1.1.1.3.2.cmml">𝟏</mn><mo id="S5.SS1.1.p1.1.m1.1.1.3.1" xref="S5.SS1.1.p1.1.m1.1.1.3.1.cmml">−</mo><mi id="S5.SS1.1.p1.1.m1.1.1.3.3" xref="S5.SS1.1.p1.1.m1.1.1.3.3.cmml">𝒖</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.1.m1.1b"><apply id="S5.SS1.1.p1.1.m1.1.1.cmml" xref="S5.SS1.1.p1.1.m1.1.1"><eq id="S5.SS1.1.p1.1.m1.1.1.1.cmml" xref="S5.SS1.1.p1.1.m1.1.1.1"></eq><ci id="S5.SS1.1.p1.1.m1.1.1.2.cmml" xref="S5.SS1.1.p1.1.m1.1.1.2">𝒑</ci><apply id="S5.SS1.1.p1.1.m1.1.1.3.cmml" xref="S5.SS1.1.p1.1.m1.1.1.3"><minus id="S5.SS1.1.p1.1.m1.1.1.3.1.cmml" xref="S5.SS1.1.p1.1.m1.1.1.3.1"></minus><cn id="S5.SS1.1.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S5.SS1.1.p1.1.m1.1.1.3.2">1</cn><ci id="S5.SS1.1.p1.1.m1.1.1.3.3.cmml" xref="S5.SS1.1.p1.1.m1.1.1.3.3">𝒖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.1.m1.1c">{\bm{p}}=\bm{1}-\bm{u}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.1.m1.1d">bold_italic_p = bold_1 - bold_italic_u</annotation></semantics></math> guarantees <math alttext="\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}=\expectationvalue{\bm{1},{\bm{x% }}}=1" class="ltx_Math" display="inline" id="S5.SS1.1.p1.2.m2.2"><semantics id="S5.SS1.1.p1.2.m2.2a"><mrow id="S5.SS1.1.p1.2.m2.2.3" xref="S5.SS1.1.p1.2.m2.2.3.cmml"><mrow id="S5.SS1.1.p1.2.m2.1.1.3" xref="S5.SS1.1.p1.2.m2.1.1.2.cmml"><mo id="S5.SS1.1.p1.2.m2.1.1.3.1" stretchy="false" xref="S5.SS1.1.p1.2.m2.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.3.cmml"><mrow id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.cmml"><mi id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.2" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.2.cmml">𝒖</mi><mo id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.1" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.1.cmml">+</mo><mi id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.3" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.3.cmml">𝒑</mi></mrow><mo id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.2" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.3.cmml">,</mo><mi id="S5.SS1.1.p1.2.m2.1.1.1.1.1.1" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.1.cmml">𝒙</mi></mrow><mo id="S5.SS1.1.p1.2.m2.1.1.3.2" stretchy="false" xref="S5.SS1.1.p1.2.m2.1.1.2.1.cmml">⟩</mo></mrow><mo id="S5.SS1.1.p1.2.m2.2.3.2" xref="S5.SS1.1.p1.2.m2.2.3.2.cmml">=</mo><mrow id="S5.SS1.1.p1.2.m2.2.2.3" xref="S5.SS1.1.p1.2.m2.2.2.2.cmml"><mo id="S5.SS1.1.p1.2.m2.2.2.3.1" xref="S5.SS1.1.p1.2.m2.2.2.2.1.cmml">⟨</mo><mrow id="S5.SS1.1.p1.2.m2.2.2.1.1.1.4" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.3.cmml"><mn id="S5.SS1.1.p1.2.m2.2.2.1.1.1.1" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.1.cmml">𝟏</mn><mo id="S5.SS1.1.p1.2.m2.2.2.1.1.1.4.1" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.3.cmml">,</mo><mi id="S5.SS1.1.p1.2.m2.2.2.1.1.1.2" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.2.cmml">𝒙</mi></mrow><mo id="S5.SS1.1.p1.2.m2.2.2.3.2" xref="S5.SS1.1.p1.2.m2.2.2.2.1.cmml">⟩</mo></mrow><mo id="S5.SS1.1.p1.2.m2.2.3.3" xref="S5.SS1.1.p1.2.m2.2.3.3.cmml">=</mo><mn id="S5.SS1.1.p1.2.m2.2.3.4" xref="S5.SS1.1.p1.2.m2.2.3.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.2.m2.2b"><apply id="S5.SS1.1.p1.2.m2.2.3.cmml" xref="S5.SS1.1.p1.2.m2.2.3"><and id="S5.SS1.1.p1.2.m2.2.3a.cmml" xref="S5.SS1.1.p1.2.m2.2.3"></and><apply id="S5.SS1.1.p1.2.m2.2.3b.cmml" xref="S5.SS1.1.p1.2.m2.2.3"><eq id="S5.SS1.1.p1.2.m2.2.3.2.cmml" xref="S5.SS1.1.p1.2.m2.2.3.2"></eq><apply id="S5.SS1.1.p1.2.m2.1.1.2.cmml" xref="S5.SS1.1.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S5.SS1.1.p1.2.m2.1.1.2.1.cmml" xref="S5.SS1.1.p1.2.m2.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2"><apply id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1"><plus id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.1.cmml" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.1"></plus><ci id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.2.cmml" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.2">𝒖</ci><ci id="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.3.cmml" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.2.1.3">𝒑</ci></apply><ci id="S5.SS1.1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.2.m2.1.1.1.1.1.1">𝒙</ci></list></apply><apply id="S5.SS1.1.p1.2.m2.2.2.2.cmml" xref="S5.SS1.1.p1.2.m2.2.2.3"><csymbol cd="latexml" id="S5.SS1.1.p1.2.m2.2.2.2.1.cmml" xref="S5.SS1.1.p1.2.m2.2.2.3.1">expectation-value</csymbol><list id="S5.SS1.1.p1.2.m2.2.2.1.1.1.3.cmml" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.4"><cn id="S5.SS1.1.p1.2.m2.2.2.1.1.1.1.cmml" type="integer" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.1">1</cn><ci id="S5.SS1.1.p1.2.m2.2.2.1.1.1.2.cmml" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.2">𝒙</ci></list></apply></apply><apply id="S5.SS1.1.p1.2.m2.2.3c.cmml" xref="S5.SS1.1.p1.2.m2.2.3"><eq id="S5.SS1.1.p1.2.m2.2.3.3.cmml" xref="S5.SS1.1.p1.2.m2.2.3.3"></eq><share href="https://arxiv.org/html/2503.01976v1#S5.SS1.1.p1.2.m2.2.2.cmml" id="S5.SS1.1.p1.2.m2.2.3d.cmml" xref="S5.SS1.1.p1.2.m2.2.3"></share><cn id="S5.SS1.1.p1.2.m2.2.3.4.cmml" type="integer" xref="S5.SS1.1.p1.2.m2.2.3.4">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.2.m2.2c">\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}=\expectationvalue{\bm{1},{\bm{x% }}}=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.2.m2.2d">⟨ start_ARG bold_italic_u + bold_italic_p , bold_italic_x end_ARG ⟩ = ⟨ start_ARG bold_1 , bold_italic_x end_ARG ⟩ = 1</annotation></semantics></math> for every <math alttext="{\bm{x}}\in\Delta(m)" class="ltx_Math" display="inline" id="S5.SS1.1.p1.3.m3.1"><semantics id="S5.SS1.1.p1.3.m3.1a"><mrow id="S5.SS1.1.p1.3.m3.1.2" xref="S5.SS1.1.p1.3.m3.1.2.cmml"><mi id="S5.SS1.1.p1.3.m3.1.2.2" xref="S5.SS1.1.p1.3.m3.1.2.2.cmml">𝒙</mi><mo id="S5.SS1.1.p1.3.m3.1.2.1" xref="S5.SS1.1.p1.3.m3.1.2.1.cmml">∈</mo><mrow id="S5.SS1.1.p1.3.m3.1.2.3" xref="S5.SS1.1.p1.3.m3.1.2.3.cmml"><mi id="S5.SS1.1.p1.3.m3.1.2.3.2" mathvariant="normal" xref="S5.SS1.1.p1.3.m3.1.2.3.2.cmml">Δ</mi><mo id="S5.SS1.1.p1.3.m3.1.2.3.1" xref="S5.SS1.1.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS1.1.p1.3.m3.1.2.3.3.2" xref="S5.SS1.1.p1.3.m3.1.2.3.cmml"><mo id="S5.SS1.1.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.SS1.1.p1.3.m3.1.2.3.cmml">(</mo><mi id="S5.SS1.1.p1.3.m3.1.1" xref="S5.SS1.1.p1.3.m3.1.1.cmml">m</mi><mo id="S5.SS1.1.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.SS1.1.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.3.m3.1b"><apply id="S5.SS1.1.p1.3.m3.1.2.cmml" xref="S5.SS1.1.p1.3.m3.1.2"><in id="S5.SS1.1.p1.3.m3.1.2.1.cmml" xref="S5.SS1.1.p1.3.m3.1.2.1"></in><ci id="S5.SS1.1.p1.3.m3.1.2.2.cmml" xref="S5.SS1.1.p1.3.m3.1.2.2">𝒙</ci><apply id="S5.SS1.1.p1.3.m3.1.2.3.cmml" xref="S5.SS1.1.p1.3.m3.1.2.3"><times id="S5.SS1.1.p1.3.m3.1.2.3.1.cmml" xref="S5.SS1.1.p1.3.m3.1.2.3.1"></times><ci id="S5.SS1.1.p1.3.m3.1.2.3.2.cmml" xref="S5.SS1.1.p1.3.m3.1.2.3.2">Δ</ci><ci id="S5.SS1.1.p1.3.m3.1.1.cmml" xref="S5.SS1.1.p1.3.m3.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.3.m3.1c">{\bm{x}}\in\Delta(m)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.3.m3.1d">bold_italic_x ∈ roman_Δ ( italic_m )</annotation></semantics></math>. Thus, in every <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.1.p1.4.m4.1"><semantics id="S5.SS1.1.p1.4.m4.1a"><mi id="S5.SS1.1.p1.4.m4.1.1" xref="S5.SS1.1.p1.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.4.m4.1b"><ci id="S5.SS1.1.p1.4.m4.1.1.cmml" xref="S5.SS1.1.p1.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.4.m4.1d">italic_ε</annotation></semantics></math>-Nash equilibrium, the agent’s utility is at most <math alttext="1+\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.1.p1.5.m5.1"><semantics id="S5.SS1.1.p1.5.m5.1a"><mrow id="S5.SS1.1.p1.5.m5.1.1" xref="S5.SS1.1.p1.5.m5.1.1.cmml"><mn id="S5.SS1.1.p1.5.m5.1.1.2" xref="S5.SS1.1.p1.5.m5.1.1.2.cmml">1</mn><mo id="S5.SS1.1.p1.5.m5.1.1.1" xref="S5.SS1.1.p1.5.m5.1.1.1.cmml">+</mo><mi id="S5.SS1.1.p1.5.m5.1.1.3" xref="S5.SS1.1.p1.5.m5.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.5.m5.1b"><apply id="S5.SS1.1.p1.5.m5.1.1.cmml" xref="S5.SS1.1.p1.5.m5.1.1"><plus id="S5.SS1.1.p1.5.m5.1.1.1.cmml" xref="S5.SS1.1.p1.5.m5.1.1.1"></plus><cn id="S5.SS1.1.p1.5.m5.1.1.2.cmml" type="integer" xref="S5.SS1.1.p1.5.m5.1.1.2">1</cn><ci id="S5.SS1.1.p1.5.m5.1.1.3.cmml" xref="S5.SS1.1.p1.5.m5.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.5.m5.1c">1+\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.5.m5.1d">1 + italic_ε</annotation></semantics></math>. Now suppose for contradiction that <math alttext="({\bm{p}},{\bm{x}})" class="ltx_Math" display="inline" id="S5.SS1.1.p1.6.m6.2"><semantics id="S5.SS1.1.p1.6.m6.2a"><mrow id="S5.SS1.1.p1.6.m6.2.3.2" xref="S5.SS1.1.p1.6.m6.2.3.1.cmml"><mo id="S5.SS1.1.p1.6.m6.2.3.2.1" stretchy="false" xref="S5.SS1.1.p1.6.m6.2.3.1.cmml">(</mo><mi id="S5.SS1.1.p1.6.m6.1.1" xref="S5.SS1.1.p1.6.m6.1.1.cmml">𝒑</mi><mo id="S5.SS1.1.p1.6.m6.2.3.2.2" xref="S5.SS1.1.p1.6.m6.2.3.1.cmml">,</mo><mi id="S5.SS1.1.p1.6.m6.2.2" xref="S5.SS1.1.p1.6.m6.2.2.cmml">𝒙</mi><mo id="S5.SS1.1.p1.6.m6.2.3.2.3" stretchy="false" xref="S5.SS1.1.p1.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.6.m6.2b"><interval closure="open" id="S5.SS1.1.p1.6.m6.2.3.1.cmml" xref="S5.SS1.1.p1.6.m6.2.3.2"><ci id="S5.SS1.1.p1.6.m6.1.1.cmml" xref="S5.SS1.1.p1.6.m6.1.1">𝒑</ci><ci id="S5.SS1.1.p1.6.m6.2.2.cmml" xref="S5.SS1.1.p1.6.m6.2.2">𝒙</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.6.m6.2c">({\bm{p}},{\bm{x}})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.6.m6.2d">( bold_italic_p , bold_italic_x )</annotation></semantics></math> is an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.1.p1.7.m7.1"><semantics id="S5.SS1.1.p1.7.m7.1a"><mi id="S5.SS1.1.p1.7.m7.1.1" xref="S5.SS1.1.p1.7.m7.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.7.m7.1b"><ci id="S5.SS1.1.p1.7.m7.1.1.cmml" xref="S5.SS1.1.p1.7.m7.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.7.m7.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.7.m7.1d">italic_ε</annotation></semantics></math>-Nash equilibrium with <math alttext="\norm*{{\bm{p}}+{\bm{u}}-\bm{1}}_{1}>4m\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.1.p1.8.m8.1"><semantics id="S5.SS1.1.p1.8.m8.1a"><mrow id="S5.SS1.1.p1.8.m8.1.2" xref="S5.SS1.1.p1.8.m8.1.2.cmml"><msub id="S5.SS1.1.p1.8.m8.1.2.2" xref="S5.SS1.1.p1.8.m8.1.2.2.cmml"><mrow id="S5.SS1.1.p1.8.m8.1.1.3" xref="S5.SS1.1.p1.8.m8.1.1.2.cmml"><mo id="S5.SS1.1.p1.8.m8.1.1.3.1" stretchy="false" xref="S5.SS1.1.p1.8.m8.1.1.2.1.cmml">‖</mo><mrow id="S5.SS1.1.p1.8.m8.1.1.1.1.1" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.cmml"><mrow id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.cmml"><mi id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.2" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.2.cmml">𝒑</mi><mo id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.1" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.1.cmml">+</mo><mi id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.3" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.3.cmml">𝒖</mi></mrow><mo id="S5.SS1.1.p1.8.m8.1.1.1.1.1.1" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.1.cmml">−</mo><mn id="S5.SS1.1.p1.8.m8.1.1.1.1.1.3" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="S5.SS1.1.p1.8.m8.1.1.3.2" stretchy="false" xref="S5.SS1.1.p1.8.m8.1.1.2.1.cmml">‖</mo></mrow><mn id="S5.SS1.1.p1.8.m8.1.2.2.2" xref="S5.SS1.1.p1.8.m8.1.2.2.2.cmml">1</mn></msub><mo id="S5.SS1.1.p1.8.m8.1.2.1" xref="S5.SS1.1.p1.8.m8.1.2.1.cmml">></mo><mrow id="S5.SS1.1.p1.8.m8.1.2.3" xref="S5.SS1.1.p1.8.m8.1.2.3.cmml"><mn id="S5.SS1.1.p1.8.m8.1.2.3.2" xref="S5.SS1.1.p1.8.m8.1.2.3.2.cmml">4</mn><mo id="S5.SS1.1.p1.8.m8.1.2.3.1" xref="S5.SS1.1.p1.8.m8.1.2.3.1.cmml">⁢</mo><mi id="S5.SS1.1.p1.8.m8.1.2.3.3" xref="S5.SS1.1.p1.8.m8.1.2.3.3.cmml">m</mi><mo id="S5.SS1.1.p1.8.m8.1.2.3.1a" xref="S5.SS1.1.p1.8.m8.1.2.3.1.cmml">⁢</mo><mi id="S5.SS1.1.p1.8.m8.1.2.3.4" xref="S5.SS1.1.p1.8.m8.1.2.3.4.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.8.m8.1b"><apply id="S5.SS1.1.p1.8.m8.1.2.cmml" xref="S5.SS1.1.p1.8.m8.1.2"><gt id="S5.SS1.1.p1.8.m8.1.2.1.cmml" xref="S5.SS1.1.p1.8.m8.1.2.1"></gt><apply id="S5.SS1.1.p1.8.m8.1.2.2.cmml" xref="S5.SS1.1.p1.8.m8.1.2.2"><csymbol cd="ambiguous" id="S5.SS1.1.p1.8.m8.1.2.2.1.cmml" xref="S5.SS1.1.p1.8.m8.1.2.2">subscript</csymbol><apply id="S5.SS1.1.p1.8.m8.1.1.2.cmml" xref="S5.SS1.1.p1.8.m8.1.1.3"><csymbol cd="latexml" id="S5.SS1.1.p1.8.m8.1.1.2.1.cmml" xref="S5.SS1.1.p1.8.m8.1.1.3.1">norm</csymbol><apply id="S5.SS1.1.p1.8.m8.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1"><minus id="S5.SS1.1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.1"></minus><apply id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2"><plus id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.1.cmml" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.1"></plus><ci id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.2.cmml" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.2">𝒑</ci><ci id="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.3.cmml" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.2.3">𝒖</ci></apply><cn id="S5.SS1.1.p1.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS1.1.p1.8.m8.1.1.1.1.1.3">1</cn></apply></apply><cn id="S5.SS1.1.p1.8.m8.1.2.2.2.cmml" type="integer" xref="S5.SS1.1.p1.8.m8.1.2.2.2">1</cn></apply><apply id="S5.SS1.1.p1.8.m8.1.2.3.cmml" xref="S5.SS1.1.p1.8.m8.1.2.3"><times id="S5.SS1.1.p1.8.m8.1.2.3.1.cmml" xref="S5.SS1.1.p1.8.m8.1.2.3.1"></times><cn id="S5.SS1.1.p1.8.m8.1.2.3.2.cmml" type="integer" xref="S5.SS1.1.p1.8.m8.1.2.3.2">4</cn><ci id="S5.SS1.1.p1.8.m8.1.2.3.3.cmml" xref="S5.SS1.1.p1.8.m8.1.2.3.3">𝑚</ci><ci id="S5.SS1.1.p1.8.m8.1.2.3.4.cmml" xref="S5.SS1.1.p1.8.m8.1.2.3.4">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.8.m8.1c">\norm*{{\bm{p}}+{\bm{u}}-\bm{1}}_{1}>4m\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.8.m8.1d">∥ start_ARG bold_italic_p + bold_italic_u - bold_1 end_ARG ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 4 italic_m italic_ε</annotation></semantics></math>. Then since <math alttext="\expectationvalue*{{\bm{p}}+{\bm{u}}-\bm{1},\bm{1}}=0" class="ltx_Math" display="inline" id="S5.SS1.1.p1.9.m9.1"><semantics id="S5.SS1.1.p1.9.m9.1a"><mrow id="S5.SS1.1.p1.9.m9.1.2" xref="S5.SS1.1.p1.9.m9.1.2.cmml"><mrow id="S5.SS1.1.p1.9.m9.1.1.3" xref="S5.SS1.1.p1.9.m9.1.1.2.cmml"><mo id="S5.SS1.1.p1.9.m9.1.1.3.1" stretchy="false" xref="S5.SS1.1.p1.9.m9.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.3.cmml"><mrow id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.cmml"><mrow id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.cmml"><mi id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.2" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.2.cmml">𝒑</mi><mo id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.1" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.1.cmml">+</mo><mi id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.3" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.3.cmml">𝒖</mi></mrow><mo id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.1" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.1.cmml">−</mo><mn id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.3" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.3.cmml">𝟏</mn></mrow><mo id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.2" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.3.cmml">,</mo><mn id="S5.SS1.1.p1.9.m9.1.1.1.1.1.1" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.1.cmml">𝟏</mn></mrow><mo id="S5.SS1.1.p1.9.m9.1.1.3.2" stretchy="false" xref="S5.SS1.1.p1.9.m9.1.1.2.1.cmml">⟩</mo></mrow><mo id="S5.SS1.1.p1.9.m9.1.2.1" xref="S5.SS1.1.p1.9.m9.1.2.1.cmml">=</mo><mn id="S5.SS1.1.p1.9.m9.1.2.2" xref="S5.SS1.1.p1.9.m9.1.2.2.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.9.m9.1b"><apply id="S5.SS1.1.p1.9.m9.1.2.cmml" xref="S5.SS1.1.p1.9.m9.1.2"><eq id="S5.SS1.1.p1.9.m9.1.2.1.cmml" xref="S5.SS1.1.p1.9.m9.1.2.1"></eq><apply id="S5.SS1.1.p1.9.m9.1.1.2.cmml" xref="S5.SS1.1.p1.9.m9.1.1.3"><csymbol cd="latexml" id="S5.SS1.1.p1.9.m9.1.1.2.1.cmml" xref="S5.SS1.1.p1.9.m9.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.1.p1.9.m9.1.1.1.1.1.3.cmml" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2"><apply id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.cmml" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1"><minus id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.1.cmml" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.1"></minus><apply id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.cmml" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2"><plus id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.1.cmml" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.1"></plus><ci id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.2.cmml" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.2">𝒑</ci><ci id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.3.cmml" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.2.3">𝒖</ci></apply><cn id="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.3.cmml" type="integer" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.2.1.3">1</cn></apply><cn id="S5.SS1.1.p1.9.m9.1.1.1.1.1.1.cmml" type="integer" xref="S5.SS1.1.p1.9.m9.1.1.1.1.1.1">1</cn></list></apply><cn id="S5.SS1.1.p1.9.m9.1.2.2.cmml" type="integer" xref="S5.SS1.1.p1.9.m9.1.2.2">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.9.m9.1c">\expectationvalue*{{\bm{p}}+{\bm{u}}-\bm{1},\bm{1}}=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.9.m9.1d">⟨ start_ARG bold_italic_p + bold_italic_u - bold_1 , bold_1 end_ARG ⟩ = 0</annotation></semantics></math> by construction, there must be an action <math alttext="a" class="ltx_Math" display="inline" id="S5.SS1.1.p1.10.m10.1"><semantics id="S5.SS1.1.p1.10.m10.1a"><mi id="S5.SS1.1.p1.10.m10.1.1" xref="S5.SS1.1.p1.10.m10.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.10.m10.1b"><ci id="S5.SS1.1.p1.10.m10.1.1.cmml" xref="S5.SS1.1.p1.10.m10.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.10.m10.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.10.m10.1d">italic_a</annotation></semantics></math> for which <math alttext="({\bm{p}}+{\bm{u}}-\bm{1})[a]>2\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.1.p1.11.m11.2"><semantics id="S5.SS1.1.p1.11.m11.2a"><mrow id="S5.SS1.1.p1.11.m11.2.2" xref="S5.SS1.1.p1.11.m11.2.2.cmml"><mrow id="S5.SS1.1.p1.11.m11.2.2.1" xref="S5.SS1.1.p1.11.m11.2.2.1.cmml"><mrow id="S5.SS1.1.p1.11.m11.2.2.1.1.1" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.cmml"><mo id="S5.SS1.1.p1.11.m11.2.2.1.1.1.2" stretchy="false" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.cmml"><mrow id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.cmml"><mi id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.2" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.2.cmml">𝒑</mi><mo id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.1" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.1.cmml">+</mo><mi id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.3" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.3.cmml">𝒖</mi></mrow><mo id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.1" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.1.cmml">−</mo><mn id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.3" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="S5.SS1.1.p1.11.m11.2.2.1.1.1.3" stretchy="false" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS1.1.p1.11.m11.2.2.1.2" xref="S5.SS1.1.p1.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS1.1.p1.11.m11.2.2.1.3.2" xref="S5.SS1.1.p1.11.m11.2.2.1.3.1.cmml"><mo id="S5.SS1.1.p1.11.m11.2.2.1.3.2.1" stretchy="false" xref="S5.SS1.1.p1.11.m11.2.2.1.3.1.1.cmml">[</mo><mi id="S5.SS1.1.p1.11.m11.1.1" xref="S5.SS1.1.p1.11.m11.1.1.cmml">a</mi><mo id="S5.SS1.1.p1.11.m11.2.2.1.3.2.2" stretchy="false" xref="S5.SS1.1.p1.11.m11.2.2.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S5.SS1.1.p1.11.m11.2.2.2" xref="S5.SS1.1.p1.11.m11.2.2.2.cmml">></mo><mrow id="S5.SS1.1.p1.11.m11.2.2.3" xref="S5.SS1.1.p1.11.m11.2.2.3.cmml"><mn id="S5.SS1.1.p1.11.m11.2.2.3.2" xref="S5.SS1.1.p1.11.m11.2.2.3.2.cmml">2</mn><mo id="S5.SS1.1.p1.11.m11.2.2.3.1" xref="S5.SS1.1.p1.11.m11.2.2.3.1.cmml">⁢</mo><mi id="S5.SS1.1.p1.11.m11.2.2.3.3" xref="S5.SS1.1.p1.11.m11.2.2.3.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.11.m11.2b"><apply id="S5.SS1.1.p1.11.m11.2.2.cmml" xref="S5.SS1.1.p1.11.m11.2.2"><gt id="S5.SS1.1.p1.11.m11.2.2.2.cmml" xref="S5.SS1.1.p1.11.m11.2.2.2"></gt><apply id="S5.SS1.1.p1.11.m11.2.2.1.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1"><times id="S5.SS1.1.p1.11.m11.2.2.1.2.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.2"></times><apply id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1"><minus id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.1"></minus><apply id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2"><plus id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.1.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.1"></plus><ci id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.2.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.2">𝒑</ci><ci id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.3.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.2.3">𝒖</ci></apply><cn id="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.3.cmml" type="integer" xref="S5.SS1.1.p1.11.m11.2.2.1.1.1.1.3">1</cn></apply><apply id="S5.SS1.1.p1.11.m11.2.2.1.3.1.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.3.2"><csymbol cd="latexml" id="S5.SS1.1.p1.11.m11.2.2.1.3.1.1.cmml" xref="S5.SS1.1.p1.11.m11.2.2.1.3.2.1">delimited-[]</csymbol><ci id="S5.SS1.1.p1.11.m11.1.1.cmml" xref="S5.SS1.1.p1.11.m11.1.1">𝑎</ci></apply></apply><apply id="S5.SS1.1.p1.11.m11.2.2.3.cmml" xref="S5.SS1.1.p1.11.m11.2.2.3"><times id="S5.SS1.1.p1.11.m11.2.2.3.1.cmml" xref="S5.SS1.1.p1.11.m11.2.2.3.1"></times><cn id="S5.SS1.1.p1.11.m11.2.2.3.2.cmml" type="integer" xref="S5.SS1.1.p1.11.m11.2.2.3.2">2</cn><ci id="S5.SS1.1.p1.11.m11.2.2.3.3.cmml" xref="S5.SS1.1.p1.11.m11.2.2.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.11.m11.2c">({\bm{p}}+{\bm{u}}-\bm{1})[a]>2\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.11.m11.2d">( bold_italic_p + bold_italic_u - bold_1 ) [ italic_a ] > 2 italic_ε</annotation></semantics></math>, <span class="ltx_text ltx_font_italic" id="S5.SS1.1.p1.14.1">i.e.</span>, <math alttext="({\bm{u}}+{\bm{p}})[a]>1+2\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.1.p1.12.m12.2"><semantics id="S5.SS1.1.p1.12.m12.2a"><mrow id="S5.SS1.1.p1.12.m12.2.2" xref="S5.SS1.1.p1.12.m12.2.2.cmml"><mrow id="S5.SS1.1.p1.12.m12.2.2.1" xref="S5.SS1.1.p1.12.m12.2.2.1.cmml"><mrow id="S5.SS1.1.p1.12.m12.2.2.1.1.1" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.cmml"><mo id="S5.SS1.1.p1.12.m12.2.2.1.1.1.2" stretchy="false" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.cmml"><mi id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.2" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.2.cmml">𝒖</mi><mo id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.1" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.1.cmml">+</mo><mi id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.3" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.3.cmml">𝒑</mi></mrow><mo id="S5.SS1.1.p1.12.m12.2.2.1.1.1.3" stretchy="false" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS1.1.p1.12.m12.2.2.1.2" xref="S5.SS1.1.p1.12.m12.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS1.1.p1.12.m12.2.2.1.3.2" xref="S5.SS1.1.p1.12.m12.2.2.1.3.1.cmml"><mo id="S5.SS1.1.p1.12.m12.2.2.1.3.2.1" stretchy="false" xref="S5.SS1.1.p1.12.m12.2.2.1.3.1.1.cmml">[</mo><mi id="S5.SS1.1.p1.12.m12.1.1" xref="S5.SS1.1.p1.12.m12.1.1.cmml">a</mi><mo id="S5.SS1.1.p1.12.m12.2.2.1.3.2.2" stretchy="false" xref="S5.SS1.1.p1.12.m12.2.2.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S5.SS1.1.p1.12.m12.2.2.2" xref="S5.SS1.1.p1.12.m12.2.2.2.cmml">></mo><mrow id="S5.SS1.1.p1.12.m12.2.2.3" xref="S5.SS1.1.p1.12.m12.2.2.3.cmml"><mn id="S5.SS1.1.p1.12.m12.2.2.3.2" xref="S5.SS1.1.p1.12.m12.2.2.3.2.cmml">1</mn><mo id="S5.SS1.1.p1.12.m12.2.2.3.1" xref="S5.SS1.1.p1.12.m12.2.2.3.1.cmml">+</mo><mrow id="S5.SS1.1.p1.12.m12.2.2.3.3" xref="S5.SS1.1.p1.12.m12.2.2.3.3.cmml"><mn id="S5.SS1.1.p1.12.m12.2.2.3.3.2" xref="S5.SS1.1.p1.12.m12.2.2.3.3.2.cmml">2</mn><mo id="S5.SS1.1.p1.12.m12.2.2.3.3.1" xref="S5.SS1.1.p1.12.m12.2.2.3.3.1.cmml">⁢</mo><mi id="S5.SS1.1.p1.12.m12.2.2.3.3.3" xref="S5.SS1.1.p1.12.m12.2.2.3.3.3.cmml">ε</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.12.m12.2b"><apply id="S5.SS1.1.p1.12.m12.2.2.cmml" xref="S5.SS1.1.p1.12.m12.2.2"><gt id="S5.SS1.1.p1.12.m12.2.2.2.cmml" xref="S5.SS1.1.p1.12.m12.2.2.2"></gt><apply id="S5.SS1.1.p1.12.m12.2.2.1.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1"><times id="S5.SS1.1.p1.12.m12.2.2.1.2.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1.2"></times><apply id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1"><plus id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.1"></plus><ci id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.2.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.2">𝒖</ci><ci id="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.3.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1.1.1.1.3">𝒑</ci></apply><apply id="S5.SS1.1.p1.12.m12.2.2.1.3.1.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1.3.2"><csymbol cd="latexml" id="S5.SS1.1.p1.12.m12.2.2.1.3.1.1.cmml" xref="S5.SS1.1.p1.12.m12.2.2.1.3.2.1">delimited-[]</csymbol><ci id="S5.SS1.1.p1.12.m12.1.1.cmml" xref="S5.SS1.1.p1.12.m12.1.1">𝑎</ci></apply></apply><apply id="S5.SS1.1.p1.12.m12.2.2.3.cmml" xref="S5.SS1.1.p1.12.m12.2.2.3"><plus id="S5.SS1.1.p1.12.m12.2.2.3.1.cmml" xref="S5.SS1.1.p1.12.m12.2.2.3.1"></plus><cn id="S5.SS1.1.p1.12.m12.2.2.3.2.cmml" type="integer" xref="S5.SS1.1.p1.12.m12.2.2.3.2">1</cn><apply id="S5.SS1.1.p1.12.m12.2.2.3.3.cmml" xref="S5.SS1.1.p1.12.m12.2.2.3.3"><times id="S5.SS1.1.p1.12.m12.2.2.3.3.1.cmml" xref="S5.SS1.1.p1.12.m12.2.2.3.3.1"></times><cn id="S5.SS1.1.p1.12.m12.2.2.3.3.2.cmml" type="integer" xref="S5.SS1.1.p1.12.m12.2.2.3.3.2">2</cn><ci id="S5.SS1.1.p1.12.m12.2.2.3.3.3.cmml" xref="S5.SS1.1.p1.12.m12.2.2.3.3.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.12.m12.2c">({\bm{u}}+{\bm{p}})[a]>1+2\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.12.m12.2d">( bold_italic_u + bold_italic_p ) [ italic_a ] > 1 + 2 italic_ε</annotation></semantics></math>. But then the agent has an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.1.p1.13.m13.1"><semantics id="S5.SS1.1.p1.13.m13.1a"><mi id="S5.SS1.1.p1.13.m13.1.1" xref="S5.SS1.1.p1.13.m13.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.13.m13.1b"><ci id="S5.SS1.1.p1.13.m13.1.1.cmml" xref="S5.SS1.1.p1.13.m13.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.13.m13.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.13.m13.1d">italic_ε</annotation></semantics></math>-profitable deviation to action <math alttext="a" class="ltx_Math" display="inline" id="S5.SS1.1.p1.14.m14.1"><semantics id="S5.SS1.1.p1.14.m14.1a"><mi id="S5.SS1.1.p1.14.m14.1.1" xref="S5.SS1.1.p1.14.m14.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.14.m14.1b"><ci id="S5.SS1.1.p1.14.m14.1.1.cmml" xref="S5.SS1.1.p1.14.m14.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.14.m14.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.14.m14.1d">italic_a</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS1.p5"> <p class="ltx_p" id="S5.SS1.p5.16">It is well known that no-regret learning algorithms converge on average to Nash equilibria in zero-sum games. In particular, if both principal and agent run no-regret algorithms, and <math alttext="R_{0}" class="ltx_Math" display="inline" id="S5.SS1.p5.1.m1.1"><semantics id="S5.SS1.p5.1.m1.1a"><msub id="S5.SS1.p5.1.m1.1.1" xref="S5.SS1.p5.1.m1.1.1.cmml"><mi id="S5.SS1.p5.1.m1.1.1.2" xref="S5.SS1.p5.1.m1.1.1.2.cmml">R</mi><mn id="S5.SS1.p5.1.m1.1.1.3" xref="S5.SS1.p5.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.1.m1.1b"><apply id="S5.SS1.p5.1.m1.1.1.cmml" xref="S5.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p5.1.m1.1.1.1.cmml" xref="S5.SS1.p5.1.m1.1.1">subscript</csymbol><ci id="S5.SS1.p5.1.m1.1.1.2.cmml" xref="S5.SS1.p5.1.m1.1.1.2">𝑅</ci><cn id="S5.SS1.p5.1.m1.1.1.3.cmml" type="integer" xref="S5.SS1.p5.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.1.m1.1c">R_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.1.m1.1d">italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the regret after <math alttext="T" class="ltx_Math" display="inline" id="S5.SS1.p5.2.m2.1"><semantics id="S5.SS1.p5.2.m2.1a"><mi id="S5.SS1.p5.2.m2.1.1" xref="S5.SS1.p5.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.2.m2.1b"><ci id="S5.SS1.p5.2.m2.1.1.cmml" xref="S5.SS1.p5.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.2.m2.1d">italic_T</annotation></semantics></math> timesteps for the principal, then the average principal strategy <math alttext="\frac{1}{T}\sum_{t=1}^{T}{\bm{p}}^{t}" class="ltx_Math" display="inline" id="S5.SS1.p5.3.m3.1"><semantics id="S5.SS1.p5.3.m3.1a"><mrow id="S5.SS1.p5.3.m3.1.1" xref="S5.SS1.p5.3.m3.1.1.cmml"><mfrac id="S5.SS1.p5.3.m3.1.1.2" xref="S5.SS1.p5.3.m3.1.1.2.cmml"><mn id="S5.SS1.p5.3.m3.1.1.2.2" xref="S5.SS1.p5.3.m3.1.1.2.2.cmml">1</mn><mi id="S5.SS1.p5.3.m3.1.1.2.3" xref="S5.SS1.p5.3.m3.1.1.2.3.cmml">T</mi></mfrac><mo id="S5.SS1.p5.3.m3.1.1.1" xref="S5.SS1.p5.3.m3.1.1.1.cmml">⁢</mo><mrow id="S5.SS1.p5.3.m3.1.1.3" xref="S5.SS1.p5.3.m3.1.1.3.cmml"><msubsup id="S5.SS1.p5.3.m3.1.1.3.1" xref="S5.SS1.p5.3.m3.1.1.3.1.cmml"><mo id="S5.SS1.p5.3.m3.1.1.3.1.2.2" xref="S5.SS1.p5.3.m3.1.1.3.1.2.2.cmml">∑</mo><mrow id="S5.SS1.p5.3.m3.1.1.3.1.2.3" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3.cmml"><mi id="S5.SS1.p5.3.m3.1.1.3.1.2.3.2" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3.2.cmml">t</mi><mo id="S5.SS1.p5.3.m3.1.1.3.1.2.3.1" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3.1.cmml">=</mo><mn id="S5.SS1.p5.3.m3.1.1.3.1.2.3.3" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S5.SS1.p5.3.m3.1.1.3.1.3" xref="S5.SS1.p5.3.m3.1.1.3.1.3.cmml">T</mi></msubsup><msup id="S5.SS1.p5.3.m3.1.1.3.2" xref="S5.SS1.p5.3.m3.1.1.3.2.cmml"><mi id="S5.SS1.p5.3.m3.1.1.3.2.2" xref="S5.SS1.p5.3.m3.1.1.3.2.2.cmml">𝒑</mi><mi id="S5.SS1.p5.3.m3.1.1.3.2.3" xref="S5.SS1.p5.3.m3.1.1.3.2.3.cmml">t</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.3.m3.1b"><apply id="S5.SS1.p5.3.m3.1.1.cmml" xref="S5.SS1.p5.3.m3.1.1"><times id="S5.SS1.p5.3.m3.1.1.1.cmml" xref="S5.SS1.p5.3.m3.1.1.1"></times><apply id="S5.SS1.p5.3.m3.1.1.2.cmml" xref="S5.SS1.p5.3.m3.1.1.2"><divide id="S5.SS1.p5.3.m3.1.1.2.1.cmml" xref="S5.SS1.p5.3.m3.1.1.2"></divide><cn id="S5.SS1.p5.3.m3.1.1.2.2.cmml" type="integer" xref="S5.SS1.p5.3.m3.1.1.2.2">1</cn><ci id="S5.SS1.p5.3.m3.1.1.2.3.cmml" xref="S5.SS1.p5.3.m3.1.1.2.3">𝑇</ci></apply><apply id="S5.SS1.p5.3.m3.1.1.3.cmml" xref="S5.SS1.p5.3.m3.1.1.3"><apply id="S5.SS1.p5.3.m3.1.1.3.1.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S5.SS1.p5.3.m3.1.1.3.1.1.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1">superscript</csymbol><apply id="S5.SS1.p5.3.m3.1.1.3.1.2.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S5.SS1.p5.3.m3.1.1.3.1.2.1.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1">subscript</csymbol><sum id="S5.SS1.p5.3.m3.1.1.3.1.2.2.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1.2.2"></sum><apply id="S5.SS1.p5.3.m3.1.1.3.1.2.3.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3"><eq id="S5.SS1.p5.3.m3.1.1.3.1.2.3.1.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3.1"></eq><ci id="S5.SS1.p5.3.m3.1.1.3.1.2.3.2.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3.2">𝑡</ci><cn id="S5.SS1.p5.3.m3.1.1.3.1.2.3.3.cmml" type="integer" xref="S5.SS1.p5.3.m3.1.1.3.1.2.3.3">1</cn></apply></apply><ci id="S5.SS1.p5.3.m3.1.1.3.1.3.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1.3">𝑇</ci></apply><apply id="S5.SS1.p5.3.m3.1.1.3.2.cmml" xref="S5.SS1.p5.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS1.p5.3.m3.1.1.3.2.1.cmml" xref="S5.SS1.p5.3.m3.1.1.3.2">superscript</csymbol><ci id="S5.SS1.p5.3.m3.1.1.3.2.2.cmml" xref="S5.SS1.p5.3.m3.1.1.3.2.2">𝒑</ci><ci id="S5.SS1.p5.3.m3.1.1.3.2.3.cmml" xref="S5.SS1.p5.3.m3.1.1.3.2.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.3.m3.1c">\frac{1}{T}\sum_{t=1}^{T}{\bm{p}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.3.m3.1d">divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_p start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> is an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS1.p5.4.m4.1"><semantics id="S5.SS1.p5.4.m4.1a"><mi id="S5.SS1.p5.4.m4.1.1" xref="S5.SS1.p5.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.4.m4.1b"><ci id="S5.SS1.p5.4.m4.1.1.cmml" xref="S5.SS1.p5.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.4.m4.1d">italic_ε</annotation></semantics></math>-Nash equilibrium for <math alttext="\varepsilon\lesssim(R_{0}+C\sqrt{T})/T" class="ltx_Math" display="inline" id="S5.SS1.p5.5.m5.1"><semantics id="S5.SS1.p5.5.m5.1a"><mrow id="S5.SS1.p5.5.m5.1.1" xref="S5.SS1.p5.5.m5.1.1.cmml"><mi id="S5.SS1.p5.5.m5.1.1.3" xref="S5.SS1.p5.5.m5.1.1.3.cmml">ε</mi><mo id="S5.SS1.p5.5.m5.1.1.2" xref="S5.SS1.p5.5.m5.1.1.2.cmml">≲</mo><mrow id="S5.SS1.p5.5.m5.1.1.1" xref="S5.SS1.p5.5.m5.1.1.1.cmml"><mrow id="S5.SS1.p5.5.m5.1.1.1.1.1" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.cmml"><mo id="S5.SS1.p5.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.p5.5.m5.1.1.1.1.1.1" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.cmml"><msub id="S5.SS1.p5.5.m5.1.1.1.1.1.1.2" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.cmml"><mi id="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.2" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.2.cmml">R</mi><mn id="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.3" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S5.SS1.p5.5.m5.1.1.1.1.1.1.1" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.cmml"><mi id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.2" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.2.cmml">C</mi><mo id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.1" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.1.cmml">⁢</mo><msqrt id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3.cmml"><mi id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3.2" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3.2.cmml">T</mi></msqrt></mrow></mrow><mo id="S5.SS1.p5.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS1.p5.5.m5.1.1.1.2" xref="S5.SS1.p5.5.m5.1.1.1.2.cmml">/</mo><mi id="S5.SS1.p5.5.m5.1.1.1.3" xref="S5.SS1.p5.5.m5.1.1.1.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.5.m5.1b"><apply id="S5.SS1.p5.5.m5.1.1.cmml" xref="S5.SS1.p5.5.m5.1.1"><csymbol cd="latexml" id="S5.SS1.p5.5.m5.1.1.2.cmml" xref="S5.SS1.p5.5.m5.1.1.2">less-than-or-similar-to</csymbol><ci id="S5.SS1.p5.5.m5.1.1.3.cmml" xref="S5.SS1.p5.5.m5.1.1.3">𝜀</ci><apply id="S5.SS1.p5.5.m5.1.1.1.cmml" xref="S5.SS1.p5.5.m5.1.1.1"><divide id="S5.SS1.p5.5.m5.1.1.1.2.cmml" xref="S5.SS1.p5.5.m5.1.1.1.2"></divide><apply id="S5.SS1.p5.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1"><plus id="S5.SS1.p5.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.1"></plus><apply id="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.2">subscript</csymbol><ci id="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.2">𝑅</ci><cn id="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.2.3">0</cn></apply><apply id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3"><times id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.1"></times><ci id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.2.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.2">𝐶</ci><apply id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3"><root id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3a.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3"></root><ci id="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3.2.cmml" xref="S5.SS1.p5.5.m5.1.1.1.1.1.1.3.3.2">𝑇</ci></apply></apply></apply><ci id="S5.SS1.p5.5.m5.1.1.1.3.cmml" xref="S5.SS1.p5.5.m5.1.1.1.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.5.m5.1c">\varepsilon\lesssim(R_{0}+C\sqrt{T})/T</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.5.m5.1d">italic_ε ≲ ( italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_C square-root start_ARG italic_T end_ARG ) / italic_T</annotation></semantics></math>. Here, we will use the projected gradient descent algorithm for the principal. Note that, although the principal’s utility function <math alttext="{\bm{p}}\mapsto\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}" class="ltx_Math" display="inline" id="S5.SS1.p5.6.m6.1"><semantics id="S5.SS1.p5.6.m6.1a"><mrow id="S5.SS1.p5.6.m6.1.2" xref="S5.SS1.p5.6.m6.1.2.cmml"><mi id="S5.SS1.p5.6.m6.1.2.2" xref="S5.SS1.p5.6.m6.1.2.2.cmml">𝒑</mi><mo id="S5.SS1.p5.6.m6.1.2.1" stretchy="false" xref="S5.SS1.p5.6.m6.1.2.1.cmml">↦</mo><mrow id="S5.SS1.p5.6.m6.1.1.3" xref="S5.SS1.p5.6.m6.1.1.2.cmml"><mo id="S5.SS1.p5.6.m6.1.1.3.1" stretchy="false" xref="S5.SS1.p5.6.m6.1.1.2.1.cmml">⟨</mo><mrow id="S5.SS1.p5.6.m6.1.1.1.1.1.2" xref="S5.SS1.p5.6.m6.1.1.1.1.1.3.cmml"><mrow id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.cmml"><mi id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.2" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.2.cmml">𝒖</mi><mo id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.1" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.1.cmml">+</mo><mi id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.3" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.3.cmml">𝒑</mi></mrow><mo id="S5.SS1.p5.6.m6.1.1.1.1.1.2.2" xref="S5.SS1.p5.6.m6.1.1.1.1.1.3.cmml">,</mo><mi id="S5.SS1.p5.6.m6.1.1.1.1.1.1" xref="S5.SS1.p5.6.m6.1.1.1.1.1.1.cmml">𝒙</mi></mrow><mo id="S5.SS1.p5.6.m6.1.1.3.2" stretchy="false" xref="S5.SS1.p5.6.m6.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.6.m6.1b"><apply id="S5.SS1.p5.6.m6.1.2.cmml" xref="S5.SS1.p5.6.m6.1.2"><csymbol cd="latexml" id="S5.SS1.p5.6.m6.1.2.1.cmml" xref="S5.SS1.p5.6.m6.1.2.1">maps-to</csymbol><ci id="S5.SS1.p5.6.m6.1.2.2.cmml" xref="S5.SS1.p5.6.m6.1.2.2">𝒑</ci><apply id="S5.SS1.p5.6.m6.1.1.2.cmml" xref="S5.SS1.p5.6.m6.1.1.3"><csymbol cd="latexml" id="S5.SS1.p5.6.m6.1.1.2.1.cmml" xref="S5.SS1.p5.6.m6.1.1.3.1">expectation-value</csymbol><list id="S5.SS1.p5.6.m6.1.1.1.1.1.3.cmml" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2"><apply id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.cmml" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1"><plus id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.1.cmml" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.1"></plus><ci id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.2.cmml" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.2">𝒖</ci><ci id="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.3.cmml" xref="S5.SS1.p5.6.m6.1.1.1.1.1.2.1.3">𝒑</ci></apply><ci id="S5.SS1.p5.6.m6.1.1.1.1.1.1.cmml" xref="S5.SS1.p5.6.m6.1.1.1.1.1.1">𝒙</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.6.m6.1c">{\bm{p}}\mapsto\expectationvalue*{{\bm{u}}+{\bm{p}},{\bm{x}}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.6.m6.1d">bold_italic_p ↦ ⟨ start_ARG bold_italic_u + bold_italic_p , bold_italic_x end_ARG ⟩</annotation></semantics></math> depends on <math alttext="{\bm{u}}" class="ltx_Math" display="inline" id="S5.SS1.p5.7.m7.1"><semantics id="S5.SS1.p5.7.m7.1a"><mi id="S5.SS1.p5.7.m7.1.1" xref="S5.SS1.p5.7.m7.1.1.cmml">𝒖</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.7.m7.1b"><ci id="S5.SS1.p5.7.m7.1.1.cmml" xref="S5.SS1.p5.7.m7.1.1">𝒖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.7.m7.1c">{\bm{u}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.7.m7.1d">bold_italic_u</annotation></semantics></math> (which the principal does not know), the gradient of the principal’s utility function is <math alttext="-{\bm{x}}" class="ltx_Math" display="inline" id="S5.SS1.p5.8.m8.1"><semantics id="S5.SS1.p5.8.m8.1a"><mrow id="S5.SS1.p5.8.m8.1.1" xref="S5.SS1.p5.8.m8.1.1.cmml"><mo id="S5.SS1.p5.8.m8.1.1a" xref="S5.SS1.p5.8.m8.1.1.cmml">−</mo><mi id="S5.SS1.p5.8.m8.1.1.2" xref="S5.SS1.p5.8.m8.1.1.2.cmml">𝒙</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.8.m8.1b"><apply id="S5.SS1.p5.8.m8.1.1.cmml" xref="S5.SS1.p5.8.m8.1.1"><minus id="S5.SS1.p5.8.m8.1.1.1.cmml" xref="S5.SS1.p5.8.m8.1.1"></minus><ci id="S5.SS1.p5.8.m8.1.1.2.cmml" xref="S5.SS1.p5.8.m8.1.1.2">𝒙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.8.m8.1c">-{\bm{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.8.m8.1d">- bold_italic_x</annotation></semantics></math>, which does not depend on <math alttext="{\bm{u}}" class="ltx_Math" display="inline" id="S5.SS1.p5.9.m9.1"><semantics id="S5.SS1.p5.9.m9.1a"><mi id="S5.SS1.p5.9.m9.1.1" xref="S5.SS1.p5.9.m9.1.1.cmml">𝒖</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.9.m9.1b"><ci id="S5.SS1.p5.9.m9.1.1.cmml" xref="S5.SS1.p5.9.m9.1.1">𝒖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.9.m9.1c">{\bm{u}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.9.m9.1d">bold_italic_u</annotation></semantics></math> and can be unbiasedly estimated by <math alttext="-{\bm{e}}_{a}" class="ltx_Math" display="inline" id="S5.SS1.p5.10.m10.1"><semantics id="S5.SS1.p5.10.m10.1a"><mrow id="S5.SS1.p5.10.m10.1.1" xref="S5.SS1.p5.10.m10.1.1.cmml"><mo id="S5.SS1.p5.10.m10.1.1a" xref="S5.SS1.p5.10.m10.1.1.cmml">−</mo><msub id="S5.SS1.p5.10.m10.1.1.2" xref="S5.SS1.p5.10.m10.1.1.2.cmml"><mi id="S5.SS1.p5.10.m10.1.1.2.2" xref="S5.SS1.p5.10.m10.1.1.2.2.cmml">𝒆</mi><mi id="S5.SS1.p5.10.m10.1.1.2.3" xref="S5.SS1.p5.10.m10.1.1.2.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.10.m10.1b"><apply id="S5.SS1.p5.10.m10.1.1.cmml" xref="S5.SS1.p5.10.m10.1.1"><minus id="S5.SS1.p5.10.m10.1.1.1.cmml" xref="S5.SS1.p5.10.m10.1.1"></minus><apply id="S5.SS1.p5.10.m10.1.1.2.cmml" xref="S5.SS1.p5.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p5.10.m10.1.1.2.1.cmml" xref="S5.SS1.p5.10.m10.1.1.2">subscript</csymbol><ci id="S5.SS1.p5.10.m10.1.1.2.2.cmml" xref="S5.SS1.p5.10.m10.1.1.2.2">𝒆</ci><ci id="S5.SS1.p5.10.m10.1.1.2.3.cmml" xref="S5.SS1.p5.10.m10.1.1.2.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.10.m10.1c">-{\bm{e}}_{a}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.10.m10.1d">- bold_italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="a" class="ltx_Math" display="inline" id="S5.SS1.p5.11.m11.1"><semantics id="S5.SS1.p5.11.m11.1a"><mi id="S5.SS1.p5.11.m11.1.1" xref="S5.SS1.p5.11.m11.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.11.m11.1b"><ci id="S5.SS1.p5.11.m11.1.1.cmml" xref="S5.SS1.p5.11.m11.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.11.m11.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.11.m11.1d">italic_a</annotation></semantics></math> is an action sampled according to <math alttext="{\bm{x}}" class="ltx_Math" display="inline" id="S5.SS1.p5.12.m12.1"><semantics id="S5.SS1.p5.12.m12.1a"><mi id="S5.SS1.p5.12.m12.1.1" xref="S5.SS1.p5.12.m12.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.12.m12.1b"><ci id="S5.SS1.p5.12.m12.1.1.cmml" xref="S5.SS1.p5.12.m12.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.12.m12.1c">{\bm{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.12.m12.1d">bold_italic_x</annotation></semantics></math> and <math alttext="{\bm{e}}_{a}" class="ltx_Math" display="inline" id="S5.SS1.p5.13.m13.1"><semantics id="S5.SS1.p5.13.m13.1a"><msub id="S5.SS1.p5.13.m13.1.1" xref="S5.SS1.p5.13.m13.1.1.cmml"><mi id="S5.SS1.p5.13.m13.1.1.2" xref="S5.SS1.p5.13.m13.1.1.2.cmml">𝒆</mi><mi id="S5.SS1.p5.13.m13.1.1.3" xref="S5.SS1.p5.13.m13.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.13.m13.1b"><apply id="S5.SS1.p5.13.m13.1.1.cmml" xref="S5.SS1.p5.13.m13.1.1"><csymbol cd="ambiguous" id="S5.SS1.p5.13.m13.1.1.1.cmml" xref="S5.SS1.p5.13.m13.1.1">subscript</csymbol><ci id="S5.SS1.p5.13.m13.1.1.2.cmml" xref="S5.SS1.p5.13.m13.1.1.2">𝒆</ci><ci id="S5.SS1.p5.13.m13.1.1.3.cmml" xref="S5.SS1.p5.13.m13.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.13.m13.1c">{\bm{e}}_{a}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.13.m13.1d">bold_italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> is the unit vector whose <math alttext="a" class="ltx_Math" display="inline" id="S5.SS1.p5.14.m14.1"><semantics id="S5.SS1.p5.14.m14.1a"><mi id="S5.SS1.p5.14.m14.1.1" xref="S5.SS1.p5.14.m14.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.14.m14.1b"><ci id="S5.SS1.p5.14.m14.1.1.cmml" xref="S5.SS1.p5.14.m14.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.14.m14.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.14.m14.1d">italic_a</annotation></semantics></math>-th component is <math alttext="1" class="ltx_Math" display="inline" id="S5.SS1.p5.15.m15.1"><semantics id="S5.SS1.p5.15.m15.1a"><mn id="S5.SS1.p5.15.m15.1.1" xref="S5.SS1.p5.15.m15.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.15.m15.1b"><cn id="S5.SS1.p5.15.m15.1.1.cmml" type="integer" xref="S5.SS1.p5.15.m15.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.15.m15.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.15.m15.1d">1</annotation></semantics></math>. Thus, the principal can run projected gradient descent without the knowledge of <math alttext="{\bm{u}}" class="ltx_Math" display="inline" id="S5.SS1.p5.16.m16.1"><semantics id="S5.SS1.p5.16.m16.1a"><mi id="S5.SS1.p5.16.m16.1.1" xref="S5.SS1.p5.16.m16.1.1.cmml">𝒖</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.16.m16.1b"><ci id="S5.SS1.p5.16.m16.1.1.cmml" xref="S5.SS1.p5.16.m16.1.1">𝒖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.16.m16.1c">{\bm{u}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.16.m16.1d">bold_italic_u</annotation></semantics></math>. The resulting algorithm is formalized in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg3" title="In 5.1 The single-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg3"> <div class="ltx_listing ltx_listing" id="alg3.2"> <div class="ltx_listingline" id="alg2.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l1.1.1.1" style="font-size:80%;">1:</span></span><math alttext="{\bm{p}}^{1}\leftarrow\bm{1}" class="ltx_Math" display="inline" id="alg2.l1.m1.1"><semantics id="alg2.l1.m1.1a"><mrow id="alg2.l1.m1.1.1" xref="alg2.l1.m1.1.1.cmml"><msup id="alg2.l1.m1.1.1.2" xref="alg2.l1.m1.1.1.2.cmml"><mi id="alg2.l1.m1.1.1.2.2" xref="alg2.l1.m1.1.1.2.2.cmml">𝒑</mi><mn id="alg2.l1.m1.1.1.2.3" xref="alg2.l1.m1.1.1.2.3.cmml">1</mn></msup><mo id="alg2.l1.m1.1.1.1" stretchy="false" xref="alg2.l1.m1.1.1.1.cmml">←</mo><mn id="alg2.l1.m1.1.1.3" xref="alg2.l1.m1.1.1.3.cmml">𝟏</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m1.1b"><apply id="alg2.l1.m1.1.1.cmml" xref="alg2.l1.m1.1.1"><ci id="alg2.l1.m1.1.1.1.cmml" xref="alg2.l1.m1.1.1.1">←</ci><apply id="alg2.l1.m1.1.1.2.cmml" xref="alg2.l1.m1.1.1.2"><csymbol cd="ambiguous" id="alg2.l1.m1.1.1.2.1.cmml" xref="alg2.l1.m1.1.1.2">superscript</csymbol><ci id="alg2.l1.m1.1.1.2.2.cmml" xref="alg2.l1.m1.1.1.2.2">𝒑</ci><cn id="alg2.l1.m1.1.1.2.3.cmml" type="integer" xref="alg2.l1.m1.1.1.2.3">1</cn></apply><cn id="alg2.l1.m1.1.1.3.cmml" type="integer" xref="alg2.l1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m1.1c">{\bm{p}}^{1}\leftarrow\bm{1}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m1.1d">bold_italic_p start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ← bold_1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l2.1.1.1" style="font-size:80%;">2:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l2.2">for</span> each time <math alttext="t=1,\dots,T" class="ltx_Math" display="inline" id="alg2.l2.m1.3"><semantics id="alg2.l2.m1.3a"><mrow id="alg2.l2.m1.3.4" xref="alg2.l2.m1.3.4.cmml"><mi id="alg2.l2.m1.3.4.2" xref="alg2.l2.m1.3.4.2.cmml">t</mi><mo id="alg2.l2.m1.3.4.1" xref="alg2.l2.m1.3.4.1.cmml">=</mo><mrow id="alg2.l2.m1.3.4.3.2" xref="alg2.l2.m1.3.4.3.1.cmml"><mn id="alg2.l2.m1.1.1" xref="alg2.l2.m1.1.1.cmml">1</mn><mo id="alg2.l2.m1.3.4.3.2.1" xref="alg2.l2.m1.3.4.3.1.cmml">,</mo><mi id="alg2.l2.m1.2.2" mathvariant="normal" xref="alg2.l2.m1.2.2.cmml">…</mi><mo id="alg2.l2.m1.3.4.3.2.2" xref="alg2.l2.m1.3.4.3.1.cmml">,</mo><mi id="alg2.l2.m1.3.3" xref="alg2.l2.m1.3.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l2.m1.3b"><apply id="alg2.l2.m1.3.4.cmml" xref="alg2.l2.m1.3.4"><eq id="alg2.l2.m1.3.4.1.cmml" xref="alg2.l2.m1.3.4.1"></eq><ci id="alg2.l2.m1.3.4.2.cmml" xref="alg2.l2.m1.3.4.2">𝑡</ci><list id="alg2.l2.m1.3.4.3.1.cmml" xref="alg2.l2.m1.3.4.3.2"><cn id="alg2.l2.m1.1.1.cmml" type="integer" xref="alg2.l2.m1.1.1">1</cn><ci id="alg2.l2.m1.2.2.cmml" xref="alg2.l2.m1.2.2">…</ci><ci id="alg2.l2.m1.3.3.cmml" xref="alg2.l2.m1.3.3">𝑇</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m1.3c">t=1,\dots,T</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m1.3d">italic_t = 1 , … , italic_T</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l2.3">do</span> </div> <div class="ltx_listingline" id="alg2.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l3.1.1.1" style="font-size:80%;">3:</span></span> principal selects payment vector <math alttext="{\bm{p}}^{t}" class="ltx_Math" display="inline" id="alg2.l3.m1.1"><semantics id="alg2.l3.m1.1a"><msup id="alg2.l3.m1.1.1" xref="alg2.l3.m1.1.1.cmml"><mi id="alg2.l3.m1.1.1.2" xref="alg2.l3.m1.1.1.2.cmml">𝒑</mi><mi id="alg2.l3.m1.1.1.3" xref="alg2.l3.m1.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="alg2.l3.m1.1b"><apply id="alg2.l3.m1.1.1.cmml" xref="alg2.l3.m1.1.1"><csymbol cd="ambiguous" id="alg2.l3.m1.1.1.1.cmml" xref="alg2.l3.m1.1.1">superscript</csymbol><ci id="alg2.l3.m1.1.1.2.cmml" xref="alg2.l3.m1.1.1.2">𝒑</ci><ci id="alg2.l3.m1.1.1.3.cmml" xref="alg2.l3.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m1.1c">{\bm{p}}^{t}</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m1.1d">bold_italic_p start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, observes action <math alttext="a^{t}" class="ltx_Math" display="inline" id="alg2.l3.m2.1"><semantics id="alg2.l3.m2.1a"><msup id="alg2.l3.m2.1.1" xref="alg2.l3.m2.1.1.cmml"><mi id="alg2.l3.m2.1.1.2" xref="alg2.l3.m2.1.1.2.cmml">a</mi><mi id="alg2.l3.m2.1.1.3" xref="alg2.l3.m2.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="alg2.l3.m2.1b"><apply id="alg2.l3.m2.1.1.cmml" xref="alg2.l3.m2.1.1"><csymbol cd="ambiguous" id="alg2.l3.m2.1.1.1.cmml" xref="alg2.l3.m2.1.1">superscript</csymbol><ci id="alg2.l3.m2.1.1.2.cmml" xref="alg2.l3.m2.1.1.2">𝑎</ci><ci id="alg2.l3.m2.1.1.3.cmml" xref="alg2.l3.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m2.1c">a^{t}</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m2.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> played by the agent </div> <div class="ltx_listingline" id="alg2.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l4.2.1.1" style="font-size:80%;">4:</span></span> principal sets 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xref="alg2.l4.m1.1.2.3.2.3.cmml">𝒫</mi></msub><mo id="alg2.l4.m1.1.2.3.1" xref="alg2.l4.m1.1.2.3.1.cmml">⁢</mo><mrow id="alg2.l4.m1.1.1.3" xref="alg2.l4.m1.1.1.1.1.1.cmml"><mo id="alg2.l4.m1.1.1.3.1" xref="alg2.l4.m1.1.1.1.1.1.cmml">[</mo><mrow id="alg2.l4.m1.1.1.1.1.1" xref="alg2.l4.m1.1.1.1.1.1.cmml"><msup id="alg2.l4.m1.1.1.1.1.1.2" xref="alg2.l4.m1.1.1.1.1.1.2.cmml"><mi id="alg2.l4.m1.1.1.1.1.1.2.2" xref="alg2.l4.m1.1.1.1.1.1.2.2.cmml">𝒑</mi><mi id="alg2.l4.m1.1.1.1.1.1.2.3" xref="alg2.l4.m1.1.1.1.1.1.2.3.cmml">t</mi></msup><mo id="alg2.l4.m1.1.1.1.1.1.1" xref="alg2.l4.m1.1.1.1.1.1.1.cmml">−</mo><mrow id="alg2.l4.m1.1.1.1.1.1.3" xref="alg2.l4.m1.1.1.1.1.1.3.cmml"><mi id="alg2.l4.m1.1.1.1.1.1.3.2" xref="alg2.l4.m1.1.1.1.1.1.3.2.cmml">η</mi><mo id="alg2.l4.m1.1.1.1.1.1.3.1" xref="alg2.l4.m1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="alg2.l4.m1.1.1.1.1.1.3.3" xref="alg2.l4.m1.1.1.1.1.1.3.3.cmml"><mi id="alg2.l4.m1.1.1.1.1.1.3.3.2" xref="alg2.l4.m1.1.1.1.1.1.3.3.2.cmml">𝒆</mi><msup 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id="alg2.l4.m1.1c">{\bm{p}}^{t+1}\leftarrow\Pi_{\mathcal{P}}\quantity[{\bm{p}}^{t}-\eta{\bm{e}}_{% a^{t}}]</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m1.1d">bold_italic_p start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ← roman_Π start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT [ start_ARG bold_italic_p start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT - italic_η bold_italic_e start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_ARG ]</annotation></semantics></math> <math alttext="\triangleright" class="ltx_Math" display="inline" id="alg2.l4.m2.1"><semantics id="alg2.l4.m2.1a"><mo id="alg2.l4.m2.1.1" mathcolor="#808080" xref="alg2.l4.m2.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg2.l4.m2.1b"><ci id="alg2.l4.m2.1.1.cmml" xref="alg2.l4.m2.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m2.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m2.1d">▷</annotation></semantics></math><span class="ltx_text" id="alg2.l4.1" style="color:#808080;"> <math alttext="\eta=\sqrt{m/T}" class="ltx_Math" display="inline" id="alg2.l4.1.m1.1"><semantics id="alg2.l4.1.m1.1a"><mrow id="alg2.l4.1.m1.1.1" xref="alg2.l4.1.m1.1.1.cmml"><mi id="alg2.l4.1.m1.1.1.2" mathcolor="#808080" xref="alg2.l4.1.m1.1.1.2.cmml">η</mi><mo id="alg2.l4.1.m1.1.1.1" mathcolor="#808080" xref="alg2.l4.1.m1.1.1.1.cmml">=</mo><msqrt id="alg2.l4.1.m1.1.1.3" mathcolor="#808080" xref="alg2.l4.1.m1.1.1.3.cmml"><mrow id="alg2.l4.1.m1.1.1.3.2" xref="alg2.l4.1.m1.1.1.3.2.cmml"><mi id="alg2.l4.1.m1.1.1.3.2.2" mathcolor="#808080" xref="alg2.l4.1.m1.1.1.3.2.2.cmml">m</mi><mo id="alg2.l4.1.m1.1.1.3.2.1" mathcolor="#808080" xref="alg2.l4.1.m1.1.1.3.2.1.cmml">/</mo><mi id="alg2.l4.1.m1.1.1.3.2.3" mathcolor="#808080" xref="alg2.l4.1.m1.1.1.3.2.3.cmml">T</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="alg2.l4.1.m1.1b"><apply id="alg2.l4.1.m1.1.1.cmml" xref="alg2.l4.1.m1.1.1"><eq id="alg2.l4.1.m1.1.1.1.cmml" xref="alg2.l4.1.m1.1.1.1"></eq><ci id="alg2.l4.1.m1.1.1.2.cmml" xref="alg2.l4.1.m1.1.1.2">𝜂</ci><apply id="alg2.l4.1.m1.1.1.3.cmml" xref="alg2.l4.1.m1.1.1.3"><root id="alg2.l4.1.m1.1.1.3a.cmml" xref="alg2.l4.1.m1.1.1.3"></root><apply id="alg2.l4.1.m1.1.1.3.2.cmml" xref="alg2.l4.1.m1.1.1.3.2"><divide id="alg2.l4.1.m1.1.1.3.2.1.cmml" xref="alg2.l4.1.m1.1.1.3.2.1"></divide><ci id="alg2.l4.1.m1.1.1.3.2.2.cmml" xref="alg2.l4.1.m1.1.1.3.2.2">𝑚</ci><ci id="alg2.l4.1.m1.1.1.3.2.3.cmml" xref="alg2.l4.1.m1.1.1.3.2.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.1.m1.1c">\eta=\sqrt{m/T}</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.1.m1.1d">italic_η = square-root start_ARG italic_m / italic_T end_ARG</annotation></semantics></math> is the step size</span> </div> <div class="ltx_listingline" id="alg2.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l5.1.1.1" style="font-size:80%;">5:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l5.2">return</span> <math alttext="-\frac{1}{T}\sum_{t=1}^{T}{\bm{p}}^{t}" class="ltx_Math" display="inline" id="alg2.l5.m1.1"><semantics id="alg2.l5.m1.1a"><mrow id="alg2.l5.m1.1.1" xref="alg2.l5.m1.1.1.cmml"><mo id="alg2.l5.m1.1.1a" xref="alg2.l5.m1.1.1.cmml">−</mo><mrow id="alg2.l5.m1.1.1.2" xref="alg2.l5.m1.1.1.2.cmml"><mfrac id="alg2.l5.m1.1.1.2.2" xref="alg2.l5.m1.1.1.2.2.cmml"><mn id="alg2.l5.m1.1.1.2.2.2" xref="alg2.l5.m1.1.1.2.2.2.cmml">1</mn><mi id="alg2.l5.m1.1.1.2.2.3" xref="alg2.l5.m1.1.1.2.2.3.cmml">T</mi></mfrac><mo id="alg2.l5.m1.1.1.2.1" xref="alg2.l5.m1.1.1.2.1.cmml">⁢</mo><mrow id="alg2.l5.m1.1.1.2.3" xref="alg2.l5.m1.1.1.2.3.cmml"><msubsup id="alg2.l5.m1.1.1.2.3.1" xref="alg2.l5.m1.1.1.2.3.1.cmml"><mo id="alg2.l5.m1.1.1.2.3.1.2.2" xref="alg2.l5.m1.1.1.2.3.1.2.2.cmml">∑</mo><mrow 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id="alg2.l5.m1.1.1.2.3.1.2.3.3.cmml" type="integer" xref="alg2.l5.m1.1.1.2.3.1.2.3.3">1</cn></apply></apply><ci id="alg2.l5.m1.1.1.2.3.1.3.cmml" xref="alg2.l5.m1.1.1.2.3.1.3">𝑇</ci></apply><apply id="alg2.l5.m1.1.1.2.3.2.cmml" xref="alg2.l5.m1.1.1.2.3.2"><csymbol cd="ambiguous" id="alg2.l5.m1.1.1.2.3.2.1.cmml" xref="alg2.l5.m1.1.1.2.3.2">superscript</csymbol><ci id="alg2.l5.m1.1.1.2.3.2.2.cmml" xref="alg2.l5.m1.1.1.2.3.2.2">𝒑</ci><ci id="alg2.l5.m1.1.1.2.3.2.3.cmml" xref="alg2.l5.m1.1.1.2.3.2.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m1.1c">-\frac{1}{T}\sum_{t=1}^{T}{\bm{p}}^{t}</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m1.1d">- divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_p start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg3.3.1.1">Algorithm 3</span> </span> Principal’s algorithm for learning a single-agent game in the no-regret model</figcaption> </figure> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.3"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2503.01976v1#alg3" title="In 5.1 The single-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem2.p1.3.3"> <math alttext="\varepsilon" 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">ε</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">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.1.m1.1d">italic_ε</annotation></semantics></math>-learns any single-agent game <math alttext="\Gamma" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.2.m2.1a"><mi id="S5.Thmtheorem2.p1.2.2.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" 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xref="S5.Thmtheorem2.p1.3.3.m3.1.1.3.2">𝜀</ci><cn id="S5.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.3.m3.1c">{\mathcal{O}}(m^{3}+C^{2}m^{2})/\varepsilon^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.3.m3.1d">caligraphic_O ( italic_m start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> rounds.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS1.2.p1"> <p class="ltx_p" id="S5.SS1.2.p1.4">Let <math alttext="\bar{\bm{p}}=\frac{1}{T}\sum_{t=1}^{T}{\bm{p}}^{t}" class="ltx_Math" display="inline" id="S5.SS1.2.p1.1.m1.1"><semantics id="S5.SS1.2.p1.1.m1.1a"><mrow id="S5.SS1.2.p1.1.m1.1.1" xref="S5.SS1.2.p1.1.m1.1.1.cmml"><mover accent="true" id="S5.SS1.2.p1.1.m1.1.1.2" xref="S5.SS1.2.p1.1.m1.1.1.2.cmml"><mi id="S5.SS1.2.p1.1.m1.1.1.2.2" xref="S5.SS1.2.p1.1.m1.1.1.2.2.cmml">𝒑</mi><mo id="S5.SS1.2.p1.1.m1.1.1.2.1" xref="S5.SS1.2.p1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S5.SS1.2.p1.1.m1.1.1.1" xref="S5.SS1.2.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S5.SS1.2.p1.1.m1.1.1.3" xref="S5.SS1.2.p1.1.m1.1.1.3.cmml"><mfrac id="S5.SS1.2.p1.1.m1.1.1.3.2" xref="S5.SS1.2.p1.1.m1.1.1.3.2.cmml"><mn id="S5.SS1.2.p1.1.m1.1.1.3.2.2" xref="S5.SS1.2.p1.1.m1.1.1.3.2.2.cmml">1</mn><mi id="S5.SS1.2.p1.1.m1.1.1.3.2.3" xref="S5.SS1.2.p1.1.m1.1.1.3.2.3.cmml">T</mi></mfrac><mo id="S5.SS1.2.p1.1.m1.1.1.3.1" xref="S5.SS1.2.p1.1.m1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.1.m1.1.1.3.3" xref="S5.SS1.2.p1.1.m1.1.1.3.3.cmml"><msubsup id="S5.SS1.2.p1.1.m1.1.1.3.3.1" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1.cmml"><mo 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xref="S5.SS1.2.p1.1.m1.1.1.3.3.1">superscript</csymbol><apply id="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1"><csymbol cd="ambiguous" id="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.1.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1">subscript</csymbol><sum id="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.2.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.2"></sum><apply id="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3"><eq id="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3.1.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3.1"></eq><ci id="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3.2.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3.2">𝑡</ci><cn id="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3.3.cmml" type="integer" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1.2.3.3">1</cn></apply></apply><ci id="S5.SS1.2.p1.1.m1.1.1.3.3.1.3.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.1.3">𝑇</ci></apply><apply id="S5.SS1.2.p1.1.m1.1.1.3.3.2.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S5.SS1.2.p1.1.m1.1.1.3.3.2.1.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.2">superscript</csymbol><ci id="S5.SS1.2.p1.1.m1.1.1.3.3.2.2.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.2.2">𝒑</ci><ci id="S5.SS1.2.p1.1.m1.1.1.3.3.2.3.cmml" xref="S5.SS1.2.p1.1.m1.1.1.3.3.2.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.1.m1.1c">\bar{\bm{p}}=\frac{1}{T}\sum_{t=1}^{T}{\bm{p}}^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.1.m1.1d">over¯ start_ARG bold_italic_p end_ARG = divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_p start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> be the average payment. From the preliminaries, the regret bound of the principal is given by <math alttext="R_{0}\leq BG\sqrt{T}" class="ltx_Math" display="inline" id="S5.SS1.2.p1.2.m2.1"><semantics id="S5.SS1.2.p1.2.m2.1a"><mrow id="S5.SS1.2.p1.2.m2.1.1" xref="S5.SS1.2.p1.2.m2.1.1.cmml"><msub id="S5.SS1.2.p1.2.m2.1.1.2" xref="S5.SS1.2.p1.2.m2.1.1.2.cmml"><mi id="S5.SS1.2.p1.2.m2.1.1.2.2" xref="S5.SS1.2.p1.2.m2.1.1.2.2.cmml">R</mi><mn id="S5.SS1.2.p1.2.m2.1.1.2.3" xref="S5.SS1.2.p1.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S5.SS1.2.p1.2.m2.1.1.1" xref="S5.SS1.2.p1.2.m2.1.1.1.cmml">≤</mo><mrow id="S5.SS1.2.p1.2.m2.1.1.3" xref="S5.SS1.2.p1.2.m2.1.1.3.cmml"><mi id="S5.SS1.2.p1.2.m2.1.1.3.2" xref="S5.SS1.2.p1.2.m2.1.1.3.2.cmml">B</mi><mo id="S5.SS1.2.p1.2.m2.1.1.3.1" xref="S5.SS1.2.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S5.SS1.2.p1.2.m2.1.1.3.3" xref="S5.SS1.2.p1.2.m2.1.1.3.3.cmml">G</mi><mo id="S5.SS1.2.p1.2.m2.1.1.3.1a" xref="S5.SS1.2.p1.2.m2.1.1.3.1.cmml">⁢</mo><msqrt id="S5.SS1.2.p1.2.m2.1.1.3.4" xref="S5.SS1.2.p1.2.m2.1.1.3.4.cmml"><mi id="S5.SS1.2.p1.2.m2.1.1.3.4.2" xref="S5.SS1.2.p1.2.m2.1.1.3.4.2.cmml">T</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.2.m2.1b"><apply id="S5.SS1.2.p1.2.m2.1.1.cmml" xref="S5.SS1.2.p1.2.m2.1.1"><leq id="S5.SS1.2.p1.2.m2.1.1.1.cmml" xref="S5.SS1.2.p1.2.m2.1.1.1"></leq><apply id="S5.SS1.2.p1.2.m2.1.1.2.cmml" xref="S5.SS1.2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.2.p1.2.m2.1.1.2.1.cmml" xref="S5.SS1.2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS1.2.p1.2.m2.1.1.2.2.cmml" xref="S5.SS1.2.p1.2.m2.1.1.2.2">𝑅</ci><cn id="S5.SS1.2.p1.2.m2.1.1.2.3.cmml" type="integer" xref="S5.SS1.2.p1.2.m2.1.1.2.3">0</cn></apply><apply id="S5.SS1.2.p1.2.m2.1.1.3.cmml" xref="S5.SS1.2.p1.2.m2.1.1.3"><times id="S5.SS1.2.p1.2.m2.1.1.3.1.cmml" xref="S5.SS1.2.p1.2.m2.1.1.3.1"></times><ci id="S5.SS1.2.p1.2.m2.1.1.3.2.cmml" xref="S5.SS1.2.p1.2.m2.1.1.3.2">𝐵</ci><ci id="S5.SS1.2.p1.2.m2.1.1.3.3.cmml" xref="S5.SS1.2.p1.2.m2.1.1.3.3">𝐺</ci><apply id="S5.SS1.2.p1.2.m2.1.1.3.4.cmml" xref="S5.SS1.2.p1.2.m2.1.1.3.4"><root id="S5.SS1.2.p1.2.m2.1.1.3.4a.cmml" xref="S5.SS1.2.p1.2.m2.1.1.3.4"></root><ci id="S5.SS1.2.p1.2.m2.1.1.3.4.2.cmml" xref="S5.SS1.2.p1.2.m2.1.1.3.4.2">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.2.m2.1c">R_{0}\leq BG\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.2.m2.1d">italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ italic_B italic_G square-root start_ARG italic_T end_ARG</annotation></semantics></math> where <math alttext="B\lesssim\sqrt{m}" class="ltx_Math" display="inline" id="S5.SS1.2.p1.3.m3.1"><semantics id="S5.SS1.2.p1.3.m3.1a"><mrow id="S5.SS1.2.p1.3.m3.1.1" xref="S5.SS1.2.p1.3.m3.1.1.cmml"><mi id="S5.SS1.2.p1.3.m3.1.1.2" xref="S5.SS1.2.p1.3.m3.1.1.2.cmml">B</mi><mo id="S5.SS1.2.p1.3.m3.1.1.1" xref="S5.SS1.2.p1.3.m3.1.1.1.cmml">≲</mo><msqrt id="S5.SS1.2.p1.3.m3.1.1.3" xref="S5.SS1.2.p1.3.m3.1.1.3.cmml"><mi id="S5.SS1.2.p1.3.m3.1.1.3.2" xref="S5.SS1.2.p1.3.m3.1.1.3.2.cmml">m</mi></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.3.m3.1b"><apply id="S5.SS1.2.p1.3.m3.1.1.cmml" xref="S5.SS1.2.p1.3.m3.1.1"><csymbol cd="latexml" id="S5.SS1.2.p1.3.m3.1.1.1.cmml" xref="S5.SS1.2.p1.3.m3.1.1.1">less-than-or-similar-to</csymbol><ci id="S5.SS1.2.p1.3.m3.1.1.2.cmml" xref="S5.SS1.2.p1.3.m3.1.1.2">𝐵</ci><apply id="S5.SS1.2.p1.3.m3.1.1.3.cmml" xref="S5.SS1.2.p1.3.m3.1.1.3"><root id="S5.SS1.2.p1.3.m3.1.1.3a.cmml" xref="S5.SS1.2.p1.3.m3.1.1.3"></root><ci id="S5.SS1.2.p1.3.m3.1.1.3.2.cmml" xref="S5.SS1.2.p1.3.m3.1.1.3.2">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.3.m3.1c">B\lesssim\sqrt{m}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.3.m3.1d">italic_B ≲ square-root start_ARG italic_m end_ARG</annotation></semantics></math> and <math alttext="G=1" class="ltx_Math" display="inline" id="S5.SS1.2.p1.4.m4.1"><semantics id="S5.SS1.2.p1.4.m4.1a"><mrow id="S5.SS1.2.p1.4.m4.1.1" xref="S5.SS1.2.p1.4.m4.1.1.cmml"><mi id="S5.SS1.2.p1.4.m4.1.1.2" xref="S5.SS1.2.p1.4.m4.1.1.2.cmml">G</mi><mo id="S5.SS1.2.p1.4.m4.1.1.1" xref="S5.SS1.2.p1.4.m4.1.1.1.cmml">=</mo><mn id="S5.SS1.2.p1.4.m4.1.1.3" xref="S5.SS1.2.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.4.m4.1b"><apply id="S5.SS1.2.p1.4.m4.1.1.cmml" xref="S5.SS1.2.p1.4.m4.1.1"><eq id="S5.SS1.2.p1.4.m4.1.1.1.cmml" xref="S5.SS1.2.p1.4.m4.1.1.1"></eq><ci id="S5.SS1.2.p1.4.m4.1.1.2.cmml" xref="S5.SS1.2.p1.4.m4.1.1.2">𝐺</ci><cn id="S5.SS1.2.p1.4.m4.1.1.3.cmml" type="integer" xref="S5.SS1.2.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.4.m4.1c">G=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.4.m4.1d">italic_G = 1</annotation></semantics></math>. Then, by <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.Thmtheorem1" title="Lemma 5.1. ‣ 5.1 The single-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.1</span></a> and the argument in the previous paragraph, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx3"> <tbody id="S5.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\varepsilon=\norm{\bar{\bm{p}}+{\bm{u}}-\bm{1}}_{\infty}\leq\norm% {\bar{\bm{p}}+{\bm{u}}-\bm{1}}_{1}\lesssim\frac{m}{T}(R_{0}+C\sqrt{T})\lesssim% \frac{m}{\sqrt{T}}(C+\sqrt{m})" class="ltx_Math" display="inline" id="S5.E5.m1.4"><semantics id="S5.E5.m1.4a"><mrow id="S5.E5.m1.4.4" xref="S5.E5.m1.4.4.cmml"><mi id="S5.E5.m1.4.4.4" xref="S5.E5.m1.4.4.4.cmml">ε</mi><mo id="S5.E5.m1.4.4.5" xref="S5.E5.m1.4.4.5.cmml">=</mo><msub id="S5.E5.m1.4.4.6" xref="S5.E5.m1.4.4.6.cmml"><mrow id="S5.E5.m1.1.1a.3" xref="S5.E5.m1.1.1a.2.cmml"><mo id="S5.E5.m1.1.1a.3.1" xref="S5.E5.m1.1.1a.2.1.cmml">‖</mo><mrow id="S5.E5.m1.1.1.1.1.1" xref="S5.E5.m1.1.1.1.1.1.cmml"><mrow id="S5.E5.m1.1.1.1.1.1.2" xref="S5.E5.m1.1.1.1.1.1.2.cmml"><mover accent="true" id="S5.E5.m1.1.1.1.1.1.2.2" xref="S5.E5.m1.1.1.1.1.1.2.2.cmml"><mi id="S5.E5.m1.1.1.1.1.1.2.2.2" xref="S5.E5.m1.1.1.1.1.1.2.2.2.cmml">𝒑</mi><mo id="S5.E5.m1.1.1.1.1.1.2.2.1" xref="S5.E5.m1.1.1.1.1.1.2.2.1.cmml">¯</mo></mover><mo id="S5.E5.m1.1.1.1.1.1.2.1" xref="S5.E5.m1.1.1.1.1.1.2.1.cmml">+</mo><mi id="S5.E5.m1.1.1.1.1.1.2.3" xref="S5.E5.m1.1.1.1.1.1.2.3.cmml">𝒖</mi></mrow><mo id="S5.E5.m1.1.1.1.1.1.1" xref="S5.E5.m1.1.1.1.1.1.1.cmml">−</mo><mn id="S5.E5.m1.1.1.1.1.1.3" xref="S5.E5.m1.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="S5.E5.m1.1.1a.3.2" xref="S5.E5.m1.1.1a.2.1.cmml">‖</mo></mrow><mi id="S5.E5.m1.4.4.6.2" mathvariant="normal" xref="S5.E5.m1.4.4.6.2.cmml">∞</mi></msub><mo id="S5.E5.m1.4.4.7" xref="S5.E5.m1.4.4.7.cmml">≤</mo><msub id="S5.E5.m1.4.4.8" xref="S5.E5.m1.4.4.8.cmml"><mrow id="S5.E5.m1.2.2a.3" xref="S5.E5.m1.2.2a.2.cmml"><mo id="S5.E5.m1.2.2a.3.1" xref="S5.E5.m1.2.2a.2.1.cmml">‖</mo><mrow id="S5.E5.m1.2.2.1.1.1" xref="S5.E5.m1.2.2.1.1.1.cmml"><mrow id="S5.E5.m1.2.2.1.1.1.2" xref="S5.E5.m1.2.2.1.1.1.2.cmml"><mover accent="true" id="S5.E5.m1.2.2.1.1.1.2.2" xref="S5.E5.m1.2.2.1.1.1.2.2.cmml"><mi id="S5.E5.m1.2.2.1.1.1.2.2.2" xref="S5.E5.m1.2.2.1.1.1.2.2.2.cmml">𝒑</mi><mo id="S5.E5.m1.2.2.1.1.1.2.2.1" xref="S5.E5.m1.2.2.1.1.1.2.2.1.cmml">¯</mo></mover><mo id="S5.E5.m1.2.2.1.1.1.2.1" xref="S5.E5.m1.2.2.1.1.1.2.1.cmml">+</mo><mi id="S5.E5.m1.2.2.1.1.1.2.3" xref="S5.E5.m1.2.2.1.1.1.2.3.cmml">𝒖</mi></mrow><mo id="S5.E5.m1.2.2.1.1.1.1" xref="S5.E5.m1.2.2.1.1.1.1.cmml">−</mo><mn id="S5.E5.m1.2.2.1.1.1.3" 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id="S5.E5.m1.4d">italic_ε = ∥ start_ARG over¯ start_ARG bold_italic_p end_ARG + bold_italic_u - bold_1 end_ARG ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≤ ∥ start_ARG over¯ start_ARG bold_italic_p end_ARG + bold_italic_u - bold_1 end_ARG ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≲ divide start_ARG italic_m end_ARG start_ARG italic_T end_ARG ( italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_C square-root start_ARG italic_T end_ARG ) ≲ divide start_ARG italic_m end_ARG start_ARG square-root start_ARG italic_T end_ARG end_ARG ( italic_C + square-root start_ARG italic_m end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS1.2.p1.5">upon which solving for <math alttext="T" class="ltx_Math" display="inline" id="S5.SS1.2.p1.5.m1.1"><semantics id="S5.SS1.2.p1.5.m1.1a"><mi id="S5.SS1.2.p1.5.m1.1.1" xref="S5.SS1.2.p1.5.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.5.m1.1b"><ci id="S5.SS1.2.p1.5.m1.1.1.cmml" xref="S5.SS1.2.p1.5.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.5.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.5.m1.1d">italic_T</annotation></semantics></math> yields the desired result. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" 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">Remark 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.4">It is possible to construct a binary search algorithm similar to <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg1" title="In 4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> for this setting, by using <math alttext="{\mathcal{O}}(m\log(1/\varepsilon))" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.m1.3"><semantics id="S5.Thmtheorem3.p1.1.m1.3a"><mrow id="S5.Thmtheorem3.p1.1.m1.3.3" xref="S5.Thmtheorem3.p1.1.m1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.1.m1.3.3.3" xref="S5.Thmtheorem3.p1.1.m1.3.3.3.cmml">𝒪</mi><mo id="S5.Thmtheorem3.p1.1.m1.3.3.2" xref="S5.Thmtheorem3.p1.1.m1.3.3.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.1.m1.3.3.1.1" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.2" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.2.cmml">m</mi><mo id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.1" lspace="0.167em" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.1.m1.2.2.4" xref="S5.Thmtheorem3.p1.1.m1.2.2.3.cmml"><mi id="S5.Thmtheorem3.p1.1.m1.2.2.2.2" xref="S5.Thmtheorem3.p1.1.m1.2.2.3.1.cmml">log</mi><mo id="S5.Thmtheorem3.p1.1.m1.2.2.4a" xref="S5.Thmtheorem3.p1.1.m1.2.2.3.1.cmml">⁡</mo><mrow id="S5.Thmtheorem3.p1.1.m1.2.2.4.1" xref="S5.Thmtheorem3.p1.1.m1.2.2.3.cmml"><mo id="S5.Thmtheorem3.p1.1.m1.2.2.4.1.1" xref="S5.Thmtheorem3.p1.1.m1.2.2.3.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.1.cmml">/</mo><mi id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Thmtheorem3.p1.1.m1.2.2.4.1.2" xref="S5.Thmtheorem3.p1.1.m1.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.m1.3b"><apply id="S5.Thmtheorem3.p1.1.m1.3.3.cmml" xref="S5.Thmtheorem3.p1.1.m1.3.3"><times id="S5.Thmtheorem3.p1.1.m1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.1.m1.3.3.2"></times><ci id="S5.Thmtheorem3.p1.1.m1.3.3.3.cmml" xref="S5.Thmtheorem3.p1.1.m1.3.3.3">𝒪</ci><apply id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1"><times id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.1"></times><ci id="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.1.m1.3.3.1.1.1.2">𝑚</ci><apply id="S5.Thmtheorem3.p1.1.m1.2.2.3.cmml" xref="S5.Thmtheorem3.p1.1.m1.2.2.4"><log id="S5.Thmtheorem3.p1.1.m1.2.2.3.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.2.2.2.2"></log><apply id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1"><divide id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.1"></divide><cn id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.1.m1.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.m1.3c">{\mathcal{O}}(m\log(1/\varepsilon))</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.m1.3d">caligraphic_O ( italic_m roman_log ( start_ARG 1 / italic_ε end_ARG ) )</annotation></semantics></math> signals, one for each step of the binary search algorithm. However, this would result in an algorithm whose dependence on <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.m2.1a"><mi id="S5.Thmtheorem3.p1.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.m2.1b"><ci id="S5.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.m2.1d">italic_ε</annotation></semantics></math> is <math alttext="1/\varepsilon^{2}\cdot\log(1/\varepsilon)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.m3.2"><semantics id="S5.Thmtheorem3.p1.3.m3.2a"><mrow id="S5.Thmtheorem3.p1.3.m3.2.3" xref="S5.Thmtheorem3.p1.3.m3.2.3.cmml"><mrow id="S5.Thmtheorem3.p1.3.m3.2.3.2" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.cmml"><mn id="S5.Thmtheorem3.p1.3.m3.2.3.2.2" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.3.m3.2.3.2.1" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.1.cmml">/</mo><msup id="S5.Thmtheorem3.p1.3.m3.2.3.2.3" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.3.cmml"><mi id="S5.Thmtheorem3.p1.3.m3.2.3.2.3.2" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.3.2.cmml">ε</mi><mn id="S5.Thmtheorem3.p1.3.m3.2.3.2.3.3" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.3.3.cmml">2</mn></msup></mrow><mo id="S5.Thmtheorem3.p1.3.m3.2.3.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmtheorem3.p1.3.m3.2.3.1.cmml">⋅</mo><mrow id="S5.Thmtheorem3.p1.3.m3.2.2.4" xref="S5.Thmtheorem3.p1.3.m3.2.2.3.cmml"><mi id="S5.Thmtheorem3.p1.3.m3.2.2.2.2" xref="S5.Thmtheorem3.p1.3.m3.2.2.3.1.cmml">log</mi><mo id="S5.Thmtheorem3.p1.3.m3.2.2.4a" xref="S5.Thmtheorem3.p1.3.m3.2.2.3.1.cmml">⁡</mo><mrow id="S5.Thmtheorem3.p1.3.m3.2.2.4.1" xref="S5.Thmtheorem3.p1.3.m3.2.2.3.cmml"><mo id="S5.Thmtheorem3.p1.3.m3.2.2.4.1.1" xref="S5.Thmtheorem3.p1.3.m3.2.2.3.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1.cmml">/</mo><mi id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Thmtheorem3.p1.3.m3.2.2.4.1.2" xref="S5.Thmtheorem3.p1.3.m3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.m3.2b"><apply id="S5.Thmtheorem3.p1.3.m3.2.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.3"><ci id="S5.Thmtheorem3.p1.3.m3.2.3.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.3.1">⋅</ci><apply id="S5.Thmtheorem3.p1.3.m3.2.3.2.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.3.2"><divide id="S5.Thmtheorem3.p1.3.m3.2.3.2.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.1"></divide><cn id="S5.Thmtheorem3.p1.3.m3.2.3.2.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.2">1</cn><apply id="S5.Thmtheorem3.p1.3.m3.2.3.2.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.m3.2.3.2.3.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.3">superscript</csymbol><ci id="S5.Thmtheorem3.p1.3.m3.2.3.2.3.2.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.3.2">𝜀</ci><cn id="S5.Thmtheorem3.p1.3.m3.2.3.2.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.m3.2.3.2.3.3">2</cn></apply></apply><apply id="S5.Thmtheorem3.p1.3.m3.2.2.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.2.4"><log id="S5.Thmtheorem3.p1.3.m3.2.2.3.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.2.2.2.2"></log><apply id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1"><divide id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1"></divide><cn id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.m3.2c">1/\varepsilon^{2}\cdot\log(1/\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.m3.2d">1 / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ roman_log ( start_ARG 1 / italic_ε end_ARG )</annotation></semantics></math>, whereas our <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg3" title="In 5.1 The single-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a> achieves better dependence <math alttext="1/\varepsilon^{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.4.m4.1"><semantics id="S5.Thmtheorem3.p1.4.m4.1a"><mrow id="S5.Thmtheorem3.p1.4.m4.1.1" xref="S5.Thmtheorem3.p1.4.m4.1.1.cmml"><mn id="S5.Thmtheorem3.p1.4.m4.1.1.2" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.4.m4.1.1.1" xref="S5.Thmtheorem3.p1.4.m4.1.1.1.cmml">/</mo><msup id="S5.Thmtheorem3.p1.4.m4.1.1.3" xref="S5.Thmtheorem3.p1.4.m4.1.1.3.cmml"><mi id="S5.Thmtheorem3.p1.4.m4.1.1.3.2" xref="S5.Thmtheorem3.p1.4.m4.1.1.3.2.cmml">ε</mi><mn id="S5.Thmtheorem3.p1.4.m4.1.1.3.3" xref="S5.Thmtheorem3.p1.4.m4.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.4.m4.1b"><apply id="S5.Thmtheorem3.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1"><divide id="S5.Thmtheorem3.p1.4.m4.1.1.1.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.1"></divide><cn id="S5.Thmtheorem3.p1.4.m4.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.4.m4.1.1.2">1</cn><apply id="S5.Thmtheorem3.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.4.m4.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p1.4.m4.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.3.2">𝜀</ci><cn id="S5.Thmtheorem3.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.4.m4.1c">1/\varepsilon^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.4.m4.1d">1 / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, saving a logarithmic term.</p> </div> </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>The multi-agent case</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.9">The no-regret learning case for multiple agents is the only case in which we take advantage of signaling. Intuitively, our algorithm uses signals to induce the action profile <math alttext="a_{-i}" 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">a</mi><mrow id="S5.SS2.p1.1.m1.1.1.3" xref="S5.SS2.p1.1.m1.1.1.3.cmml"><mo id="S5.SS2.p1.1.m1.1.1.3a" xref="S5.SS2.p1.1.m1.1.1.3.cmml">−</mo><mi id="S5.SS2.p1.1.m1.1.1.3.2" xref="S5.SS2.p1.1.m1.1.1.3.2.cmml">i</mi></mrow></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">𝑎</ci><apply id="S5.SS2.p1.1.m1.1.1.3.cmml" xref="S5.SS2.p1.1.m1.1.1.3"><minus id="S5.SS2.p1.1.m1.1.1.3.1.cmml" xref="S5.SS2.p1.1.m1.1.1.3"></minus><ci id="S5.SS2.p1.1.m1.1.1.3.2.cmml" xref="S5.SS2.p1.1.m1.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">a_{-i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> among other agents without increasing their regret by too much. More precisely, we set the signal sets as <math alttext="S_{i}:=A_{i}\sqcup\{\bot\}" class="ltx_Math" display="inline" id="S5.SS2.p1.2.m2.1"><semantics id="S5.SS2.p1.2.m2.1a"><mrow id="S5.SS2.p1.2.m2.1.2" xref="S5.SS2.p1.2.m2.1.2.cmml"><msub id="S5.SS2.p1.2.m2.1.2.2" xref="S5.SS2.p1.2.m2.1.2.2.cmml"><mi id="S5.SS2.p1.2.m2.1.2.2.2" xref="S5.SS2.p1.2.m2.1.2.2.2.cmml">S</mi><mi id="S5.SS2.p1.2.m2.1.2.2.3" xref="S5.SS2.p1.2.m2.1.2.2.3.cmml">i</mi></msub><mo id="S5.SS2.p1.2.m2.1.2.1" lspace="0.278em" rspace="0.278em" xref="S5.SS2.p1.2.m2.1.2.1.cmml">:=</mo><mrow id="S5.SS2.p1.2.m2.1.2.3" xref="S5.SS2.p1.2.m2.1.2.3.cmml"><msub id="S5.SS2.p1.2.m2.1.2.3.2" xref="S5.SS2.p1.2.m2.1.2.3.2.cmml"><mi id="S5.SS2.p1.2.m2.1.2.3.2.2" xref="S5.SS2.p1.2.m2.1.2.3.2.2.cmml">A</mi><mi id="S5.SS2.p1.2.m2.1.2.3.2.3" xref="S5.SS2.p1.2.m2.1.2.3.2.3.cmml">i</mi></msub><mo id="S5.SS2.p1.2.m2.1.2.3.1" xref="S5.SS2.p1.2.m2.1.2.3.1.cmml">⊔</mo><mrow id="S5.SS2.p1.2.m2.1.2.3.3.2" xref="S5.SS2.p1.2.m2.1.2.3.3.1.cmml"><mo id="S5.SS2.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S5.SS2.p1.2.m2.1.2.3.3.1.cmml">{</mo><mo id="S5.SS2.p1.2.m2.1.1" lspace="0em" rspace="0em" xref="S5.SS2.p1.2.m2.1.1.cmml">⊥</mo><mo id="S5.SS2.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S5.SS2.p1.2.m2.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.2.m2.1b"><apply id="S5.SS2.p1.2.m2.1.2.cmml" xref="S5.SS2.p1.2.m2.1.2"><csymbol cd="latexml" id="S5.SS2.p1.2.m2.1.2.1.cmml" xref="S5.SS2.p1.2.m2.1.2.1">assign</csymbol><apply id="S5.SS2.p1.2.m2.1.2.2.cmml" xref="S5.SS2.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.p1.2.m2.1.2.2.1.cmml" xref="S5.SS2.p1.2.m2.1.2.2">subscript</csymbol><ci id="S5.SS2.p1.2.m2.1.2.2.2.cmml" xref="S5.SS2.p1.2.m2.1.2.2.2">𝑆</ci><ci id="S5.SS2.p1.2.m2.1.2.2.3.cmml" xref="S5.SS2.p1.2.m2.1.2.2.3">𝑖</ci></apply><apply id="S5.SS2.p1.2.m2.1.2.3.cmml" xref="S5.SS2.p1.2.m2.1.2.3"><csymbol cd="latexml" id="S5.SS2.p1.2.m2.1.2.3.1.cmml" xref="S5.SS2.p1.2.m2.1.2.3.1">square-union</csymbol><apply id="S5.SS2.p1.2.m2.1.2.3.2.cmml" xref="S5.SS2.p1.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S5.SS2.p1.2.m2.1.2.3.2.1.cmml" xref="S5.SS2.p1.2.m2.1.2.3.2">subscript</csymbol><ci id="S5.SS2.p1.2.m2.1.2.3.2.2.cmml" xref="S5.SS2.p1.2.m2.1.2.3.2.2">𝐴</ci><ci id="S5.SS2.p1.2.m2.1.2.3.2.3.cmml" xref="S5.SS2.p1.2.m2.1.2.3.2.3">𝑖</ci></apply><set id="S5.SS2.p1.2.m2.1.2.3.3.1.cmml" xref="S5.SS2.p1.2.m2.1.2.3.3.2"><csymbol cd="latexml" id="S5.SS2.p1.2.m2.1.1.cmml" xref="S5.SS2.p1.2.m2.1.1">bottom</csymbol></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.2.m2.1c">S_{i}:=A_{i}\sqcup\{\bot\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.2.m2.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊔ { ⊥ }</annotation></semantics></math> where <math alttext="\bot" class="ltx_Math" display="inline" id="S5.SS2.p1.3.m3.1"><semantics id="S5.SS2.p1.3.m3.1a"><mo id="S5.SS2.p1.3.m3.1.1" xref="S5.SS2.p1.3.m3.1.1.cmml">⊥</mo><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.3.m3.1b"><csymbol cd="latexml" id="S5.SS2.p1.3.m3.1.1.cmml" xref="S5.SS2.p1.3.m3.1.1">bottom</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.3.m3.1c">\bot</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.3.m3.1d">⊥</annotation></semantics></math> is a special signal indicating that <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.p1.4.m4.1"><semantics id="S5.SS2.p1.4.m4.1a"><mi id="S5.SS2.p1.4.m4.1.1" xref="S5.SS2.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.4.m4.1b"><ci id="S5.SS2.p1.4.m4.1.1.cmml" xref="S5.SS2.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.4.m4.1d">italic_i</annotation></semantics></math>’s utility is the one being learned at the moment. Then, when learning the utility <math alttext="U_{i}(\cdot,a_{-i})" class="ltx_Math" display="inline" id="S5.SS2.p1.5.m5.2"><semantics id="S5.SS2.p1.5.m5.2a"><mrow id="S5.SS2.p1.5.m5.2.2" xref="S5.SS2.p1.5.m5.2.2.cmml"><msub id="S5.SS2.p1.5.m5.2.2.3" xref="S5.SS2.p1.5.m5.2.2.3.cmml"><mi id="S5.SS2.p1.5.m5.2.2.3.2" xref="S5.SS2.p1.5.m5.2.2.3.2.cmml">U</mi><mi id="S5.SS2.p1.5.m5.2.2.3.3" xref="S5.SS2.p1.5.m5.2.2.3.3.cmml">i</mi></msub><mo id="S5.SS2.p1.5.m5.2.2.2" xref="S5.SS2.p1.5.m5.2.2.2.cmml">⁢</mo><mrow id="S5.SS2.p1.5.m5.2.2.1.1" xref="S5.SS2.p1.5.m5.2.2.1.2.cmml"><mo id="S5.SS2.p1.5.m5.2.2.1.1.2" stretchy="false" xref="S5.SS2.p1.5.m5.2.2.1.2.cmml">(</mo><mo id="S5.SS2.p1.5.m5.1.1" lspace="0em" rspace="0em" xref="S5.SS2.p1.5.m5.1.1.cmml">⋅</mo><mo id="S5.SS2.p1.5.m5.2.2.1.1.3" xref="S5.SS2.p1.5.m5.2.2.1.2.cmml">,</mo><msub id="S5.SS2.p1.5.m5.2.2.1.1.1" xref="S5.SS2.p1.5.m5.2.2.1.1.1.cmml"><mi id="S5.SS2.p1.5.m5.2.2.1.1.1.2" xref="S5.SS2.p1.5.m5.2.2.1.1.1.2.cmml">a</mi><mrow id="S5.SS2.p1.5.m5.2.2.1.1.1.3" xref="S5.SS2.p1.5.m5.2.2.1.1.1.3.cmml"><mo id="S5.SS2.p1.5.m5.2.2.1.1.1.3a" xref="S5.SS2.p1.5.m5.2.2.1.1.1.3.cmml">−</mo><mi id="S5.SS2.p1.5.m5.2.2.1.1.1.3.2" xref="S5.SS2.p1.5.m5.2.2.1.1.1.3.2.cmml">i</mi></mrow></msub><mo id="S5.SS2.p1.5.m5.2.2.1.1.4" stretchy="false" xref="S5.SS2.p1.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.5.m5.2b"><apply id="S5.SS2.p1.5.m5.2.2.cmml" xref="S5.SS2.p1.5.m5.2.2"><times id="S5.SS2.p1.5.m5.2.2.2.cmml" xref="S5.SS2.p1.5.m5.2.2.2"></times><apply id="S5.SS2.p1.5.m5.2.2.3.cmml" xref="S5.SS2.p1.5.m5.2.2.3"><csymbol cd="ambiguous" id="S5.SS2.p1.5.m5.2.2.3.1.cmml" xref="S5.SS2.p1.5.m5.2.2.3">subscript</csymbol><ci id="S5.SS2.p1.5.m5.2.2.3.2.cmml" xref="S5.SS2.p1.5.m5.2.2.3.2">𝑈</ci><ci id="S5.SS2.p1.5.m5.2.2.3.3.cmml" xref="S5.SS2.p1.5.m5.2.2.3.3">𝑖</ci></apply><interval closure="open" id="S5.SS2.p1.5.m5.2.2.1.2.cmml" xref="S5.SS2.p1.5.m5.2.2.1.1"><ci id="S5.SS2.p1.5.m5.1.1.cmml" xref="S5.SS2.p1.5.m5.1.1">⋅</ci><apply id="S5.SS2.p1.5.m5.2.2.1.1.1.cmml" xref="S5.SS2.p1.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p1.5.m5.2.2.1.1.1.1.cmml" xref="S5.SS2.p1.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.p1.5.m5.2.2.1.1.1.2.cmml" xref="S5.SS2.p1.5.m5.2.2.1.1.1.2">𝑎</ci><apply id="S5.SS2.p1.5.m5.2.2.1.1.1.3.cmml" xref="S5.SS2.p1.5.m5.2.2.1.1.1.3"><minus id="S5.SS2.p1.5.m5.2.2.1.1.1.3.1.cmml" xref="S5.SS2.p1.5.m5.2.2.1.1.1.3"></minus><ci id="S5.SS2.p1.5.m5.2.2.1.1.1.3.2.cmml" xref="S5.SS2.p1.5.m5.2.2.1.1.1.3.2">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.5.m5.2c">U_{i}(\cdot,a_{-i})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.5.m5.2d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>, we send signal <math alttext="\bot" class="ltx_Math" display="inline" id="S5.SS2.p1.6.m6.1"><semantics id="S5.SS2.p1.6.m6.1a"><mo id="S5.SS2.p1.6.m6.1.1" xref="S5.SS2.p1.6.m6.1.1.cmml">⊥</mo><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.6.m6.1b"><csymbol cd="latexml" id="S5.SS2.p1.6.m6.1.1.cmml" xref="S5.SS2.p1.6.m6.1.1">bottom</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.6.m6.1c">\bot</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.6.m6.1d">⊥</annotation></semantics></math> to agent <math alttext="i" 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">i</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">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.7.m7.1d">italic_i</annotation></semantics></math> and the desired action profile <math alttext="a_{j}" class="ltx_Math" display="inline" id="S5.SS2.p1.8.m8.1"><semantics id="S5.SS2.p1.8.m8.1a"><msub id="S5.SS2.p1.8.m8.1.1" xref="S5.SS2.p1.8.m8.1.1.cmml"><mi id="S5.SS2.p1.8.m8.1.1.2" xref="S5.SS2.p1.8.m8.1.1.2.cmml">a</mi><mi id="S5.SS2.p1.8.m8.1.1.3" xref="S5.SS2.p1.8.m8.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.8.m8.1b"><apply id="S5.SS2.p1.8.m8.1.1.cmml" xref="S5.SS2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS2.p1.8.m8.1.1.1.cmml" xref="S5.SS2.p1.8.m8.1.1">subscript</csymbol><ci id="S5.SS2.p1.8.m8.1.1.2.cmml" xref="S5.SS2.p1.8.m8.1.1.2">𝑎</ci><ci id="S5.SS2.p1.8.m8.1.1.3.cmml" xref="S5.SS2.p1.8.m8.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.8.m8.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.8.m8.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for each agent <math alttext="j\neq i" 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">j</mi><mo id="S5.SS2.p1.9.m9.1.1.1" xref="S5.SS2.p1.9.m9.1.1.1.cmml">≠</mo><mi id="S5.SS2.p1.9.m9.1.1.3" xref="S5.SS2.p1.9.m9.1.1.3.cmml">i</mi></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"><neq id="S5.SS2.p1.9.m9.1.1.1.cmml" xref="S5.SS2.p1.9.m9.1.1.1"></neq><ci id="S5.SS2.p1.9.m9.1.1.2.cmml" xref="S5.SS2.p1.9.m9.1.1.2">𝑗</ci><ci id="S5.SS2.p1.9.m9.1.1.3.cmml" xref="S5.SS2.p1.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.9.m9.1c">j\neq i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.9.m9.1d">italic_j ≠ italic_i</annotation></semantics></math>. This idea is formalized in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg4" title="In 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg4"> <div class="ltx_listing ltx_listing" id="alg4.2"> <div class="ltx_listingline" id="alg3.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l1.1.1.1" style="font-size:80%;">1:</span></span><math alttext="t\leftarrow 1" class="ltx_Math" display="inline" id="alg3.l1.m1.1"><semantics id="alg3.l1.m1.1a"><mrow id="alg3.l1.m1.1.1" xref="alg3.l1.m1.1.1.cmml"><mi id="alg3.l1.m1.1.1.2" xref="alg3.l1.m1.1.1.2.cmml">t</mi><mo id="alg3.l1.m1.1.1.1" stretchy="false" xref="alg3.l1.m1.1.1.1.cmml">←</mo><mn id="alg3.l1.m1.1.1.3" xref="alg3.l1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m1.1b"><apply id="alg3.l1.m1.1.1.cmml" xref="alg3.l1.m1.1.1"><ci id="alg3.l1.m1.1.1.1.cmml" xref="alg3.l1.m1.1.1.1">←</ci><ci id="alg3.l1.m1.1.1.2.cmml" xref="alg3.l1.m1.1.1.2">𝑡</ci><cn id="alg3.l1.m1.1.1.3.cmml" type="integer" xref="alg3.l1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m1.1c">t\leftarrow 1</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m1.1d">italic_t ← 1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l2.1.1.1" style="font-size:80%;">2:</span></span><span class="ltx_text ltx_font_bold" id="alg3.l2.2">for</span> each agent <math alttext="i=1,\dots,n" class="ltx_Math" display="inline" id="alg3.l2.m1.3"><semantics id="alg3.l2.m1.3a"><mrow id="alg3.l2.m1.3.4" xref="alg3.l2.m1.3.4.cmml"><mi id="alg3.l2.m1.3.4.2" xref="alg3.l2.m1.3.4.2.cmml">i</mi><mo id="alg3.l2.m1.3.4.1" xref="alg3.l2.m1.3.4.1.cmml">=</mo><mrow id="alg3.l2.m1.3.4.3.2" xref="alg3.l2.m1.3.4.3.1.cmml"><mn id="alg3.l2.m1.1.1" xref="alg3.l2.m1.1.1.cmml">1</mn><mo id="alg3.l2.m1.3.4.3.2.1" xref="alg3.l2.m1.3.4.3.1.cmml">,</mo><mi id="alg3.l2.m1.2.2" mathvariant="normal" xref="alg3.l2.m1.2.2.cmml">…</mi><mo id="alg3.l2.m1.3.4.3.2.2" xref="alg3.l2.m1.3.4.3.1.cmml">,</mo><mi id="alg3.l2.m1.3.3" xref="alg3.l2.m1.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l2.m1.3b"><apply id="alg3.l2.m1.3.4.cmml" xref="alg3.l2.m1.3.4"><eq id="alg3.l2.m1.3.4.1.cmml" xref="alg3.l2.m1.3.4.1"></eq><ci id="alg3.l2.m1.3.4.2.cmml" xref="alg3.l2.m1.3.4.2">𝑖</ci><list id="alg3.l2.m1.3.4.3.1.cmml" xref="alg3.l2.m1.3.4.3.2"><cn id="alg3.l2.m1.1.1.cmml" type="integer" xref="alg3.l2.m1.1.1">1</cn><ci id="alg3.l2.m1.2.2.cmml" xref="alg3.l2.m1.2.2">…</ci><ci id="alg3.l2.m1.3.3.cmml" xref="alg3.l2.m1.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l2.m1.3c">i=1,\dots,n</annotation><annotation encoding="application/x-llamapun" id="alg3.l2.m1.3d">italic_i = 1 , … , italic_n</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg3.l2.3">do</span> </div> <div class="ltx_listingline" id="alg3.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l3.1.1.1" style="font-size:80%;">3:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l3.2">for</span> each action profile <math alttext="a_{-i}\in A_{-i}" class="ltx_Math" display="inline" id="alg3.l3.m1.1"><semantics id="alg3.l3.m1.1a"><mrow id="alg3.l3.m1.1.1" xref="alg3.l3.m1.1.1.cmml"><msub id="alg3.l3.m1.1.1.2" xref="alg3.l3.m1.1.1.2.cmml"><mi id="alg3.l3.m1.1.1.2.2" xref="alg3.l3.m1.1.1.2.2.cmml">a</mi><mrow id="alg3.l3.m1.1.1.2.3" xref="alg3.l3.m1.1.1.2.3.cmml"><mo id="alg3.l3.m1.1.1.2.3a" xref="alg3.l3.m1.1.1.2.3.cmml">−</mo><mi id="alg3.l3.m1.1.1.2.3.2" xref="alg3.l3.m1.1.1.2.3.2.cmml">i</mi></mrow></msub><mo id="alg3.l3.m1.1.1.1" xref="alg3.l3.m1.1.1.1.cmml">∈</mo><msub id="alg3.l3.m1.1.1.3" xref="alg3.l3.m1.1.1.3.cmml"><mi id="alg3.l3.m1.1.1.3.2" xref="alg3.l3.m1.1.1.3.2.cmml">A</mi><mrow id="alg3.l3.m1.1.1.3.3" xref="alg3.l3.m1.1.1.3.3.cmml"><mo id="alg3.l3.m1.1.1.3.3a" xref="alg3.l3.m1.1.1.3.3.cmml">−</mo><mi id="alg3.l3.m1.1.1.3.3.2" xref="alg3.l3.m1.1.1.3.3.2.cmml">i</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l3.m1.1b"><apply id="alg3.l3.m1.1.1.cmml" xref="alg3.l3.m1.1.1"><in id="alg3.l3.m1.1.1.1.cmml" xref="alg3.l3.m1.1.1.1"></in><apply id="alg3.l3.m1.1.1.2.cmml" xref="alg3.l3.m1.1.1.2"><csymbol cd="ambiguous" id="alg3.l3.m1.1.1.2.1.cmml" xref="alg3.l3.m1.1.1.2">subscript</csymbol><ci id="alg3.l3.m1.1.1.2.2.cmml" xref="alg3.l3.m1.1.1.2.2">𝑎</ci><apply id="alg3.l3.m1.1.1.2.3.cmml" xref="alg3.l3.m1.1.1.2.3"><minus id="alg3.l3.m1.1.1.2.3.1.cmml" xref="alg3.l3.m1.1.1.2.3"></minus><ci id="alg3.l3.m1.1.1.2.3.2.cmml" xref="alg3.l3.m1.1.1.2.3.2">𝑖</ci></apply></apply><apply id="alg3.l3.m1.1.1.3.cmml" xref="alg3.l3.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l3.m1.1.1.3.1.cmml" xref="alg3.l3.m1.1.1.3">subscript</csymbol><ci id="alg3.l3.m1.1.1.3.2.cmml" xref="alg3.l3.m1.1.1.3.2">𝐴</ci><apply id="alg3.l3.m1.1.1.3.3.cmml" xref="alg3.l3.m1.1.1.3.3"><minus id="alg3.l3.m1.1.1.3.3.1.cmml" xref="alg3.l3.m1.1.1.3.3"></minus><ci id="alg3.l3.m1.1.1.3.3.2.cmml" xref="alg3.l3.m1.1.1.3.3.2">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l3.m1.1c">a_{-i}\in A_{-i}</annotation><annotation encoding="application/x-llamapun" id="alg3.l3.m1.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg3.l3.3">do</span> </div> <div class="ltx_listingline" id="alg3.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="{\bm{p}}^{1}\leftarrow\bm{1}\in\mathbb{R}^{A_{i}}" class="ltx_Math" display="inline" id="alg3.l4.m1.1"><semantics id="alg3.l4.m1.1a"><mrow id="alg3.l4.m1.1.1" xref="alg3.l4.m1.1.1.cmml"><msup id="alg3.l4.m1.1.1.2" xref="alg3.l4.m1.1.1.2.cmml"><mi id="alg3.l4.m1.1.1.2.2" xref="alg3.l4.m1.1.1.2.2.cmml">𝒑</mi><mn id="alg3.l4.m1.1.1.2.3" xref="alg3.l4.m1.1.1.2.3.cmml">1</mn></msup><mo id="alg3.l4.m1.1.1.3" stretchy="false" xref="alg3.l4.m1.1.1.3.cmml">←</mo><mn id="alg3.l4.m1.1.1.4" xref="alg3.l4.m1.1.1.4.cmml">𝟏</mn><mo id="alg3.l4.m1.1.1.5" xref="alg3.l4.m1.1.1.5.cmml">∈</mo><msup id="alg3.l4.m1.1.1.6" xref="alg3.l4.m1.1.1.6.cmml"><mi id="alg3.l4.m1.1.1.6.2" xref="alg3.l4.m1.1.1.6.2.cmml">ℝ</mi><msub id="alg3.l4.m1.1.1.6.3" xref="alg3.l4.m1.1.1.6.3.cmml"><mi id="alg3.l4.m1.1.1.6.3.2" xref="alg3.l4.m1.1.1.6.3.2.cmml">A</mi><mi id="alg3.l4.m1.1.1.6.3.3" xref="alg3.l4.m1.1.1.6.3.3.cmml">i</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="alg3.l4.m1.1b"><apply id="alg3.l4.m1.1.1.cmml" xref="alg3.l4.m1.1.1"><and id="alg3.l4.m1.1.1a.cmml" xref="alg3.l4.m1.1.1"></and><apply id="alg3.l4.m1.1.1b.cmml" xref="alg3.l4.m1.1.1"><ci id="alg3.l4.m1.1.1.3.cmml" xref="alg3.l4.m1.1.1.3">←</ci><apply id="alg3.l4.m1.1.1.2.cmml" xref="alg3.l4.m1.1.1.2"><csymbol cd="ambiguous" id="alg3.l4.m1.1.1.2.1.cmml" xref="alg3.l4.m1.1.1.2">superscript</csymbol><ci id="alg3.l4.m1.1.1.2.2.cmml" xref="alg3.l4.m1.1.1.2.2">𝒑</ci><cn id="alg3.l4.m1.1.1.2.3.cmml" type="integer" xref="alg3.l4.m1.1.1.2.3">1</cn></apply><cn id="alg3.l4.m1.1.1.4.cmml" type="integer" xref="alg3.l4.m1.1.1.4">1</cn></apply><apply id="alg3.l4.m1.1.1c.cmml" xref="alg3.l4.m1.1.1"><in id="alg3.l4.m1.1.1.5.cmml" xref="alg3.l4.m1.1.1.5"></in><share href="https://arxiv.org/html/2503.01976v1#alg3.l4.m1.1.1.4.cmml" id="alg3.l4.m1.1.1d.cmml" xref="alg3.l4.m1.1.1"></share><apply id="alg3.l4.m1.1.1.6.cmml" xref="alg3.l4.m1.1.1.6"><csymbol cd="ambiguous" id="alg3.l4.m1.1.1.6.1.cmml" xref="alg3.l4.m1.1.1.6">superscript</csymbol><ci id="alg3.l4.m1.1.1.6.2.cmml" xref="alg3.l4.m1.1.1.6.2">ℝ</ci><apply id="alg3.l4.m1.1.1.6.3.cmml" xref="alg3.l4.m1.1.1.6.3"><csymbol cd="ambiguous" id="alg3.l4.m1.1.1.6.3.1.cmml" xref="alg3.l4.m1.1.1.6.3">subscript</csymbol><ci id="alg3.l4.m1.1.1.6.3.2.cmml" xref="alg3.l4.m1.1.1.6.3.2">𝐴</ci><ci id="alg3.l4.m1.1.1.6.3.3.cmml" xref="alg3.l4.m1.1.1.6.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l4.m1.1c">{\bm{p}}^{1}\leftarrow\bm{1}\in\mathbb{R}^{A_{i}}</annotation><annotation encoding="application/x-llamapun" id="alg3.l4.m1.1d">bold_italic_p start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ← bold_1 ∈ blackboard_R start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l5.1.1.1" style="font-size:80%;">5:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l5.2">for</span> timestep <math alttext="\ell=1,\dots,L" class="ltx_Math" display="inline" id="alg3.l5.m1.3"><semantics id="alg3.l5.m1.3a"><mrow id="alg3.l5.m1.3.4" xref="alg3.l5.m1.3.4.cmml"><mi id="alg3.l5.m1.3.4.2" mathvariant="normal" xref="alg3.l5.m1.3.4.2.cmml">ℓ</mi><mo id="alg3.l5.m1.3.4.1" xref="alg3.l5.m1.3.4.1.cmml">=</mo><mrow id="alg3.l5.m1.3.4.3.2" xref="alg3.l5.m1.3.4.3.1.cmml"><mn id="alg3.l5.m1.1.1" xref="alg3.l5.m1.1.1.cmml">1</mn><mo id="alg3.l5.m1.3.4.3.2.1" xref="alg3.l5.m1.3.4.3.1.cmml">,</mo><mi id="alg3.l5.m1.2.2" mathvariant="normal" xref="alg3.l5.m1.2.2.cmml">…</mi><mo id="alg3.l5.m1.3.4.3.2.2" xref="alg3.l5.m1.3.4.3.1.cmml">,</mo><mi id="alg3.l5.m1.3.3" xref="alg3.l5.m1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l5.m1.3b"><apply id="alg3.l5.m1.3.4.cmml" xref="alg3.l5.m1.3.4"><eq id="alg3.l5.m1.3.4.1.cmml" xref="alg3.l5.m1.3.4.1"></eq><ci id="alg3.l5.m1.3.4.2.cmml" xref="alg3.l5.m1.3.4.2">ℓ</ci><list id="alg3.l5.m1.3.4.3.1.cmml" xref="alg3.l5.m1.3.4.3.2"><cn id="alg3.l5.m1.1.1.cmml" type="integer" xref="alg3.l5.m1.1.1">1</cn><ci id="alg3.l5.m1.2.2.cmml" xref="alg3.l5.m1.2.2">…</ci><ci id="alg3.l5.m1.3.3.cmml" xref="alg3.l5.m1.3.3">𝐿</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l5.m1.3c">\ell=1,\dots,L</annotation><annotation encoding="application/x-llamapun" id="alg3.l5.m1.3d">roman_ℓ = 1 , … , italic_L</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg3.l5.3">do</span> </div> <div class="ltx_listingline" id="alg3.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l6.1.1.1" style="font-size:80%;">6:</span></span> principal sets <math alttext="P_{i}^{t}(\cdot)={\bm{p}}^{\ell}[\cdot]" class="ltx_Math" display="inline" id="alg3.l6.m1.2"><semantics id="alg3.l6.m1.2a"><mrow id="alg3.l6.m1.2.3" xref="alg3.l6.m1.2.3.cmml"><mrow id="alg3.l6.m1.2.3.2" xref="alg3.l6.m1.2.3.2.cmml"><msubsup id="alg3.l6.m1.2.3.2.2" xref="alg3.l6.m1.2.3.2.2.cmml"><mi id="alg3.l6.m1.2.3.2.2.2.2" xref="alg3.l6.m1.2.3.2.2.2.2.cmml">P</mi><mi id="alg3.l6.m1.2.3.2.2.2.3" xref="alg3.l6.m1.2.3.2.2.2.3.cmml">i</mi><mi id="alg3.l6.m1.2.3.2.2.3" xref="alg3.l6.m1.2.3.2.2.3.cmml">t</mi></msubsup><mo id="alg3.l6.m1.2.3.2.1" xref="alg3.l6.m1.2.3.2.1.cmml">⁢</mo><mrow id="alg3.l6.m1.2.3.2.3.2" xref="alg3.l6.m1.2.3.2.cmml"><mo id="alg3.l6.m1.2.3.2.3.2.1" stretchy="false" xref="alg3.l6.m1.2.3.2.cmml">(</mo><mo id="alg3.l6.m1.1.1" lspace="0em" rspace="0em" xref="alg3.l6.m1.1.1.cmml">⋅</mo><mo id="alg3.l6.m1.2.3.2.3.2.2" stretchy="false" xref="alg3.l6.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="alg3.l6.m1.2.3.1" xref="alg3.l6.m1.2.3.1.cmml">=</mo><mrow id="alg3.l6.m1.2.3.3" xref="alg3.l6.m1.2.3.3.cmml"><msup id="alg3.l6.m1.2.3.3.2" xref="alg3.l6.m1.2.3.3.2.cmml"><mi id="alg3.l6.m1.2.3.3.2.2" xref="alg3.l6.m1.2.3.3.2.2.cmml">𝒑</mi><mi id="alg3.l6.m1.2.3.3.2.3" mathvariant="normal" xref="alg3.l6.m1.2.3.3.2.3.cmml">ℓ</mi></msup><mo id="alg3.l6.m1.2.3.3.1" xref="alg3.l6.m1.2.3.3.1.cmml">⁢</mo><mrow id="alg3.l6.m1.2.3.3.3.2" xref="alg3.l6.m1.2.3.3.3.1.cmml"><mo id="alg3.l6.m1.2.3.3.3.2.1" stretchy="false" xref="alg3.l6.m1.2.3.3.3.1.1.cmml">[</mo><mo id="alg3.l6.m1.2.2" lspace="0em" rspace="0em" xref="alg3.l6.m1.2.2.cmml">⋅</mo><mo id="alg3.l6.m1.2.3.3.3.2.2" stretchy="false" xref="alg3.l6.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l6.m1.2b"><apply id="alg3.l6.m1.2.3.cmml" xref="alg3.l6.m1.2.3"><eq id="alg3.l6.m1.2.3.1.cmml" xref="alg3.l6.m1.2.3.1"></eq><apply id="alg3.l6.m1.2.3.2.cmml" xref="alg3.l6.m1.2.3.2"><times id="alg3.l6.m1.2.3.2.1.cmml" xref="alg3.l6.m1.2.3.2.1"></times><apply id="alg3.l6.m1.2.3.2.2.cmml" xref="alg3.l6.m1.2.3.2.2"><csymbol cd="ambiguous" id="alg3.l6.m1.2.3.2.2.1.cmml" xref="alg3.l6.m1.2.3.2.2">superscript</csymbol><apply id="alg3.l6.m1.2.3.2.2.2.cmml" xref="alg3.l6.m1.2.3.2.2"><csymbol cd="ambiguous" id="alg3.l6.m1.2.3.2.2.2.1.cmml" xref="alg3.l6.m1.2.3.2.2">subscript</csymbol><ci id="alg3.l6.m1.2.3.2.2.2.2.cmml" xref="alg3.l6.m1.2.3.2.2.2.2">𝑃</ci><ci id="alg3.l6.m1.2.3.2.2.2.3.cmml" xref="alg3.l6.m1.2.3.2.2.2.3">𝑖</ci></apply><ci id="alg3.l6.m1.2.3.2.2.3.cmml" xref="alg3.l6.m1.2.3.2.2.3">𝑡</ci></apply><ci id="alg3.l6.m1.1.1.cmml" xref="alg3.l6.m1.1.1">⋅</ci></apply><apply id="alg3.l6.m1.2.3.3.cmml" xref="alg3.l6.m1.2.3.3"><times id="alg3.l6.m1.2.3.3.1.cmml" xref="alg3.l6.m1.2.3.3.1"></times><apply id="alg3.l6.m1.2.3.3.2.cmml" xref="alg3.l6.m1.2.3.3.2"><csymbol cd="ambiguous" id="alg3.l6.m1.2.3.3.2.1.cmml" xref="alg3.l6.m1.2.3.3.2">superscript</csymbol><ci id="alg3.l6.m1.2.3.3.2.2.cmml" xref="alg3.l6.m1.2.3.3.2.2">𝒑</ci><ci id="alg3.l6.m1.2.3.3.2.3.cmml" xref="alg3.l6.m1.2.3.3.2.3">ℓ</ci></apply><apply id="alg3.l6.m1.2.3.3.3.1.cmml" xref="alg3.l6.m1.2.3.3.3.2"><csymbol cd="latexml" id="alg3.l6.m1.2.3.3.3.1.1.cmml" xref="alg3.l6.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="alg3.l6.m1.2.2.cmml" xref="alg3.l6.m1.2.2">⋅</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l6.m1.2c">P_{i}^{t}(\cdot)={\bm{p}}^{\ell}[\cdot]</annotation><annotation encoding="application/x-llamapun" id="alg3.l6.m1.2d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( ⋅ ) = bold_italic_p start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT [ ⋅ ]</annotation></semantics></math> and <math alttext="P_{j}^{t}(a_{j}^{\prime})=2\{a_{j}^{\prime}=a_{j}\}" class="ltx_Math" display="inline" id="alg3.l6.m2.2"><semantics id="alg3.l6.m2.2a"><mrow id="alg3.l6.m2.2.2" xref="alg3.l6.m2.2.2.cmml"><mrow id="alg3.l6.m2.1.1.1" xref="alg3.l6.m2.1.1.1.cmml"><msubsup id="alg3.l6.m2.1.1.1.3" xref="alg3.l6.m2.1.1.1.3.cmml"><mi id="alg3.l6.m2.1.1.1.3.2.2" xref="alg3.l6.m2.1.1.1.3.2.2.cmml">P</mi><mi id="alg3.l6.m2.1.1.1.3.2.3" xref="alg3.l6.m2.1.1.1.3.2.3.cmml">j</mi><mi id="alg3.l6.m2.1.1.1.3.3" xref="alg3.l6.m2.1.1.1.3.3.cmml">t</mi></msubsup><mo id="alg3.l6.m2.1.1.1.2" xref="alg3.l6.m2.1.1.1.2.cmml">⁢</mo><mrow id="alg3.l6.m2.1.1.1.1.1" xref="alg3.l6.m2.1.1.1.1.1.1.cmml"><mo id="alg3.l6.m2.1.1.1.1.1.2" stretchy="false" xref="alg3.l6.m2.1.1.1.1.1.1.cmml">(</mo><msubsup id="alg3.l6.m2.1.1.1.1.1.1" xref="alg3.l6.m2.1.1.1.1.1.1.cmml"><mi id="alg3.l6.m2.1.1.1.1.1.1.2.2" xref="alg3.l6.m2.1.1.1.1.1.1.2.2.cmml">a</mi><mi id="alg3.l6.m2.1.1.1.1.1.1.2.3" xref="alg3.l6.m2.1.1.1.1.1.1.2.3.cmml">j</mi><mo id="alg3.l6.m2.1.1.1.1.1.1.3" xref="alg3.l6.m2.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="alg3.l6.m2.1.1.1.1.1.3" stretchy="false" xref="alg3.l6.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg3.l6.m2.2.2.3" xref="alg3.l6.m2.2.2.3.cmml">=</mo><mrow id="alg3.l6.m2.2.2.2" xref="alg3.l6.m2.2.2.2.cmml"><mn id="alg3.l6.m2.2.2.2.3" xref="alg3.l6.m2.2.2.2.3.cmml">2</mn><mo id="alg3.l6.m2.2.2.2.2" xref="alg3.l6.m2.2.2.2.2.cmml">⁢</mo><mrow id="alg3.l6.m2.2.2.2.1.1" xref="alg3.l6.m2.2.2.2.1.2.cmml"><mo id="alg3.l6.m2.2.2.2.1.1.2" stretchy="false" xref="alg3.l6.m2.2.2.2.1.2.cmml">{</mo><mrow id="alg3.l6.m2.2.2.2.1.1.1" xref="alg3.l6.m2.2.2.2.1.1.1.cmml"><msubsup id="alg3.l6.m2.2.2.2.1.1.1.2" xref="alg3.l6.m2.2.2.2.1.1.1.2.cmml"><mi id="alg3.l6.m2.2.2.2.1.1.1.2.2.2" xref="alg3.l6.m2.2.2.2.1.1.1.2.2.2.cmml">a</mi><mi id="alg3.l6.m2.2.2.2.1.1.1.2.2.3" xref="alg3.l6.m2.2.2.2.1.1.1.2.2.3.cmml">j</mi><mo id="alg3.l6.m2.2.2.2.1.1.1.2.3" xref="alg3.l6.m2.2.2.2.1.1.1.2.3.cmml">′</mo></msubsup><mo id="alg3.l6.m2.2.2.2.1.1.1.1" xref="alg3.l6.m2.2.2.2.1.1.1.1.cmml">=</mo><msub id="alg3.l6.m2.2.2.2.1.1.1.3" xref="alg3.l6.m2.2.2.2.1.1.1.3.cmml"><mi id="alg3.l6.m2.2.2.2.1.1.1.3.2" xref="alg3.l6.m2.2.2.2.1.1.1.3.2.cmml">a</mi><mi id="alg3.l6.m2.2.2.2.1.1.1.3.3" xref="alg3.l6.m2.2.2.2.1.1.1.3.3.cmml">j</mi></msub></mrow><mo id="alg3.l6.m2.2.2.2.1.1.3" stretchy="false" xref="alg3.l6.m2.2.2.2.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l6.m2.2b"><apply id="alg3.l6.m2.2.2.cmml" xref="alg3.l6.m2.2.2"><eq id="alg3.l6.m2.2.2.3.cmml" xref="alg3.l6.m2.2.2.3"></eq><apply id="alg3.l6.m2.1.1.1.cmml" xref="alg3.l6.m2.1.1.1"><times id="alg3.l6.m2.1.1.1.2.cmml" xref="alg3.l6.m2.1.1.1.2"></times><apply id="alg3.l6.m2.1.1.1.3.cmml" xref="alg3.l6.m2.1.1.1.3"><csymbol cd="ambiguous" id="alg3.l6.m2.1.1.1.3.1.cmml" xref="alg3.l6.m2.1.1.1.3">superscript</csymbol><apply id="alg3.l6.m2.1.1.1.3.2.cmml" xref="alg3.l6.m2.1.1.1.3"><csymbol cd="ambiguous" id="alg3.l6.m2.1.1.1.3.2.1.cmml" xref="alg3.l6.m2.1.1.1.3">subscript</csymbol><ci id="alg3.l6.m2.1.1.1.3.2.2.cmml" xref="alg3.l6.m2.1.1.1.3.2.2">𝑃</ci><ci id="alg3.l6.m2.1.1.1.3.2.3.cmml" xref="alg3.l6.m2.1.1.1.3.2.3">𝑗</ci></apply><ci id="alg3.l6.m2.1.1.1.3.3.cmml" xref="alg3.l6.m2.1.1.1.3.3">𝑡</ci></apply><apply id="alg3.l6.m2.1.1.1.1.1.1.cmml" xref="alg3.l6.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l6.m2.1.1.1.1.1.1.1.cmml" xref="alg3.l6.m2.1.1.1.1.1">superscript</csymbol><apply id="alg3.l6.m2.1.1.1.1.1.1.2.cmml" xref="alg3.l6.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l6.m2.1.1.1.1.1.1.2.1.cmml" xref="alg3.l6.m2.1.1.1.1.1">subscript</csymbol><ci id="alg3.l6.m2.1.1.1.1.1.1.2.2.cmml" xref="alg3.l6.m2.1.1.1.1.1.1.2.2">𝑎</ci><ci id="alg3.l6.m2.1.1.1.1.1.1.2.3.cmml" xref="alg3.l6.m2.1.1.1.1.1.1.2.3">𝑗</ci></apply><ci id="alg3.l6.m2.1.1.1.1.1.1.3.cmml" xref="alg3.l6.m2.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="alg3.l6.m2.2.2.2.cmml" xref="alg3.l6.m2.2.2.2"><times id="alg3.l6.m2.2.2.2.2.cmml" xref="alg3.l6.m2.2.2.2.2"></times><cn id="alg3.l6.m2.2.2.2.3.cmml" type="integer" xref="alg3.l6.m2.2.2.2.3">2</cn><set id="alg3.l6.m2.2.2.2.1.2.cmml" xref="alg3.l6.m2.2.2.2.1.1"><apply id="alg3.l6.m2.2.2.2.1.1.1.cmml" xref="alg3.l6.m2.2.2.2.1.1.1"><eq id="alg3.l6.m2.2.2.2.1.1.1.1.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.1"></eq><apply id="alg3.l6.m2.2.2.2.1.1.1.2.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="alg3.l6.m2.2.2.2.1.1.1.2.1.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.2">superscript</csymbol><apply id="alg3.l6.m2.2.2.2.1.1.1.2.2.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="alg3.l6.m2.2.2.2.1.1.1.2.2.1.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.2">subscript</csymbol><ci id="alg3.l6.m2.2.2.2.1.1.1.2.2.2.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.2.2.2">𝑎</ci><ci id="alg3.l6.m2.2.2.2.1.1.1.2.2.3.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.2.2.3">𝑗</ci></apply><ci id="alg3.l6.m2.2.2.2.1.1.1.2.3.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.2.3">′</ci></apply><apply id="alg3.l6.m2.2.2.2.1.1.1.3.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="alg3.l6.m2.2.2.2.1.1.1.3.1.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.3">subscript</csymbol><ci id="alg3.l6.m2.2.2.2.1.1.1.3.2.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.3.2">𝑎</ci><ci id="alg3.l6.m2.2.2.2.1.1.1.3.3.cmml" xref="alg3.l6.m2.2.2.2.1.1.1.3.3">𝑗</ci></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l6.m2.2c">P_{j}^{t}(a_{j}^{\prime})=2\{a_{j}^{\prime}=a_{j}\}</annotation><annotation encoding="application/x-llamapun" id="alg3.l6.m2.2d">italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = 2 { italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT }</annotation></semantics></math> for every <math alttext="j\neq i" class="ltx_Math" display="inline" id="alg3.l6.m3.1"><semantics id="alg3.l6.m3.1a"><mrow id="alg3.l6.m3.1.1" xref="alg3.l6.m3.1.1.cmml"><mi id="alg3.l6.m3.1.1.2" xref="alg3.l6.m3.1.1.2.cmml">j</mi><mo id="alg3.l6.m3.1.1.1" xref="alg3.l6.m3.1.1.1.cmml">≠</mo><mi id="alg3.l6.m3.1.1.3" xref="alg3.l6.m3.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="alg3.l6.m3.1b"><apply id="alg3.l6.m3.1.1.cmml" xref="alg3.l6.m3.1.1"><neq id="alg3.l6.m3.1.1.1.cmml" xref="alg3.l6.m3.1.1.1"></neq><ci id="alg3.l6.m3.1.1.2.cmml" xref="alg3.l6.m3.1.1.2">𝑗</ci><ci id="alg3.l6.m3.1.1.3.cmml" xref="alg3.l6.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l6.m3.1c">j\neq i</annotation><annotation encoding="application/x-llamapun" id="alg3.l6.m3.1d">italic_j ≠ italic_i</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l7.1.1.1" style="font-size:80%;">7:</span></span> principal sends signals <math alttext="s_{i}^{t}=\bot" class="ltx_Math" display="inline" id="alg3.l7.m1.1"><semantics id="alg3.l7.m1.1a"><mrow id="alg3.l7.m1.1.1" xref="alg3.l7.m1.1.1.cmml"><msubsup id="alg3.l7.m1.1.1.2" xref="alg3.l7.m1.1.1.2.cmml"><mi id="alg3.l7.m1.1.1.2.2.2" xref="alg3.l7.m1.1.1.2.2.2.cmml">s</mi><mi id="alg3.l7.m1.1.1.2.2.3" xref="alg3.l7.m1.1.1.2.2.3.cmml">i</mi><mi id="alg3.l7.m1.1.1.2.3" xref="alg3.l7.m1.1.1.2.3.cmml">t</mi></msubsup><mo id="alg3.l7.m1.1.1.1" rspace="0em" xref="alg3.l7.m1.1.1.1.cmml">=</mo><mo id="alg3.l7.m1.1.1.3" lspace="0em" xref="alg3.l7.m1.1.1.3.cmml">⊥</mo></mrow><annotation-xml encoding="MathML-Content" id="alg3.l7.m1.1b"><apply id="alg3.l7.m1.1.1.cmml" xref="alg3.l7.m1.1.1"><eq id="alg3.l7.m1.1.1.1.cmml" xref="alg3.l7.m1.1.1.1"></eq><apply id="alg3.l7.m1.1.1.2.cmml" xref="alg3.l7.m1.1.1.2"><csymbol cd="ambiguous" id="alg3.l7.m1.1.1.2.1.cmml" xref="alg3.l7.m1.1.1.2">superscript</csymbol><apply id="alg3.l7.m1.1.1.2.2.cmml" xref="alg3.l7.m1.1.1.2"><csymbol cd="ambiguous" id="alg3.l7.m1.1.1.2.2.1.cmml" xref="alg3.l7.m1.1.1.2">subscript</csymbol><ci id="alg3.l7.m1.1.1.2.2.2.cmml" xref="alg3.l7.m1.1.1.2.2.2">𝑠</ci><ci id="alg3.l7.m1.1.1.2.2.3.cmml" xref="alg3.l7.m1.1.1.2.2.3">𝑖</ci></apply><ci id="alg3.l7.m1.1.1.2.3.cmml" xref="alg3.l7.m1.1.1.2.3">𝑡</ci></apply><csymbol cd="latexml" id="alg3.l7.m1.1.1.3.cmml" xref="alg3.l7.m1.1.1.3">bottom</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l7.m1.1c">s_{i}^{t}=\bot</annotation><annotation encoding="application/x-llamapun" id="alg3.l7.m1.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = ⊥</annotation></semantics></math> and <math alttext="s_{j}^{t}=a_{j}" class="ltx_Math" display="inline" id="alg3.l7.m2.1"><semantics id="alg3.l7.m2.1a"><mrow id="alg3.l7.m2.1.1" xref="alg3.l7.m2.1.1.cmml"><msubsup id="alg3.l7.m2.1.1.2" xref="alg3.l7.m2.1.1.2.cmml"><mi id="alg3.l7.m2.1.1.2.2.2" xref="alg3.l7.m2.1.1.2.2.2.cmml">s</mi><mi id="alg3.l7.m2.1.1.2.2.3" xref="alg3.l7.m2.1.1.2.2.3.cmml">j</mi><mi id="alg3.l7.m2.1.1.2.3" xref="alg3.l7.m2.1.1.2.3.cmml">t</mi></msubsup><mo id="alg3.l7.m2.1.1.1" xref="alg3.l7.m2.1.1.1.cmml">=</mo><msub id="alg3.l7.m2.1.1.3" xref="alg3.l7.m2.1.1.3.cmml"><mi id="alg3.l7.m2.1.1.3.2" xref="alg3.l7.m2.1.1.3.2.cmml">a</mi><mi id="alg3.l7.m2.1.1.3.3" xref="alg3.l7.m2.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l7.m2.1b"><apply id="alg3.l7.m2.1.1.cmml" xref="alg3.l7.m2.1.1"><eq id="alg3.l7.m2.1.1.1.cmml" xref="alg3.l7.m2.1.1.1"></eq><apply id="alg3.l7.m2.1.1.2.cmml" xref="alg3.l7.m2.1.1.2"><csymbol cd="ambiguous" id="alg3.l7.m2.1.1.2.1.cmml" xref="alg3.l7.m2.1.1.2">superscript</csymbol><apply id="alg3.l7.m2.1.1.2.2.cmml" xref="alg3.l7.m2.1.1.2"><csymbol cd="ambiguous" id="alg3.l7.m2.1.1.2.2.1.cmml" xref="alg3.l7.m2.1.1.2">subscript</csymbol><ci id="alg3.l7.m2.1.1.2.2.2.cmml" xref="alg3.l7.m2.1.1.2.2.2">𝑠</ci><ci id="alg3.l7.m2.1.1.2.2.3.cmml" xref="alg3.l7.m2.1.1.2.2.3">𝑗</ci></apply><ci id="alg3.l7.m2.1.1.2.3.cmml" xref="alg3.l7.m2.1.1.2.3">𝑡</ci></apply><apply id="alg3.l7.m2.1.1.3.cmml" xref="alg3.l7.m2.1.1.3"><csymbol cd="ambiguous" id="alg3.l7.m2.1.1.3.1.cmml" xref="alg3.l7.m2.1.1.3">subscript</csymbol><ci id="alg3.l7.m2.1.1.3.2.cmml" xref="alg3.l7.m2.1.1.3.2">𝑎</ci><ci id="alg3.l7.m2.1.1.3.3.cmml" xref="alg3.l7.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l7.m2.1c">s_{j}^{t}=a_{j}</annotation><annotation encoding="application/x-llamapun" id="alg3.l7.m2.1d">italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for every <math alttext="j\neq i" class="ltx_Math" display="inline" id="alg3.l7.m3.1"><semantics id="alg3.l7.m3.1a"><mrow id="alg3.l7.m3.1.1" xref="alg3.l7.m3.1.1.cmml"><mi id="alg3.l7.m3.1.1.2" xref="alg3.l7.m3.1.1.2.cmml">j</mi><mo id="alg3.l7.m3.1.1.1" xref="alg3.l7.m3.1.1.1.cmml">≠</mo><mi id="alg3.l7.m3.1.1.3" xref="alg3.l7.m3.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="alg3.l7.m3.1b"><apply id="alg3.l7.m3.1.1.cmml" xref="alg3.l7.m3.1.1"><neq id="alg3.l7.m3.1.1.1.cmml" xref="alg3.l7.m3.1.1.1"></neq><ci id="alg3.l7.m3.1.1.2.cmml" xref="alg3.l7.m3.1.1.2">𝑗</ci><ci id="alg3.l7.m3.1.1.3.cmml" xref="alg3.l7.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l7.m3.1c">j\neq i</annotation><annotation encoding="application/x-llamapun" id="alg3.l7.m3.1d">italic_j ≠ italic_i</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l8.1.1.1" style="font-size:80%;">8:</span></span> principal observes action profile <math alttext="a^{t}" class="ltx_Math" display="inline" id="alg3.l8.m1.1"><semantics id="alg3.l8.m1.1a"><msup id="alg3.l8.m1.1.1" xref="alg3.l8.m1.1.1.cmml"><mi id="alg3.l8.m1.1.1.2" xref="alg3.l8.m1.1.1.2.cmml">a</mi><mi id="alg3.l8.m1.1.1.3" xref="alg3.l8.m1.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="alg3.l8.m1.1b"><apply id="alg3.l8.m1.1.1.cmml" xref="alg3.l8.m1.1.1"><csymbol cd="ambiguous" id="alg3.l8.m1.1.1.1.cmml" xref="alg3.l8.m1.1.1">superscript</csymbol><ci id="alg3.l8.m1.1.1.2.cmml" xref="alg3.l8.m1.1.1.2">𝑎</ci><ci id="alg3.l8.m1.1.1.3.cmml" xref="alg3.l8.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l8.m1.1c">a^{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l8.m1.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> played by agents </div> <div class="ltx_listingline" id="alg3.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l9.2.1.1" style="font-size:80%;">9:</span></span> principal sets <math alttext="{\bm{p}}^{\ell+1}\leftarrow\Pi_{\mathcal{P}}\quantity[{\bm{p}}^{\ell}-\eta{\bm% {e}}_{a^{t}_{i}}]" class="ltx_Math" display="inline" id="alg3.l9.m1.1"><semantics id="alg3.l9.m1.1a"><mrow id="alg3.l9.m1.1.2" xref="alg3.l9.m1.1.2.cmml"><msup id="alg3.l9.m1.1.2.2" xref="alg3.l9.m1.1.2.2.cmml"><mi id="alg3.l9.m1.1.2.2.2" xref="alg3.l9.m1.1.2.2.2.cmml">𝒑</mi><mrow id="alg3.l9.m1.1.2.2.3" xref="alg3.l9.m1.1.2.2.3.cmml"><mi id="alg3.l9.m1.1.2.2.3.2" mathvariant="normal" xref="alg3.l9.m1.1.2.2.3.2.cmml">ℓ</mi><mo id="alg3.l9.m1.1.2.2.3.1" xref="alg3.l9.m1.1.2.2.3.1.cmml">+</mo><mn id="alg3.l9.m1.1.2.2.3.3" xref="alg3.l9.m1.1.2.2.3.3.cmml">1</mn></mrow></msup><mo id="alg3.l9.m1.1.2.1" stretchy="false" xref="alg3.l9.m1.1.2.1.cmml">←</mo><mrow id="alg3.l9.m1.1.2.3" xref="alg3.l9.m1.1.2.3.cmml"><msub id="alg3.l9.m1.1.2.3.2" xref="alg3.l9.m1.1.2.3.2.cmml"><mi id="alg3.l9.m1.1.2.3.2.2" mathvariant="normal" xref="alg3.l9.m1.1.2.3.2.2.cmml">Π</mi><mi class="ltx_font_mathcaligraphic" id="alg3.l9.m1.1.2.3.2.3" xref="alg3.l9.m1.1.2.3.2.3.cmml">𝒫</mi></msub><mo id="alg3.l9.m1.1.2.3.1" xref="alg3.l9.m1.1.2.3.1.cmml">⁢</mo><mrow id="alg3.l9.m1.1.1.3" xref="alg3.l9.m1.1.1.1.1.1.cmml"><mo id="alg3.l9.m1.1.1.3.1" xref="alg3.l9.m1.1.1.1.1.1.cmml">[</mo><mrow id="alg3.l9.m1.1.1.1.1.1" xref="alg3.l9.m1.1.1.1.1.1.cmml"><msup id="alg3.l9.m1.1.1.1.1.1.2" xref="alg3.l9.m1.1.1.1.1.1.2.cmml"><mi id="alg3.l9.m1.1.1.1.1.1.2.2" xref="alg3.l9.m1.1.1.1.1.1.2.2.cmml">𝒑</mi><mi id="alg3.l9.m1.1.1.1.1.1.2.3" mathvariant="normal" xref="alg3.l9.m1.1.1.1.1.1.2.3.cmml">ℓ</mi></msup><mo id="alg3.l9.m1.1.1.1.1.1.1" xref="alg3.l9.m1.1.1.1.1.1.1.cmml">−</mo><mrow id="alg3.l9.m1.1.1.1.1.1.3" xref="alg3.l9.m1.1.1.1.1.1.3.cmml"><mi id="alg3.l9.m1.1.1.1.1.1.3.2" xref="alg3.l9.m1.1.1.1.1.1.3.2.cmml">η</mi><mo id="alg3.l9.m1.1.1.1.1.1.3.1" xref="alg3.l9.m1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="alg3.l9.m1.1.1.1.1.1.3.3" xref="alg3.l9.m1.1.1.1.1.1.3.3.cmml"><mi id="alg3.l9.m1.1.1.1.1.1.3.3.2" xref="alg3.l9.m1.1.1.1.1.1.3.3.2.cmml">𝒆</mi><msubsup id="alg3.l9.m1.1.1.1.1.1.3.3.3" xref="alg3.l9.m1.1.1.1.1.1.3.3.3.cmml"><mi id="alg3.l9.m1.1.1.1.1.1.3.3.3.2.2" xref="alg3.l9.m1.1.1.1.1.1.3.3.3.2.2.cmml">a</mi><mi id="alg3.l9.m1.1.1.1.1.1.3.3.3.3" xref="alg3.l9.m1.1.1.1.1.1.3.3.3.3.cmml">i</mi><mi id="alg3.l9.m1.1.1.1.1.1.3.3.3.2.3" xref="alg3.l9.m1.1.1.1.1.1.3.3.3.2.3.cmml">t</mi></msubsup></msub></mrow></mrow><mo id="alg3.l9.m1.1.1.3.2" xref="alg3.l9.m1.1.1.1.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l9.m1.1b"><apply id="alg3.l9.m1.1.2.cmml" xref="alg3.l9.m1.1.2"><ci id="alg3.l9.m1.1.2.1.cmml" xref="alg3.l9.m1.1.2.1">←</ci><apply id="alg3.l9.m1.1.2.2.cmml" xref="alg3.l9.m1.1.2.2"><csymbol cd="ambiguous" id="alg3.l9.m1.1.2.2.1.cmml" xref="alg3.l9.m1.1.2.2">superscript</csymbol><ci id="alg3.l9.m1.1.2.2.2.cmml" xref="alg3.l9.m1.1.2.2.2">𝒑</ci><apply id="alg3.l9.m1.1.2.2.3.cmml" xref="alg3.l9.m1.1.2.2.3"><plus id="alg3.l9.m1.1.2.2.3.1.cmml" xref="alg3.l9.m1.1.2.2.3.1"></plus><ci id="alg3.l9.m1.1.2.2.3.2.cmml" xref="alg3.l9.m1.1.2.2.3.2">ℓ</ci><cn id="alg3.l9.m1.1.2.2.3.3.cmml" type="integer" xref="alg3.l9.m1.1.2.2.3.3">1</cn></apply></apply><apply id="alg3.l9.m1.1.2.3.cmml" xref="alg3.l9.m1.1.2.3"><times id="alg3.l9.m1.1.2.3.1.cmml" xref="alg3.l9.m1.1.2.3.1"></times><apply id="alg3.l9.m1.1.2.3.2.cmml" xref="alg3.l9.m1.1.2.3.2"><csymbol cd="ambiguous" id="alg3.l9.m1.1.2.3.2.1.cmml" xref="alg3.l9.m1.1.2.3.2">subscript</csymbol><ci id="alg3.l9.m1.1.2.3.2.2.cmml" xref="alg3.l9.m1.1.2.3.2.2">Π</ci><ci id="alg3.l9.m1.1.2.3.2.3.cmml" xref="alg3.l9.m1.1.2.3.2.3">𝒫</ci></apply><apply id="alg3.l9.m1.1.1.1.1.1.cmml" xref="alg3.l9.m1.1.1.3"><minus id="alg3.l9.m1.1.1.1.1.1.1.cmml" xref="alg3.l9.m1.1.1.1.1.1.1"></minus><apply id="alg3.l9.m1.1.1.1.1.1.2.cmml" xref="alg3.l9.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg3.l9.m1.1.1.1.1.1.2.1.cmml" xref="alg3.l9.m1.1.1.1.1.1.2">superscript</csymbol><ci id="alg3.l9.m1.1.1.1.1.1.2.2.cmml" xref="alg3.l9.m1.1.1.1.1.1.2.2">𝒑</ci><ci id="alg3.l9.m1.1.1.1.1.1.2.3.cmml" xref="alg3.l9.m1.1.1.1.1.1.2.3">ℓ</ci></apply><apply id="alg3.l9.m1.1.1.1.1.1.3.cmml" xref="alg3.l9.m1.1.1.1.1.1.3"><times id="alg3.l9.m1.1.1.1.1.1.3.1.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.1"></times><ci id="alg3.l9.m1.1.1.1.1.1.3.2.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.2">𝜂</ci><apply id="alg3.l9.m1.1.1.1.1.1.3.3.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg3.l9.m1.1.1.1.1.1.3.3.1.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3">subscript</csymbol><ci id="alg3.l9.m1.1.1.1.1.1.3.3.2.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.2">𝒆</ci><apply id="alg3.l9.m1.1.1.1.1.1.3.3.3.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="alg3.l9.m1.1.1.1.1.1.3.3.3.1.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.3">subscript</csymbol><apply id="alg3.l9.m1.1.1.1.1.1.3.3.3.2.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="alg3.l9.m1.1.1.1.1.1.3.3.3.2.1.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.3">superscript</csymbol><ci id="alg3.l9.m1.1.1.1.1.1.3.3.3.2.2.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.3.2.2">𝑎</ci><ci id="alg3.l9.m1.1.1.1.1.1.3.3.3.2.3.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.3.2.3">𝑡</ci></apply><ci id="alg3.l9.m1.1.1.1.1.1.3.3.3.3.cmml" xref="alg3.l9.m1.1.1.1.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.m1.1c">{\bm{p}}^{\ell+1}\leftarrow\Pi_{\mathcal{P}}\quantity[{\bm{p}}^{\ell}-\eta{\bm% {e}}_{a^{t}_{i}}]</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.m1.1d">bold_italic_p start_POSTSUPERSCRIPT roman_ℓ + 1 end_POSTSUPERSCRIPT ← roman_Π start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT [ start_ARG bold_italic_p start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT - italic_η bold_italic_e start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ]</annotation></semantics></math> <math alttext="\triangleright" class="ltx_Math" display="inline" id="alg3.l9.m2.1"><semantics id="alg3.l9.m2.1a"><mo id="alg3.l9.m2.1.1" mathcolor="#808080" xref="alg3.l9.m2.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg3.l9.m2.1b"><ci id="alg3.l9.m2.1.1.cmml" xref="alg3.l9.m2.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.m2.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.m2.1d">▷</annotation></semantics></math><span class="ltx_text" id="alg3.l9.1" style="color:#808080;"> <math alttext="\eta=\sqrt{m_{i}/L}" class="ltx_Math" display="inline" id="alg3.l9.1.m1.1"><semantics id="alg3.l9.1.m1.1a"><mrow id="alg3.l9.1.m1.1.1" xref="alg3.l9.1.m1.1.1.cmml"><mi id="alg3.l9.1.m1.1.1.2" mathcolor="#808080" xref="alg3.l9.1.m1.1.1.2.cmml">η</mi><mo id="alg3.l9.1.m1.1.1.1" mathcolor="#808080" xref="alg3.l9.1.m1.1.1.1.cmml">=</mo><msqrt id="alg3.l9.1.m1.1.1.3" mathcolor="#808080" xref="alg3.l9.1.m1.1.1.3.cmml"><mrow id="alg3.l9.1.m1.1.1.3.2" xref="alg3.l9.1.m1.1.1.3.2.cmml"><msub id="alg3.l9.1.m1.1.1.3.2.2" xref="alg3.l9.1.m1.1.1.3.2.2.cmml"><mi id="alg3.l9.1.m1.1.1.3.2.2.2" mathcolor="#808080" xref="alg3.l9.1.m1.1.1.3.2.2.2.cmml">m</mi><mi id="alg3.l9.1.m1.1.1.3.2.2.3" mathcolor="#808080" xref="alg3.l9.1.m1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="alg3.l9.1.m1.1.1.3.2.1" mathcolor="#808080" xref="alg3.l9.1.m1.1.1.3.2.1.cmml">/</mo><mi id="alg3.l9.1.m1.1.1.3.2.3" mathcolor="#808080" xref="alg3.l9.1.m1.1.1.3.2.3.cmml">L</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="alg3.l9.1.m1.1b"><apply id="alg3.l9.1.m1.1.1.cmml" xref="alg3.l9.1.m1.1.1"><eq id="alg3.l9.1.m1.1.1.1.cmml" xref="alg3.l9.1.m1.1.1.1"></eq><ci id="alg3.l9.1.m1.1.1.2.cmml" xref="alg3.l9.1.m1.1.1.2">𝜂</ci><apply id="alg3.l9.1.m1.1.1.3.cmml" xref="alg3.l9.1.m1.1.1.3"><root id="alg3.l9.1.m1.1.1.3a.cmml" xref="alg3.l9.1.m1.1.1.3"></root><apply id="alg3.l9.1.m1.1.1.3.2.cmml" xref="alg3.l9.1.m1.1.1.3.2"><divide id="alg3.l9.1.m1.1.1.3.2.1.cmml" xref="alg3.l9.1.m1.1.1.3.2.1"></divide><apply id="alg3.l9.1.m1.1.1.3.2.2.cmml" xref="alg3.l9.1.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="alg3.l9.1.m1.1.1.3.2.2.1.cmml" xref="alg3.l9.1.m1.1.1.3.2.2">subscript</csymbol><ci id="alg3.l9.1.m1.1.1.3.2.2.2.cmml" xref="alg3.l9.1.m1.1.1.3.2.2.2">𝑚</ci><ci id="alg3.l9.1.m1.1.1.3.2.2.3.cmml" xref="alg3.l9.1.m1.1.1.3.2.2.3">𝑖</ci></apply><ci id="alg3.l9.1.m1.1.1.3.2.3.cmml" xref="alg3.l9.1.m1.1.1.3.2.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.1.m1.1c">\eta=\sqrt{m_{i}/L}</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.1.m1.1d">italic_η = square-root start_ARG italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT / italic_L end_ARG</annotation></semantics></math> is the step size</span> </div> <div class="ltx_listingline" id="alg3.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l10.1.1.1" style="font-size:80%;">10:</span></span> <math alttext="t\leftarrow t+1" class="ltx_Math" display="inline" id="alg3.l10.m1.1"><semantics id="alg3.l10.m1.1a"><mrow id="alg3.l10.m1.1.1" xref="alg3.l10.m1.1.1.cmml"><mi id="alg3.l10.m1.1.1.2" xref="alg3.l10.m1.1.1.2.cmml">t</mi><mo id="alg3.l10.m1.1.1.1" stretchy="false" xref="alg3.l10.m1.1.1.1.cmml">←</mo><mrow id="alg3.l10.m1.1.1.3" xref="alg3.l10.m1.1.1.3.cmml"><mi id="alg3.l10.m1.1.1.3.2" xref="alg3.l10.m1.1.1.3.2.cmml">t</mi><mo id="alg3.l10.m1.1.1.3.1" xref="alg3.l10.m1.1.1.3.1.cmml">+</mo><mn id="alg3.l10.m1.1.1.3.3" xref="alg3.l10.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l10.m1.1b"><apply id="alg3.l10.m1.1.1.cmml" xref="alg3.l10.m1.1.1"><ci id="alg3.l10.m1.1.1.1.cmml" xref="alg3.l10.m1.1.1.1">←</ci><ci id="alg3.l10.m1.1.1.2.cmml" xref="alg3.l10.m1.1.1.2">𝑡</ci><apply id="alg3.l10.m1.1.1.3.cmml" xref="alg3.l10.m1.1.1.3"><plus id="alg3.l10.m1.1.1.3.1.cmml" xref="alg3.l10.m1.1.1.3.1"></plus><ci id="alg3.l10.m1.1.1.3.2.cmml" xref="alg3.l10.m1.1.1.3.2">𝑡</ci><cn id="alg3.l10.m1.1.1.3.3.cmml" type="integer" xref="alg3.l10.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l10.m1.1c">t\leftarrow t+1</annotation><annotation encoding="application/x-llamapun" id="alg3.l10.m1.1d">italic_t ← italic_t + 1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l11.1.1.1" style="font-size:80%;">11:</span></span> <math alttext="\tilde{U}_{i}(\cdot,a_{-i})=-\frac{1}{L}\sum_{\ell=1}^{L}{\bm{p}}^{\ell}" class="ltx_Math" display="inline" id="alg3.l11.m1.2"><semantics id="alg3.l11.m1.2a"><mrow id="alg3.l11.m1.2.2" xref="alg3.l11.m1.2.2.cmml"><mrow id="alg3.l11.m1.2.2.1" 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id="alg3.l11.m1.2.2.2.cmml" xref="alg3.l11.m1.2.2.2"></eq><apply id="alg3.l11.m1.2.2.1.cmml" xref="alg3.l11.m1.2.2.1"><times id="alg3.l11.m1.2.2.1.2.cmml" xref="alg3.l11.m1.2.2.1.2"></times><apply id="alg3.l11.m1.2.2.1.3.cmml" xref="alg3.l11.m1.2.2.1.3"><csymbol cd="ambiguous" id="alg3.l11.m1.2.2.1.3.1.cmml" xref="alg3.l11.m1.2.2.1.3">subscript</csymbol><apply id="alg3.l11.m1.2.2.1.3.2.cmml" xref="alg3.l11.m1.2.2.1.3.2"><ci id="alg3.l11.m1.2.2.1.3.2.1.cmml" xref="alg3.l11.m1.2.2.1.3.2.1">~</ci><ci id="alg3.l11.m1.2.2.1.3.2.2.cmml" xref="alg3.l11.m1.2.2.1.3.2.2">𝑈</ci></apply><ci id="alg3.l11.m1.2.2.1.3.3.cmml" xref="alg3.l11.m1.2.2.1.3.3">𝑖</ci></apply><interval closure="open" id="alg3.l11.m1.2.2.1.1.2.cmml" xref="alg3.l11.m1.2.2.1.1.1"><ci id="alg3.l11.m1.1.1.cmml" xref="alg3.l11.m1.1.1">⋅</ci><apply id="alg3.l11.m1.2.2.1.1.1.1.cmml" xref="alg3.l11.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l11.m1.2.2.1.1.1.1.1.cmml" xref="alg3.l11.m1.2.2.1.1.1.1">subscript</csymbol><ci 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xref="alg3.l11.m1.2.2.3.2.3"><apply id="alg3.l11.m1.2.2.3.2.3.1.cmml" xref="alg3.l11.m1.2.2.3.2.3.1"><csymbol cd="ambiguous" id="alg3.l11.m1.2.2.3.2.3.1.1.cmml" xref="alg3.l11.m1.2.2.3.2.3.1">superscript</csymbol><apply id="alg3.l11.m1.2.2.3.2.3.1.2.cmml" xref="alg3.l11.m1.2.2.3.2.3.1"><csymbol cd="ambiguous" id="alg3.l11.m1.2.2.3.2.3.1.2.1.cmml" xref="alg3.l11.m1.2.2.3.2.3.1">subscript</csymbol><sum id="alg3.l11.m1.2.2.3.2.3.1.2.2.cmml" xref="alg3.l11.m1.2.2.3.2.3.1.2.2"></sum><apply id="alg3.l11.m1.2.2.3.2.3.1.2.3.cmml" xref="alg3.l11.m1.2.2.3.2.3.1.2.3"><eq id="alg3.l11.m1.2.2.3.2.3.1.2.3.1.cmml" xref="alg3.l11.m1.2.2.3.2.3.1.2.3.1"></eq><ci id="alg3.l11.m1.2.2.3.2.3.1.2.3.2.cmml" xref="alg3.l11.m1.2.2.3.2.3.1.2.3.2">ℓ</ci><cn id="alg3.l11.m1.2.2.3.2.3.1.2.3.3.cmml" type="integer" xref="alg3.l11.m1.2.2.3.2.3.1.2.3.3">1</cn></apply></apply><ci id="alg3.l11.m1.2.2.3.2.3.1.3.cmml" xref="alg3.l11.m1.2.2.3.2.3.1.3">𝐿</ci></apply><apply id="alg3.l11.m1.2.2.3.2.3.2.cmml" xref="alg3.l11.m1.2.2.3.2.3.2"><csymbol cd="ambiguous" id="alg3.l11.m1.2.2.3.2.3.2.1.cmml" xref="alg3.l11.m1.2.2.3.2.3.2">superscript</csymbol><ci id="alg3.l11.m1.2.2.3.2.3.2.2.cmml" xref="alg3.l11.m1.2.2.3.2.3.2.2">𝒑</ci><ci id="alg3.l11.m1.2.2.3.2.3.2.3.cmml" xref="alg3.l11.m1.2.2.3.2.3.2.3">ℓ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l11.m1.2c">\tilde{U}_{i}(\cdot,a_{-i})=-\frac{1}{L}\sum_{\ell=1}^{L}{\bm{p}}^{\ell}</annotation><annotation encoding="application/x-llamapun" id="alg3.l11.m1.2d">over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) = - divide start_ARG 1 end_ARG start_ARG italic_L end_ARG ∑ start_POSTSUBSCRIPT roman_ℓ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT bold_italic_p start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l12"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l12.1.1.1" style="font-size:80%;">12:</span></span><span class="ltx_text ltx_font_bold" id="alg3.l12.2">return</span> <math alttext="\tilde{U}" class="ltx_Math" display="inline" id="alg3.l12.m1.1"><semantics id="alg3.l12.m1.1a"><mover accent="true" id="alg3.l12.m1.1.1" xref="alg3.l12.m1.1.1.cmml"><mi id="alg3.l12.m1.1.1.2" xref="alg3.l12.m1.1.1.2.cmml">U</mi><mo id="alg3.l12.m1.1.1.1" xref="alg3.l12.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="alg3.l12.m1.1b"><apply id="alg3.l12.m1.1.1.cmml" xref="alg3.l12.m1.1.1"><ci id="alg3.l12.m1.1.1.1.cmml" xref="alg3.l12.m1.1.1.1">~</ci><ci id="alg3.l12.m1.1.1.2.cmml" xref="alg3.l12.m1.1.1.2">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l12.m1.1c">\tilde{U}</annotation><annotation encoding="application/x-llamapun" id="alg3.l12.m1.1d">over~ start_ARG italic_U end_ARG</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg4.3.1.1">Algorithm 4</span> </span> Principal’s algorithm for learning a multi-agent game in the no-regret model</figcaption> </figure> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem4.p1"> <p class="ltx_p" id="S5.Thmtheorem4.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p1.3.3">For some appropriate choice of the hyperparameter <math alttext="L" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.1.1.m1.1"><semantics id="S5.Thmtheorem4.p1.1.1.m1.1a"><mi id="S5.Thmtheorem4.p1.1.1.m1.1.1" xref="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.1.1.m1.1b"><ci id="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.1.1.m1.1d">italic_L</annotation></semantics></math>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg4" title="In 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a> <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.2.2.m2.1"><semantics id="S5.Thmtheorem4.p1.2.2.m2.1a"><mi id="S5.Thmtheorem4.p1.2.2.m2.1.1" xref="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.2.2.m2.1b"><ci id="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.2.2.m2.1d">italic_ε</annotation></semantics></math>-learns any game in <math alttext="\poly(M)/\varepsilon^{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.3.3.m3.1"><semantics id="S5.Thmtheorem4.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem4.p1.3.3.m3.1.2" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.cmml"><mrow id="S5.Thmtheorem4.p1.3.3.m3.1.2.2" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2b.cmml"><mtext id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2a" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2b.cmml">\poly</mtext></merror><mo id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.1" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.3.2" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.cmml"><mo id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem4.p1.3.3.m3.1.1" xref="S5.Thmtheorem4.p1.3.3.m3.1.1.cmml">M</mi><mo id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem4.p1.3.3.m3.1.2.1" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.1.cmml">/</mo><msup id="S5.Thmtheorem4.p1.3.3.m3.1.2.3" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.3.cmml"><mi id="S5.Thmtheorem4.p1.3.3.m3.1.2.3.2" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.3.2.cmml">ε</mi><mn id="S5.Thmtheorem4.p1.3.3.m3.1.2.3.3" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.3.3.m3.1b"><apply id="S5.Thmtheorem4.p1.3.3.m3.1.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2"><divide id="S5.Thmtheorem4.p1.3.3.m3.1.2.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.1"></divide><apply id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2"><times id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.1"></times><ci id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2b.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2"><merror class="ltx_ERROR undefined undefined" id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2"><mtext id="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2a.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.2.2">\poly</mtext></merror></ci><ci id="S5.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.1">𝑀</ci></apply><apply id="S5.Thmtheorem4.p1.3.3.m3.1.2.3.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.3.3.m3.1.2.3.1.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.3">superscript</csymbol><ci id="S5.Thmtheorem4.p1.3.3.m3.1.2.3.2.cmml" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.3.2">𝜀</ci><cn id="S5.Thmtheorem4.p1.3.3.m3.1.2.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.3.3.m3.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.3.3.m3.1c">\poly(M)/\varepsilon^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.3.3.m3.1d">( italic_M ) / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> rounds.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS2.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS2.1.p1"> <p class="ltx_p" id="S5.SS2.1.p1.3">As in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.SS1" title="5.1 The single-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">5.1</span></a>, we will assume without loss of generality that <math alttext="\sum_{a_{i}\in A_{i}}U_{i}(a_{i},a_{-i})=0" class="ltx_Math" display="inline" id="S5.SS2.1.p1.1.m1.2"><semantics id="S5.SS2.1.p1.1.m1.2a"><mrow id="S5.SS2.1.p1.1.m1.2.2" xref="S5.SS2.1.p1.1.m1.2.2.cmml"><mrow id="S5.SS2.1.p1.1.m1.2.2.2" xref="S5.SS2.1.p1.1.m1.2.2.2.cmml"><msub id="S5.SS2.1.p1.1.m1.2.2.2.3" xref="S5.SS2.1.p1.1.m1.2.2.2.3.cmml"><mo id="S5.SS2.1.p1.1.m1.2.2.2.3.2" xref="S5.SS2.1.p1.1.m1.2.2.2.3.2.cmml">∑</mo><mrow id="S5.SS2.1.p1.1.m1.2.2.2.3.3" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.cmml"><msub id="S5.SS2.1.p1.1.m1.2.2.2.3.3.2" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.cmml"><mi id="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.2" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.2.cmml">a</mi><mi id="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.3" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.3.cmml">i</mi></msub><mo id="S5.SS2.1.p1.1.m1.2.2.2.3.3.1" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.1.cmml">∈</mo><msub id="S5.SS2.1.p1.1.m1.2.2.2.3.3.3" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.cmml"><mi id="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.2" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.2.cmml">A</mi><mi id="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.3" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.3.cmml">i</mi></msub></mrow></msub><mrow id="S5.SS2.1.p1.1.m1.2.2.2.2" xref="S5.SS2.1.p1.1.m1.2.2.2.2.cmml"><msub id="S5.SS2.1.p1.1.m1.2.2.2.2.4" xref="S5.SS2.1.p1.1.m1.2.2.2.2.4.cmml"><mi id="S5.SS2.1.p1.1.m1.2.2.2.2.4.2" xref="S5.SS2.1.p1.1.m1.2.2.2.2.4.2.cmml">U</mi><mi id="S5.SS2.1.p1.1.m1.2.2.2.2.4.3" xref="S5.SS2.1.p1.1.m1.2.2.2.2.4.3.cmml">i</mi></msub><mo id="S5.SS2.1.p1.1.m1.2.2.2.2.3" xref="S5.SS2.1.p1.1.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.3.cmml"><mo id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.3" stretchy="false" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.3.cmml">(</mo><msub id="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1" xref="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.2" xref="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.3" xref="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.4" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.3.cmml">,</mo><msub id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.cmml"><mi id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.2" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.2.cmml">a</mi><mrow id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.cmml"><mo id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3a" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.cmml">−</mo><mi id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.2" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.5" stretchy="false" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S5.SS2.1.p1.1.m1.2.2.3" xref="S5.SS2.1.p1.1.m1.2.2.3.cmml">=</mo><mn id="S5.SS2.1.p1.1.m1.2.2.4" xref="S5.SS2.1.p1.1.m1.2.2.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.1.p1.1.m1.2b"><apply id="S5.SS2.1.p1.1.m1.2.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2"><eq id="S5.SS2.1.p1.1.m1.2.2.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.3"></eq><apply id="S5.SS2.1.p1.1.m1.2.2.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2"><apply id="S5.SS2.1.p1.1.m1.2.2.2.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3"><csymbol cd="ambiguous" id="S5.SS2.1.p1.1.m1.2.2.2.3.1.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3">subscript</csymbol><sum id="S5.SS2.1.p1.1.m1.2.2.2.3.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.2"></sum><apply id="S5.SS2.1.p1.1.m1.2.2.2.3.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3"><in id="S5.SS2.1.p1.1.m1.2.2.2.3.3.1.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.1"></in><apply id="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.2"><csymbol cd="ambiguous" id="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.1.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.2">subscript</csymbol><ci id="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.2">𝑎</ci><ci id="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.2.3">𝑖</ci></apply><apply id="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.3"><csymbol cd="ambiguous" id="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.1.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.3">subscript</csymbol><ci id="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.2">𝐴</ci><ci id="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.3.3.3.3">𝑖</ci></apply></apply></apply><apply id="S5.SS2.1.p1.1.m1.2.2.2.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2"><times id="S5.SS2.1.p1.1.m1.2.2.2.2.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.3"></times><apply id="S5.SS2.1.p1.1.m1.2.2.2.2.4.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.4"><csymbol cd="ambiguous" id="S5.SS2.1.p1.1.m1.2.2.2.2.4.1.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.4">subscript</csymbol><ci id="S5.SS2.1.p1.1.m1.2.2.2.2.4.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.4.2">𝑈</ci><ci id="S5.SS2.1.p1.1.m1.2.2.2.2.4.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.4.3">𝑖</ci></apply><interval closure="open" id="S5.SS2.1.p1.1.m1.2.2.2.2.2.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2"><apply id="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.SS2.1.p1.1.m1.1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.1.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2">subscript</csymbol><ci id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.2">𝑎</ci><apply id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3"><minus id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.1.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3"></minus><ci id="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.2.cmml" xref="S5.SS2.1.p1.1.m1.2.2.2.2.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply></apply><cn id="S5.SS2.1.p1.1.m1.2.2.4.cmml" type="integer" xref="S5.SS2.1.p1.1.m1.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.1.p1.1.m1.2c">\sum_{a_{i}\in A_{i}}U_{i}(a_{i},a_{-i})=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.1.p1.1.m1.2d">∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> for all agents <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.1.p1.2.m2.1"><semantics id="S5.SS2.1.p1.2.m2.1a"><mi id="S5.SS2.1.p1.2.m2.1.1" xref="S5.SS2.1.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.1.p1.2.m2.1b"><ci id="S5.SS2.1.p1.2.m2.1.1.cmml" xref="S5.SS2.1.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.1.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.1.p1.2.m2.1d">italic_i</annotation></semantics></math> and opponent profiles <math alttext="a_{-i}" class="ltx_Math" display="inline" id="S5.SS2.1.p1.3.m3.1"><semantics id="S5.SS2.1.p1.3.m3.1a"><msub id="S5.SS2.1.p1.3.m3.1.1" xref="S5.SS2.1.p1.3.m3.1.1.cmml"><mi id="S5.SS2.1.p1.3.m3.1.1.2" xref="S5.SS2.1.p1.3.m3.1.1.2.cmml">a</mi><mrow id="S5.SS2.1.p1.3.m3.1.1.3" xref="S5.SS2.1.p1.3.m3.1.1.3.cmml"><mo id="S5.SS2.1.p1.3.m3.1.1.3a" xref="S5.SS2.1.p1.3.m3.1.1.3.cmml">−</mo><mi id="S5.SS2.1.p1.3.m3.1.1.3.2" xref="S5.SS2.1.p1.3.m3.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.1.p1.3.m3.1b"><apply id="S5.SS2.1.p1.3.m3.1.1.cmml" xref="S5.SS2.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS2.1.p1.3.m3.1.1.1.cmml" xref="S5.SS2.1.p1.3.m3.1.1">subscript</csymbol><ci id="S5.SS2.1.p1.3.m3.1.1.2.cmml" xref="S5.SS2.1.p1.3.m3.1.1.2">𝑎</ci><apply id="S5.SS2.1.p1.3.m3.1.1.3.cmml" xref="S5.SS2.1.p1.3.m3.1.1.3"><minus id="S5.SS2.1.p1.3.m3.1.1.3.1.cmml" xref="S5.SS2.1.p1.3.m3.1.1.3"></minus><ci id="S5.SS2.1.p1.3.m3.1.1.3.2.cmml" xref="S5.SS2.1.p1.3.m3.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.1.p1.3.m3.1c">a_{-i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.1.p1.3.m3.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.2.p2"> <p class="ltx_p" id="S5.SS2.2.p2.12">We claim first that, for any agent <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.2.p2.1.m1.1"><semantics id="S5.SS2.2.p2.1.m1.1a"><mi id="S5.SS2.2.p2.1.m1.1.1" xref="S5.SS2.2.p2.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.1.m1.1b"><ci id="S5.SS2.2.p2.1.m1.1.1.cmml" xref="S5.SS2.2.p2.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.1.m1.1d">italic_i</annotation></semantics></math> and any action <math alttext="a_{i}\in A_{i}" class="ltx_Math" display="inline" id="S5.SS2.2.p2.2.m2.1"><semantics id="S5.SS2.2.p2.2.m2.1a"><mrow id="S5.SS2.2.p2.2.m2.1.1" xref="S5.SS2.2.p2.2.m2.1.1.cmml"><msub id="S5.SS2.2.p2.2.m2.1.1.2" xref="S5.SS2.2.p2.2.m2.1.1.2.cmml"><mi id="S5.SS2.2.p2.2.m2.1.1.2.2" xref="S5.SS2.2.p2.2.m2.1.1.2.2.cmml">a</mi><mi id="S5.SS2.2.p2.2.m2.1.1.2.3" xref="S5.SS2.2.p2.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS2.2.p2.2.m2.1.1.1" xref="S5.SS2.2.p2.2.m2.1.1.1.cmml">∈</mo><msub id="S5.SS2.2.p2.2.m2.1.1.3" xref="S5.SS2.2.p2.2.m2.1.1.3.cmml"><mi id="S5.SS2.2.p2.2.m2.1.1.3.2" xref="S5.SS2.2.p2.2.m2.1.1.3.2.cmml">A</mi><mi id="S5.SS2.2.p2.2.m2.1.1.3.3" xref="S5.SS2.2.p2.2.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.2.m2.1b"><apply id="S5.SS2.2.p2.2.m2.1.1.cmml" xref="S5.SS2.2.p2.2.m2.1.1"><in id="S5.SS2.2.p2.2.m2.1.1.1.cmml" xref="S5.SS2.2.p2.2.m2.1.1.1"></in><apply id="S5.SS2.2.p2.2.m2.1.1.2.cmml" xref="S5.SS2.2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.2.m2.1.1.2.1.cmml" xref="S5.SS2.2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS2.2.p2.2.m2.1.1.2.2.cmml" xref="S5.SS2.2.p2.2.m2.1.1.2.2">𝑎</ci><ci id="S5.SS2.2.p2.2.m2.1.1.2.3.cmml" xref="S5.SS2.2.p2.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.SS2.2.p2.2.m2.1.1.3.cmml" xref="S5.SS2.2.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.2.p2.2.m2.1.1.3.1.cmml" xref="S5.SS2.2.p2.2.m2.1.1.3">subscript</csymbol><ci id="S5.SS2.2.p2.2.m2.1.1.3.2.cmml" xref="S5.SS2.2.p2.2.m2.1.1.3.2">𝐴</ci><ci id="S5.SS2.2.p2.2.m2.1.1.3.3.cmml" xref="S5.SS2.2.p2.2.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.2.m2.1c">a_{i}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the number of times that <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.2.p2.3.m3.1"><semantics id="S5.SS2.2.p2.3.m3.1a"><mi id="S5.SS2.2.p2.3.m3.1.1" xref="S5.SS2.2.p2.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.3.m3.1b"><ci id="S5.SS2.2.p2.3.m3.1.1.cmml" xref="S5.SS2.2.p2.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.3.m3.1d">italic_i</annotation></semantics></math> does <span class="ltx_text ltx_font_italic" id="S5.SS2.2.p2.12.1">not</span> play <math alttext="a_{i}^{t}=a_{i}" class="ltx_Math" display="inline" id="S5.SS2.2.p2.4.m4.1"><semantics id="S5.SS2.2.p2.4.m4.1a"><mrow id="S5.SS2.2.p2.4.m4.1.1" xref="S5.SS2.2.p2.4.m4.1.1.cmml"><msubsup id="S5.SS2.2.p2.4.m4.1.1.2" xref="S5.SS2.2.p2.4.m4.1.1.2.cmml"><mi id="S5.SS2.2.p2.4.m4.1.1.2.2.2" xref="S5.SS2.2.p2.4.m4.1.1.2.2.2.cmml">a</mi><mi id="S5.SS2.2.p2.4.m4.1.1.2.2.3" xref="S5.SS2.2.p2.4.m4.1.1.2.2.3.cmml">i</mi><mi id="S5.SS2.2.p2.4.m4.1.1.2.3" xref="S5.SS2.2.p2.4.m4.1.1.2.3.cmml">t</mi></msubsup><mo id="S5.SS2.2.p2.4.m4.1.1.1" xref="S5.SS2.2.p2.4.m4.1.1.1.cmml">=</mo><msub id="S5.SS2.2.p2.4.m4.1.1.3" xref="S5.SS2.2.p2.4.m4.1.1.3.cmml"><mi id="S5.SS2.2.p2.4.m4.1.1.3.2" xref="S5.SS2.2.p2.4.m4.1.1.3.2.cmml">a</mi><mi id="S5.SS2.2.p2.4.m4.1.1.3.3" xref="S5.SS2.2.p2.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.4.m4.1b"><apply id="S5.SS2.2.p2.4.m4.1.1.cmml" xref="S5.SS2.2.p2.4.m4.1.1"><eq id="S5.SS2.2.p2.4.m4.1.1.1.cmml" xref="S5.SS2.2.p2.4.m4.1.1.1"></eq><apply id="S5.SS2.2.p2.4.m4.1.1.2.cmml" xref="S5.SS2.2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.4.m4.1.1.2.1.cmml" xref="S5.SS2.2.p2.4.m4.1.1.2">superscript</csymbol><apply id="S5.SS2.2.p2.4.m4.1.1.2.2.cmml" xref="S5.SS2.2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.4.m4.1.1.2.2.1.cmml" xref="S5.SS2.2.p2.4.m4.1.1.2">subscript</csymbol><ci id="S5.SS2.2.p2.4.m4.1.1.2.2.2.cmml" xref="S5.SS2.2.p2.4.m4.1.1.2.2.2">𝑎</ci><ci id="S5.SS2.2.p2.4.m4.1.1.2.2.3.cmml" xref="S5.SS2.2.p2.4.m4.1.1.2.2.3">𝑖</ci></apply><ci id="S5.SS2.2.p2.4.m4.1.1.2.3.cmml" xref="S5.SS2.2.p2.4.m4.1.1.2.3">𝑡</ci></apply><apply id="S5.SS2.2.p2.4.m4.1.1.3.cmml" xref="S5.SS2.2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.2.p2.4.m4.1.1.3.1.cmml" xref="S5.SS2.2.p2.4.m4.1.1.3">subscript</csymbol><ci id="S5.SS2.2.p2.4.m4.1.1.3.2.cmml" xref="S5.SS2.2.p2.4.m4.1.1.3.2">𝑎</ci><ci id="S5.SS2.2.p2.4.m4.1.1.3.3.cmml" xref="S5.SS2.2.p2.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.4.m4.1c">a_{i}^{t}=a_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> when given signal <math alttext="s_{i}^{t}=a_{i}" class="ltx_Math" display="inline" id="S5.SS2.2.p2.5.m5.1"><semantics id="S5.SS2.2.p2.5.m5.1a"><mrow id="S5.SS2.2.p2.5.m5.1.1" xref="S5.SS2.2.p2.5.m5.1.1.cmml"><msubsup id="S5.SS2.2.p2.5.m5.1.1.2" xref="S5.SS2.2.p2.5.m5.1.1.2.cmml"><mi id="S5.SS2.2.p2.5.m5.1.1.2.2.2" xref="S5.SS2.2.p2.5.m5.1.1.2.2.2.cmml">s</mi><mi id="S5.SS2.2.p2.5.m5.1.1.2.2.3" xref="S5.SS2.2.p2.5.m5.1.1.2.2.3.cmml">i</mi><mi id="S5.SS2.2.p2.5.m5.1.1.2.3" xref="S5.SS2.2.p2.5.m5.1.1.2.3.cmml">t</mi></msubsup><mo id="S5.SS2.2.p2.5.m5.1.1.1" xref="S5.SS2.2.p2.5.m5.1.1.1.cmml">=</mo><msub id="S5.SS2.2.p2.5.m5.1.1.3" xref="S5.SS2.2.p2.5.m5.1.1.3.cmml"><mi id="S5.SS2.2.p2.5.m5.1.1.3.2" xref="S5.SS2.2.p2.5.m5.1.1.3.2.cmml">a</mi><mi id="S5.SS2.2.p2.5.m5.1.1.3.3" xref="S5.SS2.2.p2.5.m5.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.5.m5.1b"><apply id="S5.SS2.2.p2.5.m5.1.1.cmml" xref="S5.SS2.2.p2.5.m5.1.1"><eq id="S5.SS2.2.p2.5.m5.1.1.1.cmml" xref="S5.SS2.2.p2.5.m5.1.1.1"></eq><apply id="S5.SS2.2.p2.5.m5.1.1.2.cmml" xref="S5.SS2.2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.5.m5.1.1.2.1.cmml" xref="S5.SS2.2.p2.5.m5.1.1.2">superscript</csymbol><apply id="S5.SS2.2.p2.5.m5.1.1.2.2.cmml" xref="S5.SS2.2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.5.m5.1.1.2.2.1.cmml" xref="S5.SS2.2.p2.5.m5.1.1.2">subscript</csymbol><ci id="S5.SS2.2.p2.5.m5.1.1.2.2.2.cmml" xref="S5.SS2.2.p2.5.m5.1.1.2.2.2">𝑠</ci><ci id="S5.SS2.2.p2.5.m5.1.1.2.2.3.cmml" xref="S5.SS2.2.p2.5.m5.1.1.2.2.3">𝑖</ci></apply><ci id="S5.SS2.2.p2.5.m5.1.1.2.3.cmml" xref="S5.SS2.2.p2.5.m5.1.1.2.3">𝑡</ci></apply><apply id="S5.SS2.2.p2.5.m5.1.1.3.cmml" xref="S5.SS2.2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.2.p2.5.m5.1.1.3.1.cmml" xref="S5.SS2.2.p2.5.m5.1.1.3">subscript</csymbol><ci id="S5.SS2.2.p2.5.m5.1.1.3.2.cmml" xref="S5.SS2.2.p2.5.m5.1.1.3.2">𝑎</ci><ci id="S5.SS2.2.p2.5.m5.1.1.3.3.cmml" xref="S5.SS2.2.p2.5.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.5.m5.1c">s_{i}^{t}=a_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.5.m5.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is bounded by <math alttext="C\sqrt{T}" class="ltx_Math" display="inline" id="S5.SS2.2.p2.6.m6.1"><semantics id="S5.SS2.2.p2.6.m6.1a"><mrow id="S5.SS2.2.p2.6.m6.1.1" xref="S5.SS2.2.p2.6.m6.1.1.cmml"><mi id="S5.SS2.2.p2.6.m6.1.1.2" xref="S5.SS2.2.p2.6.m6.1.1.2.cmml">C</mi><mo id="S5.SS2.2.p2.6.m6.1.1.1" xref="S5.SS2.2.p2.6.m6.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.2.p2.6.m6.1.1.3" xref="S5.SS2.2.p2.6.m6.1.1.3.cmml"><mi id="S5.SS2.2.p2.6.m6.1.1.3.2" xref="S5.SS2.2.p2.6.m6.1.1.3.2.cmml">T</mi></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.6.m6.1b"><apply id="S5.SS2.2.p2.6.m6.1.1.cmml" xref="S5.SS2.2.p2.6.m6.1.1"><times id="S5.SS2.2.p2.6.m6.1.1.1.cmml" xref="S5.SS2.2.p2.6.m6.1.1.1"></times><ci id="S5.SS2.2.p2.6.m6.1.1.2.cmml" xref="S5.SS2.2.p2.6.m6.1.1.2">𝐶</ci><apply id="S5.SS2.2.p2.6.m6.1.1.3.cmml" xref="S5.SS2.2.p2.6.m6.1.1.3"><root id="S5.SS2.2.p2.6.m6.1.1.3a.cmml" xref="S5.SS2.2.p2.6.m6.1.1.3"></root><ci id="S5.SS2.2.p2.6.m6.1.1.3.2.cmml" xref="S5.SS2.2.p2.6.m6.1.1.3.2">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.6.m6.1c">C\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.6.m6.1d">italic_C square-root start_ARG italic_T end_ARG</annotation></semantics></math>. To see this, note that whenever the principal sends signal <math alttext="a_{i}" class="ltx_Math" display="inline" id="S5.SS2.2.p2.7.m7.1"><semantics id="S5.SS2.2.p2.7.m7.1a"><msub id="S5.SS2.2.p2.7.m7.1.1" xref="S5.SS2.2.p2.7.m7.1.1.cmml"><mi id="S5.SS2.2.p2.7.m7.1.1.2" xref="S5.SS2.2.p2.7.m7.1.1.2.cmml">a</mi><mi id="S5.SS2.2.p2.7.m7.1.1.3" xref="S5.SS2.2.p2.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.7.m7.1b"><apply id="S5.SS2.2.p2.7.m7.1.1.cmml" xref="S5.SS2.2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS2.2.p2.7.m7.1.1.1.cmml" xref="S5.SS2.2.p2.7.m7.1.1">subscript</csymbol><ci id="S5.SS2.2.p2.7.m7.1.1.2.cmml" xref="S5.SS2.2.p2.7.m7.1.1.2">𝑎</ci><ci id="S5.SS2.2.p2.7.m7.1.1.3.cmml" xref="S5.SS2.2.p2.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.7.m7.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.7.m7.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the payment is always set such that <math alttext="U^{t}_{i}(a_{i},a_{-i})\geq 1+U^{t}_{i}(a_{i}^{\prime},a_{-i})" class="ltx_Math" display="inline" id="S5.SS2.2.p2.8.m8.4"><semantics id="S5.SS2.2.p2.8.m8.4a"><mrow id="S5.SS2.2.p2.8.m8.4.4" xref="S5.SS2.2.p2.8.m8.4.4.cmml"><mrow id="S5.SS2.2.p2.8.m8.2.2.2" xref="S5.SS2.2.p2.8.m8.2.2.2.cmml"><msubsup id="S5.SS2.2.p2.8.m8.2.2.2.4" xref="S5.SS2.2.p2.8.m8.2.2.2.4.cmml"><mi id="S5.SS2.2.p2.8.m8.2.2.2.4.2.2" xref="S5.SS2.2.p2.8.m8.2.2.2.4.2.2.cmml">U</mi><mi id="S5.SS2.2.p2.8.m8.2.2.2.4.3" xref="S5.SS2.2.p2.8.m8.2.2.2.4.3.cmml">i</mi><mi id="S5.SS2.2.p2.8.m8.2.2.2.4.2.3" xref="S5.SS2.2.p2.8.m8.2.2.2.4.2.3.cmml">t</mi></msubsup><mo id="S5.SS2.2.p2.8.m8.2.2.2.3" xref="S5.SS2.2.p2.8.m8.2.2.2.3.cmml">⁢</mo><mrow id="S5.SS2.2.p2.8.m8.2.2.2.2.2" xref="S5.SS2.2.p2.8.m8.2.2.2.2.3.cmml"><mo id="S5.SS2.2.p2.8.m8.2.2.2.2.2.3" stretchy="false" xref="S5.SS2.2.p2.8.m8.2.2.2.2.3.cmml">(</mo><msub id="S5.SS2.2.p2.8.m8.1.1.1.1.1.1" xref="S5.SS2.2.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="S5.SS2.2.p2.8.m8.1.1.1.1.1.1.2" xref="S5.SS2.2.p2.8.m8.1.1.1.1.1.1.2.cmml">a</mi><mi id="S5.SS2.2.p2.8.m8.1.1.1.1.1.1.3" xref="S5.SS2.2.p2.8.m8.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.2.p2.8.m8.2.2.2.2.2.4" xref="S5.SS2.2.p2.8.m8.2.2.2.2.3.cmml">,</mo><msub id="S5.SS2.2.p2.8.m8.2.2.2.2.2.2" xref="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.cmml"><mi id="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.2" xref="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.2.cmml">a</mi><mrow id="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.3" xref="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.3.cmml"><mo id="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.3a" xref="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.3.cmml">−</mo><mi id="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.3.2" xref="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="S5.SS2.2.p2.8.m8.2.2.2.2.2.5" stretchy="false" xref="S5.SS2.2.p2.8.m8.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.SS2.2.p2.8.m8.4.4.5" xref="S5.SS2.2.p2.8.m8.4.4.5.cmml">≥</mo><mrow id="S5.SS2.2.p2.8.m8.4.4.4" xref="S5.SS2.2.p2.8.m8.4.4.4.cmml"><mn id="S5.SS2.2.p2.8.m8.4.4.4.4" xref="S5.SS2.2.p2.8.m8.4.4.4.4.cmml">1</mn><mo id="S5.SS2.2.p2.8.m8.4.4.4.3" xref="S5.SS2.2.p2.8.m8.4.4.4.3.cmml">+</mo><mrow id="S5.SS2.2.p2.8.m8.4.4.4.2" xref="S5.SS2.2.p2.8.m8.4.4.4.2.cmml"><msubsup id="S5.SS2.2.p2.8.m8.4.4.4.2.4" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4.cmml"><mi id="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.2" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.2.cmml">U</mi><mi id="S5.SS2.2.p2.8.m8.4.4.4.2.4.3" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4.3.cmml">i</mi><mi id="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.3" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.3.cmml">t</mi></msubsup><mo id="S5.SS2.2.p2.8.m8.4.4.4.2.3" xref="S5.SS2.2.p2.8.m8.4.4.4.2.3.cmml">⁢</mo><mrow id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.3.cmml"><mo id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.3" stretchy="false" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.3.cmml">(</mo><msubsup id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.cmml"><mi id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.2" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.2.cmml">a</mi><mi id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.3" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.3.cmml">i</mi><mo id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.3" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.3.cmml">′</mo></msubsup><mo id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.4" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.3.cmml">,</mo><msub id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.cmml"><mi id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.2" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.2.cmml">a</mi><mrow id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.cmml"><mo id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3a" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.cmml">−</mo><mi id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.2" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.5" stretchy="false" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.8.m8.4b"><apply id="S5.SS2.2.p2.8.m8.4.4.cmml" xref="S5.SS2.2.p2.8.m8.4.4"><geq id="S5.SS2.2.p2.8.m8.4.4.5.cmml" xref="S5.SS2.2.p2.8.m8.4.4.5"></geq><apply id="S5.SS2.2.p2.8.m8.2.2.2.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2"><times id="S5.SS2.2.p2.8.m8.2.2.2.3.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.3"></times><apply id="S5.SS2.2.p2.8.m8.2.2.2.4.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.4"><csymbol cd="ambiguous" id="S5.SS2.2.p2.8.m8.2.2.2.4.1.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.4">subscript</csymbol><apply id="S5.SS2.2.p2.8.m8.2.2.2.4.2.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.4"><csymbol cd="ambiguous" id="S5.SS2.2.p2.8.m8.2.2.2.4.2.1.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.4">superscript</csymbol><ci id="S5.SS2.2.p2.8.m8.2.2.2.4.2.2.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.4.2.2">𝑈</ci><ci id="S5.SS2.2.p2.8.m8.2.2.2.4.2.3.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.4.2.3">𝑡</ci></apply><ci id="S5.SS2.2.p2.8.m8.2.2.2.4.3.cmml" xref="S5.SS2.2.p2.8.m8.2.2.2.4.3">𝑖</ci></apply><interval 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xref="S5.SS2.2.p2.8.m8.2.2.2.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply><apply id="S5.SS2.2.p2.8.m8.4.4.4.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4"><plus id="S5.SS2.2.p2.8.m8.4.4.4.3.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.3"></plus><cn id="S5.SS2.2.p2.8.m8.4.4.4.4.cmml" type="integer" xref="S5.SS2.2.p2.8.m8.4.4.4.4">1</cn><apply id="S5.SS2.2.p2.8.m8.4.4.4.2.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2"><times id="S5.SS2.2.p2.8.m8.4.4.4.2.3.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.3"></times><apply id="S5.SS2.2.p2.8.m8.4.4.4.2.4.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4"><csymbol cd="ambiguous" id="S5.SS2.2.p2.8.m8.4.4.4.2.4.1.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4">subscript</csymbol><apply id="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4"><csymbol cd="ambiguous" id="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.1.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4">superscript</csymbol><ci id="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.2.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.2">𝑈</ci><ci id="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.3.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4.2.3">𝑡</ci></apply><ci id="S5.SS2.2.p2.8.m8.4.4.4.2.4.3.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.4.3">𝑖</ci></apply><interval closure="open" id="S5.SS2.2.p2.8.m8.4.4.4.2.2.3.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2"><apply id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.cmml" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.1.cmml" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1">superscript</csymbol><apply id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.cmml" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.1.cmml" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1">subscript</csymbol><ci id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.2.cmml" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.2">𝑎</ci><ci id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.3.cmml" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.2.3">𝑖</ci></apply><ci id="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.3.cmml" xref="S5.SS2.2.p2.8.m8.3.3.3.1.1.1.1.3">′</ci></apply><apply id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.1.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2">subscript</csymbol><ci id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.2.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.2">𝑎</ci><apply id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3"><minus id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.1.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3"></minus><ci id="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.2.cmml" xref="S5.SS2.2.p2.8.m8.4.4.4.2.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.8.m8.4c">U^{t}_{i}(a_{i},a_{-i})\geq 1+U^{t}_{i}(a_{i}^{\prime},a_{-i})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.8.m8.4d">italic_U start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) ≥ 1 + italic_U start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. Thus, the number of times <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.2.p2.9.m9.1"><semantics id="S5.SS2.2.p2.9.m9.1a"><mi id="S5.SS2.2.p2.9.m9.1.1" xref="S5.SS2.2.p2.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.9.m9.1b"><ci id="S5.SS2.2.p2.9.m9.1.1.cmml" xref="S5.SS2.2.p2.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.9.m9.1d">italic_i</annotation></semantics></math> does not play <math alttext="a_{i}^{t}=a_{i}" class="ltx_Math" display="inline" id="S5.SS2.2.p2.10.m10.1"><semantics id="S5.SS2.2.p2.10.m10.1a"><mrow id="S5.SS2.2.p2.10.m10.1.1" xref="S5.SS2.2.p2.10.m10.1.1.cmml"><msubsup id="S5.SS2.2.p2.10.m10.1.1.2" xref="S5.SS2.2.p2.10.m10.1.1.2.cmml"><mi id="S5.SS2.2.p2.10.m10.1.1.2.2.2" xref="S5.SS2.2.p2.10.m10.1.1.2.2.2.cmml">a</mi><mi id="S5.SS2.2.p2.10.m10.1.1.2.2.3" xref="S5.SS2.2.p2.10.m10.1.1.2.2.3.cmml">i</mi><mi id="S5.SS2.2.p2.10.m10.1.1.2.3" xref="S5.SS2.2.p2.10.m10.1.1.2.3.cmml">t</mi></msubsup><mo id="S5.SS2.2.p2.10.m10.1.1.1" xref="S5.SS2.2.p2.10.m10.1.1.1.cmml">=</mo><msub id="S5.SS2.2.p2.10.m10.1.1.3" xref="S5.SS2.2.p2.10.m10.1.1.3.cmml"><mi id="S5.SS2.2.p2.10.m10.1.1.3.2" xref="S5.SS2.2.p2.10.m10.1.1.3.2.cmml">a</mi><mi id="S5.SS2.2.p2.10.m10.1.1.3.3" xref="S5.SS2.2.p2.10.m10.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.10.m10.1b"><apply id="S5.SS2.2.p2.10.m10.1.1.cmml" xref="S5.SS2.2.p2.10.m10.1.1"><eq id="S5.SS2.2.p2.10.m10.1.1.1.cmml" xref="S5.SS2.2.p2.10.m10.1.1.1"></eq><apply id="S5.SS2.2.p2.10.m10.1.1.2.cmml" xref="S5.SS2.2.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.10.m10.1.1.2.1.cmml" xref="S5.SS2.2.p2.10.m10.1.1.2">superscript</csymbol><apply id="S5.SS2.2.p2.10.m10.1.1.2.2.cmml" xref="S5.SS2.2.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.2.p2.10.m10.1.1.2.2.1.cmml" xref="S5.SS2.2.p2.10.m10.1.1.2">subscript</csymbol><ci id="S5.SS2.2.p2.10.m10.1.1.2.2.2.cmml" xref="S5.SS2.2.p2.10.m10.1.1.2.2.2">𝑎</ci><ci id="S5.SS2.2.p2.10.m10.1.1.2.2.3.cmml" xref="S5.SS2.2.p2.10.m10.1.1.2.2.3">𝑖</ci></apply><ci id="S5.SS2.2.p2.10.m10.1.1.2.3.cmml" xref="S5.SS2.2.p2.10.m10.1.1.2.3">𝑡</ci></apply><apply id="S5.SS2.2.p2.10.m10.1.1.3.cmml" xref="S5.SS2.2.p2.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.2.p2.10.m10.1.1.3.1.cmml" xref="S5.SS2.2.p2.10.m10.1.1.3">subscript</csymbol><ci id="S5.SS2.2.p2.10.m10.1.1.3.2.cmml" xref="S5.SS2.2.p2.10.m10.1.1.3.2">𝑎</ci><ci id="S5.SS2.2.p2.10.m10.1.1.3.3.cmml" xref="S5.SS2.2.p2.10.m10.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.10.m10.1c">a_{i}^{t}=a_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.10.m10.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> quantity lower-bounds the regret <math alttext="\hat{R}(T,a_{i})" class="ltx_Math" display="inline" id="S5.SS2.2.p2.11.m11.2"><semantics id="S5.SS2.2.p2.11.m11.2a"><mrow id="S5.SS2.2.p2.11.m11.2.2" xref="S5.SS2.2.p2.11.m11.2.2.cmml"><mover accent="true" id="S5.SS2.2.p2.11.m11.2.2.3" xref="S5.SS2.2.p2.11.m11.2.2.3.cmml"><mi id="S5.SS2.2.p2.11.m11.2.2.3.2" xref="S5.SS2.2.p2.11.m11.2.2.3.2.cmml">R</mi><mo id="S5.SS2.2.p2.11.m11.2.2.3.1" xref="S5.SS2.2.p2.11.m11.2.2.3.1.cmml">^</mo></mover><mo id="S5.SS2.2.p2.11.m11.2.2.2" xref="S5.SS2.2.p2.11.m11.2.2.2.cmml">⁢</mo><mrow id="S5.SS2.2.p2.11.m11.2.2.1.1" xref="S5.SS2.2.p2.11.m11.2.2.1.2.cmml"><mo id="S5.SS2.2.p2.11.m11.2.2.1.1.2" stretchy="false" xref="S5.SS2.2.p2.11.m11.2.2.1.2.cmml">(</mo><mi id="S5.SS2.2.p2.11.m11.1.1" xref="S5.SS2.2.p2.11.m11.1.1.cmml">T</mi><mo id="S5.SS2.2.p2.11.m11.2.2.1.1.3" xref="S5.SS2.2.p2.11.m11.2.2.1.2.cmml">,</mo><msub id="S5.SS2.2.p2.11.m11.2.2.1.1.1" xref="S5.SS2.2.p2.11.m11.2.2.1.1.1.cmml"><mi id="S5.SS2.2.p2.11.m11.2.2.1.1.1.2" xref="S5.SS2.2.p2.11.m11.2.2.1.1.1.2.cmml">a</mi><mi id="S5.SS2.2.p2.11.m11.2.2.1.1.1.3" xref="S5.SS2.2.p2.11.m11.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.2.p2.11.m11.2.2.1.1.4" stretchy="false" xref="S5.SS2.2.p2.11.m11.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.11.m11.2b"><apply id="S5.SS2.2.p2.11.m11.2.2.cmml" xref="S5.SS2.2.p2.11.m11.2.2"><times id="S5.SS2.2.p2.11.m11.2.2.2.cmml" xref="S5.SS2.2.p2.11.m11.2.2.2"></times><apply id="S5.SS2.2.p2.11.m11.2.2.3.cmml" xref="S5.SS2.2.p2.11.m11.2.2.3"><ci id="S5.SS2.2.p2.11.m11.2.2.3.1.cmml" xref="S5.SS2.2.p2.11.m11.2.2.3.1">^</ci><ci id="S5.SS2.2.p2.11.m11.2.2.3.2.cmml" xref="S5.SS2.2.p2.11.m11.2.2.3.2">𝑅</ci></apply><interval closure="open" id="S5.SS2.2.p2.11.m11.2.2.1.2.cmml" xref="S5.SS2.2.p2.11.m11.2.2.1.1"><ci id="S5.SS2.2.p2.11.m11.1.1.cmml" xref="S5.SS2.2.p2.11.m11.1.1">𝑇</ci><apply id="S5.SS2.2.p2.11.m11.2.2.1.1.1.cmml" xref="S5.SS2.2.p2.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.2.p2.11.m11.2.2.1.1.1.1.cmml" xref="S5.SS2.2.p2.11.m11.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.2.p2.11.m11.2.2.1.1.1.2.cmml" xref="S5.SS2.2.p2.11.m11.2.2.1.1.1.2">𝑎</ci><ci id="S5.SS2.2.p2.11.m11.2.2.1.1.1.3.cmml" xref="S5.SS2.2.p2.11.m11.2.2.1.1.1.3">𝑖</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.11.m11.2c">\hat{R}(T,a_{i})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.11.m11.2d">over^ start_ARG italic_R end_ARG ( italic_T , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. The claim follows from the regret guarantee <math alttext="\hat{R}(T,a_{i})\leq C\sqrt{T}" class="ltx_Math" display="inline" id="S5.SS2.2.p2.12.m12.2"><semantics id="S5.SS2.2.p2.12.m12.2a"><mrow id="S5.SS2.2.p2.12.m12.2.2" xref="S5.SS2.2.p2.12.m12.2.2.cmml"><mrow id="S5.SS2.2.p2.12.m12.2.2.1" xref="S5.SS2.2.p2.12.m12.2.2.1.cmml"><mover accent="true" id="S5.SS2.2.p2.12.m12.2.2.1.3" xref="S5.SS2.2.p2.12.m12.2.2.1.3.cmml"><mi id="S5.SS2.2.p2.12.m12.2.2.1.3.2" xref="S5.SS2.2.p2.12.m12.2.2.1.3.2.cmml">R</mi><mo id="S5.SS2.2.p2.12.m12.2.2.1.3.1" xref="S5.SS2.2.p2.12.m12.2.2.1.3.1.cmml">^</mo></mover><mo id="S5.SS2.2.p2.12.m12.2.2.1.2" xref="S5.SS2.2.p2.12.m12.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS2.2.p2.12.m12.2.2.1.1.1" xref="S5.SS2.2.p2.12.m12.2.2.1.1.2.cmml"><mo id="S5.SS2.2.p2.12.m12.2.2.1.1.1.2" stretchy="false" xref="S5.SS2.2.p2.12.m12.2.2.1.1.2.cmml">(</mo><mi id="S5.SS2.2.p2.12.m12.1.1" xref="S5.SS2.2.p2.12.m12.1.1.cmml">T</mi><mo id="S5.SS2.2.p2.12.m12.2.2.1.1.1.3" xref="S5.SS2.2.p2.12.m12.2.2.1.1.2.cmml">,</mo><msub id="S5.SS2.2.p2.12.m12.2.2.1.1.1.1" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.cmml"><mi id="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.2" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.2.cmml">a</mi><mi id="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.3" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.2.p2.12.m12.2.2.1.1.1.4" stretchy="false" xref="S5.SS2.2.p2.12.m12.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.2.p2.12.m12.2.2.2" xref="S5.SS2.2.p2.12.m12.2.2.2.cmml">≤</mo><mrow id="S5.SS2.2.p2.12.m12.2.2.3" xref="S5.SS2.2.p2.12.m12.2.2.3.cmml"><mi id="S5.SS2.2.p2.12.m12.2.2.3.2" xref="S5.SS2.2.p2.12.m12.2.2.3.2.cmml">C</mi><mo id="S5.SS2.2.p2.12.m12.2.2.3.1" xref="S5.SS2.2.p2.12.m12.2.2.3.1.cmml">⁢</mo><msqrt id="S5.SS2.2.p2.12.m12.2.2.3.3" xref="S5.SS2.2.p2.12.m12.2.2.3.3.cmml"><mi id="S5.SS2.2.p2.12.m12.2.2.3.3.2" xref="S5.SS2.2.p2.12.m12.2.2.3.3.2.cmml">T</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.2.p2.12.m12.2b"><apply id="S5.SS2.2.p2.12.m12.2.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2"><leq id="S5.SS2.2.p2.12.m12.2.2.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2.2"></leq><apply id="S5.SS2.2.p2.12.m12.2.2.1.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1"><times id="S5.SS2.2.p2.12.m12.2.2.1.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.2"></times><apply id="S5.SS2.2.p2.12.m12.2.2.1.3.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.3"><ci id="S5.SS2.2.p2.12.m12.2.2.1.3.1.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.3.1">^</ci><ci id="S5.SS2.2.p2.12.m12.2.2.1.3.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.3.2">𝑅</ci></apply><interval closure="open" id="S5.SS2.2.p2.12.m12.2.2.1.1.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1"><ci id="S5.SS2.2.p2.12.m12.1.1.cmml" xref="S5.SS2.2.p2.12.m12.1.1">𝑇</ci><apply id="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.1.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1.1">subscript</csymbol><ci id="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.2">𝑎</ci><ci id="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.3.cmml" xref="S5.SS2.2.p2.12.m12.2.2.1.1.1.1.3">𝑖</ci></apply></interval></apply><apply id="S5.SS2.2.p2.12.m12.2.2.3.cmml" xref="S5.SS2.2.p2.12.m12.2.2.3"><times id="S5.SS2.2.p2.12.m12.2.2.3.1.cmml" xref="S5.SS2.2.p2.12.m12.2.2.3.1"></times><ci id="S5.SS2.2.p2.12.m12.2.2.3.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2.3.2">𝐶</ci><apply id="S5.SS2.2.p2.12.m12.2.2.3.3.cmml" xref="S5.SS2.2.p2.12.m12.2.2.3.3"><root id="S5.SS2.2.p2.12.m12.2.2.3.3a.cmml" xref="S5.SS2.2.p2.12.m12.2.2.3.3"></root><ci id="S5.SS2.2.p2.12.m12.2.2.3.3.2.cmml" xref="S5.SS2.2.p2.12.m12.2.2.3.3.2">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.2.p2.12.m12.2c">\hat{R}(T,a_{i})\leq C\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.2.p2.12.m12.2d">over^ start_ARG italic_R end_ARG ( italic_T , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≤ italic_C square-root start_ARG italic_T end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.3.p3"> <p class="ltx_p" id="S5.SS2.3.p3.10">We will refer to the iterations of the inner loop over action profiles <math alttext="a_{-i}" class="ltx_Math" display="inline" id="S5.SS2.3.p3.1.m1.1"><semantics id="S5.SS2.3.p3.1.m1.1a"><msub id="S5.SS2.3.p3.1.m1.1.1" xref="S5.SS2.3.p3.1.m1.1.1.cmml"><mi id="S5.SS2.3.p3.1.m1.1.1.2" xref="S5.SS2.3.p3.1.m1.1.1.2.cmml">a</mi><mrow id="S5.SS2.3.p3.1.m1.1.1.3" xref="S5.SS2.3.p3.1.m1.1.1.3.cmml"><mo id="S5.SS2.3.p3.1.m1.1.1.3a" xref="S5.SS2.3.p3.1.m1.1.1.3.cmml">−</mo><mi id="S5.SS2.3.p3.1.m1.1.1.3.2" xref="S5.SS2.3.p3.1.m1.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.1.m1.1b"><apply id="S5.SS2.3.p3.1.m1.1.1.cmml" xref="S5.SS2.3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.3.p3.1.m1.1.1.1.cmml" xref="S5.SS2.3.p3.1.m1.1.1">subscript</csymbol><ci id="S5.SS2.3.p3.1.m1.1.1.2.cmml" xref="S5.SS2.3.p3.1.m1.1.1.2">𝑎</ci><apply id="S5.SS2.3.p3.1.m1.1.1.3.cmml" xref="S5.SS2.3.p3.1.m1.1.1.3"><minus id="S5.SS2.3.p3.1.m1.1.1.3.1.cmml" xref="S5.SS2.3.p3.1.m1.1.1.3"></minus><ci id="S5.SS2.3.p3.1.m1.1.1.3.2.cmml" xref="S5.SS2.3.p3.1.m1.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.1.m1.1c">a_{-i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.1.m1.1d">italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as <span class="ltx_text ltx_font_italic" id="S5.SS2.3.p3.10.1">phases</span>. Fix a agent <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.3.p3.2.m2.1"><semantics id="S5.SS2.3.p3.2.m2.1a"><mi id="S5.SS2.3.p3.2.m2.1.1" xref="S5.SS2.3.p3.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.2.m2.1b"><ci id="S5.SS2.3.p3.2.m2.1.1.cmml" xref="S5.SS2.3.p3.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.2.m2.1d">italic_i</annotation></semantics></math>, and number the phases for that agent using integers <math alttext="k=1,\dots,M_{i}:=\prod_{j\neq i}m_{j}" class="ltx_Math" display="inline" id="S5.SS2.3.p3.3.m3.4"><semantics id="S5.SS2.3.p3.3.m3.4a"><mrow id="S5.SS2.3.p3.3.m3.4.4.2" xref="S5.SS2.3.p3.3.m3.4.4.3.cmml"><mrow id="S5.SS2.3.p3.3.m3.3.3.1.1" xref="S5.SS2.3.p3.3.m3.3.3.1.1.cmml"><mi id="S5.SS2.3.p3.3.m3.3.3.1.1.2" xref="S5.SS2.3.p3.3.m3.3.3.1.1.2.cmml">k</mi><mo id="S5.SS2.3.p3.3.m3.3.3.1.1.1" xref="S5.SS2.3.p3.3.m3.3.3.1.1.1.cmml">=</mo><mrow id="S5.SS2.3.p3.3.m3.3.3.1.1.3.2" xref="S5.SS2.3.p3.3.m3.3.3.1.1.3.1.cmml"><mn id="S5.SS2.3.p3.3.m3.1.1" xref="S5.SS2.3.p3.3.m3.1.1.cmml">1</mn><mo id="S5.SS2.3.p3.3.m3.3.3.1.1.3.2.1" xref="S5.SS2.3.p3.3.m3.3.3.1.1.3.1.cmml">,</mo><mi id="S5.SS2.3.p3.3.m3.2.2" mathvariant="normal" xref="S5.SS2.3.p3.3.m3.2.2.cmml">…</mi></mrow></mrow><mo id="S5.SS2.3.p3.3.m3.4.4.2.3" xref="S5.SS2.3.p3.3.m3.4.4.3a.cmml">,</mo><mrow id="S5.SS2.3.p3.3.m3.4.4.2.2" xref="S5.SS2.3.p3.3.m3.4.4.2.2.cmml"><msub id="S5.SS2.3.p3.3.m3.4.4.2.2.2" xref="S5.SS2.3.p3.3.m3.4.4.2.2.2.cmml"><mi id="S5.SS2.3.p3.3.m3.4.4.2.2.2.2" xref="S5.SS2.3.p3.3.m3.4.4.2.2.2.2.cmml">M</mi><mi id="S5.SS2.3.p3.3.m3.4.4.2.2.2.3" xref="S5.SS2.3.p3.3.m3.4.4.2.2.2.3.cmml">i</mi></msub><mo id="S5.SS2.3.p3.3.m3.4.4.2.2.1" lspace="0.278em" rspace="0.111em" xref="S5.SS2.3.p3.3.m3.4.4.2.2.1.cmml">:=</mo><mrow id="S5.SS2.3.p3.3.m3.4.4.2.2.3" xref="S5.SS2.3.p3.3.m3.4.4.2.2.3.cmml"><msub 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xref="S5.SS2.3.p3.3.m3.4.4.2.2.3.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.3.m3.4.4.2.2.3.2.1.cmml" xref="S5.SS2.3.p3.3.m3.4.4.2.2.3.2">subscript</csymbol><ci id="S5.SS2.3.p3.3.m3.4.4.2.2.3.2.2.cmml" xref="S5.SS2.3.p3.3.m3.4.4.2.2.3.2.2">𝑚</ci><ci id="S5.SS2.3.p3.3.m3.4.4.2.2.3.2.3.cmml" xref="S5.SS2.3.p3.3.m3.4.4.2.2.3.2.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.3.m3.4c">k=1,\dots,M_{i}:=\prod_{j\neq i}m_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.3.m3.4d">italic_k = 1 , … , italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := ∏ start_POSTSUBSCRIPT italic_j ≠ italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, corresponding respectively to strategy profiles <math alttext="\bar{a}_{-i}^{1},\dots,\bar{a}_{-i}^{M_{i}}\in A_{-i}" class="ltx_Math" display="inline" id="S5.SS2.3.p3.4.m4.3"><semantics 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xref="S5.SS2.3.p3.4.m4.2.2.1.1.1">superscript</csymbol><apply id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.1.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1">subscript</csymbol><apply id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.2.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.2"><ci id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.2.1.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.2.1">¯</ci><ci id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.2.2.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.2.2">𝑎</ci></apply><apply id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.3.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.3"><minus id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.3.1.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.3"></minus><ci id="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.3.2.cmml" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1.2.3.2">𝑖</ci></apply></apply><cn id="S5.SS2.3.p3.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS2.3.p3.4.m4.2.2.1.1.1.3">1</cn></apply><ci id="S5.SS2.3.p3.4.m4.1.1.cmml" xref="S5.SS2.3.p3.4.m4.1.1">…</ci><apply id="S5.SS2.3.p3.4.m4.3.3.2.2.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.4.m4.3.3.2.2.2.1.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2">superscript</csymbol><apply id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.1.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2">subscript</csymbol><apply id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.2"><ci id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.2.1.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.2.1">¯</ci><ci id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.2.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.2.2">𝑎</ci></apply><apply id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.3.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.3"><minus id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.3.1.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.3"></minus><ci id="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.3.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.2.3.2">𝑖</ci></apply></apply><apply id="S5.SS2.3.p3.4.m4.3.3.2.2.2.3.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S5.SS2.3.p3.4.m4.3.3.2.2.2.3.1.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.3">subscript</csymbol><ci id="S5.SS2.3.p3.4.m4.3.3.2.2.2.3.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.3.2">𝑀</ci><ci id="S5.SS2.3.p3.4.m4.3.3.2.2.2.3.3.cmml" xref="S5.SS2.3.p3.4.m4.3.3.2.2.2.3.3">𝑖</ci></apply></apply></list><apply id="S5.SS2.3.p3.4.m4.3.3.4.cmml" xref="S5.SS2.3.p3.4.m4.3.3.4"><csymbol cd="ambiguous" id="S5.SS2.3.p3.4.m4.3.3.4.1.cmml" xref="S5.SS2.3.p3.4.m4.3.3.4">subscript</csymbol><ci id="S5.SS2.3.p3.4.m4.3.3.4.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.4.2">𝐴</ci><apply id="S5.SS2.3.p3.4.m4.3.3.4.3.cmml" xref="S5.SS2.3.p3.4.m4.3.3.4.3"><minus id="S5.SS2.3.p3.4.m4.3.3.4.3.1.cmml" xref="S5.SS2.3.p3.4.m4.3.3.4.3"></minus><ci id="S5.SS2.3.p3.4.m4.3.3.4.3.2.cmml" xref="S5.SS2.3.p3.4.m4.3.3.4.3.2">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.4.m4.3c">\bar{a}_{-i}^{1},\dots,\bar{a}_{-i}^{M_{i}}\in A_{-i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.4.m4.3d">over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="{\mathcal{T}}_{i}(k)=\{\underaccent{\bar}{T}_{i}(k),\dots,\bar{T}_{i}(k)\}" class="ltx_Math" display="inline" id="S5.SS2.3.p3.5.m5.6"><semantics id="S5.SS2.3.p3.5.m5.6a"><mrow id="S5.SS2.3.p3.5.m5.6.6" xref="S5.SS2.3.p3.5.m5.6.6.cmml"><mrow id="S5.SS2.3.p3.5.m5.6.6.4" xref="S5.SS2.3.p3.5.m5.6.6.4.cmml"><msub id="S5.SS2.3.p3.5.m5.6.6.4.2" xref="S5.SS2.3.p3.5.m5.6.6.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.3.p3.5.m5.6.6.4.2.2" xref="S5.SS2.3.p3.5.m5.6.6.4.2.2.cmml">𝒯</mi><mi id="S5.SS2.3.p3.5.m5.6.6.4.2.3" xref="S5.SS2.3.p3.5.m5.6.6.4.2.3.cmml">i</mi></msub><mo id="S5.SS2.3.p3.5.m5.6.6.4.1" xref="S5.SS2.3.p3.5.m5.6.6.4.1.cmml">⁢</mo><mrow id="S5.SS2.3.p3.5.m5.6.6.4.3.2" xref="S5.SS2.3.p3.5.m5.6.6.4.cmml"><mo id="S5.SS2.3.p3.5.m5.6.6.4.3.2.1" stretchy="false" xref="S5.SS2.3.p3.5.m5.6.6.4.cmml">(</mo><mi id="S5.SS2.3.p3.5.m5.1.1" xref="S5.SS2.3.p3.5.m5.1.1.cmml">k</mi><mo id="S5.SS2.3.p3.5.m5.6.6.4.3.2.2" stretchy="false" xref="S5.SS2.3.p3.5.m5.6.6.4.cmml">)</mo></mrow></mrow><mo id="S5.SS2.3.p3.5.m5.6.6.3" xref="S5.SS2.3.p3.5.m5.6.6.3.cmml">=</mo><mrow id="S5.SS2.3.p3.5.m5.6.6.2.2" xref="S5.SS2.3.p3.5.m5.6.6.2.3.cmml"><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.3" stretchy="false" xref="S5.SS2.3.p3.5.m5.6.6.2.3.cmml">{</mo><mrow id="S5.SS2.3.p3.5.m5.5.5.1.1.1" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.cmml"><msub id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.cmml"><munder accentunder="true" id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.cmml"><mi id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.2" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.2.cmml">T</mi><mo id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.1" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.1.cmml">¯</mo></munder><mi id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.3" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS2.3.p3.5.m5.5.5.1.1.1.1" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS2.3.p3.5.m5.5.5.1.1.1.3.2" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.cmml"><mo id="S5.SS2.3.p3.5.m5.5.5.1.1.1.3.2.1" stretchy="false" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.cmml">(</mo><mi id="S5.SS2.3.p3.5.m5.2.2" xref="S5.SS2.3.p3.5.m5.2.2.cmml">k</mi><mo id="S5.SS2.3.p3.5.m5.5.5.1.1.1.3.2.2" stretchy="false" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.4" xref="S5.SS2.3.p3.5.m5.6.6.2.3.cmml">,</mo><mi id="S5.SS2.3.p3.5.m5.4.4" mathvariant="normal" xref="S5.SS2.3.p3.5.m5.4.4.cmml">…</mi><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.5" xref="S5.SS2.3.p3.5.m5.6.6.2.3.cmml">,</mo><mrow id="S5.SS2.3.p3.5.m5.6.6.2.2.2" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.cmml"><msub id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.cmml"><mover accent="true" id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.cmml"><mi id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.2" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.2.cmml">T</mi><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.1" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.1.cmml">¯</mo></mover><mi id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.3" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.3.cmml">i</mi></msub><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.2.1" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.3.p3.5.m5.6.6.2.2.2.3.2" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.cmml"><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.2.3.2.1" stretchy="false" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.cmml">(</mo><mi id="S5.SS2.3.p3.5.m5.3.3" xref="S5.SS2.3.p3.5.m5.3.3.cmml">k</mi><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.2.3.2.2" stretchy="false" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.3.p3.5.m5.6.6.2.2.6" stretchy="false" xref="S5.SS2.3.p3.5.m5.6.6.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.5.m5.6b"><apply id="S5.SS2.3.p3.5.m5.6.6.cmml" xref="S5.SS2.3.p3.5.m5.6.6"><eq id="S5.SS2.3.p3.5.m5.6.6.3.cmml" xref="S5.SS2.3.p3.5.m5.6.6.3"></eq><apply id="S5.SS2.3.p3.5.m5.6.6.4.cmml" xref="S5.SS2.3.p3.5.m5.6.6.4"><times id="S5.SS2.3.p3.5.m5.6.6.4.1.cmml" xref="S5.SS2.3.p3.5.m5.6.6.4.1"></times><apply id="S5.SS2.3.p3.5.m5.6.6.4.2.cmml" xref="S5.SS2.3.p3.5.m5.6.6.4.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.5.m5.6.6.4.2.1.cmml" xref="S5.SS2.3.p3.5.m5.6.6.4.2">subscript</csymbol><ci id="S5.SS2.3.p3.5.m5.6.6.4.2.2.cmml" xref="S5.SS2.3.p3.5.m5.6.6.4.2.2">𝒯</ci><ci id="S5.SS2.3.p3.5.m5.6.6.4.2.3.cmml" xref="S5.SS2.3.p3.5.m5.6.6.4.2.3">𝑖</ci></apply><ci id="S5.SS2.3.p3.5.m5.1.1.cmml" xref="S5.SS2.3.p3.5.m5.1.1">𝑘</ci></apply><set id="S5.SS2.3.p3.5.m5.6.6.2.3.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2"><apply id="S5.SS2.3.p3.5.m5.5.5.1.1.1.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1"><times id="S5.SS2.3.p3.5.m5.5.5.1.1.1.1.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.1"></times><apply id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.1.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2">subscript</csymbol><apply id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2"><ci id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.1.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.1">¯</ci><ci id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.2.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.2.2">𝑇</ci></apply><ci id="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.3.cmml" xref="S5.SS2.3.p3.5.m5.5.5.1.1.1.2.3">𝑖</ci></apply><ci id="S5.SS2.3.p3.5.m5.2.2.cmml" xref="S5.SS2.3.p3.5.m5.2.2">𝑘</ci></apply><ci id="S5.SS2.3.p3.5.m5.4.4.cmml" xref="S5.SS2.3.p3.5.m5.4.4">…</ci><apply id="S5.SS2.3.p3.5.m5.6.6.2.2.2.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2"><times id="S5.SS2.3.p3.5.m5.6.6.2.2.2.1.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.1"></times><apply id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.1.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2">subscript</csymbol><apply id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2"><ci id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.1.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.1">¯</ci><ci id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.2.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.2.2">𝑇</ci></apply><ci id="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.3.cmml" xref="S5.SS2.3.p3.5.m5.6.6.2.2.2.2.3">𝑖</ci></apply><ci id="S5.SS2.3.p3.5.m5.3.3.cmml" xref="S5.SS2.3.p3.5.m5.3.3">𝑘</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.5.m5.6c">{\mathcal{T}}_{i}(k)=\{\underaccent{\bar}{T}_{i}(k),\dots,\bar{T}_{i}(k)\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.5.m5.6d">caligraphic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) = { under¯ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) , … , over¯ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) }</annotation></semantics></math> be the set of timesteps in agent <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.3.p3.6.m6.1"><semantics id="S5.SS2.3.p3.6.m6.1a"><mi id="S5.SS2.3.p3.6.m6.1.1" xref="S5.SS2.3.p3.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.6.m6.1b"><ci id="S5.SS2.3.p3.6.m6.1.1.cmml" xref="S5.SS2.3.p3.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.6.m6.1d">italic_i</annotation></semantics></math>’s <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.3.p3.7.m7.1"><semantics id="S5.SS2.3.p3.7.m7.1a"><mi id="S5.SS2.3.p3.7.m7.1.1" xref="S5.SS2.3.p3.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.7.m7.1b"><ci id="S5.SS2.3.p3.7.m7.1.1.cmml" xref="S5.SS2.3.p3.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.7.m7.1d">italic_k</annotation></semantics></math>th phase. Let <math alttext="B_{k}" class="ltx_Math" display="inline" id="S5.SS2.3.p3.8.m8.1"><semantics id="S5.SS2.3.p3.8.m8.1a"><msub id="S5.SS2.3.p3.8.m8.1.1" xref="S5.SS2.3.p3.8.m8.1.1.cmml"><mi id="S5.SS2.3.p3.8.m8.1.1.2" xref="S5.SS2.3.p3.8.m8.1.1.2.cmml">B</mi><mi id="S5.SS2.3.p3.8.m8.1.1.3" xref="S5.SS2.3.p3.8.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.8.m8.1b"><apply id="S5.SS2.3.p3.8.m8.1.1.cmml" xref="S5.SS2.3.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS2.3.p3.8.m8.1.1.1.cmml" xref="S5.SS2.3.p3.8.m8.1.1">subscript</csymbol><ci id="S5.SS2.3.p3.8.m8.1.1.2.cmml" xref="S5.SS2.3.p3.8.m8.1.1.2">𝐵</ci><ci id="S5.SS2.3.p3.8.m8.1.1.3.cmml" xref="S5.SS2.3.p3.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.8.m8.1c">B_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.8.m8.1d">italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> be the total numer of rounds in phases <math alttext="1,\dots,k" class="ltx_Math" display="inline" id="S5.SS2.3.p3.9.m9.3"><semantics id="S5.SS2.3.p3.9.m9.3a"><mrow id="S5.SS2.3.p3.9.m9.3.4.2" xref="S5.SS2.3.p3.9.m9.3.4.1.cmml"><mn id="S5.SS2.3.p3.9.m9.1.1" xref="S5.SS2.3.p3.9.m9.1.1.cmml">1</mn><mo id="S5.SS2.3.p3.9.m9.3.4.2.1" xref="S5.SS2.3.p3.9.m9.3.4.1.cmml">,</mo><mi id="S5.SS2.3.p3.9.m9.2.2" mathvariant="normal" xref="S5.SS2.3.p3.9.m9.2.2.cmml">…</mi><mo id="S5.SS2.3.p3.9.m9.3.4.2.2" xref="S5.SS2.3.p3.9.m9.3.4.1.cmml">,</mo><mi id="S5.SS2.3.p3.9.m9.3.3" xref="S5.SS2.3.p3.9.m9.3.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.9.m9.3b"><list id="S5.SS2.3.p3.9.m9.3.4.1.cmml" xref="S5.SS2.3.p3.9.m9.3.4.2"><cn id="S5.SS2.3.p3.9.m9.1.1.cmml" type="integer" xref="S5.SS2.3.p3.9.m9.1.1">1</cn><ci id="S5.SS2.3.p3.9.m9.2.2.cmml" xref="S5.SS2.3.p3.9.m9.2.2">…</ci><ci id="S5.SS2.3.p3.9.m9.3.3.cmml" xref="S5.SS2.3.p3.9.m9.3.3">𝑘</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.9.m9.3c">1,\dots,k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.9.m9.3d">1 , … , italic_k</annotation></semantics></math> in which <math alttext="a^{t}_{-i}\neq a_{-i}" class="ltx_Math" display="inline" id="S5.SS2.3.p3.10.m10.1"><semantics id="S5.SS2.3.p3.10.m10.1a"><mrow id="S5.SS2.3.p3.10.m10.1.1" xref="S5.SS2.3.p3.10.m10.1.1.cmml"><msubsup id="S5.SS2.3.p3.10.m10.1.1.2" xref="S5.SS2.3.p3.10.m10.1.1.2.cmml"><mi id="S5.SS2.3.p3.10.m10.1.1.2.2.2" xref="S5.SS2.3.p3.10.m10.1.1.2.2.2.cmml">a</mi><mrow id="S5.SS2.3.p3.10.m10.1.1.2.3" xref="S5.SS2.3.p3.10.m10.1.1.2.3.cmml"><mo id="S5.SS2.3.p3.10.m10.1.1.2.3a" xref="S5.SS2.3.p3.10.m10.1.1.2.3.cmml">−</mo><mi id="S5.SS2.3.p3.10.m10.1.1.2.3.2" xref="S5.SS2.3.p3.10.m10.1.1.2.3.2.cmml">i</mi></mrow><mi id="S5.SS2.3.p3.10.m10.1.1.2.2.3" xref="S5.SS2.3.p3.10.m10.1.1.2.2.3.cmml">t</mi></msubsup><mo id="S5.SS2.3.p3.10.m10.1.1.1" xref="S5.SS2.3.p3.10.m10.1.1.1.cmml">≠</mo><msub id="S5.SS2.3.p3.10.m10.1.1.3" xref="S5.SS2.3.p3.10.m10.1.1.3.cmml"><mi id="S5.SS2.3.p3.10.m10.1.1.3.2" xref="S5.SS2.3.p3.10.m10.1.1.3.2.cmml">a</mi><mrow id="S5.SS2.3.p3.10.m10.1.1.3.3" xref="S5.SS2.3.p3.10.m10.1.1.3.3.cmml"><mo id="S5.SS2.3.p3.10.m10.1.1.3.3a" xref="S5.SS2.3.p3.10.m10.1.1.3.3.cmml">−</mo><mi id="S5.SS2.3.p3.10.m10.1.1.3.3.2" xref="S5.SS2.3.p3.10.m10.1.1.3.3.2.cmml">i</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.3.p3.10.m10.1b"><apply id="S5.SS2.3.p3.10.m10.1.1.cmml" xref="S5.SS2.3.p3.10.m10.1.1"><neq id="S5.SS2.3.p3.10.m10.1.1.1.cmml" xref="S5.SS2.3.p3.10.m10.1.1.1"></neq><apply id="S5.SS2.3.p3.10.m10.1.1.2.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.10.m10.1.1.2.1.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2">subscript</csymbol><apply id="S5.SS2.3.p3.10.m10.1.1.2.2.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.3.p3.10.m10.1.1.2.2.1.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2">superscript</csymbol><ci id="S5.SS2.3.p3.10.m10.1.1.2.2.2.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2.2.2">𝑎</ci><ci id="S5.SS2.3.p3.10.m10.1.1.2.2.3.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2.2.3">𝑡</ci></apply><apply id="S5.SS2.3.p3.10.m10.1.1.2.3.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2.3"><minus id="S5.SS2.3.p3.10.m10.1.1.2.3.1.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2.3"></minus><ci id="S5.SS2.3.p3.10.m10.1.1.2.3.2.cmml" xref="S5.SS2.3.p3.10.m10.1.1.2.3.2">𝑖</ci></apply></apply><apply id="S5.SS2.3.p3.10.m10.1.1.3.cmml" xref="S5.SS2.3.p3.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.3.p3.10.m10.1.1.3.1.cmml" xref="S5.SS2.3.p3.10.m10.1.1.3">subscript</csymbol><ci id="S5.SS2.3.p3.10.m10.1.1.3.2.cmml" xref="S5.SS2.3.p3.10.m10.1.1.3.2">𝑎</ci><apply id="S5.SS2.3.p3.10.m10.1.1.3.3.cmml" xref="S5.SS2.3.p3.10.m10.1.1.3.3"><minus id="S5.SS2.3.p3.10.m10.1.1.3.3.1.cmml" xref="S5.SS2.3.p3.10.m10.1.1.3.3"></minus><ci id="S5.SS2.3.p3.10.m10.1.1.3.3.2.cmml" xref="S5.SS2.3.p3.10.m10.1.1.3.3.2">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.10.m10.1c">a^{t}_{-i}\neq a_{-i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.10.m10.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ≠ italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. By the principal’s regret bound in each phase, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx4"> <tbody id="S5.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\sum_{t\in{\mathcal{T}}_{i}(k)}U_{i}^{t}(a_{i}^{t},\bar{a}_{-i}^{% k})=\sum_{t\in{\mathcal{T}}_{i}(k)}\quantity[U_{i}(a_{i}^{t},\bar{a}_{-i}^{k})% +P_{i}^{t}(a_{i}^{t})]\leq L+R_{0}" class="ltx_Math" display="inline" id="S5.E6.m1.5"><semantics id="S5.E6.m1.5a"><mrow id="S5.E6.m1.5.5" xref="S5.E6.m1.5.5.cmml"><mrow id="S5.E6.m1.5.5.2" xref="S5.E6.m1.5.5.2.cmml"><mstyle displaystyle="true" id="S5.E6.m1.5.5.2.3" xref="S5.E6.m1.5.5.2.3.cmml"><munder id="S5.E6.m1.5.5.2.3a" xref="S5.E6.m1.5.5.2.3.cmml"><mo id="S5.E6.m1.5.5.2.3.2" movablelimits="false" xref="S5.E6.m1.5.5.2.3.2.cmml">∑</mo><mrow id="S5.E6.m1.2.2.1" xref="S5.E6.m1.2.2.1.cmml"><mi id="S5.E6.m1.2.2.1.3" xref="S5.E6.m1.2.2.1.3.cmml">t</mi><mo id="S5.E6.m1.2.2.1.2" xref="S5.E6.m1.2.2.1.2.cmml">∈</mo><mrow id="S5.E6.m1.2.2.1.4" xref="S5.E6.m1.2.2.1.4.cmml"><msub id="S5.E6.m1.2.2.1.4.2" xref="S5.E6.m1.2.2.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E6.m1.2.2.1.4.2.2" xref="S5.E6.m1.2.2.1.4.2.2.cmml">𝒯</mi><mi id="S5.E6.m1.2.2.1.4.2.3" xref="S5.E6.m1.2.2.1.4.2.3.cmml">i</mi></msub><mo id="S5.E6.m1.2.2.1.4.1" xref="S5.E6.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S5.E6.m1.2.2.1.4.3.2" xref="S5.E6.m1.2.2.1.4.cmml"><mo id="S5.E6.m1.2.2.1.4.3.2.1" stretchy="false" xref="S5.E6.m1.2.2.1.4.cmml">(</mo><mi id="S5.E6.m1.2.2.1.1" xref="S5.E6.m1.2.2.1.1.cmml">k</mi><mo id="S5.E6.m1.2.2.1.4.3.2.2" stretchy="false" xref="S5.E6.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S5.E6.m1.5.5.2.2" xref="S5.E6.m1.5.5.2.2.cmml"><msubsup id="S5.E6.m1.5.5.2.2.4" xref="S5.E6.m1.5.5.2.2.4.cmml"><mi id="S5.E6.m1.5.5.2.2.4.2.2" xref="S5.E6.m1.5.5.2.2.4.2.2.cmml">U</mi><mi id="S5.E6.m1.5.5.2.2.4.2.3" xref="S5.E6.m1.5.5.2.2.4.2.3.cmml">i</mi><mi id="S5.E6.m1.5.5.2.2.4.3" xref="S5.E6.m1.5.5.2.2.4.3.cmml">t</mi></msubsup><mo id="S5.E6.m1.5.5.2.2.3" xref="S5.E6.m1.5.5.2.2.3.cmml">⁢</mo><mrow id="S5.E6.m1.5.5.2.2.2.2" xref="S5.E6.m1.5.5.2.2.2.3.cmml"><mo id="S5.E6.m1.5.5.2.2.2.2.3" stretchy="false" xref="S5.E6.m1.5.5.2.2.2.3.cmml">(</mo><msubsup id="S5.E6.m1.4.4.1.1.1.1.1" xref="S5.E6.m1.4.4.1.1.1.1.1.cmml"><mi id="S5.E6.m1.4.4.1.1.1.1.1.2.2" xref="S5.E6.m1.4.4.1.1.1.1.1.2.2.cmml">a</mi><mi id="S5.E6.m1.4.4.1.1.1.1.1.2.3" xref="S5.E6.m1.4.4.1.1.1.1.1.2.3.cmml">i</mi><mi id="S5.E6.m1.4.4.1.1.1.1.1.3" xref="S5.E6.m1.4.4.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S5.E6.m1.5.5.2.2.2.2.4" xref="S5.E6.m1.5.5.2.2.2.3.cmml">,</mo><msubsup id="S5.E6.m1.5.5.2.2.2.2.2" xref="S5.E6.m1.5.5.2.2.2.2.2.cmml"><mover accent="true" id="S5.E6.m1.5.5.2.2.2.2.2.2.2" xref="S5.E6.m1.5.5.2.2.2.2.2.2.2.cmml"><mi 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xref="S5.E6.m1.5.5.7.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E6.m1.5c">\displaystyle\sum_{t\in{\mathcal{T}}_{i}(k)}U_{i}^{t}(a_{i}^{t},\bar{a}_{-i}^{% k})=\sum_{t\in{\mathcal{T}}_{i}(k)}\quantity[U_{i}(a_{i}^{t},\bar{a}_{-i}^{k})% +P_{i}^{t}(a_{i}^{t})]\leq L+R_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.E6.m1.5d">∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG ] ≤ italic_L + italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.3.p3.11">or else the principal has a profitable deviation to <math 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xref="S5.SS2.3.p3.11.m1.3.3.3.2.2.2">𝑃</ci><ci id="S5.SS2.3.p3.11.m1.3.3.3.2.2.3.cmml" xref="S5.SS2.3.p3.11.m1.3.3.3.2.2.3">𝑖</ci></apply><ci id="S5.SS2.3.p3.11.m1.3.3.3.2.3.cmml" xref="S5.SS2.3.p3.11.m1.3.3.3.2.3">𝑡</ci></apply><ci id="S5.SS2.3.p3.11.m1.1.1.cmml" xref="S5.SS2.3.p3.11.m1.1.1">⋅</ci></apply><apply id="S5.SS2.3.p3.11.m1.3.3.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1"><minus id="S5.SS2.3.p3.11.m1.3.3.1.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.2"></minus><cn id="S5.SS2.3.p3.11.m1.3.3.1.3.cmml" type="integer" xref="S5.SS2.3.p3.11.m1.3.3.1.3">1</cn><apply id="S5.SS2.3.p3.11.m1.3.3.1.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1"><times id="S5.SS2.3.p3.11.m1.3.3.1.1.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.2"></times><apply id="S5.SS2.3.p3.11.m1.3.3.1.1.3.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.3.p3.11.m1.3.3.1.1.3.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.3">subscript</csymbol><ci id="S5.SS2.3.p3.11.m1.3.3.1.1.3.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.3.2">𝑈</ci><ci id="S5.SS2.3.p3.11.m1.3.3.1.1.3.3.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.3.3">𝑖</ci></apply><interval closure="open" id="S5.SS2.3.p3.11.m1.3.3.1.1.1.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1"><ci id="S5.SS2.3.p3.11.m1.2.2.cmml" xref="S5.SS2.3.p3.11.m1.2.2">⋅</ci><apply id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1">superscript</csymbol><apply id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1">subscript</csymbol><apply id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.2"><ci id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.2.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.2.1">¯</ci><ci id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.2.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.2.2">𝑎</ci></apply><apply id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.3.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.3"><minus id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.3.1.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.3"></minus><ci id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.3.2.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.2.3.2">𝑖</ci></apply></apply><ci id="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.3.cmml" xref="S5.SS2.3.p3.11.m1.3.3.1.1.1.1.1.3">𝑘</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.3.p3.11.m1.3c">P_{i}^{t}(\cdot)=1-U_{i}(\cdot,\bar{a}_{-i}^{k})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.3.p3.11.m1.3d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( ⋅ ) = 1 - italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.4.p4"> <p class="ltx_p" id="S5.SS2.4.p4.4">Fix some <math alttext="K\leq M_{i}" class="ltx_Math" display="inline" id="S5.SS2.4.p4.1.m1.1"><semantics id="S5.SS2.4.p4.1.m1.1a"><mrow id="S5.SS2.4.p4.1.m1.1.1" xref="S5.SS2.4.p4.1.m1.1.1.cmml"><mi id="S5.SS2.4.p4.1.m1.1.1.2" xref="S5.SS2.4.p4.1.m1.1.1.2.cmml">K</mi><mo id="S5.SS2.4.p4.1.m1.1.1.1" xref="S5.SS2.4.p4.1.m1.1.1.1.cmml">≤</mo><msub id="S5.SS2.4.p4.1.m1.1.1.3" xref="S5.SS2.4.p4.1.m1.1.1.3.cmml"><mi id="S5.SS2.4.p4.1.m1.1.1.3.2" xref="S5.SS2.4.p4.1.m1.1.1.3.2.cmml">M</mi><mi id="S5.SS2.4.p4.1.m1.1.1.3.3" xref="S5.SS2.4.p4.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.1.m1.1b"><apply id="S5.SS2.4.p4.1.m1.1.1.cmml" xref="S5.SS2.4.p4.1.m1.1.1"><leq id="S5.SS2.4.p4.1.m1.1.1.1.cmml" xref="S5.SS2.4.p4.1.m1.1.1.1"></leq><ci id="S5.SS2.4.p4.1.m1.1.1.2.cmml" xref="S5.SS2.4.p4.1.m1.1.1.2">𝐾</ci><apply id="S5.SS2.4.p4.1.m1.1.1.3.cmml" xref="S5.SS2.4.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.4.p4.1.m1.1.1.3.1.cmml" xref="S5.SS2.4.p4.1.m1.1.1.3">subscript</csymbol><ci id="S5.SS2.4.p4.1.m1.1.1.3.2.cmml" xref="S5.SS2.4.p4.1.m1.1.1.3.2">𝑀</ci><ci id="S5.SS2.4.p4.1.m1.1.1.3.3.cmml" xref="S5.SS2.4.p4.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.1.m1.1c">K\leq M_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.1.m1.1d">italic_K ≤ italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="a_{i}\in A_{i}" class="ltx_Math" display="inline" id="S5.SS2.4.p4.2.m2.1"><semantics id="S5.SS2.4.p4.2.m2.1a"><mrow id="S5.SS2.4.p4.2.m2.1.1" xref="S5.SS2.4.p4.2.m2.1.1.cmml"><msub id="S5.SS2.4.p4.2.m2.1.1.2" xref="S5.SS2.4.p4.2.m2.1.1.2.cmml"><mi id="S5.SS2.4.p4.2.m2.1.1.2.2" xref="S5.SS2.4.p4.2.m2.1.1.2.2.cmml">a</mi><mi id="S5.SS2.4.p4.2.m2.1.1.2.3" xref="S5.SS2.4.p4.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS2.4.p4.2.m2.1.1.1" xref="S5.SS2.4.p4.2.m2.1.1.1.cmml">∈</mo><msub id="S5.SS2.4.p4.2.m2.1.1.3" xref="S5.SS2.4.p4.2.m2.1.1.3.cmml"><mi id="S5.SS2.4.p4.2.m2.1.1.3.2" xref="S5.SS2.4.p4.2.m2.1.1.3.2.cmml">A</mi><mi id="S5.SS2.4.p4.2.m2.1.1.3.3" xref="S5.SS2.4.p4.2.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.2.m2.1b"><apply id="S5.SS2.4.p4.2.m2.1.1.cmml" xref="S5.SS2.4.p4.2.m2.1.1"><in id="S5.SS2.4.p4.2.m2.1.1.1.cmml" xref="S5.SS2.4.p4.2.m2.1.1.1"></in><apply id="S5.SS2.4.p4.2.m2.1.1.2.cmml" xref="S5.SS2.4.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.4.p4.2.m2.1.1.2.1.cmml" xref="S5.SS2.4.p4.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS2.4.p4.2.m2.1.1.2.2.cmml" xref="S5.SS2.4.p4.2.m2.1.1.2.2">𝑎</ci><ci id="S5.SS2.4.p4.2.m2.1.1.2.3.cmml" xref="S5.SS2.4.p4.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.SS2.4.p4.2.m2.1.1.3.cmml" xref="S5.SS2.4.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.4.p4.2.m2.1.1.3.1.cmml" xref="S5.SS2.4.p4.2.m2.1.1.3">subscript</csymbol><ci id="S5.SS2.4.p4.2.m2.1.1.3.2.cmml" xref="S5.SS2.4.p4.2.m2.1.1.3.2">𝐴</ci><ci id="S5.SS2.4.p4.2.m2.1.1.3.3.cmml" xref="S5.SS2.4.p4.2.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.2.m2.1c">a_{i}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. By the anytime regret bound of agent <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.4.p4.3.m3.1"><semantics id="S5.SS2.4.p4.3.m3.1a"><mi id="S5.SS2.4.p4.3.m3.1.1" xref="S5.SS2.4.p4.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.3.m3.1b"><ci id="S5.SS2.4.p4.3.m3.1.1.cmml" xref="S5.SS2.4.p4.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.3.m3.1d">italic_i</annotation></semantics></math> under signal <math alttext="\bot" class="ltx_Math" display="inline" id="S5.SS2.4.p4.4.m4.1"><semantics id="S5.SS2.4.p4.4.m4.1a"><mo id="S5.SS2.4.p4.4.m4.1.1" xref="S5.SS2.4.p4.4.m4.1.1.cmml">⊥</mo><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.4.m4.1b"><csymbol cd="latexml" id="S5.SS2.4.p4.4.m4.1.1.cmml" xref="S5.SS2.4.p4.4.m4.1.1">bottom</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.4.m4.1c">\bot</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.4.m4.1d">⊥</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx5"> <tbody id="S5.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\sum_{\begin{subarray}{c}k\leq K\\ t\in{\mathcal{T}}_{i}(k)\end{subarray}}U_{i}^{t}(a_{i},a_{-i}^{t})" class="ltx_Math" display="inline" id="S5.E7.m1.3"><semantics id="S5.E7.m1.3a"><mrow id="S5.E7.m1.3.3" xref="S5.E7.m1.3.3.cmml"><mstyle displaystyle="true" id="S5.E7.m1.3.3.3" xref="S5.E7.m1.3.3.3.cmml"><munder id="S5.E7.m1.3.3.3a" xref="S5.E7.m1.3.3.3.cmml"><mo id="S5.E7.m1.3.3.3.2" movablelimits="false" xref="S5.E7.m1.3.3.3.2.cmml">∑</mo><mtable id="S5.E7.m1.1.1.1.1.1.1" rowspacing="0pt" xref="S5.E7.m1.1.1.1a.2.cmml"><mtr id="S5.E7.m1.1.1.1.1.1.1a" xref="S5.E7.m1.1.1.1a.2.cmml"><mtd id="S5.E7.m1.1.1.1.1.1.1b" xref="S5.E7.m1.1.1.1a.2.cmml"><mrow id="S5.E7.m1.1.1.1.1.1.1.2.1.1" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1.cmml"><mi id="S5.E7.m1.1.1.1.1.1.1.2.1.1.2" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1.2.cmml">k</mi><mo id="S5.E7.m1.1.1.1.1.1.1.2.1.1.1" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1.1.cmml">≤</mo><mi id="S5.E7.m1.1.1.1.1.1.1.2.1.1.3" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1.3.cmml">K</mi></mrow></mtd></mtr><mtr id="S5.E7.m1.1.1.1.1.1.1c" xref="S5.E7.m1.1.1.1a.2.cmml"><mtd id="S5.E7.m1.1.1.1.1.1.1d" xref="S5.E7.m1.1.1.1a.2.cmml"><mrow id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">t</mi><mo id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">∈</mo><mrow id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.cmml"><msub id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.2" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.2.cmml">𝒯</mi><mi id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.3" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.3.cmml">i</mi></msub><mo id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.1" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.3.2" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.cmml"><mo id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.3.2.1" stretchy="false" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.cmml">(</mo><mi id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">k</mi><mo id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.3.2.2" stretchy="false" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></mtd></mtr></mtable></munder></mstyle><mrow id="S5.E7.m1.3.3.2" xref="S5.E7.m1.3.3.2.cmml"><msubsup id="S5.E7.m1.3.3.2.4" xref="S5.E7.m1.3.3.2.4.cmml"><mi id="S5.E7.m1.3.3.2.4.2.2" xref="S5.E7.m1.3.3.2.4.2.2.cmml">U</mi><mi id="S5.E7.m1.3.3.2.4.2.3" xref="S5.E7.m1.3.3.2.4.2.3.cmml">i</mi><mi id="S5.E7.m1.3.3.2.4.3" xref="S5.E7.m1.3.3.2.4.3.cmml">t</mi></msubsup><mo id="S5.E7.m1.3.3.2.3" xref="S5.E7.m1.3.3.2.3.cmml">⁢</mo><mrow id="S5.E7.m1.3.3.2.2.2" xref="S5.E7.m1.3.3.2.2.3.cmml"><mo id="S5.E7.m1.3.3.2.2.2.3" stretchy="false" xref="S5.E7.m1.3.3.2.2.3.cmml">(</mo><msub id="S5.E7.m1.2.2.1.1.1.1" xref="S5.E7.m1.2.2.1.1.1.1.cmml"><mi id="S5.E7.m1.2.2.1.1.1.1.2" xref="S5.E7.m1.2.2.1.1.1.1.2.cmml">a</mi><mi id="S5.E7.m1.2.2.1.1.1.1.3" xref="S5.E7.m1.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.E7.m1.3.3.2.2.2.4" xref="S5.E7.m1.3.3.2.2.3.cmml">,</mo><msubsup id="S5.E7.m1.3.3.2.2.2.2" xref="S5.E7.m1.3.3.2.2.2.2.cmml"><mi id="S5.E7.m1.3.3.2.2.2.2.2.2" xref="S5.E7.m1.3.3.2.2.2.2.2.2.cmml">a</mi><mrow id="S5.E7.m1.3.3.2.2.2.2.2.3" xref="S5.E7.m1.3.3.2.2.2.2.2.3.cmml"><mo id="S5.E7.m1.3.3.2.2.2.2.2.3a" xref="S5.E7.m1.3.3.2.2.2.2.2.3.cmml">−</mo><mi id="S5.E7.m1.3.3.2.2.2.2.2.3.2" xref="S5.E7.m1.3.3.2.2.2.2.2.3.2.cmml">i</mi></mrow><mi id="S5.E7.m1.3.3.2.2.2.2.3" xref="S5.E7.m1.3.3.2.2.2.2.3.cmml">t</mi></msubsup><mo id="S5.E7.m1.3.3.2.2.2.5" stretchy="false" xref="S5.E7.m1.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E7.m1.3b"><apply id="S5.E7.m1.3.3.cmml" xref="S5.E7.m1.3.3"><apply id="S5.E7.m1.3.3.3.cmml" xref="S5.E7.m1.3.3.3"><csymbol cd="ambiguous" id="S5.E7.m1.3.3.3.1.cmml" xref="S5.E7.m1.3.3.3">subscript</csymbol><sum id="S5.E7.m1.3.3.3.2.cmml" xref="S5.E7.m1.3.3.3.2"></sum><list id="S5.E7.m1.1.1.1a.2.cmml" xref="S5.E7.m1.1.1.1.1.1.1"><matrix id="S5.E7.m1.1.1.1.1.1.1.cmml" xref="S5.E7.m1.1.1.1.1.1.1"><matrixrow id="S5.E7.m1.1.1.1.1.1.1a.cmml" xref="S5.E7.m1.1.1.1.1.1.1"><apply id="S5.E7.m1.1.1.1.1.1.1.2.1.1.cmml" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1"><leq id="S5.E7.m1.1.1.1.1.1.1.2.1.1.1.cmml" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1.1"></leq><ci id="S5.E7.m1.1.1.1.1.1.1.2.1.1.2.cmml" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1.2">𝑘</ci><ci id="S5.E7.m1.1.1.1.1.1.1.2.1.1.3.cmml" xref="S5.E7.m1.1.1.1.1.1.1.2.1.1.3">𝐾</ci></apply></matrixrow><matrixrow id="S5.E7.m1.1.1.1.1.1.1b.cmml" xref="S5.E7.m1.1.1.1.1.1.1"><apply id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1"><in id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.2"></in><ci id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.3">𝑡</ci><apply id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4"><times id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.1"></times><apply id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.1.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2">subscript</csymbol><ci id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.2.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.2">𝒯</ci><ci id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.3.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.4.2.3">𝑖</ci></apply><ci id="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E7.m1.1.1.1.1.1.1.1.1.1.1.1">𝑘</ci></apply></apply></matrixrow></matrix></list></apply><apply id="S5.E7.m1.3.3.2.cmml" xref="S5.E7.m1.3.3.2"><times id="S5.E7.m1.3.3.2.3.cmml" xref="S5.E7.m1.3.3.2.3"></times><apply id="S5.E7.m1.3.3.2.4.cmml" xref="S5.E7.m1.3.3.2.4"><csymbol cd="ambiguous" id="S5.E7.m1.3.3.2.4.1.cmml" xref="S5.E7.m1.3.3.2.4">superscript</csymbol><apply id="S5.E7.m1.3.3.2.4.2.cmml" xref="S5.E7.m1.3.3.2.4"><csymbol cd="ambiguous" id="S5.E7.m1.3.3.2.4.2.1.cmml" xref="S5.E7.m1.3.3.2.4">subscript</csymbol><ci id="S5.E7.m1.3.3.2.4.2.2.cmml" xref="S5.E7.m1.3.3.2.4.2.2">𝑈</ci><ci id="S5.E7.m1.3.3.2.4.2.3.cmml" xref="S5.E7.m1.3.3.2.4.2.3">𝑖</ci></apply><ci id="S5.E7.m1.3.3.2.4.3.cmml" xref="S5.E7.m1.3.3.2.4.3">𝑡</ci></apply><interval closure="open" id="S5.E7.m1.3.3.2.2.3.cmml" xref="S5.E7.m1.3.3.2.2.2"><apply id="S5.E7.m1.2.2.1.1.1.1.cmml" xref="S5.E7.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.E7.m1.2.2.1.1.1.1.1.cmml" xref="S5.E7.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S5.E7.m1.2.2.1.1.1.1.2.cmml" xref="S5.E7.m1.2.2.1.1.1.1.2">𝑎</ci><ci id="S5.E7.m1.2.2.1.1.1.1.3.cmml" xref="S5.E7.m1.2.2.1.1.1.1.3">𝑖</ci></apply><apply id="S5.E7.m1.3.3.2.2.2.2.cmml" xref="S5.E7.m1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.E7.m1.3.3.2.2.2.2.1.cmml" xref="S5.E7.m1.3.3.2.2.2.2">superscript</csymbol><apply id="S5.E7.m1.3.3.2.2.2.2.2.cmml" xref="S5.E7.m1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.E7.m1.3.3.2.2.2.2.2.1.cmml" xref="S5.E7.m1.3.3.2.2.2.2">subscript</csymbol><ci id="S5.E7.m1.3.3.2.2.2.2.2.2.cmml" xref="S5.E7.m1.3.3.2.2.2.2.2.2">𝑎</ci><apply id="S5.E7.m1.3.3.2.2.2.2.2.3.cmml" xref="S5.E7.m1.3.3.2.2.2.2.2.3"><minus id="S5.E7.m1.3.3.2.2.2.2.2.3.1.cmml" xref="S5.E7.m1.3.3.2.2.2.2.2.3"></minus><ci id="S5.E7.m1.3.3.2.2.2.2.2.3.2.cmml" xref="S5.E7.m1.3.3.2.2.2.2.2.3.2">𝑖</ci></apply></apply><ci id="S5.E7.m1.3.3.2.2.2.2.3.cmml" xref="S5.E7.m1.3.3.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7.m1.3c">\displaystyle\sum_{\begin{subarray}{c}k\leq K\\ t\in{\mathcal{T}}_{i}(k)\end{subarray}}U_{i}^{t}(a_{i},a_{-i}^{t})</annotation><annotation encoding="application/x-llamapun" id="S5.E7.m1.3d">∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_k ≤ italic_K end_CELL end_ROW start_ROW start_CELL italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sum_{\begin{subarray}{c}k\leq K\\ t\in{\mathcal{T}}_{i}(k)\end{subarray}}U_{i}^{t}(a_{i},\bar{a}_{-i}^{k})+2% \quantity{a_{-i}^{t}\neq\bar{a}_{-i}^{k}}" class="ltx_Math" display="inline" id="S5.E7.m2.4"><semantics id="S5.E7.m2.4a"><mrow id="S5.E7.m2.4.4" xref="S5.E7.m2.4.4.cmml"><mi id="S5.E7.m2.4.4.4" xref="S5.E7.m2.4.4.4.cmml"></mi><mo id="S5.E7.m2.4.4.3" xref="S5.E7.m2.4.4.3.cmml">≤</mo><mrow id="S5.E7.m2.4.4.2" xref="S5.E7.m2.4.4.2.cmml"><mrow id="S5.E7.m2.4.4.2.2" xref="S5.E7.m2.4.4.2.2.cmml"><mstyle displaystyle="true" id="S5.E7.m2.4.4.2.2.3" xref="S5.E7.m2.4.4.2.2.3.cmml"><munder id="S5.E7.m2.4.4.2.2.3a" xref="S5.E7.m2.4.4.2.2.3.cmml"><mo id="S5.E7.m2.4.4.2.2.3.2" movablelimits="false" xref="S5.E7.m2.4.4.2.2.3.2.cmml">∑</mo><mtable id="S5.E7.m2.1.1.1.1.1.1" rowspacing="0pt" xref="S5.E7.m2.1.1.1a.2.cmml"><mtr id="S5.E7.m2.1.1.1.1.1.1a" xref="S5.E7.m2.1.1.1a.2.cmml"><mtd id="S5.E7.m2.1.1.1.1.1.1b" xref="S5.E7.m2.1.1.1a.2.cmml"><mrow 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id="S5.E7.m2.4c">\displaystyle\leq\sum_{\begin{subarray}{c}k\leq K\\ t\in{\mathcal{T}}_{i}(k)\end{subarray}}U_{i}^{t}(a_{i},\bar{a}_{-i}^{k})+2% \quantity{a_{-i}^{t}\neq\bar{a}_{-i}^{k}}</annotation><annotation encoding="application/x-llamapun" id="S5.E7.m2.4d">≤ ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_k ≤ italic_K end_CELL end_ROW start_ROW start_CELL italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) + 2 { start_ARG italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ≠ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> <tbody id="S5.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_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq LK+R_{0}K+\hat{R}_{i}(T_{i}(k),\bot)+2B_{K}" class="ltx_Math" display="inline" id="S5.E8.m1.3"><semantics id="S5.E8.m1.3a"><mrow id="S5.E8.m1.3.3" xref="S5.E8.m1.3.3.cmml"><mi id="S5.E8.m1.3.3.3" xref="S5.E8.m1.3.3.3.cmml"></mi><mo id="S5.E8.m1.3.3.2" xref="S5.E8.m1.3.3.2.cmml">≤</mo><mrow id="S5.E8.m1.3.3.1" xref="S5.E8.m1.3.3.1.cmml"><mrow id="S5.E8.m1.3.3.1.3" xref="S5.E8.m1.3.3.1.3.cmml"><mi 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xref="S5.E9.m1.4.4.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.E9.m1.4.4.2.2.1.1.1.1.cmml" xref="S5.E9.m1.4.4.2.2.1.1.1">subscript</csymbol><ci id="S5.E9.m1.4.4.2.2.1.1.1.2.cmml" xref="S5.E9.m1.4.4.2.2.1.1.1.2">𝑎</ci><ci id="S5.E9.m1.4.4.2.2.1.1.1.3.cmml" xref="S5.E9.m1.4.4.2.2.1.1.1.3">𝑖</ci></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E9.m1.4c">\displaystyle\sum_{k=1}^{K}\underbrace{\frac{1}{L}\sum_{t\in{\mathcal{T}}_{i}(% k)}[U_{i}(a_{i},a_{-i}^{t})+P_{i}^{t}(a_{i})-1]}_{:=\varepsilon_{i}(k,a_{i})}</annotation><annotation encoding="application/x-llamapun" id="S5.E9.m1.4d">∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT under⏟ start_ARG divide start_ARG 1 end_ARG start_ARG italic_L end_ARG ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) end_POSTSUBSCRIPT [ italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - 1 ] end_ARG start_POSTSUBSCRIPT := italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{1}{L}\quantity(R_{0}K+3nC\sqrt{T})." class="ltx_Math" display="inline" id="S5.E9.m2.2"><semantics id="S5.E9.m2.2a"><mrow id="S5.E9.m2.2.2.1" xref="S5.E9.m2.2.2.1.1.cmml"><mrow id="S5.E9.m2.2.2.1.1" xref="S5.E9.m2.2.2.1.1.cmml"><mi id="S5.E9.m2.2.2.1.1.2" xref="S5.E9.m2.2.2.1.1.2.cmml"></mi><mo id="S5.E9.m2.2.2.1.1.1" 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xref="S5.E9.m2.1.1.1.1.1.3.5"></root><ci id="S5.E9.m2.1.1.1.1.1.3.5.2.cmml" xref="S5.E9.m2.1.1.1.1.1.3.5.2">𝑇</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E9.m2.2c">\displaystyle\leq\frac{1}{L}\quantity(R_{0}K+3nC\sqrt{T}).</annotation><annotation encoding="application/x-llamapun" id="S5.E9.m2.2d">≤ divide start_ARG 1 end_ARG start_ARG italic_L end_ARG ( start_ARG italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K + 3 italic_n italic_C square-root start_ARG italic_T end_ARG end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.4.p4.7">The error we need to bound is <math alttext="\norm{\varepsilon_{i}(k,\cdot)}_{\infty}" class="ltx_Math" display="inline" 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xref="S5.SS2.4.p4.5.m1.1.1.1.1.1"><times id="S5.SS2.4.p4.5.m1.1.1.1.1.1.3.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.3"></times><apply id="S5.SS2.4.p4.5.m1.1.1.1.1.1.4.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S5.SS2.4.p4.5.m1.1.1.1.1.1.4.1.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.4">subscript</csymbol><ci id="S5.SS2.4.p4.5.m1.1.1.1.1.1.4.2.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.4.2">𝜀</ci><ci id="S5.SS2.4.p4.5.m1.1.1.1.1.1.4.3.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.4.3">𝑖</ci></apply><interval closure="open" id="S5.SS2.4.p4.5.m1.1.1.1.1.1.5.1.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.5.2"><ci id="S5.SS2.4.p4.5.m1.1.1.1.1.1.1.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.1">𝑘</ci><ci id="S5.SS2.4.p4.5.m1.1.1.1.1.1.2.cmml" xref="S5.SS2.4.p4.5.m1.1.1.1.1.1.2">⋅</ci></interval></apply></apply><infinity id="S5.SS2.4.p4.5.m1.1.2.2.cmml" xref="S5.SS2.4.p4.5.m1.1.2.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.5.m1.1c">\norm{\varepsilon_{i}(k,\cdot)}_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.5.m1.1d">∥ start_ARG italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k , ⋅ ) end_ARG ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>. Since this holds for any <math alttext="a_{i}" class="ltx_Math" display="inline" id="S5.SS2.4.p4.6.m2.1"><semantics id="S5.SS2.4.p4.6.m2.1a"><msub id="S5.SS2.4.p4.6.m2.1.1" xref="S5.SS2.4.p4.6.m2.1.1.cmml"><mi id="S5.SS2.4.p4.6.m2.1.1.2" xref="S5.SS2.4.p4.6.m2.1.1.2.cmml">a</mi><mi id="S5.SS2.4.p4.6.m2.1.1.3" xref="S5.SS2.4.p4.6.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.6.m2.1b"><apply id="S5.SS2.4.p4.6.m2.1.1.cmml" xref="S5.SS2.4.p4.6.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.4.p4.6.m2.1.1.1.cmml" xref="S5.SS2.4.p4.6.m2.1.1">subscript</csymbol><ci id="S5.SS2.4.p4.6.m2.1.1.2.cmml" xref="S5.SS2.4.p4.6.m2.1.1.2">𝑎</ci><ci id="S5.SS2.4.p4.6.m2.1.1.3.cmml" xref="S5.SS2.4.p4.6.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.6.m2.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.6.m2.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\sum_{a_{i}}\varepsilon_{i}(k,\cdot)=0" class="ltx_Math" display="inline" id="S5.SS2.4.p4.7.m3.2"><semantics id="S5.SS2.4.p4.7.m3.2a"><mrow id="S5.SS2.4.p4.7.m3.2.3" xref="S5.SS2.4.p4.7.m3.2.3.cmml"><mrow id="S5.SS2.4.p4.7.m3.2.3.2" xref="S5.SS2.4.p4.7.m3.2.3.2.cmml"><msub id="S5.SS2.4.p4.7.m3.2.3.2.1" xref="S5.SS2.4.p4.7.m3.2.3.2.1.cmml"><mo id="S5.SS2.4.p4.7.m3.2.3.2.1.2" xref="S5.SS2.4.p4.7.m3.2.3.2.1.2.cmml">∑</mo><msub id="S5.SS2.4.p4.7.m3.2.3.2.1.3" xref="S5.SS2.4.p4.7.m3.2.3.2.1.3.cmml"><mi id="S5.SS2.4.p4.7.m3.2.3.2.1.3.2" xref="S5.SS2.4.p4.7.m3.2.3.2.1.3.2.cmml">a</mi><mi id="S5.SS2.4.p4.7.m3.2.3.2.1.3.3" xref="S5.SS2.4.p4.7.m3.2.3.2.1.3.3.cmml">i</mi></msub></msub><mrow id="S5.SS2.4.p4.7.m3.2.3.2.2" xref="S5.SS2.4.p4.7.m3.2.3.2.2.cmml"><msub id="S5.SS2.4.p4.7.m3.2.3.2.2.2" xref="S5.SS2.4.p4.7.m3.2.3.2.2.2.cmml"><mi id="S5.SS2.4.p4.7.m3.2.3.2.2.2.2" xref="S5.SS2.4.p4.7.m3.2.3.2.2.2.2.cmml">ε</mi><mi id="S5.SS2.4.p4.7.m3.2.3.2.2.2.3" xref="S5.SS2.4.p4.7.m3.2.3.2.2.2.3.cmml">i</mi></msub><mo id="S5.SS2.4.p4.7.m3.2.3.2.2.1" xref="S5.SS2.4.p4.7.m3.2.3.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.4.p4.7.m3.2.3.2.2.3.2" xref="S5.SS2.4.p4.7.m3.2.3.2.2.3.1.cmml"><mo id="S5.SS2.4.p4.7.m3.2.3.2.2.3.2.1" stretchy="false" xref="S5.SS2.4.p4.7.m3.2.3.2.2.3.1.cmml">(</mo><mi id="S5.SS2.4.p4.7.m3.1.1" xref="S5.SS2.4.p4.7.m3.1.1.cmml">k</mi><mo id="S5.SS2.4.p4.7.m3.2.3.2.2.3.2.2" rspace="0em" xref="S5.SS2.4.p4.7.m3.2.3.2.2.3.1.cmml">,</mo><mo id="S5.SS2.4.p4.7.m3.2.2" lspace="0em" rspace="0em" xref="S5.SS2.4.p4.7.m3.2.2.cmml">⋅</mo><mo id="S5.SS2.4.p4.7.m3.2.3.2.2.3.2.3" stretchy="false" xref="S5.SS2.4.p4.7.m3.2.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.SS2.4.p4.7.m3.2.3.1" xref="S5.SS2.4.p4.7.m3.2.3.1.cmml">=</mo><mn id="S5.SS2.4.p4.7.m3.2.3.3" xref="S5.SS2.4.p4.7.m3.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.7.m3.2b"><apply id="S5.SS2.4.p4.7.m3.2.3.cmml" xref="S5.SS2.4.p4.7.m3.2.3"><eq id="S5.SS2.4.p4.7.m3.2.3.1.cmml" xref="S5.SS2.4.p4.7.m3.2.3.1"></eq><apply id="S5.SS2.4.p4.7.m3.2.3.2.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2"><apply id="S5.SS2.4.p4.7.m3.2.3.2.1.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.1"><csymbol cd="ambiguous" id="S5.SS2.4.p4.7.m3.2.3.2.1.1.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.1">subscript</csymbol><sum id="S5.SS2.4.p4.7.m3.2.3.2.1.2.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.1.2"></sum><apply id="S5.SS2.4.p4.7.m3.2.3.2.1.3.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.1.3"><csymbol cd="ambiguous" id="S5.SS2.4.p4.7.m3.2.3.2.1.3.1.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.1.3">subscript</csymbol><ci id="S5.SS2.4.p4.7.m3.2.3.2.1.3.2.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.1.3.2">𝑎</ci><ci id="S5.SS2.4.p4.7.m3.2.3.2.1.3.3.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.1.3.3">𝑖</ci></apply></apply><apply id="S5.SS2.4.p4.7.m3.2.3.2.2.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.2"><times id="S5.SS2.4.p4.7.m3.2.3.2.2.1.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.2.1"></times><apply id="S5.SS2.4.p4.7.m3.2.3.2.2.2.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.4.p4.7.m3.2.3.2.2.2.1.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.2.2">subscript</csymbol><ci id="S5.SS2.4.p4.7.m3.2.3.2.2.2.2.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.2.2.2">𝜀</ci><ci id="S5.SS2.4.p4.7.m3.2.3.2.2.2.3.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.2.2.3">𝑖</ci></apply><interval closure="open" id="S5.SS2.4.p4.7.m3.2.3.2.2.3.1.cmml" xref="S5.SS2.4.p4.7.m3.2.3.2.2.3.2"><ci id="S5.SS2.4.p4.7.m3.1.1.cmml" xref="S5.SS2.4.p4.7.m3.1.1">𝑘</ci><ci id="S5.SS2.4.p4.7.m3.2.2.cmml" xref="S5.SS2.4.p4.7.m3.2.2">⋅</ci></interval></apply></apply><cn id="S5.SS2.4.p4.7.m3.2.3.3.cmml" type="integer" xref="S5.SS2.4.p4.7.m3.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.7.m3.2c">\sum_{a_{i}}\varepsilon_{i}(k,\cdot)=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.7.m3.2d">∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k , ⋅ ) = 0</annotation></semantics></math> by definition, it follows that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx6"> <tbody id="S5.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\norm{\sum_{k=1}^{K}\varepsilon_{i}(k,a_{i})}_{\infty}\leq\frac{m% }{L}\quantity(R_{0}K+3nC\sqrt{T})." class="ltx_Math" display="inline" id="S5.E10.m1.3"><semantics id="S5.E10.m1.3a"><mrow id="S5.E10.m1.3.3.1" xref="S5.E10.m1.3.3.1.1.cmml"><mrow id="S5.E10.m1.3.3.1.1" xref="S5.E10.m1.3.3.1.1.cmml"><msub id="S5.E10.m1.3.3.1.1.2" xref="S5.E10.m1.3.3.1.1.2.cmml"><mrow id="S5.E10.m1.1.1a.3" xref="S5.E10.m1.1.1a.2.cmml"><mo id="S5.E10.m1.1.1a.3.1" xref="S5.E10.m1.1.1a.2.1.cmml">‖</mo><mrow id="S5.E10.m1.1.1.1.1.1" 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id="S5.E10.m1.3d">∥ start_ARG ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≤ divide start_ARG italic_m end_ARG start_ARG italic_L end_ARG ( start_ARG italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K + 3 italic_n italic_C square-root start_ARG italic_T end_ARG end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.4.p4.11">By an inductive application of the triangle inequality, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx7"> <tbody id="S5.E11"><tr 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xref="S5.E11.m1.3.3.1.1.3.2.cmml"><mrow id="S5.E11.m1.3.3.1.1.3.2.2" xref="S5.E11.m1.3.3.1.1.3.2.2.cmml"><mi id="S5.E11.m1.3.3.1.1.3.2.2.2" xref="S5.E11.m1.3.3.1.1.3.2.2.2.cmml">k</mi><mo id="S5.E11.m1.3.3.1.1.3.2.2.1" xref="S5.E11.m1.3.3.1.1.3.2.2.1.cmml">⁢</mo><mi id="S5.E11.m1.3.3.1.1.3.2.2.3" xref="S5.E11.m1.3.3.1.1.3.2.2.3.cmml">m</mi></mrow><mi id="S5.E11.m1.3.3.1.1.3.2.3" xref="S5.E11.m1.3.3.1.1.3.2.3.cmml">L</mi></mfrac></mstyle><mo id="S5.E11.m1.3.3.1.1.3.1" xref="S5.E11.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S5.E11.m1.2.2a.3" xref="S5.E11.m1.2.2.1.1.1.cmml"><mo id="S5.E11.m1.2.2a.3.1" xref="S5.E11.m1.2.2.1.1.1.cmml">(</mo><mrow id="S5.E11.m1.2.2.1.1.1" xref="S5.E11.m1.2.2.1.1.1.cmml"><mrow id="S5.E11.m1.2.2.1.1.1.2" xref="S5.E11.m1.2.2.1.1.1.2.cmml"><msub id="S5.E11.m1.2.2.1.1.1.2.2" xref="S5.E11.m1.2.2.1.1.1.2.2.cmml"><mi id="S5.E11.m1.2.2.1.1.1.2.2.2" xref="S5.E11.m1.2.2.1.1.1.2.2.2.cmml">R</mi><mn id="S5.E11.m1.2.2.1.1.1.2.2.3" 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xref="S5.E11.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E11.m1.3.3.1.2" lspace="0em" xref="S5.E11.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E11.m1.3b"><apply id="S5.E11.m1.3.3.1.1.cmml" xref="S5.E11.m1.3.3.1"><leq id="S5.E11.m1.3.3.1.1.1.cmml" xref="S5.E11.m1.3.3.1.1.1"></leq><apply id="S5.E11.m1.3.3.1.1.2.cmml" xref="S5.E11.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S5.E11.m1.3.3.1.1.2.1.cmml" xref="S5.E11.m1.3.3.1.1.2">subscript</csymbol><apply id="S5.E11.m1.1.1a.2.cmml" xref="S5.E11.m1.1.1a.3"><csymbol cd="latexml" id="S5.E11.m1.1.1a.2.1.cmml" xref="S5.E11.m1.1.1a.3.1">norm</csymbol><apply id="S5.E11.m1.1.1.1.1.1.cmml" xref="S5.E11.m1.1.1.1.1.1"><times id="S5.E11.m1.1.1.1.1.1.3.cmml" xref="S5.E11.m1.1.1.1.1.1.3"></times><apply id="S5.E11.m1.1.1.1.1.1.4.cmml" xref="S5.E11.m1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S5.E11.m1.1.1.1.1.1.4.1.cmml" xref="S5.E11.m1.1.1.1.1.1.4">subscript</csymbol><ci id="S5.E11.m1.1.1.1.1.1.4.2.cmml" 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id="S5.E11.m1.2.2.1.1.1.3.1.cmml" xref="S5.E11.m1.2.2.1.1.1.3.1"></times><cn id="S5.E11.m1.2.2.1.1.1.3.2.cmml" type="integer" xref="S5.E11.m1.2.2.1.1.1.3.2">3</cn><ci id="S5.E11.m1.2.2.1.1.1.3.3.cmml" xref="S5.E11.m1.2.2.1.1.1.3.3">𝑛</ci><ci id="S5.E11.m1.2.2.1.1.1.3.4.cmml" xref="S5.E11.m1.2.2.1.1.1.3.4">𝐶</ci><apply id="S5.E11.m1.2.2.1.1.1.3.5.cmml" xref="S5.E11.m1.2.2.1.1.1.3.5"><root id="S5.E11.m1.2.2.1.1.1.3.5a.cmml" xref="S5.E11.m1.2.2.1.1.1.3.5"></root><ci id="S5.E11.m1.2.2.1.1.1.3.5.2.cmml" xref="S5.E11.m1.2.2.1.1.1.3.5.2">𝑇</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E11.m1.3c">\displaystyle\norm{\varepsilon_{i}(k,\cdot)}_{\infty}\leq\frac{km}{L}\quantity% (R_{0}M+3nC\sqrt{T}).</annotation><annotation encoding="application/x-llamapun" id="S5.E11.m1.3d">∥ start_ARG italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k , ⋅ ) end_ARG ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≤ divide start_ARG italic_k italic_m end_ARG start_ARG italic_L end_ARG ( start_ARG italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_M + 3 italic_n italic_C square-root start_ARG italic_T end_ARG end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.4.p4.9">Finally, substituting <math alttext="R_{0}\lesssim\sqrt{mL}" class="ltx_Math" display="inline" id="S5.SS2.4.p4.8.m1.1"><semantics id="S5.SS2.4.p4.8.m1.1a"><mrow id="S5.SS2.4.p4.8.m1.1.1" xref="S5.SS2.4.p4.8.m1.1.1.cmml"><msub id="S5.SS2.4.p4.8.m1.1.1.2" xref="S5.SS2.4.p4.8.m1.1.1.2.cmml"><mi id="S5.SS2.4.p4.8.m1.1.1.2.2" xref="S5.SS2.4.p4.8.m1.1.1.2.2.cmml">R</mi><mn id="S5.SS2.4.p4.8.m1.1.1.2.3" xref="S5.SS2.4.p4.8.m1.1.1.2.3.cmml">0</mn></msub><mo id="S5.SS2.4.p4.8.m1.1.1.1" 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type="integer" xref="S5.SS2.4.p4.8.m1.1.1.2.3">0</cn></apply><apply id="S5.SS2.4.p4.8.m1.1.1.3.cmml" xref="S5.SS2.4.p4.8.m1.1.1.3"><root id="S5.SS2.4.p4.8.m1.1.1.3a.cmml" xref="S5.SS2.4.p4.8.m1.1.1.3"></root><apply id="S5.SS2.4.p4.8.m1.1.1.3.2.cmml" xref="S5.SS2.4.p4.8.m1.1.1.3.2"><times id="S5.SS2.4.p4.8.m1.1.1.3.2.1.cmml" xref="S5.SS2.4.p4.8.m1.1.1.3.2.1"></times><ci id="S5.SS2.4.p4.8.m1.1.1.3.2.2.cmml" xref="S5.SS2.4.p4.8.m1.1.1.3.2.2">𝑚</ci><ci id="S5.SS2.4.p4.8.m1.1.1.3.2.3.cmml" xref="S5.SS2.4.p4.8.m1.1.1.3.2.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.8.m1.1c">R_{0}\lesssim\sqrt{mL}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.8.m1.1d">italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≲ square-root start_ARG italic_m italic_L end_ARG</annotation></semantics></math> and <math alttext="T\leq nML" class="ltx_Math" display="inline" id="S5.SS2.4.p4.9.m2.1"><semantics id="S5.SS2.4.p4.9.m2.1a"><mrow id="S5.SS2.4.p4.9.m2.1.1" xref="S5.SS2.4.p4.9.m2.1.1.cmml"><mi id="S5.SS2.4.p4.9.m2.1.1.2" xref="S5.SS2.4.p4.9.m2.1.1.2.cmml">T</mi><mo id="S5.SS2.4.p4.9.m2.1.1.1" xref="S5.SS2.4.p4.9.m2.1.1.1.cmml">≤</mo><mrow id="S5.SS2.4.p4.9.m2.1.1.3" xref="S5.SS2.4.p4.9.m2.1.1.3.cmml"><mi id="S5.SS2.4.p4.9.m2.1.1.3.2" xref="S5.SS2.4.p4.9.m2.1.1.3.2.cmml">n</mi><mo id="S5.SS2.4.p4.9.m2.1.1.3.1" xref="S5.SS2.4.p4.9.m2.1.1.3.1.cmml">⁢</mo><mi id="S5.SS2.4.p4.9.m2.1.1.3.3" xref="S5.SS2.4.p4.9.m2.1.1.3.3.cmml">M</mi><mo id="S5.SS2.4.p4.9.m2.1.1.3.1a" xref="S5.SS2.4.p4.9.m2.1.1.3.1.cmml">⁢</mo><mi id="S5.SS2.4.p4.9.m2.1.1.3.4" xref="S5.SS2.4.p4.9.m2.1.1.3.4.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.9.m2.1b"><apply id="S5.SS2.4.p4.9.m2.1.1.cmml" xref="S5.SS2.4.p4.9.m2.1.1"><leq id="S5.SS2.4.p4.9.m2.1.1.1.cmml" xref="S5.SS2.4.p4.9.m2.1.1.1"></leq><ci id="S5.SS2.4.p4.9.m2.1.1.2.cmml" xref="S5.SS2.4.p4.9.m2.1.1.2">𝑇</ci><apply id="S5.SS2.4.p4.9.m2.1.1.3.cmml" xref="S5.SS2.4.p4.9.m2.1.1.3"><times id="S5.SS2.4.p4.9.m2.1.1.3.1.cmml" xref="S5.SS2.4.p4.9.m2.1.1.3.1"></times><ci id="S5.SS2.4.p4.9.m2.1.1.3.2.cmml" xref="S5.SS2.4.p4.9.m2.1.1.3.2">𝑛</ci><ci id="S5.SS2.4.p4.9.m2.1.1.3.3.cmml" xref="S5.SS2.4.p4.9.m2.1.1.3.3">𝑀</ci><ci id="S5.SS2.4.p4.9.m2.1.1.3.4.cmml" xref="S5.SS2.4.p4.9.m2.1.1.3.4">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.9.m2.1c">T\leq nML</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.9.m2.1d">italic_T ≤ italic_n italic_M italic_L</annotation></semantics></math>, we arrive at</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx8"> <tbody id="S5.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\norm{\varepsilon_{i}(k,\cdot)}_{\infty}\lesssim\frac{Mm}{\sqrt{L% }}\quantity(\sqrt{m}M+nC\sqrt{nM})" class="ltx_Math" display="inline" id="S5.E12.m1.2"><semantics id="S5.E12.m1.2a"><mrow id="S5.E12.m1.2.3" xref="S5.E12.m1.2.3.cmml"><msub id="S5.E12.m1.2.3.2" xref="S5.E12.m1.2.3.2.cmml"><mrow id="S5.E12.m1.1.1a.3" xref="S5.E12.m1.1.1a.2.cmml"><mo id="S5.E12.m1.1.1a.3.1" xref="S5.E12.m1.1.1a.2.1.cmml">‖</mo><mrow id="S5.E12.m1.1.1.1.1.1" xref="S5.E12.m1.1.1.1.1.1.cmml"><msub id="S5.E12.m1.1.1.1.1.1.4" xref="S5.E12.m1.1.1.1.1.1.4.cmml"><mi id="S5.E12.m1.1.1.1.1.1.4.2" xref="S5.E12.m1.1.1.1.1.1.4.2.cmml">ε</mi><mi id="S5.E12.m1.1.1.1.1.1.4.3" xref="S5.E12.m1.1.1.1.1.1.4.3.cmml">i</mi></msub><mo id="S5.E12.m1.1.1.1.1.1.3" xref="S5.E12.m1.1.1.1.1.1.3.cmml">⁢</mo><mrow id="S5.E12.m1.1.1.1.1.1.5.2" xref="S5.E12.m1.1.1.1.1.1.5.1.cmml"><mo id="S5.E12.m1.1.1.1.1.1.5.2.1" stretchy="false" xref="S5.E12.m1.1.1.1.1.1.5.1.cmml">(</mo><mi id="S5.E12.m1.1.1.1.1.1.1" xref="S5.E12.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S5.E12.m1.1.1.1.1.1.5.2.2" rspace="0em" xref="S5.E12.m1.1.1.1.1.1.5.1.cmml">,</mo><mo id="S5.E12.m1.1.1.1.1.1.2" lspace="0em" rspace="0em" xref="S5.E12.m1.1.1.1.1.1.2.cmml">⋅</mo><mo id="S5.E12.m1.1.1.1.1.1.5.2.3" stretchy="false" xref="S5.E12.m1.1.1.1.1.1.5.1.cmml">)</mo></mrow></mrow><mo id="S5.E12.m1.1.1a.3.2" xref="S5.E12.m1.1.1a.2.1.cmml">‖</mo></mrow><mi id="S5.E12.m1.2.3.2.2" mathvariant="normal" xref="S5.E12.m1.2.3.2.2.cmml">∞</mi></msub><mo id="S5.E12.m1.2.3.1" xref="S5.E12.m1.2.3.1.cmml">≲</mo><mrow id="S5.E12.m1.2.3.3" xref="S5.E12.m1.2.3.3.cmml"><mstyle displaystyle="true" id="S5.E12.m1.2.3.3.2" xref="S5.E12.m1.2.3.3.2.cmml"><mfrac id="S5.E12.m1.2.3.3.2a" xref="S5.E12.m1.2.3.3.2.cmml"><mrow id="S5.E12.m1.2.3.3.2.2" xref="S5.E12.m1.2.3.3.2.2.cmml"><mi id="S5.E12.m1.2.3.3.2.2.2" xref="S5.E12.m1.2.3.3.2.2.2.cmml">M</mi><mo id="S5.E12.m1.2.3.3.2.2.1" xref="S5.E12.m1.2.3.3.2.2.1.cmml">⁢</mo><mi id="S5.E12.m1.2.3.3.2.2.3" xref="S5.E12.m1.2.3.3.2.2.3.cmml">m</mi></mrow><msqrt id="S5.E12.m1.2.3.3.2.3" xref="S5.E12.m1.2.3.3.2.3.cmml"><mi id="S5.E12.m1.2.3.3.2.3.2" xref="S5.E12.m1.2.3.3.2.3.2.cmml">L</mi></msqrt></mfrac></mstyle><mo id="S5.E12.m1.2.3.3.1" xref="S5.E12.m1.2.3.3.1.cmml">⁢</mo><mrow id="S5.E12.m1.2.2a.3" xref="S5.E12.m1.2.2.1.1.1.cmml"><mo id="S5.E12.m1.2.2a.3.1" xref="S5.E12.m1.2.2.1.1.1.cmml">(</mo><mrow id="S5.E12.m1.2.2.1.1.1" xref="S5.E12.m1.2.2.1.1.1.cmml"><mrow id="S5.E12.m1.2.2.1.1.1.2" xref="S5.E12.m1.2.2.1.1.1.2.cmml"><msqrt id="S5.E12.m1.2.2.1.1.1.2.2" xref="S5.E12.m1.2.2.1.1.1.2.2.cmml"><mi id="S5.E12.m1.2.2.1.1.1.2.2.2" xref="S5.E12.m1.2.2.1.1.1.2.2.2.cmml">m</mi></msqrt><mo id="S5.E12.m1.2.2.1.1.1.2.1" xref="S5.E12.m1.2.2.1.1.1.2.1.cmml">⁢</mo><mi id="S5.E12.m1.2.2.1.1.1.2.3" xref="S5.E12.m1.2.2.1.1.1.2.3.cmml">M</mi></mrow><mo id="S5.E12.m1.2.2.1.1.1.1" xref="S5.E12.m1.2.2.1.1.1.1.cmml">+</mo><mrow id="S5.E12.m1.2.2.1.1.1.3" xref="S5.E12.m1.2.2.1.1.1.3.cmml"><mi id="S5.E12.m1.2.2.1.1.1.3.2" xref="S5.E12.m1.2.2.1.1.1.3.2.cmml">n</mi><mo id="S5.E12.m1.2.2.1.1.1.3.1" xref="S5.E12.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="S5.E12.m1.2.2.1.1.1.3.3" xref="S5.E12.m1.2.2.1.1.1.3.3.cmml">C</mi><mo id="S5.E12.m1.2.2.1.1.1.3.1a" xref="S5.E12.m1.2.2.1.1.1.3.1.cmml">⁢</mo><msqrt id="S5.E12.m1.2.2.1.1.1.3.4" xref="S5.E12.m1.2.2.1.1.1.3.4.cmml"><mrow id="S5.E12.m1.2.2.1.1.1.3.4.2" xref="S5.E12.m1.2.2.1.1.1.3.4.2.cmml"><mi id="S5.E12.m1.2.2.1.1.1.3.4.2.2" xref="S5.E12.m1.2.2.1.1.1.3.4.2.2.cmml">n</mi><mo id="S5.E12.m1.2.2.1.1.1.3.4.2.1" xref="S5.E12.m1.2.2.1.1.1.3.4.2.1.cmml">⁢</mo><mi id="S5.E12.m1.2.2.1.1.1.3.4.2.3" xref="S5.E12.m1.2.2.1.1.1.3.4.2.3.cmml">M</mi></mrow></msqrt></mrow></mrow><mo id="S5.E12.m1.2.2a.3.2" xref="S5.E12.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E12.m1.2b"><apply id="S5.E12.m1.2.3.cmml" xref="S5.E12.m1.2.3"><csymbol cd="latexml" id="S5.E12.m1.2.3.1.cmml" xref="S5.E12.m1.2.3.1">less-than-or-similar-to</csymbol><apply id="S5.E12.m1.2.3.2.cmml" xref="S5.E12.m1.2.3.2"><csymbol cd="ambiguous" id="S5.E12.m1.2.3.2.1.cmml" xref="S5.E12.m1.2.3.2">subscript</csymbol><apply id="S5.E12.m1.1.1a.2.cmml" xref="S5.E12.m1.1.1a.3"><csymbol cd="latexml" id="S5.E12.m1.1.1a.2.1.cmml" xref="S5.E12.m1.1.1a.3.1">norm</csymbol><apply id="S5.E12.m1.1.1.1.1.1.cmml" xref="S5.E12.m1.1.1.1.1.1"><times id="S5.E12.m1.1.1.1.1.1.3.cmml" xref="S5.E12.m1.1.1.1.1.1.3"></times><apply id="S5.E12.m1.1.1.1.1.1.4.cmml" xref="S5.E12.m1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S5.E12.m1.1.1.1.1.1.4.1.cmml" xref="S5.E12.m1.1.1.1.1.1.4">subscript</csymbol><ci id="S5.E12.m1.1.1.1.1.1.4.2.cmml" xref="S5.E12.m1.1.1.1.1.1.4.2">𝜀</ci><ci id="S5.E12.m1.1.1.1.1.1.4.3.cmml" xref="S5.E12.m1.1.1.1.1.1.4.3">𝑖</ci></apply><interval closure="open" id="S5.E12.m1.1.1.1.1.1.5.1.cmml" xref="S5.E12.m1.1.1.1.1.1.5.2"><ci id="S5.E12.m1.1.1.1.1.1.1.cmml" xref="S5.E12.m1.1.1.1.1.1.1">𝑘</ci><ci id="S5.E12.m1.1.1.1.1.1.2.cmml" xref="S5.E12.m1.1.1.1.1.1.2">⋅</ci></interval></apply></apply><infinity id="S5.E12.m1.2.3.2.2.cmml" xref="S5.E12.m1.2.3.2.2"></infinity></apply><apply id="S5.E12.m1.2.3.3.cmml" xref="S5.E12.m1.2.3.3"><times id="S5.E12.m1.2.3.3.1.cmml" xref="S5.E12.m1.2.3.3.1"></times><apply id="S5.E12.m1.2.3.3.2.cmml" xref="S5.E12.m1.2.3.3.2"><divide id="S5.E12.m1.2.3.3.2.1.cmml" xref="S5.E12.m1.2.3.3.2"></divide><apply id="S5.E12.m1.2.3.3.2.2.cmml" xref="S5.E12.m1.2.3.3.2.2"><times id="S5.E12.m1.2.3.3.2.2.1.cmml" xref="S5.E12.m1.2.3.3.2.2.1"></times><ci id="S5.E12.m1.2.3.3.2.2.2.cmml" xref="S5.E12.m1.2.3.3.2.2.2">𝑀</ci><ci id="S5.E12.m1.2.3.3.2.2.3.cmml" xref="S5.E12.m1.2.3.3.2.2.3">𝑚</ci></apply><apply id="S5.E12.m1.2.3.3.2.3.cmml" xref="S5.E12.m1.2.3.3.2.3"><root id="S5.E12.m1.2.3.3.2.3a.cmml" xref="S5.E12.m1.2.3.3.2.3"></root><ci id="S5.E12.m1.2.3.3.2.3.2.cmml" xref="S5.E12.m1.2.3.3.2.3.2">𝐿</ci></apply></apply><apply id="S5.E12.m1.2.2.1.1.1.cmml" xref="S5.E12.m1.2.2a.3"><plus id="S5.E12.m1.2.2.1.1.1.1.cmml" xref="S5.E12.m1.2.2.1.1.1.1"></plus><apply id="S5.E12.m1.2.2.1.1.1.2.cmml" xref="S5.E12.m1.2.2.1.1.1.2"><times id="S5.E12.m1.2.2.1.1.1.2.1.cmml" xref="S5.E12.m1.2.2.1.1.1.2.1"></times><apply id="S5.E12.m1.2.2.1.1.1.2.2.cmml" xref="S5.E12.m1.2.2.1.1.1.2.2"><root id="S5.E12.m1.2.2.1.1.1.2.2a.cmml" xref="S5.E12.m1.2.2.1.1.1.2.2"></root><ci id="S5.E12.m1.2.2.1.1.1.2.2.2.cmml" xref="S5.E12.m1.2.2.1.1.1.2.2.2">𝑚</ci></apply><ci id="S5.E12.m1.2.2.1.1.1.2.3.cmml" xref="S5.E12.m1.2.2.1.1.1.2.3">𝑀</ci></apply><apply id="S5.E12.m1.2.2.1.1.1.3.cmml" xref="S5.E12.m1.2.2.1.1.1.3"><times id="S5.E12.m1.2.2.1.1.1.3.1.cmml" xref="S5.E12.m1.2.2.1.1.1.3.1"></times><ci id="S5.E12.m1.2.2.1.1.1.3.2.cmml" xref="S5.E12.m1.2.2.1.1.1.3.2">𝑛</ci><ci id="S5.E12.m1.2.2.1.1.1.3.3.cmml" xref="S5.E12.m1.2.2.1.1.1.3.3">𝐶</ci><apply id="S5.E12.m1.2.2.1.1.1.3.4.cmml" xref="S5.E12.m1.2.2.1.1.1.3.4"><root id="S5.E12.m1.2.2.1.1.1.3.4a.cmml" xref="S5.E12.m1.2.2.1.1.1.3.4"></root><apply id="S5.E12.m1.2.2.1.1.1.3.4.2.cmml" xref="S5.E12.m1.2.2.1.1.1.3.4.2"><times id="S5.E12.m1.2.2.1.1.1.3.4.2.1.cmml" xref="S5.E12.m1.2.2.1.1.1.3.4.2.1"></times><ci id="S5.E12.m1.2.2.1.1.1.3.4.2.2.cmml" xref="S5.E12.m1.2.2.1.1.1.3.4.2.2">𝑛</ci><ci id="S5.E12.m1.2.2.1.1.1.3.4.2.3.cmml" xref="S5.E12.m1.2.2.1.1.1.3.4.2.3">𝑀</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E12.m1.2c">\displaystyle\norm{\varepsilon_{i}(k,\cdot)}_{\infty}\lesssim\frac{Mm}{\sqrt{L% }}\quantity(\sqrt{m}M+nC\sqrt{nM})</annotation><annotation encoding="application/x-llamapun" id="S5.E12.m1.2d">∥ start_ARG italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k , ⋅ ) end_ARG ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≲ divide start_ARG italic_M italic_m end_ARG start_ARG square-root start_ARG italic_L end_ARG end_ARG ( start_ARG square-root start_ARG italic_m end_ARG italic_M + italic_n italic_C square-root start_ARG italic_n italic_M end_ARG end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.4.p4.10">upon which taking <math alttext="L\leq\poly(M)/\varepsilon^{2}" class="ltx_Math" display="inline" id="S5.SS2.4.p4.10.m1.1"><semantics id="S5.SS2.4.p4.10.m1.1a"><mrow id="S5.SS2.4.p4.10.m1.1.2" xref="S5.SS2.4.p4.10.m1.1.2.cmml"><mi id="S5.SS2.4.p4.10.m1.1.2.2" xref="S5.SS2.4.p4.10.m1.1.2.2.cmml">L</mi><mo id="S5.SS2.4.p4.10.m1.1.2.1" xref="S5.SS2.4.p4.10.m1.1.2.1.cmml">≤</mo><mrow id="S5.SS2.4.p4.10.m1.1.2.3" xref="S5.SS2.4.p4.10.m1.1.2.3.cmml"><mrow id="S5.SS2.4.p4.10.m1.1.2.3.2" xref="S5.SS2.4.p4.10.m1.1.2.3.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S5.SS2.4.p4.10.m1.1.2.3.2.2" xref="S5.SS2.4.p4.10.m1.1.2.3.2.2b.cmml"><mtext id="S5.SS2.4.p4.10.m1.1.2.3.2.2a" xref="S5.SS2.4.p4.10.m1.1.2.3.2.2b.cmml">\poly</mtext></merror><mo id="S5.SS2.4.p4.10.m1.1.2.3.2.1" xref="S5.SS2.4.p4.10.m1.1.2.3.2.1.cmml">⁢</mo><mrow id="S5.SS2.4.p4.10.m1.1.2.3.2.3.2" xref="S5.SS2.4.p4.10.m1.1.2.3.2.cmml"><mo id="S5.SS2.4.p4.10.m1.1.2.3.2.3.2.1" stretchy="false" xref="S5.SS2.4.p4.10.m1.1.2.3.2.cmml">(</mo><mi id="S5.SS2.4.p4.10.m1.1.1" xref="S5.SS2.4.p4.10.m1.1.1.cmml">M</mi><mo id="S5.SS2.4.p4.10.m1.1.2.3.2.3.2.2" stretchy="false" xref="S5.SS2.4.p4.10.m1.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.4.p4.10.m1.1.2.3.1" xref="S5.SS2.4.p4.10.m1.1.2.3.1.cmml">/</mo><msup id="S5.SS2.4.p4.10.m1.1.2.3.3" xref="S5.SS2.4.p4.10.m1.1.2.3.3.cmml"><mi id="S5.SS2.4.p4.10.m1.1.2.3.3.2" xref="S5.SS2.4.p4.10.m1.1.2.3.3.2.cmml">ε</mi><mn id="S5.SS2.4.p4.10.m1.1.2.3.3.3" xref="S5.SS2.4.p4.10.m1.1.2.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.4.p4.10.m1.1b"><apply id="S5.SS2.4.p4.10.m1.1.2.cmml" xref="S5.SS2.4.p4.10.m1.1.2"><leq id="S5.SS2.4.p4.10.m1.1.2.1.cmml" xref="S5.SS2.4.p4.10.m1.1.2.1"></leq><ci id="S5.SS2.4.p4.10.m1.1.2.2.cmml" xref="S5.SS2.4.p4.10.m1.1.2.2">𝐿</ci><apply id="S5.SS2.4.p4.10.m1.1.2.3.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3"><divide id="S5.SS2.4.p4.10.m1.1.2.3.1.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.1"></divide><apply id="S5.SS2.4.p4.10.m1.1.2.3.2.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.2"><times id="S5.SS2.4.p4.10.m1.1.2.3.2.1.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.2.1"></times><ci id="S5.SS2.4.p4.10.m1.1.2.3.2.2b.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.2.2"><merror class="ltx_ERROR undefined undefined" id="S5.SS2.4.p4.10.m1.1.2.3.2.2.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.2.2"><mtext id="S5.SS2.4.p4.10.m1.1.2.3.2.2a.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.2.2">\poly</mtext></merror></ci><ci id="S5.SS2.4.p4.10.m1.1.1.cmml" xref="S5.SS2.4.p4.10.m1.1.1">𝑀</ci></apply><apply id="S5.SS2.4.p4.10.m1.1.2.3.3.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.4.p4.10.m1.1.2.3.3.1.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.3">superscript</csymbol><ci id="S5.SS2.4.p4.10.m1.1.2.3.3.2.cmml" xref="S5.SS2.4.p4.10.m1.1.2.3.3.2">𝜀</ci><cn id="S5.SS2.4.p4.10.m1.1.2.3.3.3.cmml" type="integer" xref="S5.SS2.4.p4.10.m1.1.2.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.4.p4.10.m1.1c">L\leq\poly(M)/\varepsilon^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.4.p4.10.m1.1d">italic_L ≤ ( italic_M ) / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is enough to complete the proof. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.8">Signals are vital to this analysis. Without them, it would be possible for players to incur large <span class="ltx_text ltx_font_italic" id="S5.SS2.p2.8.1">negative</span> regret, which harms the learning process because it allows the players to “delay” the learning until their regrets once again become non-negative. For example, if we were to execute our algorithm without signals, then by the time <math alttext="\underaccent{\bar}{T}_{n}(0)" 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.m1.1.2" xref="S5.SS2.p2.1.m1.1.2.cmml"><msub id="S5.SS2.p2.1.m1.1.2.2" xref="S5.SS2.p2.1.m1.1.2.2.cmml"><munder accentunder="true" id="S5.SS2.p2.1.m1.1.2.2.2" xref="S5.SS2.p2.1.m1.1.2.2.2.cmml"><mi id="S5.SS2.p2.1.m1.1.2.2.2.2" xref="S5.SS2.p2.1.m1.1.2.2.2.2.cmml">T</mi><mo id="S5.SS2.p2.1.m1.1.2.2.2.1" xref="S5.SS2.p2.1.m1.1.2.2.2.1.cmml">¯</mo></munder><mi id="S5.SS2.p2.1.m1.1.2.2.3" xref="S5.SS2.p2.1.m1.1.2.2.3.cmml">n</mi></msub><mo id="S5.SS2.p2.1.m1.1.2.1" xref="S5.SS2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.p2.1.m1.1.2.3.2" xref="S5.SS2.p2.1.m1.1.2.cmml"><mo id="S5.SS2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S5.SS2.p2.1.m1.1.2.cmml">(</mo><mn id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml">0</mn><mo id="S5.SS2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S5.SS2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.1.m1.1b"><apply id="S5.SS2.p2.1.m1.1.2.cmml" xref="S5.SS2.p2.1.m1.1.2"><times id="S5.SS2.p2.1.m1.1.2.1.cmml" xref="S5.SS2.p2.1.m1.1.2.1"></times><apply id="S5.SS2.p2.1.m1.1.2.2.cmml" xref="S5.SS2.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.p2.1.m1.1.2.2.1.cmml" xref="S5.SS2.p2.1.m1.1.2.2">subscript</csymbol><apply id="S5.SS2.p2.1.m1.1.2.2.2.cmml" xref="S5.SS2.p2.1.m1.1.2.2.2"><ci id="S5.SS2.p2.1.m1.1.2.2.2.1.cmml" xref="S5.SS2.p2.1.m1.1.2.2.2.1">¯</ci><ci id="S5.SS2.p2.1.m1.1.2.2.2.2.cmml" xref="S5.SS2.p2.1.m1.1.2.2.2.2">𝑇</ci></apply><ci id="S5.SS2.p2.1.m1.1.2.2.3.cmml" xref="S5.SS2.p2.1.m1.1.2.2.3">𝑛</ci></apply><cn id="S5.SS2.p2.1.m1.1.1.cmml" type="integer" xref="S5.SS2.p2.1.m1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">\underaccent{\bar}{T}_{n}(0)</annotation><annotation 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encoding="application/x-tex" id="S5.SS2.p2.4.m4.2c">\Omega(\underaccent{\bar}{T}_{n}(0))</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.4.m4.2d">roman_Ω ( under¯ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) )</annotation></semantics></math> regret for <span class="ltx_text ltx_font_italic" id="S5.SS2.p2.8.2">every</span> action, making it impossible to say anything about how agent <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.p2.5.m5.1"><semantics id="S5.SS2.p2.5.m5.1a"><mi id="S5.SS2.p2.5.m5.1.1" xref="S5.SS2.p2.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.5.m5.1b"><ci id="S5.SS2.p2.5.m5.1.1.cmml" xref="S5.SS2.p2.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.5.m5.1d">italic_n</annotation></semantics></math> will act for the next <math 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xref="S5.SS2.p2.6.m6.2.2.1.1.1.2.3.cmml">n</mi></msub><mo id="S5.SS2.p2.6.m6.2.2.1.1.1.1" xref="S5.SS2.p2.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS2.p2.6.m6.2.2.1.1.1.3.2" xref="S5.SS2.p2.6.m6.2.2.1.1.1.cmml"><mo id="S5.SS2.p2.6.m6.2.2.1.1.1.3.2.1" stretchy="false" xref="S5.SS2.p2.6.m6.2.2.1.1.1.cmml">(</mo><mn id="S5.SS2.p2.6.m6.1.1" xref="S5.SS2.p2.6.m6.1.1.cmml">0</mn><mo id="S5.SS2.p2.6.m6.2.2.1.1.1.3.2.2" stretchy="false" xref="S5.SS2.p2.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p2.6.m6.2.2.1.1.3" stretchy="false" xref="S5.SS2.p2.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.6.m6.2b"><apply id="S5.SS2.p2.6.m6.2.2.cmml" xref="S5.SS2.p2.6.m6.2.2"><times id="S5.SS2.p2.6.m6.2.2.2.cmml" xref="S5.SS2.p2.6.m6.2.2.2"></times><ci id="S5.SS2.p2.6.m6.2.2.3.cmml" xref="S5.SS2.p2.6.m6.2.2.3">Ω</ci><apply id="S5.SS2.p2.6.m6.2.2.1.1.1.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1"><times id="S5.SS2.p2.6.m6.2.2.1.1.1.1.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1.1.1"></times><apply id="S5.SS2.p2.6.m6.2.2.1.1.1.2.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.p2.6.m6.2.2.1.1.1.2.1.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1.1.2">subscript</csymbol><apply id="S5.SS2.p2.6.m6.2.2.1.1.1.2.2.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1.1.2.2"><ci id="S5.SS2.p2.6.m6.2.2.1.1.1.2.2.1.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1.1.2.2.1">¯</ci><ci id="S5.SS2.p2.6.m6.2.2.1.1.1.2.2.2.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1.1.2.2.2">𝑇</ci></apply><ci id="S5.SS2.p2.6.m6.2.2.1.1.1.2.3.cmml" xref="S5.SS2.p2.6.m6.2.2.1.1.1.2.3">𝑛</ci></apply><cn id="S5.SS2.p2.6.m6.1.1.cmml" type="integer" xref="S5.SS2.p2.6.m6.1.1">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.6.m6.2c">\Omega(\underaccent{\bar}{T}_{n}(0))</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.6.m6.2d">roman_Ω ( under¯ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) )</annotation></semantics></math> rounds. Using signals allows us to separate out the regret of agent <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.p2.7.m7.1"><semantics id="S5.SS2.p2.7.m7.1a"><mi id="S5.SS2.p2.7.m7.1.1" xref="S5.SS2.p2.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.7.m7.1b"><ci id="S5.SS2.p2.7.m7.1.1.cmml" xref="S5.SS2.p2.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.7.m7.1d">italic_n</annotation></semantics></math> in previous rounds from the regret of agent <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.p2.8.m8.1"><semantics id="S5.SS2.p2.8.m8.1a"><mi id="S5.SS2.p2.8.m8.1.1" xref="S5.SS2.p2.8.m8.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.8.m8.1b"><ci id="S5.SS2.p2.8.m8.1.1.cmml" xref="S5.SS2.p2.8.m8.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.8.m8.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.8.m8.1d">italic_n</annotation></semantics></math> when its own utility function is being learned.</p> </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>Lower bound</h3> <div class="ltx_para" id="S5.SS3.p1"> <p class="ltx_p" id="S5.SS3.p1.1">We now turn to lower bounds. In particular, we show a lower bound that matches <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.Thmtheorem4" title="Theorem 5.4. ‣ 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.4</span></a> up to the exponent on <math alttext="M" 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">M</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">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.1.m1.1d">italic_M</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 5.5</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem5.p1"> <p class="ltx_p" id="S5.Thmtheorem5.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem5.p1.2.2">In the no-regret model, any algorithm that <math alttext="\varepsilon" 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">ε</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">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.1.1.m1.1d">italic_ε</annotation></semantics></math>-learns a single-agent game must take at least <math alttext="\max\{\tilde{\Omega}(M)\cdot\log(1/\varepsilon),C^{2}/\varepsilon^{2}\}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.2.2.m2.6"><semantics id="S5.Thmtheorem5.p1.2.2.m2.6a"><mrow id="S5.Thmtheorem5.p1.2.2.m2.6.6.2" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.3.cmml"><mi id="S5.Thmtheorem5.p1.2.2.m2.4.4" xref="S5.Thmtheorem5.p1.2.2.m2.4.4.cmml">max</mi><mo id="S5.Thmtheorem5.p1.2.2.m2.6.6.2a" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.3.cmml">⁡</mo><mrow id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.3.cmml"><mo id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.3" stretchy="false" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.3.cmml">{</mo><mrow id="S5.Thmtheorem5.p1.2.2.m2.5.5.1.1.1" xref="S5.Thmtheorem5.p1.2.2.m2.5.5.1.1.1.cmml"><mrow id="S5.Thmtheorem5.p1.2.2.m2.5.5.1.1.1.2" xref="S5.Thmtheorem5.p1.2.2.m2.5.5.1.1.1.2.cmml"><mover accent="true" 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xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.1"></divide><apply id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2.1.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2">superscript</csymbol><ci id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2.2.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2.2">𝐶</ci><cn id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.2.3">2</cn></apply><apply id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3.1.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3">superscript</csymbol><ci id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3.2.cmml" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3.2">𝜀</ci><cn id="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.2.2.m2.6.6.2.2.2.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.2.2.m2.6c">\max\{\tilde{\Omega}(M)\cdot\log(1/\varepsilon),C^{2}/\varepsilon^{2}\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.2.2.m2.6d">roman_max { over~ start_ARG roman_Ω end_ARG ( italic_M ) ⋅ roman_log ( start_ARG 1 / italic_ε end_ARG ) , italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT }</annotation></semantics></math> rounds.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS3.1.p1"> <p class="ltx_p" id="S5.SS3.1.p1.13">Let <math alttext="Z_{i}:A\to[0,\varepsilon]" class="ltx_Math" display="inline" id="S5.SS3.1.p1.1.m1.2"><semantics id="S5.SS3.1.p1.1.m1.2a"><mrow id="S5.SS3.1.p1.1.m1.2.3" xref="S5.SS3.1.p1.1.m1.2.3.cmml"><msub id="S5.SS3.1.p1.1.m1.2.3.2" xref="S5.SS3.1.p1.1.m1.2.3.2.cmml"><mi id="S5.SS3.1.p1.1.m1.2.3.2.2" xref="S5.SS3.1.p1.1.m1.2.3.2.2.cmml">Z</mi><mi id="S5.SS3.1.p1.1.m1.2.3.2.3" xref="S5.SS3.1.p1.1.m1.2.3.2.3.cmml">i</mi></msub><mo id="S5.SS3.1.p1.1.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S5.SS3.1.p1.1.m1.2.3.1.cmml">:</mo><mrow id="S5.SS3.1.p1.1.m1.2.3.3" xref="S5.SS3.1.p1.1.m1.2.3.3.cmml"><mi id="S5.SS3.1.p1.1.m1.2.3.3.2" xref="S5.SS3.1.p1.1.m1.2.3.3.2.cmml">A</mi><mo id="S5.SS3.1.p1.1.m1.2.3.3.1" stretchy="false" xref="S5.SS3.1.p1.1.m1.2.3.3.1.cmml">→</mo><mrow id="S5.SS3.1.p1.1.m1.2.3.3.3.2" xref="S5.SS3.1.p1.1.m1.2.3.3.3.1.cmml"><mo id="S5.SS3.1.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S5.SS3.1.p1.1.m1.2.3.3.3.1.cmml">[</mo><mn id="S5.SS3.1.p1.1.m1.1.1" xref="S5.SS3.1.p1.1.m1.1.1.cmml">0</mn><mo id="S5.SS3.1.p1.1.m1.2.3.3.3.2.2" xref="S5.SS3.1.p1.1.m1.2.3.3.3.1.cmml">,</mo><mi id="S5.SS3.1.p1.1.m1.2.2" xref="S5.SS3.1.p1.1.m1.2.2.cmml">ε</mi><mo id="S5.SS3.1.p1.1.m1.2.3.3.3.2.3" stretchy="false" xref="S5.SS3.1.p1.1.m1.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.1.m1.2b"><apply id="S5.SS3.1.p1.1.m1.2.3.cmml" xref="S5.SS3.1.p1.1.m1.2.3"><ci id="S5.SS3.1.p1.1.m1.2.3.1.cmml" xref="S5.SS3.1.p1.1.m1.2.3.1">:</ci><apply id="S5.SS3.1.p1.1.m1.2.3.2.cmml" xref="S5.SS3.1.p1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S5.SS3.1.p1.1.m1.2.3.2.1.cmml" xref="S5.SS3.1.p1.1.m1.2.3.2">subscript</csymbol><ci id="S5.SS3.1.p1.1.m1.2.3.2.2.cmml" xref="S5.SS3.1.p1.1.m1.2.3.2.2">𝑍</ci><ci id="S5.SS3.1.p1.1.m1.2.3.2.3.cmml" xref="S5.SS3.1.p1.1.m1.2.3.2.3">𝑖</ci></apply><apply id="S5.SS3.1.p1.1.m1.2.3.3.cmml" xref="S5.SS3.1.p1.1.m1.2.3.3"><ci id="S5.SS3.1.p1.1.m1.2.3.3.1.cmml" xref="S5.SS3.1.p1.1.m1.2.3.3.1">→</ci><ci id="S5.SS3.1.p1.1.m1.2.3.3.2.cmml" xref="S5.SS3.1.p1.1.m1.2.3.3.2">𝐴</ci><interval closure="closed" id="S5.SS3.1.p1.1.m1.2.3.3.3.1.cmml" xref="S5.SS3.1.p1.1.m1.2.3.3.3.2"><cn id="S5.SS3.1.p1.1.m1.1.1.cmml" type="integer" xref="S5.SS3.1.p1.1.m1.1.1">0</cn><ci id="S5.SS3.1.p1.1.m1.2.2.cmml" xref="S5.SS3.1.p1.1.m1.2.2">𝜀</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.1.m1.2c">Z_{i}:A\to[0,\varepsilon]</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.1.m1.2d">italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A → [ 0 , italic_ε ]</annotation></semantics></math> be any function, and suppose that every agent plays according to utility function <math alttext="U_{i}+Z_{i}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.2.m2.1"><semantics id="S5.SS3.1.p1.2.m2.1a"><mrow id="S5.SS3.1.p1.2.m2.1.1" xref="S5.SS3.1.p1.2.m2.1.1.cmml"><msub id="S5.SS3.1.p1.2.m2.1.1.2" xref="S5.SS3.1.p1.2.m2.1.1.2.cmml"><mi id="S5.SS3.1.p1.2.m2.1.1.2.2" xref="S5.SS3.1.p1.2.m2.1.1.2.2.cmml">U</mi><mi id="S5.SS3.1.p1.2.m2.1.1.2.3" xref="S5.SS3.1.p1.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS3.1.p1.2.m2.1.1.1" xref="S5.SS3.1.p1.2.m2.1.1.1.cmml">+</mo><msub id="S5.SS3.1.p1.2.m2.1.1.3" xref="S5.SS3.1.p1.2.m2.1.1.3.cmml"><mi id="S5.SS3.1.p1.2.m2.1.1.3.2" xref="S5.SS3.1.p1.2.m2.1.1.3.2.cmml">Z</mi><mi id="S5.SS3.1.p1.2.m2.1.1.3.3" xref="S5.SS3.1.p1.2.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.2.m2.1b"><apply id="S5.SS3.1.p1.2.m2.1.1.cmml" xref="S5.SS3.1.p1.2.m2.1.1"><plus id="S5.SS3.1.p1.2.m2.1.1.1.cmml" xref="S5.SS3.1.p1.2.m2.1.1.1"></plus><apply id="S5.SS3.1.p1.2.m2.1.1.2.cmml" xref="S5.SS3.1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.1.p1.2.m2.1.1.2.1.cmml" xref="S5.SS3.1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS3.1.p1.2.m2.1.1.2.2.cmml" xref="S5.SS3.1.p1.2.m2.1.1.2.2">𝑈</ci><ci id="S5.SS3.1.p1.2.m2.1.1.2.3.cmml" xref="S5.SS3.1.p1.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.SS3.1.p1.2.m2.1.1.3.cmml" xref="S5.SS3.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.1.p1.2.m2.1.1.3.1.cmml" xref="S5.SS3.1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.SS3.1.p1.2.m2.1.1.3.2.cmml" xref="S5.SS3.1.p1.2.m2.1.1.3.2">𝑍</ci><ci id="S5.SS3.1.p1.2.m2.1.1.3.3.cmml" xref="S5.SS3.1.p1.2.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.2.m2.1c">U_{i}+Z_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> instead of <math alttext="U_{i}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.3.m3.1"><semantics id="S5.SS3.1.p1.3.m3.1a"><msub id="S5.SS3.1.p1.3.m3.1.1" xref="S5.SS3.1.p1.3.m3.1.1.cmml"><mi id="S5.SS3.1.p1.3.m3.1.1.2" xref="S5.SS3.1.p1.3.m3.1.1.2.cmml">U</mi><mi id="S5.SS3.1.p1.3.m3.1.1.3" xref="S5.SS3.1.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.3.m3.1b"><apply id="S5.SS3.1.p1.3.m3.1.1.cmml" xref="S5.SS3.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS3.1.p1.3.m3.1.1.1.cmml" xref="S5.SS3.1.p1.3.m3.1.1">subscript</csymbol><ci id="S5.SS3.1.p1.3.m3.1.1.2.cmml" xref="S5.SS3.1.p1.3.m3.1.1.2">𝑈</ci><ci id="S5.SS3.1.p1.3.m3.1.1.3.cmml" xref="S5.SS3.1.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.3.m3.1c">U_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.3.m3.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in every round. Such an agent incurs at most <math alttext="\varepsilon T" class="ltx_Math" display="inline" id="S5.SS3.1.p1.4.m4.1"><semantics id="S5.SS3.1.p1.4.m4.1a"><mrow id="S5.SS3.1.p1.4.m4.1.1" xref="S5.SS3.1.p1.4.m4.1.1.cmml"><mi id="S5.SS3.1.p1.4.m4.1.1.2" xref="S5.SS3.1.p1.4.m4.1.1.2.cmml">ε</mi><mo id="S5.SS3.1.p1.4.m4.1.1.1" xref="S5.SS3.1.p1.4.m4.1.1.1.cmml">⁢</mo><mi id="S5.SS3.1.p1.4.m4.1.1.3" xref="S5.SS3.1.p1.4.m4.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.4.m4.1b"><apply id="S5.SS3.1.p1.4.m4.1.1.cmml" xref="S5.SS3.1.p1.4.m4.1.1"><times id="S5.SS3.1.p1.4.m4.1.1.1.cmml" xref="S5.SS3.1.p1.4.m4.1.1.1"></times><ci id="S5.SS3.1.p1.4.m4.1.1.2.cmml" xref="S5.SS3.1.p1.4.m4.1.1.2">𝜀</ci><ci id="S5.SS3.1.p1.4.m4.1.1.3.cmml" xref="S5.SS3.1.p1.4.m4.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.4.m4.1c">\varepsilon T</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.4.m4.1d">italic_ε italic_T</annotation></semantics></math> regret. Therefore, to stay within the allowable regret bound of <math alttext="C\sqrt{T}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.5.m5.1"><semantics id="S5.SS3.1.p1.5.m5.1a"><mrow id="S5.SS3.1.p1.5.m5.1.1" xref="S5.SS3.1.p1.5.m5.1.1.cmml"><mi id="S5.SS3.1.p1.5.m5.1.1.2" xref="S5.SS3.1.p1.5.m5.1.1.2.cmml">C</mi><mo id="S5.SS3.1.p1.5.m5.1.1.1" xref="S5.SS3.1.p1.5.m5.1.1.1.cmml">⁢</mo><msqrt id="S5.SS3.1.p1.5.m5.1.1.3" xref="S5.SS3.1.p1.5.m5.1.1.3.cmml"><mi id="S5.SS3.1.p1.5.m5.1.1.3.2" xref="S5.SS3.1.p1.5.m5.1.1.3.2.cmml">T</mi></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.5.m5.1b"><apply id="S5.SS3.1.p1.5.m5.1.1.cmml" xref="S5.SS3.1.p1.5.m5.1.1"><times id="S5.SS3.1.p1.5.m5.1.1.1.cmml" xref="S5.SS3.1.p1.5.m5.1.1.1"></times><ci id="S5.SS3.1.p1.5.m5.1.1.2.cmml" xref="S5.SS3.1.p1.5.m5.1.1.2">𝐶</ci><apply id="S5.SS3.1.p1.5.m5.1.1.3.cmml" xref="S5.SS3.1.p1.5.m5.1.1.3"><root id="S5.SS3.1.p1.5.m5.1.1.3a.cmml" xref="S5.SS3.1.p1.5.m5.1.1.3"></root><ci id="S5.SS3.1.p1.5.m5.1.1.3.2.cmml" xref="S5.SS3.1.p1.5.m5.1.1.3.2">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.5.m5.1c">C\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.5.m5.1d">italic_C square-root start_ARG italic_T end_ARG</annotation></semantics></math>, we set <math alttext="\varepsilon=C/\sqrt{T}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.6.m6.1"><semantics id="S5.SS3.1.p1.6.m6.1a"><mrow id="S5.SS3.1.p1.6.m6.1.1" xref="S5.SS3.1.p1.6.m6.1.1.cmml"><mi id="S5.SS3.1.p1.6.m6.1.1.2" xref="S5.SS3.1.p1.6.m6.1.1.2.cmml">ε</mi><mo id="S5.SS3.1.p1.6.m6.1.1.1" xref="S5.SS3.1.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S5.SS3.1.p1.6.m6.1.1.3" xref="S5.SS3.1.p1.6.m6.1.1.3.cmml"><mi id="S5.SS3.1.p1.6.m6.1.1.3.2" xref="S5.SS3.1.p1.6.m6.1.1.3.2.cmml">C</mi><mo id="S5.SS3.1.p1.6.m6.1.1.3.1" xref="S5.SS3.1.p1.6.m6.1.1.3.1.cmml">/</mo><msqrt id="S5.SS3.1.p1.6.m6.1.1.3.3" xref="S5.SS3.1.p1.6.m6.1.1.3.3.cmml"><mi id="S5.SS3.1.p1.6.m6.1.1.3.3.2" xref="S5.SS3.1.p1.6.m6.1.1.3.3.2.cmml">T</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.6.m6.1b"><apply id="S5.SS3.1.p1.6.m6.1.1.cmml" xref="S5.SS3.1.p1.6.m6.1.1"><eq id="S5.SS3.1.p1.6.m6.1.1.1.cmml" xref="S5.SS3.1.p1.6.m6.1.1.1"></eq><ci id="S5.SS3.1.p1.6.m6.1.1.2.cmml" xref="S5.SS3.1.p1.6.m6.1.1.2">𝜀</ci><apply id="S5.SS3.1.p1.6.m6.1.1.3.cmml" xref="S5.SS3.1.p1.6.m6.1.1.3"><divide id="S5.SS3.1.p1.6.m6.1.1.3.1.cmml" xref="S5.SS3.1.p1.6.m6.1.1.3.1"></divide><ci id="S5.SS3.1.p1.6.m6.1.1.3.2.cmml" xref="S5.SS3.1.p1.6.m6.1.1.3.2">𝐶</ci><apply id="S5.SS3.1.p1.6.m6.1.1.3.3.cmml" xref="S5.SS3.1.p1.6.m6.1.1.3.3"><root id="S5.SS3.1.p1.6.m6.1.1.3.3a.cmml" xref="S5.SS3.1.p1.6.m6.1.1.3.3"></root><ci id="S5.SS3.1.p1.6.m6.1.1.3.3.2.cmml" xref="S5.SS3.1.p1.6.m6.1.1.3.3.2">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.6.m6.1c">\varepsilon=C/\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.6.m6.1d">italic_ε = italic_C / square-root start_ARG italic_T end_ARG</annotation></semantics></math>. But then such an agent is completely indistinguishable from an agent whose true utility is <math alttext="U_{i}+Z_{i}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.7.m7.1"><semantics id="S5.SS3.1.p1.7.m7.1a"><mrow id="S5.SS3.1.p1.7.m7.1.1" xref="S5.SS3.1.p1.7.m7.1.1.cmml"><msub id="S5.SS3.1.p1.7.m7.1.1.2" xref="S5.SS3.1.p1.7.m7.1.1.2.cmml"><mi id="S5.SS3.1.p1.7.m7.1.1.2.2" xref="S5.SS3.1.p1.7.m7.1.1.2.2.cmml">U</mi><mi id="S5.SS3.1.p1.7.m7.1.1.2.3" xref="S5.SS3.1.p1.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS3.1.p1.7.m7.1.1.1" xref="S5.SS3.1.p1.7.m7.1.1.1.cmml">+</mo><msub id="S5.SS3.1.p1.7.m7.1.1.3" xref="S5.SS3.1.p1.7.m7.1.1.3.cmml"><mi id="S5.SS3.1.p1.7.m7.1.1.3.2" xref="S5.SS3.1.p1.7.m7.1.1.3.2.cmml">Z</mi><mi id="S5.SS3.1.p1.7.m7.1.1.3.3" xref="S5.SS3.1.p1.7.m7.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.7.m7.1b"><apply id="S5.SS3.1.p1.7.m7.1.1.cmml" xref="S5.SS3.1.p1.7.m7.1.1"><plus id="S5.SS3.1.p1.7.m7.1.1.1.cmml" xref="S5.SS3.1.p1.7.m7.1.1.1"></plus><apply id="S5.SS3.1.p1.7.m7.1.1.2.cmml" xref="S5.SS3.1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.1.p1.7.m7.1.1.2.1.cmml" xref="S5.SS3.1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S5.SS3.1.p1.7.m7.1.1.2.2.cmml" xref="S5.SS3.1.p1.7.m7.1.1.2.2">𝑈</ci><ci id="S5.SS3.1.p1.7.m7.1.1.2.3.cmml" xref="S5.SS3.1.p1.7.m7.1.1.2.3">𝑖</ci></apply><apply id="S5.SS3.1.p1.7.m7.1.1.3.cmml" xref="S5.SS3.1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.1.p1.7.m7.1.1.3.1.cmml" xref="S5.SS3.1.p1.7.m7.1.1.3">subscript</csymbol><ci id="S5.SS3.1.p1.7.m7.1.1.3.2.cmml" xref="S5.SS3.1.p1.7.m7.1.1.3.2">𝑍</ci><ci id="S5.SS3.1.p1.7.m7.1.1.3.3.cmml" xref="S5.SS3.1.p1.7.m7.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.7.m7.1c">U_{i}+Z_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.7.m7.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and therefore the principal can never distinguish between these two possibilities. Since this is true for any <math alttext="Z_{i}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.8.m8.1"><semantics id="S5.SS3.1.p1.8.m8.1a"><msub id="S5.SS3.1.p1.8.m8.1.1" xref="S5.SS3.1.p1.8.m8.1.1.cmml"><mi id="S5.SS3.1.p1.8.m8.1.1.2" xref="S5.SS3.1.p1.8.m8.1.1.2.cmml">Z</mi><mi id="S5.SS3.1.p1.8.m8.1.1.3" xref="S5.SS3.1.p1.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.8.m8.1b"><apply id="S5.SS3.1.p1.8.m8.1.1.cmml" xref="S5.SS3.1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS3.1.p1.8.m8.1.1.1.cmml" xref="S5.SS3.1.p1.8.m8.1.1">subscript</csymbol><ci id="S5.SS3.1.p1.8.m8.1.1.2.cmml" xref="S5.SS3.1.p1.8.m8.1.1.2">𝑍</ci><ci id="S5.SS3.1.p1.8.m8.1.1.3.cmml" xref="S5.SS3.1.p1.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.8.m8.1c">Z_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.8.m8.1d">italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, this means that the principal cannot learn <math alttext="U_{i}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.9.m9.1"><semantics id="S5.SS3.1.p1.9.m9.1a"><msub id="S5.SS3.1.p1.9.m9.1.1" xref="S5.SS3.1.p1.9.m9.1.1.cmml"><mi id="S5.SS3.1.p1.9.m9.1.1.2" xref="S5.SS3.1.p1.9.m9.1.1.2.cmml">U</mi><mi id="S5.SS3.1.p1.9.m9.1.1.3" xref="S5.SS3.1.p1.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.9.m9.1b"><apply id="S5.SS3.1.p1.9.m9.1.1.cmml" xref="S5.SS3.1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS3.1.p1.9.m9.1.1.1.cmml" xref="S5.SS3.1.p1.9.m9.1.1">subscript</csymbol><ci id="S5.SS3.1.p1.9.m9.1.1.2.cmml" xref="S5.SS3.1.p1.9.m9.1.1.2">𝑈</ci><ci id="S5.SS3.1.p1.9.m9.1.1.3.cmml" xref="S5.SS3.1.p1.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.9.m9.1c">U_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.9.m9.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to accuracy better than <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS3.1.p1.10.m10.1"><semantics id="S5.SS3.1.p1.10.m10.1a"><mi id="S5.SS3.1.p1.10.m10.1.1" xref="S5.SS3.1.p1.10.m10.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.10.m10.1b"><ci id="S5.SS3.1.p1.10.m10.1.1.cmml" xref="S5.SS3.1.p1.10.m10.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.10.m10.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.10.m10.1d">italic_ε</annotation></semantics></math>. Solving for <math alttext="T" class="ltx_Math" display="inline" id="S5.SS3.1.p1.11.m11.1"><semantics id="S5.SS3.1.p1.11.m11.1a"><mi id="S5.SS3.1.p1.11.m11.1.1" xref="S5.SS3.1.p1.11.m11.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.11.m11.1b"><ci id="S5.SS3.1.p1.11.m11.1.1.cmml" xref="S5.SS3.1.p1.11.m11.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.11.m11.1c">T</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.11.m11.1d">italic_T</annotation></semantics></math> yields <math alttext="T=C^{2}/\varepsilon^{2}" class="ltx_Math" display="inline" id="S5.SS3.1.p1.12.m12.1"><semantics id="S5.SS3.1.p1.12.m12.1a"><mrow id="S5.SS3.1.p1.12.m12.1.1" xref="S5.SS3.1.p1.12.m12.1.1.cmml"><mi id="S5.SS3.1.p1.12.m12.1.1.2" xref="S5.SS3.1.p1.12.m12.1.1.2.cmml">T</mi><mo id="S5.SS3.1.p1.12.m12.1.1.1" xref="S5.SS3.1.p1.12.m12.1.1.1.cmml">=</mo><mrow id="S5.SS3.1.p1.12.m12.1.1.3" xref="S5.SS3.1.p1.12.m12.1.1.3.cmml"><msup id="S5.SS3.1.p1.12.m12.1.1.3.2" xref="S5.SS3.1.p1.12.m12.1.1.3.2.cmml"><mi id="S5.SS3.1.p1.12.m12.1.1.3.2.2" xref="S5.SS3.1.p1.12.m12.1.1.3.2.2.cmml">C</mi><mn id="S5.SS3.1.p1.12.m12.1.1.3.2.3" xref="S5.SS3.1.p1.12.m12.1.1.3.2.3.cmml">2</mn></msup><mo id="S5.SS3.1.p1.12.m12.1.1.3.1" xref="S5.SS3.1.p1.12.m12.1.1.3.1.cmml">/</mo><msup id="S5.SS3.1.p1.12.m12.1.1.3.3" xref="S5.SS3.1.p1.12.m12.1.1.3.3.cmml"><mi id="S5.SS3.1.p1.12.m12.1.1.3.3.2" xref="S5.SS3.1.p1.12.m12.1.1.3.3.2.cmml">ε</mi><mn id="S5.SS3.1.p1.12.m12.1.1.3.3.3" xref="S5.SS3.1.p1.12.m12.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.12.m12.1b"><apply id="S5.SS3.1.p1.12.m12.1.1.cmml" xref="S5.SS3.1.p1.12.m12.1.1"><eq id="S5.SS3.1.p1.12.m12.1.1.1.cmml" xref="S5.SS3.1.p1.12.m12.1.1.1"></eq><ci id="S5.SS3.1.p1.12.m12.1.1.2.cmml" xref="S5.SS3.1.p1.12.m12.1.1.2">𝑇</ci><apply id="S5.SS3.1.p1.12.m12.1.1.3.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3"><divide id="S5.SS3.1.p1.12.m12.1.1.3.1.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3.1"></divide><apply id="S5.SS3.1.p1.12.m12.1.1.3.2.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS3.1.p1.12.m12.1.1.3.2.1.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3.2">superscript</csymbol><ci id="S5.SS3.1.p1.12.m12.1.1.3.2.2.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3.2.2">𝐶</ci><cn id="S5.SS3.1.p1.12.m12.1.1.3.2.3.cmml" type="integer" xref="S5.SS3.1.p1.12.m12.1.1.3.2.3">2</cn></apply><apply id="S5.SS3.1.p1.12.m12.1.1.3.3.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS3.1.p1.12.m12.1.1.3.3.1.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3.3">superscript</csymbol><ci id="S5.SS3.1.p1.12.m12.1.1.3.3.2.cmml" xref="S5.SS3.1.p1.12.m12.1.1.3.3.2">𝜀</ci><cn id="S5.SS3.1.p1.12.m12.1.1.3.3.3.cmml" type="integer" xref="S5.SS3.1.p1.12.m12.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.12.m12.1c">T=C^{2}/\varepsilon^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.12.m12.1d">italic_T = italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, giving the second term in the maximization. The <math alttext="\tilde{\Omega}(M)\cdot\log(1/\varepsilon)" class="ltx_Math" display="inline" id="S5.SS3.1.p1.13.m13.3"><semantics id="S5.SS3.1.p1.13.m13.3a"><mrow id="S5.SS3.1.p1.13.m13.3.4" xref="S5.SS3.1.p1.13.m13.3.4.cmml"><mrow id="S5.SS3.1.p1.13.m13.3.4.2" xref="S5.SS3.1.p1.13.m13.3.4.2.cmml"><mover accent="true" id="S5.SS3.1.p1.13.m13.3.4.2.2" xref="S5.SS3.1.p1.13.m13.3.4.2.2.cmml"><mi id="S5.SS3.1.p1.13.m13.3.4.2.2.2" mathvariant="normal" xref="S5.SS3.1.p1.13.m13.3.4.2.2.2.cmml">Ω</mi><mo id="S5.SS3.1.p1.13.m13.3.4.2.2.1" xref="S5.SS3.1.p1.13.m13.3.4.2.2.1.cmml">~</mo></mover><mo id="S5.SS3.1.p1.13.m13.3.4.2.1" xref="S5.SS3.1.p1.13.m13.3.4.2.1.cmml">⁢</mo><mrow id="S5.SS3.1.p1.13.m13.3.4.2.3.2" xref="S5.SS3.1.p1.13.m13.3.4.2.cmml"><mo id="S5.SS3.1.p1.13.m13.3.4.2.3.2.1" stretchy="false" xref="S5.SS3.1.p1.13.m13.3.4.2.cmml">(</mo><mi id="S5.SS3.1.p1.13.m13.3.3" xref="S5.SS3.1.p1.13.m13.3.3.cmml">M</mi><mo id="S5.SS3.1.p1.13.m13.3.4.2.3.2.2" rspace="0.055em" stretchy="false" xref="S5.SS3.1.p1.13.m13.3.4.2.cmml">)</mo></mrow></mrow><mo id="S5.SS3.1.p1.13.m13.3.4.1" rspace="0.222em" xref="S5.SS3.1.p1.13.m13.3.4.1.cmml">⋅</mo><mrow id="S5.SS3.1.p1.13.m13.2.2.4" xref="S5.SS3.1.p1.13.m13.2.2.3.cmml"><mi id="S5.SS3.1.p1.13.m13.2.2.2.2" xref="S5.SS3.1.p1.13.m13.2.2.3.1.cmml">log</mi><mo id="S5.SS3.1.p1.13.m13.2.2.4a" xref="S5.SS3.1.p1.13.m13.2.2.3.1.cmml">⁡</mo><mrow id="S5.SS3.1.p1.13.m13.2.2.4.1" xref="S5.SS3.1.p1.13.m13.2.2.3.cmml"><mo id="S5.SS3.1.p1.13.m13.2.2.4.1.1" xref="S5.SS3.1.p1.13.m13.2.2.3.1.cmml">(</mo><mrow id="S5.SS3.1.p1.13.m13.1.1.1.1.1" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1.cmml"><mn id="S5.SS3.1.p1.13.m13.1.1.1.1.1.2" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1.2.cmml">1</mn><mo id="S5.SS3.1.p1.13.m13.1.1.1.1.1.1" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1.1.cmml">/</mo><mi id="S5.SS3.1.p1.13.m13.1.1.1.1.1.3" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.SS3.1.p1.13.m13.2.2.4.1.2" xref="S5.SS3.1.p1.13.m13.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.1.p1.13.m13.3b"><apply id="S5.SS3.1.p1.13.m13.3.4.cmml" xref="S5.SS3.1.p1.13.m13.3.4"><ci id="S5.SS3.1.p1.13.m13.3.4.1.cmml" xref="S5.SS3.1.p1.13.m13.3.4.1">⋅</ci><apply id="S5.SS3.1.p1.13.m13.3.4.2.cmml" xref="S5.SS3.1.p1.13.m13.3.4.2"><times id="S5.SS3.1.p1.13.m13.3.4.2.1.cmml" xref="S5.SS3.1.p1.13.m13.3.4.2.1"></times><apply id="S5.SS3.1.p1.13.m13.3.4.2.2.cmml" xref="S5.SS3.1.p1.13.m13.3.4.2.2"><ci id="S5.SS3.1.p1.13.m13.3.4.2.2.1.cmml" xref="S5.SS3.1.p1.13.m13.3.4.2.2.1">~</ci><ci id="S5.SS3.1.p1.13.m13.3.4.2.2.2.cmml" xref="S5.SS3.1.p1.13.m13.3.4.2.2.2">Ω</ci></apply><ci id="S5.SS3.1.p1.13.m13.3.3.cmml" xref="S5.SS3.1.p1.13.m13.3.3">𝑀</ci></apply><apply id="S5.SS3.1.p1.13.m13.2.2.3.cmml" xref="S5.SS3.1.p1.13.m13.2.2.4"><log id="S5.SS3.1.p1.13.m13.2.2.3.1.cmml" xref="S5.SS3.1.p1.13.m13.2.2.2.2"></log><apply id="S5.SS3.1.p1.13.m13.1.1.1.1.1.cmml" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1"><divide id="S5.SS3.1.p1.13.m13.1.1.1.1.1.1.cmml" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1.1"></divide><cn id="S5.SS3.1.p1.13.m13.1.1.1.1.1.2.cmml" type="integer" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1.2">1</cn><ci id="S5.SS3.1.p1.13.m13.1.1.1.1.1.3.cmml" xref="S5.SS3.1.p1.13.m13.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.1.p1.13.m13.3c">\tilde{\Omega}(M)\cdot\log(1/\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.1.p1.13.m13.3d">over~ start_ARG roman_Ω end_ARG ( italic_M ) ⋅ roman_log ( start_ARG 1 / italic_ε end_ARG )</annotation></semantics></math> term comes from <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.Thmtheorem3" title="Theorem 4.3. ‣ 4.3 Lower bound ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>, which applies to any behavioral model. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS3.p2"> <p class="ltx_p" id="S5.SS3.p2.3"><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.Thmtheorem5" title="Theorem 5.5. ‣ 5.3 Lower bound ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.5</span></a> shows that it is impossible to exponentially improve the dependence on any of the parameters in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.Thmtheorem2" title="Theorem 5.2. ‣ 5.1 The single-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.2</span></a>. For example, it implies that there can be no algorithm taking <math alttext="C^{2}/\varepsilon^{2}\cdot M^{1-\Omega(1)}" class="ltx_Math" display="inline" id="S5.SS3.p2.1.m1.1"><semantics id="S5.SS3.p2.1.m1.1a"><mrow id="S5.SS3.p2.1.m1.1.2" xref="S5.SS3.p2.1.m1.1.2.cmml"><mrow id="S5.SS3.p2.1.m1.1.2.2" xref="S5.SS3.p2.1.m1.1.2.2.cmml"><msup id="S5.SS3.p2.1.m1.1.2.2.2" xref="S5.SS3.p2.1.m1.1.2.2.2.cmml"><mi id="S5.SS3.p2.1.m1.1.2.2.2.2" xref="S5.SS3.p2.1.m1.1.2.2.2.2.cmml">C</mi><mn id="S5.SS3.p2.1.m1.1.2.2.2.3" xref="S5.SS3.p2.1.m1.1.2.2.2.3.cmml">2</mn></msup><mo id="S5.SS3.p2.1.m1.1.2.2.1" xref="S5.SS3.p2.1.m1.1.2.2.1.cmml">/</mo><msup id="S5.SS3.p2.1.m1.1.2.2.3" xref="S5.SS3.p2.1.m1.1.2.2.3.cmml"><mi id="S5.SS3.p2.1.m1.1.2.2.3.2" xref="S5.SS3.p2.1.m1.1.2.2.3.2.cmml">ε</mi><mn id="S5.SS3.p2.1.m1.1.2.2.3.3" xref="S5.SS3.p2.1.m1.1.2.2.3.3.cmml">2</mn></msup></mrow><mo id="S5.SS3.p2.1.m1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.SS3.p2.1.m1.1.2.1.cmml">⋅</mo><msup id="S5.SS3.p2.1.m1.1.2.3" xref="S5.SS3.p2.1.m1.1.2.3.cmml"><mi id="S5.SS3.p2.1.m1.1.2.3.2" xref="S5.SS3.p2.1.m1.1.2.3.2.cmml">M</mi><mrow id="S5.SS3.p2.1.m1.1.1.1" xref="S5.SS3.p2.1.m1.1.1.1.cmml"><mn id="S5.SS3.p2.1.m1.1.1.1.3" xref="S5.SS3.p2.1.m1.1.1.1.3.cmml">1</mn><mo id="S5.SS3.p2.1.m1.1.1.1.2" xref="S5.SS3.p2.1.m1.1.1.1.2.cmml">−</mo><mrow id="S5.SS3.p2.1.m1.1.1.1.4" xref="S5.SS3.p2.1.m1.1.1.1.4.cmml"><mi id="S5.SS3.p2.1.m1.1.1.1.4.2" mathvariant="normal" xref="S5.SS3.p2.1.m1.1.1.1.4.2.cmml">Ω</mi><mo id="S5.SS3.p2.1.m1.1.1.1.4.1" xref="S5.SS3.p2.1.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S5.SS3.p2.1.m1.1.1.1.4.3.2" xref="S5.SS3.p2.1.m1.1.1.1.4.cmml"><mo id="S5.SS3.p2.1.m1.1.1.1.4.3.2.1" stretchy="false" xref="S5.SS3.p2.1.m1.1.1.1.4.cmml">(</mo><mn id="S5.SS3.p2.1.m1.1.1.1.1" xref="S5.SS3.p2.1.m1.1.1.1.1.cmml">1</mn><mo id="S5.SS3.p2.1.m1.1.1.1.4.3.2.2" stretchy="false" xref="S5.SS3.p2.1.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.1.m1.1b"><apply id="S5.SS3.p2.1.m1.1.2.cmml" xref="S5.SS3.p2.1.m1.1.2"><ci id="S5.SS3.p2.1.m1.1.2.1.cmml" xref="S5.SS3.p2.1.m1.1.2.1">⋅</ci><apply id="S5.SS3.p2.1.m1.1.2.2.cmml" xref="S5.SS3.p2.1.m1.1.2.2"><divide id="S5.SS3.p2.1.m1.1.2.2.1.cmml" xref="S5.SS3.p2.1.m1.1.2.2.1"></divide><apply id="S5.SS3.p2.1.m1.1.2.2.2.cmml" xref="S5.SS3.p2.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S5.SS3.p2.1.m1.1.2.2.2.1.cmml" xref="S5.SS3.p2.1.m1.1.2.2.2">superscript</csymbol><ci id="S5.SS3.p2.1.m1.1.2.2.2.2.cmml" xref="S5.SS3.p2.1.m1.1.2.2.2.2">𝐶</ci><cn id="S5.SS3.p2.1.m1.1.2.2.2.3.cmml" type="integer" xref="S5.SS3.p2.1.m1.1.2.2.2.3">2</cn></apply><apply id="S5.SS3.p2.1.m1.1.2.2.3.cmml" xref="S5.SS3.p2.1.m1.1.2.2.3"><csymbol cd="ambiguous" id="S5.SS3.p2.1.m1.1.2.2.3.1.cmml" xref="S5.SS3.p2.1.m1.1.2.2.3">superscript</csymbol><ci id="S5.SS3.p2.1.m1.1.2.2.3.2.cmml" xref="S5.SS3.p2.1.m1.1.2.2.3.2">𝜀</ci><cn id="S5.SS3.p2.1.m1.1.2.2.3.3.cmml" type="integer" xref="S5.SS3.p2.1.m1.1.2.2.3.3">2</cn></apply></apply><apply id="S5.SS3.p2.1.m1.1.2.3.cmml" xref="S5.SS3.p2.1.m1.1.2.3"><csymbol cd="ambiguous" id="S5.SS3.p2.1.m1.1.2.3.1.cmml" xref="S5.SS3.p2.1.m1.1.2.3">superscript</csymbol><ci id="S5.SS3.p2.1.m1.1.2.3.2.cmml" xref="S5.SS3.p2.1.m1.1.2.3.2">𝑀</ci><apply id="S5.SS3.p2.1.m1.1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1.1"><minus id="S5.SS3.p2.1.m1.1.1.1.2.cmml" xref="S5.SS3.p2.1.m1.1.1.1.2"></minus><cn id="S5.SS3.p2.1.m1.1.1.1.3.cmml" type="integer" xref="S5.SS3.p2.1.m1.1.1.1.3">1</cn><apply id="S5.SS3.p2.1.m1.1.1.1.4.cmml" xref="S5.SS3.p2.1.m1.1.1.1.4"><times id="S5.SS3.p2.1.m1.1.1.1.4.1.cmml" xref="S5.SS3.p2.1.m1.1.1.1.4.1"></times><ci id="S5.SS3.p2.1.m1.1.1.1.4.2.cmml" xref="S5.SS3.p2.1.m1.1.1.1.4.2">Ω</ci><cn id="S5.SS3.p2.1.m1.1.1.1.1.cmml" type="integer" xref="S5.SS3.p2.1.m1.1.1.1.1">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.1.m1.1c">C^{2}/\varepsilon^{2}\cdot M^{1-\Omega(1)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.1.m1.1d">italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ italic_M start_POSTSUPERSCRIPT 1 - roman_Ω ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> rounds, because that would contradict the lower bound for constant <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS3.p2.2.m2.1"><semantics id="S5.SS3.p2.2.m2.1a"><mi id="S5.SS3.p2.2.m2.1.1" xref="S5.SS3.p2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.2.m2.1b"><ci id="S5.SS3.p2.2.m2.1.1.cmml" xref="S5.SS3.p2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.2.m2.1d">italic_ε</annotation></semantics></math> and suffficiently large <math alttext="M" class="ltx_Math" display="inline" id="S5.SS3.p2.3.m3.1"><semantics id="S5.SS3.p2.3.m3.1a"><mi id="S5.SS3.p2.3.m3.1.1" xref="S5.SS3.p2.3.m3.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.3.m3.1b"><ci id="S5.SS3.p2.3.m3.1.1.cmml" xref="S5.SS3.p2.3.m3.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.3.m3.1c">M</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.3.m3.1d">italic_M</annotation></semantics></math>. We leave it as an interesting open question to close the polynomial gaps between the lower and upper bounds presented here.</p> </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>Minimizing Payment</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.2">Previous sections focus on <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mi id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><ci id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">italic_ε</annotation></semantics></math>-learning a game in as few rounds as possible. This section discusses an alternative goal: <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S6.p1.2.m2.1"><semantics id="S6.p1.2.m2.1a"><mi id="S6.p1.2.m2.1.1" xref="S6.p1.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.1b"><ci id="S6.p1.2.m2.1.1.cmml" xref="S6.p1.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.1d">italic_ε</annotation></semantics></math>-learning a game while minimizing the total payment to the agents. We show that the case of a single rational agent can be solved with a small total payment, but the multi-agent case is different.</p> </div> <section class="ltx_paragraph" id="S6.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_font_bold ltx_title_paragraph">Single rational agent</h4> <div class="ltx_para" id="S6.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S6.SS0.SSS0.Px1.p1.10">First, we consider <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="S6.SS0.SSS0.Px1.p1.1.m1.1a"><mi id="S6.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S6.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.1.m1.1b"><ci id="S6.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.1.m1.1d">italic_ε</annotation></semantics></math>-learning a single-agent game under the rationalizable model (optimal action model). Section <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.SS1" title="4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">4.1</span></a> showed that <math alttext="\tilde{\Omega}(m)\cdot\log(1/\varepsilon)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.2.m2.3"><semantics id="S6.SS0.SSS0.Px1.p1.2.m2.3a"><mrow id="S6.SS0.SSS0.Px1.p1.2.m2.3.4" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.cmml"><mrow id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.cmml"><mover accent="true" id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.cmml"><mi id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.2.cmml">Ω</mi><mo id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.1" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.1.cmml">~</mo></mover><mo id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.1" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.3.2" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.cmml"><mo id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.p1.2.m2.3.3" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.3.cmml">m</mi><mo id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.3.2.2" rspace="0.055em" stretchy="false" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.1" rspace="0.222em" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.1.cmml">⋅</mo><mrow id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.4" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.cmml"><mi id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.2.2" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.1.cmml">log</mi><mo id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.4a" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.1.cmml">⁡</mo><mrow id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.4.1" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.cmml"><mo id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.4.1.1" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.cmml"><mn id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.2.cmml">1</mn><mo id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.cmml">/</mo><mi id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.4.1.2" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.2.m2.3b"><apply id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4"><ci id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.1.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.1">⋅</ci><apply id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2"><times id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.1.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.1"></times><apply id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2"><ci id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.1.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.1">~</ci><ci id="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.4.2.2.2">Ω</ci></apply><ci id="S6.SS0.SSS0.Px1.p1.2.m2.3.3.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.3.3">𝑚</ci></apply><apply id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.4"><log id="S6.SS0.SSS0.Px1.p1.2.m2.2.2.3.1.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.2.2.2.2"></log><apply id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1"><divide id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.1"></divide><cn id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.2">1</cn><ci id="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.p1.2.m2.1.1.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.2.m2.3c">\tilde{\Omega}(m)\cdot\log(1/\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.2.m2.3d">over~ start_ARG roman_Ω end_ARG ( italic_m ) ⋅ roman_log ( start_ARG 1 / italic_ε end_ARG )</annotation></semantics></math> rounds are necessary for this problem. Interestingly, we show that the total payment can be significantly smaller than the number of rounds. The algorithm is different from the binary search algorithm in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S4.SS1" title="4.1 The single-agent case ‣ 4 Learning the Utility Function in the Rationalizable Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">4.1</span></a>. 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A}U(a)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.3.m3.1d">italic_a ≠ italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT := roman_argmax start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_U ( italic_a )</annotation></semantics></math> of the agent, then try paying action <math alttext="a" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.4.m4.1"><semantics id="S6.SS0.SSS0.Px1.p1.4.m4.1a"><mi id="S6.SS0.SSS0.Px1.p1.4.m4.1.1" xref="S6.SS0.SSS0.Px1.p1.4.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.4.m4.1b"><ci id="S6.SS0.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.4.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.4.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.4.m4.1d">italic_a</annotation></semantics></math> by <math alttext="0,\varepsilon,2\varepsilon,\ldots" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.5.m5.4"><semantics id="S6.SS0.SSS0.Px1.p1.5.m5.4a"><mrow id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.2.cmml"><mn id="S6.SS0.SSS0.Px1.p1.5.m5.1.1" xref="S6.SS0.SSS0.Px1.p1.5.m5.1.1.cmml">0</mn><mo id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.2" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.2.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.p1.5.m5.2.2" xref="S6.SS0.SSS0.Px1.p1.5.m5.2.2.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.3" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.2.cmml">,</mo><mrow id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.cmml"><mn id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.2" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.2.cmml">2</mn><mo id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.1" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.1.cmml">⁢</mo><mi id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.3" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.3.cmml">ε</mi></mrow><mo id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.4" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.2.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.p1.5.m5.3.3" mathvariant="normal" xref="S6.SS0.SSS0.Px1.p1.5.m5.3.3.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.5.m5.4b"><list id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.2.cmml" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1"><cn id="S6.SS0.SSS0.Px1.p1.5.m5.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.p1.5.m5.1.1">0</cn><ci id="S6.SS0.SSS0.Px1.p1.5.m5.2.2.cmml" xref="S6.SS0.SSS0.Px1.p1.5.m5.2.2">𝜀</ci><apply id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1"><times id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.1"></times><cn id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.2.cmml" type="integer" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.2">2</cn><ci id="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.p1.5.m5.4.4.1.1.3">𝜀</ci></apply><ci id="S6.SS0.SSS0.Px1.p1.5.m5.3.3.cmml" xref="S6.SS0.SSS0.Px1.p1.5.m5.3.3">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.5.m5.4c">0,\varepsilon,2\varepsilon,\ldots</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.5.m5.4d">0 , italic_ε , 2 italic_ε , …</annotation></semantics></math>, until the agent starts to play action <math alttext="a" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.6.m6.1"><semantics id="S6.SS0.SSS0.Px1.p1.6.m6.1a"><mi id="S6.SS0.SSS0.Px1.p1.6.m6.1.1" xref="S6.SS0.SSS0.Px1.p1.6.m6.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.6.m6.1b"><ci id="S6.SS0.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.6.m6.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.6.m6.1c">a</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.6.m6.1d">italic_a</annotation></semantics></math> instead of the optimal action <math alttext="a^{*}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.7.m7.1"><semantics id="S6.SS0.SSS0.Px1.p1.7.m7.1a"><msup id="S6.SS0.SSS0.Px1.p1.7.m7.1.1" xref="S6.SS0.SSS0.Px1.p1.7.m7.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.p1.7.m7.1.1.2" xref="S6.SS0.SSS0.Px1.p1.7.m7.1.1.2.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.p1.7.m7.1.1.3" xref="S6.SS0.SSS0.Px1.p1.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.7.m7.1b"><apply id="S6.SS0.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.7.m7.1.1">superscript</csymbol><ci id="S6.SS0.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.p1.7.m7.1.1.2">𝑎</ci><times id="S6.SS0.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.p1.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.7.m7.1c">a^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.7.m7.1d">italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. Although this takes <math alttext="1/\varepsilon" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.8.m8.1"><semantics id="S6.SS0.SSS0.Px1.p1.8.m8.1a"><mrow id="S6.SS0.SSS0.Px1.p1.8.m8.1.1" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1.cmml"><mn id="S6.SS0.SSS0.Px1.p1.8.m8.1.1.2" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1.2.cmml">1</mn><mo id="S6.SS0.SSS0.Px1.p1.8.m8.1.1.1" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1.1.cmml">/</mo><mi id="S6.SS0.SSS0.Px1.p1.8.m8.1.1.3" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.8.m8.1b"><apply id="S6.SS0.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1"><divide id="S6.SS0.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1.1"></divide><cn id="S6.SS0.SSS0.Px1.p1.8.m8.1.1.2.cmml" type="integer" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1.2">1</cn><ci id="S6.SS0.SSS0.Px1.p1.8.m8.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.p1.8.m8.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.8.m8.1c">1/\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.8.m8.1d">1 / italic_ε</annotation></semantics></math> rounds, the total payment is small because we pay <math alttext="0" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.9.m9.1"><semantics id="S6.SS0.SSS0.Px1.p1.9.m9.1a"><mn id="S6.SS0.SSS0.Px1.p1.9.m9.1.1" xref="S6.SS0.SSS0.Px1.p1.9.m9.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.9.m9.1b"><cn id="S6.SS0.SSS0.Px1.p1.9.m9.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.p1.9.m9.1.1">0</cn></annotation-xml></semantics></math> whenever the agent does not play <math alttext="a" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p1.10.m10.1"><semantics id="S6.SS0.SSS0.Px1.p1.10.m10.1a"><mi id="S6.SS0.SSS0.Px1.p1.10.m10.1.1" xref="S6.SS0.SSS0.Px1.p1.10.m10.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p1.10.m10.1b"><ci id="S6.SS0.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S6.SS0.SSS0.Px1.p1.10.m10.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p1.10.m10.1c">a</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p1.10.m10.1d">italic_a</annotation></semantics></math>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg5"> <div class="ltx_listing ltx_listing" id="alg5.2"> <div class="ltx_listingline" id="alg4.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l1.1.1.1" style="font-size:80%;">1:</span></span>At round 1, give no payment and observe the agent’s optimal action <math alttext="a^{*}=\operatorname*{argmax}_{a\in A}U(a)" class="ltx_Math" display="inline" id="alg4.l1.m1.1"><semantics id="alg4.l1.m1.1a"><mrow id="alg4.l1.m1.1.2" xref="alg4.l1.m1.1.2.cmml"><msup id="alg4.l1.m1.1.2.2" xref="alg4.l1.m1.1.2.2.cmml"><mi id="alg4.l1.m1.1.2.2.2" xref="alg4.l1.m1.1.2.2.2.cmml">a</mi><mo id="alg4.l1.m1.1.2.2.3" xref="alg4.l1.m1.1.2.2.3.cmml">∗</mo></msup><mo id="alg4.l1.m1.1.2.1" rspace="0.1389em" xref="alg4.l1.m1.1.2.1.cmml">=</mo><mrow id="alg4.l1.m1.1.2.3" xref="alg4.l1.m1.1.2.3.cmml"><mrow id="alg4.l1.m1.1.2.3.2" xref="alg4.l1.m1.1.2.3.2.cmml"><msub id="alg4.l1.m1.1.2.3.2.1" xref="alg4.l1.m1.1.2.3.2.1.cmml"><mo id="alg4.l1.m1.1.2.3.2.1.2" lspace="0.1389em" rspace="0.167em" xref="alg4.l1.m1.1.2.3.2.1.2.cmml">argmax</mo><mrow id="alg4.l1.m1.1.2.3.2.1.3" xref="alg4.l1.m1.1.2.3.2.1.3.cmml"><mi id="alg4.l1.m1.1.2.3.2.1.3.2" xref="alg4.l1.m1.1.2.3.2.1.3.2.cmml">a</mi><mo id="alg4.l1.m1.1.2.3.2.1.3.1" xref="alg4.l1.m1.1.2.3.2.1.3.1.cmml">∈</mo><mi id="alg4.l1.m1.1.2.3.2.1.3.3" xref="alg4.l1.m1.1.2.3.2.1.3.3.cmml">A</mi></mrow></msub><mi id="alg4.l1.m1.1.2.3.2.2" xref="alg4.l1.m1.1.2.3.2.2.cmml">U</mi></mrow><mo id="alg4.l1.m1.1.2.3.1" xref="alg4.l1.m1.1.2.3.1.cmml">⁢</mo><mrow id="alg4.l1.m1.1.2.3.3.2" xref="alg4.l1.m1.1.2.3.cmml"><mo id="alg4.l1.m1.1.2.3.3.2.1" stretchy="false" xref="alg4.l1.m1.1.2.3.cmml">(</mo><mi id="alg4.l1.m1.1.1" xref="alg4.l1.m1.1.1.cmml">a</mi><mo id="alg4.l1.m1.1.2.3.3.2.2" stretchy="false" xref="alg4.l1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m1.1b"><apply id="alg4.l1.m1.1.2.cmml" xref="alg4.l1.m1.1.2"><eq id="alg4.l1.m1.1.2.1.cmml" xref="alg4.l1.m1.1.2.1"></eq><apply id="alg4.l1.m1.1.2.2.cmml" xref="alg4.l1.m1.1.2.2"><csymbol cd="ambiguous" id="alg4.l1.m1.1.2.2.1.cmml" xref="alg4.l1.m1.1.2.2">superscript</csymbol><ci id="alg4.l1.m1.1.2.2.2.cmml" xref="alg4.l1.m1.1.2.2.2">𝑎</ci><times id="alg4.l1.m1.1.2.2.3.cmml" xref="alg4.l1.m1.1.2.2.3"></times></apply><apply id="alg4.l1.m1.1.2.3.cmml" xref="alg4.l1.m1.1.2.3"><times id="alg4.l1.m1.1.2.3.1.cmml" xref="alg4.l1.m1.1.2.3.1"></times><apply id="alg4.l1.m1.1.2.3.2.cmml" xref="alg4.l1.m1.1.2.3.2"><apply id="alg4.l1.m1.1.2.3.2.1.cmml" xref="alg4.l1.m1.1.2.3.2.1"><csymbol cd="ambiguous" id="alg4.l1.m1.1.2.3.2.1.1.cmml" xref="alg4.l1.m1.1.2.3.2.1">subscript</csymbol><ci id="alg4.l1.m1.1.2.3.2.1.2.cmml" xref="alg4.l1.m1.1.2.3.2.1.2">argmax</ci><apply id="alg4.l1.m1.1.2.3.2.1.3.cmml" xref="alg4.l1.m1.1.2.3.2.1.3"><in id="alg4.l1.m1.1.2.3.2.1.3.1.cmml" xref="alg4.l1.m1.1.2.3.2.1.3.1"></in><ci id="alg4.l1.m1.1.2.3.2.1.3.2.cmml" xref="alg4.l1.m1.1.2.3.2.1.3.2">𝑎</ci><ci id="alg4.l1.m1.1.2.3.2.1.3.3.cmml" xref="alg4.l1.m1.1.2.3.2.1.3.3">𝐴</ci></apply></apply><ci id="alg4.l1.m1.1.2.3.2.2.cmml" xref="alg4.l1.m1.1.2.3.2.2">𝑈</ci></apply><ci id="alg4.l1.m1.1.1.cmml" xref="alg4.l1.m1.1.1">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m1.1c">a^{*}=\operatorname*{argmax}_{a\in A}U(a)</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m1.1d">italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = roman_argmax start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_U ( italic_a )</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg4.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l2.1.1.1" style="font-size:80%;">2:</span></span>let <math alttext="\tilde{U}(a^{*})=0" class="ltx_Math" display="inline" id="alg4.l2.m1.1"><semantics id="alg4.l2.m1.1a"><mrow id="alg4.l2.m1.1.1" xref="alg4.l2.m1.1.1.cmml"><mrow id="alg4.l2.m1.1.1.1" xref="alg4.l2.m1.1.1.1.cmml"><mover accent="true" id="alg4.l2.m1.1.1.1.3" xref="alg4.l2.m1.1.1.1.3.cmml"><mi id="alg4.l2.m1.1.1.1.3.2" xref="alg4.l2.m1.1.1.1.3.2.cmml">U</mi><mo id="alg4.l2.m1.1.1.1.3.1" xref="alg4.l2.m1.1.1.1.3.1.cmml">~</mo></mover><mo id="alg4.l2.m1.1.1.1.2" xref="alg4.l2.m1.1.1.1.2.cmml">⁢</mo><mrow id="alg4.l2.m1.1.1.1.1.1" xref="alg4.l2.m1.1.1.1.1.1.1.cmml"><mo id="alg4.l2.m1.1.1.1.1.1.2" stretchy="false" xref="alg4.l2.m1.1.1.1.1.1.1.cmml">(</mo><msup id="alg4.l2.m1.1.1.1.1.1.1" xref="alg4.l2.m1.1.1.1.1.1.1.cmml"><mi id="alg4.l2.m1.1.1.1.1.1.1.2" xref="alg4.l2.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="alg4.l2.m1.1.1.1.1.1.1.3" xref="alg4.l2.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="alg4.l2.m1.1.1.1.1.1.3" stretchy="false" xref="alg4.l2.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg4.l2.m1.1.1.2" xref="alg4.l2.m1.1.1.2.cmml">=</mo><mn id="alg4.l2.m1.1.1.3" xref="alg4.l2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg4.l2.m1.1b"><apply id="alg4.l2.m1.1.1.cmml" xref="alg4.l2.m1.1.1"><eq id="alg4.l2.m1.1.1.2.cmml" xref="alg4.l2.m1.1.1.2"></eq><apply id="alg4.l2.m1.1.1.1.cmml" xref="alg4.l2.m1.1.1.1"><times id="alg4.l2.m1.1.1.1.2.cmml" xref="alg4.l2.m1.1.1.1.2"></times><apply id="alg4.l2.m1.1.1.1.3.cmml" xref="alg4.l2.m1.1.1.1.3"><ci id="alg4.l2.m1.1.1.1.3.1.cmml" xref="alg4.l2.m1.1.1.1.3.1">~</ci><ci id="alg4.l2.m1.1.1.1.3.2.cmml" xref="alg4.l2.m1.1.1.1.3.2">𝑈</ci></apply><apply id="alg4.l2.m1.1.1.1.1.1.1.cmml" xref="alg4.l2.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg4.l2.m1.1.1.1.1.1.1.1.cmml" xref="alg4.l2.m1.1.1.1.1.1">superscript</csymbol><ci id="alg4.l2.m1.1.1.1.1.1.1.2.cmml" xref="alg4.l2.m1.1.1.1.1.1.1.2">𝑎</ci><times id="alg4.l2.m1.1.1.1.1.1.1.3.cmml" xref="alg4.l2.m1.1.1.1.1.1.1.3"></times></apply></apply><cn id="alg4.l2.m1.1.1.3.cmml" type="integer" xref="alg4.l2.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l2.m1.1c">\tilde{U}(a^{*})=0</annotation><annotation encoding="application/x-llamapun" id="alg4.l2.m1.1d">over~ start_ARG italic_U end_ARG ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg4.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l3.1.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg4.l3.2">for</span> each action <math alttext="a\in A\setminus\{a^{*}\}" class="ltx_Math" display="inline" id="alg4.l3.m1.1"><semantics id="alg4.l3.m1.1a"><mrow id="alg4.l3.m1.1.1" xref="alg4.l3.m1.1.1.cmml"><mi id="alg4.l3.m1.1.1.3" xref="alg4.l3.m1.1.1.3.cmml">a</mi><mo id="alg4.l3.m1.1.1.2" xref="alg4.l3.m1.1.1.2.cmml">∈</mo><mrow id="alg4.l3.m1.1.1.1" xref="alg4.l3.m1.1.1.1.cmml"><mi id="alg4.l3.m1.1.1.1.3" xref="alg4.l3.m1.1.1.1.3.cmml">A</mi><mo id="alg4.l3.m1.1.1.1.2" xref="alg4.l3.m1.1.1.1.2.cmml">∖</mo><mrow id="alg4.l3.m1.1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.2.cmml"><mo id="alg4.l3.m1.1.1.1.1.1.2" stretchy="false" xref="alg4.l3.m1.1.1.1.1.2.cmml">{</mo><msup id="alg4.l3.m1.1.1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.1.1.cmml"><mi id="alg4.l3.m1.1.1.1.1.1.1.2" xref="alg4.l3.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="alg4.l3.m1.1.1.1.1.1.1.3" xref="alg4.l3.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="alg4.l3.m1.1.1.1.1.1.3" stretchy="false" xref="alg4.l3.m1.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l3.m1.1b"><apply id="alg4.l3.m1.1.1.cmml" xref="alg4.l3.m1.1.1"><in id="alg4.l3.m1.1.1.2.cmml" xref="alg4.l3.m1.1.1.2"></in><ci id="alg4.l3.m1.1.1.3.cmml" xref="alg4.l3.m1.1.1.3">𝑎</ci><apply id="alg4.l3.m1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1"><setdiff id="alg4.l3.m1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.2"></setdiff><ci id="alg4.l3.m1.1.1.1.3.cmml" xref="alg4.l3.m1.1.1.1.3">𝐴</ci><set id="alg4.l3.m1.1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.1.1"><apply id="alg4.l3.m1.1.1.1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg4.l3.m1.1.1.1.1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1">superscript</csymbol><ci id="alg4.l3.m1.1.1.1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.2">𝑎</ci><times id="alg4.l3.m1.1.1.1.1.1.1.3.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.3"></times></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l3.m1.1c">a\in A\setminus\{a^{*}\}</annotation><annotation encoding="application/x-llamapun" id="alg4.l3.m1.1d">italic_a ∈ italic_A ∖ { italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT }</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg4.l3.3">do</span> </div> <div class="ltx_listingline" id="alg4.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l4.1.1.1" style="font-size:80%;">4:</span></span> pay <math alttext="0" class="ltx_Math" display="inline" id="alg4.l4.m1.1"><semantics id="alg4.l4.m1.1a"><mn id="alg4.l4.m1.1.1" xref="alg4.l4.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="alg4.l4.m1.1b"><cn id="alg4.l4.m1.1.1.cmml" type="integer" xref="alg4.l4.m1.1.1">0</cn></annotation-xml></semantics></math> for all actions other than <math alttext="a" class="ltx_Math" display="inline" id="alg4.l4.m2.1"><semantics id="alg4.l4.m2.1a"><mi id="alg4.l4.m2.1.1" xref="alg4.l4.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="alg4.l4.m2.1b"><ci id="alg4.l4.m2.1.1.cmml" xref="alg4.l4.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.l4.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="alg4.l4.m2.1d">italic_a</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg4.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l5.1.1.1" style="font-size:80%;">5:</span></span> try paying action <math alttext="a" class="ltx_Math" display="inline" id="alg4.l5.m1.1"><semantics id="alg4.l5.m1.1a"><mi id="alg4.l5.m1.1.1" xref="alg4.l5.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="alg4.l5.m1.1b"><ci id="alg4.l5.m1.1.1.cmml" xref="alg4.l5.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.l5.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="alg4.l5.m1.1d">italic_a</annotation></semantics></math> by <math alttext="P(a)=0,\varepsilon,2\varepsilon,\ldots,k\varepsilon" class="ltx_Math" display="inline" id="alg4.l5.m2.6"><semantics id="alg4.l5.m2.6a"><mrow id="alg4.l5.m2.6.6" xref="alg4.l5.m2.6.6.cmml"><mrow id="alg4.l5.m2.6.6.4" xref="alg4.l5.m2.6.6.4.cmml"><mi id="alg4.l5.m2.6.6.4.2" xref="alg4.l5.m2.6.6.4.2.cmml">P</mi><mo id="alg4.l5.m2.6.6.4.1" xref="alg4.l5.m2.6.6.4.1.cmml">⁢</mo><mrow id="alg4.l5.m2.6.6.4.3.2" xref="alg4.l5.m2.6.6.4.cmml"><mo id="alg4.l5.m2.6.6.4.3.2.1" stretchy="false" xref="alg4.l5.m2.6.6.4.cmml">(</mo><mi id="alg4.l5.m2.1.1" xref="alg4.l5.m2.1.1.cmml">a</mi><mo id="alg4.l5.m2.6.6.4.3.2.2" stretchy="false" xref="alg4.l5.m2.6.6.4.cmml">)</mo></mrow></mrow><mo id="alg4.l5.m2.6.6.3" xref="alg4.l5.m2.6.6.3.cmml">=</mo><mrow id="alg4.l5.m2.6.6.2.2" xref="alg4.l5.m2.6.6.2.3.cmml"><mn id="alg4.l5.m2.2.2" xref="alg4.l5.m2.2.2.cmml">0</mn><mo id="alg4.l5.m2.6.6.2.2.3" xref="alg4.l5.m2.6.6.2.3.cmml">,</mo><mi id="alg4.l5.m2.3.3" xref="alg4.l5.m2.3.3.cmml">ε</mi><mo id="alg4.l5.m2.6.6.2.2.4" xref="alg4.l5.m2.6.6.2.3.cmml">,</mo><mrow id="alg4.l5.m2.5.5.1.1.1" xref="alg4.l5.m2.5.5.1.1.1.cmml"><mn id="alg4.l5.m2.5.5.1.1.1.2" xref="alg4.l5.m2.5.5.1.1.1.2.cmml">2</mn><mo id="alg4.l5.m2.5.5.1.1.1.1" xref="alg4.l5.m2.5.5.1.1.1.1.cmml">⁢</mo><mi id="alg4.l5.m2.5.5.1.1.1.3" xref="alg4.l5.m2.5.5.1.1.1.3.cmml">ε</mi></mrow><mo id="alg4.l5.m2.6.6.2.2.5" xref="alg4.l5.m2.6.6.2.3.cmml">,</mo><mi id="alg4.l5.m2.4.4" mathvariant="normal" xref="alg4.l5.m2.4.4.cmml">…</mi><mo id="alg4.l5.m2.6.6.2.2.6" xref="alg4.l5.m2.6.6.2.3.cmml">,</mo><mrow id="alg4.l5.m2.6.6.2.2.2" xref="alg4.l5.m2.6.6.2.2.2.cmml"><mi id="alg4.l5.m2.6.6.2.2.2.2" xref="alg4.l5.m2.6.6.2.2.2.2.cmml">k</mi><mo id="alg4.l5.m2.6.6.2.2.2.1" xref="alg4.l5.m2.6.6.2.2.2.1.cmml">⁢</mo><mi id="alg4.l5.m2.6.6.2.2.2.3" xref="alg4.l5.m2.6.6.2.2.2.3.cmml">ε</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l5.m2.6b"><apply id="alg4.l5.m2.6.6.cmml" xref="alg4.l5.m2.6.6"><eq id="alg4.l5.m2.6.6.3.cmml" xref="alg4.l5.m2.6.6.3"></eq><apply id="alg4.l5.m2.6.6.4.cmml" xref="alg4.l5.m2.6.6.4"><times id="alg4.l5.m2.6.6.4.1.cmml" xref="alg4.l5.m2.6.6.4.1"></times><ci id="alg4.l5.m2.6.6.4.2.cmml" xref="alg4.l5.m2.6.6.4.2">𝑃</ci><ci id="alg4.l5.m2.1.1.cmml" xref="alg4.l5.m2.1.1">𝑎</ci></apply><list id="alg4.l5.m2.6.6.2.3.cmml" xref="alg4.l5.m2.6.6.2.2"><cn id="alg4.l5.m2.2.2.cmml" type="integer" xref="alg4.l5.m2.2.2">0</cn><ci id="alg4.l5.m2.3.3.cmml" xref="alg4.l5.m2.3.3">𝜀</ci><apply id="alg4.l5.m2.5.5.1.1.1.cmml" xref="alg4.l5.m2.5.5.1.1.1"><times id="alg4.l5.m2.5.5.1.1.1.1.cmml" xref="alg4.l5.m2.5.5.1.1.1.1"></times><cn id="alg4.l5.m2.5.5.1.1.1.2.cmml" type="integer" xref="alg4.l5.m2.5.5.1.1.1.2">2</cn><ci id="alg4.l5.m2.5.5.1.1.1.3.cmml" xref="alg4.l5.m2.5.5.1.1.1.3">𝜀</ci></apply><ci id="alg4.l5.m2.4.4.cmml" xref="alg4.l5.m2.4.4">…</ci><apply id="alg4.l5.m2.6.6.2.2.2.cmml" xref="alg4.l5.m2.6.6.2.2.2"><times id="alg4.l5.m2.6.6.2.2.2.1.cmml" xref="alg4.l5.m2.6.6.2.2.2.1"></times><ci id="alg4.l5.m2.6.6.2.2.2.2.cmml" xref="alg4.l5.m2.6.6.2.2.2.2">𝑘</ci><ci id="alg4.l5.m2.6.6.2.2.2.3.cmml" xref="alg4.l5.m2.6.6.2.2.2.3">𝜀</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l5.m2.6c">P(a)=0,\varepsilon,2\varepsilon,\ldots,k\varepsilon</annotation><annotation encoding="application/x-llamapun" id="alg4.l5.m2.6d">italic_P ( italic_a ) = 0 , italic_ε , 2 italic_ε , … , italic_k italic_ε</annotation></semantics></math> until the agent plays action <math alttext="a" class="ltx_Math" display="inline" id="alg4.l5.m3.1"><semantics id="alg4.l5.m3.1a"><mi id="alg4.l5.m3.1.1" xref="alg4.l5.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="alg4.l5.m3.1b"><ci id="alg4.l5.m3.1.1.cmml" xref="alg4.l5.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.l5.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="alg4.l5.m3.1d">italic_a</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg4.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l6.1.1.1" style="font-size:80%;">6:</span></span> let <math alttext="\tilde{U}(a)=-k\varepsilon" class="ltx_Math" display="inline" id="alg4.l6.m1.1"><semantics id="alg4.l6.m1.1a"><mrow id="alg4.l6.m1.1.2" xref="alg4.l6.m1.1.2.cmml"><mrow id="alg4.l6.m1.1.2.2" xref="alg4.l6.m1.1.2.2.cmml"><mover accent="true" id="alg4.l6.m1.1.2.2.2" xref="alg4.l6.m1.1.2.2.2.cmml"><mi id="alg4.l6.m1.1.2.2.2.2" xref="alg4.l6.m1.1.2.2.2.2.cmml">U</mi><mo id="alg4.l6.m1.1.2.2.2.1" xref="alg4.l6.m1.1.2.2.2.1.cmml">~</mo></mover><mo id="alg4.l6.m1.1.2.2.1" xref="alg4.l6.m1.1.2.2.1.cmml">⁢</mo><mrow id="alg4.l6.m1.1.2.2.3.2" xref="alg4.l6.m1.1.2.2.cmml"><mo id="alg4.l6.m1.1.2.2.3.2.1" stretchy="false" xref="alg4.l6.m1.1.2.2.cmml">(</mo><mi id="alg4.l6.m1.1.1" xref="alg4.l6.m1.1.1.cmml">a</mi><mo id="alg4.l6.m1.1.2.2.3.2.2" stretchy="false" xref="alg4.l6.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="alg4.l6.m1.1.2.1" xref="alg4.l6.m1.1.2.1.cmml">=</mo><mrow id="alg4.l6.m1.1.2.3" xref="alg4.l6.m1.1.2.3.cmml"><mo id="alg4.l6.m1.1.2.3a" xref="alg4.l6.m1.1.2.3.cmml">−</mo><mrow id="alg4.l6.m1.1.2.3.2" xref="alg4.l6.m1.1.2.3.2.cmml"><mi id="alg4.l6.m1.1.2.3.2.2" xref="alg4.l6.m1.1.2.3.2.2.cmml">k</mi><mo id="alg4.l6.m1.1.2.3.2.1" xref="alg4.l6.m1.1.2.3.2.1.cmml">⁢</mo><mi id="alg4.l6.m1.1.2.3.2.3" xref="alg4.l6.m1.1.2.3.2.3.cmml">ε</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l6.m1.1b"><apply id="alg4.l6.m1.1.2.cmml" xref="alg4.l6.m1.1.2"><eq id="alg4.l6.m1.1.2.1.cmml" xref="alg4.l6.m1.1.2.1"></eq><apply id="alg4.l6.m1.1.2.2.cmml" xref="alg4.l6.m1.1.2.2"><times id="alg4.l6.m1.1.2.2.1.cmml" xref="alg4.l6.m1.1.2.2.1"></times><apply id="alg4.l6.m1.1.2.2.2.cmml" xref="alg4.l6.m1.1.2.2.2"><ci id="alg4.l6.m1.1.2.2.2.1.cmml" xref="alg4.l6.m1.1.2.2.2.1">~</ci><ci id="alg4.l6.m1.1.2.2.2.2.cmml" xref="alg4.l6.m1.1.2.2.2.2">𝑈</ci></apply><ci id="alg4.l6.m1.1.1.cmml" xref="alg4.l6.m1.1.1">𝑎</ci></apply><apply id="alg4.l6.m1.1.2.3.cmml" xref="alg4.l6.m1.1.2.3"><minus id="alg4.l6.m1.1.2.3.1.cmml" xref="alg4.l6.m1.1.2.3"></minus><apply id="alg4.l6.m1.1.2.3.2.cmml" xref="alg4.l6.m1.1.2.3.2"><times id="alg4.l6.m1.1.2.3.2.1.cmml" xref="alg4.l6.m1.1.2.3.2.1"></times><ci id="alg4.l6.m1.1.2.3.2.2.cmml" xref="alg4.l6.m1.1.2.3.2.2">𝑘</ci><ci id="alg4.l6.m1.1.2.3.2.3.cmml" xref="alg4.l6.m1.1.2.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l6.m1.1c">\tilde{U}(a)=-k\varepsilon</annotation><annotation encoding="application/x-llamapun" id="alg4.l6.m1.1d">over~ start_ARG italic_U end_ARG ( italic_a ) = - italic_k italic_ε</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg4.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l7.1.1.1" style="font-size:80%;">7:</span></span><span class="ltx_text ltx_font_bold" id="alg4.l7.2">return</span> <math alttext="\tilde{U}" class="ltx_Math" display="inline" id="alg4.l7.m1.1"><semantics id="alg4.l7.m1.1a"><mover accent="true" id="alg4.l7.m1.1.1" xref="alg4.l7.m1.1.1.cmml"><mi id="alg4.l7.m1.1.1.2" xref="alg4.l7.m1.1.1.2.cmml">U</mi><mo id="alg4.l7.m1.1.1.1" xref="alg4.l7.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="alg4.l7.m1.1b"><apply id="alg4.l7.m1.1.1.cmml" xref="alg4.l7.m1.1.1"><ci id="alg4.l7.m1.1.1.1.cmml" xref="alg4.l7.m1.1.1.1">~</ci><ci id="alg4.l7.m1.1.1.2.cmml" xref="alg4.l7.m1.1.1.2">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l7.m1.1c">\tilde{U}</annotation><annotation encoding="application/x-llamapun" id="alg4.l7.m1.1d">over~ start_ARG italic_U end_ARG</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg5.3.1.1">Algorithm 5</span> </span> Payment-minimizing learning algorithm for a single rational agent</figcaption> </figure> <div class="ltx_theorem ltx_theorem_proposition" id="S6.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="S6.Thmtheorem1.1.1.1">Proposition 6.1</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem1.p1"> <p class="ltx_p" id="S6.Thmtheorem1.p1.3"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p1.3.3">Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg5" title="Algorithm 5 ‣ Single rational agent ‣ 6 Minimizing Payment ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">5</span></a> <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.1.1.m1.1"><semantics id="S6.Thmtheorem1.p1.1.1.m1.1a"><mi id="S6.Thmtheorem1.p1.1.1.m1.1.1" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.1.1.m1.1b"><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.1.1.m1.1d">italic_ε</annotation></semantics></math>-learns the game with total payment at most <math alttext="\Delta+m\varepsilon" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.2.2.m2.1"><semantics id="S6.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S6.Thmtheorem1.p1.2.2.m2.1.1" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1.2" mathvariant="normal" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">Δ</mi><mo id="S6.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">+</mo><mrow id="S6.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.cmml"><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml">m</mi><mo id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.1" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.2.2.m2.1b"><apply id="S6.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1"><plus id="S6.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.1"></plus><ci id="S6.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.2">Δ</ci><apply id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3"><times id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.1"></times><ci id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2">𝑚</ci><ci id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.2.2.m2.1c">\Delta+m\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.2.2.m2.1d">roman_Δ + italic_m italic_ε</annotation></semantics></math>, where <math alttext="\Delta=\sum_{a\in A}\big{(}U(a^{*})-U(a)\big{)}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.3.3.m3.2"><semantics id="S6.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S6.Thmtheorem1.p1.3.3.m3.2.2" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.cmml"><mi id="S6.Thmtheorem1.p1.3.3.m3.2.2.3" mathvariant="normal" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.3.cmml">Δ</mi><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.2" rspace="0.111em" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.cmml">=</mo><mrow id="S6.Thmtheorem1.p1.3.3.m3.2.2.1" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.cmml"><msub id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.2" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.2.cmml"><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.2.2" rspace="0em" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.2.2.cmml">∑</mo><mrow 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xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.2" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.2.cmml">U</mi><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.1" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.3.2" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.cmml"><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.cmml">(</mo><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.cmml">a</mi><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.3" maxsize="120%" minsize="120%" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.3.3.m3.2b"><apply 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xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.2.3.3">𝐴</ci></apply></apply><apply id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1"><minus id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.2"></minus><apply id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1"><times id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.2"></times><ci id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.3">𝑈</ci><apply id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.2">𝑎</ci><times id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3"><times id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.1"></times><ci id="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.3.2">𝑈</ci><ci id="S6.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.3.3.m3.2c">\Delta=\sum_{a\in A}\big{(}U(a^{*})-U(a)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.3.3.m3.2d">roman_Δ = ∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT ( italic_U ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - italic_U ( italic_a ) )</annotation></semantics></math> is a constant that depends only on the utility differences of the actions.</span></p> </div> </div> <div class="ltx_proof" id="S6.SS0.SSS0.Px1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS0.SSS0.Px1.1.p1"> <p class="ltx_p" id="S6.SS0.SSS0.Px1.1.p1.9">For each action <math alttext="a\in A\setminus\{a^{*}\}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.1.m1.1"><semantics id="S6.SS0.SSS0.Px1.1.p1.1.m1.1a"><mrow id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.3" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.3.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.2" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.2.cmml">∈</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.3" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.3.cmml">A</mi><mo 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xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.2"></in><ci id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.3">𝑎</ci><apply id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1"><setdiff id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.2"></setdiff><ci id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.3">𝐴</ci><set id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1"><apply id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1">superscript</csymbol><ci id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1.2">𝑎</ci><times id="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.1.m1.1.1.1.1.1.1.3"></times></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.1.p1.1.m1.1c">a\in A\setminus\{a^{*}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.1.m1.1d">italic_a ∈ italic_A ∖ { italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT }</annotation></semantics></math>, at each round where the payment <math alttext="P(a)=k\varepsilon<U(a^{*})-U(a)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.2.m2.3"><semantics id="S6.SS0.SSS0.Px1.1.p1.2.m2.3a"><mrow id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.cmml"><mrow id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.2" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.2.cmml">P</mi><mo id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.1" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.1.cmml">⁢</mo><mrow 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xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.2" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.2.cmml">−</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.2" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.2.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.1" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.3.2" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.cmml"><mo id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.1.p1.2.m2.2.2" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.2.2.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.3.2.2" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.1.p1.2.m2.3b"><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3"><and id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3a.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3"></and><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3b.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3"><eq id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.4.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.4"></eq><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3"><times id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.1"></times><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.3.2">𝑃</ci><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.1.1">𝑎</ci></apply><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5"><times id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.1"></times><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.2">𝑘</ci><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.3">𝜀</ci></apply></apply><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3c.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3"><lt id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.6.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.6"></lt><share href="https://arxiv.org/html/2503.01976v1#S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.5.cmml" id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3d.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3"></share><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1"><minus id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.2"></minus><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1"><times id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.2"></times><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.3">𝑈</ci><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1">superscript</csymbol><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1.1.2">𝑎</ci><times id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.1.1.1.1.3"></times></apply></apply><apply id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3"><times id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.1"></times><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.3.3.1.3.2">𝑈</ci><ci id="S6.SS0.SSS0.Px1.1.p1.2.m2.2.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.2.m2.2.2">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.1.p1.2.m2.3c">P(a)=k\varepsilon<U(a^{*})-U(a)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.2.m2.3d">italic_P ( italic_a ) = italic_k italic_ε < italic_U ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - italic_U ( italic_a )</annotation></semantics></math>, the agent will play action <math alttext="a^{*}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.3.m3.1"><semantics id="S6.SS0.SSS0.Px1.1.p1.3.m3.1a"><msup id="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1" xref="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.2" xref="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.2.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.3" xref="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.1.p1.3.m3.1b"><apply id="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1">superscript</csymbol><ci id="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.2">𝑎</ci><times id="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.1.p1.3.m3.1c">a^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.3.m3.1d">italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, so we pay <math alttext="0" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.4.m4.1"><semantics id="S6.SS0.SSS0.Px1.1.p1.4.m4.1a"><mn id="S6.SS0.SSS0.Px1.1.p1.4.m4.1.1" xref="S6.SS0.SSS0.Px1.1.p1.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.1.p1.4.m4.1b"><cn id="S6.SS0.SSS0.Px1.1.p1.4.m4.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.1.p1.4.m4.1.1">0</cn></annotation-xml></semantics></math> to the agent. At the first round where <math alttext="k\varepsilon" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.5.m5.1"><semantics id="S6.SS0.SSS0.Px1.1.p1.5.m5.1a"><mrow id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.2" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.2.cmml">k</mi><mo id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.1" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.3" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.1.p1.5.m5.1b"><apply id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1"><times id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.1"></times><ci id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.2">𝑘</ci><ci id="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.5.m5.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.1.p1.5.m5.1c">k\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.5.m5.1d">italic_k italic_ε</annotation></semantics></math> exceeds <math alttext="U(a^{*})-U(a)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.6.m6.2"><semantics id="S6.SS0.SSS0.Px1.1.p1.6.m6.2a"><mrow id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.cmml"><mrow id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.3" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.3.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.2" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.cmml">(</mo><msup id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.2.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.3" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.2" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.2.cmml">−</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.2" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.2.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.1" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.3.2" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.cmml"><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.1.p1.6.m6.1.1" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.1.1.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.3.2.2" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.1.p1.6.m6.2b"><apply id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2"><minus id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.2"></minus><apply id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1"><times id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.2"></times><ci id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.3">𝑈</ci><apply id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1">superscript</csymbol><ci id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.2">𝑎</ci><times id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.1.1.1.1.3"></times></apply></apply><apply id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3"><times id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.1"></times><ci id="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.2.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.2.2.3.2">𝑈</ci><ci id="S6.SS0.SSS0.Px1.1.p1.6.m6.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.6.m6.1.1">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.1.p1.6.m6.2c">U(a^{*})-U(a)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.6.m6.2d">italic_U ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - italic_U ( italic_a )</annotation></semantics></math>, the agent plays action <math alttext="a" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.7.m7.1"><semantics id="S6.SS0.SSS0.Px1.1.p1.7.m7.1a"><mi id="S6.SS0.SSS0.Px1.1.p1.7.m7.1.1" xref="S6.SS0.SSS0.Px1.1.p1.7.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.1.p1.7.m7.1b"><ci id="S6.SS0.SSS0.Px1.1.p1.7.m7.1.1.cmml" xref="S6.SS0.SSS0.Px1.1.p1.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.1.p1.7.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.7.m7.1d">italic_a</annotation></semantics></math>, so we pay <math alttext="P(a)=k\varepsilon\leq U(a^{*})-U(a)+\varepsilon" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.8.m8.3"><semantics id="S6.SS0.SSS0.Px1.1.p1.8.m8.3a"><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.cmml"><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.2" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.2.cmml">P</mi><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.1" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.3.2" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.cmml"><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.1.p1.8.m8.1.1" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.1.1.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.3.2.2" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.3.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.4" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.4.cmml">=</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5.2" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5.2.cmml">k</mi><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5.1" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5.1.cmml">⁢</mo><mi id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5.3" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.5.3.cmml">ε</mi></mrow><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.6" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.6.cmml">≤</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.cmml"><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.cmml"><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.3" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.3.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.2" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.1.p1.8.m8.3.3.1.1.1.1.1.1.cmml">(</mo><msup 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id="S6.SS0.SSS0.Px1.1.p1.8.m8.3c">P(a)=k\varepsilon\leq U(a^{*})-U(a)+\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.8.m8.3d">italic_P ( italic_a ) = italic_k italic_ε ≤ italic_U ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - italic_U ( italic_a ) + italic_ε</annotation></semantics></math> to the agent. Summing over all actions, the total payment is at most <math alttext="\sum_{a\in A}\big{(}U(a^{*})-U(a)\big{)}+m\varepsilon" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.1.p1.9.m9.2"><semantics id="S6.SS0.SSS0.Px1.1.p1.9.m9.2a"><mrow id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.cmml"><mrow id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.cmml"><msub id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.cmml"><mo id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.2" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.2.cmml">∑</mo><mrow id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.3" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.3.cmml"><mi id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.3.2" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.3.2.cmml">a</mi><mo id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.3.1" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.3.1.cmml">∈</mo><mi id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.1.2.3.3" 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id="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.3.3.cmml" xref="S6.SS0.SSS0.Px1.1.p1.9.m9.2.2.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.1.p1.9.m9.2c">\sum_{a\in A}\big{(}U(a^{*})-U(a)\big{)}+m\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.1.p1.9.m9.2d">∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT ( italic_U ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - italic_U ( italic_a ) ) + italic_m italic_ε</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="S6.SS0.SSS0.Px1.p2.4"><span class="ltx_text ltx_font_bold" id="S6.SS0.SSS0.Px1.p2.4.1">Multiple agents.</span> The above observation that the total payment can be significantly smaller than the number of rounds is specific to the single agent case. When there are multiple agents, the minimal achievable payment turns out to be lower bounded by the number of rounds. Formally, we define the <span class="ltx_text ltx_font_italic" id="S6.SS0.SSS0.Px1.p2.4.2">payment complexity</span> <math alttext="\mathrm{PC}(n,\varepsilon)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p2.1.m1.2"><semantics id="S6.SS0.SSS0.Px1.p2.1.m1.2a"><mrow id="S6.SS0.SSS0.Px1.p2.1.m1.2.3" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.cmml"><mi id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.2" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.2.cmml">PC</mi><mo id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.1" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.2" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.p2.1.m1.1.1" xref="S6.SS0.SSS0.Px1.p2.1.m1.1.1.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.2.2" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.p2.1.m1.2.2" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.2.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p2.1.m1.2b"><apply id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.cmml" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3"><times id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.1.cmml" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.1"></times><ci id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.2.cmml" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.2">PC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.3.3.2"><ci id="S6.SS0.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p2.1.m1.1.1">𝑛</ci><ci id="S6.SS0.SSS0.Px1.p2.1.m1.2.2.cmml" xref="S6.SS0.SSS0.Px1.p2.1.m1.2.2">𝜀</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p2.1.m1.2c">\mathrm{PC}(n,\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p2.1.m1.2d">roman_PC ( italic_n , italic_ε )</annotation></semantics></math> to be the minimal total payment required to <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p2.2.m2.1"><semantics id="S6.SS0.SSS0.Px1.p2.2.m2.1a"><mi id="S6.SS0.SSS0.Px1.p2.2.m2.1.1" xref="S6.SS0.SSS0.Px1.p2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p2.2.m2.1b"><ci id="S6.SS0.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S6.SS0.SSS0.Px1.p2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p2.2.m2.1d">italic_ε</annotation></semantics></math>-learn a game with <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p2.3.m3.1"><semantics id="S6.SS0.SSS0.Px1.p2.3.m3.1a"><mi id="S6.SS0.SSS0.Px1.p2.3.m3.1.1" xref="S6.SS0.SSS0.Px1.p2.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p2.3.m3.1b"><ci id="S6.SS0.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S6.SS0.SSS0.Px1.p2.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p2.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p2.3.m3.1d">italic_n</annotation></semantics></math> agents (in the worst case over all games), and the <span class="ltx_text ltx_font_italic" id="S6.SS0.SSS0.Px1.p2.4.3">round complexity</span> <math alttext="\mathrm{RC}(n,\varepsilon)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.p2.4.m4.2"><semantics id="S6.SS0.SSS0.Px1.p2.4.m4.2a"><mrow id="S6.SS0.SSS0.Px1.p2.4.m4.2.3" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.cmml"><mi id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.2" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.2.cmml">RC</mi><mo id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.1" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.2" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.1.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.p2.4.m4.1.1" xref="S6.SS0.SSS0.Px1.p2.4.m4.1.1.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.2.2" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.1.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.p2.4.m4.2.2" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.2.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p2.4.m4.2b"><apply id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.cmml" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3"><times id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.1.cmml" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.1"></times><ci id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.2.cmml" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.2">RC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.3.3.2"><ci id="S6.SS0.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S6.SS0.SSS0.Px1.p2.4.m4.1.1">𝑛</ci><ci id="S6.SS0.SSS0.Px1.p2.4.m4.2.2.cmml" xref="S6.SS0.SSS0.Px1.p2.4.m4.2.2">𝜀</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p2.4.m4.2c">\mathrm{RC}(n,\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.p2.4.m4.2d">roman_RC ( italic_n , italic_ε )</annotation></semantics></math> to be the minimal number of rounds to do so. Then, we have:</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.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="S6.Thmtheorem2.1.1.1">Proposition 6.2</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem2.p1"> <p class="ltx_p" id="S6.Thmtheorem2.p1.1"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem2.p1.1.1">Suppose the payment to each agent only depends on the agent’s own action. Under both rationalizable and no-regret behavioral models, we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.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="\Omega\big{(}\mathrm{RC}(n-1,\varepsilon)\big{)}\leq\mathrm{PC}(n,\varepsilon)% \leq{\mathcal{O}}\big{(}n\cdot\mathrm{RC}(n,\varepsilon)\big{)}." class="ltx_Math" display="block" id="S6.Ex3.m1.6"><semantics id="S6.Ex3.m1.6a"><mrow id="S6.Ex3.m1.6.6.1" xref="S6.Ex3.m1.6.6.1.1.cmml"><mrow id="S6.Ex3.m1.6.6.1.1" xref="S6.Ex3.m1.6.6.1.1.cmml"><mrow id="S6.Ex3.m1.6.6.1.1.1" xref="S6.Ex3.m1.6.6.1.1.1.cmml"><mi id="S6.Ex3.m1.6.6.1.1.1.3" mathvariant="normal" xref="S6.Ex3.m1.6.6.1.1.1.3.cmml">Ω</mi><mo id="S6.Ex3.m1.6.6.1.1.1.2" xref="S6.Ex3.m1.6.6.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex3.m1.6.6.1.1.1.1.1" xref="S6.Ex3.m1.6.6.1.1.1.1.1.1.cmml"><mo id="S6.Ex3.m1.6.6.1.1.1.1.1.2" maxsize="120%" 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xref="S6.Ex3.m1.1.1.cmml">ε</mi><mo id="S6.Ex3.m1.6.6.1.1.1.1.1.1.1.1.4" stretchy="false" xref="S6.Ex3.m1.6.6.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Ex3.m1.6.6.1.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S6.Ex3.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex3.m1.6.6.1.1.4" xref="S6.Ex3.m1.6.6.1.1.4.cmml">≤</mo><mrow id="S6.Ex3.m1.6.6.1.1.5" xref="S6.Ex3.m1.6.6.1.1.5.cmml"><mi id="S6.Ex3.m1.6.6.1.1.5.2" xref="S6.Ex3.m1.6.6.1.1.5.2.cmml">PC</mi><mo id="S6.Ex3.m1.6.6.1.1.5.1" xref="S6.Ex3.m1.6.6.1.1.5.1.cmml">⁢</mo><mrow id="S6.Ex3.m1.6.6.1.1.5.3.2" xref="S6.Ex3.m1.6.6.1.1.5.3.1.cmml"><mo id="S6.Ex3.m1.6.6.1.1.5.3.2.1" stretchy="false" xref="S6.Ex3.m1.6.6.1.1.5.3.1.cmml">(</mo><mi id="S6.Ex3.m1.2.2" xref="S6.Ex3.m1.2.2.cmml">n</mi><mo id="S6.Ex3.m1.6.6.1.1.5.3.2.2" xref="S6.Ex3.m1.6.6.1.1.5.3.1.cmml">,</mo><mi id="S6.Ex3.m1.3.3" xref="S6.Ex3.m1.3.3.cmml">ε</mi><mo id="S6.Ex3.m1.6.6.1.1.5.3.2.3" stretchy="false" 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xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.3.cmml">RC</mi></mrow><mo id="S6.Ex3.m1.6.6.1.1.2.1.1.1.1" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.2" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.1.cmml"><mo id="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.2.1" stretchy="false" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.1.cmml">(</mo><mi id="S6.Ex3.m1.4.4" xref="S6.Ex3.m1.4.4.cmml">n</mi><mo id="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.2.2" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.1.cmml">,</mo><mi id="S6.Ex3.m1.5.5" xref="S6.Ex3.m1.5.5.cmml">ε</mi><mo id="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.2.3" stretchy="false" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex3.m1.6.6.1.1.2.1.1.3" maxsize="120%" minsize="120%" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Ex3.m1.6.6.1.2" lspace="0em" xref="S6.Ex3.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex3.m1.6b"><apply id="S6.Ex3.m1.6.6.1.1.cmml" xref="S6.Ex3.m1.6.6.1"><and 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xref="S6.Ex3.m1.6.6.1.1.1.1.1.1.1.1.1.2">𝑛</ci><cn id="S6.Ex3.m1.6.6.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Ex3.m1.6.6.1.1.1.1.1.1.1.1.1.3">1</cn></apply><ci id="S6.Ex3.m1.1.1.cmml" xref="S6.Ex3.m1.1.1">𝜀</ci></interval></apply></apply><apply id="S6.Ex3.m1.6.6.1.1.5.cmml" xref="S6.Ex3.m1.6.6.1.1.5"><times id="S6.Ex3.m1.6.6.1.1.5.1.cmml" xref="S6.Ex3.m1.6.6.1.1.5.1"></times><ci id="S6.Ex3.m1.6.6.1.1.5.2.cmml" xref="S6.Ex3.m1.6.6.1.1.5.2">PC</ci><interval closure="open" id="S6.Ex3.m1.6.6.1.1.5.3.1.cmml" xref="S6.Ex3.m1.6.6.1.1.5.3.2"><ci id="S6.Ex3.m1.2.2.cmml" xref="S6.Ex3.m1.2.2">𝑛</ci><ci id="S6.Ex3.m1.3.3.cmml" xref="S6.Ex3.m1.3.3">𝜀</ci></interval></apply></apply><apply id="S6.Ex3.m1.6.6.1.1c.cmml" xref="S6.Ex3.m1.6.6.1"><leq id="S6.Ex3.m1.6.6.1.1.6.cmml" xref="S6.Ex3.m1.6.6.1.1.6"></leq><share href="https://arxiv.org/html/2503.01976v1#S6.Ex3.m1.6.6.1.1.5.cmml" id="S6.Ex3.m1.6.6.1.1d.cmml" xref="S6.Ex3.m1.6.6.1"></share><apply id="S6.Ex3.m1.6.6.1.1.2.cmml" xref="S6.Ex3.m1.6.6.1.1.2"><times id="S6.Ex3.m1.6.6.1.1.2.2.cmml" xref="S6.Ex3.m1.6.6.1.1.2.2"></times><ci id="S6.Ex3.m1.6.6.1.1.2.3.cmml" xref="S6.Ex3.m1.6.6.1.1.2.3">𝒪</ci><apply id="S6.Ex3.m1.6.6.1.1.2.1.1.1.cmml" xref="S6.Ex3.m1.6.6.1.1.2.1.1"><times id="S6.Ex3.m1.6.6.1.1.2.1.1.1.1.cmml" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.1"></times><apply id="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.cmml" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.2"><ci id="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.1.cmml" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.1">⋅</ci><ci id="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.2.cmml" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.2">𝑛</ci><ci id="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.3.cmml" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.2.3">RC</ci></apply><interval closure="open" id="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.1.cmml" xref="S6.Ex3.m1.6.6.1.1.2.1.1.1.3.2"><ci id="S6.Ex3.m1.4.4.cmml" xref="S6.Ex3.m1.4.4">𝑛</ci><ci id="S6.Ex3.m1.5.5.cmml" xref="S6.Ex3.m1.5.5">𝜀</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex3.m1.6c">\Omega\big{(}\mathrm{RC}(n-1,\varepsilon)\big{)}\leq\mathrm{PC}(n,\varepsilon)% \leq{\mathcal{O}}\big{(}n\cdot\mathrm{RC}(n,\varepsilon)\big{)}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex3.m1.6d">roman_Ω ( roman_RC ( italic_n - 1 , italic_ε ) ) ≤ roman_PC ( italic_n , italic_ε ) ≤ caligraphic_O ( italic_n ⋅ roman_RC ( italic_n , italic_ε ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S6.SS0.SSS0.Px1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS0.SSS0.Px1.2.p1"> <p class="ltx_p" id="S6.SS0.SSS0.Px1.2.p1.2">The inequality <math alttext="\mathrm{PC}(n,\varepsilon)\leq{\mathcal{O}}\big{(}n\cdot\mathrm{RC}(n,% \varepsilon)\big{)}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.2.p1.1.m1.5"><semantics id="S6.SS0.SSS0.Px1.2.p1.1.m1.5a"><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.cmml"><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.cmml"><mi id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.2.cmml">PC</mi><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.1" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.1.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.2.p1.1.m1.1.1" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.1.1.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.2.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.1.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.2.p1.1.m1.2.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.2.2.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.2.cmml">≤</mo><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.3" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.3.cmml">𝒪</mi><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.2" maxsize="120%" minsize="120%" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.cmml"><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.cmml"><mi id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.1.cmml">⋅</mo><mi id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.3" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.3.cmml">RC</mi></mrow><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.1.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.2.p1.1.m1.3.3" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.3.3.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.2.2" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.1.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.2.p1.1.m1.4.4" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.4.4.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.3" maxsize="120%" minsize="120%" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.2.p1.1.m1.5b"><apply id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5"><leq id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.2"></leq><apply id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3"><times id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.1"></times><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.2">PC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.3.3.2"><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.1.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.1.1">𝑛</ci><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.2.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.2.2">𝜀</ci></interval></apply><apply id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1"><times id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.2"></times><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.3.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.3">𝒪</ci><apply id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1"><times id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.1"></times><apply id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2"><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.1">⋅</ci><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.2">𝑛</ci><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.3.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.2.3">RC</ci></apply><interval closure="open" id="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.5.5.1.1.1.1.3.2"><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.3.3.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.3.3">𝑛</ci><ci id="S6.SS0.SSS0.Px1.2.p1.1.m1.4.4.cmml" xref="S6.SS0.SSS0.Px1.2.p1.1.m1.4.4">𝜀</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.2.p1.1.m1.5c">\mathrm{PC}(n,\varepsilon)\leq{\mathcal{O}}\big{(}n\cdot\mathrm{RC}(n,% \varepsilon)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.2.p1.1.m1.5d">roman_PC ( italic_n , italic_ε ) ≤ caligraphic_O ( italic_n ⋅ roman_RC ( italic_n , italic_ε ) )</annotation></semantics></math> is straightforward because the payment to each agent is bounded by <math alttext="O(1)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.2.p1.2.m2.1"><semantics id="S6.SS0.SSS0.Px1.2.p1.2.m2.1a"><mrow id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.cmml"><mi id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.2" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.2.cmml">O</mi><mo id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.1" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.3.2" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.cmml"><mo id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.cmml">(</mo><mn id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.1" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.1.cmml">1</mn><mo id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.2.p1.2.m2.1b"><apply id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2"><times id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.1.cmml" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.1"></times><ci id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.2.cmml" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.2.2">𝑂</ci><cn id="S6.SS0.SSS0.Px1.2.p1.2.m2.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.2.p1.2.m2.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.2.p1.2.m2.1c">O(1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.2.p1.2.m2.1d">italic_O ( 1 )</annotation></semantics></math> at each round.</p> </div> <div class="ltx_para" id="S6.SS0.SSS0.Px1.3.p2"> <p class="ltx_p" id="S6.SS0.SSS0.Px1.3.p2.29">To prove <math alttext="\Omega\big{(}\mathrm{RC}(n-1,\varepsilon)\big{)}\leq\mathrm{PC}(n,\varepsilon)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.1.m1.4"><semantics id="S6.SS0.SSS0.Px1.3.p2.1.m1.4a"><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.cmml"><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.3" mathvariant="normal" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.3.cmml">Ω</mi><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.2" maxsize="120%" minsize="120%" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.3.cmml">RC</mi><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.2.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.2.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.2.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.3.p2.1.m1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.1.1.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.4" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.3" maxsize="120%" minsize="120%" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.2.cmml">≤</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.2.cmml">PC</mi><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.1" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.1.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.3.p2.1.m1.2.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.2.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.2.2" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.1.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.3.p2.1.m1.3.3" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.3.3.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.1.m1.4b"><apply id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4"><leq id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.2"></leq><apply id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1"><times id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.2"></times><ci id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.3">Ω</ci><apply id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1"><times id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.2"></times><ci id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.3">RC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1"><apply id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1"><minus id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.1.1.1.1.1.1.1.3">1</cn></apply><ci id="S6.SS0.SSS0.Px1.3.p2.1.m1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.1.1">𝜀</ci></interval></apply></apply><apply id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3"><times id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.1"></times><ci id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.2">PC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.4.4.3.3.2"><ci id="S6.SS0.SSS0.Px1.3.p2.1.m1.2.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.2.2">𝑛</ci><ci id="S6.SS0.SSS0.Px1.3.p2.1.m1.3.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.1.m1.3.3">𝜀</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.1.m1.4c">\Omega\big{(}\mathrm{RC}(n-1,\varepsilon)\big{)}\leq\mathrm{PC}(n,\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.1.m1.4d">roman_Ω ( roman_RC ( italic_n - 1 , italic_ε ) ) ≤ roman_PC ( italic_n , italic_ε )</annotation></semantics></math>, we reduce the utility learning problem with <math alttext="n-1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.2.m2.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.2.m2.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.2.m2.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1"><minus id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.2.m2.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.2.m2.1d">italic_n - 1</annotation></semantics></math> agents to the problem with <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.3.m3.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.3.m3.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.3.m3.1.1" xref="S6.SS0.SSS0.Px1.3.p2.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.3.m3.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.3.m3.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.3.m3.1d">italic_n</annotation></semantics></math> agents. Let <math alttext="\Gamma_{n-1}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.4.m4.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.4.m4.1a"><msub id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.2.cmml">Γ</mi><mrow id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.2" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.1" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.3" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.4.m4.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.2">Γ</ci><apply id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3"><minus id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.4.m4.1c">\Gamma_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.4.m4.1d">roman_Γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> be an <math alttext="(n-1)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.5.m5.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.5.m5.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.3" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.5.m5.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1"><minus id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.5.m5.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.5.m5.1c">(n-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.5.m5.1d">( italic_n - 1 )</annotation></semantics></math>-agent game with utility functions <math alttext="U_{1},\ldots,U_{n-1}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.6.m6.3"><semantics id="S6.SS0.SSS0.Px1.3.p2.6.m6.3a"><mrow id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.3.cmml"><msub id="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.2.cmml">U</mi><mn id="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.3" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.3.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.3.p2.6.m6.1.1" mathvariant="normal" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.1.1.cmml">…</mi><mo id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.4" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.3.cmml">,</mo><msub id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.2" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.2.cmml">U</mi><mrow id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.2" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.1" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.3" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.6.m6.3b"><list id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2"><apply id="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.2">𝑈</ci><cn id="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.2.2.1.1.3">1</cn></apply><ci id="S6.SS0.SSS0.Px1.3.p2.6.m6.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.1.1">…</ci><apply id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.2">𝑈</ci><apply id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3"><minus id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.6.m6.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.6.m6.3c">U_{1},\ldots,U_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.6.m6.3d">italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_U start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Consider an <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.7.m7.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.7.m7.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.7.m7.1.1" xref="S6.SS0.SSS0.Px1.3.p2.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.7.m7.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.7.m7.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.7.m7.1d">italic_n</annotation></semantics></math>-agent game <math alttext="\Gamma_{n}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.8.m8.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.8.m8.1a"><msub id="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1" xref="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.2.cmml">Γ</mi><mi id="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.8.m8.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.2">Γ</ci><ci id="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.8.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.8.m8.1c">\Gamma_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.8.m8.1d">roman_Γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> where the <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.9.m9.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.9.m9.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.9.m9.1.1" xref="S6.SS0.SSS0.Px1.3.p2.9.m9.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.9.m9.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.9.m9.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.9.m9.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.9.m9.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.9.m9.1d">italic_n</annotation></semantics></math>-th agent has two actions <math alttext="A_{n}=\{0,1\}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.10.m10.2"><semantics id="S6.SS0.SSS0.Px1.3.p2.10.m10.2a"><mrow id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.cmml"><msub id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.2" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.2.cmml">A</mi><mi id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.3" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.3.cmml">n</mi></msub><mo id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.1" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.1.cmml">=</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.2" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.1.cmml">{</mo><mn id="S6.SS0.SSS0.Px1.3.p2.10.m10.1.1" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.1.1.cmml">0</mn><mo id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.2.2" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.1.cmml">,</mo><mn id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.2" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.2.cmml">1</mn><mo id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.10.m10.2b"><apply id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3"><eq id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.1"></eq><apply id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.2">𝐴</ci><ci id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.2.3">𝑛</ci></apply><set id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.3.3.2"><cn id="S6.SS0.SSS0.Px1.3.p2.10.m10.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.1.1">0</cn><cn id="S6.SS0.SSS0.Px1.3.p2.10.m10.2.2.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.10.m10.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.10.m10.2c">A_{n}=\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.10.m10.2d">italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = { 0 , 1 }</annotation></semantics></math>. If the <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.11.m11.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.11.m11.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.11.m11.1.1" xref="S6.SS0.SSS0.Px1.3.p2.11.m11.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.11.m11.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.11.m11.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.11.m11.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.11.m11.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.11.m11.1d">italic_n</annotation></semantics></math>-th agent takes action <math alttext="1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.12.m12.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.12.m12.1a"><mn id="S6.SS0.SSS0.Px1.3.p2.12.m12.1.1" xref="S6.SS0.SSS0.Px1.3.p2.12.m12.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.12.m12.1b"><cn id="S6.SS0.SSS0.Px1.3.p2.12.m12.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.12.m12.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.12.m12.1c">1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.12.m12.1d">1</annotation></semantics></math>, then all the first <math alttext="n-1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.13.m13.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.13.m13.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.13.m13.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1"><minus id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.13.m13.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.13.m13.1d">italic_n - 1</annotation></semantics></math> agents will obtain utility <math alttext="0" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.14.m14.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.14.m14.1a"><mn id="S6.SS0.SSS0.Px1.3.p2.14.m14.1.1" xref="S6.SS0.SSS0.Px1.3.p2.14.m14.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.14.m14.1b"><cn id="S6.SS0.SSS0.Px1.3.p2.14.m14.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.14.m14.1.1">0</cn></annotation-xml></semantics></math> regardless of their actions; if the <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.15.m15.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.15.m15.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.15.m15.1.1" xref="S6.SS0.SSS0.Px1.3.p2.15.m15.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.15.m15.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.15.m15.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.15.m15.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.15.m15.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.15.m15.1d">italic_n</annotation></semantics></math>-th agent takes action <math alttext="0" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.16.m16.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.16.m16.1a"><mn id="S6.SS0.SSS0.Px1.3.p2.16.m16.1.1" xref="S6.SS0.SSS0.Px1.3.p2.16.m16.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.16.m16.1b"><cn id="S6.SS0.SSS0.Px1.3.p2.16.m16.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.16.m16.1.1">0</cn></annotation-xml></semantics></math>, then the first <math alttext="n-1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.17.m17.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.17.m17.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.17.m17.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1"><minus id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.17.m17.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.17.m17.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.17.m17.1d">italic_n - 1</annotation></semantics></math> agents have the same utility functions as in game <math alttext="\Gamma_{n-1}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.18.m18.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.18.m18.1a"><msub id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.2.cmml">Γ</mi><mrow id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.2" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.1" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.3" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.18.m18.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.2">Γ</ci><apply id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3"><minus id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.18.m18.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.18.m18.1c">\Gamma_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.18.m18.1d">roman_Γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Further assume that the <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.19.m19.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.19.m19.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.19.m19.1.1" xref="S6.SS0.SSS0.Px1.3.p2.19.m19.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.19.m19.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.19.m19.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.19.m19.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.19.m19.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.19.m19.1d">italic_n</annotation></semantics></math>-th agent’s utility depends on his own action only, and in particular, <math alttext="U_{n}(a_{n}=0)=0" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.20.m20.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.20.m20.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.cmml"><mrow id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.cmml"><msub id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.2" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.2.cmml">U</mi><mi id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.3" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.3.cmml">n</mi></msub><mo id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.cmml"><msub id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.2" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.2.cmml">a</mi><mi id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.3" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.1.cmml">=</mo><mn id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.3" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.2.cmml">=</mo><mn id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.20.m20.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1"><eq id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.2"></eq><apply id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1"><times id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.2"></times><apply id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.2">𝑈</ci><ci id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.3.3">𝑛</ci></apply><apply id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1"><eq id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.1"></eq><apply id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.2">𝑎</ci><ci id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.2.3">𝑛</ci></apply><cn id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.1.1.1.1.3">0</cn></apply></apply><cn id="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.20.m20.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.20.m20.1c">U_{n}(a_{n}=0)=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.20.m20.1d">italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ) = 0</annotation></semantics></math> and <math alttext="U_{n}(a_{n}=1)=1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.21.m21.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.21.m21.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.cmml"><mrow id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.cmml"><msub id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.2" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.2.cmml">U</mi><mi id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.3" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.3.cmml">n</mi></msub><mo id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.cmml"><msub id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.2" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.2.cmml">a</mi><mi id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.3" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.1.cmml">=</mo><mn id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.3" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.2.cmml">=</mo><mn id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.21.m21.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1"><eq id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.2"></eq><apply id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1"><times id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.2"></times><apply id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.2">𝑈</ci><ci id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.3.3">𝑛</ci></apply><apply id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1"><eq id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.1"></eq><apply id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.2">𝑎</ci><ci id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.2.3">𝑛</ci></apply><cn id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.1.1.1.1.3">1</cn></apply></apply><cn id="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.21.m21.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.21.m21.1c">U_{n}(a_{n}=1)=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.21.m21.1d">italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 1 ) = 1</annotation></semantics></math>, so the <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.22.m22.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.22.m22.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.22.m22.1.1" xref="S6.SS0.SSS0.Px1.3.p2.22.m22.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.22.m22.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.22.m22.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.22.m22.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.22.m22.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.22.m22.1d">italic_n</annotation></semantics></math>-th agent takes action <math alttext="1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.23.m23.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.23.m23.1a"><mn id="S6.SS0.SSS0.Px1.3.p2.23.m23.1.1" xref="S6.SS0.SSS0.Px1.3.p2.23.m23.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.23.m23.1b"><cn id="S6.SS0.SSS0.Px1.3.p2.23.m23.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.23.m23.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.23.m23.1c">1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.23.m23.1d">1</annotation></semantics></math> by default. Intuitively, in order to learn the utility functions of game <math alttext="\Gamma_{n}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.24.m24.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.24.m24.1a"><msub id="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1" xref="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.2.cmml">Γ</mi><mi id="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.24.m24.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.2">Γ</ci><ci id="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.3.p2.24.m24.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.24.m24.1c">\Gamma_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.24.m24.1d">roman_Γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, we have to incentivize the <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.25.m25.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.25.m25.1a"><mi id="S6.SS0.SSS0.Px1.3.p2.25.m25.1.1" xref="S6.SS0.SSS0.Px1.3.p2.25.m25.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.25.m25.1b"><ci id="S6.SS0.SSS0.Px1.3.p2.25.m25.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.25.m25.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.25.m25.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.25.m25.1d">italic_n</annotation></semantics></math>-th agent to play action <math alttext="0" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.26.m26.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.26.m26.1a"><mn id="S6.SS0.SSS0.Px1.3.p2.26.m26.1.1" xref="S6.SS0.SSS0.Px1.3.p2.26.m26.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.26.m26.1b"><cn id="S6.SS0.SSS0.Px1.3.p2.26.m26.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.26.m26.1.1">0</cn></annotation-xml></semantics></math> by paying him <math alttext="1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.27.m27.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.27.m27.1a"><mn id="S6.SS0.SSS0.Px1.3.p2.27.m27.1.1" xref="S6.SS0.SSS0.Px1.3.p2.27.m27.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.27.m27.1b"><cn id="S6.SS0.SSS0.Px1.3.p2.27.m27.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.27.m27.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.27.m27.1c">1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.27.m27.1d">1</annotation></semantics></math> at each round, so that we can learn the first <math alttext="n-1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.28.m28.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.28.m28.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.28.m28.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1"><minus id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.28.m28.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.28.m28.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.28.m28.1d">italic_n - 1</annotation></semantics></math> agents’ utility functions. This means that the total payment is lower bounded by the number of rounds to learn the <math alttext="(n-1)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.3.p2.29.m29.1"><semantics id="S6.SS0.SSS0.Px1.3.p2.29.m29.1a"><mrow id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.3" stretchy="false" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.3.p2.29.m29.1b"><apply id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1"><minus id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.3.p2.29.m29.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.3.p2.29.m29.1c">(n-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.3.p2.29.m29.1d">( italic_n - 1 )</annotation></semantics></math>-agent game.</p> </div> <div class="ltx_para" id="S6.SS0.SSS0.Px1.4.p3"> <p class="ltx_p" id="S6.SS0.SSS0.Px1.4.p3.3">Formally, let ALG be an algorithm for learning <math alttext="\Gamma_{n}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.1.m1.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.1.m1.1a"><msub id="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.2.cmml">Γ</mi><mi id="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.1.m1.1b"><apply id="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.2">Γ</ci><ci id="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.1.m1.1c">\Gamma_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.1.m1.1d">roman_Γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> with payment complexity <math alttext="\mathrm{PC}(n,\varepsilon)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.2.m2.2"><semantics id="S6.SS0.SSS0.Px1.4.p3.2.m2.2a"><mrow id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.2" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.2.cmml">PC</mi><mo id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.1" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.2" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.1.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.4.p3.2.m2.1.1" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.1.1.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.2.2" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.1.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.2" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.2.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.2.m2.2b"><apply id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3"><times id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.1"></times><ci id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.2">PC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.3.3.2"><ci id="S6.SS0.SSS0.Px1.4.p3.2.m2.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.1.1">𝑛</ci><ci id="S6.SS0.SSS0.Px1.4.p3.2.m2.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.2.m2.2.2">𝜀</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.2.m2.2c">\mathrm{PC}(n,\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.2.m2.2d">roman_PC ( italic_n , italic_ε )</annotation></semantics></math>. We use ALG to construct an algorithm to learn <math alttext="\Gamma_{n-1}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.3.m3.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.3.m3.1a"><msub id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.2.cmml">Γ</mi><mrow id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.2" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.1" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.3" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.3.m3.1b"><apply id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.2">Γ</ci><apply id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3"><minus id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.1"></minus><ci id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.3.m3.1c">\Gamma_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.3.m3.1d">roman_Γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> as follows:</p> <ul class="ltx_itemize" id="S6.I1"> <li class="ltx_item" id="S6.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I1.i1.p1"> <p class="ltx_p" id="S6.I1.i1.p1.1">At each round <math alttext="t" class="ltx_Math" display="inline" id="S6.I1.i1.p1.1.m1.1"><semantics id="S6.I1.i1.p1.1.m1.1a"><mi id="S6.I1.i1.p1.1.m1.1.1" xref="S6.I1.i1.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.1.m1.1b"><ci id="S6.I1.i1.p1.1.m1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.1.m1.1d">italic_t</annotation></semantics></math>, do:</p> <ul class="ltx_itemize" id="S6.I1.i1.I1"> <li class="ltx_item" id="S6.I1.i1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S6.I1.i1.I1.i1.1.1.1">–</span></span> <div class="ltx_para" id="S6.I1.i1.I1.i1.p1"> <p class="ltx_p" id="S6.I1.i1.I1.i1.p1.1">Obtain payment functions <math alttext="P_{1}^{t},\ldots,P_{n-1}^{t},P_{n}^{t}" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i1.p1.1.m1.4"><semantics id="S6.I1.i1.I1.i1.p1.1.m1.4a"><mrow id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.4.cmml"><msubsup id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.cmml"><mi id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.2" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.2.cmml">P</mi><mn id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.3" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.3.cmml">1</mn><mi id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.3" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.4" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.4.cmml">,</mo><mi id="S6.I1.i1.I1.i1.p1.1.m1.1.1" mathvariant="normal" xref="S6.I1.i1.I1.i1.p1.1.m1.1.1.cmml">…</mi><mo id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.5" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.4.cmml">,</mo><msubsup id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.cmml"><mi id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.2" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.2.cmml">P</mi><mrow id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.cmml"><mi id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.2" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.2.cmml">n</mi><mo id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.1" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.1.cmml">−</mo><mn id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.3" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.3.cmml">1</mn></mrow><mi id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.3" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.6" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.4.cmml">,</mo><msubsup id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.cmml"><mi id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.2" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.2.cmml">P</mi><mi id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.3" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.3" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.3.cmml">t</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i1.p1.1.m1.4b"><list id="S6.I1.i1.I1.i1.p1.1.m1.4.4.4.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3"><apply id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1">superscript</csymbol><apply id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1">subscript</csymbol><ci id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.2">𝑃</ci><cn id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.3.cmml" type="integer" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.2.3">1</cn></apply><ci id="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.3.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.2.2.1.1.3">𝑡</ci></apply><ci id="S6.I1.i1.I1.i1.p1.1.m1.1.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.1.1">…</ci><apply id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2">superscript</csymbol><apply id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2">subscript</csymbol><ci id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.2">𝑃</ci><apply id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3"><minus id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.1"></minus><ci id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.2">𝑛</ci><cn id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.3.cmml" type="integer" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.2.3.3">1</cn></apply></apply><ci id="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.3.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.3.3.2.2.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3">superscript</csymbol><apply id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.1.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3">subscript</csymbol><ci id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.2.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.2">𝑃</ci><ci id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.3.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.2.3">𝑛</ci></apply><ci id="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.3.cmml" xref="S6.I1.i1.I1.i1.p1.1.m1.4.4.3.3.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i1.p1.1.m1.4c">P_{1}^{t},\ldots,P_{n-1}^{t},P_{n}^{t}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i1.p1.1.m1.4d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> from ALG.</p> </div> </li> <li class="ltx_item" id="S6.I1.i1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S6.I1.i1.I1.i2.1.1.1">–</span></span> <div class="ltx_para" id="S6.I1.i1.I1.i2.p1"> <p class="ltx_p" id="S6.I1.i1.I1.i2.p1.5">If <math alttext="P_{n}^{t}(a_{n}=0)<P_{n}^{t}(a_{n}=1)+1" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i2.p1.1.m1.2"><semantics id="S6.I1.i1.I1.i2.p1.1.m1.2a"><mrow id="S6.I1.i1.I1.i2.p1.1.m1.2.2" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.cmml"><mrow id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.cmml"><msubsup id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.cmml"><mi id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.2" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.2.cmml">P</mi><mi id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.3" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.3" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.2" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.1" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml"><msub 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xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.2"></times><apply id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3">superscript</csymbol><apply id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.2">𝑃</ci><ci id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.2.3">𝑛</ci></apply><ci id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.3.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.3.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1"><eq id="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.1.1.1.1.1.1.1"></eq><apply 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xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3">superscript</csymbol><apply id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.2.2">𝑃</ci><ci id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.2.3">𝑛</ci></apply><ci id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.3.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.3.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1"><eq id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.1"></eq><apply id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.2.3">𝑛</ci></apply><cn id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.1.1.1.1.3">1</cn></apply></apply><cn id="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.3.cmml" type="integer" xref="S6.I1.i1.I1.i2.p1.1.m1.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i2.p1.1.m1.2c">P_{n}^{t}(a_{n}=0)<P_{n}^{t}(a_{n}=1)+1</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i2.p1.1.m1.2d">italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ) < italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 1 ) + 1</annotation></semantics></math>, then let <math alttext="a^{t}_{n}=1" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i2.p1.2.m2.1"><semantics id="S6.I1.i1.I1.i2.p1.2.m2.1a"><mrow id="S6.I1.i1.I1.i2.p1.2.m2.1.1" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.cmml"><msubsup id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.cmml"><mi id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.2" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.3" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.3" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i2.p1.2.m2.1.1.1" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.1.cmml">=</mo><mn id="S6.I1.i1.I1.i2.p1.2.m2.1.1.3" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i2.p1.2.m2.1b"><apply id="S6.I1.i1.I1.i2.p1.2.m2.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1"><eq id="S6.I1.i1.I1.i2.p1.2.m2.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.1"></eq><apply id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2">subscript</csymbol><apply id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2">superscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.2.3">𝑡</ci></apply><ci id="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.2.3">𝑛</ci></apply><cn id="S6.I1.i1.I1.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S6.I1.i1.I1.i2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i2.p1.2.m2.1c">a^{t}_{n}=1</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i2.p1.2.m2.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="a^{t}_{i}=\operatorname*{argmax}_{a_{i}\in A_{i}}P_{i}^{t}(a_{i})" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i2.p1.3.m3.1"><semantics id="S6.I1.i1.I1.i2.p1.3.m3.1a"><mrow id="S6.I1.i1.I1.i2.p1.3.m3.1.1" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.cmml"><msubsup id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.cmml"><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.3.cmml">i</mi><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i2.p1.3.m3.1.1.2" rspace="0.1389em" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.2.cmml">=</mo><mrow id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.cmml"><mrow id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.cmml"><msub id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.cmml"><mo id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.2.cmml">argmax</mo><mrow id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.cmml"><msub id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.cmml"><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.3.cmml">i</mi></msub><mo id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.1" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.1.cmml">∈</mo><msub id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.cmml"><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.2.cmml">A</mi><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.3.cmml">i</mi></msub></mrow></msub><msubsup id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.cmml"><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.2.cmml">P</mi><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.3.cmml">i</mi><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.3.cmml">t</mi></msubsup></mrow><mo id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.2" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.3" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i2.p1.3.m3.1b"><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1"><eq id="S6.I1.i1.I1.i2.p1.3.m3.1.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.2"></eq><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3">subscript</csymbol><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3">superscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.2.3">𝑡</ci></apply><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.3.3">𝑖</ci></apply><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1"><times id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.2"></times><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3"><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.2">argmax</ci><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3"><in id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.1"></in><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.2.3">𝑖</ci></apply><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.2">𝐴</ci><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.1.3.3.3">𝑖</ci></apply></apply></apply><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2">superscript</csymbol><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.2">𝑃</ci><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.2.3">𝑖</ci></apply><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.3.2.3">𝑡</ci></apply></apply><apply id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S6.I1.i1.I1.i2.p1.3.m3.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i2.p1.3.m3.1c">a^{t}_{i}=\operatorname*{argmax}_{a_{i}\in A_{i}}P_{i}^{t}(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i2.p1.3.m3.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_argmax start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="i\in\{1,\ldots,n-1\}" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i2.p1.4.m4.3"><semantics id="S6.I1.i1.I1.i2.p1.4.m4.3a"><mrow id="S6.I1.i1.I1.i2.p1.4.m4.3.3" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.cmml"><mi id="S6.I1.i1.I1.i2.p1.4.m4.3.3.3" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.3.cmml">i</mi><mo id="S6.I1.i1.I1.i2.p1.4.m4.3.3.2" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.2.cmml">∈</mo><mrow id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.2.cmml"><mo id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.2" stretchy="false" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.2.cmml">{</mo><mn id="S6.I1.i1.I1.i2.p1.4.m4.1.1" xref="S6.I1.i1.I1.i2.p1.4.m4.1.1.cmml">1</mn><mo id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.3" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.2.cmml">,</mo><mi id="S6.I1.i1.I1.i2.p1.4.m4.2.2" mathvariant="normal" xref="S6.I1.i1.I1.i2.p1.4.m4.2.2.cmml">…</mi><mo id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.4" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.2.cmml">,</mo><mrow id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.cmml"><mi id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.2" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.2.cmml">n</mi><mo id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.1" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.1.cmml">−</mo><mn id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.3" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.5" stretchy="false" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i2.p1.4.m4.3b"><apply id="S6.I1.i1.I1.i2.p1.4.m4.3.3.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3"><in id="S6.I1.i1.I1.i2.p1.4.m4.3.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.2"></in><ci id="S6.I1.i1.I1.i2.p1.4.m4.3.3.3.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.3">𝑖</ci><set id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1"><cn id="S6.I1.i1.I1.i2.p1.4.m4.1.1.cmml" type="integer" xref="S6.I1.i1.I1.i2.p1.4.m4.1.1">1</cn><ci id="S6.I1.i1.I1.i2.p1.4.m4.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.2.2">…</ci><apply id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1"><minus id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.1"></minus><ci id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.2">𝑛</ci><cn id="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.3.cmml" type="integer" xref="S6.I1.i1.I1.i2.p1.4.m4.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i2.p1.4.m4.3c">i\in\{1,\ldots,n-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i2.p1.4.m4.3d">italic_i ∈ { 1 , … , italic_n - 1 }</annotation></semantics></math>. Return the action profile <math alttext="(a^{t}_{-n},a^{t}_{n})" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i2.p1.5.m5.2"><semantics id="S6.I1.i1.I1.i2.p1.5.m5.2a"><mrow id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.3.cmml"><mo id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.3" stretchy="false" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.3.cmml">(</mo><msubsup id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.cmml"><mi id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.2" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.2.cmml">a</mi><mrow id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.cmml"><mo id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3a" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.cmml">−</mo><mi id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.2" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.2.cmml">n</mi></mrow><mi id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.3" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.4" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.3.cmml">,</mo><msubsup id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.cmml"><mi id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.2" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.3" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.3" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.5" stretchy="false" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i2.p1.5.m5.2b"><interval closure="open" id="S6.I1.i1.I1.i2.p1.5.m5.2.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2"><apply id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1">subscript</csymbol><apply id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1">superscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.2.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3"><minus id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.1.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3"></minus><ci id="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.2.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.1.1.1.1.3.2">𝑛</ci></apply></apply><apply id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2">subscript</csymbol><apply id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.1.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2">superscript</csymbol><ci id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.2.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.2.3">𝑡</ci></apply><ci id="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.3.cmml" xref="S6.I1.i1.I1.i2.p1.5.m5.2.2.2.2.3">𝑛</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i2.p1.5.m5.2c">(a^{t}_{-n},a^{t}_{n})</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i2.p1.5.m5.2d">( italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - italic_n end_POSTSUBSCRIPT , italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> to ALG.</p> </div> </li> <li class="ltx_item" id="S6.I1.i1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S6.I1.i1.I1.i3.1.1.1">–</span></span> <div class="ltx_para" id="S6.I1.i1.I1.i3.p1"> <p class="ltx_p" id="S6.I1.i1.I1.i3.p1.7">If <math alttext="P_{n}^{t}(a_{n}=0)\geq P_{n}^{t}(a_{n}=1)+1" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i3.p1.1.m1.2"><semantics id="S6.I1.i1.I1.i3.p1.1.m1.2a"><mrow id="S6.I1.i1.I1.i3.p1.1.m1.2.2" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.cmml"><mrow id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.cmml"><msubsup id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.cmml"><mi id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.2.2" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.2.2.cmml">P</mi><mi id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.2.3" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.3" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.2" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml"><msub id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.cmml"><mi id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.2" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.3" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.1" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.1.cmml">=</mo><mn id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.3" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I1.i1.I1.i3.p1.1.m1.2.2.3" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.3.cmml">≥</mo><mrow id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.cmml"><mrow id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.cmml"><msubsup id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.cmml"><mi id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.2" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.2.cmml">P</mi><mi id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.3" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.3" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.2" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.2.cmml">⁢</mo><mrow id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml"><mo id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.2" stretchy="false" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml"><msub id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.cmml"><mi id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.2" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.3" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.1" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.1.cmml">=</mo><mn id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.3" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.3" stretchy="false" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.2" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.2.cmml">+</mo><mn id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.3" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i3.p1.1.m1.2b"><apply 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id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.3.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.3.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1"><eq id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.1"></eq><apply id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.2.3">𝑛</ci></apply><cn id="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.1.m1.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2"><plus id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.2"></plus><apply id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1"><times id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.2"></times><apply id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3">superscript</csymbol><apply id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3">subscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.2">𝑃</ci><ci id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.2.3">𝑛</ci></apply><ci id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.3.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.3.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1"><eq id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.1"></eq><apply id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2">subscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.2.3">𝑛</ci></apply><cn id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.1.1.1.1.3">1</cn></apply></apply><cn id="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.1.m1.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i3.p1.1.m1.2c">P_{n}^{t}(a_{n}=0)\geq P_{n}^{t}(a_{n}=1)+1</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i3.p1.1.m1.2d">italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ) ≥ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 1 ) + 1</annotation></semantics></math>, then let <math alttext="a^{t}_{n}=0" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i3.p1.2.m2.1"><semantics id="S6.I1.i1.I1.i3.p1.2.m2.1a"><mrow id="S6.I1.i1.I1.i3.p1.2.m2.1.1" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.cmml"><msubsup id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.cmml"><mi id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.2" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.3" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.3" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i3.p1.2.m2.1.1.1" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.1.cmml">=</mo><mn id="S6.I1.i1.I1.i3.p1.2.m2.1.1.3" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i3.p1.2.m2.1b"><apply id="S6.I1.i1.I1.i3.p1.2.m2.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1"><eq id="S6.I1.i1.I1.i3.p1.2.m2.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.1"></eq><apply id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2">subscript</csymbol><apply id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2">superscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.2.3">𝑡</ci></apply><ci id="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.2.3">𝑛</ci></apply><cn id="S6.I1.i1.I1.i3.p1.2.m2.1.1.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i3.p1.2.m2.1c">a^{t}_{n}=0</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i3.p1.2.m2.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0</annotation></semantics></math> and send the payment functions <math alttext="P_{1}^{t},\ldots,P_{n-1}^{t}" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i3.p1.3.m3.3"><semantics id="S6.I1.i1.I1.i3.p1.3.m3.3a"><mrow id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.3.cmml"><msubsup id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.cmml"><mi id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.2" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.2.cmml">P</mi><mn id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.3" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.3.cmml">1</mn><mi id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.3" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.3" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.3.cmml">,</mo><mi id="S6.I1.i1.I1.i3.p1.3.m3.1.1" mathvariant="normal" xref="S6.I1.i1.I1.i3.p1.3.m3.1.1.cmml">…</mi><mo id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.4" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.3.cmml">,</mo><msubsup id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.cmml"><mi id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.2" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.2.cmml">P</mi><mrow id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.cmml"><mi id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.2" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.2.cmml">n</mi><mo id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.1" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.1.cmml">−</mo><mn id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.3" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.3.cmml">1</mn></mrow><mi id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.3" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.3.cmml">t</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i3.p1.3.m3.3b"><list id="S6.I1.i1.I1.i3.p1.3.m3.3.3.3.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2"><apply id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1">superscript</csymbol><apply id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.2">𝑃</ci><cn id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.2.3">1</cn></apply><ci id="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.3.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.2.2.1.1.3">𝑡</ci></apply><ci id="S6.I1.i1.I1.i3.p1.3.m3.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.1.1">…</ci><apply id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2">superscript</csymbol><apply id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.2">𝑃</ci><apply id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3"><minus id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.1.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.1"></minus><ci id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.2.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.2">𝑛</ci><cn id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.2.3.3">1</cn></apply></apply><ci id="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.3.m3.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i3.p1.3.m3.3c">P_{1}^{t},\ldots,P_{n-1}^{t}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i3.p1.3.m3.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> to the first <math alttext="n-1" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i3.p1.4.m4.1"><semantics id="S6.I1.i1.I1.i3.p1.4.m4.1a"><mrow id="S6.I1.i1.I1.i3.p1.4.m4.1.1" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1.cmml"><mi id="S6.I1.i1.I1.i3.p1.4.m4.1.1.2" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1.2.cmml">n</mi><mo id="S6.I1.i1.I1.i3.p1.4.m4.1.1.1" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1.1.cmml">−</mo><mn id="S6.I1.i1.I1.i3.p1.4.m4.1.1.3" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i3.p1.4.m4.1b"><apply id="S6.I1.i1.I1.i3.p1.4.m4.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1"><minus id="S6.I1.i1.I1.i3.p1.4.m4.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1.1"></minus><ci id="S6.I1.i1.I1.i3.p1.4.m4.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1.2">𝑛</ci><cn id="S6.I1.i1.I1.i3.p1.4.m4.1.1.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i3.p1.4.m4.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i3.p1.4.m4.1d">italic_n - 1</annotation></semantics></math> agents. Observe their actions <math alttext="a^{t}_{-n}" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i3.p1.5.m5.1"><semantics id="S6.I1.i1.I1.i3.p1.5.m5.1a"><msubsup id="S6.I1.i1.I1.i3.p1.5.m5.1.1" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.cmml"><mi id="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.2" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.2.cmml">a</mi><mrow id="S6.I1.i1.I1.i3.p1.5.m5.1.1.3" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.cmml"><mo id="S6.I1.i1.I1.i3.p1.5.m5.1.1.3a" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.cmml">−</mo><mi id="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.2" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.2.cmml">n</mi></mrow><mi id="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.3" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i3.p1.5.m5.1b"><apply id="S6.I1.i1.I1.i3.p1.5.m5.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.5.m5.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1">subscript</csymbol><apply id="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1">superscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.2.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.3"><minus id="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.1.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.3"></minus><ci id="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.2.cmml" xref="S6.I1.i1.I1.i3.p1.5.m5.1.1.3.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i3.p1.5.m5.1c">a^{t}_{-n}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i3.p1.5.m5.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - italic_n end_POSTSUBSCRIPT</annotation></semantics></math> in game <math alttext="\Gamma_{n-1}" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i3.p1.6.m6.1"><semantics id="S6.I1.i1.I1.i3.p1.6.m6.1a"><msub id="S6.I1.i1.I1.i3.p1.6.m6.1.1" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.cmml"><mi id="S6.I1.i1.I1.i3.p1.6.m6.1.1.2" mathvariant="normal" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.2.cmml">Γ</mi><mrow id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.cmml"><mi id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.2" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.2.cmml">n</mi><mo id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.1" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.1.cmml">−</mo><mn id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.3" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i3.p1.6.m6.1b"><apply id="S6.I1.i1.I1.i3.p1.6.m6.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.6.m6.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1">subscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.6.m6.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.2">Γ</ci><apply id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.cmml" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3"><minus id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.1.cmml" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.1"></minus><ci id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.2.cmml" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.2">𝑛</ci><cn id="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S6.I1.i1.I1.i3.p1.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i3.p1.6.m6.1c">\Gamma_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i3.p1.6.m6.1d">roman_Γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Return <math alttext="(a^{t}_{-n},a^{t}_{n})" class="ltx_Math" display="inline" id="S6.I1.i1.I1.i3.p1.7.m7.2"><semantics id="S6.I1.i1.I1.i3.p1.7.m7.2a"><mrow id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.3.cmml"><mo id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.3" stretchy="false" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.3.cmml">(</mo><msubsup id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.cmml"><mi id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.2" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.2.cmml">a</mi><mrow id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.cmml"><mo id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3a" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.cmml">−</mo><mi id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.2" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.2.cmml">n</mi></mrow><mi id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.3" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.4" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.3.cmml">,</mo><msubsup id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.cmml"><mi id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.2" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.2.cmml">a</mi><mi id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.3" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.3.cmml">n</mi><mi id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.3" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.3.cmml">t</mi></msubsup><mo id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.5" stretchy="false" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.I1.i3.p1.7.m7.2b"><interval closure="open" id="S6.I1.i1.I1.i3.p1.7.m7.2.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2"><apply id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.1.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1">subscript</csymbol><apply id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1">superscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.2.3">𝑡</ci></apply><apply id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3"><minus id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.1.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3"></minus><ci id="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.2.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.1.1.1.1.3.2">𝑛</ci></apply></apply><apply id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2">subscript</csymbol><apply id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.1.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2">superscript</csymbol><ci id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.2.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.2">𝑎</ci><ci id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.2.3">𝑡</ci></apply><ci id="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.3.cmml" xref="S6.I1.i1.I1.i3.p1.7.m7.2.2.2.2.3">𝑛</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.I1.i3.p1.7.m7.2c">(a^{t}_{-n},a^{t}_{n})</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.I1.i3.p1.7.m7.2d">( italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - italic_n end_POSTSUBSCRIPT , italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> to ALG.</p> </div> </li> </ul> </div> </li> <li class="ltx_item" id="S6.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I1.i2.p1"> <p class="ltx_p" id="S6.I1.i2.p1.2">Obtain utility functions <math alttext="\tilde{U}_{1},\ldots,\tilde{U}_{n}" class="ltx_Math" display="inline" id="S6.I1.i2.p1.1.m1.3"><semantics id="S6.I1.i2.p1.1.m1.3a"><mrow id="S6.I1.i2.p1.1.m1.3.3.2" xref="S6.I1.i2.p1.1.m1.3.3.3.cmml"><msub id="S6.I1.i2.p1.1.m1.2.2.1.1" xref="S6.I1.i2.p1.1.m1.2.2.1.1.cmml"><mover accent="true" id="S6.I1.i2.p1.1.m1.2.2.1.1.2" xref="S6.I1.i2.p1.1.m1.2.2.1.1.2.cmml"><mi id="S6.I1.i2.p1.1.m1.2.2.1.1.2.2" xref="S6.I1.i2.p1.1.m1.2.2.1.1.2.2.cmml">U</mi><mo id="S6.I1.i2.p1.1.m1.2.2.1.1.2.1" xref="S6.I1.i2.p1.1.m1.2.2.1.1.2.1.cmml">~</mo></mover><mn id="S6.I1.i2.p1.1.m1.2.2.1.1.3" xref="S6.I1.i2.p1.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S6.I1.i2.p1.1.m1.3.3.2.3" xref="S6.I1.i2.p1.1.m1.3.3.3.cmml">,</mo><mi id="S6.I1.i2.p1.1.m1.1.1" mathvariant="normal" xref="S6.I1.i2.p1.1.m1.1.1.cmml">…</mi><mo id="S6.I1.i2.p1.1.m1.3.3.2.4" xref="S6.I1.i2.p1.1.m1.3.3.3.cmml">,</mo><msub id="S6.I1.i2.p1.1.m1.3.3.2.2" xref="S6.I1.i2.p1.1.m1.3.3.2.2.cmml"><mover accent="true" id="S6.I1.i2.p1.1.m1.3.3.2.2.2" xref="S6.I1.i2.p1.1.m1.3.3.2.2.2.cmml"><mi id="S6.I1.i2.p1.1.m1.3.3.2.2.2.2" xref="S6.I1.i2.p1.1.m1.3.3.2.2.2.2.cmml">U</mi><mo id="S6.I1.i2.p1.1.m1.3.3.2.2.2.1" xref="S6.I1.i2.p1.1.m1.3.3.2.2.2.1.cmml">~</mo></mover><mi id="S6.I1.i2.p1.1.m1.3.3.2.2.3" xref="S6.I1.i2.p1.1.m1.3.3.2.2.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.1.m1.3b"><list id="S6.I1.i2.p1.1.m1.3.3.3.cmml" xref="S6.I1.i2.p1.1.m1.3.3.2"><apply id="S6.I1.i2.p1.1.m1.2.2.1.1.cmml" xref="S6.I1.i2.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.2.2.1.1">subscript</csymbol><apply id="S6.I1.i2.p1.1.m1.2.2.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.2.2.1.1.2"><ci id="S6.I1.i2.p1.1.m1.2.2.1.1.2.1.cmml" xref="S6.I1.i2.p1.1.m1.2.2.1.1.2.1">~</ci><ci id="S6.I1.i2.p1.1.m1.2.2.1.1.2.2.cmml" xref="S6.I1.i2.p1.1.m1.2.2.1.1.2.2">𝑈</ci></apply><cn id="S6.I1.i2.p1.1.m1.2.2.1.1.3.cmml" type="integer" xref="S6.I1.i2.p1.1.m1.2.2.1.1.3">1</cn></apply><ci id="S6.I1.i2.p1.1.m1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1">…</ci><apply id="S6.I1.i2.p1.1.m1.3.3.2.2.cmml" xref="S6.I1.i2.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.3.3.2.2.1.cmml" xref="S6.I1.i2.p1.1.m1.3.3.2.2">subscript</csymbol><apply id="S6.I1.i2.p1.1.m1.3.3.2.2.2.cmml" xref="S6.I1.i2.p1.1.m1.3.3.2.2.2"><ci id="S6.I1.i2.p1.1.m1.3.3.2.2.2.1.cmml" xref="S6.I1.i2.p1.1.m1.3.3.2.2.2.1">~</ci><ci id="S6.I1.i2.p1.1.m1.3.3.2.2.2.2.cmml" xref="S6.I1.i2.p1.1.m1.3.3.2.2.2.2">𝑈</ci></apply><ci id="S6.I1.i2.p1.1.m1.3.3.2.2.3.cmml" xref="S6.I1.i2.p1.1.m1.3.3.2.2.3">𝑛</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.1.m1.3c">\tilde{U}_{1},\ldots,\tilde{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.1.m1.3d">over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> from ALG. Output <math alttext="\tilde{U}_{1}(\cdot,a_{n}=0),\ldots,\tilde{U}_{n-1}(\cdot,a_{n}=0)" class="ltx_Math" display="inline" id="S6.I1.i2.p1.2.m2.5"><semantics id="S6.I1.i2.p1.2.m2.5a"><mrow id="S6.I1.i2.p1.2.m2.5.5.2" xref="S6.I1.i2.p1.2.m2.5.5.3.cmml"><mrow id="S6.I1.i2.p1.2.m2.4.4.1.1" xref="S6.I1.i2.p1.2.m2.4.4.1.1.cmml"><msub id="S6.I1.i2.p1.2.m2.4.4.1.1.3" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.cmml"><mover accent="true" id="S6.I1.i2.p1.2.m2.4.4.1.1.3.2" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.cmml"><mi id="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.2" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.2.cmml">U</mi><mo id="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.1" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.1.cmml">~</mo></mover><mn id="S6.I1.i2.p1.2.m2.4.4.1.1.3.3" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.3.cmml">1</mn></msub><mo id="S6.I1.i2.p1.2.m2.4.4.1.1.2" xref="S6.I1.i2.p1.2.m2.4.4.1.1.2.cmml">⁢</mo><mrow id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.cmml"><mo id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.2" stretchy="false" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.cmml"><mrow id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.2.cmml"><mo id="S6.I1.i2.p1.2.m2.1.1" lspace="0em" rspace="0em" xref="S6.I1.i2.p1.2.m2.1.1.cmml">⋅</mo><mo id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.2" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.2.cmml">,</mo><msub id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.cmml"><mi id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.2" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.3" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.3.cmml">n</mi></msub></mrow><mo id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.2" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.2.cmml">=</mo><mn id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.3" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.3" stretchy="false" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I1.i2.p1.2.m2.5.5.2.3" xref="S6.I1.i2.p1.2.m2.5.5.3.cmml">,</mo><mi id="S6.I1.i2.p1.2.m2.3.3" mathvariant="normal" xref="S6.I1.i2.p1.2.m2.3.3.cmml">…</mi><mo id="S6.I1.i2.p1.2.m2.5.5.2.4" xref="S6.I1.i2.p1.2.m2.5.5.3.cmml">,</mo><mrow id="S6.I1.i2.p1.2.m2.5.5.2.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.cmml"><msub id="S6.I1.i2.p1.2.m2.5.5.2.2.3" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.cmml"><mover accent="true" id="S6.I1.i2.p1.2.m2.5.5.2.2.3.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.cmml"><mi id="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.2.cmml">U</mi><mo id="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.1" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.1.cmml">~</mo></mover><mrow id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.cmml"><mi id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.2.cmml">n</mi><mo id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.1" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.1.cmml">−</mo><mn id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.3" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.3.cmml">1</mn></mrow></msub><mo id="S6.I1.i2.p1.2.m2.5.5.2.2.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.2.cmml">⁢</mo><mrow id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.cmml"><mo id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.2" stretchy="false" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.cmml">(</mo><mrow id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.cmml"><mrow id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.2.cmml"><mo id="S6.I1.i2.p1.2.m2.2.2" lspace="0em" rspace="0em" xref="S6.I1.i2.p1.2.m2.2.2.cmml">⋅</mo><mo id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.2.cmml">,</mo><msub id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.cmml"><mi id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.3" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.3.cmml">n</mi></msub></mrow><mo id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.2" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.2.cmml">=</mo><mn id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.3" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.3.cmml">0</mn></mrow><mo id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.3" stretchy="false" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.2.m2.5b"><list id="S6.I1.i2.p1.2.m2.5.5.3.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2"><apply id="S6.I1.i2.p1.2.m2.4.4.1.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1"><times id="S6.I1.i2.p1.2.m2.4.4.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.2"></times><apply id="S6.I1.i2.p1.2.m2.4.4.1.1.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.4.4.1.1.3.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3">subscript</csymbol><apply id="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.2"><ci id="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.1">~</ci><ci id="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.2.2">𝑈</ci></apply><cn id="S6.I1.i2.p1.2.m2.4.4.1.1.3.3.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.4.4.1.1.3.3">1</cn></apply><apply id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1"><eq id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.2"></eq><list id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1"><ci id="S6.I1.i2.p1.2.m2.1.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1">⋅</ci><apply id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.1.1.1.3">𝑛</ci></apply></list><cn id="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.4.4.1.1.1.1.1.3">0</cn></apply></apply><ci id="S6.I1.i2.p1.2.m2.3.3.cmml" xref="S6.I1.i2.p1.2.m2.3.3">…</ci><apply id="S6.I1.i2.p1.2.m2.5.5.2.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2"><times id="S6.I1.i2.p1.2.m2.5.5.2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.2"></times><apply id="S6.I1.i2.p1.2.m2.5.5.2.2.3.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.5.5.2.2.3.1.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3">subscript</csymbol><apply id="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.2"><ci id="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.1.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.1">~</ci><ci id="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.2.2">𝑈</ci></apply><apply id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3"><minus id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.1.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.1"></minus><ci id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.2">𝑛</ci><cn id="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.3.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.5.5.2.2.3.3.3">1</cn></apply></apply><apply id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1"><eq id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.2"></eq><list id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1"><ci id="S6.I1.i2.p1.2.m2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.2.2">⋅</ci><apply id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.3.cmml" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.1.1.1.3">𝑛</ci></apply></list><cn id="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.3.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.5.5.2.2.1.1.1.3">0</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.2.m2.5c">\tilde{U}_{1}(\cdot,a_{n}=0),\ldots,\tilde{U}_{n-1}(\cdot,a_{n}=0)</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.2.m2.5d">over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ) , … , over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( ⋅ , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 )</annotation></semantics></math>.</p> </div> </li> </ul> <p class="ltx_p" id="S6.SS0.SSS0.Px1.4.p3.16">By definition, if ALG outputs <math alttext="\tilde{U}_{1},\ldots,\tilde{U}_{n}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.4.m1.3"><semantics id="S6.SS0.SSS0.Px1.4.p3.4.m1.3a"><mrow id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.3.cmml"><msub id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.cmml"><mover accent="true" id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.2" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.2.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.1" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.1.cmml">~</mo></mover><mn id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.3" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.3.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.4.p3.4.m1.1.1" mathvariant="normal" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.1.1.cmml">…</mi><mo id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.4" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.3.cmml">,</mo><msub id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.cmml"><mover accent="true" id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.2" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.2.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.1" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.1.cmml">~</mo></mover><mi id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.3" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.4.m1.3b"><list id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2"><apply id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1">subscript</csymbol><apply id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2"><ci id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.1">~</ci><ci id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.2.2">𝑈</ci></apply><cn id="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.2.2.1.1.3">1</cn></apply><ci id="S6.SS0.SSS0.Px1.4.p3.4.m1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.1.1">…</ci><apply id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2">subscript</csymbol><apply id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2"><ci id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.1">~</ci><ci id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.2.2">𝑈</ci></apply><ci id="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.4.m1.3.3.2.2.3">𝑛</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.4.m1.3c">\tilde{U}_{1},\ldots,\tilde{U}_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.4.m1.3d">over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> that are <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.5.m2.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.5.m2.1a"><mi id="S6.SS0.SSS0.Px1.4.p3.5.m2.1.1" xref="S6.SS0.SSS0.Px1.4.p3.5.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.5.m2.1b"><ci id="S6.SS0.SSS0.Px1.4.p3.5.m2.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.5.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.5.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.5.m2.1d">italic_ε</annotation></semantics></math>-close to the utility functions of <math alttext="\Gamma_{n}" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.6.m3.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.6.m3.1a"><msub id="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1" xref="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.2" mathvariant="normal" xref="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.2.cmml">Γ</mi><mi id="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.6.m3.1b"><apply id="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.2">Γ</ci><ci id="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.6.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.6.m3.1c">\Gamma_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.6.m3.1d">roman_Γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, then the outputs <math alttext="\tilde{U}_{1}(\cdot,a_{n}=0),\ldots,\tilde{U}_{n-1}(\cdot,a_{n}=0)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.7.m4.5"><semantics id="S6.SS0.SSS0.Px1.4.p3.7.m4.5a"><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.3.cmml"><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.cmml"><msub id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.cmml"><mover accent="true" id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.2.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.2.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.2.2.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.2.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.2.1.cmml">~</mo></mover><mn id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.3.3.cmml">1</mn></msub><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.cmml"><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.2.cmml"><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.1.1" lspace="0em" rspace="0em" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.1.1.cmml">⋅</mo><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.2.cmml">,</mo><msub id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.1.1.1.3.cmml">n</mi></msub></mrow><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.2.cmml">=</mo><mn id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.3" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.3.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.4.p3.7.m4.3.3" mathvariant="normal" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.3.3.cmml">…</mi><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.4" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.3.cmml">,</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.cmml"><msub id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.cmml"><mover accent="true" id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.2.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.2.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.2.2.cmml">U</mi><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.2.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.2.1.cmml">~</mo></mover><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3.3" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.3.3.3.cmml">1</mn></mrow></msub><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.2" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1.1.cmml"><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1.1.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1.1.cmml"><mrow id="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.7.m4.5.5.2.2.1.1.1.1.2.cmml"><mo id="S6.SS0.SSS0.Px1.4.p3.7.m4.2.2" lspace="0em" rspace="0em" 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xref="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1">subscript</csymbol><ci id="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.2">Γ</ci><apply id="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3"><minus id="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3.1"></minus><ci id="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.10.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.10.m7.1c">\Gamma_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.10.m7.1d">roman_Γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Each time “<math alttext="P^{t}_{n}(a_{n}=0)\geq P^{t}_{n}(a_{n}=1)+1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.11.m8.2"><semantics id="S6.SS0.SSS0.Px1.4.p3.11.m8.2a"><mrow id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.cmml"><mrow id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.cmml"><msubsup id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.2.2" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.2.2.cmml">P</mi><mi id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.3" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.3.cmml">n</mi><mi id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.2.3" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.2.3.cmml">t</mi></msubsup><mo id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.1.1.1.cmml"><mo 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id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.3" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.11.m8.2b"><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2"><geq id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.3"></geq><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1"><times id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.2"></times><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3">subscript</csymbol><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.3.2.1.cmml" 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id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.1.1.1.2.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.1.1.1.2.3">𝑛</ci></apply><cn id="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2"><plus id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.2"></plus><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1"><times id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.2"></times><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.3">subscript</csymbol><apply id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.3.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.3"><csymbol 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id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.1.1.1.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.1.1.1.2.2">𝑎</ci><ci id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.1.1.1.2.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.1.1.1.2.3">𝑛</ci></apply><cn id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.1.1.1.1.3">1</cn></apply></apply><cn id="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.11.m8.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.11.m8.2c">P^{t}_{n}(a_{n}=0)\geq P^{t}_{n}(a_{n}=1)+1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.11.m8.2d">italic_P start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ) ≥ italic_P start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 1 ) + 1</annotation></semantics></math>” happens, agent <math alttext="n" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.12.m9.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.12.m9.1a"><mi id="S6.SS0.SSS0.Px1.4.p3.12.m9.1.1" xref="S6.SS0.SSS0.Px1.4.p3.12.m9.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.12.m9.1b"><ci id="S6.SS0.SSS0.Px1.4.p3.12.m9.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.12.m9.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.12.m9.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.12.m9.1d">italic_n</annotation></semantics></math> takes action <math alttext="a^{t}_{n}=0" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.13.m10.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.13.m10.1a"><mrow id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.cmml"><msubsup id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.2" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.2.cmml">a</mi><mi id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.3" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.3.cmml">n</mi><mi id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.3" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.3.cmml">t</mi></msubsup><mo id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.1.cmml">=</mo><mn id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.13.m10.1b"><apply id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1"><eq id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.1"></eq><apply id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2">subscript</csymbol><apply id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2">superscript</csymbol><ci id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.2">𝑎</ci><ci id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.2.3">𝑡</ci></apply><ci id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.2.3">𝑛</ci></apply><cn id="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.13.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.13.m10.1c">a^{t}_{n}=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.13.m10.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0</annotation></semantics></math>, we pay at least <math alttext="1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.14.m11.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.14.m11.1a"><mn id="S6.SS0.SSS0.Px1.4.p3.14.m11.1.1" xref="S6.SS0.SSS0.Px1.4.p3.14.m11.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.14.m11.1b"><cn id="S6.SS0.SSS0.Px1.4.p3.14.m11.1.1.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.14.m11.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.14.m11.1c">1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.14.m11.1d">1</annotation></semantics></math>, and we interact with the first <math alttext="n-1" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.15.m12.1"><semantics id="S6.SS0.SSS0.Px1.4.p3.15.m12.1a"><mrow id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.15.m12.1b"><apply id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1"><minus id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.15.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.15.m12.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.15.m12.1d">italic_n - 1</annotation></semantics></math> agents once. So, the total payment is at least <math alttext="\mathrm{PC}(n,\varepsilon)\geq\mathrm{RC}(n-1,\varepsilon)" class="ltx_Math" display="inline" id="S6.SS0.SSS0.Px1.4.p3.16.m13.4"><semantics id="S6.SS0.SSS0.Px1.4.p3.16.m13.4a"><mrow id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.cmml"><mrow id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.2" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.2.cmml">PC</mi><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.1" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.1.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.2" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.1.cmml"><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.2.1" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.1.cmml">(</mo><mi id="S6.SS0.SSS0.Px1.4.p3.16.m13.1.1" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.1.1.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.2.2" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.1.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.4.p3.16.m13.2.2" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.2.2.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.2.3" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.2" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.2.cmml">≥</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.3" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.3.cmml">RC</mi><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.2" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.2.cmml">⁢</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.2.cmml"><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.2" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.2.cmml">(</mo><mrow id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.cmml"><mi id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.2" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.2.cmml">n</mi><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.1" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.1.cmml">−</mo><mn id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.3" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.2.cmml">,</mo><mi id="S6.SS0.SSS0.Px1.4.p3.16.m13.3.3" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.3.3.cmml">ε</mi><mo id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.4" stretchy="false" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.4.p3.16.m13.4b"><apply id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4"><geq id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.2"></geq><apply id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3"><times id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.1"></times><ci id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.2">PC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.3.3.2"><ci id="S6.SS0.SSS0.Px1.4.p3.16.m13.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.1.1">𝑛</ci><ci id="S6.SS0.SSS0.Px1.4.p3.16.m13.2.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.2.2">𝜀</ci></interval></apply><apply id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1"><times id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.2"></times><ci id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.3">RC</ci><interval closure="open" id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1"><apply id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1"><minus id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.1.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.1"></minus><ci id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.2.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.2">𝑛</ci><cn id="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.3.cmml" type="integer" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.4.4.1.1.1.1.3">1</cn></apply><ci id="S6.SS0.SSS0.Px1.4.p3.16.m13.3.3.cmml" xref="S6.SS0.SSS0.Px1.4.p3.16.m13.3.3">𝜀</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.4.p3.16.m13.4c">\mathrm{PC}(n,\varepsilon)\geq\mathrm{RC}(n-1,\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S6.SS0.SSS0.Px1.4.p3.16.m13.4d">roman_PC ( italic_n , italic_ε ) ≥ roman_RC ( italic_n - 1 , italic_ε )</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">7 </span>Steering No-Regret Learners by Learning Utilities</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">As mentioned in the introduction, our main motivating application of our result is to the problem of <span class="ltx_text ltx_font_italic" id="S7.p1.1.1">steering</span> no-regret learners to desirable outcomes, introduced by <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite>. In this section, we will explore this application.</p> </div> <section class="ltx_subsection" id="S7.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.1 </span>Correlated signals</h3> <div class="ltx_para" id="S7.SS1.p1"> <p class="ltx_p" id="S7.SS1.p1.12">In this section, we will make two modifications to our no-regret model. First, we will allow the signals to be <span class="ltx_text ltx_font_italic" id="S7.SS1.p1.12.1">correlated</span>. Second, we will allow payments to each agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS1.p1.1.m1.1"><semantics id="S7.SS1.p1.1.m1.1a"><mi id="S7.SS1.p1.1.m1.1.1" xref="S7.SS1.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.1.m1.1b"><ci id="S7.SS1.p1.1.m1.1.1.cmml" xref="S7.SS1.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.1.m1.1d">italic_i</annotation></semantics></math> to depend not only on agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS1.p1.2.m2.1"><semantics id="S7.SS1.p1.2.m2.1a"><mi id="S7.SS1.p1.2.m2.1.1" xref="S7.SS1.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.2.m2.1b"><ci id="S7.SS1.p1.2.m2.1.1.cmml" xref="S7.SS1.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.2.m2.1d">italic_i</annotation></semantics></math>’s action, but also the signals and actions of all the other players. Formally, each agent has a finite signal set <math alttext="S_{i}" class="ltx_Math" display="inline" id="S7.SS1.p1.3.m3.1"><semantics id="S7.SS1.p1.3.m3.1a"><msub id="S7.SS1.p1.3.m3.1.1" xref="S7.SS1.p1.3.m3.1.1.cmml"><mi id="S7.SS1.p1.3.m3.1.1.2" xref="S7.SS1.p1.3.m3.1.1.2.cmml">S</mi><mi id="S7.SS1.p1.3.m3.1.1.3" xref="S7.SS1.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.3.m3.1b"><apply id="S7.SS1.p1.3.m3.1.1.cmml" xref="S7.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.3.m3.1.1.1.cmml" xref="S7.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S7.SS1.p1.3.m3.1.1.2.cmml" xref="S7.SS1.p1.3.m3.1.1.2">𝑆</ci><ci id="S7.SS1.p1.3.m3.1.1.3.cmml" xref="S7.SS1.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.3.m3.1c">S_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. As with actions, we will write <math alttext="S=S_{1}\times\dots\times S_{n}" class="ltx_Math" display="inline" id="S7.SS1.p1.4.m4.1"><semantics id="S7.SS1.p1.4.m4.1a"><mrow id="S7.SS1.p1.4.m4.1.1" xref="S7.SS1.p1.4.m4.1.1.cmml"><mi id="S7.SS1.p1.4.m4.1.1.2" xref="S7.SS1.p1.4.m4.1.1.2.cmml">S</mi><mo id="S7.SS1.p1.4.m4.1.1.1" xref="S7.SS1.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S7.SS1.p1.4.m4.1.1.3" xref="S7.SS1.p1.4.m4.1.1.3.cmml"><msub id="S7.SS1.p1.4.m4.1.1.3.2" xref="S7.SS1.p1.4.m4.1.1.3.2.cmml"><mi id="S7.SS1.p1.4.m4.1.1.3.2.2" xref="S7.SS1.p1.4.m4.1.1.3.2.2.cmml">S</mi><mn id="S7.SS1.p1.4.m4.1.1.3.2.3" xref="S7.SS1.p1.4.m4.1.1.3.2.3.cmml">1</mn></msub><mo id="S7.SS1.p1.4.m4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S7.SS1.p1.4.m4.1.1.3.1.cmml">×</mo><mi id="S7.SS1.p1.4.m4.1.1.3.3" mathvariant="normal" xref="S7.SS1.p1.4.m4.1.1.3.3.cmml">⋯</mi><mo id="S7.SS1.p1.4.m4.1.1.3.1a" lspace="0.222em" rspace="0.222em" xref="S7.SS1.p1.4.m4.1.1.3.1.cmml">×</mo><msub id="S7.SS1.p1.4.m4.1.1.3.4" xref="S7.SS1.p1.4.m4.1.1.3.4.cmml"><mi id="S7.SS1.p1.4.m4.1.1.3.4.2" xref="S7.SS1.p1.4.m4.1.1.3.4.2.cmml">S</mi><mi id="S7.SS1.p1.4.m4.1.1.3.4.3" xref="S7.SS1.p1.4.m4.1.1.3.4.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.4.m4.1b"><apply id="S7.SS1.p1.4.m4.1.1.cmml" xref="S7.SS1.p1.4.m4.1.1"><eq id="S7.SS1.p1.4.m4.1.1.1.cmml" xref="S7.SS1.p1.4.m4.1.1.1"></eq><ci id="S7.SS1.p1.4.m4.1.1.2.cmml" xref="S7.SS1.p1.4.m4.1.1.2">𝑆</ci><apply id="S7.SS1.p1.4.m4.1.1.3.cmml" xref="S7.SS1.p1.4.m4.1.1.3"><times id="S7.SS1.p1.4.m4.1.1.3.1.cmml" xref="S7.SS1.p1.4.m4.1.1.3.1"></times><apply id="S7.SS1.p1.4.m4.1.1.3.2.cmml" xref="S7.SS1.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S7.SS1.p1.4.m4.1.1.3.2.1.cmml" xref="S7.SS1.p1.4.m4.1.1.3.2">subscript</csymbol><ci id="S7.SS1.p1.4.m4.1.1.3.2.2.cmml" xref="S7.SS1.p1.4.m4.1.1.3.2.2">𝑆</ci><cn id="S7.SS1.p1.4.m4.1.1.3.2.3.cmml" type="integer" xref="S7.SS1.p1.4.m4.1.1.3.2.3">1</cn></apply><ci id="S7.SS1.p1.4.m4.1.1.3.3.cmml" xref="S7.SS1.p1.4.m4.1.1.3.3">⋯</ci><apply id="S7.SS1.p1.4.m4.1.1.3.4.cmml" xref="S7.SS1.p1.4.m4.1.1.3.4"><csymbol cd="ambiguous" id="S7.SS1.p1.4.m4.1.1.3.4.1.cmml" xref="S7.SS1.p1.4.m4.1.1.3.4">subscript</csymbol><ci id="S7.SS1.p1.4.m4.1.1.3.4.2.cmml" xref="S7.SS1.p1.4.m4.1.1.3.4.2">𝑆</ci><ci id="S7.SS1.p1.4.m4.1.1.3.4.3.cmml" xref="S7.SS1.p1.4.m4.1.1.3.4.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.4.m4.1c">S=S_{1}\times\dots\times S_{n}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.4.m4.1d">italic_S = italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × ⋯ × italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> for the joint signal space. On each round <math alttext="t" class="ltx_Math" display="inline" id="S7.SS1.p1.5.m5.1"><semantics id="S7.SS1.p1.5.m5.1a"><mi id="S7.SS1.p1.5.m5.1.1" xref="S7.SS1.p1.5.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.5.m5.1b"><ci id="S7.SS1.p1.5.m5.1.1.cmml" xref="S7.SS1.p1.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.5.m5.1d">italic_t</annotation></semantics></math>, the principal first commits to both a signal distribution <math alttext="\mu^{t}\in\Delta(S)" class="ltx_Math" display="inline" id="S7.SS1.p1.6.m6.1"><semantics id="S7.SS1.p1.6.m6.1a"><mrow id="S7.SS1.p1.6.m6.1.2" xref="S7.SS1.p1.6.m6.1.2.cmml"><msup id="S7.SS1.p1.6.m6.1.2.2" xref="S7.SS1.p1.6.m6.1.2.2.cmml"><mi id="S7.SS1.p1.6.m6.1.2.2.2" xref="S7.SS1.p1.6.m6.1.2.2.2.cmml">μ</mi><mi id="S7.SS1.p1.6.m6.1.2.2.3" xref="S7.SS1.p1.6.m6.1.2.2.3.cmml">t</mi></msup><mo id="S7.SS1.p1.6.m6.1.2.1" xref="S7.SS1.p1.6.m6.1.2.1.cmml">∈</mo><mrow id="S7.SS1.p1.6.m6.1.2.3" xref="S7.SS1.p1.6.m6.1.2.3.cmml"><mi id="S7.SS1.p1.6.m6.1.2.3.2" mathvariant="normal" xref="S7.SS1.p1.6.m6.1.2.3.2.cmml">Δ</mi><mo id="S7.SS1.p1.6.m6.1.2.3.1" xref="S7.SS1.p1.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S7.SS1.p1.6.m6.1.2.3.3.2" xref="S7.SS1.p1.6.m6.1.2.3.cmml"><mo id="S7.SS1.p1.6.m6.1.2.3.3.2.1" stretchy="false" xref="S7.SS1.p1.6.m6.1.2.3.cmml">(</mo><mi id="S7.SS1.p1.6.m6.1.1" xref="S7.SS1.p1.6.m6.1.1.cmml">S</mi><mo id="S7.SS1.p1.6.m6.1.2.3.3.2.2" stretchy="false" xref="S7.SS1.p1.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.6.m6.1b"><apply id="S7.SS1.p1.6.m6.1.2.cmml" xref="S7.SS1.p1.6.m6.1.2"><in id="S7.SS1.p1.6.m6.1.2.1.cmml" xref="S7.SS1.p1.6.m6.1.2.1"></in><apply id="S7.SS1.p1.6.m6.1.2.2.cmml" xref="S7.SS1.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S7.SS1.p1.6.m6.1.2.2.1.cmml" xref="S7.SS1.p1.6.m6.1.2.2">superscript</csymbol><ci id="S7.SS1.p1.6.m6.1.2.2.2.cmml" xref="S7.SS1.p1.6.m6.1.2.2.2">𝜇</ci><ci id="S7.SS1.p1.6.m6.1.2.2.3.cmml" xref="S7.SS1.p1.6.m6.1.2.2.3">𝑡</ci></apply><apply id="S7.SS1.p1.6.m6.1.2.3.cmml" xref="S7.SS1.p1.6.m6.1.2.3"><times id="S7.SS1.p1.6.m6.1.2.3.1.cmml" xref="S7.SS1.p1.6.m6.1.2.3.1"></times><ci id="S7.SS1.p1.6.m6.1.2.3.2.cmml" xref="S7.SS1.p1.6.m6.1.2.3.2">Δ</ci><ci id="S7.SS1.p1.6.m6.1.1.cmml" xref="S7.SS1.p1.6.m6.1.1">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.6.m6.1c">\mu^{t}\in\Delta(S)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.6.m6.1d">italic_μ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∈ roman_Δ ( italic_S )</annotation></semantics></math> and a payment function <math alttext="P_{i}^{t}:S\times A\to\mathbb{R}_{+}" class="ltx_Math" display="inline" id="S7.SS1.p1.7.m7.1"><semantics id="S7.SS1.p1.7.m7.1a"><mrow id="S7.SS1.p1.7.m7.1.1" xref="S7.SS1.p1.7.m7.1.1.cmml"><msubsup id="S7.SS1.p1.7.m7.1.1.2" xref="S7.SS1.p1.7.m7.1.1.2.cmml"><mi id="S7.SS1.p1.7.m7.1.1.2.2.2" xref="S7.SS1.p1.7.m7.1.1.2.2.2.cmml">P</mi><mi id="S7.SS1.p1.7.m7.1.1.2.2.3" xref="S7.SS1.p1.7.m7.1.1.2.2.3.cmml">i</mi><mi id="S7.SS1.p1.7.m7.1.1.2.3" xref="S7.SS1.p1.7.m7.1.1.2.3.cmml">t</mi></msubsup><mo id="S7.SS1.p1.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S7.SS1.p1.7.m7.1.1.1.cmml">:</mo><mrow id="S7.SS1.p1.7.m7.1.1.3" xref="S7.SS1.p1.7.m7.1.1.3.cmml"><mrow id="S7.SS1.p1.7.m7.1.1.3.2" xref="S7.SS1.p1.7.m7.1.1.3.2.cmml"><mi id="S7.SS1.p1.7.m7.1.1.3.2.2" xref="S7.SS1.p1.7.m7.1.1.3.2.2.cmml">S</mi><mo id="S7.SS1.p1.7.m7.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S7.SS1.p1.7.m7.1.1.3.2.1.cmml">×</mo><mi id="S7.SS1.p1.7.m7.1.1.3.2.3" xref="S7.SS1.p1.7.m7.1.1.3.2.3.cmml">A</mi></mrow><mo id="S7.SS1.p1.7.m7.1.1.3.1" stretchy="false" xref="S7.SS1.p1.7.m7.1.1.3.1.cmml">→</mo><msub id="S7.SS1.p1.7.m7.1.1.3.3" xref="S7.SS1.p1.7.m7.1.1.3.3.cmml"><mi id="S7.SS1.p1.7.m7.1.1.3.3.2" xref="S7.SS1.p1.7.m7.1.1.3.3.2.cmml">ℝ</mi><mo id="S7.SS1.p1.7.m7.1.1.3.3.3" xref="S7.SS1.p1.7.m7.1.1.3.3.3.cmml">+</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.7.m7.1b"><apply id="S7.SS1.p1.7.m7.1.1.cmml" xref="S7.SS1.p1.7.m7.1.1"><ci id="S7.SS1.p1.7.m7.1.1.1.cmml" xref="S7.SS1.p1.7.m7.1.1.1">:</ci><apply id="S7.SS1.p1.7.m7.1.1.2.cmml" xref="S7.SS1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p1.7.m7.1.1.2.1.cmml" xref="S7.SS1.p1.7.m7.1.1.2">superscript</csymbol><apply id="S7.SS1.p1.7.m7.1.1.2.2.cmml" xref="S7.SS1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p1.7.m7.1.1.2.2.1.cmml" xref="S7.SS1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S7.SS1.p1.7.m7.1.1.2.2.2.cmml" xref="S7.SS1.p1.7.m7.1.1.2.2.2">𝑃</ci><ci id="S7.SS1.p1.7.m7.1.1.2.2.3.cmml" xref="S7.SS1.p1.7.m7.1.1.2.2.3">𝑖</ci></apply><ci id="S7.SS1.p1.7.m7.1.1.2.3.cmml" xref="S7.SS1.p1.7.m7.1.1.2.3">𝑡</ci></apply><apply id="S7.SS1.p1.7.m7.1.1.3.cmml" xref="S7.SS1.p1.7.m7.1.1.3"><ci id="S7.SS1.p1.7.m7.1.1.3.1.cmml" xref="S7.SS1.p1.7.m7.1.1.3.1">→</ci><apply id="S7.SS1.p1.7.m7.1.1.3.2.cmml" xref="S7.SS1.p1.7.m7.1.1.3.2"><times id="S7.SS1.p1.7.m7.1.1.3.2.1.cmml" xref="S7.SS1.p1.7.m7.1.1.3.2.1"></times><ci id="S7.SS1.p1.7.m7.1.1.3.2.2.cmml" xref="S7.SS1.p1.7.m7.1.1.3.2.2">𝑆</ci><ci id="S7.SS1.p1.7.m7.1.1.3.2.3.cmml" xref="S7.SS1.p1.7.m7.1.1.3.2.3">𝐴</ci></apply><apply id="S7.SS1.p1.7.m7.1.1.3.3.cmml" xref="S7.SS1.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S7.SS1.p1.7.m7.1.1.3.3.1.cmml" xref="S7.SS1.p1.7.m7.1.1.3.3">subscript</csymbol><ci id="S7.SS1.p1.7.m7.1.1.3.3.2.cmml" xref="S7.SS1.p1.7.m7.1.1.3.3.2">ℝ</ci><plus id="S7.SS1.p1.7.m7.1.1.3.3.3.cmml" xref="S7.SS1.p1.7.m7.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.7.m7.1c">P_{i}^{t}:S\times A\to\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.7.m7.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT : italic_S × italic_A → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>. The agents then select their strategies, which are functions <math alttext="\pi_{i}^{t}:S_{i}\to\Delta(A_{i})" class="ltx_Math" display="inline" id="S7.SS1.p1.8.m8.1"><semantics id="S7.SS1.p1.8.m8.1a"><mrow id="S7.SS1.p1.8.m8.1.1" xref="S7.SS1.p1.8.m8.1.1.cmml"><msubsup id="S7.SS1.p1.8.m8.1.1.3" xref="S7.SS1.p1.8.m8.1.1.3.cmml"><mi id="S7.SS1.p1.8.m8.1.1.3.2.2" xref="S7.SS1.p1.8.m8.1.1.3.2.2.cmml">π</mi><mi id="S7.SS1.p1.8.m8.1.1.3.2.3" xref="S7.SS1.p1.8.m8.1.1.3.2.3.cmml">i</mi><mi id="S7.SS1.p1.8.m8.1.1.3.3" xref="S7.SS1.p1.8.m8.1.1.3.3.cmml">t</mi></msubsup><mo id="S7.SS1.p1.8.m8.1.1.2" lspace="0.278em" rspace="0.278em" xref="S7.SS1.p1.8.m8.1.1.2.cmml">:</mo><mrow id="S7.SS1.p1.8.m8.1.1.1" xref="S7.SS1.p1.8.m8.1.1.1.cmml"><msub id="S7.SS1.p1.8.m8.1.1.1.3" xref="S7.SS1.p1.8.m8.1.1.1.3.cmml"><mi id="S7.SS1.p1.8.m8.1.1.1.3.2" xref="S7.SS1.p1.8.m8.1.1.1.3.2.cmml">S</mi><mi id="S7.SS1.p1.8.m8.1.1.1.3.3" xref="S7.SS1.p1.8.m8.1.1.1.3.3.cmml">i</mi></msub><mo id="S7.SS1.p1.8.m8.1.1.1.2" stretchy="false" xref="S7.SS1.p1.8.m8.1.1.1.2.cmml">→</mo><mrow id="S7.SS1.p1.8.m8.1.1.1.1" xref="S7.SS1.p1.8.m8.1.1.1.1.cmml"><mi id="S7.SS1.p1.8.m8.1.1.1.1.3" mathvariant="normal" xref="S7.SS1.p1.8.m8.1.1.1.1.3.cmml">Δ</mi><mo id="S7.SS1.p1.8.m8.1.1.1.1.2" xref="S7.SS1.p1.8.m8.1.1.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.p1.8.m8.1.1.1.1.1.1" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.cmml"><mo id="S7.SS1.p1.8.m8.1.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.cmml">(</mo><msub id="S7.SS1.p1.8.m8.1.1.1.1.1.1.1" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.cmml"><mi id="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.2" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.3" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S7.SS1.p1.8.m8.1.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.8.m8.1b"><apply id="S7.SS1.p1.8.m8.1.1.cmml" xref="S7.SS1.p1.8.m8.1.1"><ci id="S7.SS1.p1.8.m8.1.1.2.cmml" xref="S7.SS1.p1.8.m8.1.1.2">:</ci><apply id="S7.SS1.p1.8.m8.1.1.3.cmml" xref="S7.SS1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.8.m8.1.1.3.1.cmml" xref="S7.SS1.p1.8.m8.1.1.3">superscript</csymbol><apply id="S7.SS1.p1.8.m8.1.1.3.2.cmml" xref="S7.SS1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.8.m8.1.1.3.2.1.cmml" xref="S7.SS1.p1.8.m8.1.1.3">subscript</csymbol><ci id="S7.SS1.p1.8.m8.1.1.3.2.2.cmml" xref="S7.SS1.p1.8.m8.1.1.3.2.2">𝜋</ci><ci id="S7.SS1.p1.8.m8.1.1.3.2.3.cmml" xref="S7.SS1.p1.8.m8.1.1.3.2.3">𝑖</ci></apply><ci id="S7.SS1.p1.8.m8.1.1.3.3.cmml" xref="S7.SS1.p1.8.m8.1.1.3.3">𝑡</ci></apply><apply id="S7.SS1.p1.8.m8.1.1.1.cmml" xref="S7.SS1.p1.8.m8.1.1.1"><ci id="S7.SS1.p1.8.m8.1.1.1.2.cmml" xref="S7.SS1.p1.8.m8.1.1.1.2">→</ci><apply id="S7.SS1.p1.8.m8.1.1.1.3.cmml" xref="S7.SS1.p1.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.8.m8.1.1.1.3.1.cmml" xref="S7.SS1.p1.8.m8.1.1.1.3">subscript</csymbol><ci id="S7.SS1.p1.8.m8.1.1.1.3.2.cmml" xref="S7.SS1.p1.8.m8.1.1.1.3.2">𝑆</ci><ci id="S7.SS1.p1.8.m8.1.1.1.3.3.cmml" xref="S7.SS1.p1.8.m8.1.1.1.3.3">𝑖</ci></apply><apply id="S7.SS1.p1.8.m8.1.1.1.1.cmml" xref="S7.SS1.p1.8.m8.1.1.1.1"><times id="S7.SS1.p1.8.m8.1.1.1.1.2.cmml" xref="S7.SS1.p1.8.m8.1.1.1.1.2"></times><ci id="S7.SS1.p1.8.m8.1.1.1.1.3.cmml" xref="S7.SS1.p1.8.m8.1.1.1.1.3">Δ</ci><apply id="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.1.cmml" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.2.cmml" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.3.cmml" xref="S7.SS1.p1.8.m8.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.8.m8.1c">\pi_{i}^{t}:S_{i}\to\Delta(A_{i})</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.8.m8.1d">italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT : italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → roman_Δ ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. Then, the principal draws the joint signal <math alttext="s^{t}\sim\mu^{t}" class="ltx_Math" display="inline" id="S7.SS1.p1.9.m9.1"><semantics id="S7.SS1.p1.9.m9.1a"><mrow id="S7.SS1.p1.9.m9.1.1" xref="S7.SS1.p1.9.m9.1.1.cmml"><msup id="S7.SS1.p1.9.m9.1.1.2" xref="S7.SS1.p1.9.m9.1.1.2.cmml"><mi id="S7.SS1.p1.9.m9.1.1.2.2" xref="S7.SS1.p1.9.m9.1.1.2.2.cmml">s</mi><mi id="S7.SS1.p1.9.m9.1.1.2.3" xref="S7.SS1.p1.9.m9.1.1.2.3.cmml">t</mi></msup><mo id="S7.SS1.p1.9.m9.1.1.1" xref="S7.SS1.p1.9.m9.1.1.1.cmml">∼</mo><msup id="S7.SS1.p1.9.m9.1.1.3" xref="S7.SS1.p1.9.m9.1.1.3.cmml"><mi id="S7.SS1.p1.9.m9.1.1.3.2" xref="S7.SS1.p1.9.m9.1.1.3.2.cmml">μ</mi><mi id="S7.SS1.p1.9.m9.1.1.3.3" xref="S7.SS1.p1.9.m9.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.9.m9.1b"><apply id="S7.SS1.p1.9.m9.1.1.cmml" xref="S7.SS1.p1.9.m9.1.1"><csymbol cd="latexml" id="S7.SS1.p1.9.m9.1.1.1.cmml" xref="S7.SS1.p1.9.m9.1.1.1">similar-to</csymbol><apply id="S7.SS1.p1.9.m9.1.1.2.cmml" xref="S7.SS1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p1.9.m9.1.1.2.1.cmml" xref="S7.SS1.p1.9.m9.1.1.2">superscript</csymbol><ci id="S7.SS1.p1.9.m9.1.1.2.2.cmml" xref="S7.SS1.p1.9.m9.1.1.2.2">𝑠</ci><ci id="S7.SS1.p1.9.m9.1.1.2.3.cmml" xref="S7.SS1.p1.9.m9.1.1.2.3">𝑡</ci></apply><apply id="S7.SS1.p1.9.m9.1.1.3.cmml" xref="S7.SS1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.9.m9.1.1.3.1.cmml" xref="S7.SS1.p1.9.m9.1.1.3">superscript</csymbol><ci id="S7.SS1.p1.9.m9.1.1.3.2.cmml" xref="S7.SS1.p1.9.m9.1.1.3.2">𝜇</ci><ci id="S7.SS1.p1.9.m9.1.1.3.3.cmml" xref="S7.SS1.p1.9.m9.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.9.m9.1c">s^{t}\sim\mu^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.9.m9.1d">italic_s start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∼ italic_μ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, and each agent draws and plays an action <math alttext="a^{t}_{i}\sim\pi^{t}_{i}(s^{t}_{i})" class="ltx_Math" display="inline" id="S7.SS1.p1.10.m10.1"><semantics id="S7.SS1.p1.10.m10.1a"><mrow id="S7.SS1.p1.10.m10.1.1" xref="S7.SS1.p1.10.m10.1.1.cmml"><msubsup id="S7.SS1.p1.10.m10.1.1.3" xref="S7.SS1.p1.10.m10.1.1.3.cmml"><mi id="S7.SS1.p1.10.m10.1.1.3.2.2" xref="S7.SS1.p1.10.m10.1.1.3.2.2.cmml">a</mi><mi id="S7.SS1.p1.10.m10.1.1.3.3" xref="S7.SS1.p1.10.m10.1.1.3.3.cmml">i</mi><mi id="S7.SS1.p1.10.m10.1.1.3.2.3" xref="S7.SS1.p1.10.m10.1.1.3.2.3.cmml">t</mi></msubsup><mo id="S7.SS1.p1.10.m10.1.1.2" xref="S7.SS1.p1.10.m10.1.1.2.cmml">∼</mo><mrow id="S7.SS1.p1.10.m10.1.1.1" xref="S7.SS1.p1.10.m10.1.1.1.cmml"><msubsup id="S7.SS1.p1.10.m10.1.1.1.3" xref="S7.SS1.p1.10.m10.1.1.1.3.cmml"><mi id="S7.SS1.p1.10.m10.1.1.1.3.2.2" xref="S7.SS1.p1.10.m10.1.1.1.3.2.2.cmml">π</mi><mi id="S7.SS1.p1.10.m10.1.1.1.3.3" xref="S7.SS1.p1.10.m10.1.1.1.3.3.cmml">i</mi><mi id="S7.SS1.p1.10.m10.1.1.1.3.2.3" xref="S7.SS1.p1.10.m10.1.1.1.3.2.3.cmml">t</mi></msubsup><mo id="S7.SS1.p1.10.m10.1.1.1.2" xref="S7.SS1.p1.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.p1.10.m10.1.1.1.1.1" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.cmml"><mo id="S7.SS1.p1.10.m10.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.cmml">(</mo><msubsup id="S7.SS1.p1.10.m10.1.1.1.1.1.1" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.cmml"><mi id="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.2" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.2.cmml">s</mi><mi id="S7.SS1.p1.10.m10.1.1.1.1.1.1.3" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.3.cmml">i</mi><mi id="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.3" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.3.cmml">t</mi></msubsup><mo id="S7.SS1.p1.10.m10.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.10.m10.1b"><apply id="S7.SS1.p1.10.m10.1.1.cmml" xref="S7.SS1.p1.10.m10.1.1"><csymbol cd="latexml" id="S7.SS1.p1.10.m10.1.1.2.cmml" xref="S7.SS1.p1.10.m10.1.1.2">similar-to</csymbol><apply id="S7.SS1.p1.10.m10.1.1.3.cmml" xref="S7.SS1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.10.m10.1.1.3.1.cmml" xref="S7.SS1.p1.10.m10.1.1.3">subscript</csymbol><apply id="S7.SS1.p1.10.m10.1.1.3.2.cmml" xref="S7.SS1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.10.m10.1.1.3.2.1.cmml" xref="S7.SS1.p1.10.m10.1.1.3">superscript</csymbol><ci id="S7.SS1.p1.10.m10.1.1.3.2.2.cmml" xref="S7.SS1.p1.10.m10.1.1.3.2.2">𝑎</ci><ci id="S7.SS1.p1.10.m10.1.1.3.2.3.cmml" xref="S7.SS1.p1.10.m10.1.1.3.2.3">𝑡</ci></apply><ci id="S7.SS1.p1.10.m10.1.1.3.3.cmml" xref="S7.SS1.p1.10.m10.1.1.3.3">𝑖</ci></apply><apply id="S7.SS1.p1.10.m10.1.1.1.cmml" xref="S7.SS1.p1.10.m10.1.1.1"><times id="S7.SS1.p1.10.m10.1.1.1.2.cmml" xref="S7.SS1.p1.10.m10.1.1.1.2"></times><apply id="S7.SS1.p1.10.m10.1.1.1.3.cmml" xref="S7.SS1.p1.10.m10.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.10.m10.1.1.1.3.1.cmml" xref="S7.SS1.p1.10.m10.1.1.1.3">subscript</csymbol><apply id="S7.SS1.p1.10.m10.1.1.1.3.2.cmml" xref="S7.SS1.p1.10.m10.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.10.m10.1.1.1.3.2.1.cmml" xref="S7.SS1.p1.10.m10.1.1.1.3">superscript</csymbol><ci id="S7.SS1.p1.10.m10.1.1.1.3.2.2.cmml" xref="S7.SS1.p1.10.m10.1.1.1.3.2.2">𝜋</ci><ci id="S7.SS1.p1.10.m10.1.1.1.3.2.3.cmml" xref="S7.SS1.p1.10.m10.1.1.1.3.2.3">𝑡</ci></apply><ci id="S7.SS1.p1.10.m10.1.1.1.3.3.cmml" xref="S7.SS1.p1.10.m10.1.1.1.3.3">𝑖</ci></apply><apply id="S7.SS1.p1.10.m10.1.1.1.1.1.1.cmml" xref="S7.SS1.p1.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.10.m10.1.1.1.1.1.1.1.cmml" xref="S7.SS1.p1.10.m10.1.1.1.1.1">subscript</csymbol><apply id="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.cmml" xref="S7.SS1.p1.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.1.cmml" xref="S7.SS1.p1.10.m10.1.1.1.1.1">superscript</csymbol><ci id="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.2.cmml" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.2">𝑠</ci><ci id="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.3.cmml" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.2.3">𝑡</ci></apply><ci id="S7.SS1.p1.10.m10.1.1.1.1.1.1.3.cmml" xref="S7.SS1.p1.10.m10.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.10.m10.1c">a^{t}_{i}\sim\pi^{t}_{i}(s^{t}_{i})</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.10.m10.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ italic_π start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. As before, we assume that agents have no regret for each signal; formally, we assume for every signal <math alttext="s_{i}" class="ltx_Math" display="inline" id="S7.SS1.p1.11.m11.1"><semantics id="S7.SS1.p1.11.m11.1a"><msub id="S7.SS1.p1.11.m11.1.1" xref="S7.SS1.p1.11.m11.1.1.cmml"><mi id="S7.SS1.p1.11.m11.1.1.2" xref="S7.SS1.p1.11.m11.1.1.2.cmml">s</mi><mi id="S7.SS1.p1.11.m11.1.1.3" xref="S7.SS1.p1.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.11.m11.1b"><apply id="S7.SS1.p1.11.m11.1.1.cmml" xref="S7.SS1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.11.m11.1.1.1.cmml" xref="S7.SS1.p1.11.m11.1.1">subscript</csymbol><ci id="S7.SS1.p1.11.m11.1.1.2.cmml" xref="S7.SS1.p1.11.m11.1.1.2">𝑠</ci><ci id="S7.SS1.p1.11.m11.1.1.3.cmml" xref="S7.SS1.p1.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.11.m11.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.11.m11.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and action <math alttext="a_{i}" class="ltx_Math" display="inline" id="S7.SS1.p1.12.m12.1"><semantics id="S7.SS1.p1.12.m12.1a"><msub id="S7.SS1.p1.12.m12.1.1" xref="S7.SS1.p1.12.m12.1.1.cmml"><mi id="S7.SS1.p1.12.m12.1.1.2" xref="S7.SS1.p1.12.m12.1.1.2.cmml">a</mi><mi id="S7.SS1.p1.12.m12.1.1.3" xref="S7.SS1.p1.12.m12.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.12.m12.1b"><apply id="S7.SS1.p1.12.m12.1.1.cmml" xref="S7.SS1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.12.m12.1.1.1.cmml" xref="S7.SS1.p1.12.m12.1.1">subscript</csymbol><ci id="S7.SS1.p1.12.m12.1.1.2.cmml" xref="S7.SS1.p1.12.m12.1.1.2">𝑎</ci><ci id="S7.SS1.p1.12.m12.1.1.3.cmml" xref="S7.SS1.p1.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.12.m12.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.12.m12.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx9"> <tbody id="S7.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\hat{R}_{i}(t,s_{i}):=\max_{a_{i}\in A_{i}}\sum_{\begin{subarray}% {c}\tau\leq t\\ s_{i}^{\tau}=s_{i}\end{subarray}}\quantity[U_{i}^{\tau}(s^{\tau},a_{i},a_{-i}^% {\tau})-U_{i}^{\tau}(s^{\tau},a^{\tau})]\leq C\sqrt{T}." class="ltx_Math" display="inline" id="S7.E13.m1.4"><semantics id="S7.E13.m1.4a"><mrow id="S7.E13.m1.4.4.1" xref="S7.E13.m1.4.4.1.1.cmml"><mrow id="S7.E13.m1.4.4.1.1" xref="S7.E13.m1.4.4.1.1.cmml"><mrow id="S7.E13.m1.4.4.1.1.1" xref="S7.E13.m1.4.4.1.1.1.cmml"><msub id="S7.E13.m1.4.4.1.1.1.3" xref="S7.E13.m1.4.4.1.1.1.3.cmml"><mover accent="true" id="S7.E13.m1.4.4.1.1.1.3.2" 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s_{i}^{\tau}=s_{i}\end{subarray}}\quantity[U_{i}^{\tau}(s^{\tau},a_{i},a_{-i}^% {\tau})-U_{i}^{\tau}(s^{\tau},a^{\tau})]\leq C\sqrt{T}.</annotation><annotation encoding="application/x-llamapun" id="S7.E13.m1.4d">over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) := roman_max start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_τ ≤ italic_t end_CELL end_ROW start_ROW start_CELL italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_s start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ) - italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_s start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT , italic_a start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ) end_ARG ] ≤ italic_C square-root start_ARG italic_T end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S7.SS1.p1.13">where now <math alttext="U_{i}^{\tau}(s,a):=U_{i}(a)+P_{i}^{\tau}(s,a)" class="ltx_Math" display="inline" id="S7.SS1.p1.13.m1.5"><semantics id="S7.SS1.p1.13.m1.5a"><mrow id="S7.SS1.p1.13.m1.5.6" 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id="S7.SS1.p1.13.m1.3.3.cmml" xref="S7.SS1.p1.13.m1.3.3">𝑎</ci></apply><apply id="S7.SS1.p1.13.m1.5.6.3.3.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3"><times id="S7.SS1.p1.13.m1.5.6.3.3.1.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.1"></times><apply id="S7.SS1.p1.13.m1.5.6.3.3.2.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.2"><csymbol cd="ambiguous" id="S7.SS1.p1.13.m1.5.6.3.3.2.1.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.2">superscript</csymbol><apply id="S7.SS1.p1.13.m1.5.6.3.3.2.2.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.2"><csymbol cd="ambiguous" id="S7.SS1.p1.13.m1.5.6.3.3.2.2.1.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.2">subscript</csymbol><ci id="S7.SS1.p1.13.m1.5.6.3.3.2.2.2.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.2.2.2">𝑃</ci><ci id="S7.SS1.p1.13.m1.5.6.3.3.2.2.3.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.2.2.3">𝑖</ci></apply><ci id="S7.SS1.p1.13.m1.5.6.3.3.2.3.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.2.3">𝜏</ci></apply><interval closure="open" id="S7.SS1.p1.13.m1.5.6.3.3.3.1.cmml" xref="S7.SS1.p1.13.m1.5.6.3.3.3.2"><ci id="S7.SS1.p1.13.m1.4.4.cmml" xref="S7.SS1.p1.13.m1.4.4">𝑠</ci><ci id="S7.SS1.p1.13.m1.5.5.cmml" xref="S7.SS1.p1.13.m1.5.5">𝑎</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.13.m1.5c">U_{i}^{\tau}(s,a):=U_{i}(a)+P_{i}^{\tau}(s,a)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.13.m1.5d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_s , italic_a ) := italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_s , italic_a )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.SS1.p2"> <p class="ltx_p" id="S7.SS1.p2.4">We remark that this correlated no-regret model gives strictly more power to the principal than the no-regret model studied so far in the paper: if we restrict the principal to setting <math alttext="\mu^{t}" class="ltx_Math" display="inline" id="S7.SS1.p2.1.m1.1"><semantics id="S7.SS1.p2.1.m1.1a"><msup id="S7.SS1.p2.1.m1.1.1" xref="S7.SS1.p2.1.m1.1.1.cmml"><mi id="S7.SS1.p2.1.m1.1.1.2" xref="S7.SS1.p2.1.m1.1.1.2.cmml">μ</mi><mi id="S7.SS1.p2.1.m1.1.1.3" xref="S7.SS1.p2.1.m1.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.1.m1.1b"><apply id="S7.SS1.p2.1.m1.1.1.cmml" xref="S7.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p2.1.m1.1.1.1.cmml" xref="S7.SS1.p2.1.m1.1.1">superscript</csymbol><ci id="S7.SS1.p2.1.m1.1.1.2.cmml" xref="S7.SS1.p2.1.m1.1.1.2">𝜇</ci><ci id="S7.SS1.p2.1.m1.1.1.3.cmml" xref="S7.SS1.p2.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.1.m1.1c">\mu^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.1.m1.1d">italic_μ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> to be a deterministic distribution and <math alttext="P_{i}^{t}" class="ltx_Math" display="inline" id="S7.SS1.p2.2.m2.1"><semantics id="S7.SS1.p2.2.m2.1a"><msubsup id="S7.SS1.p2.2.m2.1.1" xref="S7.SS1.p2.2.m2.1.1.cmml"><mi id="S7.SS1.p2.2.m2.1.1.2.2" xref="S7.SS1.p2.2.m2.1.1.2.2.cmml">P</mi><mi id="S7.SS1.p2.2.m2.1.1.2.3" xref="S7.SS1.p2.2.m2.1.1.2.3.cmml">i</mi><mi id="S7.SS1.p2.2.m2.1.1.3" xref="S7.SS1.p2.2.m2.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.2.m2.1b"><apply id="S7.SS1.p2.2.m2.1.1.cmml" xref="S7.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.SS1.p2.2.m2.1.1.1.cmml" xref="S7.SS1.p2.2.m2.1.1">superscript</csymbol><apply id="S7.SS1.p2.2.m2.1.1.2.cmml" xref="S7.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.SS1.p2.2.m2.1.1.2.1.cmml" xref="S7.SS1.p2.2.m2.1.1">subscript</csymbol><ci id="S7.SS1.p2.2.m2.1.1.2.2.cmml" xref="S7.SS1.p2.2.m2.1.1.2.2">𝑃</ci><ci id="S7.SS1.p2.2.m2.1.1.2.3.cmml" xref="S7.SS1.p2.2.m2.1.1.2.3">𝑖</ci></apply><ci id="S7.SS1.p2.2.m2.1.1.3.cmml" xref="S7.SS1.p2.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.2.m2.1c">P_{i}^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> to be dependent only on agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS1.p2.3.m3.1"><semantics id="S7.SS1.p2.3.m3.1a"><mi id="S7.SS1.p2.3.m3.1.1" xref="S7.SS1.p2.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.3.m3.1b"><ci id="S7.SS1.p2.3.m3.1.1.cmml" xref="S7.SS1.p2.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.3.m3.1d">italic_i</annotation></semantics></math>’s action <math alttext="a_{i}\in A_{i}" class="ltx_Math" display="inline" id="S7.SS1.p2.4.m4.1"><semantics id="S7.SS1.p2.4.m4.1a"><mrow id="S7.SS1.p2.4.m4.1.1" xref="S7.SS1.p2.4.m4.1.1.cmml"><msub id="S7.SS1.p2.4.m4.1.1.2" xref="S7.SS1.p2.4.m4.1.1.2.cmml"><mi id="S7.SS1.p2.4.m4.1.1.2.2" xref="S7.SS1.p2.4.m4.1.1.2.2.cmml">a</mi><mi id="S7.SS1.p2.4.m4.1.1.2.3" xref="S7.SS1.p2.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S7.SS1.p2.4.m4.1.1.1" xref="S7.SS1.p2.4.m4.1.1.1.cmml">∈</mo><msub id="S7.SS1.p2.4.m4.1.1.3" xref="S7.SS1.p2.4.m4.1.1.3.cmml"><mi id="S7.SS1.p2.4.m4.1.1.3.2" xref="S7.SS1.p2.4.m4.1.1.3.2.cmml">A</mi><mi id="S7.SS1.p2.4.m4.1.1.3.3" xref="S7.SS1.p2.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.4.m4.1b"><apply id="S7.SS1.p2.4.m4.1.1.cmml" xref="S7.SS1.p2.4.m4.1.1"><in id="S7.SS1.p2.4.m4.1.1.1.cmml" xref="S7.SS1.p2.4.m4.1.1.1"></in><apply id="S7.SS1.p2.4.m4.1.1.2.cmml" xref="S7.SS1.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p2.4.m4.1.1.2.1.cmml" xref="S7.SS1.p2.4.m4.1.1.2">subscript</csymbol><ci id="S7.SS1.p2.4.m4.1.1.2.2.cmml" xref="S7.SS1.p2.4.m4.1.1.2.2">𝑎</ci><ci id="S7.SS1.p2.4.m4.1.1.2.3.cmml" xref="S7.SS1.p2.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S7.SS1.p2.4.m4.1.1.3.cmml" xref="S7.SS1.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p2.4.m4.1.1.3.1.cmml" xref="S7.SS1.p2.4.m4.1.1.3">subscript</csymbol><ci id="S7.SS1.p2.4.m4.1.1.3.2.cmml" xref="S7.SS1.p2.4.m4.1.1.3.2">𝐴</ci><ci id="S7.SS1.p2.4.m4.1.1.3.3.cmml" xref="S7.SS1.p2.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.4.m4.1c">a_{i}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the two models coincide. Therefore, all the upper bounds, most notably <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.Thmtheorem4" title="Theorem 5.4. ‣ 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.4</span></a>, apply to this model as well.</p> </div> <div class="ltx_para" id="S7.SS1.p3"> <p class="ltx_p" id="S7.SS1.p3.2">The reason for the difference in model is that the correlated signaling model makes clear in what formal sense the signals are <span class="ltx_text ltx_font_italic" id="S7.SS1.p3.2.1">private</span>: the agents’ strategies <math alttext="\pi_{i}^{t}" class="ltx_Math" display="inline" id="S7.SS1.p3.1.m1.1"><semantics id="S7.SS1.p3.1.m1.1a"><msubsup id="S7.SS1.p3.1.m1.1.1" xref="S7.SS1.p3.1.m1.1.1.cmml"><mi id="S7.SS1.p3.1.m1.1.1.2.2" xref="S7.SS1.p3.1.m1.1.1.2.2.cmml">π</mi><mi id="S7.SS1.p3.1.m1.1.1.2.3" xref="S7.SS1.p3.1.m1.1.1.2.3.cmml">i</mi><mi id="S7.SS1.p3.1.m1.1.1.3" xref="S7.SS1.p3.1.m1.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.1.m1.1b"><apply id="S7.SS1.p3.1.m1.1.1.cmml" xref="S7.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p3.1.m1.1.1.1.cmml" xref="S7.SS1.p3.1.m1.1.1">superscript</csymbol><apply id="S7.SS1.p3.1.m1.1.1.2.cmml" xref="S7.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p3.1.m1.1.1.2.1.cmml" xref="S7.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S7.SS1.p3.1.m1.1.1.2.2.cmml" xref="S7.SS1.p3.1.m1.1.1.2.2">𝜋</ci><ci id="S7.SS1.p3.1.m1.1.1.2.3.cmml" xref="S7.SS1.p3.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S7.SS1.p3.1.m1.1.1.3.cmml" xref="S7.SS1.p3.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.1.m1.1c">\pi_{i}^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.1.m1.1d">italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> can only depend on <math alttext="s_{i}" class="ltx_Math" display="inline" id="S7.SS1.p3.2.m2.1"><semantics id="S7.SS1.p3.2.m2.1a"><msub id="S7.SS1.p3.2.m2.1.1" xref="S7.SS1.p3.2.m2.1.1.cmml"><mi id="S7.SS1.p3.2.m2.1.1.2" xref="S7.SS1.p3.2.m2.1.1.2.cmml">s</mi><mi id="S7.SS1.p3.2.m2.1.1.3" xref="S7.SS1.p3.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.2.m2.1b"><apply id="S7.SS1.p3.2.m2.1.1.cmml" xref="S7.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S7.SS1.p3.2.m2.1.1.1.cmml" xref="S7.SS1.p3.2.m2.1.1">subscript</csymbol><ci id="S7.SS1.p3.2.m2.1.1.2.cmml" xref="S7.SS1.p3.2.m2.1.1.2">𝑠</ci><ci id="S7.SS1.p3.2.m2.1.1.3.cmml" xref="S7.SS1.p3.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.2.m2.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.2.m2.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, not other agents’ signals. This will allow us in turn to steer to <span class="ltx_text ltx_font_italic" id="S7.SS1.p3.2.2">correlated</span> equilibria.</p> </div> </section> <section class="ltx_subsection" id="S7.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.2 </span>What outcome should we steer to?</h3> <div class="ltx_para" id="S7.SS2.p1"> <p class="ltx_p" id="S7.SS2.p1.2">The steering problem, as defined by <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite>, stipulates for their main results that the principal knows in advance, or be able to compute, the desired outcome that we wish to induce. Of course, in our setting, such a stipulation is unreasonable: the principal does not initially know the agents’ utilities in the game <math alttext="\Gamma" class="ltx_Math" display="inline" id="S7.SS2.p1.1.m1.1"><semantics id="S7.SS2.p1.1.m1.1a"><mi id="S7.SS2.p1.1.m1.1.1" mathvariant="normal" xref="S7.SS2.p1.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S7.SS2.p1.1.m1.1b"><ci id="S7.SS2.p1.1.m1.1.1.cmml" xref="S7.SS2.p1.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p1.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p1.1.m1.1d">roman_Γ</annotation></semantics></math>, so it cannot know what outcome it wishes to induce. We thus take a more direct approach: we try to maximize the average reward, less payments, of the principal. That is, we will assume that the principal has a utility function <math alttext="U_{0}:A\to\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS2.p1.2.m2.1"><semantics id="S7.SS2.p1.2.m2.1a"><mrow id="S7.SS2.p1.2.m2.1.1" xref="S7.SS2.p1.2.m2.1.1.cmml"><msub id="S7.SS2.p1.2.m2.1.1.2" xref="S7.SS2.p1.2.m2.1.1.2.cmml"><mi id="S7.SS2.p1.2.m2.1.1.2.2" xref="S7.SS2.p1.2.m2.1.1.2.2.cmml">U</mi><mn id="S7.SS2.p1.2.m2.1.1.2.3" xref="S7.SS2.p1.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S7.SS2.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S7.SS2.p1.2.m2.1.1.1.cmml">:</mo><mrow id="S7.SS2.p1.2.m2.1.1.3" xref="S7.SS2.p1.2.m2.1.1.3.cmml"><mi id="S7.SS2.p1.2.m2.1.1.3.2" xref="S7.SS2.p1.2.m2.1.1.3.2.cmml">A</mi><mo id="S7.SS2.p1.2.m2.1.1.3.1" stretchy="false" xref="S7.SS2.p1.2.m2.1.1.3.1.cmml">→</mo><mi id="S7.SS2.p1.2.m2.1.1.3.3" xref="S7.SS2.p1.2.m2.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS2.p1.2.m2.1b"><apply id="S7.SS2.p1.2.m2.1.1.cmml" 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blackboard_R</annotation></semantics></math>, and we will attempt to optimize the principal’s objective, defined as the principal utility minus payments:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx10"> <tbody id="S7.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 F(T):=\frac{1}{T}\sum_{t=1}^{T}\quantity[U_{0}(a^{t})-\sum_{i=1}% ^{n}P_{i}^{t}(s^{t},a^{t})]." class="ltx_Math" display="inline" id="S7.E14.m1.3"><semantics id="S7.E14.m1.3a"><mrow id="S7.E14.m1.3.3.1" xref="S7.E14.m1.3.3.1.1.cmml"><mrow id="S7.E14.m1.3.3.1.1" xref="S7.E14.m1.3.3.1.1.cmml"><mrow id="S7.E14.m1.3.3.1.1.2" xref="S7.E14.m1.3.3.1.1.2.cmml"><mi id="S7.E14.m1.3.3.1.1.2.2" xref="S7.E14.m1.3.3.1.1.2.2.cmml">F</mi><mo id="S7.E14.m1.3.3.1.1.2.1" xref="S7.E14.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S7.E14.m1.3.3.1.1.2.3.2" 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id="S7.E14.m1.1.1.1.1.1.3.2.2.2.2.3.cmml" xref="S7.E14.m1.1.1.1.1.1.3.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E14.m1.3c">\displaystyle F(T):=\frac{1}{T}\sum_{t=1}^{T}\quantity[U_{0}(a^{t})-\sum_{i=1}% ^{n}P_{i}^{t}(s^{t},a^{t})].</annotation><annotation encoding="application/x-llamapun" id="S7.E14.m1.3d">italic_F ( italic_T ) := divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_s start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S7.SS2.p2"> <p class="ltx_p" id="S7.SS2.p2.1">We now introduce a solution concept which we call the <span class="ltx_text ltx_font_italic" id="S7.SS2.p2.1.1">correlated equilibrium with payments</span> (CEP). Intuitively, a CEP is a distribution of <span class="ltx_text ltx_font_italic" id="S7.SS2.p2.1.2">signals</span> and <span class="ltx_text ltx_font_italic" id="S7.SS2.p2.1.3">payment functions</span>, which may be correlated both with each other and across the agents, that satisfies the usual incentive compatibility constraints. Formally, we have the following definition.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem1.1.1.1">Definition 7.1</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem1.p1"> <p class="ltx_p" id="S7.Thmtheorem1.p1.9">A <span class="ltx_text ltx_font_italic" id="S7.Thmtheorem1.p1.9.1">correlated profile with payments</span> is a pair <math alttext="(\mu,P)\in\Delta(A)\times\mathbb{R}_{+}^{[n]\times A\times A}" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.1.m1.4"><semantics id="S7.Thmtheorem1.p1.1.m1.4a"><mrow id="S7.Thmtheorem1.p1.1.m1.4.5" xref="S7.Thmtheorem1.p1.1.m1.4.5.cmml"><mrow id="S7.Thmtheorem1.p1.1.m1.4.5.2.2" xref="S7.Thmtheorem1.p1.1.m1.4.5.2.1.cmml"><mo id="S7.Thmtheorem1.p1.1.m1.4.5.2.2.1" stretchy="false" 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xref="S7.Thmtheorem1.p1.1.m1.4.5.3.3">superscript</csymbol><apply id="S7.Thmtheorem1.p1.1.m1.4.5.3.3.2.cmml" xref="S7.Thmtheorem1.p1.1.m1.4.5.3.3"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.1.m1.4.5.3.3.2.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.4.5.3.3">subscript</csymbol><ci id="S7.Thmtheorem1.p1.1.m1.4.5.3.3.2.2.cmml" xref="S7.Thmtheorem1.p1.1.m1.4.5.3.3.2.2">ℝ</ci><plus id="S7.Thmtheorem1.p1.1.m1.4.5.3.3.2.3.cmml" xref="S7.Thmtheorem1.p1.1.m1.4.5.3.3.2.3"></plus></apply><apply id="S7.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.1"><times id="S7.Thmtheorem1.p1.1.m1.1.1.1.2.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.1.2"></times><apply id="S7.Thmtheorem1.p1.1.m1.1.1.1.3.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.1.3.2"><csymbol cd="latexml" id="S7.Thmtheorem1.p1.1.m1.1.1.1.3.1.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.1.3.2.1">delimited-[]</csymbol><ci id="S7.Thmtheorem1.p1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.1.1">𝑛</ci></apply><ci id="S7.Thmtheorem1.p1.1.m1.1.1.1.4.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.1.4">𝐴</ci><ci id="S7.Thmtheorem1.p1.1.m1.1.1.1.5.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.1.5">𝐴</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.1.m1.4c">(\mu,P)\in\Delta(A)\times\mathbb{R}_{+}^{[n]\times A\times A}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.1.m1.4d">( italic_μ , italic_P ) ∈ roman_Δ ( italic_A ) × blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_n ] × italic_A × italic_A end_POSTSUPERSCRIPT</annotation></semantics></math>. The vector <math alttext="P" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.2.m2.1"><semantics id="S7.Thmtheorem1.p1.2.m2.1a"><mi id="S7.Thmtheorem1.p1.2.m2.1.1" xref="S7.Thmtheorem1.p1.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.2.m2.1b"><ci id="S7.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem1.p1.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.2.m2.1d">italic_P</annotation></semantics></math> is to be interpreted as a collection of <math alttext="n" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.3.m3.1"><semantics id="S7.Thmtheorem1.p1.3.m3.1a"><mi id="S7.Thmtheorem1.p1.3.m3.1.1" xref="S7.Thmtheorem1.p1.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.3.m3.1b"><ci id="S7.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S7.Thmtheorem1.p1.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.3.m3.1d">italic_n</annotation></semantics></math> payment functions <math alttext="P_{i}:A\times A\to\mathbb{R}_{+}" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.4.m4.1"><semantics id="S7.Thmtheorem1.p1.4.m4.1a"><mrow id="S7.Thmtheorem1.p1.4.m4.1.1" xref="S7.Thmtheorem1.p1.4.m4.1.1.cmml"><msub id="S7.Thmtheorem1.p1.4.m4.1.1.2" xref="S7.Thmtheorem1.p1.4.m4.1.1.2.cmml"><mi id="S7.Thmtheorem1.p1.4.m4.1.1.2.2" xref="S7.Thmtheorem1.p1.4.m4.1.1.2.2.cmml">P</mi><mi id="S7.Thmtheorem1.p1.4.m4.1.1.2.3" xref="S7.Thmtheorem1.p1.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S7.Thmtheorem1.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem1.p1.4.m4.1.1.1.cmml">:</mo><mrow id="S7.Thmtheorem1.p1.4.m4.1.1.3" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.cmml"><mrow id="S7.Thmtheorem1.p1.4.m4.1.1.3.2" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2.cmml"><mi id="S7.Thmtheorem1.p1.4.m4.1.1.3.2.2" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2.2.cmml">A</mi><mo id="S7.Thmtheorem1.p1.4.m4.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2.1.cmml">×</mo><mi id="S7.Thmtheorem1.p1.4.m4.1.1.3.2.3" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2.3.cmml">A</mi></mrow><mo id="S7.Thmtheorem1.p1.4.m4.1.1.3.1" stretchy="false" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.1.cmml">→</mo><msub id="S7.Thmtheorem1.p1.4.m4.1.1.3.3" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.3.cmml"><mi id="S7.Thmtheorem1.p1.4.m4.1.1.3.3.2" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.3.2.cmml">ℝ</mi><mo id="S7.Thmtheorem1.p1.4.m4.1.1.3.3.3" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.3.3.cmml">+</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.4.m4.1b"><apply id="S7.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1"><ci id="S7.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.1">:</ci><apply id="S7.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.4.m4.1.1.2.1.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.2">subscript</csymbol><ci id="S7.Thmtheorem1.p1.4.m4.1.1.2.2.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.2.2">𝑃</ci><ci id="S7.Thmtheorem1.p1.4.m4.1.1.2.3.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S7.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3"><ci id="S7.Thmtheorem1.p1.4.m4.1.1.3.1.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.1">→</ci><apply id="S7.Thmtheorem1.p1.4.m4.1.1.3.2.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2"><times id="S7.Thmtheorem1.p1.4.m4.1.1.3.2.1.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2.1"></times><ci id="S7.Thmtheorem1.p1.4.m4.1.1.3.2.2.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2.2">𝐴</ci><ci id="S7.Thmtheorem1.p1.4.m4.1.1.3.2.3.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.2.3">𝐴</ci></apply><apply id="S7.Thmtheorem1.p1.4.m4.1.1.3.3.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.4.m4.1.1.3.3.1.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S7.Thmtheorem1.p1.4.m4.1.1.3.3.2.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.3.2">ℝ</ci><plus id="S7.Thmtheorem1.p1.4.m4.1.1.3.3.3.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.4.m4.1c">P_{i}:A\times A\to\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A × italic_A → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="P_{i}(s,a)" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.5.m5.2"><semantics id="S7.Thmtheorem1.p1.5.m5.2a"><mrow id="S7.Thmtheorem1.p1.5.m5.2.3" xref="S7.Thmtheorem1.p1.5.m5.2.3.cmml"><msub id="S7.Thmtheorem1.p1.5.m5.2.3.2" xref="S7.Thmtheorem1.p1.5.m5.2.3.2.cmml"><mi id="S7.Thmtheorem1.p1.5.m5.2.3.2.2" xref="S7.Thmtheorem1.p1.5.m5.2.3.2.2.cmml">P</mi><mi id="S7.Thmtheorem1.p1.5.m5.2.3.2.3" xref="S7.Thmtheorem1.p1.5.m5.2.3.2.3.cmml">i</mi></msub><mo id="S7.Thmtheorem1.p1.5.m5.2.3.1" xref="S7.Thmtheorem1.p1.5.m5.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem1.p1.5.m5.2.3.3.2" xref="S7.Thmtheorem1.p1.5.m5.2.3.3.1.cmml"><mo id="S7.Thmtheorem1.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem1.p1.5.m5.2.3.3.1.cmml">(</mo><mi id="S7.Thmtheorem1.p1.5.m5.1.1" xref="S7.Thmtheorem1.p1.5.m5.1.1.cmml">s</mi><mo id="S7.Thmtheorem1.p1.5.m5.2.3.3.2.2" xref="S7.Thmtheorem1.p1.5.m5.2.3.3.1.cmml">,</mo><mi id="S7.Thmtheorem1.p1.5.m5.2.2" xref="S7.Thmtheorem1.p1.5.m5.2.2.cmml">a</mi><mo id="S7.Thmtheorem1.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S7.Thmtheorem1.p1.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.5.m5.2b"><apply id="S7.Thmtheorem1.p1.5.m5.2.3.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.3"><times id="S7.Thmtheorem1.p1.5.m5.2.3.1.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.3.1"></times><apply id="S7.Thmtheorem1.p1.5.m5.2.3.2.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.5.m5.2.3.2.1.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.3.2">subscript</csymbol><ci id="S7.Thmtheorem1.p1.5.m5.2.3.2.2.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.3.2.2">𝑃</ci><ci id="S7.Thmtheorem1.p1.5.m5.2.3.2.3.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.3.2.3">𝑖</ci></apply><interval closure="open" id="S7.Thmtheorem1.p1.5.m5.2.3.3.1.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.3.3.2"><ci id="S7.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S7.Thmtheorem1.p1.5.m5.1.1">𝑠</ci><ci id="S7.Thmtheorem1.p1.5.m5.2.2.cmml" xref="S7.Thmtheorem1.p1.5.m5.2.2">𝑎</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.5.m5.2c">P_{i}(s,a)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.5.m5.2d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s , italic_a )</annotation></semantics></math> is the payment to agent <math alttext="i" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.6.m6.1"><semantics id="S7.Thmtheorem1.p1.6.m6.1a"><mi id="S7.Thmtheorem1.p1.6.m6.1.1" xref="S7.Thmtheorem1.p1.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.6.m6.1b"><ci id="S7.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S7.Thmtheorem1.p1.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.6.m6.1d">italic_i</annotation></semantics></math> given joint signal <math alttext="s" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.7.m7.1"><semantics id="S7.Thmtheorem1.p1.7.m7.1a"><mi id="S7.Thmtheorem1.p1.7.m7.1.1" xref="S7.Thmtheorem1.p1.7.m7.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.7.m7.1b"><ci id="S7.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S7.Thmtheorem1.p1.7.m7.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.7.m7.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.7.m7.1d">italic_s</annotation></semantics></math> and joint action <math alttext="a" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.8.m8.1"><semantics id="S7.Thmtheorem1.p1.8.m8.1a"><mi id="S7.Thmtheorem1.p1.8.m8.1.1" xref="S7.Thmtheorem1.p1.8.m8.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.8.m8.1b"><ci id="S7.Thmtheorem1.p1.8.m8.1.1.cmml" xref="S7.Thmtheorem1.p1.8.m8.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.8.m8.1c">a</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.8.m8.1d">italic_a</annotation></semantics></math>. Given such a distribution <math alttext="\mu" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.9.m9.1"><semantics id="S7.Thmtheorem1.p1.9.m9.1a"><mi id="S7.Thmtheorem1.p1.9.m9.1.1" xref="S7.Thmtheorem1.p1.9.m9.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.9.m9.1b"><ci id="S7.Thmtheorem1.p1.9.m9.1.1.cmml" xref="S7.Thmtheorem1.p1.9.m9.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.9.m9.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.9.m9.1d">italic_μ</annotation></semantics></math>, the <span class="ltx_text ltx_font_italic" id="S7.Thmtheorem1.p1.9.2">objective value</span> for the principal is defined as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx11"> <tbody id="S7.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 F(\mu):=\operatorname*{\mathbb{E}}_{a\sim\mu}\quantity[U_{0}(a)-% P(a)]," class="ltx_Math" display="inline" id="S7.E15.m1.3"><semantics id="S7.E15.m1.3a"><mrow id="S7.E15.m1.3.3.1" xref="S7.E15.m1.3.3.1.1.cmml"><mrow id="S7.E15.m1.3.3.1.1" xref="S7.E15.m1.3.3.1.1.cmml"><mrow id="S7.E15.m1.3.3.1.1.2" xref="S7.E15.m1.3.3.1.1.2.cmml"><mi id="S7.E15.m1.3.3.1.1.2.2" xref="S7.E15.m1.3.3.1.1.2.2.cmml">F</mi><mo id="S7.E15.m1.3.3.1.1.2.1" xref="S7.E15.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S7.E15.m1.3.3.1.1.2.3.2" xref="S7.E15.m1.3.3.1.1.2.cmml"><mo id="S7.E15.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S7.E15.m1.3.3.1.1.2.cmml">(</mo><mi id="S7.E15.m1.2.2" xref="S7.E15.m1.2.2.cmml">μ</mi><mo id="S7.E15.m1.3.3.1.1.2.3.2.2" rspace="0.278em" stretchy="false" xref="S7.E15.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S7.E15.m1.3.3.1.1.1" xref="S7.E15.m1.3.3.1.1.1.cmml">:=</mo><mrow id="S7.E15.m1.3.3.1.1.3" xref="S7.E15.m1.3.3.1.1.3.cmml"><munder id="S7.E15.m1.3.3.1.1.3.1" 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xref="S7.E15.m1.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S7.E15.m1.1.1.1.1.1.4.3.2" xref="S7.E15.m1.1.1.1.1.1.4.cmml"><mo id="S7.E15.m1.1.1.1.1.1.4.3.2.1" stretchy="false" xref="S7.E15.m1.1.1.1.1.1.4.cmml">(</mo><mi id="S7.E15.m1.1.1.1.1.1.1" xref="S7.E15.m1.1.1.1.1.1.1.cmml">a</mi><mo id="S7.E15.m1.1.1.1.1.1.4.3.2.2" stretchy="false" xref="S7.E15.m1.1.1.1.1.1.4.cmml">)</mo></mrow></mrow><mo id="S7.E15.m1.1.1.1.1.1.3" xref="S7.E15.m1.1.1.1.1.1.3.cmml">−</mo><mrow id="S7.E15.m1.1.1.1.1.1.5" xref="S7.E15.m1.1.1.1.1.1.5.cmml"><mi id="S7.E15.m1.1.1.1.1.1.5.2" xref="S7.E15.m1.1.1.1.1.1.5.2.cmml">P</mi><mo id="S7.E15.m1.1.1.1.1.1.5.1" xref="S7.E15.m1.1.1.1.1.1.5.1.cmml">⁢</mo><mrow id="S7.E15.m1.1.1.1.1.1.5.3.2" xref="S7.E15.m1.1.1.1.1.1.5.cmml"><mo id="S7.E15.m1.1.1.1.1.1.5.3.2.1" stretchy="false" xref="S7.E15.m1.1.1.1.1.1.5.cmml">(</mo><mi id="S7.E15.m1.1.1.1.1.1.2" xref="S7.E15.m1.1.1.1.1.1.2.cmml">a</mi><mo id="S7.E15.m1.1.1.1.1.1.5.3.2.2" stretchy="false" xref="S7.E15.m1.1.1.1.1.1.5.cmml">)</mo></mrow></mrow></mrow><mo id="S7.E15.m1.1.1a.3.2" xref="S7.E15.m1.1.1.1.1.1.cmml">]</mo></mrow></mrow></mrow><mo id="S7.E15.m1.3.3.1.2" xref="S7.E15.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.E15.m1.3b"><apply id="S7.E15.m1.3.3.1.1.cmml" xref="S7.E15.m1.3.3.1"><csymbol cd="latexml" id="S7.E15.m1.3.3.1.1.1.cmml" xref="S7.E15.m1.3.3.1.1.1">assign</csymbol><apply id="S7.E15.m1.3.3.1.1.2.cmml" xref="S7.E15.m1.3.3.1.1.2"><times id="S7.E15.m1.3.3.1.1.2.1.cmml" xref="S7.E15.m1.3.3.1.1.2.1"></times><ci id="S7.E15.m1.3.3.1.1.2.2.cmml" xref="S7.E15.m1.3.3.1.1.2.2">𝐹</ci><ci id="S7.E15.m1.2.2.cmml" xref="S7.E15.m1.2.2">𝜇</ci></apply><apply id="S7.E15.m1.3.3.1.1.3.cmml" xref="S7.E15.m1.3.3.1.1.3"><apply id="S7.E15.m1.3.3.1.1.3.1.cmml" xref="S7.E15.m1.3.3.1.1.3.1"><csymbol cd="ambiguous" id="S7.E15.m1.3.3.1.1.3.1.1.cmml" xref="S7.E15.m1.3.3.1.1.3.1">subscript</csymbol><ci id="S7.E15.m1.3.3.1.1.3.1.2.cmml" xref="S7.E15.m1.3.3.1.1.3.1.2">𝔼</ci><apply id="S7.E15.m1.3.3.1.1.3.1.3.cmml" xref="S7.E15.m1.3.3.1.1.3.1.3"><csymbol cd="latexml" id="S7.E15.m1.3.3.1.1.3.1.3.1.cmml" xref="S7.E15.m1.3.3.1.1.3.1.3.1">similar-to</csymbol><ci id="S7.E15.m1.3.3.1.1.3.1.3.2.cmml" xref="S7.E15.m1.3.3.1.1.3.1.3.2">𝑎</ci><ci id="S7.E15.m1.3.3.1.1.3.1.3.3.cmml" xref="S7.E15.m1.3.3.1.1.3.1.3.3">𝜇</ci></apply></apply><apply id="S7.E15.m1.1.1.1.1.1.cmml" xref="S7.E15.m1.1.1a.3"><minus id="S7.E15.m1.1.1.1.1.1.3.cmml" xref="S7.E15.m1.1.1.1.1.1.3"></minus><apply id="S7.E15.m1.1.1.1.1.1.4.cmml" xref="S7.E15.m1.1.1.1.1.1.4"><times id="S7.E15.m1.1.1.1.1.1.4.1.cmml" xref="S7.E15.m1.1.1.1.1.1.4.1"></times><apply id="S7.E15.m1.1.1.1.1.1.4.2.cmml" xref="S7.E15.m1.1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S7.E15.m1.1.1.1.1.1.4.2.1.cmml" xref="S7.E15.m1.1.1.1.1.1.4.2">subscript</csymbol><ci id="S7.E15.m1.1.1.1.1.1.4.2.2.cmml" xref="S7.E15.m1.1.1.1.1.1.4.2.2">𝑈</ci><cn id="S7.E15.m1.1.1.1.1.1.4.2.3.cmml" type="integer" xref="S7.E15.m1.1.1.1.1.1.4.2.3">0</cn></apply><ci id="S7.E15.m1.1.1.1.1.1.1.cmml" xref="S7.E15.m1.1.1.1.1.1.1">𝑎</ci></apply><apply id="S7.E15.m1.1.1.1.1.1.5.cmml" xref="S7.E15.m1.1.1.1.1.1.5"><times id="S7.E15.m1.1.1.1.1.1.5.1.cmml" xref="S7.E15.m1.1.1.1.1.1.5.1"></times><ci id="S7.E15.m1.1.1.1.1.1.5.2.cmml" xref="S7.E15.m1.1.1.1.1.1.5.2">𝑃</ci><ci id="S7.E15.m1.1.1.1.1.1.2.cmml" xref="S7.E15.m1.1.1.1.1.1.2">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E15.m1.3c">\displaystyle F(\mu):=\operatorname*{\mathbb{E}}_{a\sim\mu}\quantity[U_{0}(a)-% P(a)],</annotation><annotation encoding="application/x-llamapun" id="S7.E15.m1.3d">italic_F ( italic_μ ) := blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_μ end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a ) - italic_P ( italic_a ) end_ARG ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S7.Thmtheorem1.p1.12">where for notational simplicity we set <math alttext="P(a):=\sum_{i}P_{i}(a,a)" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.10.m1.3"><semantics id="S7.Thmtheorem1.p1.10.m1.3a"><mrow id="S7.Thmtheorem1.p1.10.m1.3.4" xref="S7.Thmtheorem1.p1.10.m1.3.4.cmml"><mrow id="S7.Thmtheorem1.p1.10.m1.3.4.2" xref="S7.Thmtheorem1.p1.10.m1.3.4.2.cmml"><mi id="S7.Thmtheorem1.p1.10.m1.3.4.2.2" xref="S7.Thmtheorem1.p1.10.m1.3.4.2.2.cmml">P</mi><mo id="S7.Thmtheorem1.p1.10.m1.3.4.2.1" xref="S7.Thmtheorem1.p1.10.m1.3.4.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem1.p1.10.m1.3.4.2.3.2" xref="S7.Thmtheorem1.p1.10.m1.3.4.2.cmml"><mo id="S7.Thmtheorem1.p1.10.m1.3.4.2.3.2.1" stretchy="false" xref="S7.Thmtheorem1.p1.10.m1.3.4.2.cmml">(</mo><mi id="S7.Thmtheorem1.p1.10.m1.1.1" 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xref="S7.Thmtheorem1.p1.10.m1.3.4.3.2"><times id="S7.Thmtheorem1.p1.10.m1.3.4.3.2.1.cmml" xref="S7.Thmtheorem1.p1.10.m1.3.4.3.2.1"></times><apply id="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2.cmml" xref="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2.1.cmml" xref="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2.2.cmml" xref="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2.2">𝑃</ci><ci id="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2.3.cmml" xref="S7.Thmtheorem1.p1.10.m1.3.4.3.2.2.3">𝑖</ci></apply><interval closure="open" id="S7.Thmtheorem1.p1.10.m1.3.4.3.2.3.1.cmml" xref="S7.Thmtheorem1.p1.10.m1.3.4.3.2.3.2"><ci id="S7.Thmtheorem1.p1.10.m1.2.2.cmml" xref="S7.Thmtheorem1.p1.10.m1.2.2">𝑎</ci><ci id="S7.Thmtheorem1.p1.10.m1.3.3.cmml" xref="S7.Thmtheorem1.p1.10.m1.3.3">𝑎</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.10.m1.3c">P(a):=\sum_{i}P_{i}(a,a)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.10.m1.3d">italic_P ( italic_a ) := ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a , italic_a )</annotation></semantics></math>. An <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.11.m2.1"><semantics id="S7.Thmtheorem1.p1.11.m2.1a"><mi id="S7.Thmtheorem1.p1.11.m2.1.1" xref="S7.Thmtheorem1.p1.11.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.11.m2.1b"><ci id="S7.Thmtheorem1.p1.11.m2.1.1.cmml" xref="S7.Thmtheorem1.p1.11.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.11.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.11.m2.1d">italic_ε</annotation></semantics></math>-<span class="ltx_text ltx_font_italic" id="S7.Thmtheorem1.p1.12.1">correlated equilibrium with payments</span> (CEP) is a pair <math alttext="(\mu,P)" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.12.m3.2"><semantics id="S7.Thmtheorem1.p1.12.m3.2a"><mrow id="S7.Thmtheorem1.p1.12.m3.2.3.2" xref="S7.Thmtheorem1.p1.12.m3.2.3.1.cmml"><mo id="S7.Thmtheorem1.p1.12.m3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem1.p1.12.m3.2.3.1.cmml">(</mo><mi id="S7.Thmtheorem1.p1.12.m3.1.1" xref="S7.Thmtheorem1.p1.12.m3.1.1.cmml">μ</mi><mo id="S7.Thmtheorem1.p1.12.m3.2.3.2.2" xref="S7.Thmtheorem1.p1.12.m3.2.3.1.cmml">,</mo><mi id="S7.Thmtheorem1.p1.12.m3.2.2" xref="S7.Thmtheorem1.p1.12.m3.2.2.cmml">P</mi><mo id="S7.Thmtheorem1.p1.12.m3.2.3.2.3" stretchy="false" xref="S7.Thmtheorem1.p1.12.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.12.m3.2b"><interval closure="open" id="S7.Thmtheorem1.p1.12.m3.2.3.1.cmml" xref="S7.Thmtheorem1.p1.12.m3.2.3.2"><ci id="S7.Thmtheorem1.p1.12.m3.1.1.cmml" xref="S7.Thmtheorem1.p1.12.m3.1.1">𝜇</ci><ci id="S7.Thmtheorem1.p1.12.m3.2.2.cmml" xref="S7.Thmtheorem1.p1.12.m3.2.2">𝑃</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.12.m3.2c">(\mu,P)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.12.m3.2d">( italic_μ , italic_P )</annotation></semantics></math> satisfying the incentive compatibility (IC) constraints</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx12"> <tbody id="S7.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{\mathbb{E}}_{a\sim\mu}\quantity[U_{i}^{P}(a,\phi_{% i}(a_{i}),a_{-i})-U_{i}^{P}(a,a)]\leq\varepsilon," class="ltx_Math" display="inline" id="S7.E16.m1.2"><semantics id="S7.E16.m1.2a"><mrow id="S7.E16.m1.2.2.1" xref="S7.E16.m1.2.2.1.1.cmml"><mrow id="S7.E16.m1.2.2.1.1" xref="S7.E16.m1.2.2.1.1.cmml"><mrow id="S7.E16.m1.2.2.1.1.2" xref="S7.E16.m1.2.2.1.1.2.cmml"><munder id="S7.E16.m1.2.2.1.1.2.1" xref="S7.E16.m1.2.2.1.1.2.1.cmml"><mo id="S7.E16.m1.2.2.1.1.2.1.2" xref="S7.E16.m1.2.2.1.1.2.1.2.cmml">𝔼</mo><mrow id="S7.E16.m1.2.2.1.1.2.1.3" xref="S7.E16.m1.2.2.1.1.2.1.3.cmml"><mi 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id="S7.E16.m1.2.2.1.1.3.cmml" xref="S7.E16.m1.2.2.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E16.m1.2c">\displaystyle\operatorname*{\mathbb{E}}_{a\sim\mu}\quantity[U_{i}^{P}(a,\phi_{% i}(a_{i}),a_{-i})-U_{i}^{P}(a,a)]\leq\varepsilon,</annotation><annotation encoding="application/x-llamapun" id="S7.E16.m1.2d">blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_μ end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( italic_a , italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) - italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( italic_a , italic_a ) end_ARG ] ≤ italic_ε ,</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">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S7.Thmtheorem1.p1.18">where <math alttext="U_{i}^{P}(s,a):=U_{i}(a)+P_{i}(s,a)" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.13.m1.5"><semantics id="S7.Thmtheorem1.p1.13.m1.5a"><mrow id="S7.Thmtheorem1.p1.13.m1.5.6" xref="S7.Thmtheorem1.p1.13.m1.5.6.cmml"><mrow id="S7.Thmtheorem1.p1.13.m1.5.6.2" xref="S7.Thmtheorem1.p1.13.m1.5.6.2.cmml"><msubsup id="S7.Thmtheorem1.p1.13.m1.5.6.2.2" xref="S7.Thmtheorem1.p1.13.m1.5.6.2.2.cmml"><mi id="S7.Thmtheorem1.p1.13.m1.5.6.2.2.2.2" xref="S7.Thmtheorem1.p1.13.m1.5.6.2.2.2.2.cmml">U</mi><mi id="S7.Thmtheorem1.p1.13.m1.5.6.2.2.2.3" xref="S7.Thmtheorem1.p1.13.m1.5.6.2.2.2.3.cmml">i</mi><mi id="S7.Thmtheorem1.p1.13.m1.5.6.2.2.3" xref="S7.Thmtheorem1.p1.13.m1.5.6.2.2.3.cmml">P</mi></msubsup><mo id="S7.Thmtheorem1.p1.13.m1.5.6.2.1" 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xref="S7.Thmtheorem1.p1.13.m1.5.6.3.3.2.3">𝑖</ci></apply><interval closure="open" id="S7.Thmtheorem1.p1.13.m1.5.6.3.3.3.1.cmml" xref="S7.Thmtheorem1.p1.13.m1.5.6.3.3.3.2"><ci id="S7.Thmtheorem1.p1.13.m1.4.4.cmml" xref="S7.Thmtheorem1.p1.13.m1.4.4">𝑠</ci><ci id="S7.Thmtheorem1.p1.13.m1.5.5.cmml" xref="S7.Thmtheorem1.p1.13.m1.5.5">𝑎</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.13.m1.5c">U_{i}^{P}(s,a):=U_{i}(a)+P_{i}(s,a)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.13.m1.5d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( italic_s , italic_a ) := italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s , italic_a )</annotation></semantics></math>, for every agent <math alttext="i" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.14.m2.1"><semantics id="S7.Thmtheorem1.p1.14.m2.1a"><mi id="S7.Thmtheorem1.p1.14.m2.1.1" xref="S7.Thmtheorem1.p1.14.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.14.m2.1b"><ci id="S7.Thmtheorem1.p1.14.m2.1.1.cmml" xref="S7.Thmtheorem1.p1.14.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.14.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.14.m2.1d">italic_i</annotation></semantics></math> and deviation function <math alttext="\phi_{i}:A_{i}\to A_{i}" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.15.m3.1"><semantics id="S7.Thmtheorem1.p1.15.m3.1a"><mrow id="S7.Thmtheorem1.p1.15.m3.1.1" xref="S7.Thmtheorem1.p1.15.m3.1.1.cmml"><msub id="S7.Thmtheorem1.p1.15.m3.1.1.2" xref="S7.Thmtheorem1.p1.15.m3.1.1.2.cmml"><mi id="S7.Thmtheorem1.p1.15.m3.1.1.2.2" xref="S7.Thmtheorem1.p1.15.m3.1.1.2.2.cmml">ϕ</mi><mi id="S7.Thmtheorem1.p1.15.m3.1.1.2.3" xref="S7.Thmtheorem1.p1.15.m3.1.1.2.3.cmml">i</mi></msub><mo id="S7.Thmtheorem1.p1.15.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem1.p1.15.m3.1.1.1.cmml">:</mo><mrow id="S7.Thmtheorem1.p1.15.m3.1.1.3" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.cmml"><msub id="S7.Thmtheorem1.p1.15.m3.1.1.3.2" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.2.cmml"><mi id="S7.Thmtheorem1.p1.15.m3.1.1.3.2.2" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.2.2.cmml">A</mi><mi id="S7.Thmtheorem1.p1.15.m3.1.1.3.2.3" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.2.3.cmml">i</mi></msub><mo id="S7.Thmtheorem1.p1.15.m3.1.1.3.1" stretchy="false" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.1.cmml">→</mo><msub id="S7.Thmtheorem1.p1.15.m3.1.1.3.3" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.3.cmml"><mi id="S7.Thmtheorem1.p1.15.m3.1.1.3.3.2" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.3.2.cmml">A</mi><mi id="S7.Thmtheorem1.p1.15.m3.1.1.3.3.3" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.15.m3.1b"><apply id="S7.Thmtheorem1.p1.15.m3.1.1.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1"><ci id="S7.Thmtheorem1.p1.15.m3.1.1.1.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.1">:</ci><apply id="S7.Thmtheorem1.p1.15.m3.1.1.2.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.2"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.15.m3.1.1.2.1.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.2">subscript</csymbol><ci id="S7.Thmtheorem1.p1.15.m3.1.1.2.2.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.2.2">italic-ϕ</ci><ci id="S7.Thmtheorem1.p1.15.m3.1.1.2.3.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.2.3">𝑖</ci></apply><apply id="S7.Thmtheorem1.p1.15.m3.1.1.3.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3"><ci id="S7.Thmtheorem1.p1.15.m3.1.1.3.1.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.1">→</ci><apply id="S7.Thmtheorem1.p1.15.m3.1.1.3.2.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.15.m3.1.1.3.2.1.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.2">subscript</csymbol><ci id="S7.Thmtheorem1.p1.15.m3.1.1.3.2.2.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.2.2">𝐴</ci><ci id="S7.Thmtheorem1.p1.15.m3.1.1.3.2.3.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.2.3">𝑖</ci></apply><apply id="S7.Thmtheorem1.p1.15.m3.1.1.3.3.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.3"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.15.m3.1.1.3.3.1.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.3">subscript</csymbol><ci id="S7.Thmtheorem1.p1.15.m3.1.1.3.3.2.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.3.2">𝐴</ci><ci id="S7.Thmtheorem1.p1.15.m3.1.1.3.3.3.cmml" xref="S7.Thmtheorem1.p1.15.m3.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.15.m3.1c">\phi_{i}:A_{i}\to A_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.15.m3.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. An <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.16.m4.1"><semantics id="S7.Thmtheorem1.p1.16.m4.1a"><mi id="S7.Thmtheorem1.p1.16.m4.1.1" xref="S7.Thmtheorem1.p1.16.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.16.m4.1b"><ci id="S7.Thmtheorem1.p1.16.m4.1.1.cmml" xref="S7.Thmtheorem1.p1.16.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.16.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.16.m4.1d">italic_ε</annotation></semantics></math>-CEP is <span class="ltx_text ltx_font_italic" id="S7.Thmtheorem1.p1.18.1">optimal</span> if it maximizes the principal objective <math alttext="F(\mu)" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.17.m5.1"><semantics id="S7.Thmtheorem1.p1.17.m5.1a"><mrow id="S7.Thmtheorem1.p1.17.m5.1.2" xref="S7.Thmtheorem1.p1.17.m5.1.2.cmml"><mi id="S7.Thmtheorem1.p1.17.m5.1.2.2" xref="S7.Thmtheorem1.p1.17.m5.1.2.2.cmml">F</mi><mo id="S7.Thmtheorem1.p1.17.m5.1.2.1" xref="S7.Thmtheorem1.p1.17.m5.1.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem1.p1.17.m5.1.2.3.2" xref="S7.Thmtheorem1.p1.17.m5.1.2.cmml"><mo id="S7.Thmtheorem1.p1.17.m5.1.2.3.2.1" stretchy="false" xref="S7.Thmtheorem1.p1.17.m5.1.2.cmml">(</mo><mi id="S7.Thmtheorem1.p1.17.m5.1.1" xref="S7.Thmtheorem1.p1.17.m5.1.1.cmml">μ</mi><mo id="S7.Thmtheorem1.p1.17.m5.1.2.3.2.2" stretchy="false" xref="S7.Thmtheorem1.p1.17.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.17.m5.1b"><apply id="S7.Thmtheorem1.p1.17.m5.1.2.cmml" xref="S7.Thmtheorem1.p1.17.m5.1.2"><times id="S7.Thmtheorem1.p1.17.m5.1.2.1.cmml" xref="S7.Thmtheorem1.p1.17.m5.1.2.1"></times><ci id="S7.Thmtheorem1.p1.17.m5.1.2.2.cmml" xref="S7.Thmtheorem1.p1.17.m5.1.2.2">𝐹</ci><ci id="S7.Thmtheorem1.p1.17.m5.1.1.cmml" xref="S7.Thmtheorem1.p1.17.m5.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.17.m5.1c">F(\mu)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.17.m5.1d">italic_F ( italic_μ )</annotation></semantics></math> over all <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.18.m6.1"><semantics id="S7.Thmtheorem1.p1.18.m6.1a"><mi id="S7.Thmtheorem1.p1.18.m6.1.1" xref="S7.Thmtheorem1.p1.18.m6.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.18.m6.1b"><ci id="S7.Thmtheorem1.p1.18.m6.1.1.cmml" xref="S7.Thmtheorem1.p1.18.m6.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.18.m6.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.18.m6.1d">italic_ε</annotation></semantics></math>-CEPs.</p> </div> </div> <div class="ltx_para" id="S7.SS2.p3"> <p class="ltx_p" id="S7.SS2.p3.2">Note that it is without loss of generality to assume that the principal never gives payments to non-equilibrium actions, <span class="ltx_text ltx_font_italic" id="S7.SS2.p3.2.1">i.e.</span>, we have <math alttext="P(s,a)=0" class="ltx_Math" display="inline" id="S7.SS2.p3.1.m1.2"><semantics id="S7.SS2.p3.1.m1.2a"><mrow id="S7.SS2.p3.1.m1.2.3" xref="S7.SS2.p3.1.m1.2.3.cmml"><mrow id="S7.SS2.p3.1.m1.2.3.2" xref="S7.SS2.p3.1.m1.2.3.2.cmml"><mi id="S7.SS2.p3.1.m1.2.3.2.2" xref="S7.SS2.p3.1.m1.2.3.2.2.cmml">P</mi><mo id="S7.SS2.p3.1.m1.2.3.2.1" xref="S7.SS2.p3.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S7.SS2.p3.1.m1.2.3.2.3.2" xref="S7.SS2.p3.1.m1.2.3.2.3.1.cmml"><mo id="S7.SS2.p3.1.m1.2.3.2.3.2.1" stretchy="false" xref="S7.SS2.p3.1.m1.2.3.2.3.1.cmml">(</mo><mi id="S7.SS2.p3.1.m1.1.1" xref="S7.SS2.p3.1.m1.1.1.cmml">s</mi><mo id="S7.SS2.p3.1.m1.2.3.2.3.2.2" xref="S7.SS2.p3.1.m1.2.3.2.3.1.cmml">,</mo><mi id="S7.SS2.p3.1.m1.2.2" xref="S7.SS2.p3.1.m1.2.2.cmml">a</mi><mo id="S7.SS2.p3.1.m1.2.3.2.3.2.3" stretchy="false" xref="S7.SS2.p3.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S7.SS2.p3.1.m1.2.3.1" xref="S7.SS2.p3.1.m1.2.3.1.cmml">=</mo><mn id="S7.SS2.p3.1.m1.2.3.3" xref="S7.SS2.p3.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS2.p3.1.m1.2b"><apply id="S7.SS2.p3.1.m1.2.3.cmml" xref="S7.SS2.p3.1.m1.2.3"><eq id="S7.SS2.p3.1.m1.2.3.1.cmml" xref="S7.SS2.p3.1.m1.2.3.1"></eq><apply id="S7.SS2.p3.1.m1.2.3.2.cmml" xref="S7.SS2.p3.1.m1.2.3.2"><times id="S7.SS2.p3.1.m1.2.3.2.1.cmml" xref="S7.SS2.p3.1.m1.2.3.2.1"></times><ci id="S7.SS2.p3.1.m1.2.3.2.2.cmml" xref="S7.SS2.p3.1.m1.2.3.2.2">𝑃</ci><interval closure="open" id="S7.SS2.p3.1.m1.2.3.2.3.1.cmml" xref="S7.SS2.p3.1.m1.2.3.2.3.2"><ci id="S7.SS2.p3.1.m1.1.1.cmml" xref="S7.SS2.p3.1.m1.1.1">𝑠</ci><ci id="S7.SS2.p3.1.m1.2.2.cmml" xref="S7.SS2.p3.1.m1.2.2">𝑎</ci></interval></apply><cn id="S7.SS2.p3.1.m1.2.3.3.cmml" type="integer" xref="S7.SS2.p3.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p3.1.m1.2c">P(s,a)=0</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p3.1.m1.2d">italic_P ( italic_s , italic_a ) = 0</annotation></semantics></math> for <math alttext="s\neq a" class="ltx_Math" display="inline" id="S7.SS2.p3.2.m2.1"><semantics id="S7.SS2.p3.2.m2.1a"><mrow id="S7.SS2.p3.2.m2.1.1" xref="S7.SS2.p3.2.m2.1.1.cmml"><mi id="S7.SS2.p3.2.m2.1.1.2" xref="S7.SS2.p3.2.m2.1.1.2.cmml">s</mi><mo id="S7.SS2.p3.2.m2.1.1.1" xref="S7.SS2.p3.2.m2.1.1.1.cmml">≠</mo><mi id="S7.SS2.p3.2.m2.1.1.3" xref="S7.SS2.p3.2.m2.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS2.p3.2.m2.1b"><apply id="S7.SS2.p3.2.m2.1.1.cmml" xref="S7.SS2.p3.2.m2.1.1"><neq id="S7.SS2.p3.2.m2.1.1.1.cmml" xref="S7.SS2.p3.2.m2.1.1.1"></neq><ci id="S7.SS2.p3.2.m2.1.1.2.cmml" xref="S7.SS2.p3.2.m2.1.1.2">𝑠</ci><ci id="S7.SS2.p3.2.m2.1.1.3.cmml" xref="S7.SS2.p3.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p3.2.m2.1c">s\neq a</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p3.2.m2.1d">italic_s ≠ italic_a</annotation></semantics></math>. Giving such payments would only decreases the principal objective and worsens incentive compatibility.</p> </div> </section> <section class="ltx_subsection" id="S7.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.3 </span>Properties of correlated equilibria with payments</h3> <div class="ltx_para" id="S7.SS3.p1"> <p class="ltx_p" id="S7.SS3.p1.1">Before proceeding with our analysis of steering learners toward CEPs, we state some simple results about CEPs in general. First, optimal (<math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.SS3.p1.1.m1.1"><semantics id="S7.SS3.p1.1.m1.1a"><mi id="S7.SS3.p1.1.m1.1.1" xref="S7.SS3.p1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p1.1.m1.1b"><ci id="S7.SS3.p1.1.m1.1.1.cmml" xref="S7.SS3.p1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p1.1.m1.1d">italic_ε</annotation></semantics></math>-)CEPs can be efficiently computed when the game is known.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S7.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="S7.Thmtheorem2.1.1.1">Proposition 7.2</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem2.p1"> <p class="ltx_p" id="S7.Thmtheorem2.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem2.p1.4.4">There is a <math alttext="\poly(M)" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.1.1.m1.1"><semantics id="S7.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem2.p1.1.1.m1.1.2" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S7.Thmtheorem2.p1.1.1.m1.1.2.2" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.2b.cmml"><mtext id="S7.Thmtheorem2.p1.1.1.m1.1.2.2a" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.2b.cmml">\poly</mtext></merror><mo id="S7.Thmtheorem2.p1.1.1.m1.1.2.1" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem2.p1.1.1.m1.1.2.3.2" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.cmml"><mo id="S7.Thmtheorem2.p1.1.1.m1.1.2.3.2.1" stretchy="false" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.cmml">(</mo><mi id="S7.Thmtheorem2.p1.1.1.m1.1.1" xref="S7.Thmtheorem2.p1.1.1.m1.1.1.cmml">M</mi><mo id="S7.Thmtheorem2.p1.1.1.m1.1.2.3.2.2" stretchy="false" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.1.1.m1.1b"><apply id="S7.Thmtheorem2.p1.1.1.m1.1.2.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.2"><times id="S7.Thmtheorem2.p1.1.1.m1.1.2.1.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.1"></times><ci id="S7.Thmtheorem2.p1.1.1.m1.1.2.2b.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.2"><merror class="ltx_ERROR undefined undefined" id="S7.Thmtheorem2.p1.1.1.m1.1.2.2.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.2"><mtext id="S7.Thmtheorem2.p1.1.1.m1.1.2.2a.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.2.2">\poly</mtext></merror></ci><ci id="S7.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.1.1.m1.1c">\poly(M)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.1.1.m1.1d">( italic_M )</annotation></semantics></math>-time algorithm for computing an optimal <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.2.2.m2.1"><semantics id="S7.Thmtheorem2.p1.2.2.m2.1a"><mi id="S7.Thmtheorem2.p1.2.2.m2.1.1" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.2.2.m2.1b"><ci id="S7.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem2.p1.2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.2.2.m2.1d">italic_ε</annotation></semantics></math>-CEP, given a game <math alttext="\Gamma" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.3.3.m3.1"><semantics id="S7.Thmtheorem2.p1.3.3.m3.1a"><mi id="S7.Thmtheorem2.p1.3.3.m3.1.1" mathvariant="normal" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.3.3.m3.1b"><ci id="S7.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.3.3.m3.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.3.3.m3.1d">roman_Γ</annotation></semantics></math> with rational utility values and rational parameter <math alttext="\varepsilon\geq 0" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.4.4.m4.1"><semantics id="S7.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S7.Thmtheorem2.p1.4.4.m4.1.1" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">ε</mi><mo id="S7.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.1.cmml">≥</mo><mn id="S7.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.4.4.m4.1b"><apply id="S7.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem2.p1.4.4.m4.1.1"><geq id="S7.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.1"></geq><ci id="S7.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.2">𝜀</ci><cn id="S7.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.4.4.m4.1c">\varepsilon\geq 0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.4.4.m4.1d">italic_ε ≥ 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S7.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS3.1.p1"> <p class="ltx_p" id="S7.SS3.1.p1.8">Use the change of variables</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx13"> <tbody id="S7.E17"><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 Q_{i}(a_{i}):=\mu(a_{i})\cdot\operatorname*{\mathbb{E}}_{a\sim% \mu|a_{i}}P_{i}(a,a)." class="ltx_Math" display="inline" id="S7.E17.m1.3"><semantics id="S7.E17.m1.3a"><mrow id="S7.E17.m1.3.3.1" xref="S7.E17.m1.3.3.1.1.cmml"><mrow id="S7.E17.m1.3.3.1.1" xref="S7.E17.m1.3.3.1.1.cmml"><mrow id="S7.E17.m1.3.3.1.1.1" xref="S7.E17.m1.3.3.1.1.1.cmml"><msub id="S7.E17.m1.3.3.1.1.1.3" xref="S7.E17.m1.3.3.1.1.1.3.cmml"><mi id="S7.E17.m1.3.3.1.1.1.3.2" xref="S7.E17.m1.3.3.1.1.1.3.2.cmml">Q</mi><mi id="S7.E17.m1.3.3.1.1.1.3.3" xref="S7.E17.m1.3.3.1.1.1.3.3.cmml">i</mi></msub><mo id="S7.E17.m1.3.3.1.1.1.2" xref="S7.E17.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S7.E17.m1.3.3.1.1.1.1.1" xref="S7.E17.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S7.E17.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S7.E17.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S7.E17.m1.3.3.1.1.1.1.1.1" xref="S7.E17.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S7.E17.m1.3.3.1.1.1.1.1.1.2" 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id="S7.E17.m1.3.3.1.1.2.1.3.2.2.cmml" xref="S7.E17.m1.3.3.1.1.2.1.3.2.2">𝑃</ci><ci id="S7.E17.m1.3.3.1.1.2.1.3.2.3.cmml" xref="S7.E17.m1.3.3.1.1.2.1.3.2.3">𝑖</ci></apply></apply></apply><interval closure="open" id="S7.E17.m1.3.3.1.1.2.3.1.cmml" xref="S7.E17.m1.3.3.1.1.2.3.2"><ci id="S7.E17.m1.1.1.cmml" xref="S7.E17.m1.1.1">𝑎</ci><ci id="S7.E17.m1.2.2.cmml" xref="S7.E17.m1.2.2">𝑎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E17.m1.3c">\displaystyle Q_{i}(a_{i}):=\mu(a_{i})\cdot\operatorname*{\mathbb{E}}_{a\sim% \mu|a_{i}}P_{i}(a,a).</annotation><annotation encoding="application/x-llamapun" id="S7.E17.m1.3d">italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) := italic_μ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ⋅ blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_μ | italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a , italic_a ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S7.SS3.1.p1.6">That is, <math alttext="Q_{i}(a_{i})" class="ltx_Math" display="inline" id="S7.SS3.1.p1.1.m1.1"><semantics id="S7.SS3.1.p1.1.m1.1a"><mrow id="S7.SS3.1.p1.1.m1.1.1" xref="S7.SS3.1.p1.1.m1.1.1.cmml"><msub id="S7.SS3.1.p1.1.m1.1.1.3" xref="S7.SS3.1.p1.1.m1.1.1.3.cmml"><mi id="S7.SS3.1.p1.1.m1.1.1.3.2" xref="S7.SS3.1.p1.1.m1.1.1.3.2.cmml">Q</mi><mi id="S7.SS3.1.p1.1.m1.1.1.3.3" xref="S7.SS3.1.p1.1.m1.1.1.3.3.cmml">i</mi></msub><mo id="S7.SS3.1.p1.1.m1.1.1.2" xref="S7.SS3.1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S7.SS3.1.p1.1.m1.1.1.1.1" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S7.SS3.1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S7.SS3.1.p1.1.m1.1.1.1.1.1" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.cmml"><mi id="S7.SS3.1.p1.1.m1.1.1.1.1.1.2" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.2.cmml">a</mi><mi id="S7.SS3.1.p1.1.m1.1.1.1.1.1.3" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S7.SS3.1.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.1.p1.1.m1.1b"><apply id="S7.SS3.1.p1.1.m1.1.1.cmml" xref="S7.SS3.1.p1.1.m1.1.1"><times id="S7.SS3.1.p1.1.m1.1.1.2.cmml" xref="S7.SS3.1.p1.1.m1.1.1.2"></times><apply id="S7.SS3.1.p1.1.m1.1.1.3.cmml" xref="S7.SS3.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.SS3.1.p1.1.m1.1.1.3.1.cmml" xref="S7.SS3.1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.SS3.1.p1.1.m1.1.1.3.2.cmml" xref="S7.SS3.1.p1.1.m1.1.1.3.2">𝑄</ci><ci id="S7.SS3.1.p1.1.m1.1.1.3.3.cmml" xref="S7.SS3.1.p1.1.m1.1.1.3.3">𝑖</ci></apply><apply id="S7.SS3.1.p1.1.m1.1.1.1.1.1.cmml" xref="S7.SS3.1.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS3.1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S7.SS3.1.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S7.SS3.1.p1.1.m1.1.1.1.1.1.2.cmml" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.2">𝑎</ci><ci id="S7.SS3.1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S7.SS3.1.p1.1.m1.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.1.p1.1.m1.1c">Q_{i}(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.1.p1.1.m1.1d">italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is the <math alttext="\mu" class="ltx_Math" display="inline" id="S7.SS3.1.p1.2.m2.1"><semantics id="S7.SS3.1.p1.2.m2.1a"><mi id="S7.SS3.1.p1.2.m2.1.1" xref="S7.SS3.1.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.1.p1.2.m2.1b"><ci id="S7.SS3.1.p1.2.m2.1.1.cmml" xref="S7.SS3.1.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.1.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.1.p1.2.m2.1d">italic_μ</annotation></semantics></math>-weighted total payment given to agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS3.1.p1.3.m3.1"><semantics id="S7.SS3.1.p1.3.m3.1a"><mi id="S7.SS3.1.p1.3.m3.1.1" xref="S7.SS3.1.p1.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.1.p1.3.m3.1b"><ci id="S7.SS3.1.p1.3.m3.1.1.cmml" xref="S7.SS3.1.p1.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.1.p1.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.1.p1.3.m3.1d">italic_i</annotation></semantics></math> across all strategy profiles on which agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS3.1.p1.4.m4.1"><semantics id="S7.SS3.1.p1.4.m4.1a"><mi id="S7.SS3.1.p1.4.m4.1.1" xref="S7.SS3.1.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.1.p1.4.m4.1b"><ci id="S7.SS3.1.p1.4.m4.1.1.cmml" xref="S7.SS3.1.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.1.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.1.p1.4.m4.1d">italic_i</annotation></semantics></math> is recommended action <math alttext="a_{i}" class="ltx_Math" display="inline" id="S7.SS3.1.p1.5.m5.1"><semantics id="S7.SS3.1.p1.5.m5.1a"><msub id="S7.SS3.1.p1.5.m5.1.1" xref="S7.SS3.1.p1.5.m5.1.1.cmml"><mi id="S7.SS3.1.p1.5.m5.1.1.2" xref="S7.SS3.1.p1.5.m5.1.1.2.cmml">a</mi><mi id="S7.SS3.1.p1.5.m5.1.1.3" xref="S7.SS3.1.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS3.1.p1.5.m5.1b"><apply id="S7.SS3.1.p1.5.m5.1.1.cmml" xref="S7.SS3.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S7.SS3.1.p1.5.m5.1.1.1.cmml" xref="S7.SS3.1.p1.5.m5.1.1">subscript</csymbol><ci id="S7.SS3.1.p1.5.m5.1.1.2.cmml" xref="S7.SS3.1.p1.5.m5.1.1.2">𝑎</ci><ci id="S7.SS3.1.p1.5.m5.1.1.3.cmml" xref="S7.SS3.1.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.1.p1.5.m5.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.1.p1.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Then the following LP precisely represents the problem of computing an optimal <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.SS3.1.p1.6.m6.1"><semantics id="S7.SS3.1.p1.6.m6.1a"><mi id="S7.SS3.1.p1.6.m6.1.1" xref="S7.SS3.1.p1.6.m6.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.1.p1.6.m6.1b"><ci id="S7.SS3.1.p1.6.m6.1.1.cmml" xref="S7.SS3.1.p1.6.m6.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.1.p1.6.m6.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.1.p1.6.m6.1d">italic_ε</annotation></semantics></math>-CEP.</p> <table class="ltx_equationgroup ltx_eqn_table" id="S7.E18"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S7.E18X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\max\quad\sum_{a\in A}\mu(a)U_{0}(a)-\sum_{\begin{subarray}{c}i% \in[n]\\ a_{i}\in 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xref="S7.E18X.2.1.1.m1.5.5.1.1.1.1.3.3">𝑖</ci></apply><interval closure="open" id="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.2.cmml" xref="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1"><apply id="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.cmml" xref="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1">subscript</csymbol><ci id="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1.2.cmml" xref="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1.3.cmml" xref="S7.E18X.2.1.1.m1.5.5.1.1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S7.E18X.2.1.1.m1.3.3.cmml" xref="S7.E18X.2.1.1.m1.3.3">𝑎</ci></interval></apply></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S7.E18X.2.1.1.m1.5c">\displaystyle\max\quad\sum_{a\in A}\mu(a)U_{0}(a)-\sum_{\begin{subarray}{c}i% \in[n]\\ a_{i}\in A_{i}\end{subarray}}Q_{i}(a_{i},a)</annotation><annotation encoding="application/x-llamapun" id="S7.E18X.2.1.1.m1.5d">roman_max ∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_μ ( italic_a ) italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a ) - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_i ∈ [ italic_n ] end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><span class="ltx_text ltx_markedasmath ltx_font_italic" id="S7.E18X.3.2.2.1"> s.t.</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padright" colspan="3"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="5"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(18)</span></td> </tr> <tr 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start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_μ ( italic_a ) [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) - italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a ) end_ARG ] - italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\varepsilon_{i}(a_{i})\mbox{\quad\quad}" class="ltx_Math" display="inline" id="S7.E18Xa.3.2.2.m1.1"><semantics id="S7.E18Xa.3.2.2.m1.1a"><mrow id="S7.E18Xa.3.2.2.m1.1.1" xref="S7.E18Xa.3.2.2.m1.1.1.cmml"><mi id="S7.E18Xa.3.2.2.m1.1.1.3" xref="S7.E18Xa.3.2.2.m1.1.1.3.cmml"></mi><mo id="S7.E18Xa.3.2.2.m1.1.1.2" xref="S7.E18Xa.3.2.2.m1.1.1.2.cmml">≤</mo><mrow 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xref="S7.E18Xa.5.1.1.m1.2.2.1.1.2.2.2.3">𝑖</ci></apply></list></apply><apply id="S7.E18Xa.5.1.1.m1.3.3.2.2.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2"><in id="S7.E18Xa.5.1.1.m1.3.3.2.2.1.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.1"></in><apply id="S7.E18Xa.5.1.1.m1.3.3.2.2.2.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S7.E18Xa.5.1.1.m1.3.3.2.2.2.1.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.2">superscript</csymbol><apply id="S7.E18Xa.5.1.1.m1.3.3.2.2.2.2.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S7.E18Xa.5.1.1.m1.3.3.2.2.2.2.1.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S7.E18Xa.5.1.1.m1.3.3.2.2.2.2.2.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.2.2.2">𝑎</ci><ci id="S7.E18Xa.5.1.1.m1.3.3.2.2.2.2.3.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.2.2.3">𝑖</ci></apply><ci id="S7.E18Xa.5.1.1.m1.3.3.2.2.2.3.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.2.3">′</ci></apply><apply id="S7.E18Xa.5.1.1.m1.3.3.2.2.3.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.3"><csymbol cd="ambiguous" id="S7.E18Xa.5.1.1.m1.3.3.2.2.3.1.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.3">subscript</csymbol><ci id="S7.E18Xa.5.1.1.m1.3.3.2.2.3.2.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.3.2">𝐴</ci><ci id="S7.E18Xa.5.1.1.m1.3.3.2.2.3.3.cmml" xref="S7.E18Xa.5.1.1.m1.3.3.2.2.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xa.5.1.1.m1.3c">\displaystyle\forall i\in[n],a_{i},a_{i}^{\prime}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xa.5.1.1.m1.3d">∀ italic_i ∈ [ italic_n ] , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S7.E18Xb"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{a_{i}\in A_{i}}\varepsilon_{i}(a_{i})" class="ltx_Math" display="inline" id="S7.E18Xb.2.1.1.m1.1"><semantics id="S7.E18Xb.2.1.1.m1.1a"><mrow id="S7.E18Xb.2.1.1.m1.1.1" xref="S7.E18Xb.2.1.1.m1.1.1.cmml"><mstyle displaystyle="true" id="S7.E18Xb.2.1.1.m1.1.1.2" xref="S7.E18Xb.2.1.1.m1.1.1.2.cmml"><munder id="S7.E18Xb.2.1.1.m1.1.1.2a" xref="S7.E18Xb.2.1.1.m1.1.1.2.cmml"><mo id="S7.E18Xb.2.1.1.m1.1.1.2.2" movablelimits="false" xref="S7.E18Xb.2.1.1.m1.1.1.2.2.cmml">∑</mo><mrow id="S7.E18Xb.2.1.1.m1.1.1.2.3" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.cmml"><msub id="S7.E18Xb.2.1.1.m1.1.1.2.3.2" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.2.cmml"><mi id="S7.E18Xb.2.1.1.m1.1.1.2.3.2.2" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.2.2.cmml">a</mi><mi id="S7.E18Xb.2.1.1.m1.1.1.2.3.2.3" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.2.3.cmml">i</mi></msub><mo id="S7.E18Xb.2.1.1.m1.1.1.2.3.1" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.1.cmml">∈</mo><msub id="S7.E18Xb.2.1.1.m1.1.1.2.3.3" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.3.cmml"><mi id="S7.E18Xb.2.1.1.m1.1.1.2.3.3.2" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.3.2.cmml">A</mi><mi id="S7.E18Xb.2.1.1.m1.1.1.2.3.3.3" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.3.3.cmml">i</mi></msub></mrow></munder></mstyle><mrow id="S7.E18Xb.2.1.1.m1.1.1.1" xref="S7.E18Xb.2.1.1.m1.1.1.1.cmml"><msub id="S7.E18Xb.2.1.1.m1.1.1.1.3" xref="S7.E18Xb.2.1.1.m1.1.1.1.3.cmml"><mi id="S7.E18Xb.2.1.1.m1.1.1.1.3.2" xref="S7.E18Xb.2.1.1.m1.1.1.1.3.2.cmml">ε</mi><mi id="S7.E18Xb.2.1.1.m1.1.1.1.3.3" xref="S7.E18Xb.2.1.1.m1.1.1.1.3.3.cmml">i</mi></msub><mo id="S7.E18Xb.2.1.1.m1.1.1.1.2" xref="S7.E18Xb.2.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S7.E18Xb.2.1.1.m1.1.1.1.1.1" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.2" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.3" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.E18Xb.2.1.1.m1.1b"><apply id="S7.E18Xb.2.1.1.m1.1.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1"><apply id="S7.E18Xb.2.1.1.m1.1.1.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S7.E18Xb.2.1.1.m1.1.1.2.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2">subscript</csymbol><sum id="S7.E18Xb.2.1.1.m1.1.1.2.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.2"></sum><apply id="S7.E18Xb.2.1.1.m1.1.1.2.3.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3"><in id="S7.E18Xb.2.1.1.m1.1.1.2.3.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.1"></in><apply id="S7.E18Xb.2.1.1.m1.1.1.2.3.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.2"><csymbol cd="ambiguous" id="S7.E18Xb.2.1.1.m1.1.1.2.3.2.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.2">subscript</csymbol><ci id="S7.E18Xb.2.1.1.m1.1.1.2.3.2.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.2.2">𝑎</ci><ci id="S7.E18Xb.2.1.1.m1.1.1.2.3.2.3.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.2.3">𝑖</ci></apply><apply id="S7.E18Xb.2.1.1.m1.1.1.2.3.3.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.3"><csymbol cd="ambiguous" id="S7.E18Xb.2.1.1.m1.1.1.2.3.3.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.3">subscript</csymbol><ci id="S7.E18Xb.2.1.1.m1.1.1.2.3.3.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.3.2">𝐴</ci><ci id="S7.E18Xb.2.1.1.m1.1.1.2.3.3.3.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.2.3.3.3">𝑖</ci></apply></apply></apply><apply id="S7.E18Xb.2.1.1.m1.1.1.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1"><times id="S7.E18Xb.2.1.1.m1.1.1.1.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.2"></times><apply id="S7.E18Xb.2.1.1.m1.1.1.1.3.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S7.E18Xb.2.1.1.m1.1.1.1.3.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.3">subscript</csymbol><ci id="S7.E18Xb.2.1.1.m1.1.1.1.3.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.3.2">𝜀</ci><ci id="S7.E18Xb.2.1.1.m1.1.1.1.3.3.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.3.3">𝑖</ci></apply><apply id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S7.E18Xb.2.1.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xb.2.1.1.m1.1c">\displaystyle\sum_{a_{i}\in A_{i}}\varepsilon_{i}(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xb.2.1.1.m1.1d">∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\varepsilon\mbox{\quad\quad}" class="ltx_Math" display="inline" id="S7.E18Xb.3.2.2.m1.1"><semantics id="S7.E18Xb.3.2.2.m1.1a"><mrow id="S7.E18Xb.3.2.2.m1.1.1" xref="S7.E18Xb.3.2.2.m1.1.1.cmml"><mi id="S7.E18Xb.3.2.2.m1.1.1.2" xref="S7.E18Xb.3.2.2.m1.1.1.2.cmml"></mi><mo id="S7.E18Xb.3.2.2.m1.1.1.1" xref="S7.E18Xb.3.2.2.m1.1.1.1.cmml">≤</mo><mrow id="S7.E18Xb.3.2.2.m1.1.1.3" xref="S7.E18Xb.3.2.2.m1.1.1.3.cmml"><mi id="S7.E18Xb.3.2.2.m1.1.1.3.2" xref="S7.E18Xb.3.2.2.m1.1.1.3.2.cmml">ε</mi><mo id="S7.E18Xb.3.2.2.m1.1.1.3.1" xref="S7.E18Xb.3.2.2.m1.1.1.3.1.cmml">⁢</mo><mrow id="S7.E18Xb.3.2.2.m1.1.1.3.3" xref="S7.E18Xb.3.2.2.m1.1.1.3.3a.cmml"></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.E18Xb.3.2.2.m1.1b"><apply id="S7.E18Xb.3.2.2.m1.1.1.cmml" xref="S7.E18Xb.3.2.2.m1.1.1"><leq id="S7.E18Xb.3.2.2.m1.1.1.1.cmml" xref="S7.E18Xb.3.2.2.m1.1.1.1"></leq><csymbol cd="latexml" id="S7.E18Xb.3.2.2.m1.1.1.2.cmml" xref="S7.E18Xb.3.2.2.m1.1.1.2">absent</csymbol><apply id="S7.E18Xb.3.2.2.m1.1.1.3.cmml" xref="S7.E18Xb.3.2.2.m1.1.1.3"><times id="S7.E18Xb.3.2.2.m1.1.1.3.1.cmml" xref="S7.E18Xb.3.2.2.m1.1.1.3.1"></times><ci id="S7.E18Xb.3.2.2.m1.1.1.3.2.cmml" xref="S7.E18Xb.3.2.2.m1.1.1.3.2">𝜀</ci><ci id="S7.E18Xb.3.2.2.m1.1.1.3.3a.cmml" xref="S7.E18Xb.3.2.2.m1.1.1.3.3"><mrow id="S7.E18Xb.3.2.2.m1.1.1.3.3.cmml" xref="S7.E18Xb.3.2.2.m1.1.1.3.3"></mrow></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xb.3.2.2.m1.1c">\displaystyle\leq\varepsilon\mbox{\quad\quad}</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xb.3.2.2.m1.1d">≤ italic_ε</annotation></semantics></math></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\forall i\in[n]" class="ltx_Math" display="inline" id="S7.E18Xb.5.1.1.m1.1"><semantics id="S7.E18Xb.5.1.1.m1.1a"><mrow id="S7.E18Xb.5.1.1.m1.1.2" xref="S7.E18Xb.5.1.1.m1.1.2.cmml"><mrow id="S7.E18Xb.5.1.1.m1.1.2.2" xref="S7.E18Xb.5.1.1.m1.1.2.2.cmml"><mo id="S7.E18Xb.5.1.1.m1.1.2.2.1" rspace="0.167em" xref="S7.E18Xb.5.1.1.m1.1.2.2.1.cmml">∀</mo><mi id="S7.E18Xb.5.1.1.m1.1.2.2.2" xref="S7.E18Xb.5.1.1.m1.1.2.2.2.cmml">i</mi></mrow><mo id="S7.E18Xb.5.1.1.m1.1.2.1" xref="S7.E18Xb.5.1.1.m1.1.2.1.cmml">∈</mo><mrow id="S7.E18Xb.5.1.1.m1.1.2.3.2" xref="S7.E18Xb.5.1.1.m1.1.2.3.1.cmml"><mo id="S7.E18Xb.5.1.1.m1.1.2.3.2.1" stretchy="false" xref="S7.E18Xb.5.1.1.m1.1.2.3.1.1.cmml">[</mo><mi id="S7.E18Xb.5.1.1.m1.1.1" xref="S7.E18Xb.5.1.1.m1.1.1.cmml">n</mi><mo id="S7.E18Xb.5.1.1.m1.1.2.3.2.2" stretchy="false" xref="S7.E18Xb.5.1.1.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.E18Xb.5.1.1.m1.1b"><apply id="S7.E18Xb.5.1.1.m1.1.2.cmml" xref="S7.E18Xb.5.1.1.m1.1.2"><in id="S7.E18Xb.5.1.1.m1.1.2.1.cmml" xref="S7.E18Xb.5.1.1.m1.1.2.1"></in><apply id="S7.E18Xb.5.1.1.m1.1.2.2.cmml" xref="S7.E18Xb.5.1.1.m1.1.2.2"><csymbol cd="latexml" id="S7.E18Xb.5.1.1.m1.1.2.2.1.cmml" xref="S7.E18Xb.5.1.1.m1.1.2.2.1">for-all</csymbol><ci id="S7.E18Xb.5.1.1.m1.1.2.2.2.cmml" xref="S7.E18Xb.5.1.1.m1.1.2.2.2">𝑖</ci></apply><apply id="S7.E18Xb.5.1.1.m1.1.2.3.1.cmml" xref="S7.E18Xb.5.1.1.m1.1.2.3.2"><csymbol cd="latexml" id="S7.E18Xb.5.1.1.m1.1.2.3.1.1.cmml" xref="S7.E18Xb.5.1.1.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S7.E18Xb.5.1.1.m1.1.1.cmml" xref="S7.E18Xb.5.1.1.m1.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xb.5.1.1.m1.1c">\displaystyle\forall i\in[n]</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xb.5.1.1.m1.1d">∀ italic_i ∈ [ italic_n ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> 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xref="S7.E18Xc.2.1.1.m1.1.2.1.3.3.cmml">A</mi></mrow></munder></mstyle><mrow id="S7.E18Xc.2.1.1.m1.1.2.2" xref="S7.E18Xc.2.1.1.m1.1.2.2.cmml"><mi id="S7.E18Xc.2.1.1.m1.1.2.2.2" xref="S7.E18Xc.2.1.1.m1.1.2.2.2.cmml">μ</mi><mo id="S7.E18Xc.2.1.1.m1.1.2.2.1" xref="S7.E18Xc.2.1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S7.E18Xc.2.1.1.m1.1.2.2.3.2" xref="S7.E18Xc.2.1.1.m1.1.2.2.cmml"><mo id="S7.E18Xc.2.1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S7.E18Xc.2.1.1.m1.1.2.2.cmml">(</mo><mi id="S7.E18Xc.2.1.1.m1.1.1" xref="S7.E18Xc.2.1.1.m1.1.1.cmml">a</mi><mo id="S7.E18Xc.2.1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S7.E18Xc.2.1.1.m1.1.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.E18Xc.2.1.1.m1.1b"><apply id="S7.E18Xc.2.1.1.m1.1.2.cmml" xref="S7.E18Xc.2.1.1.m1.1.2"><apply id="S7.E18Xc.2.1.1.m1.1.2.1.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.1"><csymbol cd="ambiguous" id="S7.E18Xc.2.1.1.m1.1.2.1.1.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.1">subscript</csymbol><sum id="S7.E18Xc.2.1.1.m1.1.2.1.2.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.1.2"></sum><apply id="S7.E18Xc.2.1.1.m1.1.2.1.3.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.1.3"><in id="S7.E18Xc.2.1.1.m1.1.2.1.3.1.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.1.3.1"></in><ci id="S7.E18Xc.2.1.1.m1.1.2.1.3.2.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.1.3.2">𝑎</ci><ci id="S7.E18Xc.2.1.1.m1.1.2.1.3.3.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.1.3.3">𝐴</ci></apply></apply><apply id="S7.E18Xc.2.1.1.m1.1.2.2.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.2"><times id="S7.E18Xc.2.1.1.m1.1.2.2.1.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.2.1"></times><ci id="S7.E18Xc.2.1.1.m1.1.2.2.2.cmml" xref="S7.E18Xc.2.1.1.m1.1.2.2.2">𝜇</ci><ci id="S7.E18Xc.2.1.1.m1.1.1.cmml" xref="S7.E18Xc.2.1.1.m1.1.1">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xc.2.1.1.m1.1c">\displaystyle\sum_{a\in A}\mu(a)</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xc.2.1.1.m1.1d">∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_μ ( italic_a )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=1" class="ltx_Math" display="inline" id="S7.E18Xc.3.2.2.m1.1"><semantics id="S7.E18Xc.3.2.2.m1.1a"><mrow id="S7.E18Xc.3.2.2.m1.1.1" xref="S7.E18Xc.3.2.2.m1.1.1.cmml"><mi id="S7.E18Xc.3.2.2.m1.1.1.2" xref="S7.E18Xc.3.2.2.m1.1.1.2.cmml"></mi><mo id="S7.E18Xc.3.2.2.m1.1.1.1" xref="S7.E18Xc.3.2.2.m1.1.1.1.cmml">=</mo><mn id="S7.E18Xc.3.2.2.m1.1.1.3" xref="S7.E18Xc.3.2.2.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.E18Xc.3.2.2.m1.1b"><apply id="S7.E18Xc.3.2.2.m1.1.1.cmml" xref="S7.E18Xc.3.2.2.m1.1.1"><eq id="S7.E18Xc.3.2.2.m1.1.1.1.cmml" xref="S7.E18Xc.3.2.2.m1.1.1.1"></eq><csymbol cd="latexml" id="S7.E18Xc.3.2.2.m1.1.1.2.cmml" xref="S7.E18Xc.3.2.2.m1.1.1.2">absent</csymbol><cn id="S7.E18Xc.3.2.2.m1.1.1.3.cmml" type="integer" xref="S7.E18Xc.3.2.2.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xc.3.2.2.m1.1c">\displaystyle=1</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xc.3.2.2.m1.1d">= 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright" colspan="3"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S7.E18Xd"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 0\leq Q_{i}(a_{i})" class="ltx_Math" display="inline" id="S7.E18Xd.2.1.1.m1.1"><semantics id="S7.E18Xd.2.1.1.m1.1a"><mrow id="S7.E18Xd.2.1.1.m1.1.1" xref="S7.E18Xd.2.1.1.m1.1.1.cmml"><mn id="S7.E18Xd.2.1.1.m1.1.1.3" xref="S7.E18Xd.2.1.1.m1.1.1.3.cmml">0</mn><mo id="S7.E18Xd.2.1.1.m1.1.1.2" xref="S7.E18Xd.2.1.1.m1.1.1.2.cmml">≤</mo><mrow id="S7.E18Xd.2.1.1.m1.1.1.1" xref="S7.E18Xd.2.1.1.m1.1.1.1.cmml"><msub id="S7.E18Xd.2.1.1.m1.1.1.1.3" xref="S7.E18Xd.2.1.1.m1.1.1.1.3.cmml"><mi id="S7.E18Xd.2.1.1.m1.1.1.1.3.2" xref="S7.E18Xd.2.1.1.m1.1.1.1.3.2.cmml">Q</mi><mi id="S7.E18Xd.2.1.1.m1.1.1.1.3.3" xref="S7.E18Xd.2.1.1.m1.1.1.1.3.3.cmml">i</mi></msub><mo id="S7.E18Xd.2.1.1.m1.1.1.1.2" xref="S7.E18Xd.2.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S7.E18Xd.2.1.1.m1.1.1.1.1.1" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.2" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.3" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.E18Xd.2.1.1.m1.1b"><apply id="S7.E18Xd.2.1.1.m1.1.1.cmml" xref="S7.E18Xd.2.1.1.m1.1.1"><leq id="S7.E18Xd.2.1.1.m1.1.1.2.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.2"></leq><cn id="S7.E18Xd.2.1.1.m1.1.1.3.cmml" type="integer" xref="S7.E18Xd.2.1.1.m1.1.1.3">0</cn><apply id="S7.E18Xd.2.1.1.m1.1.1.1.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1"><times id="S7.E18Xd.2.1.1.m1.1.1.1.2.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.2"></times><apply id="S7.E18Xd.2.1.1.m1.1.1.1.3.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S7.E18Xd.2.1.1.m1.1.1.1.3.1.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.3">subscript</csymbol><ci id="S7.E18Xd.2.1.1.m1.1.1.1.3.2.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.3.2">𝑄</ci><ci id="S7.E18Xd.2.1.1.m1.1.1.1.3.3.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.3.3">𝑖</ci></apply><apply id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S7.E18Xd.2.1.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xd.2.1.1.m1.1c">\displaystyle 0\leq Q_{i}(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xd.2.1.1.m1.1d">0 ≤ italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\mu(a_{i})" class="ltx_Math" display="inline" id="S7.E18Xd.3.2.2.m1.1"><semantics id="S7.E18Xd.3.2.2.m1.1a"><mrow id="S7.E18Xd.3.2.2.m1.1.1" xref="S7.E18Xd.3.2.2.m1.1.1.cmml"><mi id="S7.E18Xd.3.2.2.m1.1.1.3" xref="S7.E18Xd.3.2.2.m1.1.1.3.cmml"></mi><mo id="S7.E18Xd.3.2.2.m1.1.1.2" xref="S7.E18Xd.3.2.2.m1.1.1.2.cmml">≤</mo><mrow id="S7.E18Xd.3.2.2.m1.1.1.1" xref="S7.E18Xd.3.2.2.m1.1.1.1.cmml"><mi id="S7.E18Xd.3.2.2.m1.1.1.1.3" xref="S7.E18Xd.3.2.2.m1.1.1.1.3.cmml">μ</mi><mo 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xref="S7.E18Xd.5.1.1.m1.3.3.2.2.3"><csymbol cd="ambiguous" id="S7.E18Xd.5.1.1.m1.3.3.2.2.3.1.cmml" xref="S7.E18Xd.5.1.1.m1.3.3.2.2.3">subscript</csymbol><ci id="S7.E18Xd.5.1.1.m1.3.3.2.2.3.2.cmml" xref="S7.E18Xd.5.1.1.m1.3.3.2.2.3.2">𝐴</ci><ci id="S7.E18Xd.5.1.1.m1.3.3.2.2.3.3.cmml" xref="S7.E18Xd.5.1.1.m1.3.3.2.2.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E18Xd.5.1.1.m1.3c">\displaystyle\forall i\in[n],a_{i},a_{i}^{\prime}\in A_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.E18Xd.5.1.1.m1.3d">∀ italic_i ∈ [ italic_n ] , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A 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="S7.SS3.1.p1.7">This LP has <math alttext="\poly(M)" class="ltx_Math" display="inline" id="S7.SS3.1.p1.7.m1.1"><semantics id="S7.SS3.1.p1.7.m1.1a"><mrow id="S7.SS3.1.p1.7.m1.1.2" xref="S7.SS3.1.p1.7.m1.1.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S7.SS3.1.p1.7.m1.1.2.2" xref="S7.SS3.1.p1.7.m1.1.2.2b.cmml"><mtext id="S7.SS3.1.p1.7.m1.1.2.2a" xref="S7.SS3.1.p1.7.m1.1.2.2b.cmml">\poly</mtext></merror><mo id="S7.SS3.1.p1.7.m1.1.2.1" xref="S7.SS3.1.p1.7.m1.1.2.1.cmml">⁢</mo><mrow id="S7.SS3.1.p1.7.m1.1.2.3.2" xref="S7.SS3.1.p1.7.m1.1.2.cmml"><mo id="S7.SS3.1.p1.7.m1.1.2.3.2.1" stretchy="false" xref="S7.SS3.1.p1.7.m1.1.2.cmml">(</mo><mi id="S7.SS3.1.p1.7.m1.1.1" xref="S7.SS3.1.p1.7.m1.1.1.cmml">M</mi><mo id="S7.SS3.1.p1.7.m1.1.2.3.2.2" stretchy="false" xref="S7.SS3.1.p1.7.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.1.p1.7.m1.1b"><apply id="S7.SS3.1.p1.7.m1.1.2.cmml" xref="S7.SS3.1.p1.7.m1.1.2"><times id="S7.SS3.1.p1.7.m1.1.2.1.cmml" xref="S7.SS3.1.p1.7.m1.1.2.1"></times><ci id="S7.SS3.1.p1.7.m1.1.2.2b.cmml" xref="S7.SS3.1.p1.7.m1.1.2.2"><merror class="ltx_ERROR undefined undefined" id="S7.SS3.1.p1.7.m1.1.2.2.cmml" xref="S7.SS3.1.p1.7.m1.1.2.2"><mtext id="S7.SS3.1.p1.7.m1.1.2.2a.cmml" xref="S7.SS3.1.p1.7.m1.1.2.2">\poly</mtext></merror></ci><ci id="S7.SS3.1.p1.7.m1.1.1.cmml" xref="S7.SS3.1.p1.7.m1.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.1.p1.7.m1.1c">\poly(M)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.1.p1.7.m1.1d">( italic_M )</annotation></semantics></math> variables and constraints, so the proof is complete. ∎</p> </div> </div> <div class="ltx_para" id="S7.SS3.p2"> <p class="ltx_p" id="S7.SS3.p2.1">The above result also implies that it is WLOG to restrict the payment functions to output range <math alttext="[0,1]" class="ltx_Math" display="inline" id="S7.SS3.p2.1.m1.2"><semantics id="S7.SS3.p2.1.m1.2a"><mrow id="S7.SS3.p2.1.m1.2.3.2" 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incentive constraints.</p> </div> <div class="ltx_para" id="S7.SS3.p3"> <p class="ltx_p" id="S7.SS3.p3.1">Second, the fact that the payments can be signal-dependent is not innocuous, except when the payment at equilibrium is zero.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S7.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="S7.Thmtheorem3.1.1.1">Proposition 7.3</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem3.2.2"> </span>(Correlation does not help when no payments are allowed in equilibrium)<span class="ltx_text ltx_font_bold" id="S7.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem3.p1"> <p class="ltx_p" id="S7.Thmtheorem3.p1.2"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem3.p1.2.2">The <math alttext="0" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.1.1.m1.1"><semantics id="S7.Thmtheorem3.p1.1.1.m1.1a"><mn 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id="S7.Thmtheorem3.p1.2.2.m2.2.2" xref="S7.Thmtheorem3.p1.2.2.m2.2.2.cmml">a</mi><mo id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.3.2.3" stretchy="false" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem3.p1.2.2.m2.2.3.1" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.1.cmml">=</mo><mn id="S7.Thmtheorem3.p1.2.2.m2.2.3.3" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.2.2.m2.2b"><apply id="S7.Thmtheorem3.p1.2.2.m2.2.3.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3"><eq id="S7.Thmtheorem3.p1.2.2.m2.2.3.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.1"></eq><apply id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2"><times id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.1"></times><apply id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2"><apply id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1"><csymbol cd="ambiguous" id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1">subscript</csymbol><ci id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.2.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.2">𝔼</ci><apply id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3"><csymbol cd="latexml" id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3.1">similar-to</csymbol><ci id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3.2.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3.2">𝑎</ci><ci id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3.3.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.1.3.3">𝜇</ci></apply></apply><ci id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.2.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.2.2">𝑃</ci></apply><interval closure="open" id="S7.Thmtheorem3.p1.2.2.m2.2.3.2.3.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.2.3.2"><ci id="S7.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.1.1">𝑎</ci><ci id="S7.Thmtheorem3.p1.2.2.m2.2.2.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.2.2">𝑎</ci></interval></apply><cn id="S7.Thmtheorem3.p1.2.2.m2.2.3.3.cmml" type="integer" xref="S7.Thmtheorem3.p1.2.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.2.2.m2.2c">\operatorname*{\mathbb{E}}_{a\sim\mu}P(a,a)=0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.2.2.m2.2d">blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_μ end_POSTSUBSCRIPT italic_P ( italic_a , italic_a ) = 0</annotation></semantics></math> are exactly the correlated equilibria.</span></p> </div> </div> <div class="ltx_proof" id="S7.SS3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS3.2.p1"> <p class="ltx_p" id="S7.SS3.2.p1.2">In the LP (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.E18" title="Equation 18 ‣ Proof. ‣ 7.3 Properties of correlated equilibria with payments ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">18</span></a>), this is equivalent to setting <math alttext="Q_{i}(\cdot)=0" class="ltx_Math" display="inline" id="S7.SS3.2.p1.1.m1.1"><semantics id="S7.SS3.2.p1.1.m1.1a"><mrow id="S7.SS3.2.p1.1.m1.1.2" xref="S7.SS3.2.p1.1.m1.1.2.cmml"><mrow id="S7.SS3.2.p1.1.m1.1.2.2" xref="S7.SS3.2.p1.1.m1.1.2.2.cmml"><msub id="S7.SS3.2.p1.1.m1.1.2.2.2" xref="S7.SS3.2.p1.1.m1.1.2.2.2.cmml"><mi id="S7.SS3.2.p1.1.m1.1.2.2.2.2" xref="S7.SS3.2.p1.1.m1.1.2.2.2.2.cmml">Q</mi><mi id="S7.SS3.2.p1.1.m1.1.2.2.2.3" xref="S7.SS3.2.p1.1.m1.1.2.2.2.3.cmml">i</mi></msub><mo id="S7.SS3.2.p1.1.m1.1.2.2.1" xref="S7.SS3.2.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS3.2.p1.1.m1.1.2.2.3.2" xref="S7.SS3.2.p1.1.m1.1.2.2.cmml"><mo id="S7.SS3.2.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S7.SS3.2.p1.1.m1.1.2.2.cmml">(</mo><mo id="S7.SS3.2.p1.1.m1.1.1" lspace="0em" rspace="0em" xref="S7.SS3.2.p1.1.m1.1.1.cmml">⋅</mo><mo id="S7.SS3.2.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S7.SS3.2.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS3.2.p1.1.m1.1.2.1" xref="S7.SS3.2.p1.1.m1.1.2.1.cmml">=</mo><mn id="S7.SS3.2.p1.1.m1.1.2.3" xref="S7.SS3.2.p1.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.2.p1.1.m1.1b"><apply id="S7.SS3.2.p1.1.m1.1.2.cmml" xref="S7.SS3.2.p1.1.m1.1.2"><eq id="S7.SS3.2.p1.1.m1.1.2.1.cmml" xref="S7.SS3.2.p1.1.m1.1.2.1"></eq><apply id="S7.SS3.2.p1.1.m1.1.2.2.cmml" xref="S7.SS3.2.p1.1.m1.1.2.2"><times id="S7.SS3.2.p1.1.m1.1.2.2.1.cmml" xref="S7.SS3.2.p1.1.m1.1.2.2.1"></times><apply id="S7.SS3.2.p1.1.m1.1.2.2.2.cmml" xref="S7.SS3.2.p1.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS3.2.p1.1.m1.1.2.2.2.1.cmml" xref="S7.SS3.2.p1.1.m1.1.2.2.2">subscript</csymbol><ci id="S7.SS3.2.p1.1.m1.1.2.2.2.2.cmml" xref="S7.SS3.2.p1.1.m1.1.2.2.2.2">𝑄</ci><ci id="S7.SS3.2.p1.1.m1.1.2.2.2.3.cmml" xref="S7.SS3.2.p1.1.m1.1.2.2.2.3">𝑖</ci></apply><ci id="S7.SS3.2.p1.1.m1.1.1.cmml" xref="S7.SS3.2.p1.1.m1.1.1">⋅</ci></apply><cn id="S7.SS3.2.p1.1.m1.1.2.3.cmml" type="integer" xref="S7.SS3.2.p1.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.2.p1.1.m1.1c">Q_{i}(\cdot)=0</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.2.p1.1.m1.1d">italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ ) = 0</annotation></semantics></math> for every agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS3.2.p1.2.m2.1"><semantics id="S7.SS3.2.p1.2.m2.1a"><mi id="S7.SS3.2.p1.2.m2.1.1" xref="S7.SS3.2.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.2.p1.2.m2.1b"><ci id="S7.SS3.2.p1.2.m2.1.1.cmml" xref="S7.SS3.2.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.2.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.2.p1.2.m2.1d">italic_i</annotation></semantics></math>, in which case (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.E18" title="Equation 18 ‣ Proof. ‣ 7.3 Properties of correlated equilibria with payments ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">18</span></a>) is just the LP characterizing correlated equilibria. ∎</p> </div> </div> <div class="ltx_para" id="S7.SS3.p4"> <p class="ltx_p" id="S7.SS3.p4.1">However, when the payment at equilibrium is positive, it is possible for signal-dependent payments to help the principal.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S7.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="S7.Thmtheorem4.1.1.1">Proposition 7.4</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem4.2.2"> </span>(Signal-dependent payments can help in general)<span class="ltx_text ltx_font_bold" id="S7.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem4.p1"> <p class="ltx_p" id="S7.Thmtheorem4.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem4.p1.4.4">There exists a game <math alttext="\Gamma" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.1.1.m1.1"><semantics id="S7.Thmtheorem4.p1.1.1.m1.1a"><mi id="S7.Thmtheorem4.p1.1.1.m1.1.1" mathvariant="normal" xref="S7.Thmtheorem4.p1.1.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.1.1.m1.1b"><ci id="S7.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem4.p1.1.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.1.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.1.1.m1.1d">roman_Γ</annotation></semantics></math>, and pricipal utility function <math alttext="u_{0}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.2.2.m2.1"><semantics id="S7.Thmtheorem4.p1.2.2.m2.1a"><msub id="S7.Thmtheorem4.p1.2.2.m2.1.1" xref="S7.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">u</mi><mn id="S7.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.2.2.m2.1b"><apply id="S7.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.1">subscript</csymbol><ci id="S7.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.1.2">𝑢</ci><cn id="S7.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem4.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.2.2.m2.1c">u_{0}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.2.2.m2.1d">italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, such that the optimal value of (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.E18" title="Equation 18 ‣ Proof. ‣ 7.3 Properties of correlated equilibria with payments ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">18</span></a>) is greater than the objective value of the optimal CEP in which <math alttext="P(s,a)" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.3.3.m3.2"><semantics id="S7.Thmtheorem4.p1.3.3.m3.2a"><mrow id="S7.Thmtheorem4.p1.3.3.m3.2.3" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.cmml"><mi id="S7.Thmtheorem4.p1.3.3.m3.2.3.2" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.2.cmml">P</mi><mo id="S7.Thmtheorem4.p1.3.3.m3.2.3.1" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem4.p1.3.3.m3.2.3.3.2" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.3.1.cmml"><mo id="S7.Thmtheorem4.p1.3.3.m3.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.3.1.cmml">(</mo><mi id="S7.Thmtheorem4.p1.3.3.m3.1.1" xref="S7.Thmtheorem4.p1.3.3.m3.1.1.cmml">s</mi><mo id="S7.Thmtheorem4.p1.3.3.m3.2.3.3.2.2" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.3.1.cmml">,</mo><mi id="S7.Thmtheorem4.p1.3.3.m3.2.2" xref="S7.Thmtheorem4.p1.3.3.m3.2.2.cmml">a</mi><mo id="S7.Thmtheorem4.p1.3.3.m3.2.3.3.2.3" stretchy="false" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.3.3.m3.2b"><apply id="S7.Thmtheorem4.p1.3.3.m3.2.3.cmml" xref="S7.Thmtheorem4.p1.3.3.m3.2.3"><times id="S7.Thmtheorem4.p1.3.3.m3.2.3.1.cmml" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.1"></times><ci id="S7.Thmtheorem4.p1.3.3.m3.2.3.2.cmml" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.2">𝑃</ci><interval closure="open" id="S7.Thmtheorem4.p1.3.3.m3.2.3.3.1.cmml" xref="S7.Thmtheorem4.p1.3.3.m3.2.3.3.2"><ci id="S7.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem4.p1.3.3.m3.1.1">𝑠</ci><ci id="S7.Thmtheorem4.p1.3.3.m3.2.2.cmml" xref="S7.Thmtheorem4.p1.3.3.m3.2.2">𝑎</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.3.3.m3.2c">P(s,a)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.3.3.m3.2d">italic_P ( italic_s , italic_a )</annotation></semantics></math> depends only on <math alttext="a" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.4.4.m4.1"><semantics id="S7.Thmtheorem4.p1.4.4.m4.1a"><mi id="S7.Thmtheorem4.p1.4.4.m4.1.1" xref="S7.Thmtheorem4.p1.4.4.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.4.4.m4.1b"><ci id="S7.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem4.p1.4.4.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.4.4.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.4.4.m4.1d">italic_a</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S7.SS3.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof sketch.</h6> <div class="ltx_para" id="S7.SS3.3.p1"> <p class="ltx_p" id="S7.SS3.3.p1.16">In the normal-form game below, P1 and P2 play matching pennies, and the principal is willing to pay a large amount to avoid a particular pure profile.</p> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S7.SS3.3.p1.8.8"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S7.SS3.3.p1.2.2.2"> <td class="ltx_td" id="S7.SS3.3.p1.2.2.2.3"></td> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S7.SS3.3.p1.1.1.1.1"><math alttext="X" class="ltx_Math" display="inline" id="S7.SS3.3.p1.1.1.1.1.m1.1"><semantics id="S7.SS3.3.p1.1.1.1.1.m1.1a"><mi id="S7.SS3.3.p1.1.1.1.1.m1.1.1" xref="S7.SS3.3.p1.1.1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.1.1.1.1.m1.1b"><ci id="S7.SS3.3.p1.1.1.1.1.m1.1.1.cmml" xref="S7.SS3.3.p1.1.1.1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.1.1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.1.1.1.1.m1.1d">italic_X</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S7.SS3.3.p1.2.2.2.2"><math alttext="Y" class="ltx_Math" display="inline" id="S7.SS3.3.p1.2.2.2.2.m1.1"><semantics id="S7.SS3.3.p1.2.2.2.2.m1.1a"><mi id="S7.SS3.3.p1.2.2.2.2.m1.1.1" xref="S7.SS3.3.p1.2.2.2.2.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.2.2.2.2.m1.1b"><ci id="S7.SS3.3.p1.2.2.2.2.m1.1.1.cmml" xref="S7.SS3.3.p1.2.2.2.2.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.2.2.2.2.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.2.2.2.2.m1.1d">italic_Y</annotation></semantics></math></th> </tr> <tr class="ltx_tr" id="S7.SS3.3.p1.5.5.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S7.SS3.3.p1.3.3.3.1"><math alttext="X" class="ltx_Math" display="inline" id="S7.SS3.3.p1.3.3.3.1.m1.1"><semantics id="S7.SS3.3.p1.3.3.3.1.m1.1a"><mi id="S7.SS3.3.p1.3.3.3.1.m1.1.1" xref="S7.SS3.3.p1.3.3.3.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.3.3.3.1.m1.1b"><ci id="S7.SS3.3.p1.3.3.3.1.m1.1.1.cmml" xref="S7.SS3.3.p1.3.3.3.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.3.3.3.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.3.3.3.1.m1.1d">italic_X</annotation></semantics></math></th> <td class="ltx_td ltx_align_right" id="S7.SS3.3.p1.4.4.4.2"><math alttext="-\infty,0,1" class="ltx_Math" display="inline" id="S7.SS3.3.p1.4.4.4.2.m1.3"><semantics id="S7.SS3.3.p1.4.4.4.2.m1.3a"><mrow 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0</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S7.SS3.3.p1.8.8.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S7.SS3.3.p1.6.6.6.1"><math alttext="Y" class="ltx_Math" display="inline" id="S7.SS3.3.p1.6.6.6.1.m1.1"><semantics id="S7.SS3.3.p1.6.6.6.1.m1.1a"><mi id="S7.SS3.3.p1.6.6.6.1.m1.1.1" xref="S7.SS3.3.p1.6.6.6.1.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.6.6.6.1.m1.1b"><ci id="S7.SS3.3.p1.6.6.6.1.m1.1.1.cmml" xref="S7.SS3.3.p1.6.6.6.1.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.6.6.6.1.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.6.6.6.1.m1.1d">italic_Y</annotation></semantics></math></th> <td class="ltx_td ltx_align_right" id="S7.SS3.3.p1.7.7.7.2"><math alttext="0,1,0" class="ltx_Math" display="inline" id="S7.SS3.3.p1.7.7.7.2.m1.3"><semantics id="S7.SS3.3.p1.7.7.7.2.m1.3a"><mrow id="S7.SS3.3.p1.7.7.7.2.m1.3.4.2" xref="S7.SS3.3.p1.7.7.7.2.m1.3.4.1.cmml"><mn id="S7.SS3.3.p1.7.7.7.2.m1.1.1" xref="S7.SS3.3.p1.7.7.7.2.m1.1.1.cmml">0</mn><mo id="S7.SS3.3.p1.7.7.7.2.m1.3.4.2.1" xref="S7.SS3.3.p1.7.7.7.2.m1.3.4.1.cmml">,</mo><mn id="S7.SS3.3.p1.7.7.7.2.m1.2.2" xref="S7.SS3.3.p1.7.7.7.2.m1.2.2.cmml">1</mn><mo id="S7.SS3.3.p1.7.7.7.2.m1.3.4.2.2" xref="S7.SS3.3.p1.7.7.7.2.m1.3.4.1.cmml">,</mo><mn id="S7.SS3.3.p1.7.7.7.2.m1.3.3" xref="S7.SS3.3.p1.7.7.7.2.m1.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.7.7.7.2.m1.3b"><list id="S7.SS3.3.p1.7.7.7.2.m1.3.4.1.cmml" xref="S7.SS3.3.p1.7.7.7.2.m1.3.4.2"><cn id="S7.SS3.3.p1.7.7.7.2.m1.1.1.cmml" type="integer" xref="S7.SS3.3.p1.7.7.7.2.m1.1.1">0</cn><cn id="S7.SS3.3.p1.7.7.7.2.m1.2.2.cmml" type="integer" xref="S7.SS3.3.p1.7.7.7.2.m1.2.2">1</cn><cn id="S7.SS3.3.p1.7.7.7.2.m1.3.3.cmml" type="integer" xref="S7.SS3.3.p1.7.7.7.2.m1.3.3">0</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.7.7.7.2.m1.3c">0,1,0</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.7.7.7.2.m1.3d">0 , 1 , 0</annotation></semantics></math></td> <td class="ltx_td ltx_align_right" id="S7.SS3.3.p1.8.8.8.3"><math alttext="0,0,1" class="ltx_Math" display="inline" id="S7.SS3.3.p1.8.8.8.3.m1.3"><semantics id="S7.SS3.3.p1.8.8.8.3.m1.3a"><mrow id="S7.SS3.3.p1.8.8.8.3.m1.3.4.2" xref="S7.SS3.3.p1.8.8.8.3.m1.3.4.1.cmml"><mn id="S7.SS3.3.p1.8.8.8.3.m1.1.1" xref="S7.SS3.3.p1.8.8.8.3.m1.1.1.cmml">0</mn><mo id="S7.SS3.3.p1.8.8.8.3.m1.3.4.2.1" xref="S7.SS3.3.p1.8.8.8.3.m1.3.4.1.cmml">,</mo><mn id="S7.SS3.3.p1.8.8.8.3.m1.2.2" xref="S7.SS3.3.p1.8.8.8.3.m1.2.2.cmml">0</mn><mo id="S7.SS3.3.p1.8.8.8.3.m1.3.4.2.2" xref="S7.SS3.3.p1.8.8.8.3.m1.3.4.1.cmml">,</mo><mn id="S7.SS3.3.p1.8.8.8.3.m1.3.3" xref="S7.SS3.3.p1.8.8.8.3.m1.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.8.8.8.3.m1.3b"><list id="S7.SS3.3.p1.8.8.8.3.m1.3.4.1.cmml" xref="S7.SS3.3.p1.8.8.8.3.m1.3.4.2"><cn id="S7.SS3.3.p1.8.8.8.3.m1.1.1.cmml" type="integer" xref="S7.SS3.3.p1.8.8.8.3.m1.1.1">0</cn><cn id="S7.SS3.3.p1.8.8.8.3.m1.2.2.cmml" type="integer" xref="S7.SS3.3.p1.8.8.8.3.m1.2.2">0</cn><cn id="S7.SS3.3.p1.8.8.8.3.m1.3.3.cmml" type="integer" xref="S7.SS3.3.p1.8.8.8.3.m1.3.3">1</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.8.8.8.3.m1.3c">0,0,1</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.8.8.8.3.m1.3d">0 , 0 , 1</annotation></semantics></math></td> </tr> </tbody> </table> <p class="ltx_p" id="S7.SS3.3.p1.15">P1 chooses the row, P2 chooses the column. In each cell, the principal’s utility is listed first, then P1’s, then P2’s. Now consider the following CEP: The principal mixes evenly between recommending <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S7.SS3.3.p1.9.m1.2"><semantics id="S7.SS3.3.p1.9.m1.2a"><mrow id="S7.SS3.3.p1.9.m1.2.3.2" xref="S7.SS3.3.p1.9.m1.2.3.1.cmml"><mo id="S7.SS3.3.p1.9.m1.2.3.2.1" stretchy="false" xref="S7.SS3.3.p1.9.m1.2.3.1.cmml">(</mo><mi id="S7.SS3.3.p1.9.m1.1.1" xref="S7.SS3.3.p1.9.m1.1.1.cmml">X</mi><mo id="S7.SS3.3.p1.9.m1.2.3.2.2" xref="S7.SS3.3.p1.9.m1.2.3.1.cmml">,</mo><mi id="S7.SS3.3.p1.9.m1.2.2" xref="S7.SS3.3.p1.9.m1.2.2.cmml">Y</mi><mo id="S7.SS3.3.p1.9.m1.2.3.2.3" stretchy="false" xref="S7.SS3.3.p1.9.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.9.m1.2b"><interval closure="open" id="S7.SS3.3.p1.9.m1.2.3.1.cmml" xref="S7.SS3.3.p1.9.m1.2.3.2"><ci id="S7.SS3.3.p1.9.m1.1.1.cmml" xref="S7.SS3.3.p1.9.m1.1.1">𝑋</ci><ci id="S7.SS3.3.p1.9.m1.2.2.cmml" xref="S7.SS3.3.p1.9.m1.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.9.m1.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.9.m1.2d">( italic_X , italic_Y )</annotation></semantics></math>, <math alttext="(Y,X)" class="ltx_Math" display="inline" id="S7.SS3.3.p1.10.m2.2"><semantics id="S7.SS3.3.p1.10.m2.2a"><mrow id="S7.SS3.3.p1.10.m2.2.3.2" xref="S7.SS3.3.p1.10.m2.2.3.1.cmml"><mo id="S7.SS3.3.p1.10.m2.2.3.2.1" stretchy="false" xref="S7.SS3.3.p1.10.m2.2.3.1.cmml">(</mo><mi id="S7.SS3.3.p1.10.m2.1.1" xref="S7.SS3.3.p1.10.m2.1.1.cmml">Y</mi><mo id="S7.SS3.3.p1.10.m2.2.3.2.2" xref="S7.SS3.3.p1.10.m2.2.3.1.cmml">,</mo><mi id="S7.SS3.3.p1.10.m2.2.2" xref="S7.SS3.3.p1.10.m2.2.2.cmml">X</mi><mo id="S7.SS3.3.p1.10.m2.2.3.2.3" stretchy="false" xref="S7.SS3.3.p1.10.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.10.m2.2b"><interval closure="open" id="S7.SS3.3.p1.10.m2.2.3.1.cmml" xref="S7.SS3.3.p1.10.m2.2.3.2"><ci id="S7.SS3.3.p1.10.m2.1.1.cmml" xref="S7.SS3.3.p1.10.m2.1.1">𝑌</ci><ci id="S7.SS3.3.p1.10.m2.2.2.cmml" xref="S7.SS3.3.p1.10.m2.2.2">𝑋</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.10.m2.2c">(Y,X)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.10.m2.2d">( italic_Y , italic_X )</annotation></semantics></math>, and <math alttext="(Y,Y)" class="ltx_Math" display="inline" id="S7.SS3.3.p1.11.m3.2"><semantics id="S7.SS3.3.p1.11.m3.2a"><mrow id="S7.SS3.3.p1.11.m3.2.3.2" xref="S7.SS3.3.p1.11.m3.2.3.1.cmml"><mo id="S7.SS3.3.p1.11.m3.2.3.2.1" stretchy="false" xref="S7.SS3.3.p1.11.m3.2.3.1.cmml">(</mo><mi id="S7.SS3.3.p1.11.m3.1.1" xref="S7.SS3.3.p1.11.m3.1.1.cmml">Y</mi><mo id="S7.SS3.3.p1.11.m3.2.3.2.2" xref="S7.SS3.3.p1.11.m3.2.3.1.cmml">,</mo><mi id="S7.SS3.3.p1.11.m3.2.2" xref="S7.SS3.3.p1.11.m3.2.2.cmml">Y</mi><mo id="S7.SS3.3.p1.11.m3.2.3.2.3" stretchy="false" xref="S7.SS3.3.p1.11.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.11.m3.2b"><interval closure="open" id="S7.SS3.3.p1.11.m3.2.3.1.cmml" xref="S7.SS3.3.p1.11.m3.2.3.2"><ci id="S7.SS3.3.p1.11.m3.1.1.cmml" xref="S7.SS3.3.p1.11.m3.1.1">𝑌</ci><ci id="S7.SS3.3.p1.11.m3.2.2.cmml" xref="S7.SS3.3.p1.11.m3.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.11.m3.2c">(Y,Y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.11.m3.2d">( italic_Y , italic_Y )</annotation></semantics></math>. If the principal recommends <math alttext="(Y,X)" class="ltx_Math" display="inline" id="S7.SS3.3.p1.12.m4.2"><semantics id="S7.SS3.3.p1.12.m4.2a"><mrow id="S7.SS3.3.p1.12.m4.2.3.2" xref="S7.SS3.3.p1.12.m4.2.3.1.cmml"><mo id="S7.SS3.3.p1.12.m4.2.3.2.1" stretchy="false" xref="S7.SS3.3.p1.12.m4.2.3.1.cmml">(</mo><mi id="S7.SS3.3.p1.12.m4.1.1" xref="S7.SS3.3.p1.12.m4.1.1.cmml">Y</mi><mo id="S7.SS3.3.p1.12.m4.2.3.2.2" xref="S7.SS3.3.p1.12.m4.2.3.1.cmml">,</mo><mi id="S7.SS3.3.p1.12.m4.2.2" xref="S7.SS3.3.p1.12.m4.2.2.cmml">X</mi><mo id="S7.SS3.3.p1.12.m4.2.3.2.3" stretchy="false" xref="S7.SS3.3.p1.12.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.12.m4.2b"><interval closure="open" id="S7.SS3.3.p1.12.m4.2.3.1.cmml" xref="S7.SS3.3.p1.12.m4.2.3.2"><ci id="S7.SS3.3.p1.12.m4.1.1.cmml" xref="S7.SS3.3.p1.12.m4.1.1">𝑌</ci><ci id="S7.SS3.3.p1.12.m4.2.2.cmml" xref="S7.SS3.3.p1.12.m4.2.2">𝑋</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.12.m4.2c">(Y,X)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.12.m4.2d">( italic_Y , italic_X )</annotation></semantics></math>, it also promises a payment of <math alttext="1" class="ltx_Math" display="inline" id="S7.SS3.3.p1.13.m5.1"><semantics id="S7.SS3.3.p1.13.m5.1a"><mn id="S7.SS3.3.p1.13.m5.1.1" xref="S7.SS3.3.p1.13.m5.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.13.m5.1b"><cn id="S7.SS3.3.p1.13.m5.1.1.cmml" type="integer" xref="S7.SS3.3.p1.13.m5.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.13.m5.1c">1</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.13.m5.1d">1</annotation></semantics></math> to P2 if P2 follows the recommendation <math alttext="X" class="ltx_Math" display="inline" id="S7.SS3.3.p1.14.m6.1"><semantics id="S7.SS3.3.p1.14.m6.1a"><mi id="S7.SS3.3.p1.14.m6.1.1" xref="S7.SS3.3.p1.14.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.14.m6.1b"><ci id="S7.SS3.3.p1.14.m6.1.1.cmml" xref="S7.SS3.3.p1.14.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.14.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.14.m6.1d">italic_X</annotation></semantics></math>. This CEP has principal objective value <math alttext="-1/3" class="ltx_Math" display="inline" id="S7.SS3.3.p1.15.m7.1"><semantics id="S7.SS3.3.p1.15.m7.1a"><mrow id="S7.SS3.3.p1.15.m7.1.1" xref="S7.SS3.3.p1.15.m7.1.1.cmml"><mo id="S7.SS3.3.p1.15.m7.1.1a" xref="S7.SS3.3.p1.15.m7.1.1.cmml">−</mo><mrow id="S7.SS3.3.p1.15.m7.1.1.2" xref="S7.SS3.3.p1.15.m7.1.1.2.cmml"><mn id="S7.SS3.3.p1.15.m7.1.1.2.2" xref="S7.SS3.3.p1.15.m7.1.1.2.2.cmml">1</mn><mo id="S7.SS3.3.p1.15.m7.1.1.2.1" xref="S7.SS3.3.p1.15.m7.1.1.2.1.cmml">/</mo><mn id="S7.SS3.3.p1.15.m7.1.1.2.3" xref="S7.SS3.3.p1.15.m7.1.1.2.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.3.p1.15.m7.1b"><apply id="S7.SS3.3.p1.15.m7.1.1.cmml" xref="S7.SS3.3.p1.15.m7.1.1"><minus id="S7.SS3.3.p1.15.m7.1.1.1.cmml" xref="S7.SS3.3.p1.15.m7.1.1"></minus><apply id="S7.SS3.3.p1.15.m7.1.1.2.cmml" xref="S7.SS3.3.p1.15.m7.1.1.2"><divide id="S7.SS3.3.p1.15.m7.1.1.2.1.cmml" xref="S7.SS3.3.p1.15.m7.1.1.2.1"></divide><cn id="S7.SS3.3.p1.15.m7.1.1.2.2.cmml" type="integer" xref="S7.SS3.3.p1.15.m7.1.1.2.2">1</cn><cn id="S7.SS3.3.p1.15.m7.1.1.2.3.cmml" type="integer" xref="S7.SS3.3.p1.15.m7.1.1.2.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.3.p1.15.m7.1c">-1/3</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.3.p1.15.m7.1d">- 1 / 3</annotation></semantics></math>, and no signal-independent CEP can match that value. ∎</p> </div> </div> <div class="ltx_para" id="S7.SS3.p5"> <p class="ltx_p" id="S7.SS3.p5.1">The proof is formalized in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2.SS1" title="B.1 Completed proof of Proposition 7.4 ‣ Appendix B Details omitted from Section 7 ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">B.1</span></a>.</p> </div> <div class="ltx_para" id="S7.SS3.p6"> <p class="ltx_p" id="S7.SS3.p6.4">In the language of <cite class="ltx_cite ltx_citemacro_citet">Monderer and Tennenholtz (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib19" title="">2003</a>)</cite>, a CEP with <math alttext="k=\operatorname*{\mathbb{E}}_{a\sim\mu}P(a)" class="ltx_Math" display="inline" id="S7.SS3.p6.1.m1.1"><semantics id="S7.SS3.p6.1.m1.1a"><mrow id="S7.SS3.p6.1.m1.1.2" xref="S7.SS3.p6.1.m1.1.2.cmml"><mi id="S7.SS3.p6.1.m1.1.2.2" xref="S7.SS3.p6.1.m1.1.2.2.cmml">k</mi><mo id="S7.SS3.p6.1.m1.1.2.1" rspace="0.1389em" xref="S7.SS3.p6.1.m1.1.2.1.cmml">=</mo><mrow id="S7.SS3.p6.1.m1.1.2.3" xref="S7.SS3.p6.1.m1.1.2.3.cmml"><mrow id="S7.SS3.p6.1.m1.1.2.3.2" xref="S7.SS3.p6.1.m1.1.2.3.2.cmml"><msub id="S7.SS3.p6.1.m1.1.2.3.2.1" xref="S7.SS3.p6.1.m1.1.2.3.2.1.cmml"><mo id="S7.SS3.p6.1.m1.1.2.3.2.1.2" lspace="0.1389em" rspace="0.167em" xref="S7.SS3.p6.1.m1.1.2.3.2.1.2.cmml">𝔼</mo><mrow id="S7.SS3.p6.1.m1.1.2.3.2.1.3" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3.cmml"><mi id="S7.SS3.p6.1.m1.1.2.3.2.1.3.2" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3.2.cmml">a</mi><mo id="S7.SS3.p6.1.m1.1.2.3.2.1.3.1" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3.1.cmml">∼</mo><mi id="S7.SS3.p6.1.m1.1.2.3.2.1.3.3" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3.3.cmml">μ</mi></mrow></msub><mi id="S7.SS3.p6.1.m1.1.2.3.2.2" xref="S7.SS3.p6.1.m1.1.2.3.2.2.cmml">P</mi></mrow><mo id="S7.SS3.p6.1.m1.1.2.3.1" xref="S7.SS3.p6.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S7.SS3.p6.1.m1.1.2.3.3.2" xref="S7.SS3.p6.1.m1.1.2.3.cmml"><mo id="S7.SS3.p6.1.m1.1.2.3.3.2.1" stretchy="false" xref="S7.SS3.p6.1.m1.1.2.3.cmml">(</mo><mi id="S7.SS3.p6.1.m1.1.1" xref="S7.SS3.p6.1.m1.1.1.cmml">a</mi><mo id="S7.SS3.p6.1.m1.1.2.3.3.2.2" stretchy="false" xref="S7.SS3.p6.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p6.1.m1.1b"><apply id="S7.SS3.p6.1.m1.1.2.cmml" xref="S7.SS3.p6.1.m1.1.2"><eq id="S7.SS3.p6.1.m1.1.2.1.cmml" xref="S7.SS3.p6.1.m1.1.2.1"></eq><ci id="S7.SS3.p6.1.m1.1.2.2.cmml" xref="S7.SS3.p6.1.m1.1.2.2">𝑘</ci><apply id="S7.SS3.p6.1.m1.1.2.3.cmml" xref="S7.SS3.p6.1.m1.1.2.3"><times id="S7.SS3.p6.1.m1.1.2.3.1.cmml" xref="S7.SS3.p6.1.m1.1.2.3.1"></times><apply id="S7.SS3.p6.1.m1.1.2.3.2.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2"><apply id="S7.SS3.p6.1.m1.1.2.3.2.1.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.1"><csymbol cd="ambiguous" id="S7.SS3.p6.1.m1.1.2.3.2.1.1.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.1">subscript</csymbol><ci id="S7.SS3.p6.1.m1.1.2.3.2.1.2.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.1.2">𝔼</ci><apply id="S7.SS3.p6.1.m1.1.2.3.2.1.3.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3"><csymbol cd="latexml" id="S7.SS3.p6.1.m1.1.2.3.2.1.3.1.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3.1">similar-to</csymbol><ci id="S7.SS3.p6.1.m1.1.2.3.2.1.3.2.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3.2">𝑎</ci><ci id="S7.SS3.p6.1.m1.1.2.3.2.1.3.3.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.1.3.3">𝜇</ci></apply></apply><ci id="S7.SS3.p6.1.m1.1.2.3.2.2.cmml" xref="S7.SS3.p6.1.m1.1.2.3.2.2">𝑃</ci></apply><ci id="S7.SS3.p6.1.m1.1.1.cmml" xref="S7.SS3.p6.1.m1.1.1">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p6.1.m1.1c">k=\operatorname*{\mathbb{E}}_{a\sim\mu}P(a)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p6.1.m1.1d">italic_k = blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_μ end_POSTSUBSCRIPT italic_P ( italic_a )</annotation></semantics></math> is called a <math alttext="k" class="ltx_Math" display="inline" id="S7.SS3.p6.2.m2.1"><semantics id="S7.SS3.p6.2.m2.1a"><mi id="S7.SS3.p6.2.m2.1.1" xref="S7.SS3.p6.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p6.2.m2.1b"><ci id="S7.SS3.p6.2.m2.1.1.cmml" xref="S7.SS3.p6.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p6.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p6.2.m2.1d">italic_k</annotation></semantics></math>-<span class="ltx_text ltx_font_italic" id="S7.SS3.p6.4.1">implementable correlated equilibrium</span>.<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>Instead of our condition of <span class="ltx_text ltx_font_italic" id="footnote2.1">ex-interim</span> IC, <cite class="ltx_cite ltx_citemacro_citet">Monderer and Tennenholtz (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib19" title="">2003</a>)</cite> insist on <span class="ltx_text ltx_font_italic" id="footnote2.2">dominant-strategy</span> IC, that is, they insist that <math alttext="U_{i}^{P}(s,s_{i},a_{-i})\geq U_{i}^{P}(s,a)" class="ltx_Math" display="inline" id="footnote2.m1.5"><semantics id="footnote2.m1.5b"><mrow id="footnote2.m1.5.5" xref="footnote2.m1.5.5.cmml"><mrow id="footnote2.m1.5.5.2" xref="footnote2.m1.5.5.2.cmml"><msubsup id="footnote2.m1.5.5.2.4" xref="footnote2.m1.5.5.2.4.cmml"><mi id="footnote2.m1.5.5.2.4.2.2" xref="footnote2.m1.5.5.2.4.2.2.cmml">U</mi><mi id="footnote2.m1.5.5.2.4.2.3" xref="footnote2.m1.5.5.2.4.2.3.cmml">i</mi><mi id="footnote2.m1.5.5.2.4.3" xref="footnote2.m1.5.5.2.4.3.cmml">P</mi></msubsup><mo id="footnote2.m1.5.5.2.3" xref="footnote2.m1.5.5.2.3.cmml">⁢</mo><mrow id="footnote2.m1.5.5.2.2.2" xref="footnote2.m1.5.5.2.2.3.cmml"><mo id="footnote2.m1.5.5.2.2.2.3" stretchy="false" xref="footnote2.m1.5.5.2.2.3.cmml">(</mo><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">s</mi><mo id="footnote2.m1.5.5.2.2.2.4" xref="footnote2.m1.5.5.2.2.3.cmml">,</mo><msub id="footnote2.m1.4.4.1.1.1.1" xref="footnote2.m1.4.4.1.1.1.1.cmml"><mi id="footnote2.m1.4.4.1.1.1.1.2" xref="footnote2.m1.4.4.1.1.1.1.2.cmml">s</mi><mi id="footnote2.m1.4.4.1.1.1.1.3" xref="footnote2.m1.4.4.1.1.1.1.3.cmml">i</mi></msub><mo id="footnote2.m1.5.5.2.2.2.5" xref="footnote2.m1.5.5.2.2.3.cmml">,</mo><msub id="footnote2.m1.5.5.2.2.2.2" xref="footnote2.m1.5.5.2.2.2.2.cmml"><mi id="footnote2.m1.5.5.2.2.2.2.2" xref="footnote2.m1.5.5.2.2.2.2.2.cmml">a</mi><mrow id="footnote2.m1.5.5.2.2.2.2.3" xref="footnote2.m1.5.5.2.2.2.2.3.cmml"><mo id="footnote2.m1.5.5.2.2.2.2.3b" xref="footnote2.m1.5.5.2.2.2.2.3.cmml">−</mo><mi id="footnote2.m1.5.5.2.2.2.2.3.2" xref="footnote2.m1.5.5.2.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="footnote2.m1.5.5.2.2.2.6" stretchy="false" xref="footnote2.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow><mo id="footnote2.m1.5.5.3" xref="footnote2.m1.5.5.3.cmml">≥</mo><mrow id="footnote2.m1.5.5.4" xref="footnote2.m1.5.5.4.cmml"><msubsup id="footnote2.m1.5.5.4.2" xref="footnote2.m1.5.5.4.2.cmml"><mi id="footnote2.m1.5.5.4.2.2.2" xref="footnote2.m1.5.5.4.2.2.2.cmml">U</mi><mi id="footnote2.m1.5.5.4.2.2.3" xref="footnote2.m1.5.5.4.2.2.3.cmml">i</mi><mi id="footnote2.m1.5.5.4.2.3" xref="footnote2.m1.5.5.4.2.3.cmml">P</mi></msubsup><mo id="footnote2.m1.5.5.4.1" xref="footnote2.m1.5.5.4.1.cmml">⁢</mo><mrow id="footnote2.m1.5.5.4.3.2" xref="footnote2.m1.5.5.4.3.1.cmml"><mo id="footnote2.m1.5.5.4.3.2.1" stretchy="false" xref="footnote2.m1.5.5.4.3.1.cmml">(</mo><mi id="footnote2.m1.2.2" xref="footnote2.m1.2.2.cmml">s</mi><mo id="footnote2.m1.5.5.4.3.2.2" xref="footnote2.m1.5.5.4.3.1.cmml">,</mo><mi id="footnote2.m1.3.3" xref="footnote2.m1.3.3.cmml">a</mi><mo id="footnote2.m1.5.5.4.3.2.3" stretchy="false" xref="footnote2.m1.5.5.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m1.5c"><apply id="footnote2.m1.5.5.cmml" xref="footnote2.m1.5.5"><geq id="footnote2.m1.5.5.3.cmml" xref="footnote2.m1.5.5.3"></geq><apply id="footnote2.m1.5.5.2.cmml" xref="footnote2.m1.5.5.2"><times id="footnote2.m1.5.5.2.3.cmml" xref="footnote2.m1.5.5.2.3"></times><apply id="footnote2.m1.5.5.2.4.cmml" xref="footnote2.m1.5.5.2.4"><csymbol cd="ambiguous" id="footnote2.m1.5.5.2.4.1.cmml" xref="footnote2.m1.5.5.2.4">superscript</csymbol><apply id="footnote2.m1.5.5.2.4.2.cmml" xref="footnote2.m1.5.5.2.4"><csymbol cd="ambiguous" id="footnote2.m1.5.5.2.4.2.1.cmml" xref="footnote2.m1.5.5.2.4">subscript</csymbol><ci id="footnote2.m1.5.5.2.4.2.2.cmml" xref="footnote2.m1.5.5.2.4.2.2">𝑈</ci><ci id="footnote2.m1.5.5.2.4.2.3.cmml" xref="footnote2.m1.5.5.2.4.2.3">𝑖</ci></apply><ci id="footnote2.m1.5.5.2.4.3.cmml" xref="footnote2.m1.5.5.2.4.3">𝑃</ci></apply><vector id="footnote2.m1.5.5.2.2.3.cmml" xref="footnote2.m1.5.5.2.2.2"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝑠</ci><apply id="footnote2.m1.4.4.1.1.1.1.cmml" xref="footnote2.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="footnote2.m1.4.4.1.1.1.1.1.cmml" xref="footnote2.m1.4.4.1.1.1.1">subscript</csymbol><ci id="footnote2.m1.4.4.1.1.1.1.2.cmml" xref="footnote2.m1.4.4.1.1.1.1.2">𝑠</ci><ci id="footnote2.m1.4.4.1.1.1.1.3.cmml" xref="footnote2.m1.4.4.1.1.1.1.3">𝑖</ci></apply><apply id="footnote2.m1.5.5.2.2.2.2.cmml" xref="footnote2.m1.5.5.2.2.2.2"><csymbol cd="ambiguous" id="footnote2.m1.5.5.2.2.2.2.1.cmml" xref="footnote2.m1.5.5.2.2.2.2">subscript</csymbol><ci id="footnote2.m1.5.5.2.2.2.2.2.cmml" xref="footnote2.m1.5.5.2.2.2.2.2">𝑎</ci><apply id="footnote2.m1.5.5.2.2.2.2.3.cmml" xref="footnote2.m1.5.5.2.2.2.2.3"><minus id="footnote2.m1.5.5.2.2.2.2.3.1.cmml" xref="footnote2.m1.5.5.2.2.2.2.3"></minus><ci id="footnote2.m1.5.5.2.2.2.2.3.2.cmml" xref="footnote2.m1.5.5.2.2.2.2.3.2">𝑖</ci></apply></apply></vector></apply><apply id="footnote2.m1.5.5.4.cmml" xref="footnote2.m1.5.5.4"><times id="footnote2.m1.5.5.4.1.cmml" xref="footnote2.m1.5.5.4.1"></times><apply id="footnote2.m1.5.5.4.2.cmml" xref="footnote2.m1.5.5.4.2"><csymbol cd="ambiguous" id="footnote2.m1.5.5.4.2.1.cmml" xref="footnote2.m1.5.5.4.2">superscript</csymbol><apply id="footnote2.m1.5.5.4.2.2.cmml" xref="footnote2.m1.5.5.4.2"><csymbol cd="ambiguous" id="footnote2.m1.5.5.4.2.2.1.cmml" xref="footnote2.m1.5.5.4.2">subscript</csymbol><ci id="footnote2.m1.5.5.4.2.2.2.cmml" xref="footnote2.m1.5.5.4.2.2.2">𝑈</ci><ci id="footnote2.m1.5.5.4.2.2.3.cmml" xref="footnote2.m1.5.5.4.2.2.3">𝑖</ci></apply><ci id="footnote2.m1.5.5.4.2.3.cmml" xref="footnote2.m1.5.5.4.2.3">𝑃</ci></apply><interval closure="open" id="footnote2.m1.5.5.4.3.1.cmml" xref="footnote2.m1.5.5.4.3.2"><ci id="footnote2.m1.2.2.cmml" xref="footnote2.m1.2.2">𝑠</ci><ci id="footnote2.m1.3.3.cmml" xref="footnote2.m1.3.3">𝑎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.5d">U_{i}^{P}(s,s_{i},a_{-i})\geq U_{i}^{P}(s,a)</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.5e">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( italic_s , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) ≥ italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( italic_s , italic_a )</annotation></semantics></math> for <span class="ltx_text ltx_font_italic" id="footnote2.3">every</span> <math alttext="s" class="ltx_Math" display="inline" id="footnote2.m2.1"><semantics id="footnote2.m2.1b"><mi id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><ci id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">s</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.1e">italic_s</annotation></semantics></math> and <math alttext="a" class="ltx_Math" display="inline" id="footnote2.m3.1"><semantics id="footnote2.m3.1b"><mi id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="footnote2.m3.1c"><ci id="footnote2.m3.1.1.cmml" xref="footnote2.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.1d">a</annotation><annotation encoding="application/x-llamapun" id="footnote2.m3.1e">italic_a</annotation></semantics></math>. However, this requirement does not change anything in equilibrium, because one can always set <math alttext="P(s,s_{i},a_{-i})" class="ltx_Math" display="inline" id="footnote2.m4.3"><semantics id="footnote2.m4.3b"><mrow id="footnote2.m4.3.3" xref="footnote2.m4.3.3.cmml"><mi id="footnote2.m4.3.3.4" xref="footnote2.m4.3.3.4.cmml">P</mi><mo id="footnote2.m4.3.3.3" xref="footnote2.m4.3.3.3.cmml">⁢</mo><mrow id="footnote2.m4.3.3.2.2" xref="footnote2.m4.3.3.2.3.cmml"><mo id="footnote2.m4.3.3.2.2.3" stretchy="false" xref="footnote2.m4.3.3.2.3.cmml">(</mo><mi id="footnote2.m4.1.1" xref="footnote2.m4.1.1.cmml">s</mi><mo id="footnote2.m4.3.3.2.2.4" xref="footnote2.m4.3.3.2.3.cmml">,</mo><msub id="footnote2.m4.2.2.1.1.1" xref="footnote2.m4.2.2.1.1.1.cmml"><mi id="footnote2.m4.2.2.1.1.1.2" xref="footnote2.m4.2.2.1.1.1.2.cmml">s</mi><mi id="footnote2.m4.2.2.1.1.1.3" xref="footnote2.m4.2.2.1.1.1.3.cmml">i</mi></msub><mo id="footnote2.m4.3.3.2.2.5" xref="footnote2.m4.3.3.2.3.cmml">,</mo><msub id="footnote2.m4.3.3.2.2.2" xref="footnote2.m4.3.3.2.2.2.cmml"><mi id="footnote2.m4.3.3.2.2.2.2" xref="footnote2.m4.3.3.2.2.2.2.cmml">a</mi><mrow id="footnote2.m4.3.3.2.2.2.3" xref="footnote2.m4.3.3.2.2.2.3.cmml"><mo id="footnote2.m4.3.3.2.2.2.3b" xref="footnote2.m4.3.3.2.2.2.3.cmml">−</mo><mi id="footnote2.m4.3.3.2.2.2.3.2" xref="footnote2.m4.3.3.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="footnote2.m4.3.3.2.2.6" stretchy="false" xref="footnote2.m4.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m4.3c"><apply id="footnote2.m4.3.3.cmml" xref="footnote2.m4.3.3"><times id="footnote2.m4.3.3.3.cmml" xref="footnote2.m4.3.3.3"></times><ci id="footnote2.m4.3.3.4.cmml" xref="footnote2.m4.3.3.4">𝑃</ci><vector id="footnote2.m4.3.3.2.3.cmml" xref="footnote2.m4.3.3.2.2"><ci id="footnote2.m4.1.1.cmml" xref="footnote2.m4.1.1">𝑠</ci><apply id="footnote2.m4.2.2.1.1.1.cmml" xref="footnote2.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="footnote2.m4.2.2.1.1.1.1.cmml" xref="footnote2.m4.2.2.1.1.1">subscript</csymbol><ci id="footnote2.m4.2.2.1.1.1.2.cmml" xref="footnote2.m4.2.2.1.1.1.2">𝑠</ci><ci id="footnote2.m4.2.2.1.1.1.3.cmml" xref="footnote2.m4.2.2.1.1.1.3">𝑖</ci></apply><apply id="footnote2.m4.3.3.2.2.2.cmml" xref="footnote2.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="footnote2.m4.3.3.2.2.2.1.cmml" xref="footnote2.m4.3.3.2.2.2">subscript</csymbol><ci id="footnote2.m4.3.3.2.2.2.2.cmml" xref="footnote2.m4.3.3.2.2.2.2">𝑎</ci><apply id="footnote2.m4.3.3.2.2.2.3.cmml" xref="footnote2.m4.3.3.2.2.2.3"><minus id="footnote2.m4.3.3.2.2.2.3.1.cmml" xref="footnote2.m4.3.3.2.2.2.3"></minus><ci id="footnote2.m4.3.3.2.2.2.3.2.cmml" xref="footnote2.m4.3.3.2.2.2.3.2">𝑖</ci></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m4.3d">P(s,s_{i},a_{-i})</annotation><annotation encoding="application/x-llamapun" id="footnote2.m4.3e">italic_P ( italic_s , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> when <math alttext="s_{i}=a_{i}" class="ltx_Math" display="inline" id="footnote2.m5.1"><semantics id="footnote2.m5.1b"><mrow id="footnote2.m5.1.1" xref="footnote2.m5.1.1.cmml"><msub id="footnote2.m5.1.1.2" xref="footnote2.m5.1.1.2.cmml"><mi id="footnote2.m5.1.1.2.2" xref="footnote2.m5.1.1.2.2.cmml">s</mi><mi id="footnote2.m5.1.1.2.3" xref="footnote2.m5.1.1.2.3.cmml">i</mi></msub><mo id="footnote2.m5.1.1.1" xref="footnote2.m5.1.1.1.cmml">=</mo><msub id="footnote2.m5.1.1.3" xref="footnote2.m5.1.1.3.cmml"><mi id="footnote2.m5.1.1.3.2" xref="footnote2.m5.1.1.3.2.cmml">a</mi><mi id="footnote2.m5.1.1.3.3" xref="footnote2.m5.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m5.1c"><apply id="footnote2.m5.1.1.cmml" xref="footnote2.m5.1.1"><eq id="footnote2.m5.1.1.1.cmml" xref="footnote2.m5.1.1.1"></eq><apply id="footnote2.m5.1.1.2.cmml" xref="footnote2.m5.1.1.2"><csymbol cd="ambiguous" id="footnote2.m5.1.1.2.1.cmml" xref="footnote2.m5.1.1.2">subscript</csymbol><ci id="footnote2.m5.1.1.2.2.cmml" xref="footnote2.m5.1.1.2.2">𝑠</ci><ci id="footnote2.m5.1.1.2.3.cmml" xref="footnote2.m5.1.1.2.3">𝑖</ci></apply><apply id="footnote2.m5.1.1.3.cmml" xref="footnote2.m5.1.1.3"><csymbol cd="ambiguous" id="footnote2.m5.1.1.3.1.cmml" xref="footnote2.m5.1.1.3">subscript</csymbol><ci id="footnote2.m5.1.1.3.2.cmml" xref="footnote2.m5.1.1.3.2">𝑎</ci><ci id="footnote2.m5.1.1.3.3.cmml" xref="footnote2.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m5.1d">s_{i}=a_{i}</annotation><annotation encoding="application/x-llamapun" id="footnote2.m5.1e">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="s\neq a" class="ltx_Math" display="inline" id="footnote2.m6.1"><semantics id="footnote2.m6.1b"><mrow id="footnote2.m6.1.1" xref="footnote2.m6.1.1.cmml"><mi id="footnote2.m6.1.1.2" xref="footnote2.m6.1.1.2.cmml">s</mi><mo id="footnote2.m6.1.1.1" xref="footnote2.m6.1.1.1.cmml">≠</mo><mi id="footnote2.m6.1.1.3" xref="footnote2.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m6.1c"><apply id="footnote2.m6.1.1.cmml" xref="footnote2.m6.1.1"><neq id="footnote2.m6.1.1.1.cmml" xref="footnote2.m6.1.1.1"></neq><ci id="footnote2.m6.1.1.2.cmml" xref="footnote2.m6.1.1.2">𝑠</ci><ci id="footnote2.m6.1.1.3.cmml" xref="footnote2.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m6.1d">s\neq a</annotation><annotation encoding="application/x-llamapun" id="footnote2.m6.1e">italic_s ≠ italic_a</annotation></semantics></math> to be so large that playing <math alttext="a_{i}" class="ltx_Math" display="inline" id="footnote2.m7.1"><semantics id="footnote2.m7.1b"><msub id="footnote2.m7.1.1" xref="footnote2.m7.1.1.cmml"><mi id="footnote2.m7.1.1.2" xref="footnote2.m7.1.1.2.cmml">a</mi><mi id="footnote2.m7.1.1.3" xref="footnote2.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="footnote2.m7.1c"><apply id="footnote2.m7.1.1.cmml" xref="footnote2.m7.1.1"><csymbol cd="ambiguous" id="footnote2.m7.1.1.1.cmml" xref="footnote2.m7.1.1">subscript</csymbol><ci id="footnote2.m7.1.1.2.cmml" xref="footnote2.m7.1.1.2">𝑎</ci><ci id="footnote2.m7.1.1.3.cmml" xref="footnote2.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m7.1d">a_{i}</annotation><annotation encoding="application/x-llamapun" id="footnote2.m7.1e">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> becomes dominant. Indeed, <cite class="ltx_cite ltx_citemacro_citet">Monderer and Tennenholtz (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib19" title="">2003</a>)</cite> do this to establish their results on implementation; <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite> do this in their steering algorithms; and we will do the same in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7" title="7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">7</span></a>.</span></span></span> They show that all correlated equilibria are <math alttext="0" class="ltx_Math" display="inline" id="S7.SS3.p6.3.m3.1"><semantics id="S7.SS3.p6.3.m3.1a"><mn id="S7.SS3.p6.3.m3.1.1" xref="S7.SS3.p6.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S7.SS3.p6.3.m3.1b"><cn id="S7.SS3.p6.3.m3.1.1.cmml" type="integer" xref="S7.SS3.p6.3.m3.1.1">0</cn></annotation-xml></semantics></math>-implementable, but do not show the converse. Our results improve upon theirs by 1) showing the converse (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.Thmtheorem3" title="Proposition 7.3 (Correlation does not help when no payments are allowed in equilibrium). ‣ 7.3 Properties of correlated equilibria with payments ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">7.3</span></a>), and 2) analyzing the <math alttext="k>0" class="ltx_Math" display="inline" id="S7.SS3.p6.4.m4.1"><semantics id="S7.SS3.p6.4.m4.1a"><mrow id="S7.SS3.p6.4.m4.1.1" xref="S7.SS3.p6.4.m4.1.1.cmml"><mi id="S7.SS3.p6.4.m4.1.1.2" xref="S7.SS3.p6.4.m4.1.1.2.cmml">k</mi><mo id="S7.SS3.p6.4.m4.1.1.1" xref="S7.SS3.p6.4.m4.1.1.1.cmml">></mo><mn id="S7.SS3.p6.4.m4.1.1.3" xref="S7.SS3.p6.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p6.4.m4.1b"><apply id="S7.SS3.p6.4.m4.1.1.cmml" xref="S7.SS3.p6.4.m4.1.1"><gt id="S7.SS3.p6.4.m4.1.1.1.cmml" xref="S7.SS3.p6.4.m4.1.1.1"></gt><ci id="S7.SS3.p6.4.m4.1.1.2.cmml" xref="S7.SS3.p6.4.m4.1.1.2">𝑘</ci><cn id="S7.SS3.p6.4.m4.1.1.3.cmml" type="integer" xref="S7.SS3.p6.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p6.4.m4.1c">k>0</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p6.4.m4.1d">italic_k > 0</annotation></semantics></math> case, in particular, by incorporating a principal objective and showing how to compute the optimal CEP.</p> </div> <div class="ltx_para" id="S7.SS3.p7"> <p class="ltx_p" id="S7.SS3.p7.1">Finally, in <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.Thmtheorem1" title="Definition 7.1. ‣ 7.2 What outcome should we steer to? ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">7.1</span></a>, we set the signal set to be identical to the action set. In <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2.SS2" title="B.2 Revelation principle for CEPs ‣ Appendix B Details omitted from Section 7 ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">B.2</span></a> we show that this is without loss of generality, that is, a sort of <span class="ltx_text ltx_font_italic" id="S7.SS3.p7.1.1">revelation principle</span> holds for CEPs.</p> </div> </section> <section class="ltx_subsection" id="S7.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.4 </span>CEPs and optimal steering</h3> <div class="ltx_para" id="S7.SS4.p1"> <p class="ltx_p" id="S7.SS4.p1.3">We now show that the objective value of the optimal (<math alttext="0" class="ltx_Math" display="inline" id="S7.SS4.p1.1.m1.1"><semantics id="S7.SS4.p1.1.m1.1a"><mn id="S7.SS4.p1.1.m1.1.1" xref="S7.SS4.p1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S7.SS4.p1.1.m1.1b"><cn id="S7.SS4.p1.1.m1.1.1.cmml" type="integer" xref="S7.SS4.p1.1.m1.1.1">0</cn></annotation-xml></semantics></math>-)CEP is exactly the maximum value attainable (in the limit <math alttext="T\to\infty" class="ltx_Math" display="inline" id="S7.SS4.p1.2.m2.1"><semantics id="S7.SS4.p1.2.m2.1a"><mrow id="S7.SS4.p1.2.m2.1.1" xref="S7.SS4.p1.2.m2.1.1.cmml"><mi id="S7.SS4.p1.2.m2.1.1.2" xref="S7.SS4.p1.2.m2.1.1.2.cmml">T</mi><mo id="S7.SS4.p1.2.m2.1.1.1" stretchy="false" xref="S7.SS4.p1.2.m2.1.1.1.cmml">→</mo><mi id="S7.SS4.p1.2.m2.1.1.3" mathvariant="normal" xref="S7.SS4.p1.2.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.p1.2.m2.1b"><apply id="S7.SS4.p1.2.m2.1.1.cmml" xref="S7.SS4.p1.2.m2.1.1"><ci id="S7.SS4.p1.2.m2.1.1.1.cmml" xref="S7.SS4.p1.2.m2.1.1.1">→</ci><ci id="S7.SS4.p1.2.m2.1.1.2.cmml" xref="S7.SS4.p1.2.m2.1.1.2">𝑇</ci><infinity id="S7.SS4.p1.2.m2.1.1.3.cmml" xref="S7.SS4.p1.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p1.2.m2.1c">T\to\infty</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p1.2.m2.1d">italic_T → ∞</annotation></semantics></math>) by a principal in our model. We start with the upper bound. Intuitively, the upper bound holds because, for any algorithm for the principal, the agents can always compute and play a Nash equilibrium of the payment- and signal-augmented game <math alttext="\Gamma^{t}" class="ltx_Math" display="inline" id="S7.SS4.p1.3.m3.1"><semantics id="S7.SS4.p1.3.m3.1a"><msup id="S7.SS4.p1.3.m3.1.1" xref="S7.SS4.p1.3.m3.1.1.cmml"><mi id="S7.SS4.p1.3.m3.1.1.2" mathvariant="normal" xref="S7.SS4.p1.3.m3.1.1.2.cmml">Γ</mi><mi id="S7.SS4.p1.3.m3.1.1.3" xref="S7.SS4.p1.3.m3.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S7.SS4.p1.3.m3.1b"><apply id="S7.SS4.p1.3.m3.1.1.cmml" xref="S7.SS4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.SS4.p1.3.m3.1.1.1.cmml" xref="S7.SS4.p1.3.m3.1.1">superscript</csymbol><ci id="S7.SS4.p1.3.m3.1.1.2.cmml" xref="S7.SS4.p1.3.m3.1.1.2">Γ</ci><ci id="S7.SS4.p1.3.m3.1.1.3.cmml" xref="S7.SS4.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p1.3.m3.1c">\Gamma^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p1.3.m3.1d">roman_Γ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, which leads to zero regret in expectation and value bounded above by the optimal CEP value. We now formalize this argument.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S7.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="S7.Thmtheorem5.1.1.1">Theorem 7.5</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem5.p1"> <p class="ltx_p" id="S7.Thmtheorem5.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem5.p1.3.3">Let <math alttext="\Gamma" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.1.1.m1.1"><semantics id="S7.Thmtheorem5.p1.1.1.m1.1a"><mi id="S7.Thmtheorem5.p1.1.1.m1.1.1" mathvariant="normal" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.1.1.m1.1b"><ci id="S7.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem5.p1.1.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.1.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.1.1.m1.1d">roman_Γ</annotation></semantics></math> be any game, and suppose the signal sets have size <math alttext="|S_{i}|\leq\poly(m)" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.2.2.m2.2"><semantics id="S7.Thmtheorem5.p1.2.2.m2.2a"><mrow id="S7.Thmtheorem5.p1.2.2.m2.2.2" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.cmml"><mrow id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.2.cmml"><mo id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.2" stretchy="false" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.2.1.cmml">|</mo><msub id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.cmml"><mi id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.2" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.2.cmml">S</mi><mi id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.3" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.3" stretchy="false" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem5.p1.2.2.m2.2.2.2" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.2.cmml">≤</mo><mrow id="S7.Thmtheorem5.p1.2.2.m2.2.2.3" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.cmml"><merror class="ltx_ERROR undefined undefined" id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2b.cmml"><mtext id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2a" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2b.cmml">\poly</mtext></merror><mo id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.1" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.3.2" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.cmml"><mo id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.cmml">(</mo><mi id="S7.Thmtheorem5.p1.2.2.m2.1.1" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.cmml">m</mi><mo id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.2.2.m2.2b"><apply id="S7.Thmtheorem5.p1.2.2.m2.2.2.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2"><leq id="S7.Thmtheorem5.p1.2.2.m2.2.2.2.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.2"></leq><apply id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.2.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1"><abs id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.2.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.2"></abs><apply id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.2.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.2">𝑆</ci><ci id="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.3.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.3">𝑖</ci></apply></apply><apply id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3"><times id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.1"></times><ci id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2b.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2"><merror class="ltx_ERROR undefined undefined" id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2"><mtext id="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2a.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.2.2.3.2">\poly</mtext></merror></ci><ci id="S7.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.2.2.m2.2c">|S_{i}|\leq\poly(m)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.2.2.m2.2d">| italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ≤ ( italic_m )</annotation></semantics></math>. Then for any possible algorithm for the principal, there exists some algorithm that the agents can use, for which, with probability <math alttext="1-\delta" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.3.3.m3.1"><semantics id="S7.Thmtheorem5.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem5.p1.3.3.m3.1.1" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mn id="S7.Thmtheorem5.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem5.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.1.cmml">−</mo><mi id="S7.Thmtheorem5.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.3.3.m3.1b"><apply id="S7.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1"><minus id="S7.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.1"></minus><cn id="S7.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" type="integer" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.2">1</cn><ci id="S7.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.3.3.m3.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.3.3.m3.1d">1 - italic_δ</annotation></semantics></math>,</span></p> <ol class="ltx_enumerate" id="S7.I1"> <li class="ltx_item" id="S7.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S7.I1.i1.p1"> <p class="ltx_p" id="S7.I1.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S7.I1.i1.p1.1.1">no agent ever plays an action that is not rationalizable,</span></p> </div> </li> <li class="ltx_item" id="S7.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S7.I1.i2.p1"> <p class="ltx_p" id="S7.I1.i2.p1.4"><span class="ltx_text ltx_font_italic" id="S7.I1.i2.p1.4.1">each agent’s regret </span><math alttext="\hat{R}_{i}(t,s_{i})" class="ltx_Math" display="inline" id="S7.I1.i2.p1.1.m1.2"><semantics id="S7.I1.i2.p1.1.m1.2a"><mrow id="S7.I1.i2.p1.1.m1.2.2" xref="S7.I1.i2.p1.1.m1.2.2.cmml"><msub id="S7.I1.i2.p1.1.m1.2.2.3" xref="S7.I1.i2.p1.1.m1.2.2.3.cmml"><mover accent="true" id="S7.I1.i2.p1.1.m1.2.2.3.2" xref="S7.I1.i2.p1.1.m1.2.2.3.2.cmml"><mi id="S7.I1.i2.p1.1.m1.2.2.3.2.2" xref="S7.I1.i2.p1.1.m1.2.2.3.2.2.cmml">R</mi><mo id="S7.I1.i2.p1.1.m1.2.2.3.2.1" xref="S7.I1.i2.p1.1.m1.2.2.3.2.1.cmml">^</mo></mover><mi id="S7.I1.i2.p1.1.m1.2.2.3.3" xref="S7.I1.i2.p1.1.m1.2.2.3.3.cmml">i</mi></msub><mo id="S7.I1.i2.p1.1.m1.2.2.2" xref="S7.I1.i2.p1.1.m1.2.2.2.cmml">⁢</mo><mrow id="S7.I1.i2.p1.1.m1.2.2.1.1" xref="S7.I1.i2.p1.1.m1.2.2.1.2.cmml"><mo id="S7.I1.i2.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S7.I1.i2.p1.1.m1.2.2.1.2.cmml">(</mo><mi id="S7.I1.i2.p1.1.m1.1.1" xref="S7.I1.i2.p1.1.m1.1.1.cmml">t</mi><mo id="S7.I1.i2.p1.1.m1.2.2.1.1.3" xref="S7.I1.i2.p1.1.m1.2.2.1.2.cmml">,</mo><msub 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id="S7.I1.i2.p1.1.m1.2.2.3.3.cmml" xref="S7.I1.i2.p1.1.m1.2.2.3.3">𝑖</ci></apply><interval closure="open" id="S7.I1.i2.p1.1.m1.2.2.1.2.cmml" xref="S7.I1.i2.p1.1.m1.2.2.1.1"><ci id="S7.I1.i2.p1.1.m1.1.1.cmml" xref="S7.I1.i2.p1.1.m1.1.1">𝑡</ci><apply id="S7.I1.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S7.I1.i2.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S7.I1.i2.p1.1.m1.2.2.1.1.1.1.cmml" xref="S7.I1.i2.p1.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S7.I1.i2.p1.1.m1.2.2.1.1.1.2.cmml" xref="S7.I1.i2.p1.1.m1.2.2.1.1.1.2">𝑠</ci><ci id="S7.I1.i2.p1.1.m1.2.2.1.1.1.3.cmml" xref="S7.I1.i2.p1.1.m1.2.2.1.1.1.3">𝑖</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.1.m1.2c">\hat{R}_{i}(t,s_{i})</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.1.m1.2d">over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i2.p1.4.2"> is bounded by </span><math alttext="C\sqrt{T}" class="ltx_Math" display="inline" id="S7.I1.i2.p1.2.m2.1"><semantics id="S7.I1.i2.p1.2.m2.1a"><mrow id="S7.I1.i2.p1.2.m2.1.1" xref="S7.I1.i2.p1.2.m2.1.1.cmml"><mi id="S7.I1.i2.p1.2.m2.1.1.2" xref="S7.I1.i2.p1.2.m2.1.1.2.cmml">C</mi><mo id="S7.I1.i2.p1.2.m2.1.1.1" xref="S7.I1.i2.p1.2.m2.1.1.1.cmml">⁢</mo><msqrt id="S7.I1.i2.p1.2.m2.1.1.3" xref="S7.I1.i2.p1.2.m2.1.1.3.cmml"><mi id="S7.I1.i2.p1.2.m2.1.1.3.2" xref="S7.I1.i2.p1.2.m2.1.1.3.2.cmml">T</mi></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.2.m2.1b"><apply id="S7.I1.i2.p1.2.m2.1.1.cmml" xref="S7.I1.i2.p1.2.m2.1.1"><times id="S7.I1.i2.p1.2.m2.1.1.1.cmml" xref="S7.I1.i2.p1.2.m2.1.1.1"></times><ci id="S7.I1.i2.p1.2.m2.1.1.2.cmml" xref="S7.I1.i2.p1.2.m2.1.1.2">𝐶</ci><apply id="S7.I1.i2.p1.2.m2.1.1.3.cmml" xref="S7.I1.i2.p1.2.m2.1.1.3"><root id="S7.I1.i2.p1.2.m2.1.1.3a.cmml" xref="S7.I1.i2.p1.2.m2.1.1.3"></root><ci id="S7.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S7.I1.i2.p1.2.m2.1.1.3.2">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.2.m2.1c">C\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.2.m2.1d">italic_C square-root start_ARG italic_T end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i2.p1.4.3"> for every </span><math alttext="t\leq T" class="ltx_Math" display="inline" id="S7.I1.i2.p1.3.m3.1"><semantics id="S7.I1.i2.p1.3.m3.1a"><mrow id="S7.I1.i2.p1.3.m3.1.1" xref="S7.I1.i2.p1.3.m3.1.1.cmml"><mi id="S7.I1.i2.p1.3.m3.1.1.2" xref="S7.I1.i2.p1.3.m3.1.1.2.cmml">t</mi><mo id="S7.I1.i2.p1.3.m3.1.1.1" xref="S7.I1.i2.p1.3.m3.1.1.1.cmml">≤</mo><mi id="S7.I1.i2.p1.3.m3.1.1.3" xref="S7.I1.i2.p1.3.m3.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.3.m3.1b"><apply id="S7.I1.i2.p1.3.m3.1.1.cmml" xref="S7.I1.i2.p1.3.m3.1.1"><leq id="S7.I1.i2.p1.3.m3.1.1.1.cmml" xref="S7.I1.i2.p1.3.m3.1.1.1"></leq><ci id="S7.I1.i2.p1.3.m3.1.1.2.cmml" xref="S7.I1.i2.p1.3.m3.1.1.2">𝑡</ci><ci id="S7.I1.i2.p1.3.m3.1.1.3.cmml" xref="S7.I1.i2.p1.3.m3.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.3.m3.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.3.m3.1d">italic_t ≤ italic_T</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i2.p1.4.4"> and signal </span><math alttext="s_{i}" class="ltx_Math" display="inline" id="S7.I1.i2.p1.4.m4.1"><semantics id="S7.I1.i2.p1.4.m4.1a"><msub id="S7.I1.i2.p1.4.m4.1.1" xref="S7.I1.i2.p1.4.m4.1.1.cmml"><mi id="S7.I1.i2.p1.4.m4.1.1.2" xref="S7.I1.i2.p1.4.m4.1.1.2.cmml">s</mi><mi id="S7.I1.i2.p1.4.m4.1.1.3" xref="S7.I1.i2.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.4.m4.1b"><apply id="S7.I1.i2.p1.4.m4.1.1.cmml" xref="S7.I1.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.I1.i2.p1.4.m4.1.1.1.cmml" xref="S7.I1.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S7.I1.i2.p1.4.m4.1.1.2.cmml" xref="S7.I1.i2.p1.4.m4.1.1.2">𝑠</ci><ci id="S7.I1.i2.p1.4.m4.1.1.3.cmml" xref="S7.I1.i2.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.4.m4.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.4.m4.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i2.p1.4.5">, and</span></p> </div> </li> <li class="ltx_item" id="S7.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S7.I1.i3.p1"> <p class="ltx_p" id="S7.I1.i3.p1.4"><span class="ltx_text ltx_font_italic" id="S7.I1.i3.p1.4.1">the principal objective value </span><math alttext="F(T)" class="ltx_Math" display="inline" id="S7.I1.i3.p1.1.m1.1"><semantics id="S7.I1.i3.p1.1.m1.1a"><mrow id="S7.I1.i3.p1.1.m1.1.2" xref="S7.I1.i3.p1.1.m1.1.2.cmml"><mi id="S7.I1.i3.p1.1.m1.1.2.2" xref="S7.I1.i3.p1.1.m1.1.2.2.cmml">F</mi><mo id="S7.I1.i3.p1.1.m1.1.2.1" xref="S7.I1.i3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S7.I1.i3.p1.1.m1.1.2.3.2" xref="S7.I1.i3.p1.1.m1.1.2.cmml"><mo id="S7.I1.i3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S7.I1.i3.p1.1.m1.1.2.cmml">(</mo><mi id="S7.I1.i3.p1.1.m1.1.1" xref="S7.I1.i3.p1.1.m1.1.1.cmml">T</mi><mo id="S7.I1.i3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S7.I1.i3.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i3.p1.1.m1.1b"><apply id="S7.I1.i3.p1.1.m1.1.2.cmml" xref="S7.I1.i3.p1.1.m1.1.2"><times id="S7.I1.i3.p1.1.m1.1.2.1.cmml" xref="S7.I1.i3.p1.1.m1.1.2.1"></times><ci id="S7.I1.i3.p1.1.m1.1.2.2.cmml" xref="S7.I1.i3.p1.1.m1.1.2.2">𝐹</ci><ci id="S7.I1.i3.p1.1.m1.1.1.cmml" xref="S7.I1.i3.p1.1.m1.1.1">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i3.p1.1.m1.1c">F(T)</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i3.p1.1.m1.1d">italic_F ( italic_T )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i3.p1.4.2"> is bounded above by </span><math alttext="F^{*}+C/\sqrt{T}" class="ltx_Math" display="inline" id="S7.I1.i3.p1.2.m2.1"><semantics id="S7.I1.i3.p1.2.m2.1a"><mrow id="S7.I1.i3.p1.2.m2.1.1" xref="S7.I1.i3.p1.2.m2.1.1.cmml"><msup id="S7.I1.i3.p1.2.m2.1.1.2" xref="S7.I1.i3.p1.2.m2.1.1.2.cmml"><mi id="S7.I1.i3.p1.2.m2.1.1.2.2" xref="S7.I1.i3.p1.2.m2.1.1.2.2.cmml">F</mi><mo id="S7.I1.i3.p1.2.m2.1.1.2.3" xref="S7.I1.i3.p1.2.m2.1.1.2.3.cmml">∗</mo></msup><mo id="S7.I1.i3.p1.2.m2.1.1.1" xref="S7.I1.i3.p1.2.m2.1.1.1.cmml">+</mo><mrow id="S7.I1.i3.p1.2.m2.1.1.3" xref="S7.I1.i3.p1.2.m2.1.1.3.cmml"><mi id="S7.I1.i3.p1.2.m2.1.1.3.2" xref="S7.I1.i3.p1.2.m2.1.1.3.2.cmml">C</mi><mo id="S7.I1.i3.p1.2.m2.1.1.3.1" xref="S7.I1.i3.p1.2.m2.1.1.3.1.cmml">/</mo><msqrt id="S7.I1.i3.p1.2.m2.1.1.3.3" xref="S7.I1.i3.p1.2.m2.1.1.3.3.cmml"><mi id="S7.I1.i3.p1.2.m2.1.1.3.3.2" xref="S7.I1.i3.p1.2.m2.1.1.3.3.2.cmml">T</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i3.p1.2.m2.1b"><apply id="S7.I1.i3.p1.2.m2.1.1.cmml" xref="S7.I1.i3.p1.2.m2.1.1"><plus id="S7.I1.i3.p1.2.m2.1.1.1.cmml" xref="S7.I1.i3.p1.2.m2.1.1.1"></plus><apply id="S7.I1.i3.p1.2.m2.1.1.2.cmml" xref="S7.I1.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S7.I1.i3.p1.2.m2.1.1.2.1.cmml" xref="S7.I1.i3.p1.2.m2.1.1.2">superscript</csymbol><ci id="S7.I1.i3.p1.2.m2.1.1.2.2.cmml" xref="S7.I1.i3.p1.2.m2.1.1.2.2">𝐹</ci><times id="S7.I1.i3.p1.2.m2.1.1.2.3.cmml" xref="S7.I1.i3.p1.2.m2.1.1.2.3"></times></apply><apply id="S7.I1.i3.p1.2.m2.1.1.3.cmml" xref="S7.I1.i3.p1.2.m2.1.1.3"><divide id="S7.I1.i3.p1.2.m2.1.1.3.1.cmml" xref="S7.I1.i3.p1.2.m2.1.1.3.1"></divide><ci id="S7.I1.i3.p1.2.m2.1.1.3.2.cmml" xref="S7.I1.i3.p1.2.m2.1.1.3.2">𝐶</ci><apply id="S7.I1.i3.p1.2.m2.1.1.3.3.cmml" xref="S7.I1.i3.p1.2.m2.1.1.3.3"><root id="S7.I1.i3.p1.2.m2.1.1.3.3a.cmml" xref="S7.I1.i3.p1.2.m2.1.1.3.3"></root><ci id="S7.I1.i3.p1.2.m2.1.1.3.3.2.cmml" xref="S7.I1.i3.p1.2.m2.1.1.3.3.2">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i3.p1.2.m2.1c">F^{*}+C/\sqrt{T}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i3.p1.2.m2.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT + italic_C / square-root start_ARG italic_T end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i3.p1.4.3">, where </span><math alttext="F^{*}" class="ltx_Math" display="inline" id="S7.I1.i3.p1.3.m3.1"><semantics id="S7.I1.i3.p1.3.m3.1a"><msup id="S7.I1.i3.p1.3.m3.1.1" xref="S7.I1.i3.p1.3.m3.1.1.cmml"><mi id="S7.I1.i3.p1.3.m3.1.1.2" xref="S7.I1.i3.p1.3.m3.1.1.2.cmml">F</mi><mo id="S7.I1.i3.p1.3.m3.1.1.3" xref="S7.I1.i3.p1.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S7.I1.i3.p1.3.m3.1b"><apply id="S7.I1.i3.p1.3.m3.1.1.cmml" xref="S7.I1.i3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I1.i3.p1.3.m3.1.1.1.cmml" xref="S7.I1.i3.p1.3.m3.1.1">superscript</csymbol><ci id="S7.I1.i3.p1.3.m3.1.1.2.cmml" xref="S7.I1.i3.p1.3.m3.1.1.2">𝐹</ci><times id="S7.I1.i3.p1.3.m3.1.1.3.cmml" xref="S7.I1.i3.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i3.p1.3.m3.1c">F^{*}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i3.p1.3.m3.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i3.p1.4.4"> is the objective value of the optimal </span><math alttext="0" class="ltx_Math" display="inline" id="S7.I1.i3.p1.4.m4.1"><semantics id="S7.I1.i3.p1.4.m4.1a"><mn id="S7.I1.i3.p1.4.m4.1.1" xref="S7.I1.i3.p1.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S7.I1.i3.p1.4.m4.1b"><cn id="S7.I1.i3.p1.4.m4.1.1.cmml" type="integer" xref="S7.I1.i3.p1.4.m4.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i3.p1.4.5">-CEP.</span></p> </div> </li> </ol> <p class="ltx_p" id="S7.Thmtheorem5.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem5.p1.4.1">where <math alttext="C\lesssim\sqrt{\log(nm/\delta)}" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.4.1.m1.2"><semantics id="S7.Thmtheorem5.p1.4.1.m1.2a"><mrow id="S7.Thmtheorem5.p1.4.1.m1.2.3" xref="S7.Thmtheorem5.p1.4.1.m1.2.3.cmml"><mi id="S7.Thmtheorem5.p1.4.1.m1.2.3.2" xref="S7.Thmtheorem5.p1.4.1.m1.2.3.2.cmml">C</mi><mo id="S7.Thmtheorem5.p1.4.1.m1.2.3.1" xref="S7.Thmtheorem5.p1.4.1.m1.2.3.1.cmml">≲</mo><msqrt id="S7.Thmtheorem5.p1.4.1.m1.2.2" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.cmml"><mrow id="S7.Thmtheorem5.p1.4.1.m1.2.2.2.4" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.2.3.cmml"><mi id="S7.Thmtheorem5.p1.4.1.m1.2.2.2.2.2.2" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.2.3.1.cmml">log</mi><mo id="S7.Thmtheorem5.p1.4.1.m1.2.2.2.4a" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.2.3.1.cmml">⁡</mo><mrow id="S7.Thmtheorem5.p1.4.1.m1.2.2.2.4.1" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.2.3.cmml"><mo id="S7.Thmtheorem5.p1.4.1.m1.2.2.2.4.1.1" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.2.3.1.cmml">(</mo><mrow id="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1" xref="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.cmml"><mrow id="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2" xref="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2.2" xref="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2.2.cmml">n</mi><mo id="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2.1" xref="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2.3" xref="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.2.3.cmml">m</mi></mrow><mo id="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.1" xref="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.3" xref="S7.Thmtheorem5.p1.4.1.m1.1.1.1.1.1.1.1.3.cmml">δ</mi></mrow><mo id="S7.Thmtheorem5.p1.4.1.m1.2.2.2.4.1.2" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.2.3.1.cmml">)</mo></mrow></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.4.1.m1.2b"><apply id="S7.Thmtheorem5.p1.4.1.m1.2.3.cmml" xref="S7.Thmtheorem5.p1.4.1.m1.2.3"><csymbol cd="latexml" id="S7.Thmtheorem5.p1.4.1.m1.2.3.1.cmml" xref="S7.Thmtheorem5.p1.4.1.m1.2.3.1">less-than-or-similar-to</csymbol><ci id="S7.Thmtheorem5.p1.4.1.m1.2.3.2.cmml" xref="S7.Thmtheorem5.p1.4.1.m1.2.3.2">𝐶</ci><apply id="S7.Thmtheorem5.p1.4.1.m1.2.2.cmml" xref="S7.Thmtheorem5.p1.4.1.m1.2.2"><root id="S7.Thmtheorem5.p1.4.1.m1.2.2a.cmml" xref="S7.Thmtheorem5.p1.4.1.m1.2.2"></root><apply id="S7.Thmtheorem5.p1.4.1.m1.2.2.2.3.cmml" xref="S7.Thmtheorem5.p1.4.1.m1.2.2.2.4"><log 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id="S7.Thmtheorem5.p1.4.1.m1.2c">C\lesssim\sqrt{\log(nm/\delta)}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.4.1.m1.2d">italic_C ≲ square-root start_ARG roman_log ( start_ARG italic_n italic_m / italic_δ end_ARG ) end_ARG</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S7.SS4.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS4.1.p1"> <p class="ltx_p" id="S7.SS4.1.p1.6">The algorithm for the principal, on each round, selects payments function <math alttext="P^{t}_{i}" class="ltx_Math" display="inline" id="S7.SS4.1.p1.1.m1.1"><semantics id="S7.SS4.1.p1.1.m1.1a"><msubsup id="S7.SS4.1.p1.1.m1.1.1" xref="S7.SS4.1.p1.1.m1.1.1.cmml"><mi id="S7.SS4.1.p1.1.m1.1.1.2.2" xref="S7.SS4.1.p1.1.m1.1.1.2.2.cmml">P</mi><mi id="S7.SS4.1.p1.1.m1.1.1.3" xref="S7.SS4.1.p1.1.m1.1.1.3.cmml">i</mi><mi id="S7.SS4.1.p1.1.m1.1.1.2.3" xref="S7.SS4.1.p1.1.m1.1.1.2.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S7.SS4.1.p1.1.m1.1b"><apply id="S7.SS4.1.p1.1.m1.1.1.cmml" xref="S7.SS4.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS4.1.p1.1.m1.1.1.1.cmml" xref="S7.SS4.1.p1.1.m1.1.1">subscript</csymbol><apply id="S7.SS4.1.p1.1.m1.1.1.2.cmml" xref="S7.SS4.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS4.1.p1.1.m1.1.1.2.1.cmml" xref="S7.SS4.1.p1.1.m1.1.1">superscript</csymbol><ci id="S7.SS4.1.p1.1.m1.1.1.2.2.cmml" xref="S7.SS4.1.p1.1.m1.1.1.2.2">𝑃</ci><ci id="S7.SS4.1.p1.1.m1.1.1.2.3.cmml" xref="S7.SS4.1.p1.1.m1.1.1.2.3">𝑡</ci></apply><ci id="S7.SS4.1.p1.1.m1.1.1.3.cmml" xref="S7.SS4.1.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.1.p1.1.m1.1c">P^{t}_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.1.p1.1.m1.1d">italic_P start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and signal distribution <math alttext="\mu^{t}\in\Delta(S)" class="ltx_Math" display="inline" id="S7.SS4.1.p1.2.m2.1"><semantics id="S7.SS4.1.p1.2.m2.1a"><mrow id="S7.SS4.1.p1.2.m2.1.2" xref="S7.SS4.1.p1.2.m2.1.2.cmml"><msup id="S7.SS4.1.p1.2.m2.1.2.2" xref="S7.SS4.1.p1.2.m2.1.2.2.cmml"><mi id="S7.SS4.1.p1.2.m2.1.2.2.2" xref="S7.SS4.1.p1.2.m2.1.2.2.2.cmml">μ</mi><mi id="S7.SS4.1.p1.2.m2.1.2.2.3" xref="S7.SS4.1.p1.2.m2.1.2.2.3.cmml">t</mi></msup><mo id="S7.SS4.1.p1.2.m2.1.2.1" xref="S7.SS4.1.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S7.SS4.1.p1.2.m2.1.2.3" xref="S7.SS4.1.p1.2.m2.1.2.3.cmml"><mi id="S7.SS4.1.p1.2.m2.1.2.3.2" mathvariant="normal" xref="S7.SS4.1.p1.2.m2.1.2.3.2.cmml">Δ</mi><mo id="S7.SS4.1.p1.2.m2.1.2.3.1" xref="S7.SS4.1.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S7.SS4.1.p1.2.m2.1.2.3.3.2" xref="S7.SS4.1.p1.2.m2.1.2.3.cmml"><mo id="S7.SS4.1.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S7.SS4.1.p1.2.m2.1.2.3.cmml">(</mo><mi id="S7.SS4.1.p1.2.m2.1.1" xref="S7.SS4.1.p1.2.m2.1.1.cmml">S</mi><mo id="S7.SS4.1.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S7.SS4.1.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.1.p1.2.m2.1b"><apply id="S7.SS4.1.p1.2.m2.1.2.cmml" xref="S7.SS4.1.p1.2.m2.1.2"><in id="S7.SS4.1.p1.2.m2.1.2.1.cmml" xref="S7.SS4.1.p1.2.m2.1.2.1"></in><apply id="S7.SS4.1.p1.2.m2.1.2.2.cmml" xref="S7.SS4.1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S7.SS4.1.p1.2.m2.1.2.2.1.cmml" xref="S7.SS4.1.p1.2.m2.1.2.2">superscript</csymbol><ci id="S7.SS4.1.p1.2.m2.1.2.2.2.cmml" xref="S7.SS4.1.p1.2.m2.1.2.2.2">𝜇</ci><ci id="S7.SS4.1.p1.2.m2.1.2.2.3.cmml" xref="S7.SS4.1.p1.2.m2.1.2.2.3">𝑡</ci></apply><apply id="S7.SS4.1.p1.2.m2.1.2.3.cmml" xref="S7.SS4.1.p1.2.m2.1.2.3"><times id="S7.SS4.1.p1.2.m2.1.2.3.1.cmml" xref="S7.SS4.1.p1.2.m2.1.2.3.1"></times><ci id="S7.SS4.1.p1.2.m2.1.2.3.2.cmml" xref="S7.SS4.1.p1.2.m2.1.2.3.2">Δ</ci><ci id="S7.SS4.1.p1.2.m2.1.1.cmml" xref="S7.SS4.1.p1.2.m2.1.1">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.1.p1.2.m2.1c">\mu^{t}\in\Delta(S)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.1.p1.2.m2.1d">italic_μ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∈ roman_Δ ( italic_S )</annotation></semantics></math>. Together, these induce a Bayesian game <math alttext="\Gamma^{t}" class="ltx_Math" display="inline" id="S7.SS4.1.p1.3.m3.1"><semantics id="S7.SS4.1.p1.3.m3.1a"><msup id="S7.SS4.1.p1.3.m3.1.1" xref="S7.SS4.1.p1.3.m3.1.1.cmml"><mi id="S7.SS4.1.p1.3.m3.1.1.2" mathvariant="normal" xref="S7.SS4.1.p1.3.m3.1.1.2.cmml">Γ</mi><mi id="S7.SS4.1.p1.3.m3.1.1.3" xref="S7.SS4.1.p1.3.m3.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S7.SS4.1.p1.3.m3.1b"><apply id="S7.SS4.1.p1.3.m3.1.1.cmml" xref="S7.SS4.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.SS4.1.p1.3.m3.1.1.1.cmml" xref="S7.SS4.1.p1.3.m3.1.1">superscript</csymbol><ci id="S7.SS4.1.p1.3.m3.1.1.2.cmml" xref="S7.SS4.1.p1.3.m3.1.1.2">Γ</ci><ci id="S7.SS4.1.p1.3.m3.1.1.3.cmml" xref="S7.SS4.1.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.1.p1.3.m3.1c">\Gamma^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.1.p1.3.m3.1d">roman_Γ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, where the agents’ strategies correspond to functions <math alttext="\pi^{t}_{i}:S_{i}\to\Delta(A_{i})" class="ltx_Math" display="inline" id="S7.SS4.1.p1.4.m4.1"><semantics id="S7.SS4.1.p1.4.m4.1a"><mrow id="S7.SS4.1.p1.4.m4.1.1" xref="S7.SS4.1.p1.4.m4.1.1.cmml"><msubsup id="S7.SS4.1.p1.4.m4.1.1.3" xref="S7.SS4.1.p1.4.m4.1.1.3.cmml"><mi id="S7.SS4.1.p1.4.m4.1.1.3.2.2" xref="S7.SS4.1.p1.4.m4.1.1.3.2.2.cmml">π</mi><mi id="S7.SS4.1.p1.4.m4.1.1.3.3" xref="S7.SS4.1.p1.4.m4.1.1.3.3.cmml">i</mi><mi id="S7.SS4.1.p1.4.m4.1.1.3.2.3" xref="S7.SS4.1.p1.4.m4.1.1.3.2.3.cmml">t</mi></msubsup><mo id="S7.SS4.1.p1.4.m4.1.1.2" lspace="0.278em" rspace="0.278em" xref="S7.SS4.1.p1.4.m4.1.1.2.cmml">:</mo><mrow id="S7.SS4.1.p1.4.m4.1.1.1" xref="S7.SS4.1.p1.4.m4.1.1.1.cmml"><msub id="S7.SS4.1.p1.4.m4.1.1.1.3" xref="S7.SS4.1.p1.4.m4.1.1.1.3.cmml"><mi id="S7.SS4.1.p1.4.m4.1.1.1.3.2" xref="S7.SS4.1.p1.4.m4.1.1.1.3.2.cmml">S</mi><mi 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xref="S7.SS4.1.p1.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.1.p1.4.m4.1b"><apply id="S7.SS4.1.p1.4.m4.1.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1"><ci id="S7.SS4.1.p1.4.m4.1.1.2.cmml" xref="S7.SS4.1.p1.4.m4.1.1.2">:</ci><apply id="S7.SS4.1.p1.4.m4.1.1.3.cmml" xref="S7.SS4.1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.1.p1.4.m4.1.1.3.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1.3">subscript</csymbol><apply id="S7.SS4.1.p1.4.m4.1.1.3.2.cmml" xref="S7.SS4.1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.1.p1.4.m4.1.1.3.2.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1.3">superscript</csymbol><ci id="S7.SS4.1.p1.4.m4.1.1.3.2.2.cmml" xref="S7.SS4.1.p1.4.m4.1.1.3.2.2">𝜋</ci><ci id="S7.SS4.1.p1.4.m4.1.1.3.2.3.cmml" xref="S7.SS4.1.p1.4.m4.1.1.3.2.3">𝑡</ci></apply><ci id="S7.SS4.1.p1.4.m4.1.1.3.3.cmml" xref="S7.SS4.1.p1.4.m4.1.1.3.3">𝑖</ci></apply><apply id="S7.SS4.1.p1.4.m4.1.1.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1"><ci id="S7.SS4.1.p1.4.m4.1.1.1.2.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.2">→</ci><apply id="S7.SS4.1.p1.4.m4.1.1.1.3.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.1.p1.4.m4.1.1.1.3.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.3">subscript</csymbol><ci id="S7.SS4.1.p1.4.m4.1.1.1.3.2.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.3.2">𝑆</ci><ci id="S7.SS4.1.p1.4.m4.1.1.1.3.3.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.3.3">𝑖</ci></apply><apply id="S7.SS4.1.p1.4.m4.1.1.1.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.1"><times id="S7.SS4.1.p1.4.m4.1.1.1.1.2.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.1.2"></times><ci id="S7.SS4.1.p1.4.m4.1.1.1.1.3.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.1.3">Δ</ci><apply id="S7.SS4.1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS4.1.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S7.SS4.1.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S7.SS4.1.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S7.SS4.1.p1.4.m4.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.1.p1.4.m4.1c">\pi^{t}_{i}:S_{i}\to\Delta(A_{i})</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.1.p1.4.m4.1d">italic_π start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → roman_Δ ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. Suppose that the agents play according to a Nash equilibrium <math alttext="\pi^{t}" class="ltx_Math" display="inline" id="S7.SS4.1.p1.5.m5.1"><semantics id="S7.SS4.1.p1.5.m5.1a"><msup id="S7.SS4.1.p1.5.m5.1.1" xref="S7.SS4.1.p1.5.m5.1.1.cmml"><mi id="S7.SS4.1.p1.5.m5.1.1.2" xref="S7.SS4.1.p1.5.m5.1.1.2.cmml">π</mi><mi id="S7.SS4.1.p1.5.m5.1.1.3" xref="S7.SS4.1.p1.5.m5.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S7.SS4.1.p1.5.m5.1b"><apply id="S7.SS4.1.p1.5.m5.1.1.cmml" xref="S7.SS4.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S7.SS4.1.p1.5.m5.1.1.1.cmml" xref="S7.SS4.1.p1.5.m5.1.1">superscript</csymbol><ci id="S7.SS4.1.p1.5.m5.1.1.2.cmml" xref="S7.SS4.1.p1.5.m5.1.1.2">𝜋</ci><ci id="S7.SS4.1.p1.5.m5.1.1.3.cmml" xref="S7.SS4.1.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.1.p1.5.m5.1c">\pi^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.1.p1.5.m5.1d">italic_π start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="\Gamma^{t}" class="ltx_Math" display="inline" id="S7.SS4.1.p1.6.m6.1"><semantics id="S7.SS4.1.p1.6.m6.1a"><msup id="S7.SS4.1.p1.6.m6.1.1" xref="S7.SS4.1.p1.6.m6.1.1.cmml"><mi id="S7.SS4.1.p1.6.m6.1.1.2" mathvariant="normal" xref="S7.SS4.1.p1.6.m6.1.1.2.cmml">Γ</mi><mi id="S7.SS4.1.p1.6.m6.1.1.3" xref="S7.SS4.1.p1.6.m6.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S7.SS4.1.p1.6.m6.1b"><apply id="S7.SS4.1.p1.6.m6.1.1.cmml" xref="S7.SS4.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S7.SS4.1.p1.6.m6.1.1.1.cmml" xref="S7.SS4.1.p1.6.m6.1.1">superscript</csymbol><ci id="S7.SS4.1.p1.6.m6.1.1.2.cmml" xref="S7.SS4.1.p1.6.m6.1.1.2">Γ</ci><ci id="S7.SS4.1.p1.6.m6.1.1.3.cmml" xref="S7.SS4.1.p1.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.1.p1.6.m6.1c">\Gamma^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.1.p1.6.m6.1d">roman_Γ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. Then:</p> <ul class="ltx_itemize" id="S7.I2"> <li class="ltx_item" id="S7.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I2.i1.p1"> <p class="ltx_p" id="S7.I2.i1.p1.4">the deviation benefit</p> <table class="ltx_equation ltx_eqn_table" id="S7.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="\hat{R}(t,s_{i}\to a_{i}):=\sum_{\begin{subarray}{c}\tau\leq t:\\ s_{i}^{\tau}=s_{i}\end{subarray}}\quantity[U^{\tau}_{i}(s^{\tau},a_{i},a_{-i}^% {\tau})-U^{t}_{i}(s^{\tau},a^{\tau})]" class="ltx_Math" display="block" id="S7.Ex4.m1.4"><semantics id="S7.Ex4.m1.4a"><mrow id="S7.Ex4.m1.4.4" xref="S7.Ex4.m1.4.4.cmml"><mrow id="S7.Ex4.m1.4.4.1" xref="S7.Ex4.m1.4.4.1.cmml"><mover accent="true" id="S7.Ex4.m1.4.4.1.3" xref="S7.Ex4.m1.4.4.1.3.cmml"><mi id="S7.Ex4.m1.4.4.1.3.2" xref="S7.Ex4.m1.4.4.1.3.2.cmml">R</mi><mo id="S7.Ex4.m1.4.4.1.3.1" xref="S7.Ex4.m1.4.4.1.3.1.cmml">^</mo></mover><mo id="S7.Ex4.m1.4.4.1.2" xref="S7.Ex4.m1.4.4.1.2.cmml">⁢</mo><mrow id="S7.Ex4.m1.4.4.1.1.1" xref="S7.Ex4.m1.4.4.1.1.1.1.cmml"><mo id="S7.Ex4.m1.4.4.1.1.1.2" stretchy="false" xref="S7.Ex4.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="S7.Ex4.m1.4.4.1.1.1.1" xref="S7.Ex4.m1.4.4.1.1.1.1.cmml"><mrow id="S7.Ex4.m1.4.4.1.1.1.1.1.1" xref="S7.Ex4.m1.4.4.1.1.1.1.1.2.cmml"><mi id="S7.Ex4.m1.3.3" xref="S7.Ex4.m1.3.3.cmml">t</mi><mo id="S7.Ex4.m1.4.4.1.1.1.1.1.1.2" xref="S7.Ex4.m1.4.4.1.1.1.1.1.2.cmml">,</mo><msub id="S7.Ex4.m1.4.4.1.1.1.1.1.1.1" xref="S7.Ex4.m1.4.4.1.1.1.1.1.1.1.cmml"><mi id="S7.Ex4.m1.4.4.1.1.1.1.1.1.1.2" xref="S7.Ex4.m1.4.4.1.1.1.1.1.1.1.2.cmml">s</mi><mi id="S7.Ex4.m1.4.4.1.1.1.1.1.1.1.3" xref="S7.Ex4.m1.4.4.1.1.1.1.1.1.1.3.cmml">i</mi></msub></mrow><mo id="S7.Ex4.m1.4.4.1.1.1.1.2" stretchy="false" xref="S7.Ex4.m1.4.4.1.1.1.1.2.cmml">→</mo><msub id="S7.Ex4.m1.4.4.1.1.1.1.3" xref="S7.Ex4.m1.4.4.1.1.1.1.3.cmml"><mi id="S7.Ex4.m1.4.4.1.1.1.1.3.2" xref="S7.Ex4.m1.4.4.1.1.1.1.3.2.cmml">a</mi><mi id="S7.Ex4.m1.4.4.1.1.1.1.3.3" xref="S7.Ex4.m1.4.4.1.1.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S7.Ex4.m1.4.4.1.1.1.3" rspace="0.278em" stretchy="false" xref="S7.Ex4.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Ex4.m1.4.4.2" rspace="0.111em" xref="S7.Ex4.m1.4.4.2.cmml">:=</mo><mrow id="S7.Ex4.m1.4.4.3" xref="S7.Ex4.m1.4.4.3.cmml"><munder id="S7.Ex4.m1.4.4.3.1" xref="S7.Ex4.m1.4.4.3.1.cmml"><mo id="S7.Ex4.m1.4.4.3.1.2" movablelimits="false" rspace="0em" xref="S7.Ex4.m1.4.4.3.1.2.cmml">∑</mo><mtable id="S7.Ex4.m1.1.1.1.1.1.1" rowspacing="0pt" xref="S7.Ex4.m1.1.1.1.2.cmml"><mtr id="S7.Ex4.m1.1.1.1.1.1.1a" xref="S7.Ex4.m1.1.1.1.2.cmml"><mtd id="S7.Ex4.m1.1.1.1.1.1.1b" xref="S7.Ex4.m1.1.1.1.2.cmml"><mrow id="S7.Ex4.m1.1.1.1.1.1.1.1.1.1" xref="S7.Ex4.m1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S7.Ex4.m1.1.1.1.1.1.1.1.1.1.2" xref="S7.Ex4.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi 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encoding="application/x-tex" id="S7.Ex4.m1.4c">\hat{R}(t,s_{i}\to a_{i}):=\sum_{\begin{subarray}{c}\tau\leq t:\\ s_{i}^{\tau}=s_{i}\end{subarray}}\quantity[U^{\tau}_{i}(s^{\tau},a_{i},a_{-i}^% {\tau})-U^{t}_{i}(s^{\tau},a^{\tau})]</annotation><annotation encoding="application/x-llamapun" id="S7.Ex4.m1.4d">over^ start_ARG italic_R end_ARG ( italic_t , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) := ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_τ ≤ italic_t : end_CELL end_ROW start_ROW start_CELL italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ) - italic_U start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT , italic_a start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ) end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S7.I2.i1.p1.3">is a supermartingale, so by <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A1.Thmtheorem1" title="Lemma A.1. ‣ Appendix A An anytime Azuma-Hoeffding bound ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">A.1</span></a>, with probability <math alttext="1-\delta" class="ltx_Math" display="inline" id="S7.I2.i1.p1.1.m1.1"><semantics id="S7.I2.i1.p1.1.m1.1a"><mrow id="S7.I2.i1.p1.1.m1.1.1" xref="S7.I2.i1.p1.1.m1.1.1.cmml"><mn id="S7.I2.i1.p1.1.m1.1.1.2" xref="S7.I2.i1.p1.1.m1.1.1.2.cmml">1</mn><mo id="S7.I2.i1.p1.1.m1.1.1.1" xref="S7.I2.i1.p1.1.m1.1.1.1.cmml">−</mo><mi id="S7.I2.i1.p1.1.m1.1.1.3" xref="S7.I2.i1.p1.1.m1.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i1.p1.1.m1.1b"><apply id="S7.I2.i1.p1.1.m1.1.1.cmml" xref="S7.I2.i1.p1.1.m1.1.1"><minus id="S7.I2.i1.p1.1.m1.1.1.1.cmml" xref="S7.I2.i1.p1.1.m1.1.1.1"></minus><cn id="S7.I2.i1.p1.1.m1.1.1.2.cmml" type="integer" xref="S7.I2.i1.p1.1.m1.1.1.2">1</cn><ci id="S7.I2.i1.p1.1.m1.1.1.3.cmml" xref="S7.I2.i1.p1.1.m1.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.1.m1.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.1.m1.1d">1 - italic_δ</annotation></semantics></math>, it holds simultaneously for all <math alttext="t\leq T" class="ltx_Math" display="inline" id="S7.I2.i1.p1.2.m2.1"><semantics id="S7.I2.i1.p1.2.m2.1a"><mrow id="S7.I2.i1.p1.2.m2.1.1" xref="S7.I2.i1.p1.2.m2.1.1.cmml"><mi id="S7.I2.i1.p1.2.m2.1.1.2" xref="S7.I2.i1.p1.2.m2.1.1.2.cmml">t</mi><mo id="S7.I2.i1.p1.2.m2.1.1.1" xref="S7.I2.i1.p1.2.m2.1.1.1.cmml">≤</mo><mi id="S7.I2.i1.p1.2.m2.1.1.3" xref="S7.I2.i1.p1.2.m2.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i1.p1.2.m2.1b"><apply id="S7.I2.i1.p1.2.m2.1.1.cmml" xref="S7.I2.i1.p1.2.m2.1.1"><leq id="S7.I2.i1.p1.2.m2.1.1.1.cmml" xref="S7.I2.i1.p1.2.m2.1.1.1"></leq><ci id="S7.I2.i1.p1.2.m2.1.1.2.cmml" xref="S7.I2.i1.p1.2.m2.1.1.2">𝑡</ci><ci id="S7.I2.i1.p1.2.m2.1.1.3.cmml" xref="S7.I2.i1.p1.2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.2.m2.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.2.m2.1d">italic_t ≤ italic_T</annotation></semantics></math> that <math alttext="\hat{R}(t,s_{i}\to 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id="S7.I2.i1.p1.3.m3.1.1.1.1.1.1.1.3.cmml" xref="S7.I2.i1.p1.3.m3.1.1.1.1.1.1.1.3">𝛿</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.3.m3.4c">\hat{R}(t,s_{i}\to a_{i})\lesssim\sqrt{T\log(1/\delta)}</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.3.m3.4d">over^ start_ARG italic_R end_ARG ( italic_t , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≲ square-root start_ARG italic_T roman_log ( start_ARG 1 / italic_δ end_ARG ) end_ARG</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I2.i2.p1"> <p class="ltx_p" id="S7.I2.i2.p1.1">By the revelation principle (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#A2.Thmtheorem2" title="Proposition B.2 (Revelation principle for CEPs). ‣ B.2 Revelation principle for CEPs ‣ Appendix B Details omitted from Section 7 ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">B.2</span></a>), the principal objective <math alttext="\sum_{\tau\leq t}\quantity[U_{0}(a^{\tau})-F^{*}]" class="ltx_Math" display="inline" id="S7.I2.i2.p1.1.m1.1"><semantics id="S7.I2.i2.p1.1.m1.1a"><mrow id="S7.I2.i2.p1.1.m1.1.2" xref="S7.I2.i2.p1.1.m1.1.2.cmml"><msub id="S7.I2.i2.p1.1.m1.1.2.1" xref="S7.I2.i2.p1.1.m1.1.2.1.cmml"><mo id="S7.I2.i2.p1.1.m1.1.2.1.2" xref="S7.I2.i2.p1.1.m1.1.2.1.2.cmml">∑</mo><mrow id="S7.I2.i2.p1.1.m1.1.2.1.3" xref="S7.I2.i2.p1.1.m1.1.2.1.3.cmml"><mi id="S7.I2.i2.p1.1.m1.1.2.1.3.2" xref="S7.I2.i2.p1.1.m1.1.2.1.3.2.cmml">τ</mi><mo id="S7.I2.i2.p1.1.m1.1.2.1.3.1" xref="S7.I2.i2.p1.1.m1.1.2.1.3.1.cmml">≤</mo><mi id="S7.I2.i2.p1.1.m1.1.2.1.3.3" xref="S7.I2.i2.p1.1.m1.1.2.1.3.3.cmml">t</mi></mrow></msub><mrow id="S7.I2.i2.p1.1.m1.1.1.3" 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xref="S7.I2.i2.p1.1.m1.1.1.1.1.1.3.2">𝐹</ci><times id="S7.I2.i2.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S7.I2.i2.p1.1.m1.1.1.1.1.1.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i2.p1.1.m1.1c">\sum_{\tau\leq t}\quantity[U_{0}(a^{\tau})-F^{*}]</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i2.p1.1.m1.1d">∑ start_POSTSUBSCRIPT italic_τ ≤ italic_t end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG ]</annotation></semantics></math> is also a supermartingale, so the same bound holds for the principal objective.</p> </div> </li> </ul> <p class="ltx_p" id="S7.SS4.1.p1.9">Taking a union bound over the <math alttext="n" class="ltx_Math" display="inline" id="S7.SS4.1.p1.7.m1.1"><semantics id="S7.SS4.1.p1.7.m1.1a"><mi id="S7.SS4.1.p1.7.m1.1.1" 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xref="S7.SS4.1.p1.9.m3.1.2.2"><merror class="ltx_ERROR undefined undefined" id="S7.SS4.1.p1.9.m3.1.2.2.cmml" xref="S7.SS4.1.p1.9.m3.1.2.2"><mtext id="S7.SS4.1.p1.9.m3.1.2.2a.cmml" xref="S7.SS4.1.p1.9.m3.1.2.2">\poly</mtext></merror></ci><ci id="S7.SS4.1.p1.9.m3.1.1.cmml" xref="S7.SS4.1.p1.9.m3.1.1">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.1.p1.9.m3.1c">\poly(m)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.1.p1.9.m3.1d">( italic_m )</annotation></semantics></math> signals completes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S7.SS4.p2"> <p class="ltx_p" id="S7.SS4.p2.4">We now show that the principal <span class="ltx_text ltx_font_italic" id="S7.SS4.p2.4.1">can</span> achieve utility <math alttext="F^{*}" class="ltx_Math" display="inline" id="S7.SS4.p2.1.m1.1"><semantics id="S7.SS4.p2.1.m1.1a"><msup id="S7.SS4.p2.1.m1.1.1" xref="S7.SS4.p2.1.m1.1.1.cmml"><mi id="S7.SS4.p2.1.m1.1.1.2" xref="S7.SS4.p2.1.m1.1.1.2.cmml">F</mi><mo id="S7.SS4.p2.1.m1.1.1.3" xref="S7.SS4.p2.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S7.SS4.p2.1.m1.1b"><apply id="S7.SS4.p2.1.m1.1.1.cmml" xref="S7.SS4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS4.p2.1.m1.1.1.1.cmml" xref="S7.SS4.p2.1.m1.1.1">superscript</csymbol><ci id="S7.SS4.p2.1.m1.1.1.2.cmml" xref="S7.SS4.p2.1.m1.1.1.2">𝐹</ci><times id="S7.SS4.p2.1.m1.1.1.3.cmml" xref="S7.SS4.p2.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p2.1.m1.1c">F^{*}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p2.1.m1.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> in the limit <math alttext="T\to\infty" class="ltx_Math" display="inline" id="S7.SS4.p2.2.m2.1"><semantics id="S7.SS4.p2.2.m2.1a"><mrow id="S7.SS4.p2.2.m2.1.1" xref="S7.SS4.p2.2.m2.1.1.cmml"><mi id="S7.SS4.p2.2.m2.1.1.2" xref="S7.SS4.p2.2.m2.1.1.2.cmml">T</mi><mo id="S7.SS4.p2.2.m2.1.1.1" stretchy="false" xref="S7.SS4.p2.2.m2.1.1.1.cmml">→</mo><mi id="S7.SS4.p2.2.m2.1.1.3" mathvariant="normal" xref="S7.SS4.p2.2.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.p2.2.m2.1b"><apply id="S7.SS4.p2.2.m2.1.1.cmml" xref="S7.SS4.p2.2.m2.1.1"><ci id="S7.SS4.p2.2.m2.1.1.1.cmml" xref="S7.SS4.p2.2.m2.1.1.1">→</ci><ci id="S7.SS4.p2.2.m2.1.1.2.cmml" xref="S7.SS4.p2.2.m2.1.1.2">𝑇</ci><infinity id="S7.SS4.p2.2.m2.1.1.3.cmml" xref="S7.SS4.p2.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p2.2.m2.1c">T\to\infty</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p2.2.m2.1d">italic_T → ∞</annotation></semantics></math>. Intuitively, the algorithm will work in two stages. In the first stage, the principal uses <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg4" title="In 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a> to learn the utility functions of the agents. Then, the principal computes an optimal CEP and steers the agents to it. The steering algorithm is adapted from <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite>, and presented in full here for the sake of self-containment. Perhaps most notably, since the principal only knows the game up to an error <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S7.SS4.p2.3.m3.1"><semantics id="S7.SS4.p2.3.m3.1a"><mrow id="S7.SS4.p2.3.m3.1.1" xref="S7.SS4.p2.3.m3.1.1.cmml"><mi id="S7.SS4.p2.3.m3.1.1.2" xref="S7.SS4.p2.3.m3.1.1.2.cmml">ε</mi><mo id="S7.SS4.p2.3.m3.1.1.1" xref="S7.SS4.p2.3.m3.1.1.1.cmml">></mo><mn id="S7.SS4.p2.3.m3.1.1.3" xref="S7.SS4.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.p2.3.m3.1b"><apply id="S7.SS4.p2.3.m3.1.1.cmml" xref="S7.SS4.p2.3.m3.1.1"><gt id="S7.SS4.p2.3.m3.1.1.1.cmml" xref="S7.SS4.p2.3.m3.1.1.1"></gt><ci id="S7.SS4.p2.3.m3.1.1.2.cmml" xref="S7.SS4.p2.3.m3.1.1.2">𝜀</ci><cn id="S7.SS4.p2.3.m3.1.1.3.cmml" type="integer" xref="S7.SS4.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p2.3.m3.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p2.3.m3.1d">italic_ε > 0</annotation></semantics></math>, it must give extra payments of at least <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.SS4.p2.4.m4.1"><semantics id="S7.SS4.p2.4.m4.1a"><mi id="S7.SS4.p2.4.m4.1.1" xref="S7.SS4.p2.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.p2.4.m4.1b"><ci id="S7.SS4.p2.4.m4.1.1.cmml" xref="S7.SS4.p2.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p2.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p2.4.m4.1d">italic_ε</annotation></semantics></math> to ensure that agents do not deviate.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg6"> <div class="ltx_listing ltx_listing" id="alg6.2"> <div class="ltx_listingline" id="alg5.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l1.1.1.1" style="font-size:80%;">1:</span></span>using <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg4" title="In 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a>, estimate the utility functions to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="alg5.l1.m1.1"><semantics id="alg5.l1.m1.1a"><mi id="alg5.l1.m1.1.1" xref="alg5.l1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="alg5.l1.m1.1b"><ci id="alg5.l1.m1.1.1.cmml" xref="alg5.l1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m1.1d">italic_ε</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l2.1.1.1" style="font-size:80%;">2:</span></span>using the LP (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.E18" title="Equation 18 ‣ Proof. ‣ 7.3 Properties of correlated equilibria with payments ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">18</span></a>), compute an optimal <math alttext="2\varepsilon" class="ltx_Math" display="inline" id="alg5.l2.m1.1"><semantics id="alg5.l2.m1.1a"><mrow id="alg5.l2.m1.1.1" xref="alg5.l2.m1.1.1.cmml"><mn id="alg5.l2.m1.1.1.2" xref="alg5.l2.m1.1.1.2.cmml">2</mn><mo id="alg5.l2.m1.1.1.1" xref="alg5.l2.m1.1.1.1.cmml">⁢</mo><mi id="alg5.l2.m1.1.1.3" xref="alg5.l2.m1.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="alg5.l2.m1.1b"><apply id="alg5.l2.m1.1.1.cmml" xref="alg5.l2.m1.1.1"><times id="alg5.l2.m1.1.1.1.cmml" xref="alg5.l2.m1.1.1.1"></times><cn id="alg5.l2.m1.1.1.2.cmml" type="integer" xref="alg5.l2.m1.1.1.2">2</cn><ci id="alg5.l2.m1.1.1.3.cmml" xref="alg5.l2.m1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m1.1c">2\varepsilon</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m1.1d">2 italic_ε</annotation></semantics></math>-CEP <math alttext="(\mu^{*},P^{*})" class="ltx_Math" display="inline" id="alg5.l2.m2.2"><semantics id="alg5.l2.m2.2a"><mrow id="alg5.l2.m2.2.2.2" xref="alg5.l2.m2.2.2.3.cmml"><mo id="alg5.l2.m2.2.2.2.3" stretchy="false" xref="alg5.l2.m2.2.2.3.cmml">(</mo><msup id="alg5.l2.m2.1.1.1.1" xref="alg5.l2.m2.1.1.1.1.cmml"><mi id="alg5.l2.m2.1.1.1.1.2" xref="alg5.l2.m2.1.1.1.1.2.cmml">μ</mi><mo id="alg5.l2.m2.1.1.1.1.3" xref="alg5.l2.m2.1.1.1.1.3.cmml">∗</mo></msup><mo id="alg5.l2.m2.2.2.2.4" xref="alg5.l2.m2.2.2.3.cmml">,</mo><msup id="alg5.l2.m2.2.2.2.2" xref="alg5.l2.m2.2.2.2.2.cmml"><mi id="alg5.l2.m2.2.2.2.2.2" xref="alg5.l2.m2.2.2.2.2.2.cmml">P</mi><mo id="alg5.l2.m2.2.2.2.2.3" 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encoding="application/x-llamapun" id="alg5.l2.m3.1d">over~ start_ARG roman_Γ end_ARG</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l3.1.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg5.l3.2">for</span> remaining rounds <span class="ltx_text ltx_font_bold" id="alg5.l3.3">do</span> set <math alttext="\mu^{t}=\mu^{*}" class="ltx_Math" display="inline" id="alg5.l3.m1.1"><semantics id="alg5.l3.m1.1a"><mrow id="alg5.l3.m1.1.1" xref="alg5.l3.m1.1.1.cmml"><msup id="alg5.l3.m1.1.1.2" xref="alg5.l3.m1.1.1.2.cmml"><mi id="alg5.l3.m1.1.1.2.2" xref="alg5.l3.m1.1.1.2.2.cmml">μ</mi><mi id="alg5.l3.m1.1.1.2.3" xref="alg5.l3.m1.1.1.2.3.cmml">t</mi></msup><mo id="alg5.l3.m1.1.1.1" xref="alg5.l3.m1.1.1.1.cmml">=</mo><msup id="alg5.l3.m1.1.1.3" xref="alg5.l3.m1.1.1.3.cmml"><mi id="alg5.l3.m1.1.1.3.2" xref="alg5.l3.m1.1.1.3.2.cmml">μ</mi><mo 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italic_s , italic_a ) = { start_ROW start_CELL italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a , italic_a ) + 2 italic_ε + italic_ρ end_CELL start_CELL if italic_s = italic_a end_CELL end_ROW start_ROW start_CELL 2 end_CELL start_CELL if italic_s ≠ italic_a , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL otherwise end_CELL end_ROW</annotation></semantics></math>. </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg6.3.1.1">Algorithm 6</span> </span> Principal’s algorithm for steering without prior knowledge of utilities</figcaption> </figure> <div class="ltx_theorem ltx_theorem_theorem" id="S7.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="S7.Thmtheorem6.1.1.1">Theorem 7.6</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem6.p1"> <p class="ltx_p" id="S7.Thmtheorem6.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem6.p1.3.3">For appropriate choices of the hyperparameters <math alttext="L" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.1.1.m1.1"><semantics id="S7.Thmtheorem6.p1.1.1.m1.1a"><mi id="S7.Thmtheorem6.p1.1.1.m1.1.1" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.1.1.m1.1b"><ci id="S7.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.1.1.m1.1d">italic_L</annotation></semantics></math> (from <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg4" title="In 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a>) and <math alttext="\rho" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.2.2.m2.1"><semantics id="S7.Thmtheorem6.p1.2.2.m2.1a"><mi id="S7.Thmtheorem6.p1.2.2.m2.1.1" xref="S7.Thmtheorem6.p1.2.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.2.2.m2.1b"><ci id="S7.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem6.p1.2.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.2.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.2.2.m2.1d">italic_ρ</annotation></semantics></math>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg6" title="In 7.4 CEPs and optimal steering ‣ 7 Steering 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xref="S7.Thmtheorem6.p1.3.3.m3.2.3.2.2">𝐹</ci><ci id="S7.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1">𝑇</ci></apply><apply id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3"><minus id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.1"></minus><apply id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2">superscript</csymbol><ci id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2.2">𝐹</ci><times id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2.3.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.2.3"></times></apply><apply id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3"><divide id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.1"></divide><apply id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2"><times id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.1"></times><ci id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.2b.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.2"><merror class="ltx_ERROR undefined undefined" id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.2"><mtext id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.2a.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.2.2">\poly</mtext></merror></ci><ci id="S7.Thmtheorem6.p1.3.3.m3.2.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.2">𝑀</ci></apply><apply id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3">superscript</csymbol><ci id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.2">𝑇</ci><apply id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3"><divide id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3.1"></divide><cn id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3.2.cmml" type="integer" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3.2">1</cn><cn id="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3.3.cmml" type="integer" xref="S7.Thmtheorem6.p1.3.3.m3.2.3.3.3.3.3.3">4</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.3.3.m3.2c">F(T)\geq F^{*}-\poly(M)/T^{1/4}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.3.3.m3.2d">italic_F ( italic_T ) ≥ italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - ( italic_M ) / italic_T start_POSTSUPERSCRIPT 1 / 4 end_POSTSUPERSCRIPT</annotation></semantics></math> rounds.</span></p> </div> </div> <div class="ltx_proof" id="S7.SS4.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS4.2.p1"> <p class="ltx_p" id="S7.SS4.2.p1.4">From the analysis of <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.Thmtheorem4" title="Theorem 5.4. ‣ 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.4</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#alg4" title="In 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a> learns a game to precision <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.SS4.2.p1.1.m1.1"><semantics id="S7.SS4.2.p1.1.m1.1a"><mi id="S7.SS4.2.p1.1.m1.1.1" xref="S7.SS4.2.p1.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.2.p1.1.m1.1b"><ci id="S7.SS4.2.p1.1.m1.1.1.cmml" xref="S7.SS4.2.p1.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.2.p1.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.2.p1.1.m1.1d">italic_ε</annotation></semantics></math>, where <math alttext="\varepsilon=\poly(M)\sqrt{T}/L" class="ltx_Math" display="inline" id="S7.SS4.2.p1.2.m2.1"><semantics id="S7.SS4.2.p1.2.m2.1a"><mrow id="S7.SS4.2.p1.2.m2.1.2" xref="S7.SS4.2.p1.2.m2.1.2.cmml"><mi id="S7.SS4.2.p1.2.m2.1.2.2" xref="S7.SS4.2.p1.2.m2.1.2.2.cmml">ε</mi><mo id="S7.SS4.2.p1.2.m2.1.2.1" xref="S7.SS4.2.p1.2.m2.1.2.1.cmml">=</mo><mrow id="S7.SS4.2.p1.2.m2.1.2.3" xref="S7.SS4.2.p1.2.m2.1.2.3.cmml"><mrow id="S7.SS4.2.p1.2.m2.1.2.3.2" xref="S7.SS4.2.p1.2.m2.1.2.3.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S7.SS4.2.p1.2.m2.1.2.3.2.2" xref="S7.SS4.2.p1.2.m2.1.2.3.2.2b.cmml"><mtext id="S7.SS4.2.p1.2.m2.1.2.3.2.2a" xref="S7.SS4.2.p1.2.m2.1.2.3.2.2b.cmml">\poly</mtext></merror><mo id="S7.SS4.2.p1.2.m2.1.2.3.2.1" xref="S7.SS4.2.p1.2.m2.1.2.3.2.1.cmml">⁢</mo><mrow id="S7.SS4.2.p1.2.m2.1.2.3.2.3.2" xref="S7.SS4.2.p1.2.m2.1.2.3.2.cmml"><mo id="S7.SS4.2.p1.2.m2.1.2.3.2.3.2.1" stretchy="false" xref="S7.SS4.2.p1.2.m2.1.2.3.2.cmml">(</mo><mi id="S7.SS4.2.p1.2.m2.1.1" xref="S7.SS4.2.p1.2.m2.1.1.cmml">M</mi><mo id="S7.SS4.2.p1.2.m2.1.2.3.2.3.2.2" stretchy="false" xref="S7.SS4.2.p1.2.m2.1.2.3.2.cmml">)</mo></mrow><mo id="S7.SS4.2.p1.2.m2.1.2.3.2.1a" xref="S7.SS4.2.p1.2.m2.1.2.3.2.1.cmml">⁢</mo><msqrt id="S7.SS4.2.p1.2.m2.1.2.3.2.4" xref="S7.SS4.2.p1.2.m2.1.2.3.2.4.cmml"><mi id="S7.SS4.2.p1.2.m2.1.2.3.2.4.2" xref="S7.SS4.2.p1.2.m2.1.2.3.2.4.2.cmml">T</mi></msqrt></mrow><mo id="S7.SS4.2.p1.2.m2.1.2.3.1" xref="S7.SS4.2.p1.2.m2.1.2.3.1.cmml">/</mo><mi id="S7.SS4.2.p1.2.m2.1.2.3.3" xref="S7.SS4.2.p1.2.m2.1.2.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.2.p1.2.m2.1b"><apply id="S7.SS4.2.p1.2.m2.1.2.cmml" xref="S7.SS4.2.p1.2.m2.1.2"><eq id="S7.SS4.2.p1.2.m2.1.2.1.cmml" xref="S7.SS4.2.p1.2.m2.1.2.1"></eq><ci id="S7.SS4.2.p1.2.m2.1.2.2.cmml" xref="S7.SS4.2.p1.2.m2.1.2.2">𝜀</ci><apply id="S7.SS4.2.p1.2.m2.1.2.3.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3"><divide id="S7.SS4.2.p1.2.m2.1.2.3.1.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.1"></divide><apply id="S7.SS4.2.p1.2.m2.1.2.3.2.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2"><times id="S7.SS4.2.p1.2.m2.1.2.3.2.1.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2.1"></times><ci id="S7.SS4.2.p1.2.m2.1.2.3.2.2b.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2.2"><merror class="ltx_ERROR undefined undefined" id="S7.SS4.2.p1.2.m2.1.2.3.2.2.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2.2"><mtext id="S7.SS4.2.p1.2.m2.1.2.3.2.2a.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2.2">\poly</mtext></merror></ci><ci id="S7.SS4.2.p1.2.m2.1.1.cmml" xref="S7.SS4.2.p1.2.m2.1.1">𝑀</ci><apply id="S7.SS4.2.p1.2.m2.1.2.3.2.4.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2.4"><root id="S7.SS4.2.p1.2.m2.1.2.3.2.4a.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2.4"></root><ci id="S7.SS4.2.p1.2.m2.1.2.3.2.4.2.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.2.4.2">𝑇</ci></apply></apply><ci id="S7.SS4.2.p1.2.m2.1.2.3.3.cmml" xref="S7.SS4.2.p1.2.m2.1.2.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.2.p1.2.m2.1c">\varepsilon=\poly(M)\sqrt{T}/L</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.2.p1.2.m2.1d">italic_ε = ( italic_M ) square-root start_ARG italic_T end_ARG / italic_L</annotation></semantics></math>. (Notice that we cannot assume <math alttext="T=\poly(M)\cdot L" class="ltx_Math" display="inline" id="S7.SS4.2.p1.3.m3.1"><semantics id="S7.SS4.2.p1.3.m3.1a"><mrow id="S7.SS4.2.p1.3.m3.1.2" xref="S7.SS4.2.p1.3.m3.1.2.cmml"><mi id="S7.SS4.2.p1.3.m3.1.2.2" xref="S7.SS4.2.p1.3.m3.1.2.2.cmml">T</mi><mo id="S7.SS4.2.p1.3.m3.1.2.1" xref="S7.SS4.2.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S7.SS4.2.p1.3.m3.1.2.3" xref="S7.SS4.2.p1.3.m3.1.2.3.cmml"><mrow id="S7.SS4.2.p1.3.m3.1.2.3.2" xref="S7.SS4.2.p1.3.m3.1.2.3.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S7.SS4.2.p1.3.m3.1.2.3.2.2" xref="S7.SS4.2.p1.3.m3.1.2.3.2.2b.cmml"><mtext id="S7.SS4.2.p1.3.m3.1.2.3.2.2a" xref="S7.SS4.2.p1.3.m3.1.2.3.2.2b.cmml">\poly</mtext></merror><mo id="S7.SS4.2.p1.3.m3.1.2.3.2.1" xref="S7.SS4.2.p1.3.m3.1.2.3.2.1.cmml">⁢</mo><mrow id="S7.SS4.2.p1.3.m3.1.2.3.2.3.2" xref="S7.SS4.2.p1.3.m3.1.2.3.2.cmml"><mo id="S7.SS4.2.p1.3.m3.1.2.3.2.3.2.1" stretchy="false" xref="S7.SS4.2.p1.3.m3.1.2.3.2.cmml">(</mo><mi id="S7.SS4.2.p1.3.m3.1.1" xref="S7.SS4.2.p1.3.m3.1.1.cmml">M</mi><mo id="S7.SS4.2.p1.3.m3.1.2.3.2.3.2.2" rspace="0.055em" stretchy="false" xref="S7.SS4.2.p1.3.m3.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.2.p1.3.m3.1.2.3.1" rspace="0.222em" xref="S7.SS4.2.p1.3.m3.1.2.3.1.cmml">⋅</mo><mi id="S7.SS4.2.p1.3.m3.1.2.3.3" xref="S7.SS4.2.p1.3.m3.1.2.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.2.p1.3.m3.1b"><apply id="S7.SS4.2.p1.3.m3.1.2.cmml" xref="S7.SS4.2.p1.3.m3.1.2"><eq id="S7.SS4.2.p1.3.m3.1.2.1.cmml" xref="S7.SS4.2.p1.3.m3.1.2.1"></eq><ci id="S7.SS4.2.p1.3.m3.1.2.2.cmml" xref="S7.SS4.2.p1.3.m3.1.2.2">𝑇</ci><apply id="S7.SS4.2.p1.3.m3.1.2.3.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3"><ci id="S7.SS4.2.p1.3.m3.1.2.3.1.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3.1">⋅</ci><apply id="S7.SS4.2.p1.3.m3.1.2.3.2.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3.2"><times id="S7.SS4.2.p1.3.m3.1.2.3.2.1.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3.2.1"></times><ci id="S7.SS4.2.p1.3.m3.1.2.3.2.2b.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3.2.2"><merror class="ltx_ERROR undefined undefined" id="S7.SS4.2.p1.3.m3.1.2.3.2.2.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3.2.2"><mtext id="S7.SS4.2.p1.3.m3.1.2.3.2.2a.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3.2.2">\poly</mtext></merror></ci><ci id="S7.SS4.2.p1.3.m3.1.1.cmml" xref="S7.SS4.2.p1.3.m3.1.1">𝑀</ci></apply><ci id="S7.SS4.2.p1.3.m3.1.2.3.3.cmml" xref="S7.SS4.2.p1.3.m3.1.2.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.2.p1.3.m3.1c">T=\poly(M)\cdot L</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.2.p1.3.m3.1d">italic_T = ( italic_M ) ⋅ italic_L</annotation></semantics></math>, because <math alttext="T" class="ltx_Math" display="inline" id="S7.SS4.2.p1.4.m4.1"><semantics id="S7.SS4.2.p1.4.m4.1a"><mi id="S7.SS4.2.p1.4.m4.1.1" xref="S7.SS4.2.p1.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.2.p1.4.m4.1b"><ci id="S7.SS4.2.p1.4.m4.1.1.cmml" xref="S7.SS4.2.p1.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.2.p1.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.2.p1.4.m4.1d">italic_T</annotation></semantics></math> is the total number of rounds across <span class="ltx_text ltx_font_italic" id="S7.SS4.2.p1.4.1">both</span> stages of the algorithm.)</p> </div> <div class="ltx_para" id="S7.SS4.3.p2"> <p class="ltx_p" id="S7.SS4.3.p2.10">Since <math alttext="\tilde{U}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.1.m1.1"><semantics id="S7.SS4.3.p2.1.m1.1a"><mover accent="true" id="S7.SS4.3.p2.1.m1.1.1" xref="S7.SS4.3.p2.1.m1.1.1.cmml"><mi id="S7.SS4.3.p2.1.m1.1.1.2" xref="S7.SS4.3.p2.1.m1.1.1.2.cmml">U</mi><mo id="S7.SS4.3.p2.1.m1.1.1.1" xref="S7.SS4.3.p2.1.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.1.m1.1b"><apply id="S7.SS4.3.p2.1.m1.1.1.cmml" xref="S7.SS4.3.p2.1.m1.1.1"><ci id="S7.SS4.3.p2.1.m1.1.1.1.cmml" xref="S7.SS4.3.p2.1.m1.1.1.1">~</ci><ci id="S7.SS4.3.p2.1.m1.1.1.2.cmml" xref="S7.SS4.3.p2.1.m1.1.1.2">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.1.m1.1c">\tilde{U}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.1.m1.1d">over~ start_ARG italic_U end_ARG</annotation></semantics></math> and <math alttext="U" class="ltx_Math" display="inline" id="S7.SS4.3.p2.2.m2.1"><semantics id="S7.SS4.3.p2.2.m2.1a"><mi id="S7.SS4.3.p2.2.m2.1.1" xref="S7.SS4.3.p2.2.m2.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.2.m2.1b"><ci id="S7.SS4.3.p2.2.m2.1.1.cmml" xref="S7.SS4.3.p2.2.m2.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.2.m2.1c">U</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.2.m2.1d">italic_U</annotation></semantics></math> differ by only <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.SS4.3.p2.3.m3.1"><semantics id="S7.SS4.3.p2.3.m3.1a"><mi id="S7.SS4.3.p2.3.m3.1.1" xref="S7.SS4.3.p2.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.3.m3.1b"><ci id="S7.SS4.3.p2.3.m3.1.1.cmml" xref="S7.SS4.3.p2.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.3.m3.1d">italic_ε</annotation></semantics></math> (up to agent-independent terms), every CEP of <math alttext="\Gamma" class="ltx_Math" display="inline" id="S7.SS4.3.p2.4.m4.1"><semantics id="S7.SS4.3.p2.4.m4.1a"><mi id="S7.SS4.3.p2.4.m4.1.1" mathvariant="normal" xref="S7.SS4.3.p2.4.m4.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.4.m4.1b"><ci id="S7.SS4.3.p2.4.m4.1.1.cmml" xref="S7.SS4.3.p2.4.m4.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.4.m4.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.4.m4.1d">roman_Γ</annotation></semantics></math> is a <math alttext="2\varepsilon" class="ltx_Math" display="inline" id="S7.SS4.3.p2.5.m5.1"><semantics id="S7.SS4.3.p2.5.m5.1a"><mrow id="S7.SS4.3.p2.5.m5.1.1" xref="S7.SS4.3.p2.5.m5.1.1.cmml"><mn id="S7.SS4.3.p2.5.m5.1.1.2" xref="S7.SS4.3.p2.5.m5.1.1.2.cmml">2</mn><mo id="S7.SS4.3.p2.5.m5.1.1.1" xref="S7.SS4.3.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S7.SS4.3.p2.5.m5.1.1.3" xref="S7.SS4.3.p2.5.m5.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.5.m5.1b"><apply id="S7.SS4.3.p2.5.m5.1.1.cmml" xref="S7.SS4.3.p2.5.m5.1.1"><times id="S7.SS4.3.p2.5.m5.1.1.1.cmml" xref="S7.SS4.3.p2.5.m5.1.1.1"></times><cn id="S7.SS4.3.p2.5.m5.1.1.2.cmml" type="integer" xref="S7.SS4.3.p2.5.m5.1.1.2">2</cn><ci id="S7.SS4.3.p2.5.m5.1.1.3.cmml" xref="S7.SS4.3.p2.5.m5.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.5.m5.1c">2\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.5.m5.1d">2 italic_ε</annotation></semantics></math>-CEP of <math alttext="\tilde{\Gamma}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.6.m6.1"><semantics id="S7.SS4.3.p2.6.m6.1a"><mover accent="true" id="S7.SS4.3.p2.6.m6.1.1" xref="S7.SS4.3.p2.6.m6.1.1.cmml"><mi id="S7.SS4.3.p2.6.m6.1.1.2" mathvariant="normal" xref="S7.SS4.3.p2.6.m6.1.1.2.cmml">Γ</mi><mo id="S7.SS4.3.p2.6.m6.1.1.1" xref="S7.SS4.3.p2.6.m6.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.6.m6.1b"><apply id="S7.SS4.3.p2.6.m6.1.1.cmml" xref="S7.SS4.3.p2.6.m6.1.1"><ci id="S7.SS4.3.p2.6.m6.1.1.1.cmml" xref="S7.SS4.3.p2.6.m6.1.1.1">~</ci><ci id="S7.SS4.3.p2.6.m6.1.1.2.cmml" xref="S7.SS4.3.p2.6.m6.1.1.2">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.6.m6.1c">\tilde{\Gamma}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.6.m6.1d">over~ start_ARG roman_Γ end_ARG</annotation></semantics></math>. The payment function <math alttext="P_{i}^{t}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.7.m7.1"><semantics id="S7.SS4.3.p2.7.m7.1a"><msubsup id="S7.SS4.3.p2.7.m7.1.1" xref="S7.SS4.3.p2.7.m7.1.1.cmml"><mi id="S7.SS4.3.p2.7.m7.1.1.2.2" xref="S7.SS4.3.p2.7.m7.1.1.2.2.cmml">P</mi><mi id="S7.SS4.3.p2.7.m7.1.1.2.3" xref="S7.SS4.3.p2.7.m7.1.1.2.3.cmml">i</mi><mi id="S7.SS4.3.p2.7.m7.1.1.3" xref="S7.SS4.3.p2.7.m7.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.7.m7.1b"><apply id="S7.SS4.3.p2.7.m7.1.1.cmml" xref="S7.SS4.3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S7.SS4.3.p2.7.m7.1.1.1.cmml" xref="S7.SS4.3.p2.7.m7.1.1">superscript</csymbol><apply id="S7.SS4.3.p2.7.m7.1.1.2.cmml" xref="S7.SS4.3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S7.SS4.3.p2.7.m7.1.1.2.1.cmml" xref="S7.SS4.3.p2.7.m7.1.1">subscript</csymbol><ci id="S7.SS4.3.p2.7.m7.1.1.2.2.cmml" xref="S7.SS4.3.p2.7.m7.1.1.2.2">𝑃</ci><ci id="S7.SS4.3.p2.7.m7.1.1.2.3.cmml" xref="S7.SS4.3.p2.7.m7.1.1.2.3">𝑖</ci></apply><ci id="S7.SS4.3.p2.7.m7.1.1.3.cmml" xref="S7.SS4.3.p2.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.7.m7.1c">P_{i}^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.7.m7.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> for the steering stage then ensures that, when given signal <math alttext="s_{i}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.8.m8.1"><semantics id="S7.SS4.3.p2.8.m8.1a"><msub id="S7.SS4.3.p2.8.m8.1.1" xref="S7.SS4.3.p2.8.m8.1.1.cmml"><mi id="S7.SS4.3.p2.8.m8.1.1.2" xref="S7.SS4.3.p2.8.m8.1.1.2.cmml">s</mi><mi id="S7.SS4.3.p2.8.m8.1.1.3" xref="S7.SS4.3.p2.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.8.m8.1b"><apply id="S7.SS4.3.p2.8.m8.1.1.cmml" xref="S7.SS4.3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S7.SS4.3.p2.8.m8.1.1.1.cmml" xref="S7.SS4.3.p2.8.m8.1.1">subscript</csymbol><ci id="S7.SS4.3.p2.8.m8.1.1.2.cmml" xref="S7.SS4.3.p2.8.m8.1.1.2">𝑠</ci><ci id="S7.SS4.3.p2.8.m8.1.1.3.cmml" xref="S7.SS4.3.p2.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.8.m8.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.8.m8.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, it is <span class="ltx_text ltx_font_italic" id="S7.SS4.3.p2.10.1">dominant</span> for agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS4.3.p2.9.m9.1"><semantics id="S7.SS4.3.p2.9.m9.1a"><mi id="S7.SS4.3.p2.9.m9.1.1" xref="S7.SS4.3.p2.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.9.m9.1b"><ci id="S7.SS4.3.p2.9.m9.1.1.cmml" xref="S7.SS4.3.p2.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.9.m9.1d">italic_i</annotation></semantics></math> to play <math alttext="a_{i}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.10.m10.1"><semantics id="S7.SS4.3.p2.10.m10.1a"><msub id="S7.SS4.3.p2.10.m10.1.1" xref="S7.SS4.3.p2.10.m10.1.1.cmml"><mi id="S7.SS4.3.p2.10.m10.1.1.2" xref="S7.SS4.3.p2.10.m10.1.1.2.cmml">a</mi><mi id="S7.SS4.3.p2.10.m10.1.1.3" xref="S7.SS4.3.p2.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.10.m10.1b"><apply id="S7.SS4.3.p2.10.m10.1.1.cmml" xref="S7.SS4.3.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S7.SS4.3.p2.10.m10.1.1.1.cmml" xref="S7.SS4.3.p2.10.m10.1.1">subscript</csymbol><ci id="S7.SS4.3.p2.10.m10.1.1.2.cmml" xref="S7.SS4.3.p2.10.m10.1.1.2">𝑎</ci><ci id="S7.SS4.3.p2.10.m10.1.1.3.cmml" xref="S7.SS4.3.p2.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.10.m10.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.10.m10.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Formally, regardless of how other agents act, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx14"> <tbody id="S7.E19"><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 U_{i}^{t}(s,s_{i},a_{-i})-U_{i}^{t}(s,a)\geq\rho" class="ltx_Math" display="inline" id="S7.E19.m1.5"><semantics id="S7.E19.m1.5a"><mrow id="S7.E19.m1.5.5" xref="S7.E19.m1.5.5.cmml"><mrow id="S7.E19.m1.5.5.2" xref="S7.E19.m1.5.5.2.cmml"><mrow id="S7.E19.m1.5.5.2.2" xref="S7.E19.m1.5.5.2.2.cmml"><msubsup id="S7.E19.m1.5.5.2.2.4" xref="S7.E19.m1.5.5.2.2.4.cmml"><mi id="S7.E19.m1.5.5.2.2.4.2.2" xref="S7.E19.m1.5.5.2.2.4.2.2.cmml">U</mi><mi id="S7.E19.m1.5.5.2.2.4.2.3" xref="S7.E19.m1.5.5.2.2.4.2.3.cmml">i</mi><mi id="S7.E19.m1.5.5.2.2.4.3" xref="S7.E19.m1.5.5.2.2.4.3.cmml">t</mi></msubsup><mo id="S7.E19.m1.5.5.2.2.3" xref="S7.E19.m1.5.5.2.2.3.cmml">⁢</mo><mrow id="S7.E19.m1.5.5.2.2.2.2" xref="S7.E19.m1.5.5.2.2.2.3.cmml"><mo id="S7.E19.m1.5.5.2.2.2.2.3" stretchy="false" xref="S7.E19.m1.5.5.2.2.2.3.cmml">(</mo><mi id="S7.E19.m1.1.1" xref="S7.E19.m1.1.1.cmml">s</mi><mo id="S7.E19.m1.5.5.2.2.2.2.4" xref="S7.E19.m1.5.5.2.2.2.3.cmml">,</mo><msub id="S7.E19.m1.4.4.1.1.1.1.1" xref="S7.E19.m1.4.4.1.1.1.1.1.cmml"><mi id="S7.E19.m1.4.4.1.1.1.1.1.2" xref="S7.E19.m1.4.4.1.1.1.1.1.2.cmml">s</mi><mi id="S7.E19.m1.4.4.1.1.1.1.1.3" xref="S7.E19.m1.4.4.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S7.E19.m1.5.5.2.2.2.2.5" xref="S7.E19.m1.5.5.2.2.2.3.cmml">,</mo><msub id="S7.E19.m1.5.5.2.2.2.2.2" xref="S7.E19.m1.5.5.2.2.2.2.2.cmml"><mi id="S7.E19.m1.5.5.2.2.2.2.2.2" xref="S7.E19.m1.5.5.2.2.2.2.2.2.cmml">a</mi><mrow id="S7.E19.m1.5.5.2.2.2.2.2.3" xref="S7.E19.m1.5.5.2.2.2.2.2.3.cmml"><mo id="S7.E19.m1.5.5.2.2.2.2.2.3a" xref="S7.E19.m1.5.5.2.2.2.2.2.3.cmml">−</mo><mi id="S7.E19.m1.5.5.2.2.2.2.2.3.2" xref="S7.E19.m1.5.5.2.2.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="S7.E19.m1.5.5.2.2.2.2.6" stretchy="false" xref="S7.E19.m1.5.5.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S7.E19.m1.5.5.2.3" xref="S7.E19.m1.5.5.2.3.cmml">−</mo><mrow id="S7.E19.m1.5.5.2.4" xref="S7.E19.m1.5.5.2.4.cmml"><msubsup id="S7.E19.m1.5.5.2.4.2" xref="S7.E19.m1.5.5.2.4.2.cmml"><mi id="S7.E19.m1.5.5.2.4.2.2.2" xref="S7.E19.m1.5.5.2.4.2.2.2.cmml">U</mi><mi id="S7.E19.m1.5.5.2.4.2.2.3" xref="S7.E19.m1.5.5.2.4.2.2.3.cmml">i</mi><mi id="S7.E19.m1.5.5.2.4.2.3" xref="S7.E19.m1.5.5.2.4.2.3.cmml">t</mi></msubsup><mo id="S7.E19.m1.5.5.2.4.1" xref="S7.E19.m1.5.5.2.4.1.cmml">⁢</mo><mrow id="S7.E19.m1.5.5.2.4.3.2" xref="S7.E19.m1.5.5.2.4.3.1.cmml"><mo id="S7.E19.m1.5.5.2.4.3.2.1" stretchy="false" xref="S7.E19.m1.5.5.2.4.3.1.cmml">(</mo><mi id="S7.E19.m1.2.2" xref="S7.E19.m1.2.2.cmml">s</mi><mo id="S7.E19.m1.5.5.2.4.3.2.2" xref="S7.E19.m1.5.5.2.4.3.1.cmml">,</mo><mi id="S7.E19.m1.3.3" xref="S7.E19.m1.3.3.cmml">a</mi><mo id="S7.E19.m1.5.5.2.4.3.2.3" stretchy="false" xref="S7.E19.m1.5.5.2.4.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S7.E19.m1.5.5.3" xref="S7.E19.m1.5.5.3.cmml">≥</mo><mi id="S7.E19.m1.5.5.4" xref="S7.E19.m1.5.5.4.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.E19.m1.5b"><apply id="S7.E19.m1.5.5.cmml" xref="S7.E19.m1.5.5"><geq id="S7.E19.m1.5.5.3.cmml" xref="S7.E19.m1.5.5.3"></geq><apply id="S7.E19.m1.5.5.2.cmml" xref="S7.E19.m1.5.5.2"><minus id="S7.E19.m1.5.5.2.3.cmml" xref="S7.E19.m1.5.5.2.3"></minus><apply id="S7.E19.m1.5.5.2.2.cmml" xref="S7.E19.m1.5.5.2.2"><times id="S7.E19.m1.5.5.2.2.3.cmml" xref="S7.E19.m1.5.5.2.2.3"></times><apply id="S7.E19.m1.5.5.2.2.4.cmml" xref="S7.E19.m1.5.5.2.2.4"><csymbol cd="ambiguous" id="S7.E19.m1.5.5.2.2.4.1.cmml" xref="S7.E19.m1.5.5.2.2.4">superscript</csymbol><apply id="S7.E19.m1.5.5.2.2.4.2.cmml" xref="S7.E19.m1.5.5.2.2.4"><csymbol cd="ambiguous" id="S7.E19.m1.5.5.2.2.4.2.1.cmml" 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xref="S7.E19.m1.5.5.2.2.2.2.2.2">𝑎</ci><apply id="S7.E19.m1.5.5.2.2.2.2.2.3.cmml" xref="S7.E19.m1.5.5.2.2.2.2.2.3"><minus id="S7.E19.m1.5.5.2.2.2.2.2.3.1.cmml" xref="S7.E19.m1.5.5.2.2.2.2.2.3"></minus><ci id="S7.E19.m1.5.5.2.2.2.2.2.3.2.cmml" xref="S7.E19.m1.5.5.2.2.2.2.2.3.2">𝑖</ci></apply></apply></vector></apply><apply id="S7.E19.m1.5.5.2.4.cmml" xref="S7.E19.m1.5.5.2.4"><times id="S7.E19.m1.5.5.2.4.1.cmml" xref="S7.E19.m1.5.5.2.4.1"></times><apply id="S7.E19.m1.5.5.2.4.2.cmml" xref="S7.E19.m1.5.5.2.4.2"><csymbol cd="ambiguous" id="S7.E19.m1.5.5.2.4.2.1.cmml" xref="S7.E19.m1.5.5.2.4.2">superscript</csymbol><apply id="S7.E19.m1.5.5.2.4.2.2.cmml" xref="S7.E19.m1.5.5.2.4.2"><csymbol cd="ambiguous" id="S7.E19.m1.5.5.2.4.2.2.1.cmml" xref="S7.E19.m1.5.5.2.4.2">subscript</csymbol><ci id="S7.E19.m1.5.5.2.4.2.2.2.cmml" xref="S7.E19.m1.5.5.2.4.2.2.2">𝑈</ci><ci id="S7.E19.m1.5.5.2.4.2.2.3.cmml" xref="S7.E19.m1.5.5.2.4.2.2.3">𝑖</ci></apply><ci id="S7.E19.m1.5.5.2.4.2.3.cmml" xref="S7.E19.m1.5.5.2.4.2.3">𝑡</ci></apply><interval closure="open" id="S7.E19.m1.5.5.2.4.3.1.cmml" xref="S7.E19.m1.5.5.2.4.3.2"><ci id="S7.E19.m1.2.2.cmml" xref="S7.E19.m1.2.2">𝑠</ci><ci id="S7.E19.m1.3.3.cmml" xref="S7.E19.m1.3.3">𝑎</ci></interval></apply></apply><ci id="S7.E19.m1.5.5.4.cmml" xref="S7.E19.m1.5.5.4">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E19.m1.5c">\displaystyle U_{i}^{t}(s,s_{i},a_{-i})-U_{i}^{t}(s,a)\geq\rho</annotation><annotation encoding="application/x-llamapun" id="S7.E19.m1.5d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_s , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) - italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_s , italic_a ) ≥ italic_ρ</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">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S7.SS4.3.p2.21">for every <math alttext="s\in S" class="ltx_Math" display="inline" id="S7.SS4.3.p2.11.m1.1"><semantics id="S7.SS4.3.p2.11.m1.1a"><mrow id="S7.SS4.3.p2.11.m1.1.1" xref="S7.SS4.3.p2.11.m1.1.1.cmml"><mi id="S7.SS4.3.p2.11.m1.1.1.2" xref="S7.SS4.3.p2.11.m1.1.1.2.cmml">s</mi><mo id="S7.SS4.3.p2.11.m1.1.1.1" xref="S7.SS4.3.p2.11.m1.1.1.1.cmml">∈</mo><mi id="S7.SS4.3.p2.11.m1.1.1.3" xref="S7.SS4.3.p2.11.m1.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.11.m1.1b"><apply id="S7.SS4.3.p2.11.m1.1.1.cmml" xref="S7.SS4.3.p2.11.m1.1.1"><in id="S7.SS4.3.p2.11.m1.1.1.1.cmml" xref="S7.SS4.3.p2.11.m1.1.1.1"></in><ci id="S7.SS4.3.p2.11.m1.1.1.2.cmml" xref="S7.SS4.3.p2.11.m1.1.1.2">𝑠</ci><ci id="S7.SS4.3.p2.11.m1.1.1.3.cmml" xref="S7.SS4.3.p2.11.m1.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.11.m1.1c">s\in S</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.11.m1.1d">italic_s ∈ italic_S</annotation></semantics></math> and <math alttext="a\in A" class="ltx_Math" display="inline" id="S7.SS4.3.p2.12.m2.1"><semantics id="S7.SS4.3.p2.12.m2.1a"><mrow id="S7.SS4.3.p2.12.m2.1.1" xref="S7.SS4.3.p2.12.m2.1.1.cmml"><mi id="S7.SS4.3.p2.12.m2.1.1.2" xref="S7.SS4.3.p2.12.m2.1.1.2.cmml">a</mi><mo id="S7.SS4.3.p2.12.m2.1.1.1" xref="S7.SS4.3.p2.12.m2.1.1.1.cmml">∈</mo><mi id="S7.SS4.3.p2.12.m2.1.1.3" xref="S7.SS4.3.p2.12.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.12.m2.1b"><apply id="S7.SS4.3.p2.12.m2.1.1.cmml" xref="S7.SS4.3.p2.12.m2.1.1"><in id="S7.SS4.3.p2.12.m2.1.1.1.cmml" xref="S7.SS4.3.p2.12.m2.1.1.1"></in><ci id="S7.SS4.3.p2.12.m2.1.1.2.cmml" xref="S7.SS4.3.p2.12.m2.1.1.2">𝑎</ci><ci id="S7.SS4.3.p2.12.m2.1.1.3.cmml" xref="S7.SS4.3.p2.12.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.12.m2.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.12.m2.1d">italic_a ∈ italic_A</annotation></semantics></math> with <math alttext="a_{i}\neq s_{i}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.13.m3.1"><semantics id="S7.SS4.3.p2.13.m3.1a"><mrow id="S7.SS4.3.p2.13.m3.1.1" xref="S7.SS4.3.p2.13.m3.1.1.cmml"><msub id="S7.SS4.3.p2.13.m3.1.1.2" xref="S7.SS4.3.p2.13.m3.1.1.2.cmml"><mi id="S7.SS4.3.p2.13.m3.1.1.2.2" xref="S7.SS4.3.p2.13.m3.1.1.2.2.cmml">a</mi><mi id="S7.SS4.3.p2.13.m3.1.1.2.3" xref="S7.SS4.3.p2.13.m3.1.1.2.3.cmml">i</mi></msub><mo id="S7.SS4.3.p2.13.m3.1.1.1" xref="S7.SS4.3.p2.13.m3.1.1.1.cmml">≠</mo><msub id="S7.SS4.3.p2.13.m3.1.1.3" xref="S7.SS4.3.p2.13.m3.1.1.3.cmml"><mi id="S7.SS4.3.p2.13.m3.1.1.3.2" xref="S7.SS4.3.p2.13.m3.1.1.3.2.cmml">s</mi><mi id="S7.SS4.3.p2.13.m3.1.1.3.3" xref="S7.SS4.3.p2.13.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.13.m3.1b"><apply id="S7.SS4.3.p2.13.m3.1.1.cmml" xref="S7.SS4.3.p2.13.m3.1.1"><neq id="S7.SS4.3.p2.13.m3.1.1.1.cmml" xref="S7.SS4.3.p2.13.m3.1.1.1"></neq><apply id="S7.SS4.3.p2.13.m3.1.1.2.cmml" xref="S7.SS4.3.p2.13.m3.1.1.2"><csymbol cd="ambiguous" id="S7.SS4.3.p2.13.m3.1.1.2.1.cmml" xref="S7.SS4.3.p2.13.m3.1.1.2">subscript</csymbol><ci id="S7.SS4.3.p2.13.m3.1.1.2.2.cmml" xref="S7.SS4.3.p2.13.m3.1.1.2.2">𝑎</ci><ci id="S7.SS4.3.p2.13.m3.1.1.2.3.cmml" xref="S7.SS4.3.p2.13.m3.1.1.2.3">𝑖</ci></apply><apply id="S7.SS4.3.p2.13.m3.1.1.3.cmml" xref="S7.SS4.3.p2.13.m3.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.3.p2.13.m3.1.1.3.1.cmml" xref="S7.SS4.3.p2.13.m3.1.1.3">subscript</csymbol><ci id="S7.SS4.3.p2.13.m3.1.1.3.2.cmml" xref="S7.SS4.3.p2.13.m3.1.1.3.2">𝑠</ci><ci id="S7.SS4.3.p2.13.m3.1.1.3.3.cmml" xref="S7.SS4.3.p2.13.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.13.m3.1c">a_{i}\neq s_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.13.m3.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Further, from the analysis of <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S5.Thmtheorem4" title="Theorem 5.4. ‣ 5.2 The multi-agent case ‣ 5 Learning the Utility in the No-Regret Model ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.4</span></a>, agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS4.3.p2.14.m4.1"><semantics id="S7.SS4.3.p2.14.m4.1a"><mi id="S7.SS4.3.p2.14.m4.1.1" xref="S7.SS4.3.p2.14.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.14.m4.1b"><ci id="S7.SS4.3.p2.14.m4.1.1.cmml" xref="S7.SS4.3.p2.14.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.14.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.14.m4.1d">italic_i</annotation></semantics></math>’s regret against following signals <math alttext="s_{i}\neq\bot" class="ltx_Math" display="inline" id="S7.SS4.3.p2.15.m5.1"><semantics id="S7.SS4.3.p2.15.m5.1a"><mrow id="S7.SS4.3.p2.15.m5.1.1" xref="S7.SS4.3.p2.15.m5.1.1.cmml"><msub id="S7.SS4.3.p2.15.m5.1.1.2" xref="S7.SS4.3.p2.15.m5.1.1.2.cmml"><mi id="S7.SS4.3.p2.15.m5.1.1.2.2" xref="S7.SS4.3.p2.15.m5.1.1.2.2.cmml">s</mi><mi id="S7.SS4.3.p2.15.m5.1.1.2.3" xref="S7.SS4.3.p2.15.m5.1.1.2.3.cmml">i</mi></msub><mo id="S7.SS4.3.p2.15.m5.1.1.1" rspace="0em" xref="S7.SS4.3.p2.15.m5.1.1.1.cmml">≠</mo><mo id="S7.SS4.3.p2.15.m5.1.1.3" lspace="0em" xref="S7.SS4.3.p2.15.m5.1.1.3.cmml">⊥</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.15.m5.1b"><apply id="S7.SS4.3.p2.15.m5.1.1.cmml" xref="S7.SS4.3.p2.15.m5.1.1"><neq id="S7.SS4.3.p2.15.m5.1.1.1.cmml" xref="S7.SS4.3.p2.15.m5.1.1.1"></neq><apply id="S7.SS4.3.p2.15.m5.1.1.2.cmml" xref="S7.SS4.3.p2.15.m5.1.1.2"><csymbol cd="ambiguous" id="S7.SS4.3.p2.15.m5.1.1.2.1.cmml" xref="S7.SS4.3.p2.15.m5.1.1.2">subscript</csymbol><ci id="S7.SS4.3.p2.15.m5.1.1.2.2.cmml" xref="S7.SS4.3.p2.15.m5.1.1.2.2">𝑠</ci><ci id="S7.SS4.3.p2.15.m5.1.1.2.3.cmml" xref="S7.SS4.3.p2.15.m5.1.1.2.3">𝑖</ci></apply><csymbol cd="latexml" id="S7.SS4.3.p2.15.m5.1.1.3.cmml" xref="S7.SS4.3.p2.15.m5.1.1.3">bottom</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.15.m5.1c">s_{i}\neq\bot</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.15.m5.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ⊥</annotation></semantics></math> is always nonnegative. Therefore, by agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS4.3.p2.16.m6.1"><semantics id="S7.SS4.3.p2.16.m6.1a"><mi id="S7.SS4.3.p2.16.m6.1.1" xref="S7.SS4.3.p2.16.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.16.m6.1b"><ci id="S7.SS4.3.p2.16.m6.1.1.cmml" xref="S7.SS4.3.p2.16.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.16.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.16.m6.1d">italic_i</annotation></semantics></math>’s regret bound, there are at most <math alttext="C\sqrt{T}/\rho" class="ltx_Math" display="inline" id="S7.SS4.3.p2.17.m7.1"><semantics id="S7.SS4.3.p2.17.m7.1a"><mrow id="S7.SS4.3.p2.17.m7.1.1" xref="S7.SS4.3.p2.17.m7.1.1.cmml"><mrow id="S7.SS4.3.p2.17.m7.1.1.2" xref="S7.SS4.3.p2.17.m7.1.1.2.cmml"><mi id="S7.SS4.3.p2.17.m7.1.1.2.2" xref="S7.SS4.3.p2.17.m7.1.1.2.2.cmml">C</mi><mo id="S7.SS4.3.p2.17.m7.1.1.2.1" xref="S7.SS4.3.p2.17.m7.1.1.2.1.cmml">⁢</mo><msqrt id="S7.SS4.3.p2.17.m7.1.1.2.3" xref="S7.SS4.3.p2.17.m7.1.1.2.3.cmml"><mi id="S7.SS4.3.p2.17.m7.1.1.2.3.2" xref="S7.SS4.3.p2.17.m7.1.1.2.3.2.cmml">T</mi></msqrt></mrow><mo id="S7.SS4.3.p2.17.m7.1.1.1" xref="S7.SS4.3.p2.17.m7.1.1.1.cmml">/</mo><mi id="S7.SS4.3.p2.17.m7.1.1.3" xref="S7.SS4.3.p2.17.m7.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.17.m7.1b"><apply id="S7.SS4.3.p2.17.m7.1.1.cmml" xref="S7.SS4.3.p2.17.m7.1.1"><divide id="S7.SS4.3.p2.17.m7.1.1.1.cmml" xref="S7.SS4.3.p2.17.m7.1.1.1"></divide><apply id="S7.SS4.3.p2.17.m7.1.1.2.cmml" xref="S7.SS4.3.p2.17.m7.1.1.2"><times id="S7.SS4.3.p2.17.m7.1.1.2.1.cmml" xref="S7.SS4.3.p2.17.m7.1.1.2.1"></times><ci id="S7.SS4.3.p2.17.m7.1.1.2.2.cmml" xref="S7.SS4.3.p2.17.m7.1.1.2.2">𝐶</ci><apply id="S7.SS4.3.p2.17.m7.1.1.2.3.cmml" xref="S7.SS4.3.p2.17.m7.1.1.2.3"><root id="S7.SS4.3.p2.17.m7.1.1.2.3a.cmml" xref="S7.SS4.3.p2.17.m7.1.1.2.3"></root><ci id="S7.SS4.3.p2.17.m7.1.1.2.3.2.cmml" xref="S7.SS4.3.p2.17.m7.1.1.2.3.2">𝑇</ci></apply></apply><ci id="S7.SS4.3.p2.17.m7.1.1.3.cmml" xref="S7.SS4.3.p2.17.m7.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.17.m7.1c">C\sqrt{T}/\rho</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.17.m7.1d">italic_C square-root start_ARG italic_T end_ARG / italic_ρ</annotation></semantics></math> rounds on which agent <math alttext="i" class="ltx_Math" display="inline" id="S7.SS4.3.p2.18.m8.1"><semantics id="S7.SS4.3.p2.18.m8.1a"><mi id="S7.SS4.3.p2.18.m8.1.1" xref="S7.SS4.3.p2.18.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.18.m8.1b"><ci id="S7.SS4.3.p2.18.m8.1.1.cmml" xref="S7.SS4.3.p2.18.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.18.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.18.m8.1d">italic_i</annotation></semantics></math> fails to obey recommendation <math alttext="s_{i}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.19.m9.1"><semantics id="S7.SS4.3.p2.19.m9.1a"><msub id="S7.SS4.3.p2.19.m9.1.1" xref="S7.SS4.3.p2.19.m9.1.1.cmml"><mi id="S7.SS4.3.p2.19.m9.1.1.2" xref="S7.SS4.3.p2.19.m9.1.1.2.cmml">s</mi><mi id="S7.SS4.3.p2.19.m9.1.1.3" xref="S7.SS4.3.p2.19.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.19.m9.1b"><apply id="S7.SS4.3.p2.19.m9.1.1.cmml" xref="S7.SS4.3.p2.19.m9.1.1"><csymbol cd="ambiguous" id="S7.SS4.3.p2.19.m9.1.1.1.cmml" xref="S7.SS4.3.p2.19.m9.1.1">subscript</csymbol><ci id="S7.SS4.3.p2.19.m9.1.1.2.cmml" xref="S7.SS4.3.p2.19.m9.1.1.2">𝑠</ci><ci id="S7.SS4.3.p2.19.m9.1.1.3.cmml" xref="S7.SS4.3.p2.19.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.19.m9.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.19.m9.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in the steering stage. By a union bound, there are therefore <math alttext="mnC\sqrt{T}/\rho" class="ltx_Math" display="inline" id="S7.SS4.3.p2.20.m10.1"><semantics id="S7.SS4.3.p2.20.m10.1a"><mrow id="S7.SS4.3.p2.20.m10.1.1" xref="S7.SS4.3.p2.20.m10.1.1.cmml"><mrow id="S7.SS4.3.p2.20.m10.1.1.2" xref="S7.SS4.3.p2.20.m10.1.1.2.cmml"><mi id="S7.SS4.3.p2.20.m10.1.1.2.2" xref="S7.SS4.3.p2.20.m10.1.1.2.2.cmml">m</mi><mo id="S7.SS4.3.p2.20.m10.1.1.2.1" xref="S7.SS4.3.p2.20.m10.1.1.2.1.cmml">⁢</mo><mi id="S7.SS4.3.p2.20.m10.1.1.2.3" xref="S7.SS4.3.p2.20.m10.1.1.2.3.cmml">n</mi><mo id="S7.SS4.3.p2.20.m10.1.1.2.1a" xref="S7.SS4.3.p2.20.m10.1.1.2.1.cmml">⁢</mo><mi id="S7.SS4.3.p2.20.m10.1.1.2.4" xref="S7.SS4.3.p2.20.m10.1.1.2.4.cmml">C</mi><mo id="S7.SS4.3.p2.20.m10.1.1.2.1b" xref="S7.SS4.3.p2.20.m10.1.1.2.1.cmml">⁢</mo><msqrt id="S7.SS4.3.p2.20.m10.1.1.2.5" xref="S7.SS4.3.p2.20.m10.1.1.2.5.cmml"><mi id="S7.SS4.3.p2.20.m10.1.1.2.5.2" xref="S7.SS4.3.p2.20.m10.1.1.2.5.2.cmml">T</mi></msqrt></mrow><mo id="S7.SS4.3.p2.20.m10.1.1.1" xref="S7.SS4.3.p2.20.m10.1.1.1.cmml">/</mo><mi id="S7.SS4.3.p2.20.m10.1.1.3" xref="S7.SS4.3.p2.20.m10.1.1.3.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.20.m10.1b"><apply id="S7.SS4.3.p2.20.m10.1.1.cmml" xref="S7.SS4.3.p2.20.m10.1.1"><divide id="S7.SS4.3.p2.20.m10.1.1.1.cmml" xref="S7.SS4.3.p2.20.m10.1.1.1"></divide><apply id="S7.SS4.3.p2.20.m10.1.1.2.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2"><times id="S7.SS4.3.p2.20.m10.1.1.2.1.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2.1"></times><ci id="S7.SS4.3.p2.20.m10.1.1.2.2.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2.2">𝑚</ci><ci id="S7.SS4.3.p2.20.m10.1.1.2.3.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2.3">𝑛</ci><ci id="S7.SS4.3.p2.20.m10.1.1.2.4.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2.4">𝐶</ci><apply id="S7.SS4.3.p2.20.m10.1.1.2.5.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2.5"><root id="S7.SS4.3.p2.20.m10.1.1.2.5a.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2.5"></root><ci id="S7.SS4.3.p2.20.m10.1.1.2.5.2.cmml" xref="S7.SS4.3.p2.20.m10.1.1.2.5.2">𝑇</ci></apply></apply><ci id="S7.SS4.3.p2.20.m10.1.1.3.cmml" xref="S7.SS4.3.p2.20.m10.1.1.3">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.20.m10.1c">mnC\sqrt{T}/\rho</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.20.m10.1d">italic_m italic_n italic_C square-root start_ARG italic_T end_ARG / italic_ρ</annotation></semantics></math> rounds in the steering stage on which <math alttext="a^{t}\neq s^{t}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.21.m11.1"><semantics id="S7.SS4.3.p2.21.m11.1a"><mrow id="S7.SS4.3.p2.21.m11.1.1" xref="S7.SS4.3.p2.21.m11.1.1.cmml"><msup id="S7.SS4.3.p2.21.m11.1.1.2" xref="S7.SS4.3.p2.21.m11.1.1.2.cmml"><mi id="S7.SS4.3.p2.21.m11.1.1.2.2" xref="S7.SS4.3.p2.21.m11.1.1.2.2.cmml">a</mi><mi id="S7.SS4.3.p2.21.m11.1.1.2.3" xref="S7.SS4.3.p2.21.m11.1.1.2.3.cmml">t</mi></msup><mo id="S7.SS4.3.p2.21.m11.1.1.1" xref="S7.SS4.3.p2.21.m11.1.1.1.cmml">≠</mo><msup id="S7.SS4.3.p2.21.m11.1.1.3" xref="S7.SS4.3.p2.21.m11.1.1.3.cmml"><mi id="S7.SS4.3.p2.21.m11.1.1.3.2" xref="S7.SS4.3.p2.21.m11.1.1.3.2.cmml">s</mi><mi id="S7.SS4.3.p2.21.m11.1.1.3.3" xref="S7.SS4.3.p2.21.m11.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.21.m11.1b"><apply id="S7.SS4.3.p2.21.m11.1.1.cmml" xref="S7.SS4.3.p2.21.m11.1.1"><neq id="S7.SS4.3.p2.21.m11.1.1.1.cmml" xref="S7.SS4.3.p2.21.m11.1.1.1"></neq><apply id="S7.SS4.3.p2.21.m11.1.1.2.cmml" xref="S7.SS4.3.p2.21.m11.1.1.2"><csymbol cd="ambiguous" id="S7.SS4.3.p2.21.m11.1.1.2.1.cmml" xref="S7.SS4.3.p2.21.m11.1.1.2">superscript</csymbol><ci id="S7.SS4.3.p2.21.m11.1.1.2.2.cmml" xref="S7.SS4.3.p2.21.m11.1.1.2.2">𝑎</ci><ci id="S7.SS4.3.p2.21.m11.1.1.2.3.cmml" xref="S7.SS4.3.p2.21.m11.1.1.2.3">𝑡</ci></apply><apply id="S7.SS4.3.p2.21.m11.1.1.3.cmml" xref="S7.SS4.3.p2.21.m11.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.3.p2.21.m11.1.1.3.1.cmml" xref="S7.SS4.3.p2.21.m11.1.1.3">superscript</csymbol><ci id="S7.SS4.3.p2.21.m11.1.1.3.2.cmml" xref="S7.SS4.3.p2.21.m11.1.1.3.2">𝑠</ci><ci id="S7.SS4.3.p2.21.m11.1.1.3.3.cmml" xref="S7.SS4.3.p2.21.m11.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.21.m11.1c">a^{t}\neq s^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.21.m11.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ≠ italic_s start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. Thus, the principal’s suboptimality is bounded by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx15"> <tbody id="S7.E20"><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^{*}-F(T)\leq\underbrace{\frac{(2n+1)nmL}{T}}_{(1)}{}+{}% \underbrace{n(2\varepsilon+\rho)}_{(2)}{}+{}\underbrace{\frac{(2n+1)mnC}{\rho% \sqrt{T}}}_{(3)}\leq\poly(M)\cdot\quantity(\frac{L}{T}+\frac{\sqrt{T}}{L}+\rho% +\frac{1}{\rho\sqrt{T}})" class="ltx_Math" display="inline" id="S7.E20.m1.9"><semantics id="S7.E20.m1.9a"><mrow id="S7.E20.m1.9.10" xref="S7.E20.m1.9.10.cmml"><mrow id="S7.E20.m1.9.10.2" xref="S7.E20.m1.9.10.2.cmml"><msup id="S7.E20.m1.9.10.2.2" xref="S7.E20.m1.9.10.2.2.cmml"><mi id="S7.E20.m1.9.10.2.2.2" xref="S7.E20.m1.9.10.2.2.2.cmml">F</mi><mo id="S7.E20.m1.9.10.2.2.3" xref="S7.E20.m1.9.10.2.2.3.cmml">∗</mo></msup><mo id="S7.E20.m1.9.10.2.1" xref="S7.E20.m1.9.10.2.1.cmml">−</mo><mrow id="S7.E20.m1.9.10.2.3" xref="S7.E20.m1.9.10.2.3.cmml"><mi id="S7.E20.m1.9.10.2.3.2" xref="S7.E20.m1.9.10.2.3.2.cmml">F</mi><mo id="S7.E20.m1.9.10.2.3.1" xref="S7.E20.m1.9.10.2.3.1.cmml">⁢</mo><mrow id="S7.E20.m1.9.10.2.3.3.2" xref="S7.E20.m1.9.10.2.3.cmml"><mo id="S7.E20.m1.9.10.2.3.3.2.1" stretchy="false" xref="S7.E20.m1.9.10.2.3.cmml">(</mo><mi id="S7.E20.m1.8.8" xref="S7.E20.m1.8.8.cmml">T</mi><mo id="S7.E20.m1.9.10.2.3.3.2.2" stretchy="false" xref="S7.E20.m1.9.10.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S7.E20.m1.9.10.3" xref="S7.E20.m1.9.10.3.cmml">≤</mo><mrow id="S7.E20.m1.9.10.4" xref="S7.E20.m1.9.10.4.cmml"><munder id="S7.E20.m1.9.10.4.2" xref="S7.E20.m1.9.10.4.2.cmml"><munder accentunder="true" id="S7.E20.m1.2.2" xref="S7.E20.m1.2.2.cmml"><mstyle displaystyle="true" id="S7.E20.m1.2.2.1" xref="S7.E20.m1.2.2.1.cmml"><mfrac id="S7.E20.m1.2.2.1a" xref="S7.E20.m1.2.2.1.cmml"><mrow id="S7.E20.m1.2.2.1.1.1" xref="S7.E20.m1.2.2.1.1.1.cmml"><mrow id="S7.E20.m1.2.2.1.1.1.1.1" xref="S7.E20.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S7.E20.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S7.E20.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.E20.m1.2.2.1.1.1.1.1.1" xref="S7.E20.m1.2.2.1.1.1.1.1.1.cmml"><mrow id="S7.E20.m1.2.2.1.1.1.1.1.1.2" xref="S7.E20.m1.2.2.1.1.1.1.1.1.2.cmml"><mn id="S7.E20.m1.2.2.1.1.1.1.1.1.2.2" xref="S7.E20.m1.2.2.1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S7.E20.m1.2.2.1.1.1.1.1.1.2.1" xref="S7.E20.m1.2.2.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S7.E20.m1.2.2.1.1.1.1.1.1.2.3" xref="S7.E20.m1.2.2.1.1.1.1.1.1.2.3.cmml">n</mi></mrow><mo id="S7.E20.m1.2.2.1.1.1.1.1.1.1" xref="S7.E20.m1.2.2.1.1.1.1.1.1.1.cmml">+</mo><mn id="S7.E20.m1.2.2.1.1.1.1.1.1.3" xref="S7.E20.m1.2.2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S7.E20.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S7.E20.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S7.E20.m1.2.2.1.1.1.2" xref="S7.E20.m1.2.2.1.1.1.2.cmml">⁢</mo><mi id="S7.E20.m1.2.2.1.1.1.3" xref="S7.E20.m1.2.2.1.1.1.3.cmml">n</mi><mo id="S7.E20.m1.2.2.1.1.1.2a" xref="S7.E20.m1.2.2.1.1.1.2.cmml">⁢</mo><mi id="S7.E20.m1.2.2.1.1.1.4" xref="S7.E20.m1.2.2.1.1.1.4.cmml">m</mi><mo id="S7.E20.m1.2.2.1.1.1.2b" xref="S7.E20.m1.2.2.1.1.1.2.cmml">⁢</mo><mi id="S7.E20.m1.2.2.1.1.1.5" xref="S7.E20.m1.2.2.1.1.1.5.cmml">L</mi></mrow><mi id="S7.E20.m1.2.2.1.3" xref="S7.E20.m1.2.2.1.3.cmml">T</mi></mfrac></mstyle><mo id="S7.E20.m1.2.2.2" xref="S7.E20.m1.2.2.2.cmml">⏟</mo></munder><mrow id="S7.E20.m1.3.3.1.3" xref="S7.E20.m1.9.10.4.2.cmml"><mo id="S7.E20.m1.3.3.1.3.1" stretchy="false" xref="S7.E20.m1.9.10.4.2.cmml">(</mo><mn id="S7.E20.m1.3.3.1.1" xref="S7.E20.m1.3.3.1.1.cmml">1</mn><mo id="S7.E20.m1.3.3.1.3.2" stretchy="false" xref="S7.E20.m1.9.10.4.2.cmml">)</mo></mrow></munder><mo id="S7.E20.m1.9.10.4.1" xref="S7.E20.m1.9.10.4.1.cmml">+</mo><munder id="S7.E20.m1.9.10.4.3" xref="S7.E20.m1.9.10.4.3.cmml"><munder accentunder="true" id="S7.E20.m1.4.4" xref="S7.E20.m1.4.4.cmml"><mrow 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id="S7.E20.m1.1.1.1.1.1.5.3.3.2.cmml" xref="S7.E20.m1.1.1.1.1.1.5.3.3.2">𝑇</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.E20.m1.9c">\displaystyle F^{*}-F(T)\leq\underbrace{\frac{(2n+1)nmL}{T}}_{(1)}{}+{}% \underbrace{n(2\varepsilon+\rho)}_{(2)}{}+{}\underbrace{\frac{(2n+1)mnC}{\rho% \sqrt{T}}}_{(3)}\leq\poly(M)\cdot\quantity(\frac{L}{T}+\frac{\sqrt{T}}{L}+\rho% +\frac{1}{\rho\sqrt{T}})</annotation><annotation encoding="application/x-llamapun" id="S7.E20.m1.9d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_F ( italic_T ) ≤ under⏟ start_ARG divide start_ARG ( 2 italic_n + 1 ) italic_n italic_m italic_L end_ARG start_ARG italic_T end_ARG end_ARG start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT + under⏟ start_ARG italic_n ( 2 italic_ε + italic_ρ ) end_ARG start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT + under⏟ start_ARG divide start_ARG ( 2 italic_n + 1 ) italic_m italic_n italic_C end_ARG start_ARG italic_ρ square-root start_ARG italic_T end_ARG end_ARG end_ARG start_POSTSUBSCRIPT ( 3 ) end_POSTSUBSCRIPT ≤ ( italic_M ) ⋅ ( start_ARG divide start_ARG italic_L end_ARG start_ARG italic_T end_ARG + divide start_ARG square-root start_ARG italic_T end_ARG end_ARG start_ARG italic_L end_ARG + italic_ρ + divide start_ARG 1 end_ARG start_ARG italic_ρ square-root start_ARG italic_T end_ARG end_ARG end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S7.SS4.3.p2.24">where the three terms are:</p> <ol class="ltx_enumerate" id="S7.I3"> <li class="ltx_item" id="S7.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S7.I3.i1.p1"> <p class="ltx_p" id="S7.I3.i1.p1.1">the suboptimality and payments in the utility learning stage,</p> </div> </li> <li class="ltx_item" id="S7.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S7.I3.i2.p1"> <p class="ltx_p" id="S7.I3.i2.p1.1">the bonus payments to ensure strict incentive compatibility in the steering stage, and</p> </div> </li> <li class="ltx_item" id="S7.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S7.I3.i3.p1"> <p class="ltx_p" id="S7.I3.i3.p1.1">the suboptimality and payments in rounds on which <math alttext="a^{t}\neq s^{t}" class="ltx_Math" display="inline" id="S7.I3.i3.p1.1.m1.1"><semantics id="S7.I3.i3.p1.1.m1.1a"><mrow id="S7.I3.i3.p1.1.m1.1.1" xref="S7.I3.i3.p1.1.m1.1.1.cmml"><msup id="S7.I3.i3.p1.1.m1.1.1.2" xref="S7.I3.i3.p1.1.m1.1.1.2.cmml"><mi id="S7.I3.i3.p1.1.m1.1.1.2.2" xref="S7.I3.i3.p1.1.m1.1.1.2.2.cmml">a</mi><mi id="S7.I3.i3.p1.1.m1.1.1.2.3" xref="S7.I3.i3.p1.1.m1.1.1.2.3.cmml">t</mi></msup><mo id="S7.I3.i3.p1.1.m1.1.1.1" xref="S7.I3.i3.p1.1.m1.1.1.1.cmml">≠</mo><msup id="S7.I3.i3.p1.1.m1.1.1.3" xref="S7.I3.i3.p1.1.m1.1.1.3.cmml"><mi id="S7.I3.i3.p1.1.m1.1.1.3.2" xref="S7.I3.i3.p1.1.m1.1.1.3.2.cmml">s</mi><mi id="S7.I3.i3.p1.1.m1.1.1.3.3" xref="S7.I3.i3.p1.1.m1.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.I3.i3.p1.1.m1.1b"><apply id="S7.I3.i3.p1.1.m1.1.1.cmml" xref="S7.I3.i3.p1.1.m1.1.1"><neq id="S7.I3.i3.p1.1.m1.1.1.1.cmml" xref="S7.I3.i3.p1.1.m1.1.1.1"></neq><apply id="S7.I3.i3.p1.1.m1.1.1.2.cmml" xref="S7.I3.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S7.I3.i3.p1.1.m1.1.1.2.1.cmml" xref="S7.I3.i3.p1.1.m1.1.1.2">superscript</csymbol><ci id="S7.I3.i3.p1.1.m1.1.1.2.2.cmml" xref="S7.I3.i3.p1.1.m1.1.1.2.2">𝑎</ci><ci id="S7.I3.i3.p1.1.m1.1.1.2.3.cmml" xref="S7.I3.i3.p1.1.m1.1.1.2.3">𝑡</ci></apply><apply id="S7.I3.i3.p1.1.m1.1.1.3.cmml" xref="S7.I3.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I3.i3.p1.1.m1.1.1.3.1.cmml" xref="S7.I3.i3.p1.1.m1.1.1.3">superscript</csymbol><ci id="S7.I3.i3.p1.1.m1.1.1.3.2.cmml" xref="S7.I3.i3.p1.1.m1.1.1.3.2">𝑠</ci><ci id="S7.I3.i3.p1.1.m1.1.1.3.3.cmml" xref="S7.I3.i3.p1.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i3.p1.1.m1.1c">a^{t}\neq s^{t}</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i3.p1.1.m1.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ≠ italic_s start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S7.SS4.3.p2.23">Setting <math alttext="\rho=T^{-1/4}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.22.m1.1"><semantics id="S7.SS4.3.p2.22.m1.1a"><mrow id="S7.SS4.3.p2.22.m1.1.1" xref="S7.SS4.3.p2.22.m1.1.1.cmml"><mi id="S7.SS4.3.p2.22.m1.1.1.2" xref="S7.SS4.3.p2.22.m1.1.1.2.cmml">ρ</mi><mo id="S7.SS4.3.p2.22.m1.1.1.1" xref="S7.SS4.3.p2.22.m1.1.1.1.cmml">=</mo><msup id="S7.SS4.3.p2.22.m1.1.1.3" xref="S7.SS4.3.p2.22.m1.1.1.3.cmml"><mi id="S7.SS4.3.p2.22.m1.1.1.3.2" xref="S7.SS4.3.p2.22.m1.1.1.3.2.cmml">T</mi><mrow id="S7.SS4.3.p2.22.m1.1.1.3.3" xref="S7.SS4.3.p2.22.m1.1.1.3.3.cmml"><mo id="S7.SS4.3.p2.22.m1.1.1.3.3a" xref="S7.SS4.3.p2.22.m1.1.1.3.3.cmml">−</mo><mrow id="S7.SS4.3.p2.22.m1.1.1.3.3.2" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2.cmml"><mn id="S7.SS4.3.p2.22.m1.1.1.3.3.2.2" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2.2.cmml">1</mn><mo id="S7.SS4.3.p2.22.m1.1.1.3.3.2.1" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2.1.cmml">/</mo><mn id="S7.SS4.3.p2.22.m1.1.1.3.3.2.3" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2.3.cmml">4</mn></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.22.m1.1b"><apply id="S7.SS4.3.p2.22.m1.1.1.cmml" xref="S7.SS4.3.p2.22.m1.1.1"><eq id="S7.SS4.3.p2.22.m1.1.1.1.cmml" xref="S7.SS4.3.p2.22.m1.1.1.1"></eq><ci id="S7.SS4.3.p2.22.m1.1.1.2.cmml" xref="S7.SS4.3.p2.22.m1.1.1.2">𝜌</ci><apply id="S7.SS4.3.p2.22.m1.1.1.3.cmml" xref="S7.SS4.3.p2.22.m1.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.3.p2.22.m1.1.1.3.1.cmml" xref="S7.SS4.3.p2.22.m1.1.1.3">superscript</csymbol><ci id="S7.SS4.3.p2.22.m1.1.1.3.2.cmml" xref="S7.SS4.3.p2.22.m1.1.1.3.2">𝑇</ci><apply id="S7.SS4.3.p2.22.m1.1.1.3.3.cmml" xref="S7.SS4.3.p2.22.m1.1.1.3.3"><minus id="S7.SS4.3.p2.22.m1.1.1.3.3.1.cmml" xref="S7.SS4.3.p2.22.m1.1.1.3.3"></minus><apply id="S7.SS4.3.p2.22.m1.1.1.3.3.2.cmml" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2"><divide id="S7.SS4.3.p2.22.m1.1.1.3.3.2.1.cmml" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2.1"></divide><cn id="S7.SS4.3.p2.22.m1.1.1.3.3.2.2.cmml" type="integer" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2.2">1</cn><cn id="S7.SS4.3.p2.22.m1.1.1.3.3.2.3.cmml" type="integer" xref="S7.SS4.3.p2.22.m1.1.1.3.3.2.3">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.22.m1.1c">\rho=T^{-1/4}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.22.m1.1d">italic_ρ = italic_T start_POSTSUPERSCRIPT - 1 / 4 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="L=T^{3/4}" class="ltx_Math" display="inline" id="S7.SS4.3.p2.23.m2.1"><semantics id="S7.SS4.3.p2.23.m2.1a"><mrow id="S7.SS4.3.p2.23.m2.1.1" xref="S7.SS4.3.p2.23.m2.1.1.cmml"><mi id="S7.SS4.3.p2.23.m2.1.1.2" xref="S7.SS4.3.p2.23.m2.1.1.2.cmml">L</mi><mo id="S7.SS4.3.p2.23.m2.1.1.1" xref="S7.SS4.3.p2.23.m2.1.1.1.cmml">=</mo><msup id="S7.SS4.3.p2.23.m2.1.1.3" xref="S7.SS4.3.p2.23.m2.1.1.3.cmml"><mi id="S7.SS4.3.p2.23.m2.1.1.3.2" xref="S7.SS4.3.p2.23.m2.1.1.3.2.cmml">T</mi><mrow id="S7.SS4.3.p2.23.m2.1.1.3.3" xref="S7.SS4.3.p2.23.m2.1.1.3.3.cmml"><mn id="S7.SS4.3.p2.23.m2.1.1.3.3.2" xref="S7.SS4.3.p2.23.m2.1.1.3.3.2.cmml">3</mn><mo id="S7.SS4.3.p2.23.m2.1.1.3.3.1" xref="S7.SS4.3.p2.23.m2.1.1.3.3.1.cmml">/</mo><mn id="S7.SS4.3.p2.23.m2.1.1.3.3.3" xref="S7.SS4.3.p2.23.m2.1.1.3.3.3.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.3.p2.23.m2.1b"><apply id="S7.SS4.3.p2.23.m2.1.1.cmml" xref="S7.SS4.3.p2.23.m2.1.1"><eq id="S7.SS4.3.p2.23.m2.1.1.1.cmml" xref="S7.SS4.3.p2.23.m2.1.1.1"></eq><ci id="S7.SS4.3.p2.23.m2.1.1.2.cmml" xref="S7.SS4.3.p2.23.m2.1.1.2">𝐿</ci><apply id="S7.SS4.3.p2.23.m2.1.1.3.cmml" xref="S7.SS4.3.p2.23.m2.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.3.p2.23.m2.1.1.3.1.cmml" xref="S7.SS4.3.p2.23.m2.1.1.3">superscript</csymbol><ci id="S7.SS4.3.p2.23.m2.1.1.3.2.cmml" xref="S7.SS4.3.p2.23.m2.1.1.3.2">𝑇</ci><apply id="S7.SS4.3.p2.23.m2.1.1.3.3.cmml" xref="S7.SS4.3.p2.23.m2.1.1.3.3"><divide id="S7.SS4.3.p2.23.m2.1.1.3.3.1.cmml" xref="S7.SS4.3.p2.23.m2.1.1.3.3.1"></divide><cn id="S7.SS4.3.p2.23.m2.1.1.3.3.2.cmml" type="integer" xref="S7.SS4.3.p2.23.m2.1.1.3.3.2">3</cn><cn id="S7.SS4.3.p2.23.m2.1.1.3.3.3.cmml" type="integer" xref="S7.SS4.3.p2.23.m2.1.1.3.3.3">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.3.p2.23.m2.1c">L=T^{3/4}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.3.p2.23.m2.1d">italic_L = italic_T start_POSTSUPERSCRIPT 3 / 4 end_POSTSUPERSCRIPT</annotation></semantics></math> then completes the proof. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S8"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">8 </span>Conclusions and Future Research</h2> <div class="ltx_para" id="S8.p1"> <p class="ltx_p" id="S8.p1.1">We have shown that a principal can efficiently learn the utility functions of agents in games through only payments. We have shown that—except in the case of a single player—minimizing payments and minimizing the number of rounds are essentially equivalent. Finally, we have shown how to apply our algorithms to achieve optimal steering in games without prior knowledge of the utility functions. We have given upper- and lower-bounds on all of these problems, which in many cases are near-tight.</p> </div> <div class="ltx_para" id="S8.p2"> <p class="ltx_p" id="S8.p2.1">We leave a number of interesting questions for future research.</p> </div> <section class="ltx_paragraph" id="S8.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">- Closing the polynomial gaps</h4> <div class="ltx_para" id="S8.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S8.SS0.SSS0.Px1.p1.2">We made little attempt throughout the paper to optimize polynomial dependencies on <math alttext="M" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="S8.SS0.SSS0.Px1.p1.1.m1.1a"><mi id="S8.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S8.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px1.p1.1.m1.1b"><ci id="S8.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S8.SS0.SSS0.Px1.p1.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px1.p1.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px1.p1.1.m1.1d">italic_M</annotation></semantics></math>. As such, our upper- and lower-bounds in the no-regret setting are off by <math alttext="\poly(M)" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px1.p1.2.m2.1"><semantics id="S8.SS0.SSS0.Px1.p1.2.m2.1a"><mrow id="S8.SS0.SSS0.Px1.p1.2.m2.1.2" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.cmml"><merror class="ltx_ERROR undefined undefined" id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2b.cmml"><mtext id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2a" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2b.cmml">\poly</mtext></merror><mo id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.1" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.3.2" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.cmml"><mo id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.cmml">(</mo><mi id="S8.SS0.SSS0.Px1.p1.2.m2.1.1" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.1.cmml">M</mi><mo id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px1.p1.2.m2.1b"><apply id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.cmml" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2"><times id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.1.cmml" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.1"></times><ci id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2b.cmml" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2"><merror class="ltx_ERROR undefined undefined" id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2.cmml" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2"><mtext id="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2a.cmml" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.2.2">\poly</mtext></merror></ci><ci id="S8.SS0.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S8.SS0.SSS0.Px1.p1.2.m2.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px1.p1.2.m2.1c">\poly(M)</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px1.p1.2.m2.1d">( italic_M )</annotation></semantics></math> factors. We leave it as an interesting open problem to close these gaps.</p> </div> </section> <section class="ltx_paragraph" id="S8.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">- Extending beyond normal-form games</h4> <div class="ltx_para" id="S8.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S8.SS0.SSS0.Px2.p1.1">Our techniques are specialized to normal-form games, and require, for example, that the principal observe the action <math alttext="a^{t}" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px2.p1.1.m1.1"><semantics id="S8.SS0.SSS0.Px2.p1.1.m1.1a"><msup id="S8.SS0.SSS0.Px2.p1.1.m1.1.1" xref="S8.SS0.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S8.SS0.SSS0.Px2.p1.1.m1.1.1.2" xref="S8.SS0.SSS0.Px2.p1.1.m1.1.1.2.cmml">a</mi><mi id="S8.SS0.SSS0.Px2.p1.1.m1.1.1.3" xref="S8.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px2.p1.1.m1.1b"><apply id="S8.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S8.SS0.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S8.SS0.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S8.SS0.SSS0.Px2.p1.1.m1.1.1">superscript</csymbol><ci id="S8.SS0.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S8.SS0.SSS0.Px2.p1.1.m1.1.1.2">𝑎</ci><ci id="S8.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S8.SS0.SSS0.Px2.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px2.p1.1.m1.1c">a^{t}</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px2.p1.1.m1.1d">italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> chosen by the agents at every timestep. Beyond normal-form games, for example, in <span class="ltx_text ltx_font_italic" id="S8.SS0.SSS0.Px2.p1.1.1">extensive-form games</span>, this may no longer be a reasonable assumption: one may wish instead to assume that we only observe <span class="ltx_text ltx_font_italic" id="S8.SS0.SSS0.Px2.p1.1.2">on-path</span> agent actions. We leave it as future work to extend our results to such settings.</p> </div> </section> <section class="ltx_paragraph" id="S8.SS0.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">- Round complexity of steering without utilities</h4> <div class="ltx_para" id="S8.SS0.SSS0.Px3.p1"> <p class="ltx_p" id="S8.SS0.SSS0.Px3.p1.13">An examination of the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.Thmtheorem6" title="Theorem 7.6. ‣ 7.4 CEPs and optimal steering ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">7.6</span></a> reveals that the polynomial dependence on <math alttext="M" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.1.m1.1"><semantics id="S8.SS0.SSS0.Px3.p1.1.m1.1a"><mi id="S8.SS0.SSS0.Px3.p1.1.m1.1.1" xref="S8.SS0.SSS0.Px3.p1.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.1.m1.1b"><ci id="S8.SS0.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.1.m1.1d">italic_M</annotation></semantics></math> (rather than <math alttext="m" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.2.m2.1"><semantics id="S8.SS0.SSS0.Px3.p1.2.m2.1a"><mi id="S8.SS0.SSS0.Px3.p1.2.m2.1.1" xref="S8.SS0.SSS0.Px3.p1.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.2.m2.1b"><ci id="S8.SS0.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.2.m2.1d">italic_m</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.3.m3.1"><semantics id="S8.SS0.SSS0.Px3.p1.3.m3.1a"><mi id="S8.SS0.SSS0.Px3.p1.3.m3.1.1" xref="S8.SS0.SSS0.Px3.p1.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.3.m3.1b"><ci id="S8.SS0.SSS0.Px3.p1.3.m3.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.3.m3.1d">italic_n</annotation></semantics></math>) is only due to the need for the principal to learn the utilities first. In other words, if the principal already knew the utility function, it would be possible achieve <math alttext="F(T)\geq F^{*}-\varepsilon" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.4.m4.1"><semantics id="S8.SS0.SSS0.Px3.p1.4.m4.1a"><mrow id="S8.SS0.SSS0.Px3.p1.4.m4.1.2" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.cmml"><mrow id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml"><mi id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.2" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.2.cmml">F</mi><mo id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.1" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.3.2" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml"><mo id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml">(</mo><mi id="S8.SS0.SSS0.Px3.p1.4.m4.1.1" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.1.cmml">T</mi><mo id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.1" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.1.cmml">≥</mo><mrow id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.cmml"><msup id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.cmml"><mi id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.2" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.2.cmml">F</mi><mo id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.3" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.3.cmml">∗</mo></msup><mo id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.1" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.1.cmml">−</mo><mi id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.3" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.4.m4.1b"><apply id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2"><geq id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.1.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.1"></geq><apply id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2"><times id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.1.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.1"></times><ci id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.2.2">𝐹</ci><ci id="S8.SS0.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.1">𝑇</ci></apply><apply id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3"><minus id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.1.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.1"></minus><apply id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.1.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2">superscript</csymbol><ci id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.2">𝐹</ci><times id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.3.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.2.3"></times></apply><ci id="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.3.cmml" xref="S8.SS0.SSS0.Px3.p1.4.m4.1.2.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.4.m4.1c">F(T)\geq F^{*}-\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.4.m4.1d">italic_F ( italic_T ) ≥ italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_ε</annotation></semantics></math> in <math alttext="\poly(m,n,1/\varepsilon)" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.5.m5.3"><semantics id="S8.SS0.SSS0.Px3.p1.5.m5.3a"><mrow id="S8.SS0.SSS0.Px3.p1.5.m5.3.3" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.cmml"><merror class="ltx_ERROR undefined undefined" id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3b.cmml"><mtext id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3a" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3b.cmml">\poly</mtext></merror><mo id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.2" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.2.cmml">⁢</mo><mrow id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.2.cmml"><mo id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.2.cmml">(</mo><mi id="S8.SS0.SSS0.Px3.p1.5.m5.1.1" xref="S8.SS0.SSS0.Px3.p1.5.m5.1.1.cmml">m</mi><mo id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.3" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.2.cmml">,</mo><mi id="S8.SS0.SSS0.Px3.p1.5.m5.2.2" xref="S8.SS0.SSS0.Px3.p1.5.m5.2.2.cmml">n</mi><mo id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.4" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.2.cmml">,</mo><mrow id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.cmml"><mn id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.2" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.2.cmml">1</mn><mo id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.1" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.1.cmml">/</mo><mi id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.3" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.3.cmml">ε</mi></mrow><mo id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.5" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.5.m5.3b"><apply id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3"><times id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.2.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.2"></times><ci id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3b.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3"><merror class="ltx_ERROR undefined undefined" id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3"><mtext id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3a.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.3">\poly</mtext></merror></ci><vector id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1"><ci id="S8.SS0.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.1.1">𝑚</ci><ci id="S8.SS0.SSS0.Px3.p1.5.m5.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.2.2">𝑛</ci><apply id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1"><divide id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.1"></divide><cn id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.2.cmml" type="integer" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.2">1</cn><ci id="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.3.cmml" xref="S8.SS0.SSS0.Px3.p1.5.m5.3.3.1.1.1.3">𝜀</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.5.m5.3c">\poly(m,n,1/\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.5.m5.3d">( italic_m , italic_n , 1 / italic_ε )</annotation></semantics></math> rounds.<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>Although they do not state this explicitly, the analysis of <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib24" title="">2024</a>)</cite> also implies this.</span></span></span> This begs the question: is it possible to steer <span class="ltx_text ltx_font_italic" id="S8.SS0.SSS0.Px3.p1.13.1">without</span> expending so much effort to learn the whole utility function? More precisely, is it possible to steer in worst-case <math alttext="o(M)" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.6.m6.1"><semantics id="S8.SS0.SSS0.Px3.p1.6.m6.1a"><mrow id="S8.SS0.SSS0.Px3.p1.6.m6.1.2" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.cmml"><mi id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.2" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.2.cmml">o</mi><mo id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.1" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.3.2" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.cmml"><mo id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.cmml">(</mo><mi id="S8.SS0.SSS0.Px3.p1.6.m6.1.1" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.1.cmml">M</mi><mo id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.6.m6.1b"><apply id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2"><times id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.1.cmml" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.1"></times><ci id="S8.SS0.SSS0.Px3.p1.6.m6.1.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.2.2">𝑜</ci><ci id="S8.SS0.SSS0.Px3.p1.6.m6.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.6.m6.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.6.m6.1c">o(M)</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.6.m6.1d">italic_o ( italic_M )</annotation></semantics></math> time, even to constant accuracy <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.7.m7.1"><semantics id="S8.SS0.SSS0.Px3.p1.7.m7.1a"><mi id="S8.SS0.SSS0.Px3.p1.7.m7.1.1" xref="S8.SS0.SSS0.Px3.p1.7.m7.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.7.m7.1b"><ci id="S8.SS0.SSS0.Px3.p1.7.m7.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.7.m7.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.7.m7.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.7.m7.1d">italic_ε</annotation></semantics></math>? Clearly this is impossible if “time” is to be interpreted in the computational sense, as merely reading the principal’s utility function <math alttext="U_{0}:A\to\mathbb{R}" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.8.m8.1"><semantics id="S8.SS0.SSS0.Px3.p1.8.m8.1a"><mrow id="S8.SS0.SSS0.Px3.p1.8.m8.1.1" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.cmml"><msub id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.cmml"><mi id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.2" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.2.cmml">U</mi><mn id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.3" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.3.cmml">0</mn></msub><mo id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.1.cmml">:</mo><mrow id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.cmml"><mi id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.2" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.2.cmml">A</mi><mo id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.1" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.1.cmml">→</mo><mi id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.3" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.8.m8.1b"><apply id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1"><ci id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.1">:</ci><apply id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.1.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2">subscript</csymbol><ci id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.2">𝑈</ci><cn id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.3.cmml" type="integer" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.2.3">0</cn></apply><apply id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3"><ci id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.1.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.1">→</ci><ci id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.2.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.2">𝐴</ci><ci id="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.3.cmml" xref="S8.SS0.SSS0.Px3.p1.8.m8.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.8.m8.1c">U_{0}:A\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.8.m8.1d">italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : italic_A → blackboard_R</annotation></semantics></math> takes <math alttext="O(M)" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.9.m9.1"><semantics id="S8.SS0.SSS0.Px3.p1.9.m9.1a"><mrow id="S8.SS0.SSS0.Px3.p1.9.m9.1.2" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.cmml"><mi id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.2" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.2.cmml">O</mi><mo id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.1" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.3.2" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.cmml"><mo id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.cmml">(</mo><mi id="S8.SS0.SSS0.Px3.p1.9.m9.1.1" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.1.cmml">M</mi><mo id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.9.m9.1b"><apply id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2"><times id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.1.cmml" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.1"></times><ci id="S8.SS0.SSS0.Px3.p1.9.m9.1.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.2.2">𝑂</ci><ci id="S8.SS0.SSS0.Px3.p1.9.m9.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.9.m9.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.9.m9.1c">O(M)</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.9.m9.1d">italic_O ( italic_M )</annotation></semantics></math> time. As such, we will not concern ourselves too much with this question in the present paper; however, an interesting question for future research is to answer whether it is (information-theoretically) possible to steer to optimal CEPs in <math alttext="o(M)" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.10.m10.1"><semantics id="S8.SS0.SSS0.Px3.p1.10.m10.1a"><mrow id="S8.SS0.SSS0.Px3.p1.10.m10.1.2" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.cmml"><mi id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.2" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.2.cmml">o</mi><mo id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.1" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.3.2" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.cmml"><mo id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.cmml">(</mo><mi id="S8.SS0.SSS0.Px3.p1.10.m10.1.1" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.1.cmml">M</mi><mo id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.10.m10.1b"><apply id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2"><times id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.1.cmml" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.1"></times><ci id="S8.SS0.SSS0.Px3.p1.10.m10.1.2.2.cmml" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.2.2">𝑜</ci><ci id="S8.SS0.SSS0.Px3.p1.10.m10.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.10.m10.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.10.m10.1c">o(M)</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.10.m10.1d">italic_o ( italic_M )</annotation></semantics></math> <span class="ltx_text ltx_font_italic" id="S8.SS0.SSS0.Px3.p1.13.2">rounds</span>. Establishing this lower bound may seem trivial, but it is not: since the principal knows its own utility <math alttext="U_{0}" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.11.m11.1"><semantics id="S8.SS0.SSS0.Px3.p1.11.m11.1a"><msub id="S8.SS0.SSS0.Px3.p1.11.m11.1.1" xref="S8.SS0.SSS0.Px3.p1.11.m11.1.1.cmml"><mi id="S8.SS0.SSS0.Px3.p1.11.m11.1.1.2" xref="S8.SS0.SSS0.Px3.p1.11.m11.1.1.2.cmml">U</mi><mn id="S8.SS0.SSS0.Px3.p1.11.m11.1.1.3" xref="S8.SS0.SSS0.Px3.p1.11.m11.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.11.m11.1b"><apply id="S8.SS0.SSS0.Px3.p1.11.m11.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S8.SS0.SSS0.Px3.p1.11.m11.1.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.11.m11.1.1">subscript</csymbol><ci id="S8.SS0.SSS0.Px3.p1.11.m11.1.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.11.m11.1.1.2">𝑈</ci><cn id="S8.SS0.SSS0.Px3.p1.11.m11.1.1.3.cmml" type="integer" xref="S8.SS0.SSS0.Px3.p1.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.11.m11.1c">U_{0}</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.11.m11.1d">italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, one cannot, for example, construct an identical-interest game with <math alttext="U_{0}(a)=U_{1}(a)=\dots=U_{n}(a)=\quantity{a=a^{*}}" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.12.m12.4"><semantics id="S8.SS0.SSS0.Px3.p1.12.m12.4a"><mrow id="S8.SS0.SSS0.Px3.p1.12.m12.4.5" xref="S8.SS0.SSS0.Px3.p1.12.m12.4.5.cmml"><mrow id="S8.SS0.SSS0.Px3.p1.12.m12.4.5.2" xref="S8.SS0.SSS0.Px3.p1.12.m12.4.5.2.cmml"><msub id="S8.SS0.SSS0.Px3.p1.12.m12.4.5.2.2" xref="S8.SS0.SSS0.Px3.p1.12.m12.4.5.2.2.cmml"><mi id="S8.SS0.SSS0.Px3.p1.12.m12.4.5.2.2.2" xref="S8.SS0.SSS0.Px3.p1.12.m12.4.5.2.2.2.cmml">U</mi><mn id="S8.SS0.SSS0.Px3.p1.12.m12.4.5.2.2.3" 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because then the principal’s utility <math alttext="U_{0}" class="ltx_Math" display="inline" id="S8.SS0.SSS0.Px3.p1.13.m13.1"><semantics id="S8.SS0.SSS0.Px3.p1.13.m13.1a"><msub id="S8.SS0.SSS0.Px3.p1.13.m13.1.1" xref="S8.SS0.SSS0.Px3.p1.13.m13.1.1.cmml"><mi id="S8.SS0.SSS0.Px3.p1.13.m13.1.1.2" xref="S8.SS0.SSS0.Px3.p1.13.m13.1.1.2.cmml">U</mi><mn id="S8.SS0.SSS0.Px3.p1.13.m13.1.1.3" xref="S8.SS0.SSS0.Px3.p1.13.m13.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S8.SS0.SSS0.Px3.p1.13.m13.1b"><apply id="S8.SS0.SSS0.Px3.p1.13.m13.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S8.SS0.SSS0.Px3.p1.13.m13.1.1.1.cmml" xref="S8.SS0.SSS0.Px3.p1.13.m13.1.1">subscript</csymbol><ci id="S8.SS0.SSS0.Px3.p1.13.m13.1.1.2.cmml" xref="S8.SS0.SSS0.Px3.p1.13.m13.1.1.2">𝑈</ci><cn id="S8.SS0.SSS0.Px3.p1.13.m13.1.1.3.cmml" type="integer" xref="S8.SS0.SSS0.Px3.p1.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS0.SSS0.Px3.p1.13.m13.1c">U_{0}</annotation><annotation encoding="application/x-llamapun" id="S8.SS0.SSS0.Px3.p1.13.m13.1d">italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> would immediately reveal every other agent’s utility as well.</p> </div> </section> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">T.S. is supported by the Vannevar Bush Faculty Fellowship ONR N00014-23-1-2876, National Science Foundation grants RI-2312342 and RI-1901403, ARO award W911NF2210266, and NIH award A240108S001. B.H.Z. is supported by the CMU Computer Science Department Hans Berliner PhD Student Fellowship. Y.C. is supported by National Science Foundation grant IIS-2147187 and by Amazon. T.L. is supported by a Siebel Scholarship.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Aumann [1974]</span> <span class="ltx_bibblock"> Robert Aumann. </span> <span class="ltx_bibblock">Subjectivity and correlation in randomized strategies. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">Journal of Mathematical Economics</em>, 1(1):67–96, 1974. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Azuma [1967]</span> <span class="ltx_bibblock"> Kazuoki Azuma. </span> <span class="ltx_bibblock">Weighted sums of certain dependent random variables. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib2.1.1">Tohoku Mathematical Journal</em>, 19(3):357–367, 1967. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bacchiocchi et al. [2024]</span> <span class="ltx_bibblock"> Francesco Bacchiocchi, Matteo Bollini, Matteo Castiglioni, Alberto Marchesi, and Nicola Gatti. </span> <span class="ltx_bibblock">Online bayesian persuasion without a clue. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib3.1.1">Neural Information Processing Systems (NeurIPS)</em>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Beigman and Vohra [2006]</span> <span class="ltx_bibblock"> Eyal Beigman and Rakesh Vohra. </span> <span class="ltx_bibblock">Learning from revealed preference. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib4.1.1">Proceedings of the 7th ACM conference on Electronic commerce</em>, pages 36–42, Ann Arbor Michigan USA, June 2006. ACM. </span> <span class="ltx_bibblock">ISBN 978-1-59593-236-5. </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bernheim [1984]</span> <span class="ltx_bibblock"> B Douglas Bernheim. </span> <span class="ltx_bibblock">Rationalizable strategic behavior. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib5.1.1">Econometrica</em>, 52(4):1007–28, 1984. </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Cai et al. [2024]</span> <span class="ltx_bibblock"> Yang Cai, Gabriele Farina, Julien Grand-Clément, Christian Kroer, Chung-Wei Lee, Haipeng Luo, and Weiqiang Zheng. </span> <span class="ltx_bibblock">Fast last-iterate convergence of learning in games requires forgetful algorithms. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib6.1.1">The Thirty-eighth Annual Conference on Neural Information Processing Systems</em>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Chen et al. [2009]</span> <span class="ltx_bibblock"> Xi Chen, Xiaotie Deng, and Shang-Hua Teng. </span> <span class="ltx_bibblock">Settling the complexity of computing two-player Nash equilibria. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib7.1.1">Journal of the ACM</em>, 56(3):14, 2009. </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Daskalakis et al. [2006]</span> <span class="ltx_bibblock"> Constantinos Daskalakis, Paul Goldberg, and Christos Papadimitriou. </span> <span class="ltx_bibblock">The complexity of computing a Nash equilibrium. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib8.1.1">Symposium on Theory of Computing (STOC)</em>, 2006. </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Deng et al. [2019]</span> <span class="ltx_bibblock"> Yuan Deng, Jon Schneider, and Balasubramanian Sivan. </span> <span class="ltx_bibblock">Strategizing against no-regret learners. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib9.1.1">Neural Information Processing Systems (NeurIPS)</em>, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Feng et al. [2022]</span> <span class="ltx_bibblock"> Yiding Feng, Wei Tang, and Haifeng Xu. </span> <span class="ltx_bibblock">Online Bayesian Recommendation with No Regret. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib10.1.1">Proceedings of the 23rd ACM Conference on Economics and Computation</em>, pages 818–819, Boulder CO USA, July 2022. ACM. </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Freund and Schapire [1999]</span> <span class="ltx_bibblock"> Yoav Freund and Robert Schapire. </span> <span class="ltx_bibblock">Adaptive game playing using multiplicative weights. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib11.1.1">Games and Economic Behavior</em>, 29:79–103, 1999. </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Haghtalab et al. [2022]</span> <span class="ltx_bibblock"> Nika Haghtalab, Thodoris Lykouris, Sloan Nietert, and Alexander Wei. </span> <span class="ltx_bibblock">Learning in stackelberg games with non-myopic agents. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib12.1.1">ACM Conference on Economics and Computation (EC)</em>, pages 917–918, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Hoeffding [1963]</span> <span class="ltx_bibblock"> Wassily Hoeffding. </span> <span class="ltx_bibblock">Probability inequalities for sums of bounded random variables. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib13.1.1">Journal of the American Statistical Association</em>, 58(301):13–30, 1963. </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kamenica and Gentzkow [2011]</span> <span class="ltx_bibblock"> Emir Kamenica and Matthew Gentzkow. </span> <span class="ltx_bibblock">Bayesian Persuasion. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib14.1.1">American Economic Review</em>, 101(6):2590–2615, October 2011. </span> <span class="ltx_bibblock">ISSN 0002-8282. </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kuleshov and Schrijvers [2015]</span> <span class="ltx_bibblock"> Volodymyr Kuleshov and Okke Schrijvers. </span> <span class="ltx_bibblock">Inverse game theory: Learning utilities in succinct games. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib15.1.1">International Workshop On Internet And Network Economics (WINE)</em>, pages 413–427. Springer, 2015. </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Letchford et al. [2009]</span> <span class="ltx_bibblock"> Joshua Letchford, Vincent Conitzer, and Kamesh Munagala. </span> <span class="ltx_bibblock">Learning and approximating the optimal strategy to commit to. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib16.1.1">International Symposium on Algorithmic Game Theory</em>, pages 250–262. Springer, 2009. </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Lin and Chen [2025]</span> <span class="ltx_bibblock"> Tao Lin and Yiling Chen. </span> <span class="ltx_bibblock">Generalized principal-agent problem with a learning agent. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib17.1.1">The Thirteenth International Conference on Learning Representations</em>, 2025. </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Mansour et al. [2022]</span> <span class="ltx_bibblock"> Yishay Mansour, Mehryar Mohri, Jon Schneider, and Balasubramanian Sivan. </span> <span class="ltx_bibblock">Strategizing against learners in bayesian games. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib18.1.1">Conference on Learning Theory (COLT)</em>, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Monderer and Tennenholtz [2003]</span> <span class="ltx_bibblock"> Dov Monderer and Moshe Tennenholtz. </span> <span class="ltx_bibblock">k-Implementation. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib19.1.1">ACM Conference on Electronic Commerce (ACM-EC)</em>, pages 19–28, San Diego, CA, 2003. </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Pearce [1984]</span> <span class="ltx_bibblock"> David G. Pearce. </span> <span class="ltx_bibblock">Rationalizable strategic behavior and the problem of perfection. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib20.1.1">Econometrica</em>, 52:1029–1050, 1984. </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Scheid et al. [2024]</span> <span class="ltx_bibblock"> Antoine Scheid, Aymeric Capitaine, Etienne Boursier, Eric Moulines, Michael Jordan, and Alain Oliviero Durmus. </span> <span class="ltx_bibblock">Learning to mitigate externalities: the coase theorem with hindsight rationality. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib21.1.1">The Thirty-eighth Annual Conference on Neural Information Processing Systems</em>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Wu et al. [2022]</span> <span class="ltx_bibblock"> Jibang Wu, Haifeng Xu, and Fan Yao. </span> <span class="ltx_bibblock">Multi-Agent Learning for Iterative Dominance Elimination: Formal Barriers and New Algorithms. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib22.1.1">Proceedings of Thirty Fifth Conference on Learning Theory</em>, volume 178 of <em class="ltx_emph ltx_font_italic" id="bib.bib22.2.2">Proceedings of Machine Learning Research</em>, pages 543–543. PMLR, July 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Zadimoghaddam and Roth [2012]</span> <span class="ltx_bibblock"> Morteza Zadimoghaddam and Aaron Roth. </span> <span class="ltx_bibblock">Efficiently Learning from Revealed Preference. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib23.1.1">Internet and Network Economics</em>, volume 7695, pages 114–127. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. </span> <span class="ltx_bibblock">ISBN 978-3-642-35310-9 978-3-642-35311-6. </span> <span class="ltx_bibblock">Series Title: Lecture Notes in Computer Science. </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Zhang et al. [2024]</span> <span class="ltx_bibblock"> Brian Hu Zhang, Gabriele Farina, Ioannis Anagnostides, Federico Cacciamani, Stephen Marcus McAleer, Andreas Alexander Haupt, Andrea Celli, Nicola Gatti, Vincent Conitzer, and Tuomas Sandholm. </span> <span class="ltx_bibblock">Steering no-regret learners to a desired equilibrium. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib24.1.1">ACM Conference on Economics and Computation (EC)</em>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Zinkevich [2003]</span> <span class="ltx_bibblock"> Martin Zinkevich. </span> <span class="ltx_bibblock">Online convex programming and generalized infinitesimal gradient ascent. </span> <span class="ltx_bibblock">In <em class="ltx_emph ltx_font_italic" id="bib.bib25.1.1">Proceedings of the 20th International Conference on Machine Learning (ICML-03)</em>, pages 928–936, 2003. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>An anytime Azuma-Hoeffding bound</h2> <div class="ltx_theorem ltx_theorem_lemma" id="A1.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="A1.Thmtheorem1.1.1.1">Lemma A.1</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem1.p1"> <p class="ltx_p" id="A1.Thmtheorem1.p1.7"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem1.p1.7.7">Let <math alttext="\{X_{t}\}_{t=0}^{T}" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.1.1.m1.1"><semantics id="A1.Thmtheorem1.p1.1.1.m1.1a"><msubsup id="A1.Thmtheorem1.p1.1.1.m1.1.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mrow id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.2.cmml"><mo id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.2.cmml">{</mo><msub id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.cmml"><mi id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.2" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.2.cmml">X</mi><mi id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.3" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3.cmml"><mi id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3.2" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3.2.cmml">t</mi><mo id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3.1.cmml">=</mo><mn id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3.3" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.3.3.cmml">0</mn></mrow><mi id="A1.Thmtheorem1.p1.1.1.m1.1.1.3" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">T</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.1.1.m1.1b"><apply id="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1">superscript</csymbol><apply id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1">subscript</csymbol><set id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1"><apply id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.1.1.1.2.cmml" 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start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> be a (super)martingale with <math alttext="X_{0}=0" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.2.2.m2.1"><semantics id="A1.Thmtheorem1.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem1.p1.2.2.m2.1.1" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml"><msub id="A1.Thmtheorem1.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml"><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.2" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.2.cmml">X</mi><mn id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.3" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="A1.Thmtheorem1.p1.2.2.m2.1.1.1" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">=</mo><mn id="A1.Thmtheorem1.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.2.2.m2.1b"><apply id="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1"><eq id="A1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.1"></eq><apply id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.2.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.2">𝑋</ci><cn id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.3.cmml" type="integer" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.3">0</cn></apply><cn id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" type="integer" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.2.2.m2.1c">X_{0}=0</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.2.2.m2.1d">italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math> and <math alttext="|X_{t+1}-X_{t}|\leq 1" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.3.3.m3.1"><semantics id="A1.Thmtheorem1.p1.3.3.m3.1a"><mrow id="A1.Thmtheorem1.p1.3.3.m3.1.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mrow id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.2.cmml"><mo id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.2.1.cmml">|</mo><mrow id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml"><msub id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml"><mi id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2.cmml">X</mi><mrow id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.cmml"><mi id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2.cmml">t</mi><mo id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.3" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml">−</mo><msub id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml"><mi id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2.cmml">X</mi><mi id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.3" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.3.cmml">t</mi></msub></mrow><mo id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="A1.Thmtheorem1.p1.3.3.m3.1.1.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.2.cmml">≤</mo><mn id="A1.Thmtheorem1.p1.3.3.m3.1.1.3" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.3.3.m3.1b"><apply id="A1.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1"><leq id="A1.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.2"></leq><apply id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1"><abs id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.2.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.2"></abs><apply id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"><minus id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1"></minus><apply id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2">subscript</csymbol><ci id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2">𝑋</ci><apply id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3"><plus id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.1"></plus><ci id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2">𝑡</ci><cn id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.3.cmml" type="integer" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2">𝑋</ci><ci id="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.3.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><cn id="A1.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" type="integer" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.3.3.m3.1c">|X_{t+1}-X_{t}|\leq 1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.3.3.m3.1d">| italic_X start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT - italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | ≤ 1</annotation></semantics></math> for all <math alttext="t" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.4.4.m4.1"><semantics id="A1.Thmtheorem1.p1.4.4.m4.1a"><mi id="A1.Thmtheorem1.p1.4.4.m4.1.1" xref="A1.Thmtheorem1.p1.4.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.4.4.m4.1b"><ci id="A1.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem1.p1.4.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.4.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.4.4.m4.1d">italic_t</annotation></semantics></math>. Then, with probability <math alttext="1-\delta" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.5.5.m5.1"><semantics id="A1.Thmtheorem1.p1.5.5.m5.1a"><mrow id="A1.Thmtheorem1.p1.5.5.m5.1.1" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.cmml"><mn id="A1.Thmtheorem1.p1.5.5.m5.1.1.2" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem1.p1.5.5.m5.1.1.1" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.1.cmml">−</mo><mi id="A1.Thmtheorem1.p1.5.5.m5.1.1.3" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.5.5.m5.1b"><apply id="A1.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem1.p1.5.5.m5.1.1"><minus id="A1.Thmtheorem1.p1.5.5.m5.1.1.1.cmml" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.1"></minus><cn id="A1.Thmtheorem1.p1.5.5.m5.1.1.2.cmml" type="integer" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.2">1</cn><ci id="A1.Thmtheorem1.p1.5.5.m5.1.1.3.cmml" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.5.5.m5.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.5.5.m5.1d">1 - italic_δ</annotation></semantics></math>, it holds simultaneously for all <math alttext="t\leq T" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.6.6.m6.1"><semantics id="A1.Thmtheorem1.p1.6.6.m6.1a"><mrow id="A1.Thmtheorem1.p1.6.6.m6.1.1" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.cmml"><mi id="A1.Thmtheorem1.p1.6.6.m6.1.1.2" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.2.cmml">t</mi><mo id="A1.Thmtheorem1.p1.6.6.m6.1.1.1" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.1.cmml">≤</mo><mi id="A1.Thmtheorem1.p1.6.6.m6.1.1.3" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.6.6.m6.1b"><apply id="A1.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.1"><leq id="A1.Thmtheorem1.p1.6.6.m6.1.1.1.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.1"></leq><ci id="A1.Thmtheorem1.p1.6.6.m6.1.1.2.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.2">𝑡</ci><ci id="A1.Thmtheorem1.p1.6.6.m6.1.1.3.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.6.6.m6.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.6.6.m6.1d">italic_t ≤ italic_T</annotation></semantics></math> that <math alttext="X_{t}\leq\sqrt{2T\log(1/\delta)}" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.7.7.m7.2"><semantics id="A1.Thmtheorem1.p1.7.7.m7.2a"><mrow id="A1.Thmtheorem1.p1.7.7.m7.2.3" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.cmml"><msub id="A1.Thmtheorem1.p1.7.7.m7.2.3.2" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.2.cmml"><mi id="A1.Thmtheorem1.p1.7.7.m7.2.3.2.2" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.2.2.cmml">X</mi><mi id="A1.Thmtheorem1.p1.7.7.m7.2.3.2.3" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.2.3.cmml">t</mi></msub><mo id="A1.Thmtheorem1.p1.7.7.m7.2.3.1" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.1.cmml">≤</mo><msqrt id="A1.Thmtheorem1.p1.7.7.m7.2.2" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.cmml"><mrow id="A1.Thmtheorem1.p1.7.7.m7.2.2.2" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.cmml"><mn id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.4" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.4.cmml">2</mn><mo id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.3" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.3.cmml">⁢</mo><mi id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.5" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.5.cmml">T</mi><mo id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.3a" lspace="0.167em" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.3.cmml">⁢</mo><mrow id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.4" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.cmml"><mi id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.4a" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.1.cmml">⁡</mo><mrow id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.4.1" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.cmml"><mo id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.4.1.1" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.1.cmml">(</mo><mrow id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.cmml"><mn id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.2" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.1" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.3" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.3.cmml">δ</mi></mrow><mo id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.4.1.2" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.7.7.m7.2b"><apply id="A1.Thmtheorem1.p1.7.7.m7.2.3.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.3"><leq id="A1.Thmtheorem1.p1.7.7.m7.2.3.1.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.1"></leq><apply id="A1.Thmtheorem1.p1.7.7.m7.2.3.2.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.7.7.m7.2.3.2.1.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.2">subscript</csymbol><ci id="A1.Thmtheorem1.p1.7.7.m7.2.3.2.2.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.2.2">𝑋</ci><ci id="A1.Thmtheorem1.p1.7.7.m7.2.3.2.3.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.3.2.3">𝑡</ci></apply><apply id="A1.Thmtheorem1.p1.7.7.m7.2.2.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.2"><root id="A1.Thmtheorem1.p1.7.7.m7.2.2a.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.2"></root><apply id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2"><times id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.3.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.3"></times><cn id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.4.cmml" type="integer" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.4">2</cn><ci id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.5.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.5">𝑇</ci><apply id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.4"><log id="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.3.1.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2"></log><apply id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1"><divide id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.1"></divide><cn id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.2">1</cn><ci id="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.1.1.3">𝛿</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.7.7.m7.2c">X_{t}\leq\sqrt{2T\log(1/\delta)}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.7.7.m7.2d">italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≤ square-root start_ARG 2 italic_T roman_log ( start_ARG 1 / italic_δ end_ARG ) end_ARG</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="A1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.1.p1"> <p class="ltx_p" id="A1.1.p1.9">Let <math alttext="B=\sqrt{2T\log(1/\delta)}" class="ltx_Math" display="inline" id="A1.1.p1.1.m1.2"><semantics id="A1.1.p1.1.m1.2a"><mrow id="A1.1.p1.1.m1.2.3" xref="A1.1.p1.1.m1.2.3.cmml"><mi id="A1.1.p1.1.m1.2.3.2" xref="A1.1.p1.1.m1.2.3.2.cmml">B</mi><mo id="A1.1.p1.1.m1.2.3.1" xref="A1.1.p1.1.m1.2.3.1.cmml">=</mo><msqrt id="A1.1.p1.1.m1.2.2" xref="A1.1.p1.1.m1.2.2.cmml"><mrow id="A1.1.p1.1.m1.2.2.2" xref="A1.1.p1.1.m1.2.2.2.cmml"><mn id="A1.1.p1.1.m1.2.2.2.4" xref="A1.1.p1.1.m1.2.2.2.4.cmml">2</mn><mo id="A1.1.p1.1.m1.2.2.2.3" xref="A1.1.p1.1.m1.2.2.2.3.cmml">⁢</mo><mi id="A1.1.p1.1.m1.2.2.2.5" xref="A1.1.p1.1.m1.2.2.2.5.cmml">T</mi><mo id="A1.1.p1.1.m1.2.2.2.3a" lspace="0.167em" xref="A1.1.p1.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="A1.1.p1.1.m1.2.2.2.2.4" xref="A1.1.p1.1.m1.2.2.2.2.3.cmml"><mi id="A1.1.p1.1.m1.2.2.2.2.2.2" xref="A1.1.p1.1.m1.2.2.2.2.3.1.cmml">log</mi><mo id="A1.1.p1.1.m1.2.2.2.2.4a" xref="A1.1.p1.1.m1.2.2.2.2.3.1.cmml">⁡</mo><mrow id="A1.1.p1.1.m1.2.2.2.2.4.1" xref="A1.1.p1.1.m1.2.2.2.2.3.cmml"><mo id="A1.1.p1.1.m1.2.2.2.2.4.1.1" xref="A1.1.p1.1.m1.2.2.2.2.3.1.cmml">(</mo><mrow id="A1.1.p1.1.m1.1.1.1.1.1.1.1" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1.cmml"><mn id="A1.1.p1.1.m1.1.1.1.1.1.1.1.2" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A1.1.p1.1.m1.1.1.1.1.1.1.1.1" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="A1.1.p1.1.m1.1.1.1.1.1.1.1.3" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1.3.cmml">δ</mi></mrow><mo id="A1.1.p1.1.m1.2.2.2.2.4.1.2" xref="A1.1.p1.1.m1.2.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.1.m1.2b"><apply id="A1.1.p1.1.m1.2.3.cmml" xref="A1.1.p1.1.m1.2.3"><eq id="A1.1.p1.1.m1.2.3.1.cmml" xref="A1.1.p1.1.m1.2.3.1"></eq><ci id="A1.1.p1.1.m1.2.3.2.cmml" xref="A1.1.p1.1.m1.2.3.2">𝐵</ci><apply id="A1.1.p1.1.m1.2.2.cmml" xref="A1.1.p1.1.m1.2.2"><root id="A1.1.p1.1.m1.2.2a.cmml" xref="A1.1.p1.1.m1.2.2"></root><apply id="A1.1.p1.1.m1.2.2.2.cmml" xref="A1.1.p1.1.m1.2.2.2"><times id="A1.1.p1.1.m1.2.2.2.3.cmml" xref="A1.1.p1.1.m1.2.2.2.3"></times><cn id="A1.1.p1.1.m1.2.2.2.4.cmml" type="integer" xref="A1.1.p1.1.m1.2.2.2.4">2</cn><ci id="A1.1.p1.1.m1.2.2.2.5.cmml" xref="A1.1.p1.1.m1.2.2.2.5">𝑇</ci><apply id="A1.1.p1.1.m1.2.2.2.2.3.cmml" xref="A1.1.p1.1.m1.2.2.2.2.4"><log id="A1.1.p1.1.m1.2.2.2.2.3.1.cmml" xref="A1.1.p1.1.m1.2.2.2.2.2.2"></log><apply id="A1.1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1"><divide id="A1.1.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1.1"></divide><cn id="A1.1.p1.1.m1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1.2">1</cn><ci id="A1.1.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="A1.1.p1.1.m1.1.1.1.1.1.1.1.3">𝛿</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.1.m1.2c">B=\sqrt{2T\log(1/\delta)}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.1.m1.2d">italic_B = square-root start_ARG 2 italic_T roman_log ( start_ARG 1 / italic_δ end_ARG ) end_ARG</annotation></semantics></math>. Consider the stopped supermartingale <math alttext="Y_{t}" class="ltx_Math" display="inline" id="A1.1.p1.2.m2.1"><semantics id="A1.1.p1.2.m2.1a"><msub id="A1.1.p1.2.m2.1.1" xref="A1.1.p1.2.m2.1.1.cmml"><mi id="A1.1.p1.2.m2.1.1.2" xref="A1.1.p1.2.m2.1.1.2.cmml">Y</mi><mi id="A1.1.p1.2.m2.1.1.3" xref="A1.1.p1.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.1.p1.2.m2.1b"><apply id="A1.1.p1.2.m2.1.1.cmml" xref="A1.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.1.p1.2.m2.1.1.1.cmml" xref="A1.1.p1.2.m2.1.1">subscript</csymbol><ci id="A1.1.p1.2.m2.1.1.2.cmml" xref="A1.1.p1.2.m2.1.1.2">𝑌</ci><ci id="A1.1.p1.2.m2.1.1.3.cmml" xref="A1.1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.2.m2.1c">Y_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.2.m2.1d">italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> defined by <math alttext="Y_{t}=X_{t\land\tau}" class="ltx_Math" display="inline" id="A1.1.p1.3.m3.1"><semantics id="A1.1.p1.3.m3.1a"><mrow id="A1.1.p1.3.m3.1.1" xref="A1.1.p1.3.m3.1.1.cmml"><msub id="A1.1.p1.3.m3.1.1.2" xref="A1.1.p1.3.m3.1.1.2.cmml"><mi id="A1.1.p1.3.m3.1.1.2.2" xref="A1.1.p1.3.m3.1.1.2.2.cmml">Y</mi><mi id="A1.1.p1.3.m3.1.1.2.3" xref="A1.1.p1.3.m3.1.1.2.3.cmml">t</mi></msub><mo id="A1.1.p1.3.m3.1.1.1" xref="A1.1.p1.3.m3.1.1.1.cmml">=</mo><msub id="A1.1.p1.3.m3.1.1.3" xref="A1.1.p1.3.m3.1.1.3.cmml"><mi id="A1.1.p1.3.m3.1.1.3.2" xref="A1.1.p1.3.m3.1.1.3.2.cmml">X</mi><mrow id="A1.1.p1.3.m3.1.1.3.3" xref="A1.1.p1.3.m3.1.1.3.3.cmml"><mi id="A1.1.p1.3.m3.1.1.3.3.2" xref="A1.1.p1.3.m3.1.1.3.3.2.cmml">t</mi><mo id="A1.1.p1.3.m3.1.1.3.3.1" xref="A1.1.p1.3.m3.1.1.3.3.1.cmml">∧</mo><mi id="A1.1.p1.3.m3.1.1.3.3.3" xref="A1.1.p1.3.m3.1.1.3.3.3.cmml">τ</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.3.m3.1b"><apply id="A1.1.p1.3.m3.1.1.cmml" xref="A1.1.p1.3.m3.1.1"><eq id="A1.1.p1.3.m3.1.1.1.cmml" xref="A1.1.p1.3.m3.1.1.1"></eq><apply id="A1.1.p1.3.m3.1.1.2.cmml" xref="A1.1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.3.m3.1.1.2.1.cmml" xref="A1.1.p1.3.m3.1.1.2">subscript</csymbol><ci id="A1.1.p1.3.m3.1.1.2.2.cmml" xref="A1.1.p1.3.m3.1.1.2.2">𝑌</ci><ci id="A1.1.p1.3.m3.1.1.2.3.cmml" xref="A1.1.p1.3.m3.1.1.2.3">𝑡</ci></apply><apply id="A1.1.p1.3.m3.1.1.3.cmml" xref="A1.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.3.m3.1.1.3.1.cmml" xref="A1.1.p1.3.m3.1.1.3">subscript</csymbol><ci id="A1.1.p1.3.m3.1.1.3.2.cmml" xref="A1.1.p1.3.m3.1.1.3.2">𝑋</ci><apply id="A1.1.p1.3.m3.1.1.3.3.cmml" xref="A1.1.p1.3.m3.1.1.3.3"><and id="A1.1.p1.3.m3.1.1.3.3.1.cmml" xref="A1.1.p1.3.m3.1.1.3.3.1"></and><ci id="A1.1.p1.3.m3.1.1.3.3.2.cmml" xref="A1.1.p1.3.m3.1.1.3.3.2">𝑡</ci><ci id="A1.1.p1.3.m3.1.1.3.3.3.cmml" xref="A1.1.p1.3.m3.1.1.3.3.3">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.3.m3.1c">Y_{t}=X_{t\land\tau}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.3.m3.1d">italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_X start_POSTSUBSCRIPT italic_t ∧ italic_τ end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\tau=\min\{t:X_{t}>B\}" class="ltx_Math" display="inline" id="A1.1.p1.4.m4.2"><semantics id="A1.1.p1.4.m4.2a"><mrow id="A1.1.p1.4.m4.2.2" xref="A1.1.p1.4.m4.2.2.cmml"><mi id="A1.1.p1.4.m4.2.2.3" xref="A1.1.p1.4.m4.2.2.3.cmml">τ</mi><mo id="A1.1.p1.4.m4.2.2.2" xref="A1.1.p1.4.m4.2.2.2.cmml">=</mo><mrow id="A1.1.p1.4.m4.2.2.1.1" xref="A1.1.p1.4.m4.2.2.1.2.cmml"><mi id="A1.1.p1.4.m4.1.1" xref="A1.1.p1.4.m4.1.1.cmml">min</mi><mo id="A1.1.p1.4.m4.2.2.1.1a" xref="A1.1.p1.4.m4.2.2.1.2.cmml">⁡</mo><mrow id="A1.1.p1.4.m4.2.2.1.1.1" xref="A1.1.p1.4.m4.2.2.1.2.cmml"><mo id="A1.1.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="A1.1.p1.4.m4.2.2.1.2.cmml">{</mo><mrow id="A1.1.p1.4.m4.2.2.1.1.1.1" xref="A1.1.p1.4.m4.2.2.1.1.1.1.cmml"><mi id="A1.1.p1.4.m4.2.2.1.1.1.1.2" xref="A1.1.p1.4.m4.2.2.1.1.1.1.2.cmml">t</mi><mo id="A1.1.p1.4.m4.2.2.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="A1.1.p1.4.m4.2.2.1.1.1.1.1.cmml">:</mo><mrow id="A1.1.p1.4.m4.2.2.1.1.1.1.3" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.cmml"><msub id="A1.1.p1.4.m4.2.2.1.1.1.1.3.2" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.cmml"><mi id="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.2" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.2.cmml">X</mi><mi id="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.3" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.3.cmml">t</mi></msub><mo id="A1.1.p1.4.m4.2.2.1.1.1.1.3.1" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.1.cmml">></mo><mi id="A1.1.p1.4.m4.2.2.1.1.1.1.3.3" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.3.cmml">B</mi></mrow></mrow><mo id="A1.1.p1.4.m4.2.2.1.1.1.3" stretchy="false" xref="A1.1.p1.4.m4.2.2.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.4.m4.2b"><apply id="A1.1.p1.4.m4.2.2.cmml" xref="A1.1.p1.4.m4.2.2"><eq id="A1.1.p1.4.m4.2.2.2.cmml" xref="A1.1.p1.4.m4.2.2.2"></eq><ci id="A1.1.p1.4.m4.2.2.3.cmml" xref="A1.1.p1.4.m4.2.2.3">𝜏</ci><apply id="A1.1.p1.4.m4.2.2.1.2.cmml" xref="A1.1.p1.4.m4.2.2.1.1"><min id="A1.1.p1.4.m4.1.1.cmml" xref="A1.1.p1.4.m4.1.1"></min><apply id="A1.1.p1.4.m4.2.2.1.1.1.1.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1"><ci id="A1.1.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.1">:</ci><ci id="A1.1.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.2">𝑡</ci><apply id="A1.1.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3"><gt id="A1.1.p1.4.m4.2.2.1.1.1.1.3.1.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.1"></gt><apply id="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.1.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.2">subscript</csymbol><ci id="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.2.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.2">𝑋</ci><ci id="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.3.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.2.3">𝑡</ci></apply><ci id="A1.1.p1.4.m4.2.2.1.1.1.1.3.3.cmml" xref="A1.1.p1.4.m4.2.2.1.1.1.1.3.3">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.4.m4.2c">\tau=\min\{t:X_{t}>B\}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.4.m4.2d">italic_τ = roman_min { italic_t : italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT > italic_B }</annotation></semantics></math>. Then <math alttext="Y_{t}" class="ltx_Math" display="inline" id="A1.1.p1.5.m5.1"><semantics id="A1.1.p1.5.m5.1a"><msub id="A1.1.p1.5.m5.1.1" xref="A1.1.p1.5.m5.1.1.cmml"><mi id="A1.1.p1.5.m5.1.1.2" xref="A1.1.p1.5.m5.1.1.2.cmml">Y</mi><mi id="A1.1.p1.5.m5.1.1.3" xref="A1.1.p1.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.1.p1.5.m5.1b"><apply id="A1.1.p1.5.m5.1.1.cmml" xref="A1.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A1.1.p1.5.m5.1.1.1.cmml" xref="A1.1.p1.5.m5.1.1">subscript</csymbol><ci id="A1.1.p1.5.m5.1.1.2.cmml" xref="A1.1.p1.5.m5.1.1.2">𝑌</ci><ci id="A1.1.p1.5.m5.1.1.3.cmml" xref="A1.1.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.5.m5.1c">Y_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.5.m5.1d">italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is a bounded supermartingale, so by the Azuma-Hoeffding inequality <cite class="ltx_cite ltx_citemacro_cite">Azuma [<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib2" title="">1967</a>], Hoeffding [<a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#bib.bib13" title="">1963</a>]</cite>, we have <math alttext="\Pr[Y_{T}>B]\leq\delta" class="ltx_Math" display="inline" id="A1.1.p1.6.m6.2"><semantics id="A1.1.p1.6.m6.2a"><mrow id="A1.1.p1.6.m6.2.2" xref="A1.1.p1.6.m6.2.2.cmml"><mrow id="A1.1.p1.6.m6.2.2.1.1" xref="A1.1.p1.6.m6.2.2.1.2.cmml"><mi id="A1.1.p1.6.m6.1.1" xref="A1.1.p1.6.m6.1.1.cmml">Pr</mi><mo id="A1.1.p1.6.m6.2.2.1.1a" xref="A1.1.p1.6.m6.2.2.1.2.cmml">⁡</mo><mrow id="A1.1.p1.6.m6.2.2.1.1.1" xref="A1.1.p1.6.m6.2.2.1.2.cmml"><mo id="A1.1.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="A1.1.p1.6.m6.2.2.1.2.cmml">[</mo><mrow id="A1.1.p1.6.m6.2.2.1.1.1.1" xref="A1.1.p1.6.m6.2.2.1.1.1.1.cmml"><msub id="A1.1.p1.6.m6.2.2.1.1.1.1.2" xref="A1.1.p1.6.m6.2.2.1.1.1.1.2.cmml"><mi id="A1.1.p1.6.m6.2.2.1.1.1.1.2.2" xref="A1.1.p1.6.m6.2.2.1.1.1.1.2.2.cmml">Y</mi><mi id="A1.1.p1.6.m6.2.2.1.1.1.1.2.3" xref="A1.1.p1.6.m6.2.2.1.1.1.1.2.3.cmml">T</mi></msub><mo id="A1.1.p1.6.m6.2.2.1.1.1.1.1" xref="A1.1.p1.6.m6.2.2.1.1.1.1.1.cmml">></mo><mi id="A1.1.p1.6.m6.2.2.1.1.1.1.3" xref="A1.1.p1.6.m6.2.2.1.1.1.1.3.cmml">B</mi></mrow><mo id="A1.1.p1.6.m6.2.2.1.1.1.3" stretchy="false" xref="A1.1.p1.6.m6.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="A1.1.p1.6.m6.2.2.2" xref="A1.1.p1.6.m6.2.2.2.cmml">≤</mo><mi id="A1.1.p1.6.m6.2.2.3" xref="A1.1.p1.6.m6.2.2.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.6.m6.2b"><apply id="A1.1.p1.6.m6.2.2.cmml" xref="A1.1.p1.6.m6.2.2"><leq id="A1.1.p1.6.m6.2.2.2.cmml" xref="A1.1.p1.6.m6.2.2.2"></leq><apply id="A1.1.p1.6.m6.2.2.1.2.cmml" xref="A1.1.p1.6.m6.2.2.1.1"><csymbol cd="latexml" id="A1.1.p1.6.m6.1.1.cmml" xref="A1.1.p1.6.m6.1.1">probability</csymbol><apply id="A1.1.p1.6.m6.2.2.1.1.1.1.cmml" xref="A1.1.p1.6.m6.2.2.1.1.1.1"><gt id="A1.1.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="A1.1.p1.6.m6.2.2.1.1.1.1.1"></gt><apply id="A1.1.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="A1.1.p1.6.m6.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.6.m6.2.2.1.1.1.1.2.1.cmml" xref="A1.1.p1.6.m6.2.2.1.1.1.1.2">subscript</csymbol><ci id="A1.1.p1.6.m6.2.2.1.1.1.1.2.2.cmml" xref="A1.1.p1.6.m6.2.2.1.1.1.1.2.2">𝑌</ci><ci id="A1.1.p1.6.m6.2.2.1.1.1.1.2.3.cmml" xref="A1.1.p1.6.m6.2.2.1.1.1.1.2.3">𝑇</ci></apply><ci id="A1.1.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="A1.1.p1.6.m6.2.2.1.1.1.1.3">𝐵</ci></apply></apply><ci id="A1.1.p1.6.m6.2.2.3.cmml" xref="A1.1.p1.6.m6.2.2.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.6.m6.2c">\Pr[Y_{T}>B]\leq\delta</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.6.m6.2d">roman_Pr [ italic_Y start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT > italic_B ] ≤ italic_δ</annotation></semantics></math>. But, by construction, we have <math alttext="Y_{T}>B" class="ltx_Math" display="inline" id="A1.1.p1.7.m7.1"><semantics id="A1.1.p1.7.m7.1a"><mrow id="A1.1.p1.7.m7.1.1" xref="A1.1.p1.7.m7.1.1.cmml"><msub id="A1.1.p1.7.m7.1.1.2" xref="A1.1.p1.7.m7.1.1.2.cmml"><mi id="A1.1.p1.7.m7.1.1.2.2" xref="A1.1.p1.7.m7.1.1.2.2.cmml">Y</mi><mi id="A1.1.p1.7.m7.1.1.2.3" xref="A1.1.p1.7.m7.1.1.2.3.cmml">T</mi></msub><mo id="A1.1.p1.7.m7.1.1.1" xref="A1.1.p1.7.m7.1.1.1.cmml">></mo><mi id="A1.1.p1.7.m7.1.1.3" xref="A1.1.p1.7.m7.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.7.m7.1b"><apply id="A1.1.p1.7.m7.1.1.cmml" xref="A1.1.p1.7.m7.1.1"><gt id="A1.1.p1.7.m7.1.1.1.cmml" xref="A1.1.p1.7.m7.1.1.1"></gt><apply id="A1.1.p1.7.m7.1.1.2.cmml" xref="A1.1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.7.m7.1.1.2.1.cmml" xref="A1.1.p1.7.m7.1.1.2">subscript</csymbol><ci id="A1.1.p1.7.m7.1.1.2.2.cmml" xref="A1.1.p1.7.m7.1.1.2.2">𝑌</ci><ci id="A1.1.p1.7.m7.1.1.2.3.cmml" xref="A1.1.p1.7.m7.1.1.2.3">𝑇</ci></apply><ci id="A1.1.p1.7.m7.1.1.3.cmml" xref="A1.1.p1.7.m7.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.7.m7.1c">Y_{T}>B</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.7.m7.1d">italic_Y start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT > italic_B</annotation></semantics></math> if and only there exists some <math alttext="t\leq T" class="ltx_Math" display="inline" id="A1.1.p1.8.m8.1"><semantics id="A1.1.p1.8.m8.1a"><mrow id="A1.1.p1.8.m8.1.1" xref="A1.1.p1.8.m8.1.1.cmml"><mi id="A1.1.p1.8.m8.1.1.2" xref="A1.1.p1.8.m8.1.1.2.cmml">t</mi><mo id="A1.1.p1.8.m8.1.1.1" xref="A1.1.p1.8.m8.1.1.1.cmml">≤</mo><mi id="A1.1.p1.8.m8.1.1.3" xref="A1.1.p1.8.m8.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.8.m8.1b"><apply id="A1.1.p1.8.m8.1.1.cmml" xref="A1.1.p1.8.m8.1.1"><leq id="A1.1.p1.8.m8.1.1.1.cmml" xref="A1.1.p1.8.m8.1.1.1"></leq><ci id="A1.1.p1.8.m8.1.1.2.cmml" xref="A1.1.p1.8.m8.1.1.2">𝑡</ci><ci id="A1.1.p1.8.m8.1.1.3.cmml" xref="A1.1.p1.8.m8.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.8.m8.1c">t\leq T</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.8.m8.1d">italic_t ≤ italic_T</annotation></semantics></math> for which <math alttext="X_{t}>B" class="ltx_Math" display="inline" id="A1.1.p1.9.m9.1"><semantics id="A1.1.p1.9.m9.1a"><mrow id="A1.1.p1.9.m9.1.1" xref="A1.1.p1.9.m9.1.1.cmml"><msub id="A1.1.p1.9.m9.1.1.2" xref="A1.1.p1.9.m9.1.1.2.cmml"><mi id="A1.1.p1.9.m9.1.1.2.2" xref="A1.1.p1.9.m9.1.1.2.2.cmml">X</mi><mi id="A1.1.p1.9.m9.1.1.2.3" xref="A1.1.p1.9.m9.1.1.2.3.cmml">t</mi></msub><mo id="A1.1.p1.9.m9.1.1.1" xref="A1.1.p1.9.m9.1.1.1.cmml">></mo><mi id="A1.1.p1.9.m9.1.1.3" xref="A1.1.p1.9.m9.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.9.m9.1b"><apply id="A1.1.p1.9.m9.1.1.cmml" xref="A1.1.p1.9.m9.1.1"><gt id="A1.1.p1.9.m9.1.1.1.cmml" xref="A1.1.p1.9.m9.1.1.1"></gt><apply id="A1.1.p1.9.m9.1.1.2.cmml" xref="A1.1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.9.m9.1.1.2.1.cmml" xref="A1.1.p1.9.m9.1.1.2">subscript</csymbol><ci id="A1.1.p1.9.m9.1.1.2.2.cmml" xref="A1.1.p1.9.m9.1.1.2.2">𝑋</ci><ci id="A1.1.p1.9.m9.1.1.2.3.cmml" xref="A1.1.p1.9.m9.1.1.2.3">𝑡</ci></apply><ci id="A1.1.p1.9.m9.1.1.3.cmml" xref="A1.1.p1.9.m9.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.9.m9.1c">X_{t}>B</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.9.m9.1d">italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT > italic_B</annotation></semantics></math>. The claim follows. ∎</p> </div> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Details omitted from <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7" title="7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">7</span></a> </h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.1">In this section, we fill in details omitted from <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7" title="7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">7</span></a>.</p> </div> <section class="ltx_subsection" id="A2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">B.1 </span>Completed proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.Thmtheorem4" title="Proposition 7.4 (Signal-dependent payments can help in general). ‣ 7.3 Properties of correlated equilibria with payments ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">7.4</span></a> </h3> <div class="ltx_para" id="A2.SS1.p1"> <p class="ltx_p" id="A2.SS1.p1.9">We first show that the claimed CEP is actually a CEP.</p> <ul class="ltx_itemize" id="A2.I1"> <li class="ltx_item" id="A2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A2.I1.i1.p1"> <p class="ltx_p" id="A2.I1.i1.p1.3">Conditioned on P1 being recommended <math alttext="X" class="ltx_Math" display="inline" id="A2.I1.i1.p1.1.m1.1"><semantics id="A2.I1.i1.p1.1.m1.1a"><mi id="A2.I1.i1.p1.1.m1.1.1" xref="A2.I1.i1.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.1.m1.1b"><ci id="A2.I1.i1.p1.1.m1.1.1.cmml" xref="A2.I1.i1.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.1.m1.1d">italic_X</annotation></semantics></math>, P2’s action is deterministically <math alttext="Y" class="ltx_Math" display="inline" id="A2.I1.i1.p1.2.m2.1"><semantics id="A2.I1.i1.p1.2.m2.1a"><mi id="A2.I1.i1.p1.2.m2.1.1" xref="A2.I1.i1.p1.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.2.m2.1b"><ci id="A2.I1.i1.p1.2.m2.1.1.cmml" xref="A2.I1.i1.p1.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.2.m2.1d">italic_Y</annotation></semantics></math>, against which <math alttext="X" class="ltx_Math" display="inline" id="A2.I1.i1.p1.3.m3.1"><semantics id="A2.I1.i1.p1.3.m3.1a"><mi id="A2.I1.i1.p1.3.m3.1.1" xref="A2.I1.i1.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.3.m3.1b"><ci id="A2.I1.i1.p1.3.m3.1.1.cmml" xref="A2.I1.i1.p1.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.3.m3.1d">italic_X</annotation></semantics></math> is the best response for P1.</p> </div> </li> <li class="ltx_item" id="A2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A2.I1.i2.p1"> <p class="ltx_p" id="A2.I1.i2.p1.2">Conditioned on P1 being recommended <math alttext="Y" class="ltx_Math" display="inline" id="A2.I1.i2.p1.1.m1.1"><semantics id="A2.I1.i2.p1.1.m1.1a"><mi id="A2.I1.i2.p1.1.m1.1.1" xref="A2.I1.i2.p1.1.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.1.m1.1b"><ci id="A2.I1.i2.p1.1.m1.1.1.cmml" xref="A2.I1.i2.p1.1.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.1.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.1.m1.1d">italic_Y</annotation></semantics></math>, P2’s action is uniform random, against which <math alttext="Y" class="ltx_Math" display="inline" id="A2.I1.i2.p1.2.m2.1"><semantics id="A2.I1.i2.p1.2.m2.1a"><mi id="A2.I1.i2.p1.2.m2.1.1" xref="A2.I1.i2.p1.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.2.m2.1b"><ci id="A2.I1.i2.p1.2.m2.1.1.cmml" xref="A2.I1.i2.p1.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.2.m2.1d">italic_Y</annotation></semantics></math> is a best response for P1.</p> </div> </li> <li class="ltx_item" id="A2.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A2.I1.i3.p1"> <p class="ltx_p" id="A2.I1.i3.p1.4">Conditioned on P2 being recommended <math alttext="X" class="ltx_Math" display="inline" id="A2.I1.i3.p1.1.m1.1"><semantics id="A2.I1.i3.p1.1.m1.1a"><mi id="A2.I1.i3.p1.1.m1.1.1" xref="A2.I1.i3.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i3.p1.1.m1.1b"><ci id="A2.I1.i3.p1.1.m1.1.1.cmml" xref="A2.I1.i3.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i3.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i3.p1.1.m1.1d">italic_X</annotation></semantics></math>, P1’s action is deterministically <math alttext="Y" class="ltx_Math" display="inline" id="A2.I1.i3.p1.2.m2.1"><semantics id="A2.I1.i3.p1.2.m2.1a"><mi id="A2.I1.i3.p1.2.m2.1.1" xref="A2.I1.i3.p1.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i3.p1.2.m2.1b"><ci id="A2.I1.i3.p1.2.m2.1.1.cmml" xref="A2.I1.i3.p1.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i3.p1.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i3.p1.2.m2.1d">italic_Y</annotation></semantics></math>, against which the principal’s promised payment of <math alttext="1" class="ltx_Math" display="inline" id="A2.I1.i3.p1.3.m3.1"><semantics id="A2.I1.i3.p1.3.m3.1a"><mn id="A2.I1.i3.p1.3.m3.1.1" xref="A2.I1.i3.p1.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A2.I1.i3.p1.3.m3.1b"><cn id="A2.I1.i3.p1.3.m3.1.1.cmml" type="integer" xref="A2.I1.i3.p1.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i3.p1.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i3.p1.3.m3.1d">1</annotation></semantics></math> makes <math alttext="X" class="ltx_Math" display="inline" id="A2.I1.i3.p1.4.m4.1"><semantics id="A2.I1.i3.p1.4.m4.1a"><mi id="A2.I1.i3.p1.4.m4.1.1" xref="A2.I1.i3.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i3.p1.4.m4.1b"><ci id="A2.I1.i3.p1.4.m4.1.1.cmml" xref="A2.I1.i3.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i3.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i3.p1.4.m4.1d">italic_X</annotation></semantics></math> a best response for P2.</p> </div> </li> <li class="ltx_item" id="A2.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A2.I1.i4.p1"> <p class="ltx_p" id="A2.I1.i4.p1.2">Conditioned on P2 being recommended <math alttext="Y" class="ltx_Math" display="inline" id="A2.I1.i4.p1.1.m1.1"><semantics id="A2.I1.i4.p1.1.m1.1a"><mi id="A2.I1.i4.p1.1.m1.1.1" xref="A2.I1.i4.p1.1.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i4.p1.1.m1.1b"><ci id="A2.I1.i4.p1.1.m1.1.1.cmml" xref="A2.I1.i4.p1.1.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i4.p1.1.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i4.p1.1.m1.1d">italic_Y</annotation></semantics></math>, P1’s action is uniform random, against which <math alttext="Y" class="ltx_Math" display="inline" id="A2.I1.i4.p1.2.m2.1"><semantics id="A2.I1.i4.p1.2.m2.1a"><mi id="A2.I1.i4.p1.2.m2.1.1" xref="A2.I1.i4.p1.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I1.i4.p1.2.m2.1b"><ci id="A2.I1.i4.p1.2.m2.1.1.cmml" xref="A2.I1.i4.p1.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i4.p1.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i4.p1.2.m2.1d">italic_Y</annotation></semantics></math> is a best response for P2.</p> </div> </li> </ul> <p class="ltx_p" id="A2.SS1.p1.8">It remains only to show that objective value <math alttext="-1/3" class="ltx_Math" display="inline" id="A2.SS1.p1.1.m1.1"><semantics id="A2.SS1.p1.1.m1.1a"><mrow id="A2.SS1.p1.1.m1.1.1" xref="A2.SS1.p1.1.m1.1.1.cmml"><mo id="A2.SS1.p1.1.m1.1.1a" xref="A2.SS1.p1.1.m1.1.1.cmml">−</mo><mrow id="A2.SS1.p1.1.m1.1.1.2" xref="A2.SS1.p1.1.m1.1.1.2.cmml"><mn id="A2.SS1.p1.1.m1.1.1.2.2" xref="A2.SS1.p1.1.m1.1.1.2.2.cmml">1</mn><mo id="A2.SS1.p1.1.m1.1.1.2.1" xref="A2.SS1.p1.1.m1.1.1.2.1.cmml">/</mo><mn id="A2.SS1.p1.1.m1.1.1.2.3" xref="A2.SS1.p1.1.m1.1.1.2.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.1.m1.1b"><apply id="A2.SS1.p1.1.m1.1.1.cmml" xref="A2.SS1.p1.1.m1.1.1"><minus id="A2.SS1.p1.1.m1.1.1.1.cmml" xref="A2.SS1.p1.1.m1.1.1"></minus><apply id="A2.SS1.p1.1.m1.1.1.2.cmml" xref="A2.SS1.p1.1.m1.1.1.2"><divide id="A2.SS1.p1.1.m1.1.1.2.1.cmml" xref="A2.SS1.p1.1.m1.1.1.2.1"></divide><cn id="A2.SS1.p1.1.m1.1.1.2.2.cmml" type="integer" xref="A2.SS1.p1.1.m1.1.1.2.2">1</cn><cn id="A2.SS1.p1.1.m1.1.1.2.3.cmml" type="integer" xref="A2.SS1.p1.1.m1.1.1.2.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.1.m1.1c">-1/3</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.1.m1.1d">- 1 / 3</annotation></semantics></math> cannot be achieved by any CEP in which payments are signal-independent. Intuitively, this will follow because, in the above <math alttext="\mu" class="ltx_Math" display="inline" id="A2.SS1.p1.2.m2.1"><semantics id="A2.SS1.p1.2.m2.1a"><mi id="A2.SS1.p1.2.m2.1.1" xref="A2.SS1.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.2.m2.1b"><ci id="A2.SS1.p1.2.m2.1.1.cmml" xref="A2.SS1.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.2.m2.1d">italic_μ</annotation></semantics></math>, it is vital that, if the principal recommends <math alttext="(Y,Y)" class="ltx_Math" display="inline" id="A2.SS1.p1.3.m3.2"><semantics id="A2.SS1.p1.3.m3.2a"><mrow id="A2.SS1.p1.3.m3.2.3.2" xref="A2.SS1.p1.3.m3.2.3.1.cmml"><mo id="A2.SS1.p1.3.m3.2.3.2.1" stretchy="false" xref="A2.SS1.p1.3.m3.2.3.1.cmml">(</mo><mi id="A2.SS1.p1.3.m3.1.1" xref="A2.SS1.p1.3.m3.1.1.cmml">Y</mi><mo id="A2.SS1.p1.3.m3.2.3.2.2" xref="A2.SS1.p1.3.m3.2.3.1.cmml">,</mo><mi id="A2.SS1.p1.3.m3.2.2" xref="A2.SS1.p1.3.m3.2.2.cmml">Y</mi><mo id="A2.SS1.p1.3.m3.2.3.2.3" stretchy="false" xref="A2.SS1.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.3.m3.2b"><interval closure="open" id="A2.SS1.p1.3.m3.2.3.1.cmml" xref="A2.SS1.p1.3.m3.2.3.2"><ci id="A2.SS1.p1.3.m3.1.1.cmml" xref="A2.SS1.p1.3.m3.1.1">𝑌</ci><ci id="A2.SS1.p1.3.m3.2.2.cmml" xref="A2.SS1.p1.3.m3.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.3.m3.2c">(Y,Y)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.3.m3.2d">( italic_Y , italic_Y )</annotation></semantics></math>, P2 <span class="ltx_text ltx_font_italic" id="A2.SS1.p1.8.1">not</span> be paid for deviating to <math alttext="(Y,X)" class="ltx_Math" display="inline" id="A2.SS1.p1.4.m4.2"><semantics id="A2.SS1.p1.4.m4.2a"><mrow id="A2.SS1.p1.4.m4.2.3.2" xref="A2.SS1.p1.4.m4.2.3.1.cmml"><mo id="A2.SS1.p1.4.m4.2.3.2.1" stretchy="false" xref="A2.SS1.p1.4.m4.2.3.1.cmml">(</mo><mi id="A2.SS1.p1.4.m4.1.1" xref="A2.SS1.p1.4.m4.1.1.cmml">Y</mi><mo id="A2.SS1.p1.4.m4.2.3.2.2" xref="A2.SS1.p1.4.m4.2.3.1.cmml">,</mo><mi id="A2.SS1.p1.4.m4.2.2" xref="A2.SS1.p1.4.m4.2.2.cmml">X</mi><mo id="A2.SS1.p1.4.m4.2.3.2.3" stretchy="false" xref="A2.SS1.p1.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.4.m4.2b"><interval closure="open" id="A2.SS1.p1.4.m4.2.3.1.cmml" xref="A2.SS1.p1.4.m4.2.3.2"><ci id="A2.SS1.p1.4.m4.1.1.cmml" xref="A2.SS1.p1.4.m4.1.1">𝑌</ci><ci id="A2.SS1.p1.4.m4.2.2.cmml" xref="A2.SS1.p1.4.m4.2.2">𝑋</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.4.m4.2c">(Y,X)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.4.m4.2d">( italic_Y , italic_X )</annotation></semantics></math>. We now show this formally. Let <math alttext="(\mu,P)" class="ltx_Math" display="inline" id="A2.SS1.p1.5.m5.2"><semantics id="A2.SS1.p1.5.m5.2a"><mrow id="A2.SS1.p1.5.m5.2.3.2" xref="A2.SS1.p1.5.m5.2.3.1.cmml"><mo id="A2.SS1.p1.5.m5.2.3.2.1" stretchy="false" xref="A2.SS1.p1.5.m5.2.3.1.cmml">(</mo><mi id="A2.SS1.p1.5.m5.1.1" xref="A2.SS1.p1.5.m5.1.1.cmml">μ</mi><mo id="A2.SS1.p1.5.m5.2.3.2.2" xref="A2.SS1.p1.5.m5.2.3.1.cmml">,</mo><mi id="A2.SS1.p1.5.m5.2.2" xref="A2.SS1.p1.5.m5.2.2.cmml">P</mi><mo id="A2.SS1.p1.5.m5.2.3.2.3" stretchy="false" xref="A2.SS1.p1.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.5.m5.2b"><interval closure="open" id="A2.SS1.p1.5.m5.2.3.1.cmml" xref="A2.SS1.p1.5.m5.2.3.2"><ci id="A2.SS1.p1.5.m5.1.1.cmml" xref="A2.SS1.p1.5.m5.1.1">𝜇</ci><ci id="A2.SS1.p1.5.m5.2.2.cmml" xref="A2.SS1.p1.5.m5.2.2">𝑃</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.5.m5.2c">(\mu,P)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.5.m5.2d">( italic_μ , italic_P )</annotation></semantics></math> satisfy the independence property. Equivalently, <math alttext="\mu\in\Delta(A)" class="ltx_Math" display="inline" id="A2.SS1.p1.6.m6.1"><semantics id="A2.SS1.p1.6.m6.1a"><mrow id="A2.SS1.p1.6.m6.1.2" xref="A2.SS1.p1.6.m6.1.2.cmml"><mi id="A2.SS1.p1.6.m6.1.2.2" xref="A2.SS1.p1.6.m6.1.2.2.cmml">μ</mi><mo id="A2.SS1.p1.6.m6.1.2.1" xref="A2.SS1.p1.6.m6.1.2.1.cmml">∈</mo><mrow id="A2.SS1.p1.6.m6.1.2.3" xref="A2.SS1.p1.6.m6.1.2.3.cmml"><mi id="A2.SS1.p1.6.m6.1.2.3.2" mathvariant="normal" xref="A2.SS1.p1.6.m6.1.2.3.2.cmml">Δ</mi><mo id="A2.SS1.p1.6.m6.1.2.3.1" xref="A2.SS1.p1.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="A2.SS1.p1.6.m6.1.2.3.3.2" xref="A2.SS1.p1.6.m6.1.2.3.cmml"><mo id="A2.SS1.p1.6.m6.1.2.3.3.2.1" stretchy="false" xref="A2.SS1.p1.6.m6.1.2.3.cmml">(</mo><mi id="A2.SS1.p1.6.m6.1.1" xref="A2.SS1.p1.6.m6.1.1.cmml">A</mi><mo id="A2.SS1.p1.6.m6.1.2.3.3.2.2" stretchy="false" xref="A2.SS1.p1.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.6.m6.1b"><apply id="A2.SS1.p1.6.m6.1.2.cmml" xref="A2.SS1.p1.6.m6.1.2"><in id="A2.SS1.p1.6.m6.1.2.1.cmml" xref="A2.SS1.p1.6.m6.1.2.1"></in><ci id="A2.SS1.p1.6.m6.1.2.2.cmml" xref="A2.SS1.p1.6.m6.1.2.2">𝜇</ci><apply id="A2.SS1.p1.6.m6.1.2.3.cmml" xref="A2.SS1.p1.6.m6.1.2.3"><times id="A2.SS1.p1.6.m6.1.2.3.1.cmml" xref="A2.SS1.p1.6.m6.1.2.3.1"></times><ci id="A2.SS1.p1.6.m6.1.2.3.2.cmml" xref="A2.SS1.p1.6.m6.1.2.3.2">Δ</ci><ci id="A2.SS1.p1.6.m6.1.1.cmml" xref="A2.SS1.p1.6.m6.1.1">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.6.m6.1c">\mu\in\Delta(A)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.6.m6.1d">italic_μ ∈ roman_Δ ( italic_A )</annotation></semantics></math> is a correlated equilibrium of the game with utility function <math alttext="U+P" class="ltx_Math" display="inline" id="A2.SS1.p1.7.m7.1"><semantics id="A2.SS1.p1.7.m7.1a"><mrow id="A2.SS1.p1.7.m7.1.1" xref="A2.SS1.p1.7.m7.1.1.cmml"><mi id="A2.SS1.p1.7.m7.1.1.2" xref="A2.SS1.p1.7.m7.1.1.2.cmml">U</mi><mo id="A2.SS1.p1.7.m7.1.1.1" xref="A2.SS1.p1.7.m7.1.1.1.cmml">+</mo><mi id="A2.SS1.p1.7.m7.1.1.3" xref="A2.SS1.p1.7.m7.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.7.m7.1b"><apply id="A2.SS1.p1.7.m7.1.1.cmml" xref="A2.SS1.p1.7.m7.1.1"><plus id="A2.SS1.p1.7.m7.1.1.1.cmml" xref="A2.SS1.p1.7.m7.1.1.1"></plus><ci id="A2.SS1.p1.7.m7.1.1.2.cmml" xref="A2.SS1.p1.7.m7.1.1.2">𝑈</ci><ci id="A2.SS1.p1.7.m7.1.1.3.cmml" xref="A2.SS1.p1.7.m7.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.7.m7.1c">U+P</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.7.m7.1d">italic_U + italic_P</annotation></semantics></math>. Assume for contradiction that this is the case, and that the expected payment is <math alttext="\sum_{a}\mu(a)P(a)\leq 1/3" class="ltx_Math" display="inline" id="A2.SS1.p1.8.m8.2"><semantics id="A2.SS1.p1.8.m8.2a"><mrow id="A2.SS1.p1.8.m8.2.3" xref="A2.SS1.p1.8.m8.2.3.cmml"><mrow id="A2.SS1.p1.8.m8.2.3.2" xref="A2.SS1.p1.8.m8.2.3.2.cmml"><msub id="A2.SS1.p1.8.m8.2.3.2.1" xref="A2.SS1.p1.8.m8.2.3.2.1.cmml"><mo id="A2.SS1.p1.8.m8.2.3.2.1.2" xref="A2.SS1.p1.8.m8.2.3.2.1.2.cmml">∑</mo><mi id="A2.SS1.p1.8.m8.2.3.2.1.3" xref="A2.SS1.p1.8.m8.2.3.2.1.3.cmml">a</mi></msub><mrow id="A2.SS1.p1.8.m8.2.3.2.2" xref="A2.SS1.p1.8.m8.2.3.2.2.cmml"><mi id="A2.SS1.p1.8.m8.2.3.2.2.2" xref="A2.SS1.p1.8.m8.2.3.2.2.2.cmml">μ</mi><mo id="A2.SS1.p1.8.m8.2.3.2.2.1" xref="A2.SS1.p1.8.m8.2.3.2.2.1.cmml">⁢</mo><mrow id="A2.SS1.p1.8.m8.2.3.2.2.3.2" xref="A2.SS1.p1.8.m8.2.3.2.2.cmml"><mo id="A2.SS1.p1.8.m8.2.3.2.2.3.2.1" stretchy="false" xref="A2.SS1.p1.8.m8.2.3.2.2.cmml">(</mo><mi id="A2.SS1.p1.8.m8.1.1" xref="A2.SS1.p1.8.m8.1.1.cmml">a</mi><mo id="A2.SS1.p1.8.m8.2.3.2.2.3.2.2" stretchy="false" xref="A2.SS1.p1.8.m8.2.3.2.2.cmml">)</mo></mrow><mo id="A2.SS1.p1.8.m8.2.3.2.2.1a" xref="A2.SS1.p1.8.m8.2.3.2.2.1.cmml">⁢</mo><mi id="A2.SS1.p1.8.m8.2.3.2.2.4" xref="A2.SS1.p1.8.m8.2.3.2.2.4.cmml">P</mi><mo id="A2.SS1.p1.8.m8.2.3.2.2.1b" xref="A2.SS1.p1.8.m8.2.3.2.2.1.cmml">⁢</mo><mrow id="A2.SS1.p1.8.m8.2.3.2.2.5.2" xref="A2.SS1.p1.8.m8.2.3.2.2.cmml"><mo id="A2.SS1.p1.8.m8.2.3.2.2.5.2.1" stretchy="false" xref="A2.SS1.p1.8.m8.2.3.2.2.cmml">(</mo><mi id="A2.SS1.p1.8.m8.2.2" xref="A2.SS1.p1.8.m8.2.2.cmml">a</mi><mo id="A2.SS1.p1.8.m8.2.3.2.2.5.2.2" stretchy="false" xref="A2.SS1.p1.8.m8.2.3.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A2.SS1.p1.8.m8.2.3.1" xref="A2.SS1.p1.8.m8.2.3.1.cmml">≤</mo><mrow id="A2.SS1.p1.8.m8.2.3.3" xref="A2.SS1.p1.8.m8.2.3.3.cmml"><mn id="A2.SS1.p1.8.m8.2.3.3.2" xref="A2.SS1.p1.8.m8.2.3.3.2.cmml">1</mn><mo id="A2.SS1.p1.8.m8.2.3.3.1" xref="A2.SS1.p1.8.m8.2.3.3.1.cmml">/</mo><mn id="A2.SS1.p1.8.m8.2.3.3.3" xref="A2.SS1.p1.8.m8.2.3.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.8.m8.2b"><apply id="A2.SS1.p1.8.m8.2.3.cmml" xref="A2.SS1.p1.8.m8.2.3"><leq id="A2.SS1.p1.8.m8.2.3.1.cmml" xref="A2.SS1.p1.8.m8.2.3.1"></leq><apply id="A2.SS1.p1.8.m8.2.3.2.cmml" xref="A2.SS1.p1.8.m8.2.3.2"><apply id="A2.SS1.p1.8.m8.2.3.2.1.cmml" xref="A2.SS1.p1.8.m8.2.3.2.1"><csymbol cd="ambiguous" id="A2.SS1.p1.8.m8.2.3.2.1.1.cmml" xref="A2.SS1.p1.8.m8.2.3.2.1">subscript</csymbol><sum id="A2.SS1.p1.8.m8.2.3.2.1.2.cmml" xref="A2.SS1.p1.8.m8.2.3.2.1.2"></sum><ci id="A2.SS1.p1.8.m8.2.3.2.1.3.cmml" xref="A2.SS1.p1.8.m8.2.3.2.1.3">𝑎</ci></apply><apply id="A2.SS1.p1.8.m8.2.3.2.2.cmml" xref="A2.SS1.p1.8.m8.2.3.2.2"><times id="A2.SS1.p1.8.m8.2.3.2.2.1.cmml" xref="A2.SS1.p1.8.m8.2.3.2.2.1"></times><ci id="A2.SS1.p1.8.m8.2.3.2.2.2.cmml" xref="A2.SS1.p1.8.m8.2.3.2.2.2">𝜇</ci><ci id="A2.SS1.p1.8.m8.1.1.cmml" xref="A2.SS1.p1.8.m8.1.1">𝑎</ci><ci id="A2.SS1.p1.8.m8.2.3.2.2.4.cmml" xref="A2.SS1.p1.8.m8.2.3.2.2.4">𝑃</ci><ci id="A2.SS1.p1.8.m8.2.2.cmml" xref="A2.SS1.p1.8.m8.2.2">𝑎</ci></apply></apply><apply id="A2.SS1.p1.8.m8.2.3.3.cmml" xref="A2.SS1.p1.8.m8.2.3.3"><divide id="A2.SS1.p1.8.m8.2.3.3.1.cmml" xref="A2.SS1.p1.8.m8.2.3.3.1"></divide><cn id="A2.SS1.p1.8.m8.2.3.3.2.cmml" type="integer" xref="A2.SS1.p1.8.m8.2.3.3.2">1</cn><cn id="A2.SS1.p1.8.m8.2.3.3.3.cmml" type="integer" xref="A2.SS1.p1.8.m8.2.3.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.8.m8.2c">\sum_{a}\mu(a)P(a)\leq 1/3</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p1.8.m8.2d">∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_μ ( italic_a ) italic_P ( italic_a ) ≤ 1 / 3</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.SS1.p2"> <p class="ltx_p" id="A2.SS1.p2.2">Since <math alttext="U_{P}(X,X)=-\infty" class="ltx_Math" display="inline" id="A2.SS1.p2.1.m1.2"><semantics id="A2.SS1.p2.1.m1.2a"><mrow id="A2.SS1.p2.1.m1.2.3" xref="A2.SS1.p2.1.m1.2.3.cmml"><mrow id="A2.SS1.p2.1.m1.2.3.2" xref="A2.SS1.p2.1.m1.2.3.2.cmml"><msub id="A2.SS1.p2.1.m1.2.3.2.2" xref="A2.SS1.p2.1.m1.2.3.2.2.cmml"><mi id="A2.SS1.p2.1.m1.2.3.2.2.2" xref="A2.SS1.p2.1.m1.2.3.2.2.2.cmml">U</mi><mi id="A2.SS1.p2.1.m1.2.3.2.2.3" xref="A2.SS1.p2.1.m1.2.3.2.2.3.cmml">P</mi></msub><mo id="A2.SS1.p2.1.m1.2.3.2.1" xref="A2.SS1.p2.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.p2.1.m1.2.3.2.3.2" xref="A2.SS1.p2.1.m1.2.3.2.3.1.cmml"><mo id="A2.SS1.p2.1.m1.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.p2.1.m1.2.3.2.3.1.cmml">(</mo><mi id="A2.SS1.p2.1.m1.1.1" xref="A2.SS1.p2.1.m1.1.1.cmml">X</mi><mo id="A2.SS1.p2.1.m1.2.3.2.3.2.2" xref="A2.SS1.p2.1.m1.2.3.2.3.1.cmml">,</mo><mi id="A2.SS1.p2.1.m1.2.2" xref="A2.SS1.p2.1.m1.2.2.cmml">X</mi><mo id="A2.SS1.p2.1.m1.2.3.2.3.2.3" stretchy="false" xref="A2.SS1.p2.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.p2.1.m1.2.3.1" xref="A2.SS1.p2.1.m1.2.3.1.cmml">=</mo><mrow id="A2.SS1.p2.1.m1.2.3.3" xref="A2.SS1.p2.1.m1.2.3.3.cmml"><mo id="A2.SS1.p2.1.m1.2.3.3a" xref="A2.SS1.p2.1.m1.2.3.3.cmml">−</mo><mi id="A2.SS1.p2.1.m1.2.3.3.2" mathvariant="normal" xref="A2.SS1.p2.1.m1.2.3.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p2.1.m1.2b"><apply id="A2.SS1.p2.1.m1.2.3.cmml" xref="A2.SS1.p2.1.m1.2.3"><eq id="A2.SS1.p2.1.m1.2.3.1.cmml" xref="A2.SS1.p2.1.m1.2.3.1"></eq><apply id="A2.SS1.p2.1.m1.2.3.2.cmml" xref="A2.SS1.p2.1.m1.2.3.2"><times id="A2.SS1.p2.1.m1.2.3.2.1.cmml" xref="A2.SS1.p2.1.m1.2.3.2.1"></times><apply id="A2.SS1.p2.1.m1.2.3.2.2.cmml" xref="A2.SS1.p2.1.m1.2.3.2.2"><csymbol cd="ambiguous" id="A2.SS1.p2.1.m1.2.3.2.2.1.cmml" xref="A2.SS1.p2.1.m1.2.3.2.2">subscript</csymbol><ci id="A2.SS1.p2.1.m1.2.3.2.2.2.cmml" xref="A2.SS1.p2.1.m1.2.3.2.2.2">𝑈</ci><ci id="A2.SS1.p2.1.m1.2.3.2.2.3.cmml" xref="A2.SS1.p2.1.m1.2.3.2.2.3">𝑃</ci></apply><interval closure="open" id="A2.SS1.p2.1.m1.2.3.2.3.1.cmml" xref="A2.SS1.p2.1.m1.2.3.2.3.2"><ci id="A2.SS1.p2.1.m1.1.1.cmml" xref="A2.SS1.p2.1.m1.1.1">𝑋</ci><ci id="A2.SS1.p2.1.m1.2.2.cmml" xref="A2.SS1.p2.1.m1.2.2">𝑋</ci></interval></apply><apply id="A2.SS1.p2.1.m1.2.3.3.cmml" xref="A2.SS1.p2.1.m1.2.3.3"><minus id="A2.SS1.p2.1.m1.2.3.3.1.cmml" xref="A2.SS1.p2.1.m1.2.3.3"></minus><infinity id="A2.SS1.p2.1.m1.2.3.3.2.cmml" xref="A2.SS1.p2.1.m1.2.3.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p2.1.m1.2c">U_{P}(X,X)=-\infty</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p2.1.m1.2d">italic_U start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_X , italic_X ) = - ∞</annotation></semantics></math>, we must have <math alttext="\mu(X,X)=0" class="ltx_Math" display="inline" id="A2.SS1.p2.2.m2.2"><semantics id="A2.SS1.p2.2.m2.2a"><mrow id="A2.SS1.p2.2.m2.2.3" xref="A2.SS1.p2.2.m2.2.3.cmml"><mrow id="A2.SS1.p2.2.m2.2.3.2" xref="A2.SS1.p2.2.m2.2.3.2.cmml"><mi id="A2.SS1.p2.2.m2.2.3.2.2" xref="A2.SS1.p2.2.m2.2.3.2.2.cmml">μ</mi><mo id="A2.SS1.p2.2.m2.2.3.2.1" xref="A2.SS1.p2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.p2.2.m2.2.3.2.3.2" xref="A2.SS1.p2.2.m2.2.3.2.3.1.cmml"><mo id="A2.SS1.p2.2.m2.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.p2.2.m2.2.3.2.3.1.cmml">(</mo><mi id="A2.SS1.p2.2.m2.1.1" xref="A2.SS1.p2.2.m2.1.1.cmml">X</mi><mo id="A2.SS1.p2.2.m2.2.3.2.3.2.2" xref="A2.SS1.p2.2.m2.2.3.2.3.1.cmml">,</mo><mi id="A2.SS1.p2.2.m2.2.2" xref="A2.SS1.p2.2.m2.2.2.cmml">X</mi><mo id="A2.SS1.p2.2.m2.2.3.2.3.2.3" stretchy="false" xref="A2.SS1.p2.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.p2.2.m2.2.3.1" xref="A2.SS1.p2.2.m2.2.3.1.cmml">=</mo><mn id="A2.SS1.p2.2.m2.2.3.3" xref="A2.SS1.p2.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p2.2.m2.2b"><apply id="A2.SS1.p2.2.m2.2.3.cmml" xref="A2.SS1.p2.2.m2.2.3"><eq id="A2.SS1.p2.2.m2.2.3.1.cmml" xref="A2.SS1.p2.2.m2.2.3.1"></eq><apply id="A2.SS1.p2.2.m2.2.3.2.cmml" xref="A2.SS1.p2.2.m2.2.3.2"><times id="A2.SS1.p2.2.m2.2.3.2.1.cmml" xref="A2.SS1.p2.2.m2.2.3.2.1"></times><ci id="A2.SS1.p2.2.m2.2.3.2.2.cmml" xref="A2.SS1.p2.2.m2.2.3.2.2">𝜇</ci><interval closure="open" id="A2.SS1.p2.2.m2.2.3.2.3.1.cmml" xref="A2.SS1.p2.2.m2.2.3.2.3.2"><ci id="A2.SS1.p2.2.m2.1.1.cmml" xref="A2.SS1.p2.2.m2.1.1">𝑋</ci><ci id="A2.SS1.p2.2.m2.2.2.cmml" xref="A2.SS1.p2.2.m2.2.2">𝑋</ci></interval></apply><cn id="A2.SS1.p2.2.m2.2.3.3.cmml" type="integer" xref="A2.SS1.p2.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p2.2.m2.2c">\mu(X,X)=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p2.2.m2.2d">italic_μ ( italic_X , italic_X ) = 0</annotation></semantics></math>. We consider two cases.</p> <ul class="ltx_itemize" id="A2.I2"> <li class="ltx_item" id="A2.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A2.I2.i1.p1"> <p class="ltx_p" id="A2.I2.i1.p1.11"><span class="ltx_text ltx_font_italic" id="A2.I2.i1.p1.11.1">Case 1:</span> <math alttext="\mu(Y,X)>0" class="ltx_Math" display="inline" id="A2.I2.i1.p1.1.m1.2"><semantics id="A2.I2.i1.p1.1.m1.2a"><mrow id="A2.I2.i1.p1.1.m1.2.3" xref="A2.I2.i1.p1.1.m1.2.3.cmml"><mrow id="A2.I2.i1.p1.1.m1.2.3.2" xref="A2.I2.i1.p1.1.m1.2.3.2.cmml"><mi id="A2.I2.i1.p1.1.m1.2.3.2.2" xref="A2.I2.i1.p1.1.m1.2.3.2.2.cmml">μ</mi><mo id="A2.I2.i1.p1.1.m1.2.3.2.1" xref="A2.I2.i1.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.1.m1.2.3.2.3.2" xref="A2.I2.i1.p1.1.m1.2.3.2.3.1.cmml"><mo id="A2.I2.i1.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="A2.I2.i1.p1.1.m1.2.3.2.3.1.cmml">(</mo><mi id="A2.I2.i1.p1.1.m1.1.1" xref="A2.I2.i1.p1.1.m1.1.1.cmml">Y</mi><mo id="A2.I2.i1.p1.1.m1.2.3.2.3.2.2" xref="A2.I2.i1.p1.1.m1.2.3.2.3.1.cmml">,</mo><mi id="A2.I2.i1.p1.1.m1.2.2" xref="A2.I2.i1.p1.1.m1.2.2.cmml">X</mi><mo id="A2.I2.i1.p1.1.m1.2.3.2.3.2.3" stretchy="false" xref="A2.I2.i1.p1.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i1.p1.1.m1.2.3.1" xref="A2.I2.i1.p1.1.m1.2.3.1.cmml">></mo><mn id="A2.I2.i1.p1.1.m1.2.3.3" xref="A2.I2.i1.p1.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.1.m1.2b"><apply id="A2.I2.i1.p1.1.m1.2.3.cmml" xref="A2.I2.i1.p1.1.m1.2.3"><gt id="A2.I2.i1.p1.1.m1.2.3.1.cmml" xref="A2.I2.i1.p1.1.m1.2.3.1"></gt><apply id="A2.I2.i1.p1.1.m1.2.3.2.cmml" xref="A2.I2.i1.p1.1.m1.2.3.2"><times id="A2.I2.i1.p1.1.m1.2.3.2.1.cmml" xref="A2.I2.i1.p1.1.m1.2.3.2.1"></times><ci id="A2.I2.i1.p1.1.m1.2.3.2.2.cmml" xref="A2.I2.i1.p1.1.m1.2.3.2.2">𝜇</ci><interval closure="open" id="A2.I2.i1.p1.1.m1.2.3.2.3.1.cmml" xref="A2.I2.i1.p1.1.m1.2.3.2.3.2"><ci id="A2.I2.i1.p1.1.m1.1.1.cmml" xref="A2.I2.i1.p1.1.m1.1.1">𝑌</ci><ci id="A2.I2.i1.p1.1.m1.2.2.cmml" xref="A2.I2.i1.p1.1.m1.2.2">𝑋</ci></interval></apply><cn id="A2.I2.i1.p1.1.m1.2.3.3.cmml" type="integer" xref="A2.I2.i1.p1.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.1.m1.2c">\mu(Y,X)>0</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.1.m1.2d">italic_μ ( italic_Y , italic_X ) > 0</annotation></semantics></math>. Then <math alttext="U_{2}^{P}(Y,X)\geq U_{2}^{P}(Y,Y)" class="ltx_Math" display="inline" id="A2.I2.i1.p1.2.m2.4"><semantics id="A2.I2.i1.p1.2.m2.4a"><mrow id="A2.I2.i1.p1.2.m2.4.5" xref="A2.I2.i1.p1.2.m2.4.5.cmml"><mrow id="A2.I2.i1.p1.2.m2.4.5.2" xref="A2.I2.i1.p1.2.m2.4.5.2.cmml"><msubsup id="A2.I2.i1.p1.2.m2.4.5.2.2" xref="A2.I2.i1.p1.2.m2.4.5.2.2.cmml"><mi id="A2.I2.i1.p1.2.m2.4.5.2.2.2.2" xref="A2.I2.i1.p1.2.m2.4.5.2.2.2.2.cmml">U</mi><mn id="A2.I2.i1.p1.2.m2.4.5.2.2.2.3" xref="A2.I2.i1.p1.2.m2.4.5.2.2.2.3.cmml">2</mn><mi id="A2.I2.i1.p1.2.m2.4.5.2.2.3" xref="A2.I2.i1.p1.2.m2.4.5.2.2.3.cmml">P</mi></msubsup><mo id="A2.I2.i1.p1.2.m2.4.5.2.1" xref="A2.I2.i1.p1.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.2.m2.4.5.2.3.2" xref="A2.I2.i1.p1.2.m2.4.5.2.3.1.cmml"><mo id="A2.I2.i1.p1.2.m2.4.5.2.3.2.1" stretchy="false" xref="A2.I2.i1.p1.2.m2.4.5.2.3.1.cmml">(</mo><mi id="A2.I2.i1.p1.2.m2.1.1" xref="A2.I2.i1.p1.2.m2.1.1.cmml">Y</mi><mo id="A2.I2.i1.p1.2.m2.4.5.2.3.2.2" xref="A2.I2.i1.p1.2.m2.4.5.2.3.1.cmml">,</mo><mi id="A2.I2.i1.p1.2.m2.2.2" xref="A2.I2.i1.p1.2.m2.2.2.cmml">X</mi><mo id="A2.I2.i1.p1.2.m2.4.5.2.3.2.3" stretchy="false" xref="A2.I2.i1.p1.2.m2.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i1.p1.2.m2.4.5.1" xref="A2.I2.i1.p1.2.m2.4.5.1.cmml">≥</mo><mrow id="A2.I2.i1.p1.2.m2.4.5.3" xref="A2.I2.i1.p1.2.m2.4.5.3.cmml"><msubsup id="A2.I2.i1.p1.2.m2.4.5.3.2" xref="A2.I2.i1.p1.2.m2.4.5.3.2.cmml"><mi id="A2.I2.i1.p1.2.m2.4.5.3.2.2.2" xref="A2.I2.i1.p1.2.m2.4.5.3.2.2.2.cmml">U</mi><mn id="A2.I2.i1.p1.2.m2.4.5.3.2.2.3" xref="A2.I2.i1.p1.2.m2.4.5.3.2.2.3.cmml">2</mn><mi id="A2.I2.i1.p1.2.m2.4.5.3.2.3" xref="A2.I2.i1.p1.2.m2.4.5.3.2.3.cmml">P</mi></msubsup><mo id="A2.I2.i1.p1.2.m2.4.5.3.1" xref="A2.I2.i1.p1.2.m2.4.5.3.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.2.m2.4.5.3.3.2" xref="A2.I2.i1.p1.2.m2.4.5.3.3.1.cmml"><mo id="A2.I2.i1.p1.2.m2.4.5.3.3.2.1" stretchy="false" xref="A2.I2.i1.p1.2.m2.4.5.3.3.1.cmml">(</mo><mi id="A2.I2.i1.p1.2.m2.3.3" xref="A2.I2.i1.p1.2.m2.3.3.cmml">Y</mi><mo id="A2.I2.i1.p1.2.m2.4.5.3.3.2.2" xref="A2.I2.i1.p1.2.m2.4.5.3.3.1.cmml">,</mo><mi id="A2.I2.i1.p1.2.m2.4.4" xref="A2.I2.i1.p1.2.m2.4.4.cmml">Y</mi><mo id="A2.I2.i1.p1.2.m2.4.5.3.3.2.3" stretchy="false" xref="A2.I2.i1.p1.2.m2.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.2.m2.4b"><apply id="A2.I2.i1.p1.2.m2.4.5.cmml" xref="A2.I2.i1.p1.2.m2.4.5"><geq id="A2.I2.i1.p1.2.m2.4.5.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.1"></geq><apply id="A2.I2.i1.p1.2.m2.4.5.2.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2"><times id="A2.I2.i1.p1.2.m2.4.5.2.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.1"></times><apply id="A2.I2.i1.p1.2.m2.4.5.2.2.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.2"><csymbol cd="ambiguous" id="A2.I2.i1.p1.2.m2.4.5.2.2.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.2">superscript</csymbol><apply id="A2.I2.i1.p1.2.m2.4.5.2.2.2.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.2"><csymbol cd="ambiguous" id="A2.I2.i1.p1.2.m2.4.5.2.2.2.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.2">subscript</csymbol><ci id="A2.I2.i1.p1.2.m2.4.5.2.2.2.2.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.2.2.2">𝑈</ci><cn id="A2.I2.i1.p1.2.m2.4.5.2.2.2.3.cmml" type="integer" xref="A2.I2.i1.p1.2.m2.4.5.2.2.2.3">2</cn></apply><ci id="A2.I2.i1.p1.2.m2.4.5.2.2.3.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.2.3">𝑃</ci></apply><interval closure="open" id="A2.I2.i1.p1.2.m2.4.5.2.3.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.2.3.2"><ci id="A2.I2.i1.p1.2.m2.1.1.cmml" xref="A2.I2.i1.p1.2.m2.1.1">𝑌</ci><ci id="A2.I2.i1.p1.2.m2.2.2.cmml" xref="A2.I2.i1.p1.2.m2.2.2">𝑋</ci></interval></apply><apply id="A2.I2.i1.p1.2.m2.4.5.3.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3"><times id="A2.I2.i1.p1.2.m2.4.5.3.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.1"></times><apply id="A2.I2.i1.p1.2.m2.4.5.3.2.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.2"><csymbol cd="ambiguous" id="A2.I2.i1.p1.2.m2.4.5.3.2.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.2">superscript</csymbol><apply id="A2.I2.i1.p1.2.m2.4.5.3.2.2.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.2"><csymbol cd="ambiguous" id="A2.I2.i1.p1.2.m2.4.5.3.2.2.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.2">subscript</csymbol><ci id="A2.I2.i1.p1.2.m2.4.5.3.2.2.2.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.2.2.2">𝑈</ci><cn id="A2.I2.i1.p1.2.m2.4.5.3.2.2.3.cmml" type="integer" xref="A2.I2.i1.p1.2.m2.4.5.3.2.2.3">2</cn></apply><ci id="A2.I2.i1.p1.2.m2.4.5.3.2.3.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.2.3">𝑃</ci></apply><interval closure="open" id="A2.I2.i1.p1.2.m2.4.5.3.3.1.cmml" xref="A2.I2.i1.p1.2.m2.4.5.3.3.2"><ci id="A2.I2.i1.p1.2.m2.3.3.cmml" xref="A2.I2.i1.p1.2.m2.3.3">𝑌</ci><ci id="A2.I2.i1.p1.2.m2.4.4.cmml" xref="A2.I2.i1.p1.2.m2.4.4">𝑌</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.2.m2.4c">U_{2}^{P}(Y,X)\geq U_{2}^{P}(Y,Y)</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.2.m2.4d">italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( italic_Y , italic_X ) ≥ italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( italic_Y , italic_Y )</annotation></semantics></math>, or else P2 has profitable deviation <math alttext="X\to Y" class="ltx_Math" display="inline" id="A2.I2.i1.p1.3.m3.1"><semantics id="A2.I2.i1.p1.3.m3.1a"><mrow id="A2.I2.i1.p1.3.m3.1.1" xref="A2.I2.i1.p1.3.m3.1.1.cmml"><mi id="A2.I2.i1.p1.3.m3.1.1.2" xref="A2.I2.i1.p1.3.m3.1.1.2.cmml">X</mi><mo id="A2.I2.i1.p1.3.m3.1.1.1" stretchy="false" xref="A2.I2.i1.p1.3.m3.1.1.1.cmml">→</mo><mi id="A2.I2.i1.p1.3.m3.1.1.3" xref="A2.I2.i1.p1.3.m3.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.3.m3.1b"><apply id="A2.I2.i1.p1.3.m3.1.1.cmml" xref="A2.I2.i1.p1.3.m3.1.1"><ci id="A2.I2.i1.p1.3.m3.1.1.1.cmml" xref="A2.I2.i1.p1.3.m3.1.1.1">→</ci><ci id="A2.I2.i1.p1.3.m3.1.1.2.cmml" xref="A2.I2.i1.p1.3.m3.1.1.2">𝑋</ci><ci id="A2.I2.i1.p1.3.m3.1.1.3.cmml" xref="A2.I2.i1.p1.3.m3.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.3.m3.1c">X\to Y</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.3.m3.1d">italic_X → italic_Y</annotation></semantics></math>. Thus, we have <math alttext="P_{2}(Y,X)\geq 1" class="ltx_Math" display="inline" id="A2.I2.i1.p1.4.m4.2"><semantics id="A2.I2.i1.p1.4.m4.2a"><mrow id="A2.I2.i1.p1.4.m4.2.3" xref="A2.I2.i1.p1.4.m4.2.3.cmml"><mrow id="A2.I2.i1.p1.4.m4.2.3.2" xref="A2.I2.i1.p1.4.m4.2.3.2.cmml"><msub id="A2.I2.i1.p1.4.m4.2.3.2.2" xref="A2.I2.i1.p1.4.m4.2.3.2.2.cmml"><mi id="A2.I2.i1.p1.4.m4.2.3.2.2.2" xref="A2.I2.i1.p1.4.m4.2.3.2.2.2.cmml">P</mi><mn id="A2.I2.i1.p1.4.m4.2.3.2.2.3" xref="A2.I2.i1.p1.4.m4.2.3.2.2.3.cmml">2</mn></msub><mo id="A2.I2.i1.p1.4.m4.2.3.2.1" xref="A2.I2.i1.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.4.m4.2.3.2.3.2" xref="A2.I2.i1.p1.4.m4.2.3.2.3.1.cmml"><mo id="A2.I2.i1.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="A2.I2.i1.p1.4.m4.2.3.2.3.1.cmml">(</mo><mi id="A2.I2.i1.p1.4.m4.1.1" xref="A2.I2.i1.p1.4.m4.1.1.cmml">Y</mi><mo id="A2.I2.i1.p1.4.m4.2.3.2.3.2.2" xref="A2.I2.i1.p1.4.m4.2.3.2.3.1.cmml">,</mo><mi id="A2.I2.i1.p1.4.m4.2.2" xref="A2.I2.i1.p1.4.m4.2.2.cmml">X</mi><mo id="A2.I2.i1.p1.4.m4.2.3.2.3.2.3" stretchy="false" xref="A2.I2.i1.p1.4.m4.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i1.p1.4.m4.2.3.1" xref="A2.I2.i1.p1.4.m4.2.3.1.cmml">≥</mo><mn id="A2.I2.i1.p1.4.m4.2.3.3" xref="A2.I2.i1.p1.4.m4.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.4.m4.2b"><apply id="A2.I2.i1.p1.4.m4.2.3.cmml" xref="A2.I2.i1.p1.4.m4.2.3"><geq id="A2.I2.i1.p1.4.m4.2.3.1.cmml" xref="A2.I2.i1.p1.4.m4.2.3.1"></geq><apply id="A2.I2.i1.p1.4.m4.2.3.2.cmml" xref="A2.I2.i1.p1.4.m4.2.3.2"><times id="A2.I2.i1.p1.4.m4.2.3.2.1.cmml" xref="A2.I2.i1.p1.4.m4.2.3.2.1"></times><apply id="A2.I2.i1.p1.4.m4.2.3.2.2.cmml" xref="A2.I2.i1.p1.4.m4.2.3.2.2"><csymbol cd="ambiguous" id="A2.I2.i1.p1.4.m4.2.3.2.2.1.cmml" xref="A2.I2.i1.p1.4.m4.2.3.2.2">subscript</csymbol><ci id="A2.I2.i1.p1.4.m4.2.3.2.2.2.cmml" xref="A2.I2.i1.p1.4.m4.2.3.2.2.2">𝑃</ci><cn id="A2.I2.i1.p1.4.m4.2.3.2.2.3.cmml" type="integer" xref="A2.I2.i1.p1.4.m4.2.3.2.2.3">2</cn></apply><interval closure="open" id="A2.I2.i1.p1.4.m4.2.3.2.3.1.cmml" xref="A2.I2.i1.p1.4.m4.2.3.2.3.2"><ci id="A2.I2.i1.p1.4.m4.1.1.cmml" xref="A2.I2.i1.p1.4.m4.1.1">𝑌</ci><ci id="A2.I2.i1.p1.4.m4.2.2.cmml" xref="A2.I2.i1.p1.4.m4.2.2">𝑋</ci></interval></apply><cn id="A2.I2.i1.p1.4.m4.2.3.3.cmml" type="integer" xref="A2.I2.i1.p1.4.m4.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.4.m4.2c">P_{2}(Y,X)\geq 1</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.4.m4.2d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_Y , italic_X ) ≥ 1</annotation></semantics></math>, and thus <math alttext="\mu(Y,X)\leq 1/3" class="ltx_Math" display="inline" id="A2.I2.i1.p1.5.m5.2"><semantics id="A2.I2.i1.p1.5.m5.2a"><mrow id="A2.I2.i1.p1.5.m5.2.3" xref="A2.I2.i1.p1.5.m5.2.3.cmml"><mrow id="A2.I2.i1.p1.5.m5.2.3.2" xref="A2.I2.i1.p1.5.m5.2.3.2.cmml"><mi id="A2.I2.i1.p1.5.m5.2.3.2.2" xref="A2.I2.i1.p1.5.m5.2.3.2.2.cmml">μ</mi><mo id="A2.I2.i1.p1.5.m5.2.3.2.1" xref="A2.I2.i1.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.5.m5.2.3.2.3.2" xref="A2.I2.i1.p1.5.m5.2.3.2.3.1.cmml"><mo id="A2.I2.i1.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="A2.I2.i1.p1.5.m5.2.3.2.3.1.cmml">(</mo><mi id="A2.I2.i1.p1.5.m5.1.1" xref="A2.I2.i1.p1.5.m5.1.1.cmml">Y</mi><mo id="A2.I2.i1.p1.5.m5.2.3.2.3.2.2" xref="A2.I2.i1.p1.5.m5.2.3.2.3.1.cmml">,</mo><mi id="A2.I2.i1.p1.5.m5.2.2" xref="A2.I2.i1.p1.5.m5.2.2.cmml">X</mi><mo id="A2.I2.i1.p1.5.m5.2.3.2.3.2.3" stretchy="false" xref="A2.I2.i1.p1.5.m5.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i1.p1.5.m5.2.3.1" xref="A2.I2.i1.p1.5.m5.2.3.1.cmml">≤</mo><mrow id="A2.I2.i1.p1.5.m5.2.3.3" xref="A2.I2.i1.p1.5.m5.2.3.3.cmml"><mn id="A2.I2.i1.p1.5.m5.2.3.3.2" xref="A2.I2.i1.p1.5.m5.2.3.3.2.cmml">1</mn><mo id="A2.I2.i1.p1.5.m5.2.3.3.1" xref="A2.I2.i1.p1.5.m5.2.3.3.1.cmml">/</mo><mn id="A2.I2.i1.p1.5.m5.2.3.3.3" xref="A2.I2.i1.p1.5.m5.2.3.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.5.m5.2b"><apply id="A2.I2.i1.p1.5.m5.2.3.cmml" xref="A2.I2.i1.p1.5.m5.2.3"><leq id="A2.I2.i1.p1.5.m5.2.3.1.cmml" xref="A2.I2.i1.p1.5.m5.2.3.1"></leq><apply id="A2.I2.i1.p1.5.m5.2.3.2.cmml" xref="A2.I2.i1.p1.5.m5.2.3.2"><times id="A2.I2.i1.p1.5.m5.2.3.2.1.cmml" xref="A2.I2.i1.p1.5.m5.2.3.2.1"></times><ci id="A2.I2.i1.p1.5.m5.2.3.2.2.cmml" xref="A2.I2.i1.p1.5.m5.2.3.2.2">𝜇</ci><interval closure="open" id="A2.I2.i1.p1.5.m5.2.3.2.3.1.cmml" xref="A2.I2.i1.p1.5.m5.2.3.2.3.2"><ci id="A2.I2.i1.p1.5.m5.1.1.cmml" xref="A2.I2.i1.p1.5.m5.1.1">𝑌</ci><ci id="A2.I2.i1.p1.5.m5.2.2.cmml" xref="A2.I2.i1.p1.5.m5.2.2">𝑋</ci></interval></apply><apply id="A2.I2.i1.p1.5.m5.2.3.3.cmml" xref="A2.I2.i1.p1.5.m5.2.3.3"><divide id="A2.I2.i1.p1.5.m5.2.3.3.1.cmml" xref="A2.I2.i1.p1.5.m5.2.3.3.1"></divide><cn id="A2.I2.i1.p1.5.m5.2.3.3.2.cmml" type="integer" xref="A2.I2.i1.p1.5.m5.2.3.3.2">1</cn><cn id="A2.I2.i1.p1.5.m5.2.3.3.3.cmml" type="integer" xref="A2.I2.i1.p1.5.m5.2.3.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.5.m5.2c">\mu(Y,X)\leq 1/3</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.5.m5.2d">italic_μ ( italic_Y , italic_X ) ≤ 1 / 3</annotation></semantics></math> by the expected payment constraint. We now use P1’s IC constraint: conditioned on P1 being recommended <math alttext="Y" class="ltx_Math" display="inline" id="A2.I2.i1.p1.6.m6.1"><semantics id="A2.I2.i1.p1.6.m6.1a"><mi id="A2.I2.i1.p1.6.m6.1.1" xref="A2.I2.i1.p1.6.m6.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.6.m6.1b"><ci id="A2.I2.i1.p1.6.m6.1.1.cmml" xref="A2.I2.i1.p1.6.m6.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.6.m6.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.6.m6.1d">italic_Y</annotation></semantics></math>, P2’s action is <math alttext="Y" class="ltx_Math" display="inline" id="A2.I2.i1.p1.7.m7.1"><semantics id="A2.I2.i1.p1.7.m7.1a"><mi id="A2.I2.i1.p1.7.m7.1.1" xref="A2.I2.i1.p1.7.m7.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.7.m7.1b"><ci id="A2.I2.i1.p1.7.m7.1.1.cmml" xref="A2.I2.i1.p1.7.m7.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.7.m7.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.7.m7.1d">italic_Y</annotation></semantics></math> with probability at least <math alttext="1-\mu(Y,X)" class="ltx_Math" display="inline" id="A2.I2.i1.p1.8.m8.2"><semantics id="A2.I2.i1.p1.8.m8.2a"><mrow id="A2.I2.i1.p1.8.m8.2.3" xref="A2.I2.i1.p1.8.m8.2.3.cmml"><mn id="A2.I2.i1.p1.8.m8.2.3.2" xref="A2.I2.i1.p1.8.m8.2.3.2.cmml">1</mn><mo id="A2.I2.i1.p1.8.m8.2.3.1" xref="A2.I2.i1.p1.8.m8.2.3.1.cmml">−</mo><mrow id="A2.I2.i1.p1.8.m8.2.3.3" xref="A2.I2.i1.p1.8.m8.2.3.3.cmml"><mi id="A2.I2.i1.p1.8.m8.2.3.3.2" xref="A2.I2.i1.p1.8.m8.2.3.3.2.cmml">μ</mi><mo id="A2.I2.i1.p1.8.m8.2.3.3.1" xref="A2.I2.i1.p1.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.8.m8.2.3.3.3.2" xref="A2.I2.i1.p1.8.m8.2.3.3.3.1.cmml"><mo id="A2.I2.i1.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="A2.I2.i1.p1.8.m8.2.3.3.3.1.cmml">(</mo><mi id="A2.I2.i1.p1.8.m8.1.1" xref="A2.I2.i1.p1.8.m8.1.1.cmml">Y</mi><mo id="A2.I2.i1.p1.8.m8.2.3.3.3.2.2" xref="A2.I2.i1.p1.8.m8.2.3.3.3.1.cmml">,</mo><mi id="A2.I2.i1.p1.8.m8.2.2" xref="A2.I2.i1.p1.8.m8.2.2.cmml">X</mi><mo id="A2.I2.i1.p1.8.m8.2.3.3.3.2.3" stretchy="false" xref="A2.I2.i1.p1.8.m8.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.8.m8.2b"><apply id="A2.I2.i1.p1.8.m8.2.3.cmml" xref="A2.I2.i1.p1.8.m8.2.3"><minus id="A2.I2.i1.p1.8.m8.2.3.1.cmml" xref="A2.I2.i1.p1.8.m8.2.3.1"></minus><cn id="A2.I2.i1.p1.8.m8.2.3.2.cmml" type="integer" xref="A2.I2.i1.p1.8.m8.2.3.2">1</cn><apply id="A2.I2.i1.p1.8.m8.2.3.3.cmml" xref="A2.I2.i1.p1.8.m8.2.3.3"><times id="A2.I2.i1.p1.8.m8.2.3.3.1.cmml" xref="A2.I2.i1.p1.8.m8.2.3.3.1"></times><ci id="A2.I2.i1.p1.8.m8.2.3.3.2.cmml" xref="A2.I2.i1.p1.8.m8.2.3.3.2">𝜇</ci><interval closure="open" id="A2.I2.i1.p1.8.m8.2.3.3.3.1.cmml" xref="A2.I2.i1.p1.8.m8.2.3.3.3.2"><ci id="A2.I2.i1.p1.8.m8.1.1.cmml" xref="A2.I2.i1.p1.8.m8.1.1">𝑌</ci><ci id="A2.I2.i1.p1.8.m8.2.2.cmml" xref="A2.I2.i1.p1.8.m8.2.2">𝑋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.8.m8.2c">1-\mu(Y,X)</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.8.m8.2d">1 - italic_μ ( italic_Y , italic_X )</annotation></semantics></math>, so, to prevent the deviation <math alttext="Y\to X" class="ltx_Math" display="inline" id="A2.I2.i1.p1.9.m9.1"><semantics id="A2.I2.i1.p1.9.m9.1a"><mrow id="A2.I2.i1.p1.9.m9.1.1" xref="A2.I2.i1.p1.9.m9.1.1.cmml"><mi id="A2.I2.i1.p1.9.m9.1.1.2" xref="A2.I2.i1.p1.9.m9.1.1.2.cmml">Y</mi><mo id="A2.I2.i1.p1.9.m9.1.1.1" stretchy="false" xref="A2.I2.i1.p1.9.m9.1.1.1.cmml">→</mo><mi id="A2.I2.i1.p1.9.m9.1.1.3" xref="A2.I2.i1.p1.9.m9.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.9.m9.1b"><apply id="A2.I2.i1.p1.9.m9.1.1.cmml" xref="A2.I2.i1.p1.9.m9.1.1"><ci id="A2.I2.i1.p1.9.m9.1.1.1.cmml" xref="A2.I2.i1.p1.9.m9.1.1.1">→</ci><ci id="A2.I2.i1.p1.9.m9.1.1.2.cmml" xref="A2.I2.i1.p1.9.m9.1.1.2">𝑌</ci><ci id="A2.I2.i1.p1.9.m9.1.1.3.cmml" xref="A2.I2.i1.p1.9.m9.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.9.m9.1c">Y\to X</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.9.m9.1d">italic_Y → italic_X</annotation></semantics></math>, the principal must expect to pay at least <math alttext="1-2\mu(Y,X)" class="ltx_Math" display="inline" id="A2.I2.i1.p1.10.m10.2"><semantics id="A2.I2.i1.p1.10.m10.2a"><mrow id="A2.I2.i1.p1.10.m10.2.3" xref="A2.I2.i1.p1.10.m10.2.3.cmml"><mn id="A2.I2.i1.p1.10.m10.2.3.2" xref="A2.I2.i1.p1.10.m10.2.3.2.cmml">1</mn><mo id="A2.I2.i1.p1.10.m10.2.3.1" xref="A2.I2.i1.p1.10.m10.2.3.1.cmml">−</mo><mrow id="A2.I2.i1.p1.10.m10.2.3.3" xref="A2.I2.i1.p1.10.m10.2.3.3.cmml"><mn id="A2.I2.i1.p1.10.m10.2.3.3.2" xref="A2.I2.i1.p1.10.m10.2.3.3.2.cmml">2</mn><mo id="A2.I2.i1.p1.10.m10.2.3.3.1" xref="A2.I2.i1.p1.10.m10.2.3.3.1.cmml">⁢</mo><mi id="A2.I2.i1.p1.10.m10.2.3.3.3" xref="A2.I2.i1.p1.10.m10.2.3.3.3.cmml">μ</mi><mo id="A2.I2.i1.p1.10.m10.2.3.3.1a" xref="A2.I2.i1.p1.10.m10.2.3.3.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.10.m10.2.3.3.4.2" xref="A2.I2.i1.p1.10.m10.2.3.3.4.1.cmml"><mo id="A2.I2.i1.p1.10.m10.2.3.3.4.2.1" stretchy="false" xref="A2.I2.i1.p1.10.m10.2.3.3.4.1.cmml">(</mo><mi id="A2.I2.i1.p1.10.m10.1.1" xref="A2.I2.i1.p1.10.m10.1.1.cmml">Y</mi><mo id="A2.I2.i1.p1.10.m10.2.3.3.4.2.2" xref="A2.I2.i1.p1.10.m10.2.3.3.4.1.cmml">,</mo><mi id="A2.I2.i1.p1.10.m10.2.2" xref="A2.I2.i1.p1.10.m10.2.2.cmml">X</mi><mo id="A2.I2.i1.p1.10.m10.2.3.3.4.2.3" stretchy="false" xref="A2.I2.i1.p1.10.m10.2.3.3.4.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.10.m10.2b"><apply id="A2.I2.i1.p1.10.m10.2.3.cmml" xref="A2.I2.i1.p1.10.m10.2.3"><minus id="A2.I2.i1.p1.10.m10.2.3.1.cmml" xref="A2.I2.i1.p1.10.m10.2.3.1"></minus><cn id="A2.I2.i1.p1.10.m10.2.3.2.cmml" type="integer" xref="A2.I2.i1.p1.10.m10.2.3.2">1</cn><apply id="A2.I2.i1.p1.10.m10.2.3.3.cmml" xref="A2.I2.i1.p1.10.m10.2.3.3"><times id="A2.I2.i1.p1.10.m10.2.3.3.1.cmml" xref="A2.I2.i1.p1.10.m10.2.3.3.1"></times><cn id="A2.I2.i1.p1.10.m10.2.3.3.2.cmml" type="integer" xref="A2.I2.i1.p1.10.m10.2.3.3.2">2</cn><ci id="A2.I2.i1.p1.10.m10.2.3.3.3.cmml" xref="A2.I2.i1.p1.10.m10.2.3.3.3">𝜇</ci><interval closure="open" id="A2.I2.i1.p1.10.m10.2.3.3.4.1.cmml" xref="A2.I2.i1.p1.10.m10.2.3.3.4.2"><ci id="A2.I2.i1.p1.10.m10.1.1.cmml" xref="A2.I2.i1.p1.10.m10.1.1">𝑌</ci><ci id="A2.I2.i1.p1.10.m10.2.2.cmml" xref="A2.I2.i1.p1.10.m10.2.2">𝑋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.10.m10.2c">1-2\mu(Y,X)</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.10.m10.2d">1 - 2 italic_μ ( italic_Y , italic_X )</annotation></semantics></math> to P1. But then the principal’s total expected payment is at least <math alttext="1-\mu(Y,X)\geq 2/3" class="ltx_Math" display="inline" id="A2.I2.i1.p1.11.m11.2"><semantics id="A2.I2.i1.p1.11.m11.2a"><mrow id="A2.I2.i1.p1.11.m11.2.3" xref="A2.I2.i1.p1.11.m11.2.3.cmml"><mrow id="A2.I2.i1.p1.11.m11.2.3.2" xref="A2.I2.i1.p1.11.m11.2.3.2.cmml"><mn id="A2.I2.i1.p1.11.m11.2.3.2.2" xref="A2.I2.i1.p1.11.m11.2.3.2.2.cmml">1</mn><mo id="A2.I2.i1.p1.11.m11.2.3.2.1" xref="A2.I2.i1.p1.11.m11.2.3.2.1.cmml">−</mo><mrow id="A2.I2.i1.p1.11.m11.2.3.2.3" xref="A2.I2.i1.p1.11.m11.2.3.2.3.cmml"><mi id="A2.I2.i1.p1.11.m11.2.3.2.3.2" xref="A2.I2.i1.p1.11.m11.2.3.2.3.2.cmml">μ</mi><mo id="A2.I2.i1.p1.11.m11.2.3.2.3.1" xref="A2.I2.i1.p1.11.m11.2.3.2.3.1.cmml">⁢</mo><mrow id="A2.I2.i1.p1.11.m11.2.3.2.3.3.2" xref="A2.I2.i1.p1.11.m11.2.3.2.3.3.1.cmml"><mo id="A2.I2.i1.p1.11.m11.2.3.2.3.3.2.1" stretchy="false" xref="A2.I2.i1.p1.11.m11.2.3.2.3.3.1.cmml">(</mo><mi id="A2.I2.i1.p1.11.m11.1.1" xref="A2.I2.i1.p1.11.m11.1.1.cmml">Y</mi><mo id="A2.I2.i1.p1.11.m11.2.3.2.3.3.2.2" xref="A2.I2.i1.p1.11.m11.2.3.2.3.3.1.cmml">,</mo><mi id="A2.I2.i1.p1.11.m11.2.2" xref="A2.I2.i1.p1.11.m11.2.2.cmml">X</mi><mo id="A2.I2.i1.p1.11.m11.2.3.2.3.3.2.3" stretchy="false" xref="A2.I2.i1.p1.11.m11.2.3.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="A2.I2.i1.p1.11.m11.2.3.1" xref="A2.I2.i1.p1.11.m11.2.3.1.cmml">≥</mo><mrow id="A2.I2.i1.p1.11.m11.2.3.3" xref="A2.I2.i1.p1.11.m11.2.3.3.cmml"><mn id="A2.I2.i1.p1.11.m11.2.3.3.2" xref="A2.I2.i1.p1.11.m11.2.3.3.2.cmml">2</mn><mo id="A2.I2.i1.p1.11.m11.2.3.3.1" xref="A2.I2.i1.p1.11.m11.2.3.3.1.cmml">/</mo><mn id="A2.I2.i1.p1.11.m11.2.3.3.3" xref="A2.I2.i1.p1.11.m11.2.3.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i1.p1.11.m11.2b"><apply id="A2.I2.i1.p1.11.m11.2.3.cmml" xref="A2.I2.i1.p1.11.m11.2.3"><geq id="A2.I2.i1.p1.11.m11.2.3.1.cmml" xref="A2.I2.i1.p1.11.m11.2.3.1"></geq><apply id="A2.I2.i1.p1.11.m11.2.3.2.cmml" xref="A2.I2.i1.p1.11.m11.2.3.2"><minus id="A2.I2.i1.p1.11.m11.2.3.2.1.cmml" xref="A2.I2.i1.p1.11.m11.2.3.2.1"></minus><cn id="A2.I2.i1.p1.11.m11.2.3.2.2.cmml" type="integer" xref="A2.I2.i1.p1.11.m11.2.3.2.2">1</cn><apply id="A2.I2.i1.p1.11.m11.2.3.2.3.cmml" xref="A2.I2.i1.p1.11.m11.2.3.2.3"><times id="A2.I2.i1.p1.11.m11.2.3.2.3.1.cmml" xref="A2.I2.i1.p1.11.m11.2.3.2.3.1"></times><ci id="A2.I2.i1.p1.11.m11.2.3.2.3.2.cmml" xref="A2.I2.i1.p1.11.m11.2.3.2.3.2">𝜇</ci><interval closure="open" id="A2.I2.i1.p1.11.m11.2.3.2.3.3.1.cmml" xref="A2.I2.i1.p1.11.m11.2.3.2.3.3.2"><ci id="A2.I2.i1.p1.11.m11.1.1.cmml" xref="A2.I2.i1.p1.11.m11.1.1">𝑌</ci><ci id="A2.I2.i1.p1.11.m11.2.2.cmml" xref="A2.I2.i1.p1.11.m11.2.2">𝑋</ci></interval></apply></apply><apply id="A2.I2.i1.p1.11.m11.2.3.3.cmml" xref="A2.I2.i1.p1.11.m11.2.3.3"><divide id="A2.I2.i1.p1.11.m11.2.3.3.1.cmml" xref="A2.I2.i1.p1.11.m11.2.3.3.1"></divide><cn id="A2.I2.i1.p1.11.m11.2.3.3.2.cmml" type="integer" xref="A2.I2.i1.p1.11.m11.2.3.3.2">2</cn><cn id="A2.I2.i1.p1.11.m11.2.3.3.3.cmml" type="integer" xref="A2.I2.i1.p1.11.m11.2.3.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i1.p1.11.m11.2c">1-\mu(Y,X)\geq 2/3</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i1.p1.11.m11.2d">1 - italic_μ ( italic_Y , italic_X ) ≥ 2 / 3</annotation></semantics></math>, a contradiction.</p> </div> </li> <li class="ltx_item" id="A2.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A2.I2.i2.p1"> <p class="ltx_p" id="A2.I2.i2.p1.14"><span class="ltx_text ltx_font_italic" id="A2.I2.i2.p1.14.1">Case 2:</span> <math alttext="\mu(Y,X)=0" class="ltx_Math" display="inline" id="A2.I2.i2.p1.1.m1.2"><semantics id="A2.I2.i2.p1.1.m1.2a"><mrow id="A2.I2.i2.p1.1.m1.2.3" xref="A2.I2.i2.p1.1.m1.2.3.cmml"><mrow id="A2.I2.i2.p1.1.m1.2.3.2" xref="A2.I2.i2.p1.1.m1.2.3.2.cmml"><mi id="A2.I2.i2.p1.1.m1.2.3.2.2" xref="A2.I2.i2.p1.1.m1.2.3.2.2.cmml">μ</mi><mo id="A2.I2.i2.p1.1.m1.2.3.2.1" xref="A2.I2.i2.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="A2.I2.i2.p1.1.m1.2.3.2.3.2" xref="A2.I2.i2.p1.1.m1.2.3.2.3.1.cmml"><mo id="A2.I2.i2.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="A2.I2.i2.p1.1.m1.2.3.2.3.1.cmml">(</mo><mi id="A2.I2.i2.p1.1.m1.1.1" xref="A2.I2.i2.p1.1.m1.1.1.cmml">Y</mi><mo id="A2.I2.i2.p1.1.m1.2.3.2.3.2.2" xref="A2.I2.i2.p1.1.m1.2.3.2.3.1.cmml">,</mo><mi id="A2.I2.i2.p1.1.m1.2.2" xref="A2.I2.i2.p1.1.m1.2.2.cmml">X</mi><mo id="A2.I2.i2.p1.1.m1.2.3.2.3.2.3" stretchy="false" xref="A2.I2.i2.p1.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i2.p1.1.m1.2.3.1" xref="A2.I2.i2.p1.1.m1.2.3.1.cmml">=</mo><mn id="A2.I2.i2.p1.1.m1.2.3.3" xref="A2.I2.i2.p1.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.1.m1.2b"><apply id="A2.I2.i2.p1.1.m1.2.3.cmml" xref="A2.I2.i2.p1.1.m1.2.3"><eq id="A2.I2.i2.p1.1.m1.2.3.1.cmml" xref="A2.I2.i2.p1.1.m1.2.3.1"></eq><apply id="A2.I2.i2.p1.1.m1.2.3.2.cmml" xref="A2.I2.i2.p1.1.m1.2.3.2"><times id="A2.I2.i2.p1.1.m1.2.3.2.1.cmml" xref="A2.I2.i2.p1.1.m1.2.3.2.1"></times><ci id="A2.I2.i2.p1.1.m1.2.3.2.2.cmml" xref="A2.I2.i2.p1.1.m1.2.3.2.2">𝜇</ci><interval closure="open" id="A2.I2.i2.p1.1.m1.2.3.2.3.1.cmml" xref="A2.I2.i2.p1.1.m1.2.3.2.3.2"><ci id="A2.I2.i2.p1.1.m1.1.1.cmml" xref="A2.I2.i2.p1.1.m1.1.1">𝑌</ci><ci id="A2.I2.i2.p1.1.m1.2.2.cmml" xref="A2.I2.i2.p1.1.m1.2.2">𝑋</ci></interval></apply><cn id="A2.I2.i2.p1.1.m1.2.3.3.cmml" type="integer" xref="A2.I2.i2.p1.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.1.m1.2c">\mu(Y,X)=0</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.1.m1.2d">italic_μ ( italic_Y , italic_X ) = 0</annotation></semantics></math>. Thus, under <math alttext="\mu" class="ltx_Math" display="inline" id="A2.I2.i2.p1.2.m2.1"><semantics id="A2.I2.i2.p1.2.m2.1a"><mi id="A2.I2.i2.p1.2.m2.1.1" xref="A2.I2.i2.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.2.m2.1b"><ci id="A2.I2.i2.p1.2.m2.1.1.cmml" xref="A2.I2.i2.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.2.m2.1d">italic_μ</annotation></semantics></math>, P2 is always recommended <math alttext="Y" class="ltx_Math" display="inline" id="A2.I2.i2.p1.3.m3.1"><semantics id="A2.I2.i2.p1.3.m3.1a"><mi id="A2.I2.i2.p1.3.m3.1.1" xref="A2.I2.i2.p1.3.m3.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.3.m3.1b"><ci id="A2.I2.i2.p1.3.m3.1.1.cmml" xref="A2.I2.i2.p1.3.m3.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.3.m3.1c">Y</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.3.m3.1d">italic_Y</annotation></semantics></math>. Then <math alttext="\mu(Y,Y)>0" class="ltx_Math" display="inline" id="A2.I2.i2.p1.4.m4.2"><semantics id="A2.I2.i2.p1.4.m4.2a"><mrow id="A2.I2.i2.p1.4.m4.2.3" xref="A2.I2.i2.p1.4.m4.2.3.cmml"><mrow id="A2.I2.i2.p1.4.m4.2.3.2" xref="A2.I2.i2.p1.4.m4.2.3.2.cmml"><mi id="A2.I2.i2.p1.4.m4.2.3.2.2" xref="A2.I2.i2.p1.4.m4.2.3.2.2.cmml">μ</mi><mo id="A2.I2.i2.p1.4.m4.2.3.2.1" xref="A2.I2.i2.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="A2.I2.i2.p1.4.m4.2.3.2.3.2" xref="A2.I2.i2.p1.4.m4.2.3.2.3.1.cmml"><mo id="A2.I2.i2.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="A2.I2.i2.p1.4.m4.2.3.2.3.1.cmml">(</mo><mi id="A2.I2.i2.p1.4.m4.1.1" xref="A2.I2.i2.p1.4.m4.1.1.cmml">Y</mi><mo id="A2.I2.i2.p1.4.m4.2.3.2.3.2.2" xref="A2.I2.i2.p1.4.m4.2.3.2.3.1.cmml">,</mo><mi id="A2.I2.i2.p1.4.m4.2.2" xref="A2.I2.i2.p1.4.m4.2.2.cmml">Y</mi><mo id="A2.I2.i2.p1.4.m4.2.3.2.3.2.3" stretchy="false" xref="A2.I2.i2.p1.4.m4.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i2.p1.4.m4.2.3.1" xref="A2.I2.i2.p1.4.m4.2.3.1.cmml">></mo><mn id="A2.I2.i2.p1.4.m4.2.3.3" xref="A2.I2.i2.p1.4.m4.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.4.m4.2b"><apply id="A2.I2.i2.p1.4.m4.2.3.cmml" xref="A2.I2.i2.p1.4.m4.2.3"><gt id="A2.I2.i2.p1.4.m4.2.3.1.cmml" xref="A2.I2.i2.p1.4.m4.2.3.1"></gt><apply id="A2.I2.i2.p1.4.m4.2.3.2.cmml" xref="A2.I2.i2.p1.4.m4.2.3.2"><times id="A2.I2.i2.p1.4.m4.2.3.2.1.cmml" xref="A2.I2.i2.p1.4.m4.2.3.2.1"></times><ci id="A2.I2.i2.p1.4.m4.2.3.2.2.cmml" xref="A2.I2.i2.p1.4.m4.2.3.2.2">𝜇</ci><interval closure="open" id="A2.I2.i2.p1.4.m4.2.3.2.3.1.cmml" xref="A2.I2.i2.p1.4.m4.2.3.2.3.2"><ci id="A2.I2.i2.p1.4.m4.1.1.cmml" xref="A2.I2.i2.p1.4.m4.1.1">𝑌</ci><ci id="A2.I2.i2.p1.4.m4.2.2.cmml" xref="A2.I2.i2.p1.4.m4.2.2">𝑌</ci></interval></apply><cn id="A2.I2.i2.p1.4.m4.2.3.3.cmml" type="integer" xref="A2.I2.i2.p1.4.m4.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.4.m4.2c">\mu(Y,Y)>0</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.4.m4.2d">italic_μ ( italic_Y , italic_Y ) > 0</annotation></semantics></math>, or else <math alttext="\mu" class="ltx_Math" display="inline" id="A2.I2.i2.p1.5.m5.1"><semantics id="A2.I2.i2.p1.5.m5.1a"><mi id="A2.I2.i2.p1.5.m5.1.1" xref="A2.I2.i2.p1.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.5.m5.1b"><ci id="A2.I2.i2.p1.5.m5.1.1.cmml" xref="A2.I2.i2.p1.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.5.m5.1d">italic_μ</annotation></semantics></math> is deterministically <math alttext="(X,Y)" class="ltx_Math" display="inline" id="A2.I2.i2.p1.6.m6.2"><semantics id="A2.I2.i2.p1.6.m6.2a"><mrow id="A2.I2.i2.p1.6.m6.2.3.2" xref="A2.I2.i2.p1.6.m6.2.3.1.cmml"><mo id="A2.I2.i2.p1.6.m6.2.3.2.1" stretchy="false" xref="A2.I2.i2.p1.6.m6.2.3.1.cmml">(</mo><mi id="A2.I2.i2.p1.6.m6.1.1" xref="A2.I2.i2.p1.6.m6.1.1.cmml">X</mi><mo id="A2.I2.i2.p1.6.m6.2.3.2.2" xref="A2.I2.i2.p1.6.m6.2.3.1.cmml">,</mo><mi id="A2.I2.i2.p1.6.m6.2.2" xref="A2.I2.i2.p1.6.m6.2.2.cmml">Y</mi><mo id="A2.I2.i2.p1.6.m6.2.3.2.3" stretchy="false" xref="A2.I2.i2.p1.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.6.m6.2b"><interval closure="open" id="A2.I2.i2.p1.6.m6.2.3.1.cmml" xref="A2.I2.i2.p1.6.m6.2.3.2"><ci id="A2.I2.i2.p1.6.m6.1.1.cmml" xref="A2.I2.i2.p1.6.m6.1.1">𝑋</ci><ci id="A2.I2.i2.p1.6.m6.2.2.cmml" xref="A2.I2.i2.p1.6.m6.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.6.m6.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.6.m6.2d">( italic_X , italic_Y )</annotation></semantics></math>, in which case the principal would have to pay <math alttext="1" class="ltx_Math" display="inline" id="A2.I2.i2.p1.7.m7.1"><semantics id="A2.I2.i2.p1.7.m7.1a"><mn id="A2.I2.i2.p1.7.m7.1.1" xref="A2.I2.i2.p1.7.m7.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.7.m7.1b"><cn id="A2.I2.i2.p1.7.m7.1.1.cmml" type="integer" xref="A2.I2.i2.p1.7.m7.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.7.m7.1c">1</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.7.m7.1d">1</annotation></semantics></math> to P2 to prevent the deviation <math alttext="Y\to X" class="ltx_Math" display="inline" id="A2.I2.i2.p1.8.m8.1"><semantics id="A2.I2.i2.p1.8.m8.1a"><mrow id="A2.I2.i2.p1.8.m8.1.1" xref="A2.I2.i2.p1.8.m8.1.1.cmml"><mi id="A2.I2.i2.p1.8.m8.1.1.2" xref="A2.I2.i2.p1.8.m8.1.1.2.cmml">Y</mi><mo id="A2.I2.i2.p1.8.m8.1.1.1" stretchy="false" xref="A2.I2.i2.p1.8.m8.1.1.1.cmml">→</mo><mi id="A2.I2.i2.p1.8.m8.1.1.3" xref="A2.I2.i2.p1.8.m8.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.8.m8.1b"><apply id="A2.I2.i2.p1.8.m8.1.1.cmml" xref="A2.I2.i2.p1.8.m8.1.1"><ci id="A2.I2.i2.p1.8.m8.1.1.1.cmml" xref="A2.I2.i2.p1.8.m8.1.1.1">→</ci><ci id="A2.I2.i2.p1.8.m8.1.1.2.cmml" xref="A2.I2.i2.p1.8.m8.1.1.2">𝑌</ci><ci id="A2.I2.i2.p1.8.m8.1.1.3.cmml" xref="A2.I2.i2.p1.8.m8.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.8.m8.1c">Y\to X</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.8.m8.1d">italic_Y → italic_X</annotation></semantics></math>. Then by P1’s IC constraint against deviation <math alttext="Y\to X" class="ltx_Math" display="inline" id="A2.I2.i2.p1.9.m9.1"><semantics id="A2.I2.i2.p1.9.m9.1a"><mrow id="A2.I2.i2.p1.9.m9.1.1" xref="A2.I2.i2.p1.9.m9.1.1.cmml"><mi id="A2.I2.i2.p1.9.m9.1.1.2" xref="A2.I2.i2.p1.9.m9.1.1.2.cmml">Y</mi><mo id="A2.I2.i2.p1.9.m9.1.1.1" stretchy="false" xref="A2.I2.i2.p1.9.m9.1.1.1.cmml">→</mo><mi id="A2.I2.i2.p1.9.m9.1.1.3" xref="A2.I2.i2.p1.9.m9.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.9.m9.1b"><apply id="A2.I2.i2.p1.9.m9.1.1.cmml" xref="A2.I2.i2.p1.9.m9.1.1"><ci id="A2.I2.i2.p1.9.m9.1.1.1.cmml" xref="A2.I2.i2.p1.9.m9.1.1.1">→</ci><ci id="A2.I2.i2.p1.9.m9.1.1.2.cmml" xref="A2.I2.i2.p1.9.m9.1.1.2">𝑌</ci><ci id="A2.I2.i2.p1.9.m9.1.1.3.cmml" xref="A2.I2.i2.p1.9.m9.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.9.m9.1c">Y\to X</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.9.m9.1d">italic_Y → italic_X</annotation></semantics></math>, we have <math alttext="P_{1}(Y,Y)\geq 1" class="ltx_Math" display="inline" id="A2.I2.i2.p1.10.m10.2"><semantics id="A2.I2.i2.p1.10.m10.2a"><mrow id="A2.I2.i2.p1.10.m10.2.3" xref="A2.I2.i2.p1.10.m10.2.3.cmml"><mrow id="A2.I2.i2.p1.10.m10.2.3.2" xref="A2.I2.i2.p1.10.m10.2.3.2.cmml"><msub id="A2.I2.i2.p1.10.m10.2.3.2.2" xref="A2.I2.i2.p1.10.m10.2.3.2.2.cmml"><mi id="A2.I2.i2.p1.10.m10.2.3.2.2.2" xref="A2.I2.i2.p1.10.m10.2.3.2.2.2.cmml">P</mi><mn id="A2.I2.i2.p1.10.m10.2.3.2.2.3" xref="A2.I2.i2.p1.10.m10.2.3.2.2.3.cmml">1</mn></msub><mo id="A2.I2.i2.p1.10.m10.2.3.2.1" xref="A2.I2.i2.p1.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="A2.I2.i2.p1.10.m10.2.3.2.3.2" xref="A2.I2.i2.p1.10.m10.2.3.2.3.1.cmml"><mo id="A2.I2.i2.p1.10.m10.2.3.2.3.2.1" stretchy="false" xref="A2.I2.i2.p1.10.m10.2.3.2.3.1.cmml">(</mo><mi id="A2.I2.i2.p1.10.m10.1.1" xref="A2.I2.i2.p1.10.m10.1.1.cmml">Y</mi><mo id="A2.I2.i2.p1.10.m10.2.3.2.3.2.2" xref="A2.I2.i2.p1.10.m10.2.3.2.3.1.cmml">,</mo><mi id="A2.I2.i2.p1.10.m10.2.2" xref="A2.I2.i2.p1.10.m10.2.2.cmml">Y</mi><mo id="A2.I2.i2.p1.10.m10.2.3.2.3.2.3" stretchy="false" xref="A2.I2.i2.p1.10.m10.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i2.p1.10.m10.2.3.1" xref="A2.I2.i2.p1.10.m10.2.3.1.cmml">≥</mo><mn id="A2.I2.i2.p1.10.m10.2.3.3" xref="A2.I2.i2.p1.10.m10.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.10.m10.2b"><apply id="A2.I2.i2.p1.10.m10.2.3.cmml" xref="A2.I2.i2.p1.10.m10.2.3"><geq id="A2.I2.i2.p1.10.m10.2.3.1.cmml" xref="A2.I2.i2.p1.10.m10.2.3.1"></geq><apply id="A2.I2.i2.p1.10.m10.2.3.2.cmml" xref="A2.I2.i2.p1.10.m10.2.3.2"><times id="A2.I2.i2.p1.10.m10.2.3.2.1.cmml" xref="A2.I2.i2.p1.10.m10.2.3.2.1"></times><apply id="A2.I2.i2.p1.10.m10.2.3.2.2.cmml" xref="A2.I2.i2.p1.10.m10.2.3.2.2"><csymbol cd="ambiguous" id="A2.I2.i2.p1.10.m10.2.3.2.2.1.cmml" xref="A2.I2.i2.p1.10.m10.2.3.2.2">subscript</csymbol><ci id="A2.I2.i2.p1.10.m10.2.3.2.2.2.cmml" xref="A2.I2.i2.p1.10.m10.2.3.2.2.2">𝑃</ci><cn id="A2.I2.i2.p1.10.m10.2.3.2.2.3.cmml" type="integer" xref="A2.I2.i2.p1.10.m10.2.3.2.2.3">1</cn></apply><interval closure="open" id="A2.I2.i2.p1.10.m10.2.3.2.3.1.cmml" xref="A2.I2.i2.p1.10.m10.2.3.2.3.2"><ci id="A2.I2.i2.p1.10.m10.1.1.cmml" xref="A2.I2.i2.p1.10.m10.1.1">𝑌</ci><ci id="A2.I2.i2.p1.10.m10.2.2.cmml" xref="A2.I2.i2.p1.10.m10.2.2">𝑌</ci></interval></apply><cn id="A2.I2.i2.p1.10.m10.2.3.3.cmml" type="integer" xref="A2.I2.i2.p1.10.m10.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.10.m10.2c">P_{1}(Y,Y)\geq 1</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.10.m10.2d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_Y , italic_Y ) ≥ 1</annotation></semantics></math> and thus <math alttext="\mu(Y,Y)\leq 1/3" class="ltx_Math" display="inline" id="A2.I2.i2.p1.11.m11.2"><semantics id="A2.I2.i2.p1.11.m11.2a"><mrow id="A2.I2.i2.p1.11.m11.2.3" xref="A2.I2.i2.p1.11.m11.2.3.cmml"><mrow id="A2.I2.i2.p1.11.m11.2.3.2" xref="A2.I2.i2.p1.11.m11.2.3.2.cmml"><mi id="A2.I2.i2.p1.11.m11.2.3.2.2" xref="A2.I2.i2.p1.11.m11.2.3.2.2.cmml">μ</mi><mo id="A2.I2.i2.p1.11.m11.2.3.2.1" xref="A2.I2.i2.p1.11.m11.2.3.2.1.cmml">⁢</mo><mrow id="A2.I2.i2.p1.11.m11.2.3.2.3.2" xref="A2.I2.i2.p1.11.m11.2.3.2.3.1.cmml"><mo id="A2.I2.i2.p1.11.m11.2.3.2.3.2.1" stretchy="false" xref="A2.I2.i2.p1.11.m11.2.3.2.3.1.cmml">(</mo><mi id="A2.I2.i2.p1.11.m11.1.1" xref="A2.I2.i2.p1.11.m11.1.1.cmml">Y</mi><mo id="A2.I2.i2.p1.11.m11.2.3.2.3.2.2" xref="A2.I2.i2.p1.11.m11.2.3.2.3.1.cmml">,</mo><mi id="A2.I2.i2.p1.11.m11.2.2" xref="A2.I2.i2.p1.11.m11.2.2.cmml">Y</mi><mo id="A2.I2.i2.p1.11.m11.2.3.2.3.2.3" stretchy="false" xref="A2.I2.i2.p1.11.m11.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.I2.i2.p1.11.m11.2.3.1" xref="A2.I2.i2.p1.11.m11.2.3.1.cmml">≤</mo><mrow id="A2.I2.i2.p1.11.m11.2.3.3" xref="A2.I2.i2.p1.11.m11.2.3.3.cmml"><mn id="A2.I2.i2.p1.11.m11.2.3.3.2" xref="A2.I2.i2.p1.11.m11.2.3.3.2.cmml">1</mn><mo id="A2.I2.i2.p1.11.m11.2.3.3.1" xref="A2.I2.i2.p1.11.m11.2.3.3.1.cmml">/</mo><mn id="A2.I2.i2.p1.11.m11.2.3.3.3" xref="A2.I2.i2.p1.11.m11.2.3.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.11.m11.2b"><apply id="A2.I2.i2.p1.11.m11.2.3.cmml" xref="A2.I2.i2.p1.11.m11.2.3"><leq id="A2.I2.i2.p1.11.m11.2.3.1.cmml" xref="A2.I2.i2.p1.11.m11.2.3.1"></leq><apply id="A2.I2.i2.p1.11.m11.2.3.2.cmml" xref="A2.I2.i2.p1.11.m11.2.3.2"><times id="A2.I2.i2.p1.11.m11.2.3.2.1.cmml" xref="A2.I2.i2.p1.11.m11.2.3.2.1"></times><ci id="A2.I2.i2.p1.11.m11.2.3.2.2.cmml" xref="A2.I2.i2.p1.11.m11.2.3.2.2">𝜇</ci><interval closure="open" id="A2.I2.i2.p1.11.m11.2.3.2.3.1.cmml" xref="A2.I2.i2.p1.11.m11.2.3.2.3.2"><ci id="A2.I2.i2.p1.11.m11.1.1.cmml" xref="A2.I2.i2.p1.11.m11.1.1">𝑌</ci><ci id="A2.I2.i2.p1.11.m11.2.2.cmml" xref="A2.I2.i2.p1.11.m11.2.2">𝑌</ci></interval></apply><apply id="A2.I2.i2.p1.11.m11.2.3.3.cmml" xref="A2.I2.i2.p1.11.m11.2.3.3"><divide id="A2.I2.i2.p1.11.m11.2.3.3.1.cmml" xref="A2.I2.i2.p1.11.m11.2.3.3.1"></divide><cn id="A2.I2.i2.p1.11.m11.2.3.3.2.cmml" type="integer" xref="A2.I2.i2.p1.11.m11.2.3.3.2">1</cn><cn id="A2.I2.i2.p1.11.m11.2.3.3.3.cmml" type="integer" xref="A2.I2.i2.p1.11.m11.2.3.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.11.m11.2c">\mu(Y,Y)\leq 1/3</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.11.m11.2d">italic_μ ( italic_Y , italic_Y ) ≤ 1 / 3</annotation></semantics></math>. We now again use P2’s IC constraint: to prevent the deviation <math alttext="Y\to X" class="ltx_Math" display="inline" id="A2.I2.i2.p1.12.m12.1"><semantics id="A2.I2.i2.p1.12.m12.1a"><mrow id="A2.I2.i2.p1.12.m12.1.1" xref="A2.I2.i2.p1.12.m12.1.1.cmml"><mi id="A2.I2.i2.p1.12.m12.1.1.2" xref="A2.I2.i2.p1.12.m12.1.1.2.cmml">Y</mi><mo id="A2.I2.i2.p1.12.m12.1.1.1" stretchy="false" xref="A2.I2.i2.p1.12.m12.1.1.1.cmml">→</mo><mi id="A2.I2.i2.p1.12.m12.1.1.3" xref="A2.I2.i2.p1.12.m12.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.12.m12.1b"><apply id="A2.I2.i2.p1.12.m12.1.1.cmml" xref="A2.I2.i2.p1.12.m12.1.1"><ci id="A2.I2.i2.p1.12.m12.1.1.1.cmml" xref="A2.I2.i2.p1.12.m12.1.1.1">→</ci><ci id="A2.I2.i2.p1.12.m12.1.1.2.cmml" xref="A2.I2.i2.p1.12.m12.1.1.2">𝑌</ci><ci id="A2.I2.i2.p1.12.m12.1.1.3.cmml" xref="A2.I2.i2.p1.12.m12.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.12.m12.1c">Y\to X</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.12.m12.1d">italic_Y → italic_X</annotation></semantics></math>, P2 must receive expected payment at least <math alttext="1-2\mu(Y,Y)" class="ltx_Math" display="inline" id="A2.I2.i2.p1.13.m13.2"><semantics id="A2.I2.i2.p1.13.m13.2a"><mrow id="A2.I2.i2.p1.13.m13.2.3" xref="A2.I2.i2.p1.13.m13.2.3.cmml"><mn id="A2.I2.i2.p1.13.m13.2.3.2" xref="A2.I2.i2.p1.13.m13.2.3.2.cmml">1</mn><mo id="A2.I2.i2.p1.13.m13.2.3.1" xref="A2.I2.i2.p1.13.m13.2.3.1.cmml">−</mo><mrow id="A2.I2.i2.p1.13.m13.2.3.3" xref="A2.I2.i2.p1.13.m13.2.3.3.cmml"><mn id="A2.I2.i2.p1.13.m13.2.3.3.2" xref="A2.I2.i2.p1.13.m13.2.3.3.2.cmml">2</mn><mo id="A2.I2.i2.p1.13.m13.2.3.3.1" xref="A2.I2.i2.p1.13.m13.2.3.3.1.cmml">⁢</mo><mi id="A2.I2.i2.p1.13.m13.2.3.3.3" xref="A2.I2.i2.p1.13.m13.2.3.3.3.cmml">μ</mi><mo id="A2.I2.i2.p1.13.m13.2.3.3.1a" xref="A2.I2.i2.p1.13.m13.2.3.3.1.cmml">⁢</mo><mrow id="A2.I2.i2.p1.13.m13.2.3.3.4.2" xref="A2.I2.i2.p1.13.m13.2.3.3.4.1.cmml"><mo id="A2.I2.i2.p1.13.m13.2.3.3.4.2.1" stretchy="false" xref="A2.I2.i2.p1.13.m13.2.3.3.4.1.cmml">(</mo><mi id="A2.I2.i2.p1.13.m13.1.1" xref="A2.I2.i2.p1.13.m13.1.1.cmml">Y</mi><mo id="A2.I2.i2.p1.13.m13.2.3.3.4.2.2" xref="A2.I2.i2.p1.13.m13.2.3.3.4.1.cmml">,</mo><mi id="A2.I2.i2.p1.13.m13.2.2" xref="A2.I2.i2.p1.13.m13.2.2.cmml">Y</mi><mo id="A2.I2.i2.p1.13.m13.2.3.3.4.2.3" stretchy="false" xref="A2.I2.i2.p1.13.m13.2.3.3.4.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.13.m13.2b"><apply id="A2.I2.i2.p1.13.m13.2.3.cmml" xref="A2.I2.i2.p1.13.m13.2.3"><minus id="A2.I2.i2.p1.13.m13.2.3.1.cmml" xref="A2.I2.i2.p1.13.m13.2.3.1"></minus><cn id="A2.I2.i2.p1.13.m13.2.3.2.cmml" type="integer" xref="A2.I2.i2.p1.13.m13.2.3.2">1</cn><apply id="A2.I2.i2.p1.13.m13.2.3.3.cmml" xref="A2.I2.i2.p1.13.m13.2.3.3"><times id="A2.I2.i2.p1.13.m13.2.3.3.1.cmml" xref="A2.I2.i2.p1.13.m13.2.3.3.1"></times><cn id="A2.I2.i2.p1.13.m13.2.3.3.2.cmml" type="integer" xref="A2.I2.i2.p1.13.m13.2.3.3.2">2</cn><ci id="A2.I2.i2.p1.13.m13.2.3.3.3.cmml" xref="A2.I2.i2.p1.13.m13.2.3.3.3">𝜇</ci><interval closure="open" id="A2.I2.i2.p1.13.m13.2.3.3.4.1.cmml" xref="A2.I2.i2.p1.13.m13.2.3.3.4.2"><ci id="A2.I2.i2.p1.13.m13.1.1.cmml" xref="A2.I2.i2.p1.13.m13.1.1">𝑌</ci><ci id="A2.I2.i2.p1.13.m13.2.2.cmml" xref="A2.I2.i2.p1.13.m13.2.2">𝑌</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.13.m13.2c">1-2\mu(Y,Y)</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.13.m13.2d">1 - 2 italic_μ ( italic_Y , italic_Y )</annotation></semantics></math>. But then the total payment is once again at least <math alttext="1-\mu(Y,Y)\geq 2/3" class="ltx_Math" display="inline" id="A2.I2.i2.p1.14.m14.2"><semantics id="A2.I2.i2.p1.14.m14.2a"><mrow id="A2.I2.i2.p1.14.m14.2.3" xref="A2.I2.i2.p1.14.m14.2.3.cmml"><mrow id="A2.I2.i2.p1.14.m14.2.3.2" xref="A2.I2.i2.p1.14.m14.2.3.2.cmml"><mn id="A2.I2.i2.p1.14.m14.2.3.2.2" xref="A2.I2.i2.p1.14.m14.2.3.2.2.cmml">1</mn><mo id="A2.I2.i2.p1.14.m14.2.3.2.1" xref="A2.I2.i2.p1.14.m14.2.3.2.1.cmml">−</mo><mrow id="A2.I2.i2.p1.14.m14.2.3.2.3" xref="A2.I2.i2.p1.14.m14.2.3.2.3.cmml"><mi id="A2.I2.i2.p1.14.m14.2.3.2.3.2" xref="A2.I2.i2.p1.14.m14.2.3.2.3.2.cmml">μ</mi><mo id="A2.I2.i2.p1.14.m14.2.3.2.3.1" xref="A2.I2.i2.p1.14.m14.2.3.2.3.1.cmml">⁢</mo><mrow id="A2.I2.i2.p1.14.m14.2.3.2.3.3.2" xref="A2.I2.i2.p1.14.m14.2.3.2.3.3.1.cmml"><mo id="A2.I2.i2.p1.14.m14.2.3.2.3.3.2.1" stretchy="false" xref="A2.I2.i2.p1.14.m14.2.3.2.3.3.1.cmml">(</mo><mi id="A2.I2.i2.p1.14.m14.1.1" xref="A2.I2.i2.p1.14.m14.1.1.cmml">Y</mi><mo id="A2.I2.i2.p1.14.m14.2.3.2.3.3.2.2" xref="A2.I2.i2.p1.14.m14.2.3.2.3.3.1.cmml">,</mo><mi id="A2.I2.i2.p1.14.m14.2.2" xref="A2.I2.i2.p1.14.m14.2.2.cmml">Y</mi><mo id="A2.I2.i2.p1.14.m14.2.3.2.3.3.2.3" stretchy="false" xref="A2.I2.i2.p1.14.m14.2.3.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="A2.I2.i2.p1.14.m14.2.3.1" xref="A2.I2.i2.p1.14.m14.2.3.1.cmml">≥</mo><mrow id="A2.I2.i2.p1.14.m14.2.3.3" xref="A2.I2.i2.p1.14.m14.2.3.3.cmml"><mn id="A2.I2.i2.p1.14.m14.2.3.3.2" xref="A2.I2.i2.p1.14.m14.2.3.3.2.cmml">2</mn><mo id="A2.I2.i2.p1.14.m14.2.3.3.1" xref="A2.I2.i2.p1.14.m14.2.3.3.1.cmml">/</mo><mn id="A2.I2.i2.p1.14.m14.2.3.3.3" xref="A2.I2.i2.p1.14.m14.2.3.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I2.i2.p1.14.m14.2b"><apply id="A2.I2.i2.p1.14.m14.2.3.cmml" xref="A2.I2.i2.p1.14.m14.2.3"><geq id="A2.I2.i2.p1.14.m14.2.3.1.cmml" xref="A2.I2.i2.p1.14.m14.2.3.1"></geq><apply id="A2.I2.i2.p1.14.m14.2.3.2.cmml" xref="A2.I2.i2.p1.14.m14.2.3.2"><minus id="A2.I2.i2.p1.14.m14.2.3.2.1.cmml" xref="A2.I2.i2.p1.14.m14.2.3.2.1"></minus><cn id="A2.I2.i2.p1.14.m14.2.3.2.2.cmml" type="integer" xref="A2.I2.i2.p1.14.m14.2.3.2.2">1</cn><apply id="A2.I2.i2.p1.14.m14.2.3.2.3.cmml" xref="A2.I2.i2.p1.14.m14.2.3.2.3"><times id="A2.I2.i2.p1.14.m14.2.3.2.3.1.cmml" xref="A2.I2.i2.p1.14.m14.2.3.2.3.1"></times><ci id="A2.I2.i2.p1.14.m14.2.3.2.3.2.cmml" xref="A2.I2.i2.p1.14.m14.2.3.2.3.2">𝜇</ci><interval closure="open" id="A2.I2.i2.p1.14.m14.2.3.2.3.3.1.cmml" xref="A2.I2.i2.p1.14.m14.2.3.2.3.3.2"><ci id="A2.I2.i2.p1.14.m14.1.1.cmml" xref="A2.I2.i2.p1.14.m14.1.1">𝑌</ci><ci id="A2.I2.i2.p1.14.m14.2.2.cmml" xref="A2.I2.i2.p1.14.m14.2.2">𝑌</ci></interval></apply></apply><apply id="A2.I2.i2.p1.14.m14.2.3.3.cmml" xref="A2.I2.i2.p1.14.m14.2.3.3"><divide id="A2.I2.i2.p1.14.m14.2.3.3.1.cmml" xref="A2.I2.i2.p1.14.m14.2.3.3.1"></divide><cn id="A2.I2.i2.p1.14.m14.2.3.3.2.cmml" type="integer" xref="A2.I2.i2.p1.14.m14.2.3.3.2">2</cn><cn id="A2.I2.i2.p1.14.m14.2.3.3.3.cmml" type="integer" xref="A2.I2.i2.p1.14.m14.2.3.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I2.i2.p1.14.m14.2c">1-\mu(Y,Y)\geq 2/3</annotation><annotation encoding="application/x-llamapun" id="A2.I2.i2.p1.14.m14.2d">1 - italic_μ ( italic_Y , italic_Y ) ≥ 2 / 3</annotation></semantics></math>, which is a contradiction. ∎</p> </div> </li> </ul> </div> </section> <section class="ltx_subsection" id="A2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">B.2 </span>Revelation principle for CEPs</h3> <div class="ltx_theorem ltx_theorem_definition" id="A2.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="A2.Thmtheorem1.1.1.1">Definition B.1</span></span><span class="ltx_text ltx_font_bold" id="A2.Thmtheorem1.2.2"> </span>(Non-canonical CEP)<span class="ltx_text ltx_font_bold" id="A2.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="A2.Thmtheorem1.p1"> <p class="ltx_p" id="A2.Thmtheorem1.p1.5">A correlated profile with payments is a tuple <math alttext="(\mu,P,\pi)" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.1.m1.3"><semantics id="A2.Thmtheorem1.p1.1.m1.3a"><mrow id="A2.Thmtheorem1.p1.1.m1.3.4.2" xref="A2.Thmtheorem1.p1.1.m1.3.4.1.cmml"><mo id="A2.Thmtheorem1.p1.1.m1.3.4.2.1" stretchy="false" xref="A2.Thmtheorem1.p1.1.m1.3.4.1.cmml">(</mo><mi id="A2.Thmtheorem1.p1.1.m1.1.1" xref="A2.Thmtheorem1.p1.1.m1.1.1.cmml">μ</mi><mo id="A2.Thmtheorem1.p1.1.m1.3.4.2.2" xref="A2.Thmtheorem1.p1.1.m1.3.4.1.cmml">,</mo><mi id="A2.Thmtheorem1.p1.1.m1.2.2" xref="A2.Thmtheorem1.p1.1.m1.2.2.cmml">P</mi><mo id="A2.Thmtheorem1.p1.1.m1.3.4.2.3" xref="A2.Thmtheorem1.p1.1.m1.3.4.1.cmml">,</mo><mi id="A2.Thmtheorem1.p1.1.m1.3.3" xref="A2.Thmtheorem1.p1.1.m1.3.3.cmml">π</mi><mo id="A2.Thmtheorem1.p1.1.m1.3.4.2.4" stretchy="false" xref="A2.Thmtheorem1.p1.1.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.1.m1.3b"><vector id="A2.Thmtheorem1.p1.1.m1.3.4.1.cmml" xref="A2.Thmtheorem1.p1.1.m1.3.4.2"><ci id="A2.Thmtheorem1.p1.1.m1.1.1.cmml" xref="A2.Thmtheorem1.p1.1.m1.1.1">𝜇</ci><ci id="A2.Thmtheorem1.p1.1.m1.2.2.cmml" xref="A2.Thmtheorem1.p1.1.m1.2.2">𝑃</ci><ci id="A2.Thmtheorem1.p1.1.m1.3.3.cmml" xref="A2.Thmtheorem1.p1.1.m1.3.3">𝜋</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.1.m1.3c">(\mu,P,\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.1.m1.3d">( italic_μ , italic_P , italic_π )</annotation></semantics></math>, where <math alttext="\mu\in\Delta(S)" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.2.m2.1"><semantics id="A2.Thmtheorem1.p1.2.m2.1a"><mrow id="A2.Thmtheorem1.p1.2.m2.1.2" xref="A2.Thmtheorem1.p1.2.m2.1.2.cmml"><mi id="A2.Thmtheorem1.p1.2.m2.1.2.2" xref="A2.Thmtheorem1.p1.2.m2.1.2.2.cmml">μ</mi><mo id="A2.Thmtheorem1.p1.2.m2.1.2.1" xref="A2.Thmtheorem1.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="A2.Thmtheorem1.p1.2.m2.1.2.3" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.cmml"><mi id="A2.Thmtheorem1.p1.2.m2.1.2.3.2" mathvariant="normal" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.2.cmml">Δ</mi><mo id="A2.Thmtheorem1.p1.2.m2.1.2.3.1" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="A2.Thmtheorem1.p1.2.m2.1.2.3.3.2" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.cmml"><mo id="A2.Thmtheorem1.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.cmml">(</mo><mi id="A2.Thmtheorem1.p1.2.m2.1.1" xref="A2.Thmtheorem1.p1.2.m2.1.1.cmml">S</mi><mo id="A2.Thmtheorem1.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.2.m2.1b"><apply id="A2.Thmtheorem1.p1.2.m2.1.2.cmml" xref="A2.Thmtheorem1.p1.2.m2.1.2"><in id="A2.Thmtheorem1.p1.2.m2.1.2.1.cmml" xref="A2.Thmtheorem1.p1.2.m2.1.2.1"></in><ci id="A2.Thmtheorem1.p1.2.m2.1.2.2.cmml" xref="A2.Thmtheorem1.p1.2.m2.1.2.2">𝜇</ci><apply id="A2.Thmtheorem1.p1.2.m2.1.2.3.cmml" xref="A2.Thmtheorem1.p1.2.m2.1.2.3"><times id="A2.Thmtheorem1.p1.2.m2.1.2.3.1.cmml" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.1"></times><ci id="A2.Thmtheorem1.p1.2.m2.1.2.3.2.cmml" xref="A2.Thmtheorem1.p1.2.m2.1.2.3.2">Δ</ci><ci id="A2.Thmtheorem1.p1.2.m2.1.1.cmml" xref="A2.Thmtheorem1.p1.2.m2.1.1">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.2.m2.1c">\mu\in\Delta(S)</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.2.m2.1d">italic_μ ∈ roman_Δ ( italic_S )</annotation></semantics></math> is a correlated signal distribution, <math alttext="P_{i}:S\times A\to\mathbb{R}_{+}" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.3.m3.1"><semantics id="A2.Thmtheorem1.p1.3.m3.1a"><mrow id="A2.Thmtheorem1.p1.3.m3.1.1" xref="A2.Thmtheorem1.p1.3.m3.1.1.cmml"><msub id="A2.Thmtheorem1.p1.3.m3.1.1.2" xref="A2.Thmtheorem1.p1.3.m3.1.1.2.cmml"><mi id="A2.Thmtheorem1.p1.3.m3.1.1.2.2" xref="A2.Thmtheorem1.p1.3.m3.1.1.2.2.cmml">P</mi><mi id="A2.Thmtheorem1.p1.3.m3.1.1.2.3" xref="A2.Thmtheorem1.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="A2.Thmtheorem1.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="A2.Thmtheorem1.p1.3.m3.1.1.1.cmml">:</mo><mrow id="A2.Thmtheorem1.p1.3.m3.1.1.3" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.cmml"><mrow id="A2.Thmtheorem1.p1.3.m3.1.1.3.2" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2.cmml"><mi id="A2.Thmtheorem1.p1.3.m3.1.1.3.2.2" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2.2.cmml">S</mi><mo id="A2.Thmtheorem1.p1.3.m3.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2.1.cmml">×</mo><mi id="A2.Thmtheorem1.p1.3.m3.1.1.3.2.3" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2.3.cmml">A</mi></mrow><mo id="A2.Thmtheorem1.p1.3.m3.1.1.3.1" stretchy="false" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.1.cmml">→</mo><msub id="A2.Thmtheorem1.p1.3.m3.1.1.3.3" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.3.cmml"><mi id="A2.Thmtheorem1.p1.3.m3.1.1.3.3.2" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.3.2.cmml">ℝ</mi><mo id="A2.Thmtheorem1.p1.3.m3.1.1.3.3.3" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.3.3.cmml">+</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.3.m3.1b"><apply id="A2.Thmtheorem1.p1.3.m3.1.1.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1"><ci id="A2.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.1">:</ci><apply id="A2.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.3.m3.1.1.2.1.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.2">subscript</csymbol><ci id="A2.Thmtheorem1.p1.3.m3.1.1.2.2.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.2.2">𝑃</ci><ci id="A2.Thmtheorem1.p1.3.m3.1.1.2.3.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="A2.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3"><ci id="A2.Thmtheorem1.p1.3.m3.1.1.3.1.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.1">→</ci><apply id="A2.Thmtheorem1.p1.3.m3.1.1.3.2.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2"><times id="A2.Thmtheorem1.p1.3.m3.1.1.3.2.1.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2.1"></times><ci id="A2.Thmtheorem1.p1.3.m3.1.1.3.2.2.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2.2">𝑆</ci><ci id="A2.Thmtheorem1.p1.3.m3.1.1.3.2.3.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.2.3">𝐴</ci></apply><apply id="A2.Thmtheorem1.p1.3.m3.1.1.3.3.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.3.m3.1.1.3.3.1.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="A2.Thmtheorem1.p1.3.m3.1.1.3.3.2.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.3.2">ℝ</ci><plus id="A2.Thmtheorem1.p1.3.m3.1.1.3.3.3.cmml" xref="A2.Thmtheorem1.p1.3.m3.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.3.m3.1c">P_{i}:S\times A\to\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_S × italic_A → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> is the payment function, and <math alttext="\pi_{i}:S_{i}\to\Delta(A_{i})" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.4.m4.1"><semantics id="A2.Thmtheorem1.p1.4.m4.1a"><mrow id="A2.Thmtheorem1.p1.4.m4.1.1" xref="A2.Thmtheorem1.p1.4.m4.1.1.cmml"><msub id="A2.Thmtheorem1.p1.4.m4.1.1.3" xref="A2.Thmtheorem1.p1.4.m4.1.1.3.cmml"><mi id="A2.Thmtheorem1.p1.4.m4.1.1.3.2" xref="A2.Thmtheorem1.p1.4.m4.1.1.3.2.cmml">π</mi><mi id="A2.Thmtheorem1.p1.4.m4.1.1.3.3" xref="A2.Thmtheorem1.p1.4.m4.1.1.3.3.cmml">i</mi></msub><mo id="A2.Thmtheorem1.p1.4.m4.1.1.2" lspace="0.278em" rspace="0.278em" xref="A2.Thmtheorem1.p1.4.m4.1.1.2.cmml">:</mo><mrow id="A2.Thmtheorem1.p1.4.m4.1.1.1" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.cmml"><msub id="A2.Thmtheorem1.p1.4.m4.1.1.1.3" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.3.cmml"><mi id="A2.Thmtheorem1.p1.4.m4.1.1.1.3.2" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.3.2.cmml">S</mi><mi id="A2.Thmtheorem1.p1.4.m4.1.1.1.3.3" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.3.3.cmml">i</mi></msub><mo id="A2.Thmtheorem1.p1.4.m4.1.1.1.2" stretchy="false" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.2.cmml">→</mo><mrow id="A2.Thmtheorem1.p1.4.m4.1.1.1.1" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.cmml"><mi id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.3" mathvariant="normal" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.3.cmml">Δ</mi><mo id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.2" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.cmml"><mo id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.cmml">(</mo><msub id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.cmml"><mi id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.2" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.2.cmml">A</mi><mi id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.3" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.4.m4.1b"><apply id="A2.Thmtheorem1.p1.4.m4.1.1.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1"><ci id="A2.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.2">:</ci><apply id="A2.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.4.m4.1.1.3.1.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.3">subscript</csymbol><ci id="A2.Thmtheorem1.p1.4.m4.1.1.3.2.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.3.2">𝜋</ci><ci id="A2.Thmtheorem1.p1.4.m4.1.1.3.3.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.3.3">𝑖</ci></apply><apply id="A2.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1"><ci id="A2.Thmtheorem1.p1.4.m4.1.1.1.2.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.2">→</ci><apply id="A2.Thmtheorem1.p1.4.m4.1.1.1.3.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.4.m4.1.1.1.3.1.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.3">subscript</csymbol><ci id="A2.Thmtheorem1.p1.4.m4.1.1.1.3.2.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.3.2">𝑆</ci><ci id="A2.Thmtheorem1.p1.4.m4.1.1.1.3.3.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.3.3">𝑖</ci></apply><apply id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1"><times id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.2.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.2"></times><ci id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.3.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.3">Δ</ci><apply id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.2">𝐴</ci><ci id="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="A2.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.4.m4.1c">\pi_{i}:S_{i}\to\Delta(A_{i})</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.4.m4.1d">italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → roman_Δ ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is agent <math alttext="i" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.5.m5.1"><semantics id="A2.Thmtheorem1.p1.5.m5.1a"><mi id="A2.Thmtheorem1.p1.5.m5.1.1" xref="A2.Thmtheorem1.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.5.m5.1b"><ci id="A2.Thmtheorem1.p1.5.m5.1.1.cmml" xref="A2.Thmtheorem1.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.5.m5.1d">italic_i</annotation></semantics></math>’s strategy. The objective value is given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx16"> <tbody id="A2.E21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{\mathbb{E}}_{\begin{subarray}{c}s\sim\mu,\\ a\sim\pi(s)\end{subarray}}\quantity[U_{0}(\pi(s))-P(s,\pi(s))]." class="ltx_Math" display="inline" id="A2.E21.m1.3"><semantics id="A2.E21.m1.3a"><mrow id="A2.E21.m1.3.3.1" xref="A2.E21.m1.3.3.1.1.cmml"><mrow id="A2.E21.m1.3.3.1.1" xref="A2.E21.m1.3.3.1.1.cmml"><munder id="A2.E21.m1.3.3.1.1.1" xref="A2.E21.m1.3.3.1.1.1.cmml"><mo id="A2.E21.m1.3.3.1.1.1.2" xref="A2.E21.m1.3.3.1.1.1.2.cmml">𝔼</mo><mtable id="A2.E21.m1.1.1.1.1.1.1" rowspacing="0pt" xref="A2.E21.m1.1.1.1a.2.cmml"><mtr id="A2.E21.m1.1.1.1.1.1.1a" xref="A2.E21.m1.1.1.1a.2.cmml"><mtd id="A2.E21.m1.1.1.1.1.1.1b" xref="A2.E21.m1.1.1.1a.2.cmml"><mrow id="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1" xref="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">s</mi><mo id="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">∼</mo><mi id="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A2.E21.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">,</mo></mrow></mtd></mtr><mtr id="A2.E21.m1.1.1.1.1.1.1c" xref="A2.E21.m1.1.1.1a.2.cmml"><mtd id="A2.E21.m1.1.1.1.1.1.1d" xref="A2.E21.m1.1.1.1a.2.cmml"><mrow id="A2.E21.m1.1.1.1.1.1.1.2.2.1.1" xref="A2.E21.m1.1.1.1.1.1.1.2.2.1.1.cmml"><mi id="A2.E21.m1.1.1.1.1.1.1.2.2.1.1.3" xref="A2.E21.m1.1.1.1.1.1.1.2.2.1.1.3.cmml">a</mi><mo id="A2.E21.m1.1.1.1.1.1.1.2.2.1.1.2" xref="A2.E21.m1.1.1.1.1.1.1.2.2.1.1.2.cmml">∼</mo><mrow 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start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_π ( italic_s ) ) - italic_P ( italic_s , italic_π ( italic_s ) ) end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.Thmtheorem1.p1.10"><math alttext="(\mu,P,\pi)" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.6.m1.3"><semantics id="A2.Thmtheorem1.p1.6.m1.3a"><mrow id="A2.Thmtheorem1.p1.6.m1.3.4.2" xref="A2.Thmtheorem1.p1.6.m1.3.4.1.cmml"><mo id="A2.Thmtheorem1.p1.6.m1.3.4.2.1" stretchy="false" xref="A2.Thmtheorem1.p1.6.m1.3.4.1.cmml">(</mo><mi id="A2.Thmtheorem1.p1.6.m1.1.1" xref="A2.Thmtheorem1.p1.6.m1.1.1.cmml">μ</mi><mo id="A2.Thmtheorem1.p1.6.m1.3.4.2.2" xref="A2.Thmtheorem1.p1.6.m1.3.4.1.cmml">,</mo><mi id="A2.Thmtheorem1.p1.6.m1.2.2" xref="A2.Thmtheorem1.p1.6.m1.2.2.cmml">P</mi><mo id="A2.Thmtheorem1.p1.6.m1.3.4.2.3" xref="A2.Thmtheorem1.p1.6.m1.3.4.1.cmml">,</mo><mi id="A2.Thmtheorem1.p1.6.m1.3.3" xref="A2.Thmtheorem1.p1.6.m1.3.3.cmml">π</mi><mo id="A2.Thmtheorem1.p1.6.m1.3.4.2.4" stretchy="false" xref="A2.Thmtheorem1.p1.6.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.6.m1.3b"><vector id="A2.Thmtheorem1.p1.6.m1.3.4.1.cmml" xref="A2.Thmtheorem1.p1.6.m1.3.4.2"><ci id="A2.Thmtheorem1.p1.6.m1.1.1.cmml" xref="A2.Thmtheorem1.p1.6.m1.1.1">𝜇</ci><ci id="A2.Thmtheorem1.p1.6.m1.2.2.cmml" xref="A2.Thmtheorem1.p1.6.m1.2.2">𝑃</ci><ci id="A2.Thmtheorem1.p1.6.m1.3.3.cmml" xref="A2.Thmtheorem1.p1.6.m1.3.3">𝜋</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.6.m1.3c">(\mu,P,\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.6.m1.3d">( italic_μ , italic_P , italic_π )</annotation></semantics></math> is a CEP if <math alttext="\pi:=(\pi_{1},\dots,\pi_{n})" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.7.m2.3"><semantics id="A2.Thmtheorem1.p1.7.m2.3a"><mrow id="A2.Thmtheorem1.p1.7.m2.3.3" xref="A2.Thmtheorem1.p1.7.m2.3.3.cmml"><mi id="A2.Thmtheorem1.p1.7.m2.3.3.4" xref="A2.Thmtheorem1.p1.7.m2.3.3.4.cmml">π</mi><mo id="A2.Thmtheorem1.p1.7.m2.3.3.3" lspace="0.278em" rspace="0.278em" xref="A2.Thmtheorem1.p1.7.m2.3.3.3.cmml">:=</mo><mrow id="A2.Thmtheorem1.p1.7.m2.3.3.2.2" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.3.cmml"><mo id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.3" stretchy="false" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.3.cmml">(</mo><msub id="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1" xref="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.cmml"><mi id="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.2" xref="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.2.cmml">π</mi><mn id="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.3" xref="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.4" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.3.cmml">,</mo><mi id="A2.Thmtheorem1.p1.7.m2.1.1" mathvariant="normal" xref="A2.Thmtheorem1.p1.7.m2.1.1.cmml">…</mi><mo id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.5" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.3.cmml">,</mo><msub id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.cmml"><mi id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.2" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.2.cmml">π</mi><mi id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.3" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.3.cmml">n</mi></msub><mo id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.6" stretchy="false" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.7.m2.3b"><apply id="A2.Thmtheorem1.p1.7.m2.3.3.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3"><csymbol cd="latexml" id="A2.Thmtheorem1.p1.7.m2.3.3.3.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3.3">assign</csymbol><ci id="A2.Thmtheorem1.p1.7.m2.3.3.4.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3.4">𝜋</ci><vector id="A2.Thmtheorem1.p1.7.m2.3.3.2.3.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2"><apply id="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.cmml" xref="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.1.cmml" xref="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1">subscript</csymbol><ci id="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.2.cmml" xref="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.2">𝜋</ci><cn id="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.3.cmml" type="integer" xref="A2.Thmtheorem1.p1.7.m2.2.2.1.1.1.3">1</cn></apply><ci id="A2.Thmtheorem1.p1.7.m2.1.1.cmml" xref="A2.Thmtheorem1.p1.7.m2.1.1">…</ci><apply id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.1.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2">subscript</csymbol><ci id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.2.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.2">𝜋</ci><ci id="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.3.cmml" xref="A2.Thmtheorem1.p1.7.m2.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.7.m2.3c">\pi:=(\pi_{1},\dots,\pi_{n})</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.7.m2.3d">italic_π := ( italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_π start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> is an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.8.m3.1"><semantics id="A2.Thmtheorem1.p1.8.m3.1a"><mi id="A2.Thmtheorem1.p1.8.m3.1.1" xref="A2.Thmtheorem1.p1.8.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.8.m3.1b"><ci id="A2.Thmtheorem1.p1.8.m3.1.1.cmml" xref="A2.Thmtheorem1.p1.8.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.8.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.8.m3.1d">italic_ε</annotation></semantics></math>-Nash equilibrium of the game <math alttext="\Gamma^{\mu,P}" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.9.m4.2"><semantics id="A2.Thmtheorem1.p1.9.m4.2a"><msup id="A2.Thmtheorem1.p1.9.m4.2.3" xref="A2.Thmtheorem1.p1.9.m4.2.3.cmml"><mi id="A2.Thmtheorem1.p1.9.m4.2.3.2" mathvariant="normal" xref="A2.Thmtheorem1.p1.9.m4.2.3.2.cmml">Γ</mi><mrow id="A2.Thmtheorem1.p1.9.m4.2.2.2.4" xref="A2.Thmtheorem1.p1.9.m4.2.2.2.3.cmml"><mi id="A2.Thmtheorem1.p1.9.m4.1.1.1.1" xref="A2.Thmtheorem1.p1.9.m4.1.1.1.1.cmml">μ</mi><mo id="A2.Thmtheorem1.p1.9.m4.2.2.2.4.1" xref="A2.Thmtheorem1.p1.9.m4.2.2.2.3.cmml">,</mo><mi id="A2.Thmtheorem1.p1.9.m4.2.2.2.2" xref="A2.Thmtheorem1.p1.9.m4.2.2.2.2.cmml">P</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.9.m4.2b"><apply id="A2.Thmtheorem1.p1.9.m4.2.3.cmml" xref="A2.Thmtheorem1.p1.9.m4.2.3"><csymbol cd="ambiguous" id="A2.Thmtheorem1.p1.9.m4.2.3.1.cmml" xref="A2.Thmtheorem1.p1.9.m4.2.3">superscript</csymbol><ci id="A2.Thmtheorem1.p1.9.m4.2.3.2.cmml" xref="A2.Thmtheorem1.p1.9.m4.2.3.2">Γ</ci><list id="A2.Thmtheorem1.p1.9.m4.2.2.2.3.cmml" xref="A2.Thmtheorem1.p1.9.m4.2.2.2.4"><ci id="A2.Thmtheorem1.p1.9.m4.1.1.1.1.cmml" xref="A2.Thmtheorem1.p1.9.m4.1.1.1.1">𝜇</ci><ci id="A2.Thmtheorem1.p1.9.m4.2.2.2.2.cmml" xref="A2.Thmtheorem1.p1.9.m4.2.2.2.2">𝑃</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.9.m4.2c">\Gamma^{\mu,P}</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.9.m4.2d">roman_Γ start_POSTSUPERSCRIPT italic_μ , italic_P end_POSTSUPERSCRIPT</annotation></semantics></math> in which each agent <math alttext="i" class="ltx_Math" display="inline" id="A2.Thmtheorem1.p1.10.m5.1"><semantics id="A2.Thmtheorem1.p1.10.m5.1a"><mi id="A2.Thmtheorem1.p1.10.m5.1.1" xref="A2.Thmtheorem1.p1.10.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem1.p1.10.m5.1b"><ci id="A2.Thmtheorem1.p1.10.m5.1.1.cmml" xref="A2.Thmtheorem1.p1.10.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem1.p1.10.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem1.p1.10.m5.1d">italic_i</annotation></semantics></math>’s utility function is given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx17"> <tbody id="A2.E22"><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 U_{i}^{\mu,P}(\pi)=\operatorname*{\mathbb{E}}_{\begin{subarray}{% c}s\sim\mu,\\ a\sim\pi(s)\end{subarray}}\quantity[U_{i}(a)+P_{i}(s,a)]." class="ltx_Math" display="inline" id="A2.E22.m1.6"><semantics id="A2.E22.m1.6a"><mrow id="A2.E22.m1.6.6.1" xref="A2.E22.m1.6.6.1.1.cmml"><mrow id="A2.E22.m1.6.6.1.1" xref="A2.E22.m1.6.6.1.1.cmml"><mrow id="A2.E22.m1.6.6.1.1.2" xref="A2.E22.m1.6.6.1.1.2.cmml"><msubsup id="A2.E22.m1.6.6.1.1.2.2" xref="A2.E22.m1.6.6.1.1.2.2.cmml"><mi id="A2.E22.m1.6.6.1.1.2.2.2.2" xref="A2.E22.m1.6.6.1.1.2.2.2.2.cmml">U</mi><mi id="A2.E22.m1.6.6.1.1.2.2.2.3" xref="A2.E22.m1.6.6.1.1.2.2.2.3.cmml">i</mi><mrow id="A2.E22.m1.4.4.2.4" xref="A2.E22.m1.4.4.2.3.cmml"><mi id="A2.E22.m1.3.3.1.1" xref="A2.E22.m1.3.3.1.1.cmml">μ</mi><mo id="A2.E22.m1.4.4.2.4.1" xref="A2.E22.m1.4.4.2.3.cmml">,</mo><mi id="A2.E22.m1.4.4.2.2" xref="A2.E22.m1.4.4.2.2.cmml">P</mi></mrow></msubsup><mo id="A2.E22.m1.6.6.1.1.2.1" xref="A2.E22.m1.6.6.1.1.2.1.cmml">⁢</mo><mrow id="A2.E22.m1.6.6.1.1.2.3.2" xref="A2.E22.m1.6.6.1.1.2.cmml"><mo id="A2.E22.m1.6.6.1.1.2.3.2.1" stretchy="false" xref="A2.E22.m1.6.6.1.1.2.cmml">(</mo><mi id="A2.E22.m1.5.5" xref="A2.E22.m1.5.5.cmml">π</mi><mo id="A2.E22.m1.6.6.1.1.2.3.2.2" stretchy="false" xref="A2.E22.m1.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo 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xref="A2.E22.m1.2.2.1.1.1.1">𝑎</ci></apply><apply id="A2.E22.m1.2.2.1.1.1.6.cmml" xref="A2.E22.m1.2.2.1.1.1.6"><times id="A2.E22.m1.2.2.1.1.1.6.1.cmml" xref="A2.E22.m1.2.2.1.1.1.6.1"></times><apply id="A2.E22.m1.2.2.1.1.1.6.2.cmml" xref="A2.E22.m1.2.2.1.1.1.6.2"><csymbol cd="ambiguous" id="A2.E22.m1.2.2.1.1.1.6.2.1.cmml" xref="A2.E22.m1.2.2.1.1.1.6.2">subscript</csymbol><ci id="A2.E22.m1.2.2.1.1.1.6.2.2.cmml" xref="A2.E22.m1.2.2.1.1.1.6.2.2">𝑃</ci><ci id="A2.E22.m1.2.2.1.1.1.6.2.3.cmml" xref="A2.E22.m1.2.2.1.1.1.6.2.3">𝑖</ci></apply><interval closure="open" id="A2.E22.m1.2.2.1.1.1.6.3.1.cmml" xref="A2.E22.m1.2.2.1.1.1.6.3.2"><ci id="A2.E22.m1.2.2.1.1.1.2.cmml" xref="A2.E22.m1.2.2.1.1.1.2">𝑠</ci><ci id="A2.E22.m1.2.2.1.1.1.3.cmml" xref="A2.E22.m1.2.2.1.1.1.3">𝑎</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E22.m1.6c">\displaystyle U_{i}^{\mu,P}(\pi)=\operatorname*{\mathbb{E}}_{\begin{subarray}{% c}s\sim\mu,\\ a\sim\pi(s)\end{subarray}}\quantity[U_{i}(a)+P_{i}(s,a)].</annotation><annotation encoding="application/x-llamapun" id="A2.E22.m1.6d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ , italic_P end_POSTSUPERSCRIPT ( italic_π ) = blackboard_E start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_s ∼ italic_μ , end_CELL end_ROW start_ROW start_CELL italic_a ∼ italic_π ( italic_s ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s , italic_a ) end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="A2.SS2.p1"> <p class="ltx_p" id="A2.SS2.p1.4">We say that <math alttext="(\mu,P,\pi)" class="ltx_Math" display="inline" id="A2.SS2.p1.1.m1.3"><semantics id="A2.SS2.p1.1.m1.3a"><mrow id="A2.SS2.p1.1.m1.3.4.2" xref="A2.SS2.p1.1.m1.3.4.1.cmml"><mo id="A2.SS2.p1.1.m1.3.4.2.1" stretchy="false" xref="A2.SS2.p1.1.m1.3.4.1.cmml">(</mo><mi id="A2.SS2.p1.1.m1.1.1" xref="A2.SS2.p1.1.m1.1.1.cmml">μ</mi><mo id="A2.SS2.p1.1.m1.3.4.2.2" xref="A2.SS2.p1.1.m1.3.4.1.cmml">,</mo><mi id="A2.SS2.p1.1.m1.2.2" xref="A2.SS2.p1.1.m1.2.2.cmml">P</mi><mo id="A2.SS2.p1.1.m1.3.4.2.3" xref="A2.SS2.p1.1.m1.3.4.1.cmml">,</mo><mi id="A2.SS2.p1.1.m1.3.3" xref="A2.SS2.p1.1.m1.3.3.cmml">π</mi><mo id="A2.SS2.p1.1.m1.3.4.2.4" stretchy="false" xref="A2.SS2.p1.1.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.1.m1.3b"><vector id="A2.SS2.p1.1.m1.3.4.1.cmml" xref="A2.SS2.p1.1.m1.3.4.2"><ci id="A2.SS2.p1.1.m1.1.1.cmml" xref="A2.SS2.p1.1.m1.1.1">𝜇</ci><ci id="A2.SS2.p1.1.m1.2.2.cmml" xref="A2.SS2.p1.1.m1.2.2">𝑃</ci><ci id="A2.SS2.p1.1.m1.3.3.cmml" xref="A2.SS2.p1.1.m1.3.3">𝜋</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.1.m1.3c">(\mu,P,\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.1.m1.3d">( italic_μ , italic_P , italic_π )</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="A2.SS2.p1.4.1">canonical</span> if, for every player <math alttext="i" class="ltx_Math" display="inline" id="A2.SS2.p1.2.m2.1"><semantics id="A2.SS2.p1.2.m2.1a"><mi id="A2.SS2.p1.2.m2.1.1" xref="A2.SS2.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.2.m2.1b"><ci id="A2.SS2.p1.2.m2.1.1.cmml" xref="A2.SS2.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.2.m2.1d">italic_i</annotation></semantics></math>, <math alttext="S_{i}=A_{i}" class="ltx_Math" display="inline" id="A2.SS2.p1.3.m3.1"><semantics id="A2.SS2.p1.3.m3.1a"><mrow id="A2.SS2.p1.3.m3.1.1" xref="A2.SS2.p1.3.m3.1.1.cmml"><msub id="A2.SS2.p1.3.m3.1.1.2" xref="A2.SS2.p1.3.m3.1.1.2.cmml"><mi id="A2.SS2.p1.3.m3.1.1.2.2" xref="A2.SS2.p1.3.m3.1.1.2.2.cmml">S</mi><mi id="A2.SS2.p1.3.m3.1.1.2.3" xref="A2.SS2.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="A2.SS2.p1.3.m3.1.1.1" xref="A2.SS2.p1.3.m3.1.1.1.cmml">=</mo><msub id="A2.SS2.p1.3.m3.1.1.3" xref="A2.SS2.p1.3.m3.1.1.3.cmml"><mi id="A2.SS2.p1.3.m3.1.1.3.2" xref="A2.SS2.p1.3.m3.1.1.3.2.cmml">A</mi><mi id="A2.SS2.p1.3.m3.1.1.3.3" xref="A2.SS2.p1.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.3.m3.1b"><apply id="A2.SS2.p1.3.m3.1.1.cmml" xref="A2.SS2.p1.3.m3.1.1"><eq id="A2.SS2.p1.3.m3.1.1.1.cmml" xref="A2.SS2.p1.3.m3.1.1.1"></eq><apply id="A2.SS2.p1.3.m3.1.1.2.cmml" xref="A2.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p1.3.m3.1.1.2.1.cmml" xref="A2.SS2.p1.3.m3.1.1.2">subscript</csymbol><ci id="A2.SS2.p1.3.m3.1.1.2.2.cmml" xref="A2.SS2.p1.3.m3.1.1.2.2">𝑆</ci><ci id="A2.SS2.p1.3.m3.1.1.2.3.cmml" xref="A2.SS2.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="A2.SS2.p1.3.m3.1.1.3.cmml" xref="A2.SS2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p1.3.m3.1.1.3.1.cmml" xref="A2.SS2.p1.3.m3.1.1.3">subscript</csymbol><ci id="A2.SS2.p1.3.m3.1.1.3.2.cmml" xref="A2.SS2.p1.3.m3.1.1.3.2">𝐴</ci><ci id="A2.SS2.p1.3.m3.1.1.3.3.cmml" xref="A2.SS2.p1.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.3.m3.1c">S_{i}=A_{i}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\pi_{i}" class="ltx_Math" display="inline" id="A2.SS2.p1.4.m4.1"><semantics id="A2.SS2.p1.4.m4.1a"><msub id="A2.SS2.p1.4.m4.1.1" xref="A2.SS2.p1.4.m4.1.1.cmml"><mi id="A2.SS2.p1.4.m4.1.1.2" xref="A2.SS2.p1.4.m4.1.1.2.cmml">π</mi><mi id="A2.SS2.p1.4.m4.1.1.3" xref="A2.SS2.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.4.m4.1b"><apply id="A2.SS2.p1.4.m4.1.1.cmml" xref="A2.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.4.m4.1.1.1.cmml" xref="A2.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="A2.SS2.p1.4.m4.1.1.2.cmml" xref="A2.SS2.p1.4.m4.1.1.2">𝜋</ci><ci id="A2.SS2.p1.4.m4.1.1.3.cmml" xref="A2.SS2.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.4.m4.1c">\pi_{i}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.4.m4.1d">italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the identity map. Note that canonical equilibria are precisely the equilibria defined by <a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S7.Thmtheorem1" title="Definition 7.1. ‣ 7.2 What outcome should we steer to? ‣ 7 Steering No-Regret Learners by Learning Utilities ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">7.1</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="A2.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="A2.Thmtheorem2.1.1.1">Proposition B.2</span></span><span class="ltx_text ltx_font_bold" id="A2.Thmtheorem2.2.2"> </span>(Revelation principle for CEPs)<span class="ltx_text ltx_font_bold" id="A2.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="A2.Thmtheorem2.p1"> <p class="ltx_p" id="A2.Thmtheorem2.p1.2"><span class="ltx_text ltx_font_italic" id="A2.Thmtheorem2.p1.2.2">Every CEP is equivalent to a canonical CEP, in the sense that, for every CEP <math alttext="(\mu,P,\pi)" class="ltx_Math" display="inline" id="A2.Thmtheorem2.p1.1.1.m1.3"><semantics id="A2.Thmtheorem2.p1.1.1.m1.3a"><mrow id="A2.Thmtheorem2.p1.1.1.m1.3.4.2" xref="A2.Thmtheorem2.p1.1.1.m1.3.4.1.cmml"><mo id="A2.Thmtheorem2.p1.1.1.m1.3.4.2.1" stretchy="false" xref="A2.Thmtheorem2.p1.1.1.m1.3.4.1.cmml">(</mo><mi id="A2.Thmtheorem2.p1.1.1.m1.1.1" xref="A2.Thmtheorem2.p1.1.1.m1.1.1.cmml">μ</mi><mo id="A2.Thmtheorem2.p1.1.1.m1.3.4.2.2" xref="A2.Thmtheorem2.p1.1.1.m1.3.4.1.cmml">,</mo><mi id="A2.Thmtheorem2.p1.1.1.m1.2.2" xref="A2.Thmtheorem2.p1.1.1.m1.2.2.cmml">P</mi><mo id="A2.Thmtheorem2.p1.1.1.m1.3.4.2.3" xref="A2.Thmtheorem2.p1.1.1.m1.3.4.1.cmml">,</mo><mi id="A2.Thmtheorem2.p1.1.1.m1.3.3" xref="A2.Thmtheorem2.p1.1.1.m1.3.3.cmml">π</mi><mo id="A2.Thmtheorem2.p1.1.1.m1.3.4.2.4" stretchy="false" xref="A2.Thmtheorem2.p1.1.1.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem2.p1.1.1.m1.3b"><vector id="A2.Thmtheorem2.p1.1.1.m1.3.4.1.cmml" xref="A2.Thmtheorem2.p1.1.1.m1.3.4.2"><ci id="A2.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="A2.Thmtheorem2.p1.1.1.m1.1.1">𝜇</ci><ci id="A2.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="A2.Thmtheorem2.p1.1.1.m1.2.2">𝑃</ci><ci id="A2.Thmtheorem2.p1.1.1.m1.3.3.cmml" xref="A2.Thmtheorem2.p1.1.1.m1.3.3">𝜋</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem2.p1.1.1.m1.3c">(\mu,P,\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem2.p1.1.1.m1.3d">( italic_μ , italic_P , italic_π )</annotation></semantics></math>, there is a canonical CEP <math alttext="(\mu^{\prime},P^{\prime},\operatorname{Id})" class="ltx_Math" display="inline" id="A2.Thmtheorem2.p1.2.2.m2.3"><semantics id="A2.Thmtheorem2.p1.2.2.m2.3a"><mrow id="A2.Thmtheorem2.p1.2.2.m2.3.3.2" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.3.cmml"><mo id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.3" stretchy="false" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.3.cmml">(</mo><msup id="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1" xref="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.cmml"><mi id="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2" xref="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2.cmml">μ</mi><mo id="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3" xref="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3.cmml">′</mo></msup><mo id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.4" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.3.cmml">,</mo><msup id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.cmml"><mi id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.2" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.2.cmml">P</mi><mo id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.3" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.3.cmml">′</mo></msup><mo id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.5" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.3.cmml">,</mo><mi id="A2.Thmtheorem2.p1.2.2.m2.1.1" xref="A2.Thmtheorem2.p1.2.2.m2.1.1.cmml">Id</mi><mo id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.6" stretchy="false" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Thmtheorem2.p1.2.2.m2.3b"><vector id="A2.Thmtheorem2.p1.2.2.m2.3.3.3.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2"><apply id="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.1.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1">superscript</csymbol><ci id="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2">𝜇</ci><ci id="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3">′</ci></apply><apply id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.1.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2">superscript</csymbol><ci id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.2.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.2">𝑃</ci><ci id="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.3.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.3.3.2.2.3">′</ci></apply><ci id="A2.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="A2.Thmtheorem2.p1.2.2.m2.1.1">Id</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.Thmtheorem2.p1.2.2.m2.3c">(\mu^{\prime},P^{\prime},\operatorname{Id})</annotation><annotation encoding="application/x-llamapun" id="A2.Thmtheorem2.p1.2.2.m2.3d">( italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , roman_Id )</annotation></semantics></math> achieving the same principal objective value.</span></p> </div> </div> <div class="ltx_proof" id="A2.SS2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A2.SS2.1.p1"> <p class="ltx_p" id="A2.SS2.1.p1.5">Given a CEP <math alttext="(\mu,P,\pi)" class="ltx_Math" display="inline" id="A2.SS2.1.p1.1.m1.3"><semantics id="A2.SS2.1.p1.1.m1.3a"><mrow id="A2.SS2.1.p1.1.m1.3.4.2" xref="A2.SS2.1.p1.1.m1.3.4.1.cmml"><mo id="A2.SS2.1.p1.1.m1.3.4.2.1" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.1.cmml">(</mo><mi id="A2.SS2.1.p1.1.m1.1.1" xref="A2.SS2.1.p1.1.m1.1.1.cmml">μ</mi><mo id="A2.SS2.1.p1.1.m1.3.4.2.2" xref="A2.SS2.1.p1.1.m1.3.4.1.cmml">,</mo><mi id="A2.SS2.1.p1.1.m1.2.2" xref="A2.SS2.1.p1.1.m1.2.2.cmml">P</mi><mo id="A2.SS2.1.p1.1.m1.3.4.2.3" xref="A2.SS2.1.p1.1.m1.3.4.1.cmml">,</mo><mi id="A2.SS2.1.p1.1.m1.3.3" xref="A2.SS2.1.p1.1.m1.3.3.cmml">π</mi><mo id="A2.SS2.1.p1.1.m1.3.4.2.4" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.1.m1.3b"><vector id="A2.SS2.1.p1.1.m1.3.4.1.cmml" xref="A2.SS2.1.p1.1.m1.3.4.2"><ci id="A2.SS2.1.p1.1.m1.1.1.cmml" xref="A2.SS2.1.p1.1.m1.1.1">𝜇</ci><ci id="A2.SS2.1.p1.1.m1.2.2.cmml" xref="A2.SS2.1.p1.1.m1.2.2">𝑃</ci><ci id="A2.SS2.1.p1.1.m1.3.3.cmml" xref="A2.SS2.1.p1.1.m1.3.3">𝜋</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.1.m1.3c">(\mu,P,\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.1.m1.3d">( italic_μ , italic_P , italic_π )</annotation></semantics></math>, set <math alttext="\mu^{\prime}\in\Delta(A)" class="ltx_Math" display="inline" id="A2.SS2.1.p1.2.m2.1"><semantics id="A2.SS2.1.p1.2.m2.1a"><mrow id="A2.SS2.1.p1.2.m2.1.2" xref="A2.SS2.1.p1.2.m2.1.2.cmml"><msup id="A2.SS2.1.p1.2.m2.1.2.2" xref="A2.SS2.1.p1.2.m2.1.2.2.cmml"><mi id="A2.SS2.1.p1.2.m2.1.2.2.2" xref="A2.SS2.1.p1.2.m2.1.2.2.2.cmml">μ</mi><mo id="A2.SS2.1.p1.2.m2.1.2.2.3" xref="A2.SS2.1.p1.2.m2.1.2.2.3.cmml">′</mo></msup><mo id="A2.SS2.1.p1.2.m2.1.2.1" xref="A2.SS2.1.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="A2.SS2.1.p1.2.m2.1.2.3" xref="A2.SS2.1.p1.2.m2.1.2.3.cmml"><mi id="A2.SS2.1.p1.2.m2.1.2.3.2" mathvariant="normal" xref="A2.SS2.1.p1.2.m2.1.2.3.2.cmml">Δ</mi><mo id="A2.SS2.1.p1.2.m2.1.2.3.1" xref="A2.SS2.1.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.2.m2.1.2.3.3.2" xref="A2.SS2.1.p1.2.m2.1.2.3.cmml"><mo id="A2.SS2.1.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="A2.SS2.1.p1.2.m2.1.2.3.cmml">(</mo><mi id="A2.SS2.1.p1.2.m2.1.1" xref="A2.SS2.1.p1.2.m2.1.1.cmml">A</mi><mo id="A2.SS2.1.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="A2.SS2.1.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.2.m2.1b"><apply id="A2.SS2.1.p1.2.m2.1.2.cmml" xref="A2.SS2.1.p1.2.m2.1.2"><in id="A2.SS2.1.p1.2.m2.1.2.1.cmml" xref="A2.SS2.1.p1.2.m2.1.2.1"></in><apply id="A2.SS2.1.p1.2.m2.1.2.2.cmml" xref="A2.SS2.1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.2.m2.1.2.2.1.cmml" xref="A2.SS2.1.p1.2.m2.1.2.2">superscript</csymbol><ci id="A2.SS2.1.p1.2.m2.1.2.2.2.cmml" xref="A2.SS2.1.p1.2.m2.1.2.2.2">𝜇</ci><ci id="A2.SS2.1.p1.2.m2.1.2.2.3.cmml" xref="A2.SS2.1.p1.2.m2.1.2.2.3">′</ci></apply><apply id="A2.SS2.1.p1.2.m2.1.2.3.cmml" xref="A2.SS2.1.p1.2.m2.1.2.3"><times id="A2.SS2.1.p1.2.m2.1.2.3.1.cmml" xref="A2.SS2.1.p1.2.m2.1.2.3.1"></times><ci id="A2.SS2.1.p1.2.m2.1.2.3.2.cmml" xref="A2.SS2.1.p1.2.m2.1.2.3.2">Δ</ci><ci id="A2.SS2.1.p1.2.m2.1.1.cmml" xref="A2.SS2.1.p1.2.m2.1.1">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.2.m2.1c">\mu^{\prime}\in\Delta(A)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.2.m2.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Δ ( italic_A )</annotation></semantics></math> to be the distribution that samples <math alttext="s\sim\mu" class="ltx_Math" display="inline" id="A2.SS2.1.p1.3.m3.1"><semantics id="A2.SS2.1.p1.3.m3.1a"><mrow id="A2.SS2.1.p1.3.m3.1.1" xref="A2.SS2.1.p1.3.m3.1.1.cmml"><mi id="A2.SS2.1.p1.3.m3.1.1.2" xref="A2.SS2.1.p1.3.m3.1.1.2.cmml">s</mi><mo id="A2.SS2.1.p1.3.m3.1.1.1" xref="A2.SS2.1.p1.3.m3.1.1.1.cmml">∼</mo><mi id="A2.SS2.1.p1.3.m3.1.1.3" xref="A2.SS2.1.p1.3.m3.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.3.m3.1b"><apply id="A2.SS2.1.p1.3.m3.1.1.cmml" xref="A2.SS2.1.p1.3.m3.1.1"><csymbol cd="latexml" id="A2.SS2.1.p1.3.m3.1.1.1.cmml" xref="A2.SS2.1.p1.3.m3.1.1.1">similar-to</csymbol><ci id="A2.SS2.1.p1.3.m3.1.1.2.cmml" xref="A2.SS2.1.p1.3.m3.1.1.2">𝑠</ci><ci id="A2.SS2.1.p1.3.m3.1.1.3.cmml" xref="A2.SS2.1.p1.3.m3.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.3.m3.1c">s\sim\mu</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.3.m3.1d">italic_s ∼ italic_μ</annotation></semantics></math> and then samples and outputs <math alttext="a\sim\pi(s)" class="ltx_Math" display="inline" id="A2.SS2.1.p1.4.m4.1"><semantics id="A2.SS2.1.p1.4.m4.1a"><mrow id="A2.SS2.1.p1.4.m4.1.2" xref="A2.SS2.1.p1.4.m4.1.2.cmml"><mi id="A2.SS2.1.p1.4.m4.1.2.2" xref="A2.SS2.1.p1.4.m4.1.2.2.cmml">a</mi><mo id="A2.SS2.1.p1.4.m4.1.2.1" xref="A2.SS2.1.p1.4.m4.1.2.1.cmml">∼</mo><mrow id="A2.SS2.1.p1.4.m4.1.2.3" xref="A2.SS2.1.p1.4.m4.1.2.3.cmml"><mi id="A2.SS2.1.p1.4.m4.1.2.3.2" xref="A2.SS2.1.p1.4.m4.1.2.3.2.cmml">π</mi><mo id="A2.SS2.1.p1.4.m4.1.2.3.1" xref="A2.SS2.1.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.4.m4.1.2.3.3.2" xref="A2.SS2.1.p1.4.m4.1.2.3.cmml"><mo id="A2.SS2.1.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="A2.SS2.1.p1.4.m4.1.2.3.cmml">(</mo><mi id="A2.SS2.1.p1.4.m4.1.1" xref="A2.SS2.1.p1.4.m4.1.1.cmml">s</mi><mo id="A2.SS2.1.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="A2.SS2.1.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.4.m4.1b"><apply id="A2.SS2.1.p1.4.m4.1.2.cmml" xref="A2.SS2.1.p1.4.m4.1.2"><csymbol cd="latexml" id="A2.SS2.1.p1.4.m4.1.2.1.cmml" xref="A2.SS2.1.p1.4.m4.1.2.1">similar-to</csymbol><ci id="A2.SS2.1.p1.4.m4.1.2.2.cmml" xref="A2.SS2.1.p1.4.m4.1.2.2">𝑎</ci><apply id="A2.SS2.1.p1.4.m4.1.2.3.cmml" xref="A2.SS2.1.p1.4.m4.1.2.3"><times id="A2.SS2.1.p1.4.m4.1.2.3.1.cmml" xref="A2.SS2.1.p1.4.m4.1.2.3.1"></times><ci id="A2.SS2.1.p1.4.m4.1.2.3.2.cmml" xref="A2.SS2.1.p1.4.m4.1.2.3.2">𝜋</ci><ci id="A2.SS2.1.p1.4.m4.1.1.cmml" xref="A2.SS2.1.p1.4.m4.1.1">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.4.m4.1c">a\sim\pi(s)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.4.m4.1d">italic_a ∼ italic_π ( italic_s )</annotation></semantics></math>. Then define <math alttext="P^{\prime}_{i}:A\times A\to\mathbb{R}_{+}" class="ltx_Math" display="inline" id="A2.SS2.1.p1.5.m5.1"><semantics id="A2.SS2.1.p1.5.m5.1a"><mrow id="A2.SS2.1.p1.5.m5.1.1" xref="A2.SS2.1.p1.5.m5.1.1.cmml"><msubsup id="A2.SS2.1.p1.5.m5.1.1.2" xref="A2.SS2.1.p1.5.m5.1.1.2.cmml"><mi id="A2.SS2.1.p1.5.m5.1.1.2.2.2" xref="A2.SS2.1.p1.5.m5.1.1.2.2.2.cmml">P</mi><mi id="A2.SS2.1.p1.5.m5.1.1.2.3" xref="A2.SS2.1.p1.5.m5.1.1.2.3.cmml">i</mi><mo id="A2.SS2.1.p1.5.m5.1.1.2.2.3" xref="A2.SS2.1.p1.5.m5.1.1.2.2.3.cmml">′</mo></msubsup><mo id="A2.SS2.1.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="A2.SS2.1.p1.5.m5.1.1.1.cmml">:</mo><mrow id="A2.SS2.1.p1.5.m5.1.1.3" xref="A2.SS2.1.p1.5.m5.1.1.3.cmml"><mrow id="A2.SS2.1.p1.5.m5.1.1.3.2" xref="A2.SS2.1.p1.5.m5.1.1.3.2.cmml"><mi id="A2.SS2.1.p1.5.m5.1.1.3.2.2" xref="A2.SS2.1.p1.5.m5.1.1.3.2.2.cmml">A</mi><mo id="A2.SS2.1.p1.5.m5.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="A2.SS2.1.p1.5.m5.1.1.3.2.1.cmml">×</mo><mi id="A2.SS2.1.p1.5.m5.1.1.3.2.3" xref="A2.SS2.1.p1.5.m5.1.1.3.2.3.cmml">A</mi></mrow><mo id="A2.SS2.1.p1.5.m5.1.1.3.1" stretchy="false" xref="A2.SS2.1.p1.5.m5.1.1.3.1.cmml">→</mo><msub id="A2.SS2.1.p1.5.m5.1.1.3.3" xref="A2.SS2.1.p1.5.m5.1.1.3.3.cmml"><mi id="A2.SS2.1.p1.5.m5.1.1.3.3.2" xref="A2.SS2.1.p1.5.m5.1.1.3.3.2.cmml">ℝ</mi><mo id="A2.SS2.1.p1.5.m5.1.1.3.3.3" xref="A2.SS2.1.p1.5.m5.1.1.3.3.3.cmml">+</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.5.m5.1b"><apply id="A2.SS2.1.p1.5.m5.1.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1"><ci id="A2.SS2.1.p1.5.m5.1.1.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1.1">:</ci><apply id="A2.SS2.1.p1.5.m5.1.1.2.cmml" xref="A2.SS2.1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.5.m5.1.1.2.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1.2">subscript</csymbol><apply id="A2.SS2.1.p1.5.m5.1.1.2.2.cmml" xref="A2.SS2.1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.5.m5.1.1.2.2.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1.2">superscript</csymbol><ci id="A2.SS2.1.p1.5.m5.1.1.2.2.2.cmml" xref="A2.SS2.1.p1.5.m5.1.1.2.2.2">𝑃</ci><ci id="A2.SS2.1.p1.5.m5.1.1.2.2.3.cmml" xref="A2.SS2.1.p1.5.m5.1.1.2.2.3">′</ci></apply><ci id="A2.SS2.1.p1.5.m5.1.1.2.3.cmml" xref="A2.SS2.1.p1.5.m5.1.1.2.3">𝑖</ci></apply><apply id="A2.SS2.1.p1.5.m5.1.1.3.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3"><ci id="A2.SS2.1.p1.5.m5.1.1.3.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.1">→</ci><apply id="A2.SS2.1.p1.5.m5.1.1.3.2.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.2"><times id="A2.SS2.1.p1.5.m5.1.1.3.2.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.2.1"></times><ci id="A2.SS2.1.p1.5.m5.1.1.3.2.2.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.2.2">𝐴</ci><ci id="A2.SS2.1.p1.5.m5.1.1.3.2.3.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.2.3">𝐴</ci></apply><apply id="A2.SS2.1.p1.5.m5.1.1.3.3.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="A2.SS2.1.p1.5.m5.1.1.3.3.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="A2.SS2.1.p1.5.m5.1.1.3.3.2.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.3.2">ℝ</ci><plus id="A2.SS2.1.p1.5.m5.1.1.3.3.3.cmml" xref="A2.SS2.1.p1.5.m5.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.5.m5.1c">P^{\prime}_{i}:A\times A\to\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.5.m5.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A × italic_A → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> by</p> <table class="ltx_equation ltx_eqn_table" id="A2.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="P^{\prime}_{i}(a,a^{\prime})=\operatorname*{\mathbb{E}}_{s\sim\mu|a}P_{i}(s,a^% {\prime})," class="ltx_Math" display="block" id="A2.Ex5.m1.3"><semantics id="A2.Ex5.m1.3a"><mrow id="A2.Ex5.m1.3.3.1" xref="A2.Ex5.m1.3.3.1.1.cmml"><mrow id="A2.Ex5.m1.3.3.1.1" xref="A2.Ex5.m1.3.3.1.1.cmml"><mrow id="A2.Ex5.m1.3.3.1.1.1" xref="A2.Ex5.m1.3.3.1.1.1.cmml"><msubsup id="A2.Ex5.m1.3.3.1.1.1.3" xref="A2.Ex5.m1.3.3.1.1.1.3.cmml"><mi id="A2.Ex5.m1.3.3.1.1.1.3.2.2" xref="A2.Ex5.m1.3.3.1.1.1.3.2.2.cmml">P</mi><mi id="A2.Ex5.m1.3.3.1.1.1.3.3" xref="A2.Ex5.m1.3.3.1.1.1.3.3.cmml">i</mi><mo id="A2.Ex5.m1.3.3.1.1.1.3.2.3" xref="A2.Ex5.m1.3.3.1.1.1.3.2.3.cmml">′</mo></msubsup><mo id="A2.Ex5.m1.3.3.1.1.1.2" xref="A2.Ex5.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex5.m1.3.3.1.1.1.1.1" xref="A2.Ex5.m1.3.3.1.1.1.1.2.cmml"><mo id="A2.Ex5.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="A2.Ex5.m1.3.3.1.1.1.1.2.cmml">(</mo><mi id="A2.Ex5.m1.1.1" xref="A2.Ex5.m1.1.1.cmml">a</mi><mo id="A2.Ex5.m1.3.3.1.1.1.1.1.3" xref="A2.Ex5.m1.3.3.1.1.1.1.2.cmml">,</mo><msup id="A2.Ex5.m1.3.3.1.1.1.1.1.1" xref="A2.Ex5.m1.3.3.1.1.1.1.1.1.cmml"><mi id="A2.Ex5.m1.3.3.1.1.1.1.1.1.2" xref="A2.Ex5.m1.3.3.1.1.1.1.1.1.2.cmml">a</mi><mo id="A2.Ex5.m1.3.3.1.1.1.1.1.1.3" xref="A2.Ex5.m1.3.3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="A2.Ex5.m1.3.3.1.1.1.1.1.4" stretchy="false" xref="A2.Ex5.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.Ex5.m1.3.3.1.1.3" rspace="0.1389em" xref="A2.Ex5.m1.3.3.1.1.3.cmml">=</mo><mrow id="A2.Ex5.m1.3.3.1.1.2" xref="A2.Ex5.m1.3.3.1.1.2.cmml"><mrow id="A2.Ex5.m1.3.3.1.1.2.3" xref="A2.Ex5.m1.3.3.1.1.2.3.cmml"><munder id="A2.Ex5.m1.3.3.1.1.2.3.1" xref="A2.Ex5.m1.3.3.1.1.2.3.1.cmml"><mo id="A2.Ex5.m1.3.3.1.1.2.3.1.2" lspace="0.1389em" rspace="0.167em" xref="A2.Ex5.m1.3.3.1.1.2.3.1.2.cmml">𝔼</mo><mrow id="A2.Ex5.m1.3.3.1.1.2.3.1.3" xref="A2.Ex5.m1.3.3.1.1.2.3.1.3.cmml"><mi id="A2.Ex5.m1.3.3.1.1.2.3.1.3.2" xref="A2.Ex5.m1.3.3.1.1.2.3.1.3.2.cmml">s</mi><mo id="A2.Ex5.m1.3.3.1.1.2.3.1.3.1" xref="A2.Ex5.m1.3.3.1.1.2.3.1.3.1.cmml">∼</mo><mrow id="A2.Ex5.m1.3.3.1.1.2.3.1.3.3" xref="A2.Ex5.m1.3.3.1.1.2.3.1.3.3.cmml"><mi id="A2.Ex5.m1.3.3.1.1.2.3.1.3.3.2" xref="A2.Ex5.m1.3.3.1.1.2.3.1.3.3.2.cmml">μ</mi><mo fence="false" id="A2.Ex5.m1.3.3.1.1.2.3.1.3.3.1" xref="A2.Ex5.m1.3.3.1.1.2.3.1.3.3.1.cmml">|</mo><mi id="A2.Ex5.m1.3.3.1.1.2.3.1.3.3.3" xref="A2.Ex5.m1.3.3.1.1.2.3.1.3.3.3.cmml">a</mi></mrow></mrow></munder><msub id="A2.Ex5.m1.3.3.1.1.2.3.2" xref="A2.Ex5.m1.3.3.1.1.2.3.2.cmml"><mi id="A2.Ex5.m1.3.3.1.1.2.3.2.2" xref="A2.Ex5.m1.3.3.1.1.2.3.2.2.cmml">P</mi><mi id="A2.Ex5.m1.3.3.1.1.2.3.2.3" xref="A2.Ex5.m1.3.3.1.1.2.3.2.3.cmml">i</mi></msub></mrow><mo id="A2.Ex5.m1.3.3.1.1.2.2" xref="A2.Ex5.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="A2.Ex5.m1.3.3.1.1.2.1.1" xref="A2.Ex5.m1.3.3.1.1.2.1.2.cmml"><mo id="A2.Ex5.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="A2.Ex5.m1.3.3.1.1.2.1.2.cmml">(</mo><mi id="A2.Ex5.m1.2.2" xref="A2.Ex5.m1.2.2.cmml">s</mi><mo id="A2.Ex5.m1.3.3.1.1.2.1.1.3" xref="A2.Ex5.m1.3.3.1.1.2.1.2.cmml">,</mo><msup id="A2.Ex5.m1.3.3.1.1.2.1.1.1" xref="A2.Ex5.m1.3.3.1.1.2.1.1.1.cmml"><mi 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id="A2.Ex5.m1.3c">P^{\prime}_{i}(a,a^{\prime})=\operatorname*{\mathbb{E}}_{s\sim\mu|a}P_{i}(s,a^% {\prime}),</annotation><annotation encoding="application/x-llamapun" id="A2.Ex5.m1.3d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = blackboard_E start_POSTSUBSCRIPT italic_s ∼ italic_μ | italic_a end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s , italic_a 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="A2.SS2.1.p1.9">where <math alttext="s\sim\mu|a" class="ltx_Math" display="inline" id="A2.SS2.1.p1.6.m1.1"><semantics id="A2.SS2.1.p1.6.m1.1a"><mrow id="A2.SS2.1.p1.6.m1.1.1" xref="A2.SS2.1.p1.6.m1.1.1.cmml"><mi id="A2.SS2.1.p1.6.m1.1.1.2" xref="A2.SS2.1.p1.6.m1.1.1.2.cmml">s</mi><mo id="A2.SS2.1.p1.6.m1.1.1.1" xref="A2.SS2.1.p1.6.m1.1.1.1.cmml">∼</mo><mrow id="A2.SS2.1.p1.6.m1.1.1.3" xref="A2.SS2.1.p1.6.m1.1.1.3.cmml"><mi id="A2.SS2.1.p1.6.m1.1.1.3.2" xref="A2.SS2.1.p1.6.m1.1.1.3.2.cmml">μ</mi><mo fence="false" id="A2.SS2.1.p1.6.m1.1.1.3.1" xref="A2.SS2.1.p1.6.m1.1.1.3.1.cmml">|</mo><mi id="A2.SS2.1.p1.6.m1.1.1.3.3" xref="A2.SS2.1.p1.6.m1.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.6.m1.1b"><apply id="A2.SS2.1.p1.6.m1.1.1.cmml" xref="A2.SS2.1.p1.6.m1.1.1"><csymbol cd="latexml" id="A2.SS2.1.p1.6.m1.1.1.1.cmml" xref="A2.SS2.1.p1.6.m1.1.1.1">similar-to</csymbol><ci id="A2.SS2.1.p1.6.m1.1.1.2.cmml" xref="A2.SS2.1.p1.6.m1.1.1.2">𝑠</ci><apply id="A2.SS2.1.p1.6.m1.1.1.3.cmml" xref="A2.SS2.1.p1.6.m1.1.1.3"><csymbol cd="latexml" id="A2.SS2.1.p1.6.m1.1.1.3.1.cmml" xref="A2.SS2.1.p1.6.m1.1.1.3.1">conditional</csymbol><ci id="A2.SS2.1.p1.6.m1.1.1.3.2.cmml" xref="A2.SS2.1.p1.6.m1.1.1.3.2">𝜇</ci><ci id="A2.SS2.1.p1.6.m1.1.1.3.3.cmml" xref="A2.SS2.1.p1.6.m1.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.6.m1.1c">s\sim\mu|a</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.6.m1.1d">italic_s ∼ italic_μ | italic_a</annotation></semantics></math> denotes sampling <math alttext="s" class="ltx_Math" display="inline" id="A2.SS2.1.p1.7.m2.1"><semantics id="A2.SS2.1.p1.7.m2.1a"><mi id="A2.SS2.1.p1.7.m2.1.1" xref="A2.SS2.1.p1.7.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.7.m2.1b"><ci id="A2.SS2.1.p1.7.m2.1.1.cmml" xref="A2.SS2.1.p1.7.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.7.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.7.m2.1d">italic_s</annotation></semantics></math> with probability proportional to <math alttext="\mu(s)\cdot\pi(a|s)" class="ltx_Math" display="inline" id="A2.SS2.1.p1.8.m3.2"><semantics id="A2.SS2.1.p1.8.m3.2a"><mrow id="A2.SS2.1.p1.8.m3.2.2" xref="A2.SS2.1.p1.8.m3.2.2.cmml"><mrow id="A2.SS2.1.p1.8.m3.2.2.3" xref="A2.SS2.1.p1.8.m3.2.2.3.cmml"><mrow id="A2.SS2.1.p1.8.m3.2.2.3.2" xref="A2.SS2.1.p1.8.m3.2.2.3.2.cmml"><mi id="A2.SS2.1.p1.8.m3.2.2.3.2.2" xref="A2.SS2.1.p1.8.m3.2.2.3.2.2.cmml">μ</mi><mo id="A2.SS2.1.p1.8.m3.2.2.3.2.1" xref="A2.SS2.1.p1.8.m3.2.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.8.m3.2.2.3.2.3.2" xref="A2.SS2.1.p1.8.m3.2.2.3.2.cmml"><mo id="A2.SS2.1.p1.8.m3.2.2.3.2.3.2.1" stretchy="false" xref="A2.SS2.1.p1.8.m3.2.2.3.2.cmml">(</mo><mi id="A2.SS2.1.p1.8.m3.1.1" xref="A2.SS2.1.p1.8.m3.1.1.cmml">s</mi><mo id="A2.SS2.1.p1.8.m3.2.2.3.2.3.2.2" rspace="0.055em" stretchy="false" xref="A2.SS2.1.p1.8.m3.2.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.8.m3.2.2.3.1" rspace="0.222em" xref="A2.SS2.1.p1.8.m3.2.2.3.1.cmml">⋅</mo><mi id="A2.SS2.1.p1.8.m3.2.2.3.3" xref="A2.SS2.1.p1.8.m3.2.2.3.3.cmml">π</mi></mrow><mo id="A2.SS2.1.p1.8.m3.2.2.2" xref="A2.SS2.1.p1.8.m3.2.2.2.cmml">⁢</mo><mrow id="A2.SS2.1.p1.8.m3.2.2.1.1" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.cmml"><mo id="A2.SS2.1.p1.8.m3.2.2.1.1.2" stretchy="false" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.cmml">(</mo><mrow id="A2.SS2.1.p1.8.m3.2.2.1.1.1" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.cmml"><mi id="A2.SS2.1.p1.8.m3.2.2.1.1.1.2" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.2.cmml">a</mi><mo fence="false" id="A2.SS2.1.p1.8.m3.2.2.1.1.1.1" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.1.cmml">|</mo><mi id="A2.SS2.1.p1.8.m3.2.2.1.1.1.3" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.3.cmml">s</mi></mrow><mo id="A2.SS2.1.p1.8.m3.2.2.1.1.3" stretchy="false" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.8.m3.2b"><apply id="A2.SS2.1.p1.8.m3.2.2.cmml" xref="A2.SS2.1.p1.8.m3.2.2"><times id="A2.SS2.1.p1.8.m3.2.2.2.cmml" xref="A2.SS2.1.p1.8.m3.2.2.2"></times><apply id="A2.SS2.1.p1.8.m3.2.2.3.cmml" xref="A2.SS2.1.p1.8.m3.2.2.3"><ci id="A2.SS2.1.p1.8.m3.2.2.3.1.cmml" xref="A2.SS2.1.p1.8.m3.2.2.3.1">⋅</ci><apply id="A2.SS2.1.p1.8.m3.2.2.3.2.cmml" xref="A2.SS2.1.p1.8.m3.2.2.3.2"><times id="A2.SS2.1.p1.8.m3.2.2.3.2.1.cmml" xref="A2.SS2.1.p1.8.m3.2.2.3.2.1"></times><ci id="A2.SS2.1.p1.8.m3.2.2.3.2.2.cmml" xref="A2.SS2.1.p1.8.m3.2.2.3.2.2">𝜇</ci><ci id="A2.SS2.1.p1.8.m3.1.1.cmml" xref="A2.SS2.1.p1.8.m3.1.1">𝑠</ci></apply><ci id="A2.SS2.1.p1.8.m3.2.2.3.3.cmml" xref="A2.SS2.1.p1.8.m3.2.2.3.3">𝜋</ci></apply><apply id="A2.SS2.1.p1.8.m3.2.2.1.1.1.cmml" xref="A2.SS2.1.p1.8.m3.2.2.1.1"><csymbol cd="latexml" id="A2.SS2.1.p1.8.m3.2.2.1.1.1.1.cmml" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.1">conditional</csymbol><ci id="A2.SS2.1.p1.8.m3.2.2.1.1.1.2.cmml" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.2">𝑎</ci><ci id="A2.SS2.1.p1.8.m3.2.2.1.1.1.3.cmml" xref="A2.SS2.1.p1.8.m3.2.2.1.1.1.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.8.m3.2c">\mu(s)\cdot\pi(a|s)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.8.m3.2d">italic_μ ( italic_s ) ⋅ italic_π ( italic_a | italic_s )</annotation></semantics></math>. Then note that, for any <math alttext="\phi_{i}:A_{i}\to A_{i}" class="ltx_Math" display="inline" id="A2.SS2.1.p1.9.m4.1"><semantics id="A2.SS2.1.p1.9.m4.1a"><mrow id="A2.SS2.1.p1.9.m4.1.1" xref="A2.SS2.1.p1.9.m4.1.1.cmml"><msub id="A2.SS2.1.p1.9.m4.1.1.2" xref="A2.SS2.1.p1.9.m4.1.1.2.cmml"><mi id="A2.SS2.1.p1.9.m4.1.1.2.2" xref="A2.SS2.1.p1.9.m4.1.1.2.2.cmml">ϕ</mi><mi id="A2.SS2.1.p1.9.m4.1.1.2.3" xref="A2.SS2.1.p1.9.m4.1.1.2.3.cmml">i</mi></msub><mo id="A2.SS2.1.p1.9.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="A2.SS2.1.p1.9.m4.1.1.1.cmml">:</mo><mrow id="A2.SS2.1.p1.9.m4.1.1.3" xref="A2.SS2.1.p1.9.m4.1.1.3.cmml"><msub id="A2.SS2.1.p1.9.m4.1.1.3.2" xref="A2.SS2.1.p1.9.m4.1.1.3.2.cmml"><mi id="A2.SS2.1.p1.9.m4.1.1.3.2.2" xref="A2.SS2.1.p1.9.m4.1.1.3.2.2.cmml">A</mi><mi id="A2.SS2.1.p1.9.m4.1.1.3.2.3" xref="A2.SS2.1.p1.9.m4.1.1.3.2.3.cmml">i</mi></msub><mo id="A2.SS2.1.p1.9.m4.1.1.3.1" stretchy="false" xref="A2.SS2.1.p1.9.m4.1.1.3.1.cmml">→</mo><msub id="A2.SS2.1.p1.9.m4.1.1.3.3" xref="A2.SS2.1.p1.9.m4.1.1.3.3.cmml"><mi id="A2.SS2.1.p1.9.m4.1.1.3.3.2" xref="A2.SS2.1.p1.9.m4.1.1.3.3.2.cmml">A</mi><mi id="A2.SS2.1.p1.9.m4.1.1.3.3.3" xref="A2.SS2.1.p1.9.m4.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.9.m4.1b"><apply id="A2.SS2.1.p1.9.m4.1.1.cmml" xref="A2.SS2.1.p1.9.m4.1.1"><ci id="A2.SS2.1.p1.9.m4.1.1.1.cmml" xref="A2.SS2.1.p1.9.m4.1.1.1">:</ci><apply id="A2.SS2.1.p1.9.m4.1.1.2.cmml" xref="A2.SS2.1.p1.9.m4.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.9.m4.1.1.2.1.cmml" xref="A2.SS2.1.p1.9.m4.1.1.2">subscript</csymbol><ci id="A2.SS2.1.p1.9.m4.1.1.2.2.cmml" xref="A2.SS2.1.p1.9.m4.1.1.2.2">italic-ϕ</ci><ci id="A2.SS2.1.p1.9.m4.1.1.2.3.cmml" xref="A2.SS2.1.p1.9.m4.1.1.2.3">𝑖</ci></apply><apply id="A2.SS2.1.p1.9.m4.1.1.3.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3"><ci id="A2.SS2.1.p1.9.m4.1.1.3.1.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.1">→</ci><apply id="A2.SS2.1.p1.9.m4.1.1.3.2.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.9.m4.1.1.3.2.1.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.2">subscript</csymbol><ci id="A2.SS2.1.p1.9.m4.1.1.3.2.2.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.2.2">𝐴</ci><ci id="A2.SS2.1.p1.9.m4.1.1.3.2.3.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.2.3">𝑖</ci></apply><apply id="A2.SS2.1.p1.9.m4.1.1.3.3.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.3"><csymbol cd="ambiguous" id="A2.SS2.1.p1.9.m4.1.1.3.3.1.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.3">subscript</csymbol><ci id="A2.SS2.1.p1.9.m4.1.1.3.3.2.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.3.2">𝐴</ci><ci id="A2.SS2.1.p1.9.m4.1.1.3.3.3.cmml" xref="A2.SS2.1.p1.9.m4.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.9.m4.1c">\phi_{i}:A_{i}\to A_{i}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.9.m4.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx18"> <tbody id="A2.E23"><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 U_{i}^{\mu^{\prime},P^{\prime}}(\phi_{i},\operatorname{Id}_{-i})" class="ltx_Math" display="inline" id="A2.E23.m1.4"><semantics id="A2.E23.m1.4a"><mrow id="A2.E23.m1.4.4" xref="A2.E23.m1.4.4.cmml"><msubsup id="A2.E23.m1.4.4.4" xref="A2.E23.m1.4.4.4.cmml"><mi id="A2.E23.m1.4.4.4.2.2" xref="A2.E23.m1.4.4.4.2.2.cmml">U</mi><mi id="A2.E23.m1.4.4.4.2.3" xref="A2.E23.m1.4.4.4.2.3.cmml">i</mi><mrow id="A2.E23.m1.2.2.2.2" xref="A2.E23.m1.2.2.2.3.cmml"><msup id="A2.E23.m1.1.1.1.1.1" xref="A2.E23.m1.1.1.1.1.1.cmml"><mi id="A2.E23.m1.1.1.1.1.1.2" xref="A2.E23.m1.1.1.1.1.1.2.cmml">μ</mi><mo id="A2.E23.m1.1.1.1.1.1.3" xref="A2.E23.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="A2.E23.m1.2.2.2.2.3" xref="A2.E23.m1.2.2.2.3.cmml">,</mo><msup id="A2.E23.m1.2.2.2.2.2" xref="A2.E23.m1.2.2.2.2.2.cmml"><mi id="A2.E23.m1.2.2.2.2.2.2" xref="A2.E23.m1.2.2.2.2.2.2.cmml">P</mi><mo id="A2.E23.m1.2.2.2.2.2.3" xref="A2.E23.m1.2.2.2.2.2.3.cmml">′</mo></msup></mrow></msubsup><mo id="A2.E23.m1.4.4.3" xref="A2.E23.m1.4.4.3.cmml">⁢</mo><mrow id="A2.E23.m1.4.4.2.2" xref="A2.E23.m1.4.4.2.3.cmml"><mo id="A2.E23.m1.4.4.2.2.3" stretchy="false" xref="A2.E23.m1.4.4.2.3.cmml">(</mo><msub id="A2.E23.m1.3.3.1.1.1" xref="A2.E23.m1.3.3.1.1.1.cmml"><mi id="A2.E23.m1.3.3.1.1.1.2" xref="A2.E23.m1.3.3.1.1.1.2.cmml">ϕ</mi><mi id="A2.E23.m1.3.3.1.1.1.3" xref="A2.E23.m1.3.3.1.1.1.3.cmml">i</mi></msub><mo id="A2.E23.m1.4.4.2.2.4" xref="A2.E23.m1.4.4.2.3.cmml">,</mo><msub id="A2.E23.m1.4.4.2.2.2" xref="A2.E23.m1.4.4.2.2.2.cmml"><mi id="A2.E23.m1.4.4.2.2.2.2" xref="A2.E23.m1.4.4.2.2.2.2.cmml">Id</mi><mrow id="A2.E23.m1.4.4.2.2.2.3" xref="A2.E23.m1.4.4.2.2.2.3.cmml"><mo id="A2.E23.m1.4.4.2.2.2.3a" xref="A2.E23.m1.4.4.2.2.2.3.cmml">−</mo><mi id="A2.E23.m1.4.4.2.2.2.3.2" xref="A2.E23.m1.4.4.2.2.2.3.2.cmml">i</mi></mrow></msub><mo id="A2.E23.m1.4.4.2.2.5" stretchy="false" xref="A2.E23.m1.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E23.m1.4b"><apply id="A2.E23.m1.4.4.cmml" xref="A2.E23.m1.4.4"><times id="A2.E23.m1.4.4.3.cmml" xref="A2.E23.m1.4.4.3"></times><apply id="A2.E23.m1.4.4.4.cmml" xref="A2.E23.m1.4.4.4"><csymbol cd="ambiguous" id="A2.E23.m1.4.4.4.1.cmml" xref="A2.E23.m1.4.4.4">superscript</csymbol><apply id="A2.E23.m1.4.4.4.2.cmml" xref="A2.E23.m1.4.4.4"><csymbol cd="ambiguous" id="A2.E23.m1.4.4.4.2.1.cmml" xref="A2.E23.m1.4.4.4">subscript</csymbol><ci id="A2.E23.m1.4.4.4.2.2.cmml" xref="A2.E23.m1.4.4.4.2.2">𝑈</ci><ci id="A2.E23.m1.4.4.4.2.3.cmml" xref="A2.E23.m1.4.4.4.2.3">𝑖</ci></apply><list id="A2.E23.m1.2.2.2.3.cmml" xref="A2.E23.m1.2.2.2.2"><apply id="A2.E23.m1.1.1.1.1.1.cmml" xref="A2.E23.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.E23.m1.1.1.1.1.1.1.cmml" xref="A2.E23.m1.1.1.1.1.1">superscript</csymbol><ci id="A2.E23.m1.1.1.1.1.1.2.cmml" xref="A2.E23.m1.1.1.1.1.1.2">𝜇</ci><ci id="A2.E23.m1.1.1.1.1.1.3.cmml" xref="A2.E23.m1.1.1.1.1.1.3">′</ci></apply><apply id="A2.E23.m1.2.2.2.2.2.cmml" xref="A2.E23.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.E23.m1.2.2.2.2.2.1.cmml" xref="A2.E23.m1.2.2.2.2.2">superscript</csymbol><ci id="A2.E23.m1.2.2.2.2.2.2.cmml" xref="A2.E23.m1.2.2.2.2.2.2">𝑃</ci><ci id="A2.E23.m1.2.2.2.2.2.3.cmml" xref="A2.E23.m1.2.2.2.2.2.3">′</ci></apply></list></apply><interval closure="open" id="A2.E23.m1.4.4.2.3.cmml" xref="A2.E23.m1.4.4.2.2"><apply id="A2.E23.m1.3.3.1.1.1.cmml" xref="A2.E23.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="A2.E23.m1.3.3.1.1.1.1.cmml" xref="A2.E23.m1.3.3.1.1.1">subscript</csymbol><ci id="A2.E23.m1.3.3.1.1.1.2.cmml" xref="A2.E23.m1.3.3.1.1.1.2">italic-ϕ</ci><ci id="A2.E23.m1.3.3.1.1.1.3.cmml" xref="A2.E23.m1.3.3.1.1.1.3">𝑖</ci></apply><apply id="A2.E23.m1.4.4.2.2.2.cmml" xref="A2.E23.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="A2.E23.m1.4.4.2.2.2.1.cmml" xref="A2.E23.m1.4.4.2.2.2">subscript</csymbol><ci id="A2.E23.m1.4.4.2.2.2.2.cmml" xref="A2.E23.m1.4.4.2.2.2.2">Id</ci><apply id="A2.E23.m1.4.4.2.2.2.3.cmml" xref="A2.E23.m1.4.4.2.2.2.3"><minus id="A2.E23.m1.4.4.2.2.2.3.1.cmml" xref="A2.E23.m1.4.4.2.2.2.3"></minus><ci id="A2.E23.m1.4.4.2.2.2.3.2.cmml" xref="A2.E23.m1.4.4.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E23.m1.4c">\displaystyle U_{i}^{\mu^{\prime},P^{\prime}}(\phi_{i},\operatorname{Id}_{-i})</annotation><annotation encoding="application/x-llamapun" id="A2.E23.m1.4d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , roman_Id start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname*{\mathbb{E}}_{a\sim\mu^{\prime}}\quantity[U_{i}(% \phi_{i}(a_{i}),a_{-i})+P_{i}^{\prime}(a,\phi_{i}(a_{i}),a_{-i})]" class="ltx_Math" display="inline" id="A2.E23.m2.1"><semantics id="A2.E23.m2.1a"><mrow id="A2.E23.m2.1.2" xref="A2.E23.m2.1.2.cmml"><mi id="A2.E23.m2.1.2.2" xref="A2.E23.m2.1.2.2.cmml"></mi><mo id="A2.E23.m2.1.2.1" rspace="0.1389em" xref="A2.E23.m2.1.2.1.cmml">=</mo><mrow id="A2.E23.m2.1.2.3" xref="A2.E23.m2.1.2.3.cmml"><munder id="A2.E23.m2.1.2.3.1" xref="A2.E23.m2.1.2.3.1.cmml"><mo id="A2.E23.m2.1.2.3.1.2" lspace="0.1389em" rspace="0em" xref="A2.E23.m2.1.2.3.1.2.cmml">𝔼</mo><mrow id="A2.E23.m2.1.2.3.1.3" xref="A2.E23.m2.1.2.3.1.3.cmml"><mi id="A2.E23.m2.1.2.3.1.3.2" 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xref="A2.E23.m2.1.1.1.1.1.4.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="A2.E23.m2.1.1.1.1.1.5.2.2.2.cmml" xref="A2.E23.m2.1.1.1.1.1.5.2.2.2"><csymbol cd="ambiguous" id="A2.E23.m2.1.1.1.1.1.5.2.2.2.1.cmml" xref="A2.E23.m2.1.1.1.1.1.5.2.2.2">subscript</csymbol><ci id="A2.E23.m2.1.1.1.1.1.5.2.2.2.2.cmml" xref="A2.E23.m2.1.1.1.1.1.5.2.2.2.2">𝑎</ci><apply id="A2.E23.m2.1.1.1.1.1.5.2.2.2.3.cmml" xref="A2.E23.m2.1.1.1.1.1.5.2.2.2.3"><minus id="A2.E23.m2.1.1.1.1.1.5.2.2.2.3.1.cmml" xref="A2.E23.m2.1.1.1.1.1.5.2.2.2.3"></minus><ci id="A2.E23.m2.1.1.1.1.1.5.2.2.2.3.2.cmml" xref="A2.E23.m2.1.1.1.1.1.5.2.2.2.3.2">𝑖</ci></apply></apply></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E23.m2.1c">\displaystyle=\operatorname*{\mathbb{E}}_{a\sim\mu^{\prime}}\quantity[U_{i}(% \phi_{i}(a_{i}),a_{-i})+P_{i}^{\prime}(a,\phi_{i}(a_{i}),a_{-i})]</annotation><annotation encoding="application/x-llamapun" id="A2.E23.m2.1d">= blackboard_E start_POSTSUBSCRIPT italic_a ∼ italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a , italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> <tbody id="A2.E24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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start_CELL italic_a ∼ italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_s ∼ italic_μ | italic_a end_CELL end_ROW end_ARG end_POSTSUBSCRIPT [ start_ARG italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s , italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> <tbody id="A2.E25"><tr class="ltx_equation 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italic_i end_POSTSUBSCRIPT ) , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) + italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s , italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_a start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ) end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> <tbody id="A2.E26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=U_{i}^{\mu,P}(\phi_{i}\circ\pi_{i},\pi_{-i})." class="ltx_Math" display="inline" id="A2.E26.m1.3"><semantics id="A2.E26.m1.3a"><mrow 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xref="A2.E26.m1.3.3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.E26.m1.3.3.1.1.1.1.1.1.2.1.cmml" xref="A2.E26.m1.3.3.1.1.1.1.1.1.2">subscript</csymbol><ci id="A2.E26.m1.3.3.1.1.1.1.1.1.2.2.cmml" xref="A2.E26.m1.3.3.1.1.1.1.1.1.2.2">italic-ϕ</ci><ci id="A2.E26.m1.3.3.1.1.1.1.1.1.2.3.cmml" xref="A2.E26.m1.3.3.1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="A2.E26.m1.3.3.1.1.1.1.1.1.3.cmml" xref="A2.E26.m1.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.E26.m1.3.3.1.1.1.1.1.1.3.1.cmml" xref="A2.E26.m1.3.3.1.1.1.1.1.1.3">subscript</csymbol><ci id="A2.E26.m1.3.3.1.1.1.1.1.1.3.2.cmml" xref="A2.E26.m1.3.3.1.1.1.1.1.1.3.2">𝜋</ci><ci id="A2.E26.m1.3.3.1.1.1.1.1.1.3.3.cmml" xref="A2.E26.m1.3.3.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="A2.E26.m1.3.3.1.1.2.2.2.2.cmml" xref="A2.E26.m1.3.3.1.1.2.2.2.2"><csymbol cd="ambiguous" id="A2.E26.m1.3.3.1.1.2.2.2.2.1.cmml" xref="A2.E26.m1.3.3.1.1.2.2.2.2">subscript</csymbol><ci id="A2.E26.m1.3.3.1.1.2.2.2.2.2.cmml" xref="A2.E26.m1.3.3.1.1.2.2.2.2.2">𝜋</ci><apply id="A2.E26.m1.3.3.1.1.2.2.2.2.3.cmml" xref="A2.E26.m1.3.3.1.1.2.2.2.2.3"><minus id="A2.E26.m1.3.3.1.1.2.2.2.2.3.1.cmml" xref="A2.E26.m1.3.3.1.1.2.2.2.2.3"></minus><ci id="A2.E26.m1.3.3.1.1.2.2.2.2.3.2.cmml" xref="A2.E26.m1.3.3.1.1.2.2.2.2.3.2">𝑖</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E26.m1.3c">\displaystyle=U_{i}^{\mu,P}(\phi_{i}\circ\pi_{i},\pi_{-i}).</annotation><annotation encoding="application/x-llamapun" id="A2.E26.m1.3d">= italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ , italic_P end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∘ italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_π 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">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.1.p1.15">Thus, if <math alttext="\phi_{i}" class="ltx_Math" display="inline" id="A2.SS2.1.p1.10.m1.1"><semantics id="A2.SS2.1.p1.10.m1.1a"><msub id="A2.SS2.1.p1.10.m1.1.1" xref="A2.SS2.1.p1.10.m1.1.1.cmml"><mi id="A2.SS2.1.p1.10.m1.1.1.2" xref="A2.SS2.1.p1.10.m1.1.1.2.cmml">ϕ</mi><mi id="A2.SS2.1.p1.10.m1.1.1.3" xref="A2.SS2.1.p1.10.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.10.m1.1b"><apply id="A2.SS2.1.p1.10.m1.1.1.cmml" xref="A2.SS2.1.p1.10.m1.1.1"><csymbol cd="ambiguous" id="A2.SS2.1.p1.10.m1.1.1.1.cmml" xref="A2.SS2.1.p1.10.m1.1.1">subscript</csymbol><ci id="A2.SS2.1.p1.10.m1.1.1.2.cmml" xref="A2.SS2.1.p1.10.m1.1.1.2">italic-ϕ</ci><ci id="A2.SS2.1.p1.10.m1.1.1.3.cmml" xref="A2.SS2.1.p1.10.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.10.m1.1c">\phi_{i}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.10.m1.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a profitable deviation for agent <math alttext="i" class="ltx_Math" display="inline" id="A2.SS2.1.p1.11.m2.1"><semantics id="A2.SS2.1.p1.11.m2.1a"><mi id="A2.SS2.1.p1.11.m2.1.1" xref="A2.SS2.1.p1.11.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.11.m2.1b"><ci id="A2.SS2.1.p1.11.m2.1.1.cmml" xref="A2.SS2.1.p1.11.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.11.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.11.m2.1d">italic_i</annotation></semantics></math> in from <math alttext="(\mu^{\prime},P^{\prime},\operatorname{Id})" class="ltx_Math" display="inline" id="A2.SS2.1.p1.12.m3.3"><semantics id="A2.SS2.1.p1.12.m3.3a"><mrow id="A2.SS2.1.p1.12.m3.3.3.2" xref="A2.SS2.1.p1.12.m3.3.3.3.cmml"><mo id="A2.SS2.1.p1.12.m3.3.3.2.3" stretchy="false" xref="A2.SS2.1.p1.12.m3.3.3.3.cmml">(</mo><msup id="A2.SS2.1.p1.12.m3.2.2.1.1" xref="A2.SS2.1.p1.12.m3.2.2.1.1.cmml"><mi id="A2.SS2.1.p1.12.m3.2.2.1.1.2" xref="A2.SS2.1.p1.12.m3.2.2.1.1.2.cmml">μ</mi><mo id="A2.SS2.1.p1.12.m3.2.2.1.1.3" xref="A2.SS2.1.p1.12.m3.2.2.1.1.3.cmml">′</mo></msup><mo id="A2.SS2.1.p1.12.m3.3.3.2.4" xref="A2.SS2.1.p1.12.m3.3.3.3.cmml">,</mo><msup id="A2.SS2.1.p1.12.m3.3.3.2.2" xref="A2.SS2.1.p1.12.m3.3.3.2.2.cmml"><mi id="A2.SS2.1.p1.12.m3.3.3.2.2.2" xref="A2.SS2.1.p1.12.m3.3.3.2.2.2.cmml">P</mi><mo id="A2.SS2.1.p1.12.m3.3.3.2.2.3" xref="A2.SS2.1.p1.12.m3.3.3.2.2.3.cmml">′</mo></msup><mo id="A2.SS2.1.p1.12.m3.3.3.2.5" xref="A2.SS2.1.p1.12.m3.3.3.3.cmml">,</mo><mi id="A2.SS2.1.p1.12.m3.1.1" xref="A2.SS2.1.p1.12.m3.1.1.cmml">Id</mi><mo id="A2.SS2.1.p1.12.m3.3.3.2.6" stretchy="false" xref="A2.SS2.1.p1.12.m3.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.12.m3.3b"><vector id="A2.SS2.1.p1.12.m3.3.3.3.cmml" xref="A2.SS2.1.p1.12.m3.3.3.2"><apply id="A2.SS2.1.p1.12.m3.2.2.1.1.cmml" xref="A2.SS2.1.p1.12.m3.2.2.1.1"><csymbol cd="ambiguous" id="A2.SS2.1.p1.12.m3.2.2.1.1.1.cmml" xref="A2.SS2.1.p1.12.m3.2.2.1.1">superscript</csymbol><ci id="A2.SS2.1.p1.12.m3.2.2.1.1.2.cmml" xref="A2.SS2.1.p1.12.m3.2.2.1.1.2">𝜇</ci><ci id="A2.SS2.1.p1.12.m3.2.2.1.1.3.cmml" xref="A2.SS2.1.p1.12.m3.2.2.1.1.3">′</ci></apply><apply id="A2.SS2.1.p1.12.m3.3.3.2.2.cmml" xref="A2.SS2.1.p1.12.m3.3.3.2.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.12.m3.3.3.2.2.1.cmml" xref="A2.SS2.1.p1.12.m3.3.3.2.2">superscript</csymbol><ci id="A2.SS2.1.p1.12.m3.3.3.2.2.2.cmml" xref="A2.SS2.1.p1.12.m3.3.3.2.2.2">𝑃</ci><ci id="A2.SS2.1.p1.12.m3.3.3.2.2.3.cmml" xref="A2.SS2.1.p1.12.m3.3.3.2.2.3">′</ci></apply><ci id="A2.SS2.1.p1.12.m3.1.1.cmml" xref="A2.SS2.1.p1.12.m3.1.1">Id</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.12.m3.3c">(\mu^{\prime},P^{\prime},\operatorname{Id})</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.12.m3.3d">( italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , roman_Id )</annotation></semantics></math>, then <math alttext="\phi_{i}\circ\pi_{i}" class="ltx_Math" display="inline" id="A2.SS2.1.p1.13.m4.1"><semantics id="A2.SS2.1.p1.13.m4.1a"><mrow id="A2.SS2.1.p1.13.m4.1.1" xref="A2.SS2.1.p1.13.m4.1.1.cmml"><msub id="A2.SS2.1.p1.13.m4.1.1.2" xref="A2.SS2.1.p1.13.m4.1.1.2.cmml"><mi id="A2.SS2.1.p1.13.m4.1.1.2.2" xref="A2.SS2.1.p1.13.m4.1.1.2.2.cmml">ϕ</mi><mi id="A2.SS2.1.p1.13.m4.1.1.2.3" xref="A2.SS2.1.p1.13.m4.1.1.2.3.cmml">i</mi></msub><mo id="A2.SS2.1.p1.13.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="A2.SS2.1.p1.13.m4.1.1.1.cmml">∘</mo><msub id="A2.SS2.1.p1.13.m4.1.1.3" xref="A2.SS2.1.p1.13.m4.1.1.3.cmml"><mi id="A2.SS2.1.p1.13.m4.1.1.3.2" xref="A2.SS2.1.p1.13.m4.1.1.3.2.cmml">π</mi><mi id="A2.SS2.1.p1.13.m4.1.1.3.3" xref="A2.SS2.1.p1.13.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.13.m4.1b"><apply id="A2.SS2.1.p1.13.m4.1.1.cmml" xref="A2.SS2.1.p1.13.m4.1.1"><compose id="A2.SS2.1.p1.13.m4.1.1.1.cmml" xref="A2.SS2.1.p1.13.m4.1.1.1"></compose><apply id="A2.SS2.1.p1.13.m4.1.1.2.cmml" xref="A2.SS2.1.p1.13.m4.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.13.m4.1.1.2.1.cmml" xref="A2.SS2.1.p1.13.m4.1.1.2">subscript</csymbol><ci id="A2.SS2.1.p1.13.m4.1.1.2.2.cmml" xref="A2.SS2.1.p1.13.m4.1.1.2.2">italic-ϕ</ci><ci id="A2.SS2.1.p1.13.m4.1.1.2.3.cmml" xref="A2.SS2.1.p1.13.m4.1.1.2.3">𝑖</ci></apply><apply id="A2.SS2.1.p1.13.m4.1.1.3.cmml" xref="A2.SS2.1.p1.13.m4.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.1.p1.13.m4.1.1.3.1.cmml" xref="A2.SS2.1.p1.13.m4.1.1.3">subscript</csymbol><ci id="A2.SS2.1.p1.13.m4.1.1.3.2.cmml" xref="A2.SS2.1.p1.13.m4.1.1.3.2">𝜋</ci><ci id="A2.SS2.1.p1.13.m4.1.1.3.3.cmml" xref="A2.SS2.1.p1.13.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.13.m4.1c">\phi_{i}\circ\pi_{i}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.13.m4.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∘ italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a profitable deviation from <math alttext="(\mu,P,\pi)" class="ltx_Math" display="inline" id="A2.SS2.1.p1.14.m5.3"><semantics id="A2.SS2.1.p1.14.m5.3a"><mrow id="A2.SS2.1.p1.14.m5.3.4.2" xref="A2.SS2.1.p1.14.m5.3.4.1.cmml"><mo id="A2.SS2.1.p1.14.m5.3.4.2.1" stretchy="false" xref="A2.SS2.1.p1.14.m5.3.4.1.cmml">(</mo><mi id="A2.SS2.1.p1.14.m5.1.1" xref="A2.SS2.1.p1.14.m5.1.1.cmml">μ</mi><mo id="A2.SS2.1.p1.14.m5.3.4.2.2" xref="A2.SS2.1.p1.14.m5.3.4.1.cmml">,</mo><mi id="A2.SS2.1.p1.14.m5.2.2" xref="A2.SS2.1.p1.14.m5.2.2.cmml">P</mi><mo id="A2.SS2.1.p1.14.m5.3.4.2.3" xref="A2.SS2.1.p1.14.m5.3.4.1.cmml">,</mo><mi id="A2.SS2.1.p1.14.m5.3.3" xref="A2.SS2.1.p1.14.m5.3.3.cmml">π</mi><mo id="A2.SS2.1.p1.14.m5.3.4.2.4" stretchy="false" xref="A2.SS2.1.p1.14.m5.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.14.m5.3b"><vector id="A2.SS2.1.p1.14.m5.3.4.1.cmml" xref="A2.SS2.1.p1.14.m5.3.4.2"><ci id="A2.SS2.1.p1.14.m5.1.1.cmml" xref="A2.SS2.1.p1.14.m5.1.1">𝜇</ci><ci id="A2.SS2.1.p1.14.m5.2.2.cmml" xref="A2.SS2.1.p1.14.m5.2.2">𝑃</ci><ci id="A2.SS2.1.p1.14.m5.3.3.cmml" xref="A2.SS2.1.p1.14.m5.3.3">𝜋</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.14.m5.3c">(\mu,P,\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.14.m5.3d">( italic_μ , italic_P , italic_π )</annotation></semantics></math>, so <math alttext="(\mu^{\prime},P^{\prime},\operatorname{Id})" class="ltx_Math" display="inline" id="A2.SS2.1.p1.15.m6.3"><semantics id="A2.SS2.1.p1.15.m6.3a"><mrow id="A2.SS2.1.p1.15.m6.3.3.2" xref="A2.SS2.1.p1.15.m6.3.3.3.cmml"><mo id="A2.SS2.1.p1.15.m6.3.3.2.3" stretchy="false" xref="A2.SS2.1.p1.15.m6.3.3.3.cmml">(</mo><msup id="A2.SS2.1.p1.15.m6.2.2.1.1" xref="A2.SS2.1.p1.15.m6.2.2.1.1.cmml"><mi id="A2.SS2.1.p1.15.m6.2.2.1.1.2" xref="A2.SS2.1.p1.15.m6.2.2.1.1.2.cmml">μ</mi><mo id="A2.SS2.1.p1.15.m6.2.2.1.1.3" xref="A2.SS2.1.p1.15.m6.2.2.1.1.3.cmml">′</mo></msup><mo id="A2.SS2.1.p1.15.m6.3.3.2.4" xref="A2.SS2.1.p1.15.m6.3.3.3.cmml">,</mo><msup id="A2.SS2.1.p1.15.m6.3.3.2.2" xref="A2.SS2.1.p1.15.m6.3.3.2.2.cmml"><mi id="A2.SS2.1.p1.15.m6.3.3.2.2.2" xref="A2.SS2.1.p1.15.m6.3.3.2.2.2.cmml">P</mi><mo id="A2.SS2.1.p1.15.m6.3.3.2.2.3" xref="A2.SS2.1.p1.15.m6.3.3.2.2.3.cmml">′</mo></msup><mo id="A2.SS2.1.p1.15.m6.3.3.2.5" xref="A2.SS2.1.p1.15.m6.3.3.3.cmml">,</mo><mi id="A2.SS2.1.p1.15.m6.1.1" xref="A2.SS2.1.p1.15.m6.1.1.cmml">Id</mi><mo id="A2.SS2.1.p1.15.m6.3.3.2.6" stretchy="false" xref="A2.SS2.1.p1.15.m6.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.15.m6.3b"><vector id="A2.SS2.1.p1.15.m6.3.3.3.cmml" xref="A2.SS2.1.p1.15.m6.3.3.2"><apply id="A2.SS2.1.p1.15.m6.2.2.1.1.cmml" xref="A2.SS2.1.p1.15.m6.2.2.1.1"><csymbol cd="ambiguous" id="A2.SS2.1.p1.15.m6.2.2.1.1.1.cmml" xref="A2.SS2.1.p1.15.m6.2.2.1.1">superscript</csymbol><ci id="A2.SS2.1.p1.15.m6.2.2.1.1.2.cmml" xref="A2.SS2.1.p1.15.m6.2.2.1.1.2">𝜇</ci><ci id="A2.SS2.1.p1.15.m6.2.2.1.1.3.cmml" xref="A2.SS2.1.p1.15.m6.2.2.1.1.3">′</ci></apply><apply id="A2.SS2.1.p1.15.m6.3.3.2.2.cmml" xref="A2.SS2.1.p1.15.m6.3.3.2.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.15.m6.3.3.2.2.1.cmml" xref="A2.SS2.1.p1.15.m6.3.3.2.2">superscript</csymbol><ci id="A2.SS2.1.p1.15.m6.3.3.2.2.2.cmml" xref="A2.SS2.1.p1.15.m6.3.3.2.2.2">𝑃</ci><ci id="A2.SS2.1.p1.15.m6.3.3.2.2.3.cmml" xref="A2.SS2.1.p1.15.m6.3.3.2.2.3">′</ci></apply><ci id="A2.SS2.1.p1.15.m6.1.1.cmml" xref="A2.SS2.1.p1.15.m6.1.1">Id</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.15.m6.3c">(\mu^{\prime},P^{\prime},\operatorname{Id})</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.15.m6.3d">( italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , roman_Id )</annotation></semantics></math> is a CEP. ∎</p> </div> </div> </section> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>Other omitted proofs</h2> <section class="ltx_subsection" id="A3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">C.1 </span><a class="ltx_ref" href="https://arxiv.org/html/2503.01976v1#S2.Thmtheorem1" title="Proposition 2.1. ‣ 2.3 Realized and in-expectation regret ‣ 2 Notation and Preliminaries ‣ Learning a Game by Paying the Agents"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">2.1</span></a> </h3> <div class="ltx_proof" id="A3.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A3.SS1.1.p1"> <p class="ltx_p" id="A3.SS1.1.p1.5">We have:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx19"> <tbody id="A3.E27"><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\hat{R}(T)" class="ltx_Math" display="inline" id="A3.E27.m1.1"><semantics id="A3.E27.m1.1a"><mrow id="A3.E27.m1.1.2" xref="A3.E27.m1.1.2.cmml"><mover accent="true" id="A3.E27.m1.1.2.2" xref="A3.E27.m1.1.2.2.cmml"><mi id="A3.E27.m1.1.2.2.2" xref="A3.E27.m1.1.2.2.2.cmml">R</mi><mo id="A3.E27.m1.1.2.2.1" xref="A3.E27.m1.1.2.2.1.cmml">^</mo></mover><mo id="A3.E27.m1.1.2.1" xref="A3.E27.m1.1.2.1.cmml">⁢</mo><mrow id="A3.E27.m1.1.2.3.2" xref="A3.E27.m1.1.2.cmml"><mo id="A3.E27.m1.1.2.3.2.1" stretchy="false" xref="A3.E27.m1.1.2.cmml">(</mo><mi id="A3.E27.m1.1.1" xref="A3.E27.m1.1.1.cmml">T</mi><mo id="A3.E27.m1.1.2.3.2.2" stretchy="false" xref="A3.E27.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E27.m1.1b"><apply id="A3.E27.m1.1.2.cmml" xref="A3.E27.m1.1.2"><times id="A3.E27.m1.1.2.1.cmml" xref="A3.E27.m1.1.2.1"></times><apply id="A3.E27.m1.1.2.2.cmml" xref="A3.E27.m1.1.2.2"><ci id="A3.E27.m1.1.2.2.1.cmml" xref="A3.E27.m1.1.2.2.1">^</ci><ci id="A3.E27.m1.1.2.2.2.cmml" xref="A3.E27.m1.1.2.2.2">𝑅</ci></apply><ci id="A3.E27.m1.1.1.cmml" xref="A3.E27.m1.1.1">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E27.m1.1c">\displaystyle\hat{R}(T)</annotation><annotation encoding="application/x-llamapun" id="A3.E27.m1.1d">over^ start_ARG italic_R end_ARG ( italic_T )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\max_{a\in A}\sum_{t=1}^{T}\quantity({\bm{u}}^{t}[a]-{\bm{u}}^{t% }[a^{t}])=\max_{{\bm{x}}\in{\mathcal{X}}}\sum_{t=1}^{T}\expectationvalue{{\bm{% u}}^{t},{\bm{x}}-{\bm{x}}^{t}}+\sum_{t=1}^{T}\quantity(\expectationvalue{{\bm{% u}}^{t},{\bm{x}}^{t}}-{\bm{u}}^{t}[a^{t}])." class="ltx_Math" display="inline" id="A3.E27.m2.4"><semantics id="A3.E27.m2.4a"><mrow id="A3.E27.m2.4.4.1" xref="A3.E27.m2.4.4.1.1.cmml"><mrow id="A3.E27.m2.4.4.1.1" xref="A3.E27.m2.4.4.1.1.cmml"><mi id="A3.E27.m2.4.4.1.1.2" xref="A3.E27.m2.4.4.1.1.2.cmml"></mi><mo id="A3.E27.m2.4.4.1.1.3" xref="A3.E27.m2.4.4.1.1.3.cmml">=</mo><mrow id="A3.E27.m2.4.4.1.1.4" xref="A3.E27.m2.4.4.1.1.4.cmml"><munder id="A3.E27.m2.4.4.1.1.4.2" xref="A3.E27.m2.4.4.1.1.4.2.cmml"><mi id="A3.E27.m2.4.4.1.1.4.2.2" xref="A3.E27.m2.4.4.1.1.4.2.2.cmml">max</mi><mrow id="A3.E27.m2.4.4.1.1.4.2.3" xref="A3.E27.m2.4.4.1.1.4.2.3.cmml"><mi id="A3.E27.m2.4.4.1.1.4.2.3.2" xref="A3.E27.m2.4.4.1.1.4.2.3.2.cmml">a</mi><mo id="A3.E27.m2.4.4.1.1.4.2.3.1" 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start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_a ] - bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ] end_ARG ) = roman_max start_POSTSUBSCRIPT bold_italic_x ∈ caligraphic_X end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ⟨ start_ARG bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , bold_italic_x - bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_ARG ⟩ + ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( start_ARG ⟨ start_ARG bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , bold_italic_x start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_ARG ⟩ - bold_italic_u start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_a start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ] end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.1.p1.4">The first term is the regret of <math alttext="{\mathcal{R}}" class="ltx_Math" display="inline" id="A3.SS1.1.p1.1.m1.1"><semantics id="A3.SS1.1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="A3.SS1.1.p1.1.m1.1.1" xref="A3.SS1.1.p1.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.1.m1.1b"><ci id="A3.SS1.1.p1.1.m1.1.1.cmml" xref="A3.SS1.1.p1.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.1.m1.1c">{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.1.m1.1d">caligraphic_R</annotation></semantics></math>, which is bounded by <math alttext="R(T)" class="ltx_Math" display="inline" id="A3.SS1.1.p1.2.m2.1"><semantics id="A3.SS1.1.p1.2.m2.1a"><mrow id="A3.SS1.1.p1.2.m2.1.2" xref="A3.SS1.1.p1.2.m2.1.2.cmml"><mi id="A3.SS1.1.p1.2.m2.1.2.2" xref="A3.SS1.1.p1.2.m2.1.2.2.cmml">R</mi><mo id="A3.SS1.1.p1.2.m2.1.2.1" xref="A3.SS1.1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="A3.SS1.1.p1.2.m2.1.2.3.2" xref="A3.SS1.1.p1.2.m2.1.2.cmml"><mo id="A3.SS1.1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="A3.SS1.1.p1.2.m2.1.2.cmml">(</mo><mi id="A3.SS1.1.p1.2.m2.1.1" xref="A3.SS1.1.p1.2.m2.1.1.cmml">T</mi><mo id="A3.SS1.1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="A3.SS1.1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.2.m2.1b"><apply id="A3.SS1.1.p1.2.m2.1.2.cmml" xref="A3.SS1.1.p1.2.m2.1.2"><times id="A3.SS1.1.p1.2.m2.1.2.1.cmml" xref="A3.SS1.1.p1.2.m2.1.2.1"></times><ci id="A3.SS1.1.p1.2.m2.1.2.2.cmml" xref="A3.SS1.1.p1.2.m2.1.2.2">𝑅</ci><ci id="A3.SS1.1.p1.2.m2.1.1.cmml" xref="A3.SS1.1.p1.2.m2.1.1">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.2.m2.1c">R(T)</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.2.m2.1d">italic_R ( italic_T )</annotation></semantics></math>. The second term is a martingale difference sequence, and thus by the Azuma-Hoeffding inequality it is bounded by <math alttext="{\mathcal{O}}\quantity(\sqrt{T\log(1/\delta)})" class="ltx_Math" display="inline" id="A3.SS1.1.p1.3.m3.1"><semantics id="A3.SS1.1.p1.3.m3.1a"><mrow id="A3.SS1.1.p1.3.m3.1.2" xref="A3.SS1.1.p1.3.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS1.1.p1.3.m3.1.2.2" xref="A3.SS1.1.p1.3.m3.1.2.2.cmml">𝒪</mi><mo id="A3.SS1.1.p1.3.m3.1.2.1" xref="A3.SS1.1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="A3.SS1.1.p1.3.m3.1.1.3" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.cmml"><mo id="A3.SS1.1.p1.3.m3.1.1.3.1" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.cmml">(</mo><msqrt id="A3.SS1.1.p1.3.m3.1.1.1.1.1" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.cmml"><mrow id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.cmml"><mi id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.4" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.4.cmml">T</mi><mo id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.3" lspace="0.167em" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.3.cmml">⁢</mo><mrow id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.4" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.cmml"><mi id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.2.2" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.1.cmml">log</mi><mo id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.4a" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.1.cmml">⁡</mo><mrow id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.4.1" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.cmml"><mo id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.4.1.1" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.1.cmml">(</mo><mrow id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.cmml"><mn id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.2" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.1" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.3" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.3.cmml">δ</mi></mrow><mo id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.4.1.2" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></msqrt><mo id="A3.SS1.1.p1.3.m3.1.1.3.2" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.3.m3.1b"><apply id="A3.SS1.1.p1.3.m3.1.2.cmml" xref="A3.SS1.1.p1.3.m3.1.2"><times id="A3.SS1.1.p1.3.m3.1.2.1.cmml" xref="A3.SS1.1.p1.3.m3.1.2.1"></times><ci id="A3.SS1.1.p1.3.m3.1.2.2.cmml" xref="A3.SS1.1.p1.3.m3.1.2.2">𝒪</ci><apply id="A3.SS1.1.p1.3.m3.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1.3"><root id="A3.SS1.1.p1.3.m3.1.1.1.1.1a.cmml" xref="A3.SS1.1.p1.3.m3.1.1.3"></root><apply id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2"><times id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.3.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.3"></times><ci id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.4.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.4">𝑇</ci><apply id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.4"><log id="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.3.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.2.2.2.2.2"></log><apply id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1"><divide id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.1"></divide><cn id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.2">1</cn><ci id="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1.1.1.1.1.1.1.1.1.3">𝛿</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.3.m3.1c">{\mathcal{O}}\quantity(\sqrt{T\log(1/\delta)})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.3.m3.1d">caligraphic_O ( start_ARG square-root start_ARG italic_T roman_log ( start_ARG 1 / italic_δ end_ARG ) end_ARG end_ARG )</annotation></semantics></math> with probability at least <math alttext="1-\delta" class="ltx_Math" display="inline" id="A3.SS1.1.p1.4.m4.1"><semantics id="A3.SS1.1.p1.4.m4.1a"><mrow id="A3.SS1.1.p1.4.m4.1.1" xref="A3.SS1.1.p1.4.m4.1.1.cmml"><mn id="A3.SS1.1.p1.4.m4.1.1.2" xref="A3.SS1.1.p1.4.m4.1.1.2.cmml">1</mn><mo id="A3.SS1.1.p1.4.m4.1.1.1" xref="A3.SS1.1.p1.4.m4.1.1.1.cmml">−</mo><mi id="A3.SS1.1.p1.4.m4.1.1.3" xref="A3.SS1.1.p1.4.m4.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.4.m4.1b"><apply id="A3.SS1.1.p1.4.m4.1.1.cmml" xref="A3.SS1.1.p1.4.m4.1.1"><minus id="A3.SS1.1.p1.4.m4.1.1.1.cmml" xref="A3.SS1.1.p1.4.m4.1.1.1"></minus><cn id="A3.SS1.1.p1.4.m4.1.1.2.cmml" type="integer" xref="A3.SS1.1.p1.4.m4.1.1.2">1</cn><ci id="A3.SS1.1.p1.4.m4.1.1.3.cmml" xref="A3.SS1.1.p1.4.m4.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.4.m4.1c">1-\delta</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.4.m4.1d">1 - italic_δ</annotation></semantics></math>. The result follows. ∎</p> </div> </div> </section> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Mon Mar 3 19:01:34 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>